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A micro approach to mathematical arms race analysis Aboughoushe, Adam 1992

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A MICRO APPROACH TO MATHEMATICALARMS RACE ANALYSISBYADAM ABOUGHOUSHEB.A. (Hon.), The University of Alberta, 1987M.A., The University of Alberta, 1988A THESIS SUBMHTED INPARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYINTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF POLITICAL SCIENCEINTERNATIONAL RELATIONSWe accept this thesis asconforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril, 1992Adam Aboughoushe, 1992Signature(s) removed to protect privacyNational Libraryof CanadaBibhotheque nationaedu CanadaCanadian Theses Service Service des (hès canadiennesOttawa. CanadaKIAON4The author has granted an irrevocable non-exclusive licence allowing the National Ubraiyof Canada to reproduce, loan. cfistiibute or sellcopies of his/her thesis by any means and inany form or format, making this thesis availableto interested persons.The author retains ownership of the copyrightin his/her thesis. Neither the thesis norsubstantial extracts from it may be printed orotherwise reproduced without his/her permission.L’auteur a accordé une licence irrevocable etnon exdusive permettant a Ia Bibliothéquenatiönale du Canada de reproduire, préter.distribuer ou vendre des copies de sa thesede quelque maniére et sous quelque formequo ce soit pour mettre des exemplaires decette these a Ia disposition des personnesintéressées.-L’auteur conserve Ia propnété du droit d’auteurqui protege sa these. Ni Ia these ni des extraitssubstantiels de celle-ci ne doiverit êtreimprimés ou autrement reproduits sans sonautorisation.CaiiadISBN 0-315-75402-8In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of POLF-r I CAL 5 E’JCEThe University of British ColumbiaVancouver, CanadaDate APRIL-. zDE-6 (2/88)Signature(s) removed to protect privacyTHESIS ABSTRACTEven with the end of the Cold War, the question, Were the United States and theSoviet Union engaged in an action-reaction arms race? remains important and controversial.The bulk of empirical mathematical arms race research suggests that the US and USSR werenot so engaged. Indeed, most such research into the matter suggests that US arms acquisitionswere driven overwhelmingly by internal or domestic forces, as were Soviet arms acquisitions.Given the longstanding political, economic and military rivalry, between the US and USSR,the finding that they were not engaged in an arms race is perplexing. This is particularly sowith respect to nuclear weapons acquisitions. Orthodox nuclear deterrence theory clearlyposits that the attempt by each side to maintain a balance of nuclear forces with the other andhence deter the other from launching a first-strike should result in an action-reaction nucleararms race. Why, then, does the overwhelming mass of quantitative research suggest that theopposite was true, in practice, in the US-Soviet case?The problem, in part, has been that researchers have been using underspecifiedmathematical models of action-reaction arms race interaction. The most famous of thesemodels is Richardson’s 1960 action-reaction model. Researchers have long been aware thatRichardson’s model is underspecified and as such that it may not be capable of revealing thetrue nature of US-Soviet military interaction. Since the late 1960s, arms race researchers have11THESIS ABSTRACT (CONT)attempted to move beyond Richardson’s simple arms race specification. Several newapproaches to arms race analysis have subsequently emerged: the game theoretic approach,the economic (stock adjustment) approach, and the expectations (adaptive, extrapolative, andrational) approach. Taken individually, neither of these approaches has, however, yieldedmuch fruit.In this dissertation, the game, stock adjustment, and rational expectations approacheswere combined for the first time into a single, more comprehensive, analytical approach anda new action-reaction arms race model was derived, which we have named the GSR Model.In addition, it was argued that a new approach was needed for testing arms race models.Arms races are generally seen as competitions of total armed versus total armed might. Armsrace models have, accordingly, been tested against data on states’ annual militaryexpenditures. We argued instead that an arms race is made of several subraces, the object ofeach subrace being a specific weapons system and a specific counter weapons system,deployed by an opponent and designed to thwart the former’s political and military effect.Models should, then, be tested for each subrace in a given arms race, that is, against data onweapons system-counter weapons system deployment levels. Time frames for the analysis of111THESIS ABSTRACT (CONT)a given weapons system-counter weapons system competition should be set to accord withthe period in which those systems were dominant in the military calculations of thecompeting states.In effect, we have specified an alternative approach to mathematical arms raceanalysis, the micro approach to mathematical arms race analysis. The GSR Model was testedagainst data on annual US and Soviet strategic nuclear warhead deployment levels,— specifically, those onboard ICBMs (1960-71) and submarines (1972-87). The GSR model wasalso tested against annual US-Soviet aggregate strategic nuclear warhead deployment data(ICBM, SLBM and bomber based totals), 1967-84. Estimates of the GSR model suggest thatthe US and USSR were in fact engaged in an action-reaction arms race over submarinelaunched nuclear warheads. Regression analysis also indicates that the US and USSR stronglyinteracted, asymmetrically, over ICBM based nuclear warheads. There appears to have beenno interaction over aggregate warhead deployments. Finally, the implications of these findingsfor the maintenance of a stable nuclear deterrent were discussed.ivTABLE OF CONTENTSABSTRACT.. iiTABLE OF CONTENTS vLIST OF TABLES viiLIST OF FIGURES viiiACKNOWLEDGEMENT ixPREFACE 1INTRODUCTION. 3CHAPTER 1: REVIEW OF RICHARDSON’S MODEL 19CHAPTER 2: EMPIRICAL TESTS OF RICHARDSON’S AND OTHERA-R MODELS 42CHAPTER 3: RECONCEPTUALIZING THE ARMS RACE PHENOMENON... 57CHAPTER 4: A GSR ACTION-REACTION ARMS RACE MODEL 77CHAPTER 5: DERIVING TESTABLE PROPOSITIONS FROM THEGSR MODEL 112VTABLE OF CONTENTS (CONT)CHAPTER 6: AN EMPIRICAL TEST OF THE GSR MODEL: THEUS-SOVIET NUCLEAR ARMS RACE 145CHAPTER 7: SUMMARY 198BIBLIOGRAPHY 208APPENDIX A: US-SOVIET NUCLEAR ARMS CONTROL AGREEMENTS.. 242APPENDIX B: MULTILATERAL NUCLEAR ARMS CONTROLAGREEMENTS 244APPENDIX C: ANNUAL US-SOVIET SLBM NUCLEAR WARHEADDEPLOYMENTS, 1972-1987 245APPENDIX D: ANNUAL US-SOVIET ICBM NUCLEAR WARHEADDEPLOYMENTS, 1960-197 1 246APPENDIX E: ANNUAL AGGREGATE US-SOVIET STRATEGICNUCLEAR WARHEAD DEPLOYMENTS, 1967-1984 247APPENDIX F: ALTERNATIVE TESTS OF THE GSR MODEL 248viLIST OF TABLES2.1 Action-reaction in weapons system development: US-USSR 446.1 Autocorrelation test results for the GSR model. 1616.2 Parameter estimates for the US-Soviet strategicnuclear warhead race 1956.2 Continued 196viiLIST OF FIGURES1.1 Stability in Richardson’s model:all parameters > 0 281.2 Instability in Richardson’s model:all parameters > 0 291.3 Stability in Richardson’s model:not all parameters > 0, Case I 341.4 Stability in Richardson’s model:not all parameters > 0, Case II 351.5 Stability in Richardson’s model:not all parameters > 0, Case ifi 361.6 The dynamics of Richardson’ssubmissiveness model 394.1 Stable equilibrium in the GSR model 1084.2 Unstable equiilbrium in the GSR model 1096.1 Stable equilibrium in the US-USSR SLBM warhead race 1716.2 Stable equilibrium in the US-USSR ICBM warhead race 183vi”ACKNOWLEDGEMENTI would, foremost, like to express my gratitude to my research supervisor, Dr. BrianL. Job of the Department of Political Science, University of British Columbia, for his patientand always positive guidance.I would also like to thank Dr. Donald E. Blake, Dr. J.A. Brander, Dr. B.E. Eckbo, Dr.Richard G.C. Johnston, Dr. Michael D. Wallace, all of the University of British Columbia,and Dr. Stephen J. Majeski of the University of Washington for serving on my dissertationexamination committee. Their input was thoughtful and greatly appreciated.Finally, I would like to thank Dr. Michael D. McGinnis of Indiana University for anumber of important comments he made on an early draft of this dissertation.r accept sole responsibility for any remaining errors or omissions In this final draft.x1PREFACEThe care of human life and happiness, and not their destruction, is the first and onlylegitimate object of good government.Thomas Jefferson, US, 1809For many years, politicians, journalists, and scholars alike have debated the merits ofarms racing. Some argue the ancient Roman dictum “if you want peace, then prepare forwar.” By stockpiling armaments a nation can avoid war, because, knowing of that stockpile,its adversaries will fear it, and hence, avoid war with it. Others contend that arms races, onthe contrary, lead to, or culminate in, war. Historically, proponents of this view argue, moststates which have engaged each other in an arms race have also gone to war with each other.Neither side in the debate, however, has provided a completely convincing systematic analysisof cause and effect.Yet in today’s world, war, or peace, are not the only potential by-products of armsracing. Even a cursory look at available social and military expenditure statistics shows thatthroughout the world military armament programs are draining limited economic andindustrial resources to the detriment of human development programs (Sivard, 1985). To citeonly two examples, in 1985, developed nations expended 5.4 percent (on average) of theirGNPs on military purposes. The same countries, in 1985, allocated only 0.3 percent (onaverage) of their GNPs to development aid to Third World countries. In 1983, an average of2US$45 per capita was spent by the governments of the world on military research while onlyUS$11 was spent per capita on medical research.Because arms races may lead to war, because arms races may help us to avoid war,because arms races drain away resources from other, potentially more useful, pursuits, effortsmust be made to study them and to understand how they work. That is the purpose of mydissertation.3INTRODUCTIONIn her Presidential Address to the International Studies Association, Zinnes (1980)issued a challenge to members of the international relations research community to developa mathematical model, supported by data, of action-reaction arms race interaction among rivalstates in world politics. In this dissertation, I will take up that challenge. Indeed, in thisdissertation, I will specify and apply an alternative approach to mathematical arms raceanalysis.Mathematical arms race modelling goes back to 1960 when Richardson (1960a)presented his now famous linear differential arms race equations in his seminal work Armsand Insecurity. The assumptions behind his model were simple and intuitively appealing. Setin difference equation form, Richardson’s model is as follows:X = kY1 + (1 - a)X1 + g (R28)= 1X, + (1- b)Y1 + h (R29)State X’s armaments at time t, denoted XL, depend positively upon the level of armspossessed by State X’s rival, State Y, in time t-1, denoted Y, times State X’s defencecoefficient, k, negatively upon the level of annaments possessed by State X at time t-1,4denoted X1, times State X’s economic fatigue term (1-a) and on some constant amount, g,which reflects the degree of grievance State X has against State Y.Richardson’s work was indeed seminal. Until today, virtually every mathematical armsrace study begins with some reference to Richardson and his model, the present study beingno exception. Since Richardson presented his model, several other authors (e.g., Caspary,1967; Majeski, 1983a, 1985; Ostrom, 1978b; Ward, 1984a; Gillespie et al, 1977c) haveadvanced mathematical models of action-reaction arms race interaction. Most of these modelshave been presented as alternatives to or as modifications of Richardson’s model. Why, then,do we need another mathematical arms race model?1. BACKGROUND: THE MACRO APPROACH TO ARMS RACE ANALYSISNo attempt at mathematically modelling the arms race phenomenon has yet met withwidespread approval in the international relations community, Richardson’s model being noexception (Wallace, 1980a). Developing a model is one thing. Models must, however, betested against empirical data.International relations researchers have, for example, attempted to estimate theparameters of Richardson’s model for a number of different arms races including the ArabIraeli race and the Iran-Iraq race. The most important application of Richardson’s model,5however, was to the US-Soviet arms race. There are a number of compelling reasons tobelieve that the US and Soviet Union are engaged in an action reaction arms race, the mostcompelling being the need for each side to maintain a balance of power with the other.Indeed both sides often justify their arms acquisitions on that basis.Most researchers take the view that the US and Soviets wage their arms competitionon the basis of total armed might versus total armed might. A state’s annual aggregatemilitary expenditures, it is felt, reflects its total military strength. Thus most researchers testRichardson’s model for the US-Soviet arms race against annual US-Soviet aggregate militaryexpenditure data. I will term this approach to mathematical arms race analysis the macroapproach to emphasize its aggregate orientation.Time and time again, in such instances, Richardson’s model was disconfirmed byempirical data: estimates of Richardson’s arms race model for the US-Soviet race suggest thateach side’s military expenditures are internally driven. There is no action-reaction interaction.In this study, I will term this result the non-interaction outcome. Still other research suggeststhat current US military expenditures depend upon previous period expenditures in the USSR,whereas Soviet military expenditures are internally driven (Majeski, 1985). I will term thisresult the asymmetric outcome.62. NEW DIRECTIONS IN ARMS RACE RESEARCH METHODOLOGY: ACRITIQUE OF THE MACRO APPROACHThese counterintuitive findings, the non-interaction and asymmetric outcomes, havegenerated a great deal of debate within the mathematical arms race research community(Gillespie and Zinnes, 1982), culminating in Zinnes’ challenge. Were US-Soviet armsacquisitions, as suggested by estimates of Richardson’s and other’s models, really independentof one another? Or have we yet to develop a methodology--a model, and model testingprocedure--which can uncover the interactive component of that competition?Those who have answered the latter question in the affirmative cite a number ofdeficiencies in conventional arms race research methodology which could account for thefinding of non-interaction in quantitative studies of the US-Soviet arms race. These critiquescan be grouped as those which focus on deficiencies in existing mathematical arms raceformulations and those, which focus on deficiencies in the way those models have been tested.Some who argue that US-Soviet arms acquisitions were not independent of oneanother suggest that Richardson’s model is somehow flawed, oversimplified, and as such thatit could not be used to uncover the true nature of the US-Soviet arms race. Numerous factors,other than those suggested by Richardson, impact on a state’s armament calculus. It wasnecessary, then, to expand if not totally rework Richardson’s conceptualization of action-7reaction arms interaction. The most important ideas presented in this rethink include the ideathat state’s base their armament calculations as much, if not more, on what they expect a rivalwill possess in the future as on what he held in the past (e.g., Majeski, 1985), a factor whichRichardson’s model does not take account of. Still others argue that states engaged in an armsrace do so with a particular objective in mind such as maintaining a balance of power witha rival. Arms racing states, moreover, design and implement strategies to achieve their arminggoals. Such strategies, if they are to be successful, must take into account the arming goalsand strategies of rival states. Richardson’s model does not contain any such decision calculus(Gillespie et al, 1977c). Others point out that those armament goals must be subject toeconomic, political and institutional constraints. Indeed there may be a disjuncture, in the end,between the level a state may wish to arm to and the level it can arm to (Caspary, 1967).Calculations such as these would need to be factored into any model of action-reaction armsrace interaction. Indeed work was well underway on reconceptualizing action-reaction armsrace interaction long before Zinnes issued her formal challenge to the international relationsresearch community.Still, there are those who have rejected, outright, the notion that US-Soviet militaryacquisitions are action-reaction driven. Some, taking their lead from developments inorganizational theory (Davis, Dempster and Wildavsky, 1966; Kanter, 1973), have suggestedthat US-Soviet arms acquisitions could, in large measure, be independently driven. AsWallace (1979) notes, the organizational approach, as applied to the US-Soviet arms race, is8two pronged. Some (Ostrom, 1977a, 1978b, Majeski, 1983a) see arms accumulation as theend product of a complex internal bureaucratic process. Still others (e.g. Nincic and Cusack,1979) see arms accumulation as a product of internal political and economic forces. Underthis perspective, military production is used to maintain or increase internal economic activity.Thus far, the organizational approach has met with little success in explaining the variancein US arms acquisitions (the focus of this approach, with few exceptions, notably Johnsonand Wells, 1986, has been on the US).An attempt, incidentally, was made to determine empirically which approach, theaction-reaction or the organizational process, accounts for more variation in US-Soviet aimsacquisitions. Ostrom (1977a) conducted a systematic comparison of a Richardson type actionreaction model with a Davis, Dempster and Wildavsky type organizational process model andfound that forecasts generated by these models for the US were indistinguishable from oneanother.One of the most sophisticated attempts made thus far to uncover the basis of USSoviet arms acquisitions is Williams and McGinnis (1988). They specified a set of conditionsunder which one could expect to find interaction in US-Soviet arms acquisitions and a set ofconditions under which one could expect to find no interaction in US-Soviet armsacquisitions. They argued that two basic factors would determine a state’s militaryexpenditures for time t. The first s what it expected at time t its opponent would expend on9armaments at time t. Under the Rational Expectations hypothesis, the expectation made byState K at time t, the current time period, regarding its opponent’s, State J’s, expenditures attime t is equal to State K’s expectation at time t- 1 of its opponent’s arms expenditures at timet- 1 and a prediction error showing, at time t, how far State K’s current expectation was off.The second factor impacting a state’s military expenditure decision at time t is its expectationat time t of its own economic condition at time t. This expectation would, likewise, beinfluenced by a state’s previous period expectations and a prediction error.Williams and McGinnis (1988) argue that in these circumstances, State J would, attime t, react not to the absolute level of expenditures which were observed to be made by itsopponent, State K, at time t, but rather, State J would react at time t to the value of State K’sprediction errors for time t. If, in the space leading up to time period t, State K were to findthat the economic cost of a preferred level of military expenditure was too high and itsubsequently reduced its expenditure level for time t, a negative error would then obtain inState K’s time period t military expenditure calculus. Williams and McGinnis call suchshocks innovations. If State J were vigilant in collecting information on innovations in StateK’s arming behaviour, as it would be under the Rational Expectations hypothesis, then StateJ should also lower its time period t military expenditures, seeing State K’s reduction in itsmilitary expenditures as a reduction in threat. This would produce a negative error in Statel’s military expenditure calculus at time t.10States, then, react to errors in each other’s military expenditure forecasts. If each sideis efficient in its collection and use of information regarding innovations in the other’sexpenditure behaviour, then forecast errors should drive an arms race. Past expenditure levelseffected by each side are important in determining current expenditures only if neither sideis very efficient in its collection and use of information on innovations in the other’s armingbehaviour.Where such inefficiencies do exist, and they likely would in any empirical context,it would then be important to have a model describing the sort of connection or interactionwhich could occur in the armament processes of rival states.Other critiques of conventional arms race research methodology focus on the question,What exactly do the US and Soviets race over, if they race at all? The assumption that armsracing states compete on the basis of total armed might versus total anned might is widelyaccepted in the arms race research community. Indeed, most studies aimed at estimating theparameters of Richardson’s model for the US-Soviet arms race are based on that veryassumption. A new line of thought suggests that states engaged in an arms race, in particularthe US and Soviet Union, do not compete on the basis of total armed might versus totalarmed might (McCubbins, 1983). The alternative view, the micro view, is that states, instead,compete thusly: when one side deploys a particular weapons systems, the other responds bydeploying a system designed to counter the latter’s military and political effect. An arms race11between two rivals, then, is made of several subraces, the object of each particular subracebeing a particular weapons systems and a particular counter weapons system. Theoretically,rival states could engage one another in several different subraces each with its own particularproperties. Models, then, would have to be tested against each of these subcompetitions.If states, in particular the US and USSR, did not compete on the basis of total armedmight versus total armed might, then it should be no surprise to find that US and Sovietaggregate military expenditure series, which reflect total military capabilities, are independentof one another. In a given rivalry, some subraces could heat up while others cool down. Inaggregate military expenditure data, peaks and valleys in individual weapons systemacquisitions could cancel each other Out. Indeed, evidence of action-reaction interaction in thearms acquisitions of two rivals might be totally distorted by aggregate military data.McCubbins (1983) application of the subrace approach to the US-Soviet arms race has showna great deal of promise in demonstrating the existence of action-reaction interaction in theirarms acquisitions.Still others have focused on the question of time frames for the analysis of the USSoviet arms competition. Lucier (1979) argues that arms race parameters do not remainconstant in value over time. Parameters may, over time, increase or decrease in valuedepending upon factors such as changes in leadership in the competing countries. Moststudies which have shown no interaction in US-Soviet arms acquisitions spanned the period121950-60 to the present. Did the parameters of the US-Soviet competition remain constant invalue over that time. If not, then an attempt to estimate the parameters of Richardson’s, orany other, model from 1950 to the present might yield misleading results.3. A NEW APPROACH TO ARMS RACE ANALYSIS: A MICRO APPROACHIt is my contention that the findings of non-interaction and asymmetric reaction instudies of the US-Soviet arms race are due more to deficiencies in conventional arms racemethodology stemming from an insufficient understanding of what drives an arms race thanto the genuine absence of interaction in US-Soviet arms acquisitions. The variousmodifications to conventional arms race research methodology suggested by Majeski (1985),Gillespie et al (1977c), McCubbins (1983) and Lucier (1979) and others, however, dorepresent a growth in the understanding of the factors which drive an arms race.Unfortunately, never before have all of the modifications to conventional arms racemethodology discussed above been employed all at once in a single study. Majeski (1985),for example, made the important point that a state engaged in an arms race focuses itsattention more on what it expects its rival will possess in terms of arms in the future than onwhat its rival possessed in the past. After having derived a mathematical model whichexpressed this reality, he then estimated it against US and Soviet aggregate militaryexpenditure data running 1949-8 1. The estimate of his model showed an asymmetric arms13race. The US reacted to Soviet expenditures, but Soviet expenditures were internally driven.It is my position that, at a minimum, a study of the US-Soviet arms race, one which couldreveal the so far illusive two-way interactive component of that competition, would requirethe amalgamation of all of the suggested modifications to conventional arms racemethodology specified above into a single approach. That is what I propose to do in thisdissertation.More specifically, in this dissertation, I will specify and apply a micro approach, ordisaggregate approach, to arms race analysis. There are two aspects to this approach: a newaction-reaction arms race model and a specific approach to testing it.The model I will derive in this thesis is an optimal behaviour model. It will give theconditions under which it is optimal for actors to engage each other in an action-reactionarms race. It will also give the conditions under which it is optimal for actors to engage eachother in an asymmetric or non-interactive competition. Under my formulation of the arms racephenomenon, actors can both have non-zero defence coefficients and still engage each otherin asymmetric or non-interactive competitions.My model will be based on three key assumptions drawn from the rethink in armsrace research which followed the unsuccessful testing of Richardson’s model: (1) each stateis assumed to arm in accordance with some goal, in particular, the goal of maintaining a14balance of power with its rival (I will assume that weapons acquisitions are determined ona weapons systems versus cross-purpose weapons system basis), (2) that economic, political,and institutional constraints limit the extent to which each particular state is capable ofrealizing its armament goal, and (3) that each state forms and uses expectations of the other’sfuture arming behaviour in calculating its own future armament requirements. I will formalizethese assumptions by drawing on the methods of game theory, economic theory and RationalExpectations theory.Secondly, I will test this model, not against military expenditure data, but against dataon individual weapons systems with cross-purposes. More specifically, I will apply it in ananalysis of the US-Soviet arms race. I will, of course, not examine every subrace in the largerUS-Soviet military competition. I will focus my efforts on subraces within the US-Sovietstrategic nuclear competition. More specifically, I will conduct three tests of my model:against data on US-Soviet SLBM warhead deployments, against data on US-Soviet ICBMwarhead deployments, and against data on total US-Soviet strategic nuclear warheaddeployments.I will set the time frame for each test to reflect a period in which each systemcounter system was dominant in US-Soviet strategic calculations. More specifically, I will setthe time span of the US-Soviet SLBM warhead deployment analysis to run from 1972 to1987 and the time span of the US-Soviet ICBM warhead deployment analysis to run from151960 to 1971.Previous attempts to uncover evidence of action-reaction interaction in US-Sovietnuclear weapons acquisitions, using the macro approach, have been largely unsuccessful (e.g.,Kugler et al, 1980; McGuire, 1977). My micro approach to mathematical arms race analysiswill yield strong evidence that the US and Soviets were engaged in an action-reactionstrategic nuclear arms race. More specifically, I will conclude that the US and Soviets wereengaged in an action-reaction competition over SLBM warhead deployments. This finding isimportant for it suggests that the logical conditions for the maintenance of strategic deterrencewere being met. Maintaining a nuclear balance of power, and hence, maintaining nucleardeterrence, requires competing powers to engage one another in an action-reaction arms race.Indeed, it is reasonable to find that the US and Soviets did race over SLBM warheaddeployments especially since both the US and the Soviets have viewed their SLBM basedwarheads as reserve second strike forces.Secondly, from my analysis, we will be able to gain a deeper understanding of howarms races work. In particular, I will provide evidence which suggests that both the US andSoviets formed and used expectations of their own future arming requirements andexpectations of each other’s future arming requirements in accordance with the RationalExpectations hypothesis. The implication of this fmding is that US and Soviet militarycalculations were intimately linked. My work will reveal several dimensions of that linkage.164. ORGANIZATION OF THE DISSERTATIONI will organize this thesis in accordance with the five sequential steps of themathematical modelling process. In that process, one must first abstract the phenomenon onewishes to model down to its main essential characteristics or elements. Second, one mustformalize the substance and interrelationships among these elements using mathematicalsymbols. In this step, the model takes on its structure and its substance. One must then derivetestable hypotheses concerning the sort of behaviour, mathematical properties, that the modelobtained in the previous step predicts. As fourth step in the modelling process, one mustdesign and ii plement tests of those predictions against observed reality. Finally, based onthe degree to which one’s model’s predictions are confirmed by empirical observation, onemust return to the first step of the modelling process and amend what one first thoughtconstituted the basic elements malcing up the phenomenon in question. Therewith, steps two,three, four and five would be repeated.I will begin my study with a review of the arms race literature. The arms raceliterature can be divided into two main categories, those which look at arms building modelsand those which look at arms using models (Moll and Lubbert (1980)). Arms building modelsdescribe the forces which determine the arms expenditures of states. Arms using models, asthe name suggests, describe the rate at which a state’s forces may be consumed in battle. Thearms building category can be further divided into Richardson and non-Richardson type17models. This latter category includes organizational process models. In Chapter 1, I will givea detailed review of Richardson’s model. I will present a full derivation of that model andstudy its various properties. This will correspond to steps one, two and three of the modellingprocess where the arms race phenomenon will be abstracted down to its main essentialelements and mathematized. In Chapter 2, I will report on the outcome of various attemptsto estimate Richardson’s model for the US-Soviet arms race. I will focus here on the typesof data against which Richardson’s and other’s models have been tested for the US-Sovietarms race. Chapter two, then, corresponds to step four of the modelling process. In Chapter3, I will look at the efforts that have been made to move beyond Richardson’s simpleconceptual framework. I will look at how game-control theory, economic theory andexpectations theory have been recently employed in an effort to reconceptualize the arms racephenomenon. Chapter 3, then, corresponds to step five of the modelling process. In Chapter4, I will return to steps one and two of the modelling process where I will present thederivation of my own arms race model. In deriving my model, I will draw on the conceptspresented in the previous chapter, melding them into a single framework. Chapter 5 will bea chapter on old problems, new solutions. Here, I will use my model to derive testablepropositions concerning the nature of the action-reaction, non-interaction and asymmetricoutcomes. Chapter 5, then, corresponds to step three of the modelling process. In Chapter 6,I will test my model against data on the US-Soviet strategic nuclear arms race. I will, on theone hand, show that the US and Soviet Union do not race when it comes to each side’saggregate strategic warhead counts (the sum of ICBM, SLBM and bomber based nuclear18warheads). In that same chapter, I disaggregate my US-Soviet nuclear warhead data into itscomponent parts: SLBM, bomber and ICBM warhead deployments. I will present and discussthe results of an analysis made of the SLBM and ICBM warhead data sets. In the case of USand Soviet SLBM warhead deployments (1972-1987), I find clear cut evidence of action-reaction interaction between the US and USSR. Over the period 1960-7 1, I find that the USand Soviet Union were engaged in an asymmetric arms race over ICBM based nuclearwarhead deployments. Both the SLBM warhead race and the ICBM based warhead are,moreover, stable. Finally, in Chapter 7, which corresponds to the final step in the modellingprocess, I will consider the degree to which my study has advanced the search for a model,confirmed by data, of action-reaction arms race interaction among rival states in worldpolitics.19CHAPTER I: A REVIEWOF RICHARDSON’S MODELMathematical arms race modelling and analysis began with Richardson’s Arms andInsecurity (1960a) where he presented his now famous linear differential action-reaction armsrace equations. In the subsequent literature on quantitative arms race analysis, one candiscern, since Richardson’s original work, a steady and logical progression in themethodology--model type, model operationalization, and model testing techniques--of thefield. My own work is simply the next stepping stone in that progression. Accordingly, it isimportant to understand, in some detail, that progression if the significance of my work is tobe understood. I will therefore begin my study with a detailed review of Richardson’s model.This review of Richardson’s model will take us through the first three steps of themodelling process. More specifically, in this chapter, I will ask, How did Richardson cometo his specific action-reaction arms race formulation? What properties does Richardson’smodel have? How have Richardson’s assumptions been altered by others? What other typesof models did Richardson construct besides his basic linear model? In the next chapter, I willask, How well does Richardson’s linear model account for the US-Soviet arms race? In short,the answer to this question is that Richardson’s model is not well supported in the US-Sovietcase. Why? This is one of the principal questions in mathematical arms race research today.In the third chapter, I will look at how arms race modelling has progressed since Arms and20Insecurity.1. RICHARDSON’S DERIVATIONThe insurance theory of armament stockpiling lies at the heart of most debates aboutarms racing. A nation can avoid war by stockpiling arms because, knowing of that stockpile,its potential adversaries will fear its strength, and hence, avoid war with it. Stockpiling canalso benefit a nation should it become engaged in war. Richardson doubted the validity of thetheory, but nevertheless based his model upon it. Regardless of its actual validity, Richardsonthought, statesmen tend to believe the insurance theory and indeed act in accordance with it.How, then, did Richardson come to his specific action-reaction arms race formulation?Richardson (1960a), in formulating his model, asked us to consider the arguments ofa number of imaginary and well known real life public figures concerning the issues involvedin national defence. A physicist by training, Richardson then converted these arguments intomathematical statements. First, he quoted a speech made by the Minister of Defence ofJedesland, a mythical country. The Minister stated:The intentions of our country are entirely pacific. We have given ampleevidence of this by the treaties which we have recently concluded with ourneighbors. Yet, when we consider the state of unrest in the world at large andthe menaces by which we are surrounded, we should be failing in our duty asa government if we did not take adequate steps to increase the defence of ourbeloved land [p. 14].21There are a number of ways in which the Minister’s statement could be converted intoa mathematical statement. Richardson, limiting himself to an analysis of the rivafry betweentwo states, X and Y, translated the Minister’s words thusly.dX/dt = kY (Ri)dY/dt = 1X (R2)Here, X represents State X’s armaments and Y, State Y’s. k represents the degree of threator menace which State X perceives State Y’s armaments pose to t. I, similarly, representsthe degree of threat or menace which State Y perceives State X’s armaments pose to it. k and1 have come to be known, respectively, as State X and Y’ s defence coefficients. The change,then, in State X’s armaments per unit change in time (dX/dt) depends upon the level of armspossessed by State Y (Y) weighted by the threat which State X perceives those armamentsto pose to it (k). State Y’s armament level is similarly determined. The problem with thisformulation is that if States X and Y each possess some positive level of annaments and ifStates X and Y each perceive the other’s armaments to be some positive threat to it then eachstate will, as per the specification of dXldt and dY/dt, be spurred on to ever increasing armsexpenditures. But, for economic reasons, no state could ever effect ever increasing armsexpenditures. Richardson was well aware of this fact.22He thus quoted Churchill’s address to the British cabinet in 1923 concerningGermany’s naval armament program:Believing that there are practically no checks upon German naval expansionexcept those imposed by the increasing difficulties of getting money, I havehad the enclosed report prepared with a view to showing how far thoselimitations are becoming effective. It is clear that they are becoming terriblyeffective [p. 15].Richardson thus amended his preliminary specification as follows.dX/dt = kY - aX (R3)dY/dt = lx - bY (R4)The parameter a shows the economic burden to State X of its existing armament stockpile,X. The parameter b, similarly, shows the economic burden to State Y of its existingarmament stockpile, Y. Richardson assumed both a and b to be positive. The products aX andbY, Richardson thought, should be subtracted from dX/dt and dY/dt respectively. Considerwhat this means for State X. While State Y’s armaments act to stimulate growth in State X’sarmaments over time, State X’s existing stockpile tends to dampen that growth by a factorof a. bY has the same sort of effect in the case of State Y.23Richardson was still not satisfied with his model. He considered that British ForeignSecretary Grey wrote in 1925 thatThe increase of armaments that is intended in each nation to produceconsciousness of strength, and a sense of security, does not produce theseeffects. On the contrary, it produces a consciousness of the strength of othernations and a sense of fear .... The enormous growth of armaments in Europe,the sense of insecurity and fear caused by them--it was these that made warinevitable .... This is the real and final account of the origin of the Great War[p. 15].Richardson has already accounted for the sense of fear that Grey said nations experience fromthe armaments of their neighbors in his model via the parameters k and 1, State X and Y’sdefence coefficients. But consider now British Member of Parliament Amery’s challenge ofGrey’s view, in 1936, in an address to the House:With all respect to the memory of [the] eminent statesman, [Secretary Grey]I believe [his] statement to be entirely mistaken. The armaments were only thesymptoms of the conflict of ambitions and ideals, of those nationalist forces,which created the war. The War was brought about because Serbia, Italy,Rumania passionately desired the incorporation in their States of territorieswhich at the time belonged to the Austrian Empire and which the AustrianGovernment were not prepared to abandon without a struggle. France wasprepared if the opportunity ever came to make an effort to recover AlsaceLorraine. It was in those facts, in those insoluble conflicts of ambitions andnot in the armaments themselves that the cause of the War lay [p. 15-16].Richardson took account of Amery’s retort not by eliminating State X and Y’s defencecoefficients from his model, but by working in new terms.24dX/dt = kY - aX + g (R5)dY/dt = lx - bY + h (R6)The parameters g and h are, respectively, State X and State Y’s grievance terms. Richardsonassumed g and h to be positive. g thus could positively impact the rate of change in State X’sarmaments per unit change in time independently of the level of armaments possessed byState X or the level possessed by State Y. g’s impact on the rate of change in State X’sarmament level over time reflects the magnitude of State X’s grievances against State Y. Theparameter h has a similar interpretation with respect to State Y. This, then, is Richardsonbasic model.2. PROPERTIES OF RICHARDSON’S MODEL: EQUILIBRIUM CONDITIONSRichardson was also interested in the sorts of consequences he could deduce from hismodel (Caspary, 1967). His model is dynamic: State X and State Y’s arms competition isspecified in terms of changes in each respective state’s armaments per unit change in time.Richardson thus asked, under what conditions would an arms race, as he had formulated it,stop? Mathematically, under what conditions willdXldt = 0 = dY/dt (R7)25If we setO=kY-aX+g (R8)O=1X-bY+h (R9)the answer becomes readily apparent. dXldt = 0 = dY/dt, equilibrium, occurs inRichardson’s model whenX = (kh + bg)/(ab - kl) (RiO)and= (ig + ah)I(ab - ki) (Ri 1)Thus (X*, Y*) is a general equilibrium solution for Richardson’s arms race model. Mostimportantly, the ,int (X*, yS) depends upon State X and Y’s defence, economic fatigue andgrievance terms. If, however,(ab - kl) = 0 (R12)26then no equilibrium solution will exist for Richardson’s arms race system. Yet even if theequilibrium point (X*, y*) exists, it may be unstable. Richardson’s arms race system is stableif, after reaching the point (X*, y*) States X and Y return to it if some exogenousdisturbance impacts on the system which causes them to move to some other point.Otherwise, the point (X*, y*) is unstable. For Richardson, the distinction between unstableand stable equilibrium points was no small matter. Richardson believed that States X and Y’sarming competition, if unstable, could culminate in war. We must, therefore, distinguishbetween stable and unstable equilibrium points.To determine the stability conditions for Richardson’s model, we first plot the locusof points in (X, Y) space where, for State X, dX/dt = 0 and where, for State Y, dY/dt = 0.The point of intersection between these lines will give the equilibrium solution (X*, y*) Therelative slopes of the dX/dt = 0 and dY/dt = 0 curves will, on the other hand, show whetheror not the point (X*, y*) is stable. The dX/dt = 0 line and the dY/dt = 0 line in Graphs 1.1and 1.2 were plotted as follows. From0=kY-aX+g (R8)we obtainY = (a/k)X-(gfk) (R13)27and fromO=1X-bY+h (R9)we obtainY = (lIb)X - (h/b) (R14)Graphs 1.1 and 1.2 illustrate the dynamics of Richardson’s model.Graph 1.1 shows a stable arms race. The dX/dt = 0 line and the dY/dt 0 line weredrawn under the assumption that k, 1, a, b, g and h are all positive. In Graph 1.1, there existsan equilibrium point, the point of intersection between the dX/dt = 0 line and the dY/dt = 0line, and States X and Y’s armaments are tending toward it. Graph 1.2 shows an unstablearms race. In Graph 1.2, the dX/dt = 0 line and the dY/dt = 0 lines were also drawn underthe assumption that k, 1, a, b, g and h are all positive. Here there is also an equilibriumarmament point. However, State X and Y’s armaments are tending away from it.Whether the slope arrangement given by Graph 1.1 or by Graph 1.2 prevails for anygiven arms race depends upon the relative values of k, 1, a, b, g and h. Comparing the slopesof the dX/dt = 0 line and the dY/dt = 0 in Graphs 1.1 and 1.2, we see that an arms race, as28GRAPH 1.1: Stability in Richardson’s arms race system when all parameters are positiveYdX/dt = 0dY/dt = 0(X, Y)x29GRAPH 1.2: Instability in Richardson’s anns race system when all parameters are positiveYx30Richardson has specified it, is stable if(a/k) > (1/b) (R15)orab > ki (R16)Stability, then, in Richardson’s model, depends on the relative values of State X and Y’ seconomic fatigue terms and the values of their defence coefficients. For an arms race to bestable, by Richardson’s formulation, the product of the racing states’ domestic economicfatigue must exceed the product of their fear or defence needs.3. VARYING RICHARDSON’S ASSUMPTIONSRichardson had assumed that the parameters a, b, k, 1, g and h all take on positivevalues. This assumption is not unreasonable, but it does limit the applicability of his modelin the real world. Zinnes et al (1976a) argue that a, b, k, 1, g and h need not and, in certainempirical contexts might not, all be positive.For example, a state’s arming program need not always constitute a drain on its31economy. When countries’ productive capacities are already heavily geared toward weaponsproduction, one of or both of the economic fatigue terms, a and b, for a pair of rivals, maybe positive. Secondly, negative values for the grievance terms g and h might indicatecooperation between two rivals. One question immediately arises, however. Can some of a,b, k, 1, g and h be positive while others are negative?In short, the answer is yes. Because the economic fatigue coefficients, a and b, referto internal considerations in the rival states, their values, positive or negative, can standindependently of the values of k and 1 or g and h. k and 1, the defence coefficients and g andh, the grievance terms, are relational parameters which take on their values in accordancewith the nature of the rivalry between the states in question. Yet the relationship between theelements of the pair k and g and the elements of the pair 1 and h is not a simple one. Bothelements of given pair, either k and g or 1 and h, need not be positive. For example, a nationcould fear the armaments of a newly emerging rival, thus making its defence coefficient kpositive, but have no history of conflict with that new rival, thus making its grievance termg negative.It is important to note that if, in a given empirical context, a, b, k, 1, g and h are notall positive, then the equilibrium conditions specified above may not hold. Indeed, theequilibrium dynamics of State X and Y’s competition will vary as the signs of a, b, k, 1, gand h vary.32If we know the values, positive or negative, of k, 1, a, b, g and h for any pair of StatesX and Y, we can then plot the course of their competition as they move toward or away fromequilibrium. Here, I will focus on the dynamics of State X and Y’s arming process as theymove toward equilibrium. Zinnes et al (1976a) show that the key to uncovering these patternsis to determine the sign of the discriminant D which, for Richardson’s model, is given byD=(a-b)2+4k1 (R17)The discriminant D can take on any one of three values: D > 0, D <0, or D 0. In additionto calculating D, we should also calculate the eigenvalues E1 and E2 for Richardson’s model.E1 = [-(a+b)+[(a+b)2-4(a -lk)]”]/2 (R18)= [-(a+b)-[(a+b)2-4( b-llc)]‘a112 (Ri 9)E1 and E2 can either be(i) distinct real numbers: E, E2(ii) distinct complex numbers: E, u + iv and E2 = u - iv33(iii) equal real numbers: E1 = E2If D > 0, then E1 and E2 will be distinct real numbers. In this case, the arming patternbetween States X and Y around equilibrium is given by Graph 1.3 (Note: Graphs 1.3 to 1.5taken from Zinnes et al, 1976a). Graph 1.3 shows that when D > 0, State X and Y arm in away that leads them nearly directly to their race’s equilibrium point. If D <0, however, thematter is less clear cut. Two arming patterns are possible when D <0. When D <0, E1 andE2 are distinct complex numbers. The arming dynamic between States X and Y around theirrace’s equilibrium will depend upon the value of u. If u = 0, States X and Y will arm tolevels that continually revolve around their race’s equilibrium point, but they never approachit. This process is shown in Graph 1.4. If, on the other hand, u <0, States X and Y will armin a way which spirals in toward their race’s equilibrium value. This result is shown in Graph1.5. In the final case, D = 0. If, in addition to D = 0, State X and Y’s defence coefficients,k and 1, both equal zero, then, States X and Y will arm in pattern which tends quite stronglytoward their race’s equilibrium in a nearly straight line fashion. The dynamics of this outcomeare similar to those portrayed in Graph 1.1. If instead 1 equals zero but k does not, then StatesX and Y will still arm in a pattern which leads them toward equilibrium in a straight line afashion. The arming pattern in this case is similar to that shown in Graph 1.1. Finally, ifk equals zero but 1 does not, then States X and Y will arm at levels that approach their race’sequilibrium point in, again, nearly straight line fashion as shown in Graph 1.1. Zinnes andher colleagues have thus shed an important new light on arms race dynamics with their34GRAPH 1.3: Stability in Richardson’s arms race system when all parameters are not positive,Case IY/(X, Y)/x35GRAPH 1.4: Stability in Richardson’s arms race system when all parameters are not positive,Case IIY(X, Y)Y36GRAPH 1.5: Stability in Richardson’s arms race system when all parameters are not positive,Case IllY(X, Ye)x37analysis of Richardson’s model.4. OTHER MODELS: RICHARDSON’S RIVALRY MODELRichardson proposed still other arms race models. Two of the more interesting onesare the rivalry and the submissiveness models. The rivalry model is a simple extension of thehis basic linear model. He suggests that a state will be concerned by a rival’s armaments onlyif that rival possesses more armaments than it does. It is, then, the discrepancy between onestate’s arms levels and those of its adversary that drives an arms race. Richardson thusamends his basic model as follows.dX/dt = k(Y - X) - aX + g (R20)dY/dt = l(X - Y) - bY + h (R21)The parameters k, 1, a, b, g and h continue to be interpreted as before. Here, too, Richardsoncalculated stability conditions. Richardson found thatk+a+l+b>O (R22)and38kb+la+ab>O (R23)together constitute the stability conditions for his rivalry model. But k, 1, a, b, g and h should,Richardson thought, all be positive. This, then, would mean that the rivalry model wouldalways predict a givenarms race to end up being stable. Because he was unwilling to varyhis assumptions regarding the values of k, 1, a, b, g and h and because he knew that,empirically, not all arms races are stable, Richardson abandoned the rivalry model.5. THE SUBMISSIVENESS MODELWith the submissiveness model, Richardson drops his assumption that arms racesbehave in a linear fashion. In region B of Graph 1.6, Richardson argued that the threat posedto State X by State Y’ s armaments would be such as to impel State X to increase the rate ofgrowth of its armaments: dX/dt> 1 (Zinnes, 1976). In region A, the threat posed to StateX by State Y’s annaments, Richardson thought, would be so small as to impel State X toreduce the rate of growth in its armaments and, in region C, to be so great as to overwhelmState X such that it could not hope to counter State Y and thus gives in by reducing the rateof growth in its armaments. In regions A and C, dX/dt < 1. Richardson could have chosento formalize this analysis in any one of a number of different ways. He chosedX/dt = kY(1 - s(Y - X)) - aX + g (R24)39GRAPH 1.6: The dynamics of Richardson’s submissiveness modelYREGION CREGION BREGION Ax40dY/dt = lX(1 - p(X - Y)) - bY + h (R25)In this model, k, 1, a, b, g and h continue to hold the same meaning as before. The parameterss and p are new. Their meaning is not clear. Indeed the model itself is not easy to understand.But Zinnes (1976) shows that if States X and Y begin their race from an initial point ofparity, the submissiveness model breaks down to Richardson’s original formulation. If StateY initially possesses substantially more armaments than State X, then State X in effectbecomes submissive and begins to disarm. If State X initially possesses substantially morearmaments than State Y, then State X will continue to arm but at a decreasing rate as timeprogresses. Richardson did not, however, fully work out the stability conditions for thesubmissiveness model. The task would have been an extremely difficult one because themodel is so complicated.It is, finally, interesting to note that Hollist (1 977b) tested Richardson’s rivalry andsubmissiveness models against US and Soviet annual military expenditure data running 1948-70. Hollist’s analysis supported neither model.This, then, concludes the derivation of Richardson’s basic linear action-reaction armsrace model, and the discussion of its various properties. The model takes the formdX/dt = kY - aX + g (R5)41dY/dt = lx - bY + h (R6)with equilibrium occurring at the pointX’ (kh + bg)/(ab - ki) (RiO)and= (ig + ah)/(ab- ki) (Ri 1)In the next chapter, I will turn to step four in the modelling process, where I will analyze thevarious attempts that have been made to test Richardson’s and other’s arms race models forthe US-Soviet arms race. I will make several recommendations, based on that analysis,regarding testing procedures for action-reaction arms race models.42CHAPTER II: EMPIRICAL TESTSOF RICHARDSON’S AND OThERA-R ARMS RACE MODELSRichardson’s basic linear model has been tested in hundreds of quantitative arms racestudies. Attempts have been made to estimate it for many races, including the Arab-Israelirace, the India-Pakistan race and, most importantly, the US-Soviet race. Despite its basicallysound conceptual framework, however, Zinnes (1980) was able to cite instance after instancewhere Richardsons linear action-reaction model was disconfirmed by empirical data.The most perplexing of such examples given by Zinnes (1980) concerns the US-Sovietarms race. Most quantitative studies aimed at estimating Richardson’s model for the US-Soviet military competition have concluded that the US and Soviet Union have not beenengaged in an action-reaction arms race. In this chapter, I will look at the various attemptsthat have been made to estimate the parameters of Richardson’s and others’ models for theUS-Soviet arms race with a special emphasis on the types of data that they have been testedagainst. Clear and logical advances have been made in this particular dimension of arms raceresearch methodology. Indeed, much has changed since the earliest quantitative studies of theUS-Soviet arms race.43In this chapter, I will begin with some introductory notes on the nature of the US-Soviet military competition. Secondly, I will discuss the general procedure for estimatingRichardson’s model. Third, I will look at the initial attempts that were made to estimateRichardson’s model for the US-USSR. Fourth, I will discuss what researchers had learnedfrom this initial round and where they went with their work. Finally, I will discuss the currentstudies of the US-Soviet arms race.1. THE US-SOVIET MiLITARY COMPETITIONMany professional observers of military affairs contend that the US and Soviet Unionhave been engaged in an action-reaction arms race. Arms racing is one way to maintain abalance of power between rivals. Indeed, a US Defence Department document entitled SovietMilitary Power, dated 1981, states, in fact, that the aim behind America’s armament programis to counter the threat posed by the build-up of armaments in the Soviet Union. A similardocument, issued by the Soviet Ministry of Defence, entitled, From whence the threat topeace, dated 1982, states that the aim behind Soviet Union’s armament program is to counterthe threat posed by the build-up of armaments in the United States. Sivard’s (1985) data,shown in Table 2.1, on weapons system development in the US and Soviet Union shows justhow closely a weapons development in one country is matched by the other. Scholars have,accordingly, tried to estimate the parameters of Richardson’s action-reaction model for theUS-Soviet arms race. What did they find?44TABLE 2.1: Action-reaction in weapons system development in the US and USSR.Action System ReactionUSA 1945 atomic bomb 1949 USSRUSA 1948 intercontinental bomber 1955 USSRUSA 1952 thermonuclear bomb 1953 USSRUSSR 1957 ICBMs 1958 USAUSSR 1957 man-made satellite 1958 USAUSA 1960 SLBMs 1968 USSRUSA 1966 multiple warhead 1968 USSRUSSR 1968 anti-ballistic missile - 19’72 USAUSA 1970 MIR.V system 1975 USSRUSA 1982 long-range cruise missile 1984 USSRUSA 1983 neutron bomb 198? USSRSource: Savard (1985)452. ESTIMATING RICHARDSON’S MODEL FOR THE US-USSR: SOMEINTRODUCTORY NOTESFirst, a few methodological notes are in order. Before estimating Richardson’s model,for any arms race, it must converted from differential equation form to difference equationform. Data on the US-Soviet race compiled by various research institutes is specified indiscrete rather than continuous form. FromdX/dt = kY - aX + g (R5)dY/dt = 1X - bY + h (R6)we write the following equivalent statement:X, - X1 = kY1 - aX1 + g (R26)-Y.1 = lX1 - bY1 + h (R27)We can simplify these statements as follows:46X = kYLI + (1- a)X1 + g (R28)= lX.1 + (1 - b)Y1 + h (R29)Secondly, Richardson had assumed that hostile states compete with each other on the basisof each side’s total armed might. He thought that an individual state’s annual aggregatemilitary expenditures would accurately reflect its total armed might. A state’s annualaggregate military expenditures would include all monies budgeted to its armed services ina given year. I have term this perspective the macro perspective.3. THE EARLY ATTEMPTS AT ESTIMATING RICHARDSON’S MODEL FOR THEUS-USSRMost studies which have attempted to estimate Richardson’s model for the US-Sovietarms race test it against US and Soviet aggregate annual military expenditures. These macrostudies (Abolfathi (1978), Chatterji (1969), Gillespie et al (1977a), Gregory (1974), Hamblinet al (1977), Roust (1977a,b), Hollist and Guetzkow (1978), Lambelet (1973), Lambelet etal (1979), Luterbacher (1974, 1975), Majeski and Jones (1981), Milstein (1972), Rattinger(1976b), Saris and Middendorp (1980), Shisko (1977), Strauss (1972, 1978), Taagepera et al(1975), Wagner et al (1973), Wallace (1979, 1980a), and Zinnes and Gillespie (1973); seealso Schrodt (1978b) for a discussion of the problems one faces when attempting to estimate47Richardson’s model) generally show that the US and Soviet are not engaged in an action-reaction arms race. American military expenditures at time period t depend only uponAmerican military expenditures at time t-l, the previous time period. Soviet previous periodmilitary expenditures do not seem to enter into the American expenditure calculus. Sovietmilitary expenditures at time period t, likewise, seem to be self-driven.More technically, many have found that the US defence coefficient, k, is zero and thatthe Soviet defence coefficient, 1, is zero. The US, by custom, has been labeled as nation X,and the Soviet Union as nation Y. Substantively, neither the US nor the Soviet Union,Richardson’s model accordingly suggests, feels threatened by the military expenditures of theother. This result will be termed, throughout this study, as the non-interaction outcome.A smaller number of researchers (e.g., Majeski, 1985; Majeski and Jones, 1981;Ashley, 1980), on the other hand, have reported finding statistical evidence that suggests thatk, the US defence coefficient, is in fact non-zero, but, curiously, that 1, the Soviet Union’sdefence coefficient, is zero. This suggests, according to Richardson’s model, that the US doesfeel threatened by the military expenditures of the Soviet Union, but that the Soviet Uniondoes not feel threatened by the military expenditures of the US. This result will be termedthe asymmetric outcome. Were US and Soviet military expenditures really independent ofeach other? Or were the non-interaction and asymmetric outcomes the result of somemethodological deficiency.484. THE US-SOVIET ARMS RACE REOPERATIONALIZED: THE USE OFWEAPONS STOCKS.Serious doubts were raised about the propriety of testing Richardson’s model for theUS-Soviet arms race against each side’s annual aggregate military expenditures. Theassumption underlying the use of aggregate military expenditures in the US-Soviet case hasbeen that increases in aggregate military expenditures reflect increases in armament levels(Hollist (1977b)). In fact, increases in military budgets may simply reflect increased costs ofproduction without any increase in capability. Those who do use aggregate militaryexpenditure data in their work must, moreover, assume that military cost-effectivenessremains constant not only over time, but over space as well (Wallace (1979)). Yet this is notlikely to be true. Thus some researchers (for example, Lambelet (1973), Lambelet,Luterbacher, and Allan (1979), Allan and Luterbacher (1981), McGuire (1977), Desai andBlake (1981), and Squires (1982)) began to consider the prospect of testing Richardson’smodel against total US and Soviet weapons stockpile data rather than military budget data.Weapons stocks might more accurately reflect a nation’s military strength than would itsmilitary expenditures.While the unit of analysis for these researchers has changed from US and Sovietaggregate military expenditures to US and Soviet total weapons stocks, their outlook remainsmacro oriented. Lambelet (1973), for example, created an index of US total conventional49weapons capability and an index for total Soviet conventional weapons capability. Each indexwas structured by multiplying manpower by firepower by mobility of forces for each state.McGuire (1977), conversely, tested Richardson’s model against data on total US and Sovietstrategic warhead deployment levels, total US and Soviet megatonnage levels, and total USand Soviet nuclear missile and bomber deployment levels running 1960-73. He found littleevidence to suggest that the US and Soviet Union were engaged in an action-reaction armsrace. Indeed, for the most part, such studies have not provided conclusive empirical evidencefor the existence of an action-reaction arms race between the US and Soviet Union.5. THE US-SOVIET ARMS RACE REOPERATIONALIZED: THE USE OFWEAPONS STOCKS AND MILITARY BUDGETS.Another macro perspective developed in light of the finding of non-interaction in US-Soviet military expenditures was that evidence of action-reaction interaction in the US-Sovietarms race could be uncovered if both stocks of weapons and military budgets were analyzedtogether. Taagepera (1979-1980) first raised this question. This idea is just beginning toreceive attention in the arms race literature. To date only Organski and Kugler (1980),Luterbacher, Allan and Imhoff (1979) and Ward (1984a) have conducted systematic analysesof the weapons stocks-military budget hypothesis. In his study, Ward developed and testedhis own weapons stock-budget model. From his tests, he concluded (1984a: 309):50The United States and the USSR do appear to be reactive to one another, yetnot through budgets alone. Rather they each try to achieve or maintain a leadover the other in terms of the stocks of weapons, both strategic andconventional, for which the budget is spent.Despite Ward’s tentative successes, there are sound theoretical and methodological reasonsto question the validity of his findings. In the next section, I will question the validity of themacro approach to US-Soviet arms race analysis.6. THE US-SOVIET ARMS RACE REOPERATIONALIZED: A LOOK ATINDIVIDUAL WEAPONS SYSTEMS WITH CROSS PURPOSES.McCubbins (1983) took arms race research methodology into a new direction. Heargues that rival nations do not, as Richardson suggested, engage each other in a single raceover “total armed might.” The US and Soviets do not, as McCubbins argues, compete witheach other, in an action-reaction mode, when it comes to total armed might because neitherside in the US-Soviet case has the economic capability or institutional where-with-all to reactto the full range of the other’s military capability, as a single package, with a single responseeven though each side may fear the other’s total armed might.McCubbins (1983) instead argues for a disaggregate approach to US-Soviet arms raceanalysis. He argues that the US and Soviet Union compete with each other, in an actionreaction mode, when it comes to specific, individual weapons systems with cross-purposes.51For example, if one side steps up its deployments of heavy bombers, the other would respondby stepping up its deployments of jet interceptors. Conceivably, he furthermore argues, theUS and Soviet Union could engage each other in several individual races, the object of anyone race being a particular weapons systems and a particular counter weapons systems. Atany given time, some sub-races, as we may call them, may be heating up while others arecooling down. Accordingly, testing an arms race model against aggregate militaryexpenditures (the sum of all monies expended by each side on all weapons systems andgeneral upkeep) could lead to misleading parameter estimates. A model would have to betested against data on individual subraces.McCubbins developed and then tested a model which set State X’s deployment of aparticular weapons system to be a function of State Y’s deployment of a counter-purposeweapons system and State X’s own GNP. He found, for example, that US deployment levelsof strategic bombers, as he had expected, depend upon Soviet deployments of strategicinterceptors. He, nevertheless, also looked for evidence of a system to system competition.For example he tested to see if US deployments of strategic bombers depended on Sovietdeployments of strategic bombers. He found very little evidence of system to systemcompetition.To summarize, McCubbins’ study is important in that it specifies a sound theoreticalapproach to the US-Soviet arms race analysis and in that it does show that the US and52Soviets are engaged in an action-reaction arms race. Rather than competing over aggregatecapability levels, the US and Soviet Union compete when it comes to individual weaponssystems with cross purposes.7. SYNTHESISBeyond the question of what sort of data Richardson’s or others’ arms race modelsshould be tested against in the US-Soviet case, there is the question of what constitutes anappropriate time frame for the analysis of the US-Soviet arms race. Lucier (1979) has arguedthat arms race parameters, for example k, a, g, 1, b, and h from Richardson’s model, do notremain constant over time. Changes in leadership, changes in standard operating proceduresand even the signing by two adversaries of an arms control agreement can cause arms raceparameters to change. If a data set spans a number of changes in the value of a particulararms race parameter, some complementary, some running in the opposite direction, thenresults obtained from statistical tests aimed at estimating its value might be misleading. Totest his hypothesis, Lucier used a variant of Richardson’s model, the organizational processmodel:X =He found, using data on the naval expenditures of Britain, Japan and the US during the53interwar period, that indeed the value of q changed over time.Few, if any, arms race researchers studying the US-Soviet race had considered thepossibility that the parameters of that race might not be constant over time. Hollist (1977b),for example, did not consider this possibility. Hollist’s analysis of the US-Soviet arms racewas based on a military expenditure data set which spanned twenty-two years running 1948to 1970. He tested Richardson’s model against this data set and found no interaction in US-Soviet military expenditures. Surely things happened over this twenty-two year period whichmight have caused the values of the parameters of Richardson’s model to change. Forexample the sub-period 1962-1970 must have seen a change in the value of the SovietUnion’s defence coefficient. This was a period of unprecedented nuclear build-up for theSoviet Union, a build-up which historians tell us was incited by the Soviet Union’s weaknessin the face of America during the Cuban Missile Crisis. Since Hollist’s data set spanned thislikely parameter change we must be careful how much weight we attribute to his findings.US-Soviet arms race researchers could take a lesson from Rattinger’s (1976: 502)study of the Arab-Israeli arms race. Rattinger wrote, formost nations in the area, military spending is a rather meaningless indicatorof capability because of the complex interplay of military aid, regular armsprocurements, gifts and nonmaterial forms of payment by political allegianceto arms donors, and the like. This fact--together with the high degree ofhostility and comparatively good information on hardware levels--makesdefense spending less important as a perceptual variable in the Middle East54context.Rattinger thus created indexes of strategic capability for individual weapons systems, e.g. fortanks, for each side in the Middle East conflict which included information on numbers andquality. He wanted to see, for example, if the Arabs and Israels were competing when it cometo individual weapons systems such as tanks, or missile boats. On top of that, Rattinger brokedown his data sets into two periods, one running 1956 to 1967, the other running 1967 to1973. The reasoning behind this break-down involves a series of key military and politicalchanges which beset the Middle East after the 1967 war. So Rattinger used, to summarize,desegregated data. In addition, he recognized that the parameters in the model that he wastesting likely did not remain constant over time. His analysis, accordingly, showed evidenceof a Richardsonian-type action-reaction race between the Arabs and Israelis, particularly inthe period 1956-1967.From this discussion, two important methodological points have been made. Theprocess of estimating an action-reaction model, be it Richardson’s or any other model, for theUS-Soviet arms race, must be preceded by a careful consideration of the factors which drivethat race. There is strong theoretical and empirical evidence to suggest that the US-Sovietarms race is made up of a number of subraces, the object of each being some weaponssystem, deployed by one side, and a weapons system with a cross-purpose deployed by theother. It would be necessary, then, to conduct several tests of a model, each test55corresponding to a particular race, or subrace, over a particular weapons systems and aparticular counter weapons system. Models should be tested against deployment levels ofthese systems. A picture of the overall US-Soviet military competition could then be sketchedon the basis of those analyses. There is a need also to consider analytical time frames. Anyparticular subrace will have its own start and end point and these points may lie well withinthe start and end points of the overall military competition between two rivals. The start andend points of a subrace correspond with the period in which the weapons systems which werethe object of the subrace were dominant in the arms race participants’ strategic or tacticalcalculations.--I will term this approach to arms race model testing, the micro approach tomathematical arms race analysis.While the micro approach may be theoretically sound, it is not without its ownproblems. In particular, sample sizes, under the micro approach, could end up being verysmall (less than 30 points). On the one hand, one must be concerned with choosing a validapproach to US-Soviet arms race analysis. Still, one must be concerned with the technicalrequirements of statistical estimation methods. Small sample sizes, for example, pose anumber of technical problems when it comes to model estimation. In Chapter 5, I willconsider, in detail, the technical problems associated with the micro approach to arms raceanalysis and suggest a way to put it into practise.56In the next chapter, I will examine an alternative approach to explaining the non-interaction and asymmetric outcomes. A number of attempts have been made toreconceptualize the arms race phenomenon since Richardson’s original work. The motivatingforce behind these efforts has been the idea that Richardson’s model has oversimplifiedimportant elements of the arms race phenomenon and as such that it could not capture,empirically, the true dynamics of real world races such as the US-Soviet race. What wasneeded, then, was a new action-reaction arms race model. Several new approaches to armsrace modelling were subsequently to emerge, each with its own set of behaviorial assumptionsand analytical techniques. In essence, arms race research returned to step one of themodelling process.57CHAPTER III: RECONCEPTUALIZINGTHE ARMS RACE PHENOMENONBy the mid 1970s serious questions were being raised about the validity and utilityof Richardson’s model. Richardson’s model, many were to argue, oversimplified the armsrace phenomenon, and as such could not be properly applied in the effort to determine the“true” nature of the US-Soviet military competition. A number of new mathematicalconceptualizations of the arms race phenomenon were subsequently offered as alternativesto Richardson’s modeL Several excellent comprehensive reviews of this material have beenoffered including Moll and Luebbert, 1980; Isard and Anderton, 1988; Busch, 1970; Zinnes,1976; Intrilligator and Brito, 1976. In this chapter, I, too, will offer a review of post-Richardson arms race modelling efforts. This review, however, will be selective, featuringthose developments which are directly relevant to my own arms race modelling effort. Morespecifically, I will focus my review on the key works which have drawn on game theory,control theory (dynamic game theory), expectations theory and economic theory in modellingthe arms race phenomenon. In what follows, I will outline the basic assumptions of each ofthese approaches and I will show the models which were derived from them. I will alsoconsider the extent to which each of these models does capture the action-reaction elementof the US-Soviet arms competition. We have returned, then, to step one of a second loop inthe mathematical modelling process.581. THE GAME-CONTROL THEORY APPROACHGame theory is based on two key behaviorial assumptions (see for example, Zinneset al 1978b; Lichbach, 1989, 1990; Shubik, 1982, 1984; Simman and Cruz, 1973, 1975a;Smale, 1980; Harasanyi, 1965; Brams and Wittman, 1981; Brito and Intrilligator, 1980;Gillespie et al, 1975, 1977a; Brams, 1975; Brams and Kilgour, 1988). First, players or statesact in accordance with some goal. Second, a goal cannot, for the most part, be realized unlessit is pursued in accordance with some sort of strategy. For that strategy to be successful, itmust, to one degree or another, take into account the goals strategies of opposing players orstates. The concepts of goal and strategy apply to a wide range of human activity andcertainly to the activity of arms racing. The main criticism against the application of gametheory to international relations research in general and arms race analysis in particular is thatit assumes that the play in question is a one shot affair with costs and payoffs fixed over thelife of the game. In reality, however, the game of international relations is continuous withcosts and payoffs for the players involved constantly changing. It is on this score that armsrace research turned to optimal control theory. Optimal control theory has provided one wayfor researchers to model state goals and strategies in an arms race situation taking intoaccount the dynamic, continuous nature of international competition.How, specifically, has game-optimal control theory been applied to arms race analysis.Like game theory, control theory views arms racing as a game where each player (or state)59sets itself a goal and a strategy for obtaining that goal. An arms race can be conceptualizedas a game to the extent that each participant takes into account the goal/strategy of itsopponent when formulating its own goal oriented strategy. The concepts involved in gametheory can be formalized using optimal control theory mathematics. Under optimal controltheory, goals can be defined by a control variable, e.g., u(t), and the environment by a statevariable, e.g., x(t). State, control and time variables together define a complete and closedsystem:d.x/dt = F(x(t), u(t), t) (CT1)Equation CT1 is known as the plant equation. Control theory assumes that at any point intime, a system is in some state. Control theory mathematics allows a nation to make thesystem respond in different ways at different time periods by manipulating the value of itscontrol variable. Indeed a set of alternative “future histories” for a system can be plotted bya state by considering different control variable values. The desired yield from a system atany point in time can be mathematically expressed a function of the state and controlvariables and time:J(x, u, t) (CT2)Equation CT2 is known as the objective function. In particular the optimization problem will60take the general form= jTL[x(t), u(t), t]dt (CT3)where u*(t) is selected to optimize I subject todx/dt = F(x(t), u(t), t) (CT1)The optimal value for the control variable, u*(t), can be found by first defming a functionknown as the Hamiltonian, H, and then solving for u*(t). The Hamiltonian combines theobject and plant equations through the use of auxiliary variables, namely, p(t). The auxiliaryvariables are called the adjoint vector.H(x,u,p,t) = L(x,u,t) + pF(x,u,t) (CT4)In Equation CT4, T is the transpose of the adjoint vector and it satisfies the condition-= dH/dX (CT5)u*(t), then, is calculated by findingdH/du = 0 (CT6)61Once u*(t), the optimal strategy, is calculated, then it can be substituted into the plantequation CT1 to obtain what is referred to as an optimal trajectory. An optimal trajectory issimply an optimal pattern of behavior given one’s goals and one’s operating environment.Brito (1972), Brito and Intrilligator (1973) and Simaan and Cruz (1973) were the firstto apply control theory mathematics to the study of arms races. Brito (1972), for example,conceptualized an arms race as a problem in welfare maximization. Each state, in an armsrace dyad, seeks to find some optimal balance between consumption on the one hand, anddefence on the other hand. State A’s defence index, DA, depends upon the stock of weaponscurrently in its possession, MA, and the stock of weapons currently possessed by its opponent,State B, MB. Mathematically,DA = DA(MA, MB) (Bi)DA, State A’s defence index, measures State A’s well being in terms of security at any pointin time. DA increases with MA and decreases with MB. State A’s utility, UA, depends upon itsdefence index, DA, and its level of consumption, CA.UA = UA[CA, DA(MA, MB)] (B2)62State A will seek to maximize its welfare over all time periods given the present value of allfuture utility levels discounted over time. Mathematically, Brito expressed this assumptionin the following way.WA = J e UA[CA, DA(MA, MB)ldt (B3)Brito then imposes constraints on the values of the defence and utility functions. He solvesthe welfare maximization problem by first specifying a parallel set of equations for State Band then applying control theory mathematics. The result is as given by the set of equationsdMA/dt = FA(MA, MB) (B4)dM/dt = FB(MA, M) (B5)In the special case where FA and FB are linear, the optimal solution to Brito’s consumptiondefence model is given by Richardson’s arms race model. Brito was, thus, able to show oneset of conditions under which Richardson’s model might constitute an optimal trajectory fortwo nations engaged in an arms race. This was an important accomplishment because, asGillespie et al (1977c: 226) write, “While Richardson’s equations can be interpreted asdescribing how nations will react to one another in an arms race, they do not explain how the63nations arrived at such a strategy.’ The principal shortcoming of Brito’s work, however, isthat he chose to discuss national goals and environmental conditions only in the most generalof terms. Thus one could only draw general conclusions from his model.Gillespie et al’s (1977c) control theory study was the first to attempt a specificstatement of national goals and environmental contexts for actors involved in an arms race.Nation U was posited to have the dual objective of, one, maintaining a balance of power withits rival, nation X, and, two, minimizing the total number of armaments possessed by it andnation X. The first goal was mathematized by Gillespie et al as followsii = LI [(u(t) - ax(t))2]dt (Gi)The second goal took the form= jT {c(u(t) + x(t))]dt (G2)In Equations G 1 and G2, u(t) represents State U’s armaments and x(t), State X’s annaments.Equations Gi and G2 were combined to give a comprehensive statement of State U’s goals= ST [(u(t) - ax(t))2 + c(u(t) + x(t))jdt (G4)64The operating environment for nation U is defined in terms of State X’s arming behavior.Gillespie et al defined State X’s arming behavior by Richardson’s model wheredx/dt = lu(t) - bx(t) + h (G5)From Equations G 1 to G5, Gillespie et al were able to calculate an optimal trajectory forStates U and X and to calculate arms race stability conditions for State U and State X. Theyapplied this entire framework to the US-Soviet arms race. Gillespie et al’s study, based onannual aggregate military expenditure data, showed that the US-USSR arms race is unstable.Both sides in this race, they argue, would likely continue to arm to ever increasing levels.2. THE EXPECTATIONS APPROACHOver the years, international relations research has borrowed insights from a numberof related academic disciplines. In the 1970s and 1980s arms race researchers began turningto economics for help in their efforts to develop more realistic models of arms races. Inparticular, arms race researchers had begun to consider that states calculate their armsrequirements as much on the basis of what they expect an opponent’s future armament levelwill be as they do on what an opponent’s current and past arrament levels are or were.65Fortunately, economists had long recognized the role that people’s expectations of thefuture values of different economic variables play in the functioning of an economic system.Accordingly, many conceptually deep, but mathematically simple ways were developed tomodel people’s expectations, including the extrapolative expectations approach, the adaptiveexpectations approach and the Rational Expectations hypothesis. These approaches are generaland can be applied to a wide range of economic and social problems, including the study ofarms races.Majeski (1985), for example, argued that certain aspects of a state’s militaryexpenditure behavior could be modelled by reference to the theories of extrapolative andadaptive expectations. Majeski began by assuming that State X’s military expenditures at timeperiod t should, first, depend upon internal bureaucratic momentum within that state. Majeski-modelled the impact of bureaucratic processes in State X on its armament expenditurecalculus as the sum of its past period expenditures as shown in Equation Ml.X = a1X (Ml)Yt = (M2)Equation M2 shows the impact of bureaucratic processes in State Y on its armamentexpenditure calculus. Next, when formulating its armament requirements for time period t,66State X would consider the question, at what level would State Y’s armaments at time periodt be? State X must, in this case, formulate an expectation regarding State Y’s futurearmament level. Extrapolative expectations theory provides that State X could formulate thatexpectation simply by taking a weighted average of State Y’s past period deployment levels.Majeski denoted State X’s expectation by YP, whereYP, = b1Y (M3)Thus Equation Ml is rewritten as followsX, = + bYP, (M4)A parallel set of equations can also be specified for State YXP c1X,, (M5)Thus Equation M2 can be rewritten as follows= dY,1 + cXP, (M6)67Finally, Majeski drew on the adaptive expectations hypothesis. He posited that State X wouldalso wish to take account of any errors it had made in the past when calculating itsexpectations of State Y’ s future armament levels and accordingly take compensatory action.Adaptive expectations theory provides a simple solution to this problem. Majeski simplyincorporated an error term into his model which represented the difference between State Y’ sactual deployment level at time period t and State X’s expectation of State Y’s time periodt deployment level. Majeski denoted State X’s prediction error by YE whereYE-YP (M7)State Y’s prediction error is, similarly, given byXEL = X - XP (M8)Thus Equations M4 and M6 can be amended as followsX = aX1 + bYPL + g1YE1 (M9)= + cXP + hIXEU (M1O)68Equations M9 and MlO constitute the final form of Majeski’s expectations model.Majeski applied his expectations model to the US-Soviet arms expenditure race. Hefound evidence that the US and Soviet Union are engaged in an asymmetric reaction armsrace. US military expenditures at time period t, he found, depend upon, one, previous periodUS military expenditures and, two, previous period Soviet military expenditures. Sovietmilitary expenditures at time period t, in contrast, depend only upon Soviet previous periodmilitary expenditures. Majeski interpreted this result to have come about as a consequenceof the disparity between US and Soviet economic-industrial capacity. The US, having a largereconomic-industrial base than the Soviet Union, could more easily afford the luxury ofmatching Soviet expenditures while the Soviets, with their weaker economic-industrial base,could not do likewise.3. ECONOMIC CONSTRAINT THEORYCaspary (1967), dissatisfied with the way in which Richardson had formalized StateX and Y’s economic constraints, offered a new economic constraint formulation. Richardson,he felt, had oversimplified the way in which economic constraints could impact each nation’sarmament program. Caspary argued that the constraint should be non-linear rather than linear.He began by rewriting Richardson’s equations. From69dX/dt=kY-aX+g (R5)dY/dt = lx - bY + h (R6)He wroteD = dX/dt = a(kY/a - X + g/a) (Cl)D’ = dY/dt = b(1X/b - Y + h/b) (C2)This formulation is similar to Richardson’s rivalry formulation in that it states that a nationwill increase its military strength per unit time in response to a difference between somefraction of its existing force level and some fraction of its opponent existing force level. D,then, shows State A’s desired arms increase per unit time. D’ shows the same for State Y.Caspary made the point, however, that neither State A nor State B could arm without dueconsideration to its available resources. More specifically, he argues that there are diminishingreturns to military expenditures. For example, additional expenditures by a state for militarycapability are likely to yield less security when expenditures are already high fraction of itstotal available resources for military items than if they are low. Caspary formalized theseassumptions with the following equations.70p(dXldt) = a(C - MX)(l- (C3)p’(dY/dt) = a(C’ - M’X)(l- e”’ (C4)In Equations C3 and C4,C = the total resources that State X has available for military itemsD =as defined aboveN = the per unit cost of new armaments in State XM = the cost to State X of maintaining one unit of old military equipmentp = a dimensional constant used by Caspary to convert from armament levels/time todollars/time in State X.What Equation C3 shows is that if ND/C is large, which means that State X’s desired militaryexpenditures are set at a point which is a large fraction of its total available resources formilitary expenditures, then, in practice, the actual increase in State X’s military expenditureswill be less than its desired increase in military expenditures. Caspary never did71operationalize his model. He was content to show how resource constraints could theoreticallyaffect the levels of arms accumulation in states engaged in an arms race. That in itself wasan important contribution.3.1 STOCK ADJUSTMENT THEORYA much simpler way to formalize the idea that each state must face economic andinstitutional constraints on its armament programs is suggested by the work of Nerlove(1958). Intrilligator (1975) suggested the potential of Nerlove’s (1958) stock adjustmenthypothesis in arms race research and so far, it has seen a fair amount of use (see Brito(1972), Gillespie and Zinnes (1975), Gillespie et all (1977c), Majeski (1983a, b) and Ward(1984)). There are different variations of the hypothesis, but essentially it states that thechange that actually occurs in some agent’s stock level from time t-1 to time t will be somefraction of what that agent wished to effect in his stock levels over that same time period.4. SUMMARY: WHERE TO FROM HERE?Arms race research has made important conceptual and analytical advances sinceRichardson’s original work. Here, then, is where we stand today. Researchers now assumethat a state engaged in an arms race with a rival power arms in accordance with some goal,such as maintaining a balance of power with that rival. Moreover, some assume that a state72will design and implement a strategy to achieve its arming goal. For that strategy to beeffective, a state must, to one degree or another, take into account the goals and strategiesof its rival. In an effort to formalize the concepts of goal and strategy, researchers haveemployed the methodologies of game-control theory. Expectations theory has been employedto formalize the assumption that states calculate their arms requirements as much on the basisof what they expect an opponent’s future armament levels will be as they do on what anopponent’s current and past armament levels are or were. The idea that no state can armwithout due regard to its available resources is important. The stock adjustment hypothesisshows a great deal of promise in facilitating the formalization of the economic constraintassumption.Despite these conceptual advances in mathematical arms race modelling, however,researchers still have not been able to uncover any strong evidence of action-reaction in theUS-Soviet arms race. Why? Each approach, game theory, expectations theory, stockadjustment theory, in practise, has been treated as a way to examine arms races in and ofitself. Some studies focus only on the gaming aspects of arms racing. Some focus only onthe expectations aspects of arms racing. Still others focus only the economic aspects of armsracing. But arms races have gaming dimensions, expectations dimensions and economicdimensions all at the same time. Focusing on one approach, then, amounts to anoversimplification of the arms race phenomenon. This could explain why researchers havestill not been able to uncover any strong evidence of action-reaction interaction in the US-73Soviet military competition. An intuitively appealing strategy, then, would be to combine thegame-control, expectations and economic approaches into a single, more comprehensiveapproach to arms race analysis. Indeed, three major attempts have already been made in thisregard, namely, Ostrom (1978b) Majeski (1983a), and Ostrom and Marra (1986).Each, first, argued that an arms expenditure process, particularly the US armsexpenditure process, must be desegregated before it can be analyzed and modelled. Insteadof developing a single model describing US military expenditures at time t, Ostrom (1978b),Majeski (1983a) and Ostrom and Marra (1986) developed four separate but interconnectedmodels of the US arms expenditure process, each describing the behavior of a different policymaking group in the process. The four models describe the expenditure decision rules of theDefence Services Agency, the President, the Congress, and the Department of Defence, eachof which plays a crucial role in the determination of the US defence budget. In modelling thebehavior of these groups, Ostrom (1978b), Majeski (1983a), and Ostrom and Marra (1986)drew on the theory of adaptive expectations, a theory which describes how actors mightformulate forecasts of future conditions which have a bearing on the policy problems ofconcern to them. They also drew on game theory insofar as they argued that each of thepolicy making groups in question has a specific goal in mind with respect to the defencebudget. They, finally, also drew on Nerlove’s stock adjustment hypothesis as a way toformalize the real world fact that there will be a disjunction between what each policy groupwould like to have as a defence budget and what political and economic conditions will in74the end allow as the budget. Majeski’s (1983a) study was successful in many ways, but hewas unable to find any statistical evidence that external threat, as measured by Soviet militaryexpenditures, has any impact on the US military expenditure decision process over the period1953-79. He concluded that this was further evidence for the growing belief that US militaryexpenditures are driven principally, if not entirely, by domestic, rather than external, forces.Ostrom (1978b) and Ostrom and Marra (1986) uncovered evidence to the contrary.My work, like Ostrom’s (1978b), Majeski’s (1983a), and Ostrom and Marra (1986)is an attempt to meld together some of the main assumptions concerning the arming behaviorof states which have arisen since Richardson’s original work. I have, however, approachedthis task from a much different angle. Unlike Ostrom’s (1978), Majeski’s (1983), and Ostromand Marra (1986), I will approach my study of the arms race behavior of states from anaggregate perspective, as though states behaved as unitary rational actors. I do not mean tosay here that a single person makes the arming decisions in any given state. Indeed thearmament decision process in each country is competitive, involving many persons and manygroups. Nevertheless, a single arming decision ultimately emerges from this competition. Iwill assume that the aggregate outcome of an armament decision process can be modelled asthough it were made by a unitary rational actor (Williams and McGinnis, 1988). Secondly,I will draw on the Rational Expectations hypothesis rather than the adaptive expectations inmodelling state forecasts of future conditions. The adaptive expectations hypothesis whichMajeski (1983) used differs in critically important ways from the Rational Expectations75hypothesis which Williams and McGinnis (1988) and McGinnis and Williams (1989a) andI have used. Under the Rational Expectations hypothesis, actors, be they states or individuals,can form unbiased expectations of future variables. Actors can and do make mistakenestimates from time to time, but their mistakes are not systematic. Under the adaptiveexpectations hypothesis which Majeski used in his work, actors, in contrast, can makesystematic errors in their expectations. It seems reasonable, however, to assume that actorsdo pay attention to how well they have done in predicting future conditions. If the expectationrule that an actor happens to be using systematically has yielded over or under estimates, thenit is reasonable to assume that actor will modify his expectation rule such that his expectationerrors become less systematic. Hence my use of the Rational Expectations hypothesis.More specifically, I will derive an action-reaction arms race model from the followingset of assumptions, drawn from game-control theory, Rational Expectations theory and stockadjustment theory:ASSUMPTION 1: The behavior of each of two states engaged in arms race is goal oriented.In my model, each state seeks to maintain some balance of forces between itself and itsopponent in each successive time period throughout the life span of the race.ASSUMPTION 2: Each state must face economic and institutional constraints on itsarmament program. The impact of political, economic, and institutional constraints on a76state’s arming program are such that the change that actually occurs in each state’s force levelfrom time t- 1 to time t will be some fraction (between zero and one inclusive) of the changethat each state wanted (desired) to effect in its force level over that one time period. This isessentially Nerlove’s (1958) stock adjustment hypothesis.ASSUMPTION 3: Each state takes account of its rival’s future armament program whenformulating its own armament program. But no state knows exactly what its rival’s armingstrategy will be. Each state is assumed, however, to be able to act in accordance with theRational Expectations hypothesis. That is, each state can form an informed, unbiased estimateof its adversary’s armament plans, and hence, each state can form an expectation of what itsadversary’s future armament level will be.In essence, in the next chapter, I will return to step one of the modelling process. From myassumptions, I will be able to derive a new action-reaction arms race model. In a laterchapter, I will demonstrate the utility of my model by testing it for the US-Soviet arms race.That test will reveal the illusive action-reaction component of the US-Soviet militarycompetition.77CHAPTER IV: A GSR ACTIONREACTION ARMS RACE MODELArms race researchers have long been engaged in a quest to model the arms racephenomenon mathematically. Today, the quest continues. Many approaches have been appliedalong the way, notably, game-control theory, expectations theory and the economic-stockadjustment approach since Richardson’s original work. Taken individually, none of these newapproaches has, however, faired much better than Richardson’s: no model has yet been ableto adequately reveal, empirically, the dynamics of real world races such as the US-Sovietrace, as scholars intuitively understand those dynamics. The problem is that arms races, likemost social phenomena, are multi-dimensional. They have gaming dimensions, economicdimensions, and expectational dimensions all at once. I propose, therefore, to combine theassumptions and insights of the game theoretic, expectations, and economic approaches intoa more comprehensive framework, and then derive there from my own action reaction armsrace model. In effect, I will, in this chapter, carry out steps one and two of the modellingprocess.I will begin by, (1), defining the phenomenon to be modeled, namely, an arms race,then, (2), consider various questions concerning the level of analysis at which my study will78be conducted, (3), specify the unit of analysis, (4), discuss the behavioral assumptions behindmy model, (5), formalize assumption 1: state goal formation, (6), formalize assumption 2:economic constraints on goals, (7), formalize assumption 3: forming a strategy, (8), presentmy new arms race model, then, (9), ask, does my model conform to my definition of an armsrace?, and finally, (10), discuss the equilibrium properties of my model.1. DEFINING THE PHENOMENON TO BE MODELEDIn this chapter, I will derive an action-reaction arms race model. An arms race has twokey defining features. First, two rivals can be said, at a minimum, to be engaged in an armsrace to the extent that each side takes into account its rival’s armaments, past, current, orfuture when deciding on its own future annament level (Gray, 1971, 1976). Further, for anarms race to be an anns race, it must be driven by the pressure of the military rivalrybetween the states involved and not by forces exogenous to that rivalry, such as domesticpressures in either or both of the states involved to promote economic growth throughmilitary development programs (Wallace, 1979). What specific decisions and processes couldgenerate behavior consistent with this arms race definition?792. LEVEL OF ANALYSISMost mathematical arms race research is conducted at a high level of aggregation.That is, analysis has mostly centered on the state. Conducting arms race research at the levelof the state as actor is reasonable, for, despite arguments to the contrary, principally, thoseput forward by the Globalist school of international relations, the state is and remains theprime mover in international politics (for the Globalist perspective, see Mendlovitz (1975)and for the state centric perspective, see Holsti (1985)). States compete. States arm. Statesfight. That is not to take away from the fact that what a state does, to a degree, is theconsequence of internal competition and bargaining between groups and people (Allison,1971).In the military expenditure field, two major works, Ostrom (1978b) and Majeski(1983a), analyze the military expenditure process in the US in terms of the competingindividual groups which most directly influence it. Initial research by Stromberg (1970) andOstrom (1978b) suggests that the military expenditure process in the US involves,sequentially, four groups: the Defense Services Agency which makes a budget request onbehalf of the armed services, the President who submits a formal budget to Congress, theCongress who appropriates monies for defence, and finally the Department of Defense which80expends those mOnies. Each of these groups has its own interests and its own concerns. Butit remains that the competition between them ends in a single budgetary decision. Somehow,all of the various interests within the US (and elsewhere) aggregate and a single armingdecision or policy emerges. In this study, I will focus on that aggregate decision, on howstates act, on how states arm. This immediately raises the question of how to go aboutconceptualizing state behavior.In this thesis, I will draw on the Rational Expectations hypothesis in my effort tomodel the outcome of states’ military allocation processes. Originally developed byeconomists to model macro-economic phenomena (Muth, 1961; Begg, 1982), the RationalExpections hypothesis “refers to the aggregate result of private economic actors utilizingrelevant information in forming unbiased expectations of the future behavior of the economyas a whole [Williams and McGinnis, 1988: 973].” The theory of Rational Expectations hasits detractors. Some of its assumptions are indeed controversial. But as a theory, it isconcise and internally consistent.Although the Rational Expectations hypothesis was originally developed to modelmacro economic phenomena, Williams and McGinnis (1988) have argued its utility inmodelling aggregate political processes and outcomes. They argue that its principal virtue,in contrast to alternative theories of expectations formation, is its emphasis on a sophisticateduse of information by actors. The basic tennents of the Rational Expectations hypothesis as81applied to political processes and outcomes generally and to military allocation processes andoutcomes specifically are as follows.1. Many actors are involved in the formation of public policy, be it defence policy orotherwise. The process of formulating public policy is competitive even in such countries asthe USSR. Ministries, as they must in all countries, compete for a share of their nation’slimited resources. The aggregate outcome of the competition between bureaucrats overalternative policy choices can be modeled as though it were set by a unitary rational actor(see Williams, 1988).2. Bureaucratic actors will draw on a wide range of information in developing and defendingtheir policy positions. Actors are assumed to know the relevant variables which pertain totheir policy concerns and they are also assumed to know the connections between thosevariables. In effect, actors have some model in mind.3. Actors all have access to the same quality of information at the same cost.4. The process of gathering and processing information is highly efficient as is the processof transforming new information into changed behavior.5. Defence policy today, by necessity, must be based on the political, strategic, economic and82technological conditions which are expected to prevail tomorrow. Under the RationalExpectations Hypothesis, actors may make mistaken forecasts regarding future conditions, butthey must be able to correct for their mistakes when realized so that successive forecasts willnot be systematically wrong. More specifically, the aggregate expectation’s of these actorsare assumed to be unbiased.Williams and McGinnis (1988) note that both the US and the Soviet Union do putforth a great deal of effort to learn as much as possible about each other’s positions andintentions. Because each is the other’s principal rival, and thus because both are sointerdependent, such information is crucial to defence policy formulation processes in eachcountry. A good deal of US defence policy is based on information provided by itsintelligence services, namely, the CIA. The CIA has, however, provided erroneousinformation to US policy makers in the past. In the 1960s, for example, the bomber andmissile gaps were grossly overestimated. In contrast, Soviet ICBM deployment levels wereunderestimated by the CIA during the SALT negotiations. But, note Williams and McGinnis(1988), the pattern of overestimation-underestimation in CIA estimates is neither consistentnor systematic. The works of Freedman (1986) and Prados (1986) support this conclusion.6. Following Begg (1982: 29), the hypothesis of Rational Expectations asserts that theunobservable subjective expectations of actors or states would be exactly the true83mathematical conditional expectations implied by the models in question.3. UNIT OF ANALYSISThe unit of analysis issue, put in the form of a question, is this: what, specifically, doarms race participants compete over? Traditionally, mathematical arms race analysis has beenbased on a macro approach. The unit of analysis in mathematical arms race studies has,traditionally, been the annual aggregate military expenditures of competing states, that is, allof the monies expended by competing states on maintaining previous and building new forcesin each year. In this thesis, I will break with that tradition. The methodological basis, as Ihave argued previously, for the macro-approach to arms race analysis is weak. FollowingWard (1984) and particularly McCubbins (1983), I will assume that arms race participantscompete with each over individual weapons systems with cross-purposes. For example, if oneside in a race increased its heavy bomber deployments, the other would increase its jetinterceptor deployments. And my focus will not be on how much each side spends on thesesystems, but rather on the amount of system deployed per year. McCubbins (1983) presentsa strong argument for this perspective which I have set out elsewhere in this thesis. Thus indeveloping my model, I will speak of states countering each other’s deployments of particularweapons systems rather than of states countering each other’s expenditures.844. BEHAVIORAL ASSUMPTIONSWhat specific assumptions, then, do I malce regarding the behavior of states in thecontext of an arming competition? My model is based on assumptions woven together fromgame theory, stock adjustment theory and the Rational Expectations hypothesis. They are asfollows:ASSUMPTION 1: The behavior of each of two states engaged in an arms race is goaloriented. In my model, each state seeks to maintain a balance of forces between itself and itsopponent in each successive time period throughout the life span of the race. That is not tosay that each side seeks to maintain a one-to-one numerical correspondence between its ownforces and those of its adversary. What any given state considers a balance to be will depend,inter alia, upon technological, geographic, and economic factors.ASSUMPTION 2: Each state must face political, economic and institutional constraints onits armament program. The impact of political, economic and institutional constraints on astate’s arming program are such that the change that actually occurs in a state’s force levelfrom time t- 1 to time t will be some fraction (between zero and one inclusive) of the changethat state wanted (desired) to effect in its force level over that one time period.85ASSUMPTION 3: Each state takes account of its rival’s future armament program whenformulating its own future armament program. But no state knows exactly what its rival’sfuture armament plans will be. Each state is assumed, however, to be able to act inaccordance with the Rational Expectations hypothesis. That is, each state can form aninformed, unbiased estimate of its adversary’s armament plans, and hence, each state canform an expectation of what its adversary’s future armament level will be.In what follows, I will formalize each of these assumptions into simple mathematicalstatements. I will then combine these statements in order to obtain my model.5. FORMALIZING ASSUMPTION 1: STATE GOAL FORMATIONIn my model, States K and X are military rivals. One purpose for arms racing,Caspary (1967) notes, is to be militarily prepared vis-a-vis an opponent should war breakout.In order to be so prepared, a state must seek to maintain some sort of balance or proportionbetween its own forces and those of its adversary. The question as to what constitutes abalance in any given situation must be left open. As afready stated, balance need not meana one-to-one numerical correspondence between opposing forces. Factors other than simplenumbers are considered by policy makers when determining balance points between their ownmilitary forces and those of an adversary. For example, Dougherty and Pfaltzgraff (1981:404) write:86Rough parity [under SALT I] was arrived at by political intuition, rather thanby the computation of strict mathematical equality of the superpowers’ missilearsenals. The Soviets were assigned a 40 percent margin of superiority in thenumber of land-based ICBMs and about a one-third margin the number ofocean-based SLBMs. The agreement was widely criticized in the United Statesfor conceding to the Soviets a substantial advantage in missile payload, butratification was justified by the Nixon Administration on the grounds that theUnited States possessed several compensating advantages (such as overseas airand submarine bases, the total number of warheads deployed, the number oflong-range bombers, and qualitative superiority in a variety of importanttechnological dimensions)As a first approximation, State K, then, is simply assumed to set its desired deployment levelfor time period t, denoted K, to be a linear function of its expectation at time t- 1 of StateX’s actual deployment level at time period t, denoted (E’1XJ, times some balance or defencecoefficient, B1. K*t is assumed to also depend upon a grievance term B0, and a predictionerror U, as per Equation 1. State K is assumed to form its expectations of State X’supcoming armament level on the basis of information contained in its defence informationsetK*,= B0 + Bi(EkL1XJ + U, B0, B1 E R (1)X = B’0 + Bi(EXt.iKt) + V, B’0, B’1 e R (2)Equation 1 is, then, State K’s goal equation and Equation 2 is State X’s goal equation.87Equation 1 and Equation 2 are set on the basis of the principle that arms acquisitiondynamics in today’s world are, necessarily, anticipatory. According to Majeski (1985: 220),the arms race participantrecognizes that expenditure decisions to obtain security ... are directly affectedby the choices and behavior of his opponent. In fact, what his opponent doesor will do is probably more important and relevant to the arms race participantin terms of meeting his objectives than his opponent’s prior behavior.Expectations of current and future behavior play a crucial role in this context.The value range of B1, in Equation 1, is left open in recognition of the fact that states do notnecessarily calculate balances in strict one-to-one terms. In a specific context, one couldhypothesize a value for B1, but otherwise, its value must be determined empirically. State K’sdesired deployment level at time t is also influenced by its grievance term B0. One could alsohypothesize a value for B0 in a specific context, but otherwise, its value, like B1, must bedetermined empirically. Finally, State K’s desired deployment level at time t depends on theerror term Ut. In accordance with the Rational Expectations hypothesis, U is assumed to havea mean of zero: E(U) = 0 and it is assumed to be uncorrelated with the expectation termE’1X. U, by definition, reflects any unanticipated developments in State X’s armingbehavior in State K’s armament decision calculus. State X’s desired deployment goal isassumed to be set in a similar fashion based on information contained in its defenceinformation set I’. Finally, in accordance with the Rational Expectations hypothesis, corr(U,VJ is assumed not equal to zero.886. FORMALIZING ASSUMPTION 2: ECONOMIC CONSTRAINTS ON GOALSEach state must weigh its annament objectives in light of political, economic,industrial and institutional realities. Ostrom (1978b) and Majeski (1983a) recognized this factand made compensation by formalizing the potential impact of these factors, the political, theeconomic, the industhal, and the institutional, on a state’s armament decision process byusing Nerlove’s (1958) stock adjustment hypothesis. Ostrom, for example, noted that defenceexpenditures cannot be made without regard to public opinion and that public opinion mayhave a dampening effect on the final amount expended.-The stock adjustment hypothesis systematically relates a state’s desired deploymentlevel at time t, its actual deployment level at time t and its actual deployment level at timet-1. More specifically, the stock adjustment hypothesis is this: the change that actually occursin State K’s force level from time t- 1 to time t will be some fraction, namely S, of the changethat State K wanted (desired) to effect in its force level over that one time period. The samehypothesis can be applied to State X. S and S’ (for State X) are called stock adjustmentparameters and they take values in the range 0 S,S’ 1. They show the extent to whicheach respective state is capable, politically, economically and institutionally, of achievingsome predefined desired armament goal. Mathematically,K - K1 = S(K*L - K1), 0 S 1 (3)89X-X1 = S(X*L- X1), 0 S’ 1 (4)Equations 3 and 4 are general and can accommodate any number of definitions for K*t andX respectively. In general, S = 1, in Equation 3, implies that State K’s actual deploymentlevel will fully adjust to some predefined desired deployment level in each t. 0 < S < 1implies State K’s actual deployment level will only partially adjust to some predefineddesired deployment level in each t. S = 0 implies that there will be no adjustment, that is,change, in State K’s actual deployment level over time t.Having specified for State K its desired deployment goal equation and its stockadjustment equation, I then substitute the former into the latter. I do the same for State X.These substitutions yield a preliminary description of each state’s arming behavior. I termthese equations, State K and X’s constrained goal equations. Mathematically,- K = S(B0 +B1(E’XL) + U, - K1.) (5)By multiplying out Equation 5 and setting the result in terms of K, we obtain,K = SB0 + SBi(EILiXL) + (1 - S)K1 + SU, (6)Equation 6 gives State K’s actual deployment level at time period t, K,, which is a function90of its expectation at time t- 1 of State X’s actual deployment level at time period t, EkIX,State K’s own deployment level at time t- 1, K, and State K’s prediction error term U. S,State K’s stock adjustment coefficient, now refers specifically to State K’s political, economicand institutional capability to effect a desired balance, as defined by Equation 1, between itsforces and those it expects its rival, State X, to hold in each successive time period over thelife span their race. S = 1 would indicate that State K had the economic and institutionalcapacity to fully adjust its actual deployment level for time t to a level where a desiredbalance between its own forces and those it expected State X was going to deploy at time twould obtain. If 0 <S < 1, then that adjustment would only be partial and if S = 0 no suchadjustment would occur over time t.When S = 1, Equation 6 reduces to Equation 1. That is, State K’s actual deploymentlevel at time t will equal its desired deployment level for time t. When 0 < S < 1, the term(1- S)KL1, in Equation 6, comes into play. (1 - S)K1 is State K’s carry-over term. Toreiterate, when 0 < S < 1, State K can only partially adjust its actual deployment level at timet to a level where a balance between its own forces and those it expected State X woulddeploy at time t would obtain. (1 - S)K1 shows the extent to which State K, accordingly,underdeployed forces in time t-1. State K would, thus, need to compensate at each successivetime t by factoring into its armament balancing calculus, as given by Equation 6, the amount(1 - S)K1.91A solution can, similarly, be found for XL, State K’s actual deployment level. Bysubstitution,X-X1 = S’(B’0 + Bi(EXtiKt) + V - X11) (7)Therefore, State B’s actual deployment level in time t is given by,X = S’B’0 + S’B’1(EK) + (1- S’)X1 + S’V, (8)Equation 8 can be read as Equation 6 was read. -7. FORMALIZING ASSUMPTION 3: FORMING A STRATEGYThus State K and X are assumed to each have a particular arming goal, balancing theother’s future military capabilities, subject to political, economic and institutional constraints.When developing an arming policy for itself, a state would realize that policy, if it is to besuccessful, must, to one degree or another, take into account the expected arming goals andstrategies of its opponent. The Rational Expectations hypothesis provides a concise andsystematic way to model such expectations.92Under the Rational Expectations hypothesis, States K and X may form theirexpectations of each other’s arming strategies thusly. We can assume that State K’s defenceinformation set I contains Equations 1 to 8. Similarly let us assume that State X’s defenceinformation set also contains Equations 1 to 8. Placing Equations 1 through 8 into eachstates information set is completely consistent with the Rational Expectations hypothesis. AsWillams and McGinnis (1988: 977) note there should be aconsiderable overlap in the two states’ information sets. This restriction isjustified, indeed required, by the presumption that these two states are lockedinto a competitive relationship in which each devotes considerable effort todetermining the likely behavior of the other. Thus, each should obtain accessto essentially the same information to predict each other’s militaryexpenditures. For if one state finds a certain piece of information useful informing expectations of some factor relevant to their joint securitycompetition, even if it relates to domestic constraints, then the other state mustnecessarily find this same piece of information useful.Thus State K would find all of the equations in its information set which pertain to State X’sbehavior useful. State X would, similarly, find the equations which pertain to State K’sarming behavior useful. Let us assume further that State K routinely takes measurements,expectations, of those equations pertaining to State X, contained in its defence informationset, at time period t-1. Let us, likewise, assume the same with respect to State X. Let E1denote an expectation made by State K at time period t-1 and let E1 denote an expectationmade by State X at time period t-1.93The problem now for State K is that its constrained goal equation contains anunknown variable, E1XL, its expectation at time t-l of State X’s actual deployment level attime period t. In order to determine a value for E’1XL, State K would need to be able to forman expectation as to what State X’s arming strategy might be.K SB0 + SBL(EkL1XJ + (1 - S)K + SU (6)State X must, similarly, try to solve for EXL.IKL, its expectation at time t-l of State K’s actualdeployment level at time period t from Equation 8, its constrained goal equation.X = S’B’ + SBi(EXtIKt) + (1- S’)X1 + S’V (8)Under the Rational expectations hypothesis, State K can obtain a value for E’1XL by takingthe expectation at time t-1 of Equation 8, State X’s constrained goal equation, which iscontained in its information set. This course would be an efficient use of information in thatEquation 8 contains a great deal of information regarding State X’s arming behavior.Accordingly, the Rational Expectations hypothesis would assert that State K’s expectation ofEquation 8 would be given byEkL1X = S’B’0 + S’B’1(EK)) + (1 - S’)X1 +94(9)Equation 9 gives State K’s counter strategy equation. It gives State K’s expectation at timet- 1 of State X’s actual deployment level at time period t, E1X. EkiXL depends uponE1(EiK), which is State K’s expectation at time t-1 of State X’s expectation at time t-lof State K’s actual deployment level at time t. E’IX also depends upon X1, State X’s actualdeployment level at time t-l and EkL1Vt, State K’s expectation at time t-l of State X’sprediction error at time t. State X can, similarly, solve for EXL1K by taking the expectationat time t- 1 of Equation 6, State K’s constrained goal equation, which is contained in itsinformation set. In this instance, Equation 10 would obtain.EXLiK = SB0 + SB1(E’X)) + (1 .- S)K +S(E1U) (10)Equation 10 gives State X’s counter strategy equation. Thus State K can directly substituteEquation 9 into 6 and State X can directly substitute Equation 10 into Equation 8. The resultof these substitutions is given, respectively, by Equations 11 and 12.K, = SB0 + SB1’ ’0+SB1’ ’(EK))+95SB1(l- S’)XLl + SBiS(Ek1V) + (1- S)KL1 + SUL (11)X = S’B’0 + S’B’10+SB1(EXLkX))+S’B’1(1 - S)K1 + SBiS(EXiU) + (1 - S’)X1 + S’V (12)Equations 11 and 12 give a preliminary description of State K and X’s arming behavior. Butthey each still contain unknown expectational variables. Equations 11 and 12 can, however,be substantially simplified by referring back to the basic assumptions which underlie theRational Expectations hypothesis.First, State K must solve for the term EXL1KL, State X’s expectation at time t-1 of StateK’s actual deployment level for time t, in Equation 11. State K could figure out the structureof State X’s counter-strategy equation and use it to solve for EXt1KL. But State X’s counterstrategy equation contains information which State K has already used since State X’s counterstrategy equation is simply State X’s expectation at time t-1 of State K’s constrained goalequation. State K would do better to refer to the Rational Expectations assumption that anyexpectation will equal the real value in question plus some error term. State K can thusassume that96EXC1Kt= K + M, where E(M) = 0 (13)Equation 13’s value to State K is that it is concise and that it does contain new information,namely, the error term M. The variable EkL1Xt, State K’s expectation at time t-1 of State X’sactual deployment level for time t, from Equation 12 can, similarly, be simplified. State Xcan assume thatE1X = X + N, where E(N) = 0 (14)The relationships specified in Equations 13 and 14 can be used to simplify Equations 11 and12. By substituting Equation 13 into Equation 11 and Equation 14 into Equation 12 weobtain.K, = SB0 + SB1’ ’0+SB1’ ’(EK)) +SB1’ ’(EM)) + 5B1(1 - S’)X1 +SBISkiV) +97(1- S)K1 + SU, (15)X, = S’B’0 + S’B’10+ S’B’1(E’,X,)) +SBiSBi(EX,i(Nt)) + S’B’1(1-S)K,1 +S’B’1(EUL) +(1-S’)X,1 + S’V, (16)Equations 15 and 16 can be further simplified. Again, we note that under the RationalExpectations Hypothesis, the term E1(K,), State K’s expectation at time t- 1 of its owndeployment level for time t, from Equation 15, will be that value plus some error term. Thesame holds true forE1(X), State X’s expectation at time t-1 of its own deployment levelfor time t, from Equation 16.Ek,1K = K, + P,, where E(P,) = 0 (17)EXL1X, = X, + Q,, where E(Q,) = 0 (18)98With Equations 17 and 18, Equations 15 and 16 reduce as followsK = SB0 + SB1’B’Q + SB1’B’LKL + SB1’ ’PL +SB1Bi(EkCMJ + SB1(1- S’)X1 +SB1’(EV) + (1 - S)KL1 + SUL (19)X = S’B’0 + S’B’10+ S’B’1XL + S’B’1Q, +SBiSBI(EXtiNJ + S’B’1(1- S)K1 +SBiS(EXLiUL) + (1 - S’)X1 + S’V (20)Finally, if States K and X set Equations 19 and 20 in terms of K and X, respectively, thenEquations 21 and 22 would result. Equations 21 and 22 give, finally, my GSR action reactionarms race model (GSR is an acronym for Game, Stock Adjustment, Rational Expectations).998. A NEW ARMS RACE MODELMy formalization of Assumptions 1, 2 and 3 ultimately yields the arms race Equations21 and 22. What is particularly interesting about Equations 21 and 22 is that they arespecified solely in terms of observed, measurable variables. State K’s actual militaryarmament requirement for time period t becomes a function of its own armaments at timeperiod t- 1, the previous period, the armaments of State X at time period t- 1 and some errorYL. State X’s actual military armament requirements for time period t are similarly driven.Equations 21 and 22, given immediately below, constitute the final form of my action-reaction arms race model.(SB0 + SB1’ ’0) SB1(1- S’) (1- 5)K = + X1 + K + Y (21)1 - SB1’ ’ 1 - SB1’ ’ 1 - SB1’ ’(S’B’0 + S’B’10) S’B’1(1 - 5) (1— 5’)X = + K + X + Z, (22)1 - SB1’ ’ 1 - SB1’ ’ 1- SB1’ ’where Y and Z are error terms which take the form100SBiSEkLiV, + SU, + SB1’ ’(P, +E1M,)= (21a)1 - SB1’ ’SB1EXU,+ S’V, + S’B’1(Q, + EXL1N,)Z = (22a)1- SB1’ ’The parameters B0 and B1 are, respectively, State K’s Richardsonian gricvaice and defencecoefficients. S is State K’s stock adjustment coefficient. The parameters B’0, B’1 and 5’ are,respectively, State X’s Richardsonian grievance, defence and stock adjustment coefficients.In what follows, I will refer to the parameter arrangements [(SB0 + SB1’ ’0)I1- SB1’ ’]and [(S’B’0 + S’B’10)/1 - SB1’ ’] from Equations 21 and 22 as State K and X’s GSRgrievance terms. I will refer to the arrangements [SB1(1 - S’)/l- SB1’ ’] and [S’B’1(1 -S)/1 - SB1’ ’] as State K and X’s GSR defence coefficients. Finally, I will refer to thearrangements [(1 - 5)/i - SB1’ ’]and [(1 - S’)/l - SB1’ ’]as State K and X’s GSR stockadjustment terms. (GSR is an acronym for game, stock adjustment, Rational Expectations.)1019. DOES MY GSR MODEL CONFORM TO MY DEFINITION OF AN ARMS RACE?The structure of my arms race model is determined by how one chooses to specifyeach state’s desired deployment goal. By specifying each state’s goal as one of maintainingsome balance of forces with its rival in each successive time period, as given by Equations1 and 2,= B0 +B1(EX) + U, B0, B1 € R (1)X = B’0 +B1(EXIKJ + V, B’0, B’1 e R (2)the final step in my arms race derivation will yield a model which consists of two differenceequations which take the form: K = f(X1,K11, Y) and X = g(K1,X, Zr). The structuralspecification K = f(X1,K, Y) and X= g(K1.,X.1, Z) is important insofar as it constitutesan irreducible minimum specification for an arms race as I have defined it.First, I defined an arms race to be a situation where, at a minimum, each side takesinto account its rival’s armaments, past, current, or future, when deciding on its own currentarmament level. In the final step of the derivation of my model, each state takes into accountits adversary’s past period annament level when calculating its own current armamentrequirements. Secondly, I stated that for an arms race to be an arms race, it must be driven102by the pressure of the military rivalry between the states involved and not by forcesexogenous to that rivalry. Operationally, this means that each state’s arming behavior mustbe described solely on the basis of perceived external threat, and not domestic imperativessuch as employment. Why then does each state, as posited by my model, take account of isown past period armament level when calculating its future armament requirements? InRichardson’s model, the fact that a nation considers its own previous armament level indetermining its future armament level has less to do with pressures emanating from its rivalrythan with pressures emanating from within itself, namely, its own economy and its owninstitutions. Richardson believed that a state’s own previous armament expenditures wouldhave a dampening effect on its future expenditure levels insofar as those previous armamentshad drained resources from its economy. In fact, however, the opposite can also be true.Previous expenditures can spur on future expenditures for purely domestic reasons such asthe need to maintain employment levels. Neither of these instances would, however, meshwith my definition of what an arms race is. What I have done is to alter, radically, the natureof the impact that a state’s own previous armament levels has on its future armament calculusthrough my use of the stock adjustment hypothesis. Through the stock adjustment hypothesis,each state takes account of its own previous period deployment level when calculating itsfuture armament requirements only to the extent that its previous period armament level wasbelow the level necessary to achieve its goal of maintaining a desired balance of forcesbetween itself and its rival in that period. Thus the structure of my model is consistent withmy definition of an arms race.10310. EQUILIBRIUM PROPERTIESLoosely speaking, the term arms race equilibrium refers to a point, if it exists, ofmutually agreeable arms balance between two rivals engaged in an arms race, though it neednot necessarily refer to the occurrence of one-to-one balance. Equilibrium points can be stableor unstable. If, after having reached equilibrium, two adversaries return to it whenever someexogenous shock causes them to move to a different armament level, then that equilibriumpoint can be said to be stable. Otherwise, such a point would be said to be unstable. In thissection, I will set out the equilibrium and stability-instability conditions for my GSR model.Equilibrium analysis has long held an important place in mathematical arms raceanalysis. Richardson (1951, 1960a) contended, though never systematically demonstrated, thatarms races, particularly unstable ones, lead to war. A great deal of energy has been expendedin determining the validity of that position (Caspary, 1967; Saaty, 1964; Gray, 1971, 1973,1974, 1975, 1976; Smoker, 1965; Wohlstetter, 1974; Lambelet, 1971, 1975; Wallace, 1979,1980a, 1980b, 1980c, 1982; Weede, 1980; Smith, 1980; Siverson and Diehl, 1990; Diehl,1983; Altfeld, 1983; Morrow, 1989). Still, the question has not yet been settled. Leidy andStaiger (1985) suggest three other practical reasons why arms race equilibrium analysis isimportant. First, the existence of a stable equilibrium point for a given arms race suggests thatthe race will have a finite duration. This in turn suggests that there will be a bound on thedrain of resources expended in executing the race. Second, the atmosphere attending a race104which has a stable equilibrium point might be less tense, making it easier for the participantsto negotiate their way out of the race. Thirdly, the character of an arms race can be alteredby changing its parameters. One could, for example, suggest through equilibrium analysishow to turn an unstable race into a stable race.Under what conditions, then, will State K and X’s arms race behavior, as describedby Equations 21 and 22, tend toward equilibrium? Under what conditions will thatequilibrium be stable or unstable? I will begin by rewriting Equations 21 and 22 in asimplified form. This form will make easier the task of calculating equilibrium and stabilityconditions.K = L0 + L1X + L2K + Y (23)X = L’ + L’1K + L’2X1 + Z (24)Equation 23 corresponds to Equation 21 and Equation 24 corresponds to Equation 22. L0 andL’0 represent, respectively, State K and X’s GSR grievance terms, L1 and L’1 State K and X’sGSR defence terms and L2 and L’2 represent State K and X’s GSR stock adjustment terms.In my model, equilibrium occurs when the difference between State K’s actual deploymentlevel at time t and State K’s actual deployment level at time t-1 and State X’s actualdeployment level at time t and State X’s actual deployment level at time t-1 is,105simultaneously, zero. MathematicallyK - K1 = 0 (25)X-X1 = 0 (26)Thus, if we disregard the error terms Vt and Z, we can rewrite Equations 23 and 24 asfollows:K - K = L0 + L1X + LK1 - K1 = 0 (27)X- X1 = L’ + L’1K + L’2XL1 - X.1 = 0 (28)Equilibrium in my arms race model, then, occurs at the pointL0(1 - L’2) + L1’0K, = (29)(1 - LJ(1- L’2) - L’1- L2) + L’10(30)(1 - L)(1 - L’2) - L’1106Under what conditions will the arms race equilibrium point (K1,X1) be stable? Twomethods can be applied here, the graphical method and the characteristic root method. First,from Equations 27 and 28, we can obtain the following demarcation curves. Equation 31represents the locus of points in the (Ku, X) space where the condition K - K1 = 0 issatisfied. Equation 32, similarly, represents the locus of points in the (K1,X) space wherethe condition X-X1 0 is satisfied.L0 (L2-1)X1 = K (31)L1 L1L’0 L’1X•1 = + K1 (32)(1-L’2) (1-L’2)What, then, are the arms race stability conditions for my model as implied by Equations 31and 32? It should, first of all, be noted that the parameters L0, L1, L2, L’0, L’1 and L’2 cantake on a wide range of values, from negative to positive. This means that there is a largenumber of possible stability conditions. The equilibrium point (K1,X1) can be graphicallydetermined to be stable or unstable by plotting Equations 31 and 32 and determining theirphase trajectories as was done in Chapter 1 using Richardson’s model.107Basic examples of a stable and an unstable race are given below in Graphs 4.1 and4.2. In Graph 4.1, stability obtains: (L2 - 1)1L <0 and L’1/(1 - L’2) > 0 where I(L2 - 1)1L> IL’/(1 - L’2)I. Here, the dynamics of State K and X’s arms race are such that both side’sarmaments tend nearly directly toward their race’s equilibrium point. Graph 4.2 shows anunstable arms race: (L2 - 1)/L <0 and L’1/(1- L’2) > 0 where I(L2 - 1)/L < PL’1/(1 -L’2)I. In this instance, States K and X arm in a way which always leads them away from theirrace’s equilibrium point.Alternatively, one could determine the equilibrium characteristics of an arms race bycalculating the Eigen values or characteristic roots, A1 and A2, of the characteristic equationfor Equations 23 and 24. A1 and A2 are as follows.(L1 + L’1) + [(L1)2 - 2L1L’ + 4L2L’ + (L1)2]mA1 = (33)2(L1 + L’1) - 2L1L’ + 4L2L’ + (L’1)2]’A2 = (34)2In the case where estimates of L1, L2, L’1 and L’2 yield values for A1 and A2 which areboth less than unity in absolute value, the arms race in question whose parameters arerepresented by L1, L2, L’1 and L’2 can be said to be stable. If either A1 or A2 or both areGRAPH 4.1: Stable equilibrium in the GSR modelXL1XL - XL1 = 0108KL - K1 = 0(K X1)K1GRAPH 4.2: Unstable equilibrium in the GSR modelxt-1 XL - XL1 = 0109\K - K1 = 0(TJ yet-1, 1k\K1110greater than unity in absolute value, then the race in question has an unstable equilibriumpoint.The conditions for the existence of a stable equilibrium point can be further refmed.Smith (1980) argued that if an arms race did have a stable equilibrium point, the practicalimpact on the outcome of a race (e.g., war-no war) may be no more different than if the racehad been unstable. Specifically, she drew a distinction between interesting anduninterestingly stable amis race equilibria. She defined an interestingly stable arms race asone where its solution, 90 percent of its equilibrium value, is realized within the lifespan ofthe race. For example, she estimated that the Russo-Japanese race which lasted for 18 yearswould not have approached equilibrium, given its dynamics, for about 90 years after its start.In this context, the fact that the Russo-Japanese race had a stable equilibrium point has littlemeaning. An uninterestingly stable race will likely occur when equilibrium occurs at someextraordinarily high armament level (as compared to the levels which obtained at the start ofa race). The time it takes for a race to approach 90 percent of its equilibrium solution, Smithstates, can be calculated using the equationt =1og0(0.1)/1ogA)where A is the largest of the characteristic roots A1 or A2.111In this chapter, I have derived a new action-reaction arms race model based onassumptions taken from game theory, Rational Expectations theory and stock adjustmenttheory. This model was, further, derived under assumption that states compete over individualweapons systems with cross-purposes. It is, therefore, a micro arms race model and as suchconstitutes a key additional component of the micro approach to mathematical arms raceanalysis which I have been developing in this thesis. Having derived my GSR model andhaving demonstrated its equilibrium properties, I will, in the next chapter, move on to discussfour testable propositions concerning the arms race behavior of states which derive from mymodel. In the chapter which follows that, I will apply my micro approach to mathematicalarms race analysis to the US-Soviet arms race where I have been able to uncover strongevidence of action-reaction interaction in the US-Soviet strategic nuclear warheadcompetition.112CHAPTER V: DERiVING TESTABLEPROPOSITIONS FROMTHE GSR MODELIn the last chapter, I derived my GSR action-reaction arms race model. My modelspecifies optimal arming behaviors to be followed by States K and X, two arms raceparticipants, given that each state has the goal of maintaining a balance of forces with theother over each successive time period over the lifespan of their race, given that each state’sarming goal must be subject to economic- constraints, and given that each staten takes- accountof the arming strategy of the other. State K’s actual deployment level for time period tbecomes a function of its own deployment level at time period t- 1, the previous period, StateX’s deployment level at time period t-1, and an error Y. State X’s actual deployment levelfor time period t is similarly driven. My model is, accordingly, structurally identical toRichardson’s action-reaction arms race model. Given immediately below is the final form ofmy action-reaction anns race model(SB0 + SB1’ ’0) SB1(1 - S’) (1- S)K= + X+ Kt1+YL (21)1 - SB1’ ’ 1 - SB1’ ’ 1 - SB1’ ’113(S’B’0 + S’B’10) S’B’1(1- 5) (1- S’)X,= - + -K1+ -X1+Z (22)1 - SB1’ ’ 1- SB1’ ’ 1 - SB1’ ’where Y1 and Z, are error termsSBiSEkLiVL + SU + SB1’B’l(PL + EiM)Y = (21a)1 - SB1’ ’SBiSEXLiUI + S’V + S’B’1(Q, + EXLiN)Z = (22a)1SBS’E’ -The parameters B0 and B1 are, respectively, State K’s Richardsonian grievance and defencecoefficients. S is State K’s stock adjustment coefficient. The parameters B ‘Corline Scott B0,B‘are, respectively, State X’s Richardsonian grievance and defence coefficients. S’is StateX’s stock adjustment coefficient. (SB0 + SB1’ ’0)/1 - SB1’ ’ from Equation 21 is StateK’s GSR grievance term. Moreover, from Equation 21, 5B1(1- S’)/l - SB1’ ’ is State K’sGSR defence term and (1 - S)/i - SB1’ ’ is State K’s GSR economic term. From Equation22, (S’B’0 + S’B’10)/1 - SB1’ ’ is State X’s GSR grievance term, S’B’1(1- S)/1 -SB1’ ’ is State X’s GSR defence term and (1 - S’)/l - SB1’ ’ is State X’s GSReconomic term.114In this chapter, I will continue the modelling process sequence by discussing fourmain propositions concerning the arming behavior of states which derive from that model.Firstly, I will derive testable propositions which follow from my application of the RationalExpectations hypothesis to arms race modelling. These propositions concern the extent towhich aims race actors, in given empirical context, do form their expectations of theirarmament requirements in accordance with the dictates of the Rational Expectationshypothesis. I will then determine the conditions under which State K and X’s armscompetition will manifest itself in the form of an action-reaction competition and under whatconditions it will manifest asymmetric reaction or non-interaction. Finally, I will discuss theprocedure to be followed in estimating my GSR model.1. DERIVING TESTABLE PROPOSITIONS: RATIONAL EXPECTATIONS ANDARMS RACE MODELLING.A principal assumption underlying my GSR arms race model is that states formexpectations of each other’s arming goals and strategies and use those expectations indetermining their own force requirements. Most importantly, my model is based on theassumption that states form these expectations in accordance with the Rational Expectationshypothesis. How valid is that assumption? Can its validity be empirically established in aspecific arms race context? In their premiere work with the Rational Expectations hypothesisand the US-Soviet military competition, Williams and McGinnis (1988) suggested that if the115US and Soviet Union do calculate their military expenditure requirements in accordance withthe Rational Expectations hypothesis, then certain implications or propositions can be derivedregarding the dynamics of those expenditures over time. The validity of using RationalExpectations in arms race modelling could be assessed by analyzing US and Soviet militaryexpenditure series and determining if those propositions hold.Williams and McGinnis’ (1988) case has already been set out in an early chapter, butessentially, they argued that each side in a dyad would base its military expenditurecalculations on what it expected the other would be expending, on its expected domesticeconomic fatigue, and on some prediction error. These expectations would be based oninformation which both held in common (a basic assumption of the Rational Expectationshypothesis). Suppose, then, argue Williams and McGinnis, that some exogenous shock causedState K to deploy armaments at a level which fell below the level it calculated that it woulddeploy for time t. This would translate into a negative error in State K’s actual deploymentlevel at time t. If State J were vigilant in collecting and analyzing data on State K’s armingbehavior, as the Rational Expectations hypothesis would assert, then State J would havebecome aware of this upcoming deviation between State K’s planned deployment level fortime t and State K’s actual deployment at time t. State J should take this as a decrease, albeittemporary, in the security threat posed to it by State K. Accordingly, State 3 should lower itsdeployment level for time t below what it had planned. This would lead to a negative errorin State 3’s actual deployment level at time t. The reverse of this case would also be true if116States K and J did form the expectations in accordance with the Rational Expectationshypothesis. If State J were highly efficient in transforming newly obtained information onexogenous shocks in State K’s arming behavior into changes in its own behavior and if StateK were likewise efficient, then one should expect to find a strong contemporaneous crosscorrelation in State K and State J’ s prediction errors. This is a key proposition in Williamsand McGinnis and it did hold using US-Soviet military expenditure data over the years 1954-1987 (using a simple distributed lag model).My GSR model, similarly, suggests that if States K and X are highly efficient inaltering their arming behavior on the basis of new information on exogenous shocks in theother’s arming behavior, then one should find a high contemporaneous cross correlation intheir prediction errors Y and ZL In an empirical context, it would be necessary then todetermine if the following proposition holds or not in order for one to be able to concludethat States K and X do indeed form their expectations of each other’s arming behavior inaccordance with the Rational Expectations Hypothesis.PROPOSITION 1.0: If States K and X do in fact form their expectations of each other’sarming behavior in accordance with the Rational Expectations hypothesis, then the errors inState K and X’s arming behavior, as modelled by Equations 21a and 22a, should be stronglycorrelated. Mathematically corr(Y, Z) 0. This result, it must be noted, is conditional uponthe information contained in State K and X’s defence information sets.117Proposition 1.0, to reiterate, is identical to the one advanced by Williams and McGinnis(1988) in their study of Rational Expectations and reaction in the US-Soviet arms race. Buta high contemporaneous cross correlation in the prediction errors of rival states, they correctlynote, may also result if rivals form their expectations adaptively. An adaptive expectationsprocess could generate such a correlation if expectations formed under that process equaledexpectations formed under the Rational Expectations process. Thus, the result corr(Y, Z)does not equal zero is only a causally necessary, but not sufficient, condition for RationalExpectations formation in the armament processes of rival states.Secondly, Williams and McGinnis argued that prediction errors drive an arms race andnot absolute expenditure levels. Williams and McGinnis (1988: 980) predicted that “In anarms race under Rational Expectations, neither military expenditure series will Granger causethe other.” That is, a knowledge of State K’s military expenditure series will not help one topredict State 3’s expenditure series and vice-versa. Both this prediction and the predictionconcerning State K and 3’s error terms were borne out in Williams and McGinnis. They note,however, thatsystematic delay in observing or acting upon previously available information[regarding an opponent’s error term] may have the consequence that one seriesdoes indeed help to improve predictions of the other series (p. 981).More specifically, they argue, a Rational Expectations arms race model would allow118for the possibility that one state may successfully keep some informationsecret and that the time needed to develop new weapons or to overcomebureaucratic inertia may delay any response to a change in expectations. Allof these factors may be interpreted as ‘inefficiencies’ in the transformation ofnew information into changed behavior .... If one state manages to keep somerelevant information secret, the rival’s reaction to this specific event may bedelayed, but the overall correlation [in prediction errors] would be eliminatedonly in the presence of a continuing pattern of successful deception orconcealment. ... If bureaucratic or technical inefficiencies delay each state’sresponse to new information approximately the same length of time, then theirjoint reaction would be distributed across adjacent time periods, reducing thesize of any particular year’s [error] correlation.... Only if there is a pronouncedasymmetry in the two state’s adjustment times would contemporaneousresiduals no longer be correlated, whereas lagged ones would be (p. 981).Accordingly, a companion proposition to Proposition could be specified as followsPROPOSITION 1.1: If States K and X do in fact form their expectations of each other’sarming behavior in accordance with the Rational Expectations hypothesis, but if there aresubstantial inefficiencies in the transmission and transformation of information between StatesK and X, then the errors in State K and X’s arming behavior, as modelled by Equations 21aand 22a, should show a weak contemporaneous correlation or a strong non-contemporaneouscorrelation. Mathematically corr(Y, 4) = 0 or corr(Y1,Z) 0, where i and j > 0 andi=j or i j. This result is conditional upon the information contained in State K and X’sdefence information sets.Proposition 1.1, like Proposition 1.0, must be qualified. A very weakcontemporaneous, or a strong non-contemporaneous cross correlation in State K and X’s119errors is a causally necessary, but not sufficient, condition for Rational Expecations. Suchcorrelations would also be consistent with an adaptive expectations formation process.Inefficiency in the transformation of new information into new or changed behavior isconsistent with both Rational Expectations formation and adaptive expectations formation.But this fact would not eliminate the key theoretical distinction between Rational andadaptive expections formation processes, namely, that an actor’s prediction errors would beunbiased under Rational Expectations and biased under adaptive expectations.- In the instance where States K andX do experience inefficiencies in the transmissionand transformation of information about the other, then, as Williams and McGinnis suggest,a knowledge of one state’s military deployment series should help to predict the other’sdeployment series. In the next sections, I will spell out, in detail, the predictions my modelmakes with respect to interaction between State K and X’s deployment series.2. DERIVING TESTABLE PROPOSITIONS: THE CAUSE OF THE ASYMMETRICARMS RACEAshley (1980), Freeman (1983), Majeski (1985), Majeski and Jones (1981), andWilliams and McGinnis (1988) have studied the asymmetric outcome. Ashley (1980), Majeski(1985), and Majeski and Jones’ (1981) studies of the USA-USSR military expenditure120competition indicate that the US and Soviets are engaged in an asymmetric arms race.American military expenditures, at time period t, show up as a function of Soviet militaryexpenditures at time t- 1 and US military expenditures at time t- 1, just as the classicalRichardson model would predict. The Soviets, on the other hand, do not seem to beresponding to American military expenditures. Soviet military expenditures at time period thave been found to be a function of Soviet military expenditures at time t-1 only. The US-Soviet arms competition is defined here, then, by a zero defence coefficient for the SovietUnion and a non-zero defence coefficient for the US.When Majeski (1985) found statistical evidence that the US and USSR were engagedin an asymmetric arms race, he adopted Ashley’s (1980) explanation. Ashley had himselffound a similar result. Ashley writes: the asymmetry isthe result of an ‘ordered relationship.’ The United States is a more rapidlygrowing society with greater resources and latitude to invest in militarycapability to respond to the activities of others than the Soviet Union. ... [TjheSoviet Union has sought to extract the maximum resources from its economyto build its military instruments to overcome obstacles and threats posed notso much by U.S. arms but by ‘sustained American lateral pressure.’ Havingalready committed their resources to the fullest to meet the external threat, theSoviet Union has little latitude to react to changes in U.S. capabilities[Majeski, 1985: 239].Ashley’s reasoning can, in fact, be formalized through my GSR model.If, accordingly, we take the United States to be State K we could in fact assign its121coefficient of adjustment (S) the maximum value, namely, one. The coefficient ofadjustment, to reiterate, is a measure of institutional or technical rigidities, or cost of change.America seems little constrained. But what about the Soviet Union, State X? Its coefficientof adjustment (S’) should, accordingly, be assigned a value of less than one, but more thanzero. The Soviets seem to be facing, at a minimum, institutional rigidities which prevent itfrom mirroring US deployments. High opportunity costs would seem, too, to be a factorpushing the Soviet Union’s coefficient of adjustment downward. In short, these valueassignments, S = 1 and 0 < 5’ < 1, if substituted into Equations 21 and 22, will yield anasymmetric arms race outcome. Given, now, is the resultant, Equations 33 and 34, an- asymmetric arms race outcome.(B0 + B1S’B’0) B1(1- S’)K = + X.1 + YY (33)1-BS’ ’ 1-BS’ ’(S’B’0 + S’B’1B0) (1- 5’)X = + X1 + ZZ, (34)1-BS’ ’ 1-BS’ ’where YY and ZZ are error terms.122B1SEkL.iVL + U +B1S’B’(P, + E’1M)= (33a)1-BS’ ’SBiEXCiUL + S’V, + S’B’B(Q + EXL1NL)ZZ = (34a)1-BS’ ’Clearly this result conforms with Ashley (1980). A deeper cause for asymmetric reaction is,- however, suggested by the Rational Expectations hypothesis.Consider that my GSR model, Equations 21 and 22, was derived on the basis that eachof the states in question, States K and X, each formed an informed expectation of the other’sarming goal and arming strategy and used that expectation in coming to a final determinationas to what its time period t armament requirements should be, as given by Equation 21 forState K and Equation 22 State X. Under the Rational Expectations hypothesis, State K wouldknow what the values of State X’s defence policy parameters, B’0, B’1 and 5’ are. Similarly,State X should know the values of State K’s defence policy parameters, B0, B1 and S are (thisinformation should be contained in their defence information sets). If State X knows that S,State K’s stock adjustment coefficient is equal to one, and that its own lies between zero andone, then State X would know that State K should have no carry over term in its GSR123Equation 21. That is, the term (1 - S)/1 - SB1’ ’ times State K’s own previous perioddeployment level, K, would disappear from Equation 21. State K would, under S = 1, 0 <S’ < 1, only take account of State X’s previous period deployment level when determiningits current deployment needs. Indeed State X should know that if S = 1 and 0 < S’ <1, thenState K’s GSR Equation 21 should break down to Equation 33.Since State X’s GSR Equation 22 was derived on the basis that it had formed andused an accurate expectation of State K’s arming goal, and arming strategy, combined witha knowledge of the values of State K’s defence policy parameters, it follows then that StateX’s GSR Equation 22 should break down to Equation 34. If State K is not talcing account ofits own previous period armament level when determining its current period requirement, thenwhy should State X If State K is only talcing account of State X’s previous period armamentlevel when determining its current period requirement, then State X must use thatinformation, and attempt to balance State K’s current period armament level on that basis.This accounting of asymmetric reaction, I will show, directly contradicts the workdone by Williams and McGinnis (1988: 990). They consider one-way or asymmetric armsrace formulations to be of rather dubious validity.fundamental deficienc[yj in any undirectional causation formulation. Althoughan international dispute may indeed have been originally caused by one state’saggressive intentions or behavior, once the target state has learned of theother’s continued animosity and has decided to resist its advances, then both124should form and act upon their expectations of each other’s future behavior.Any attribution of most of the contemporaneous correlation to one side or theother must be justified by reference to a persistent asymmetry in theirrespective capabilities to form and act upon expectations of the other’sbehavior, such as a Soviet inability to predict the outcomes of the complexU.S. domestic policy process or the inability of the U.S. analysts to penetratethe veil of secrecy surrounding Soviet decision making.Ironically, what I have shown with my Rational Expectations arms race model is that one-way causation or asymmetric reaction can indeed come about as a consequence of both sidesforming and acting “upon their expectations of each other’s future behavior.” But mostimportantly, I have shown that, under the Rational Expectations hypothesis, asymmetricreaction can occur even when their is no asymmetry in “their respective capabilities to formand act upon expectations of the other’s behavior . .. .“ Indeed asymmetric reaction can occurjust because each side in an arms race is forming efficient expectations of the other’s armingbehavior.Can an asymmetric arms race really be called an arms race? Wallace (1979: 5) writes:at a minimum, we can only speak of an ‘arms race’ between nations whoseforeign and defense policies are heavily interdependent; the behavior andcapabilities of each nation must be highly salient to the other nations.In Equations 33 and 34, we see the possibility that asymmetric arms races can be highly125interactive affairs. Consider, accordingly, that Equation 34, which gives States X’s actualdeployment level at time t, contains B1, State K’s defence coefficient and B0, State K’sgrievance term. This suggests that State X’s own deployment behavior is affected by StateK’s defence policy parameters. Equation 33 which gives State’s K’s actual deployment levelat time t, similarly, depends upon State X’s defence policy parameters. More specifically, itdepends upon State X’s grievance term B’0 and State X’s defence coefficient B’1.This is important to note because the term asymmetric reaction suggests that one sideis reacting and the other is not. Indeed, Equation 34 shows that X, State X’s actualdeployment level at time t, can be a function of B’1, State X’s own defence coefficient eventhough it is not taking explicit account of State K’s past period armament level whencalculating its deployment level for time t. And, as one might expect, as the value of B’1increases, the value of X increases. This is a significantly new way to look at asymmetricarms races. The traditional Richardsonian explanation of why asymmetric arms races occurinvolved reasoning that State X’s defence coefficient was zero. This was never really veryconvincing especially when two very antagonistic states were the object of the analysis. Fromthis analysis, the following proposition emerges.PROPOSITION 2.0: If, for any given arms race dyad, estimates of Equations 21 and 22 showan asymmetric arms race, that outcome will be due the conditions summarized by either S= 1, 0 < 5’ <1 and B1, B’1 do not equal zero or S’ = 1, 0 <S < 1 and B1, B’1 do not equal126zero.It should also be noted that my GSR model will show an asymmetric arms race obtains if,for any pair of states, one state’s Richardsonian defence coefficient is zero, the other’s is nonzero and both have stock adjustment coefficients which lie between zero and one. If, forexample, B1 0, B’1 does not equal zero and 0 < S,S’ <1, then my GSR model breaks downas followsK = SB0 + (1- S)K1 + (21b)X, = (S’B’1SB) + S’B’1(l - S)K1 + (1- S’)X1 + (22b)Equation 21b shows that State K’s reaction to State X is effected only through itsRichardsonian grievance term B0. That is, State K’s reaction to State X is constant despitevariations in State X’s armament level. State X’s reaction to State K’s, in contrast, is variable.State X’s armament decision calculus at time t, as given by Equation 22b, depends, inter alia,on State K’s previous period armament deployment level. A companion proposition toProposition 2.0, then, can be put as follows.PROPOSITION 2.1: If, for any given arms race dyad, estimates of Equations 21 and 22 showan asymmetric arms race, that outcome will be due the conditions summarized by 0 <S, S’127< 1 and either B1 = 0 and B’1 does not equal zero, or B1 does not equal zero and B’1= 0.3. DERIVING TESTABLE PROPOSITIONS: CONDITIONS FOR AN ACTION-REACTION ARMS RACEMy GSR model specifies that States K and X will engage each other in an action-reaction arms race only under certain conditions.PROPOSITION 3.0 If, for any given arms race dyad, estimates of Equations 21 and 22 showan action-reaction arms race, that outcome will be due the conditions summarized by 0 < S,S’<1 and B1, B’1 do not equal zero.That is, each side must have a non-zero Richardsonian defence coefficient and each side musthave a stock adjustment coefficient which is greater than zero but less than one in order foran action-reaction amis race to occur between States K and X. Why? Under the RationalExpectations hypothesis, State K would know that State X’s Richardsonian defencecoefficient is non-zero and that its stock adjustment coefficient lies between zero and one.State X would, similarly, know that State K’s Richardsonian defence coefficient is non-zeroand that its stock adjustment coefficient lies between zero and one. This information wouldbe contained in their defence information sets.128State K thus could draw two basic conclusions regarding State X’s arming behavior.First, State K would have to conclude that in an effort to realize its arming goal ofmaintaining a balance between its own forces at time t and those it expects that State K willdeploy at time t, State X will take State K’s previous period deployment level into accountwhen calculating its own time t period deployment level (since S’B’1(1 - S)/1 - SB1’ ’times Ku). But, because State X’s stock adjustment term is less than one (but greater thanzero), State K would also have to conclude that State X would not be able to adjust fully itsactual deployment level at time t to a level where balance between its own forces at time tand those it expected State K’s was going to deploy at time t obtained. State X would haveto make up for the shortfall in each successive time period t through its carry over term inthe amount of [(1 - S’)/l - SB1’ ’] times its previous deployment level X1. That is, inaddition to taking into account State K’s previous period deployment level when calculatingits own current period deployment needs, State X would also take into account its ownprevious period period deployment level. State X would come to the same sorts ofconclusions with respect to State K. Because each state would take into account the other’sarming calculus when determining its own armament requirements under RationalExpectations, the net effect will be an action-reaction arms race.1294. DERIVING TESTABLE PROPOSITIONS: THE CAUSE OF THE NONINTERACTION OUTCOME.A special case of arming behavior occurs when S = S’ = 0, that is, when both StateK and State X have zero stock adjustment coefficients. It happens that when S = S’ = 0, theform of interaction posited by my GSR model between States K and X will break-down fromaction-reaction interaction, as given by Equations 21 and 22, to non-interaction, as given byEquations 37 and 38 below.K, = (1)K (37)X = (1)X (38)The significance of Equations 37 and 38 becomes more apparent when they are rewritten asfollowsK - K1 = 0 (37a)X - X,1 = 0 (38a)130Equations 37a and 38a indicate that when both states have zero stock adjustment coefficients,no change occurs from time t- 1 to time t in either side’s deployment level. In effect, thereis no arms race. However, it is interesting to note that Equations 21 and 22 will break downto Equations 37 and 38 when S = S’ = 0 irrespective of the values of State K’s defence andgrievance terms and State X’s defence and grievance terms. That is, the form of arms raceinteraction in my GSR model can break down from action-reaction to non-interaction evenif States K and X have non-zero defence coefficients, that is, even States K and X do feareach other’s military deployments.This property of my GSR model can be used to shed some light on results obtainedin previous research on the US-Soviet arms race. Mathematical arms race analysis began withRichardson’s (1960a) linear action reaction arms race model. His model is given immediatelybelow, specified in difference equation form.X = kY1 + (1 - a)X1 + g (R28)= 1Y. + (1- b)Y1 + h (R29)In Richardson’s model, States X and Y will engage each other in an action-reaction arms raceif k and 1, respectively, State X and Y’s defence coefficients, are both non-zero. If, for anygiven arms race, k and 1 take on values of zero, then Richardson’s model breaks down to an131organizational process model, as shown immediately below.X = (1- a)XL1 + g (R30)= (1- b)Y1 + h (R31)In this instance, State X and Y’s military expenditures are internally driven. It is interestingto note that most studies aimed at estimating the parameters of Richardson’s model for theUS-Soviet arms race (see citations in Chapter 2) suggest that both the US and Soviet defencecoefficients are zero, indeed, that their military expenditures are internally driven. Given thelongstanding political, economic and military rivalry between the US and Soviet Union, thisconclusion is, to the least, counter-intuitive, for it suggests, according to Richardson’s model,that neither side really fears, nor even pays attention to, the military expenditures of the other.As discussed elsewhere in this dissertation in more detail, Richardson had assumedthat hostile states engaged in an arms race would compete with each other on the basis ofeach side’s total armed might. He suggested that a state’s total armed might could berepresented by its annual aggregate military expenditure figures. Richardson’s assumptionimplies, rather dubiously, that each state is capable of reacting to the other’s militarycapability as a single package with a single response. A more reasonable assumption,however, is that while states may fear each other’s total military capability, they react to each132other on a weapons-system by weapons-system basis (McCubbins, 1983). In particular, theycompete over individual weapons systems with cross-purposes. This suggests that many sub-races can occur within the context of a larger military competition. With some sub-racesheating up while others cool down, one could well find no-interaction between the militaryexpenditure series of two rivals even thought each side may really fear the other’s total armedmight. In effect, evidence of action-reaction interaction between two rivals might be maskedby measures of their aggregate military strength. This could well explain why mostmathematical arms race research shows no interaction in the US-Soviet aggregate militaryexpenditure race.One should, then, expect to find no interaction in the aggregate military expenditureseries, or indeed in any aggregate measure of military capability, of rival states. Institutionsin State K and X simply may not be capable of reacting to each other’s total armed mightas a single package with a single response, even though each might fear the total might ofthe other. In this context, State K would know that State X’s stock adjustment would be zeroand State X would know that State K’s stock adjustment coefficient is zero. That informationwould be contained in their defence information sets. With that information, the form of armsrace interaction formalized by my GSR model will break down from action-reaction to noninteraction if S = 5’ = 0 irrespective of the values of B1 and B’1. Substantively, B1 and B’1both equal zero simply means that neither of the rivals in question fears the armaments of theother. But in the case of a true rivalry, B1 and B’1 should not, then, both equal zero. This133analysis suggests the following propositionPROPOSITION 4.0: if, for any given arms race dyad, estimates of Equations 21 and 22 usingdata showing the aggregate military strength of each of the dyad members shows a noninteraction outcome, that outcome will be due the conditions summarized by S = S’ = 0 andB1 and B’1 do not equal zero.Put in terms of my GSR formulation, non-interaction in the US-Soviet aggregate expenditurecompetition is more palatable. It should, finally, be noted that interaction will break fromaction-reaction to non-interaction in my GSR model if State K and X have zero defencecoefficients (i.e., B1 = B’1 = 0), irrespective of the values of their stock adjustmentcoefficients. But where two truly competitive states are the object of the analysis, B1 and B’1should be non-zero.PROPOSITION 4.1: if, for any given arms race dyad, estimates of Equations 21 and 22 usingdata showing the aggregate military strength of each of the dyad members shows a noninteraction outcome, that outcome will be due the conditions summarized by B1 =‘=0 and0 S S’ 1.Alternatively, if for any pair of rival states, both have zero stock adjustment coefficients, thenmy GSR model will break down to a non-interaction model even if one of the states in134question has a zero Richardsonian defence term.PROPOSITION 4.2: If, for any given arms race dyad, estimates of Equations 21 and 22 usingdata showing the aggregate military strength of each of the dyad members shows a non-interaction outcome, that outcome will be due the conditions summarized by S = 5’ = 0 andeither B1 = 0 and B’1 does not equal zero, or B1 does not equal zero and B’1 = 0.5. MODEL AND PROPOSITION TESTING PROCEDURESEquations 21 and 22 are specified in a form which can be easily and readily estimated.Estimates of these equations would give an indication, through the values obtained for eachGSR parameter, what sort of arms race States K and X happen to be engaged in: actionreaction, asymmetric, non-interactive. But in order to determine which of Propositions 1-4hold in a given case, estimates would have to be obtained for the Richardsonian parametersB1, 5, B’1 and S’. Given the complexity of each GSR parameter, however, how could we tell,in a specific case study, what the values of B0, B1, S, B’0, B’1 and S’ were? The values ofthese parameters could, with a little work, be obtained by also estimating Equations 6 and 8,State K and X’s constrained goal equations.KL = SB0 + SB1(E’X)+ (1 - S)K1 + SU, (6)135X, = S’B’0 + SBi(EXiKt) + (1- S’)X1 + S’V, (8)First, Equations 6 and 8 could not be estimated as they stand. Each contains an unobservedvariable. Equation 6 contains EkiX, State K’s expectation at time t-1 of State X’s actualdeployment level for time t. Equation 8 contains EXL1KL, State X’s expectation at time t- 1 ofState K’s actual deployment level for time t. But by referring back to the RationalExpectations hypothesis, we could assume thatEk1X = X + R, where E(R) = 0 (37)EXiKt = K + S, where E(S) = 0 (38)Thus, Equations 6 an& 8 could be rewritten asK = SB0 + SB1X + (1 - S)K1 + S(U, + B1R) (6a)X S’B’0 + S’B’1K + (1 - S’)X..1 + S’(V + B’1S) (8a)Then, from Equation 6a, we could obtain an estimate for the parameter (1- S) and from thatestimate, calculate the value of S. With a value for S in hand, we could then the divide the136estimate obtained for the parameter SB1 by S to obtain a value for B1. Similarly, we coulddivide the value obtained for SB0 by S to obtain B0. The same procedure could be followedwith Equation 8a to obtain values for S’, B’0 and B’1.Note, however, that if a value of zero is obtained for S from Equation 6a and if eitherone or both of the parameters SB1 and SB0 estimates is zero, then it may be impossible to tellfrom Equation 6a whether or not B1 and B0 have non-zero values. One would know if B1,B0,B’1, and B’0 were non-zero by estimating Equations 1 and 2. Estimating Equations 1 and 2is, however, a complicated task. In addition to containing the variables EkZ1X, and EL.1K ,Equations 1 and 2 also contain the variables K*t and K*t and X are, as you will recallfrom my earlier specifications, theoretical, unobserved variables. K*t refers to State K’sdesired deployment level at time period t and X refers to State X’s desired deployment levelat time period t. K and K can, nevertheless, be calculated. Equations 1 and 2 arereproduced immediately below.K*1 B0 +B1(Ek.IX) + U, B0, B1 R (1)X = B’0 + Bi(EXiK) + V, B’0, B’1 e R (2)First, Equation 37 can be substituted into Equation 1 and Equation 38 can be substituted into137Equation 2 in order to eliminate the unknowns ELlXt and EXE1KL respectively. The specifiedsubstitutions will giveK*t= B0 + B1X, + (B1R + U) (la)X = B’0 + B’1K + (B1SL + V) (2a)How can the variables and X*t be calculated?K*t and X are theoretical, unobserved variables. How then can their values beobtained? The key to estimating Equations la and 2a is to first estimate Equations 6a and 8aand extract from them, respectively, the values S and S’. Next, rewrite Equations 3 and 4,displayed immediately below, setting them in terms, respectively, of K’ and X’. FromK - K = S(K’, - K1) (3)X, - X = S(X*,- X1) (4)we obtain,138K - (1- S)KLIK*t = (39)SX - (1- S’)X1X’= (40)s,As set, Equations 39 and 40 can yield, on the basis of known values, the theoretical values,respectively, K*t and X. With these values, Equations la and 2a can readily be estimated.It is important to note that the procedure just specified is valid, methodologically, onlyif the error terms S(UL + B1R) from Equation 6a and S’(V + B’1S) from Equation 8a are noteach autocorrelated. Why? Specifically, if S(UL + B1R) and S’(V + B’1S) are eachautocorrelated, the standard errors of the estimates obtained for (1- S) and (1- S’) wouldlikely be underestimated. This could lead to misleading conclusions about the statisticalsignificance of the estimates obtained for (1 - S) and (1- S’). In that eventuality, thesubsequent calculation of K*t and X and hence the estimation of B0, B1, B’0, and B’1 wouldbe correspondingly misleading. This leads, finally, to two questions: which regressiontechnique should be used to estimate Equations la, 2a, 6a, 8a, 21 and 22 and what methodshould be used to test the regression results for autocorrelation?One of the three pillars of the micro approach to mathematical arms race analysis139which I have developed in this thesis is that an arms race between two rivals is made up ofa series of subraces. Each subrace has its own focus: each involves a competition wherein oneside in the dyad in question deploys a particular weapons system and the other responds bydeploying a weapons systems designed to counter the former’s effectiveness or purpose.Secondly, these subraces have a start and an end point which may lie well within the startand end points of the overall militaiy rivalry. This is the second pillar of the micro approach.When preparing to study a given arms race, one must, therefore, make a decision as to whatweapons system-counter weapons system subcompetition to study and determine its start andend points. More specifically, a sample used to estimate my GSR model and its componentequations should be set, timewise, to correspond with a historical, economic, and politicalperiod in which parameter values for that subrace are constant (see Lucier (1979)). Thisapproach to parameter estimation could lead to very small sample sizes (less than 30 points).Basic statistical theory is, though, large sample based. However, small sample (n =10) experimental studies by Kmenta (1971) suggest that parameter estimates obtained for Y= a + bX + U would still be unbiased. Standard error estimates would, though, increase asthe sample size decreased under both OLS and GLS. This simply means that it would bemore difficult to obtain significant estimates for the parameters in my GSR componentEquations la and 2a.Equations la and 2a could first be estimated by OLS. The standard Durbin Watson140DW test could then be applied to the results to test for autocorrelation. If autocorrelation isfound, then Equations la and 2a could be reestimated using GLS.Small sample experimental studies by Malinvaud (1970) suggest that parameterestimates for autoregressive equations, such as my GSR component Equations 6a and 8a andmy GSR Equations 21 and 22, will be biased in a downward direction (about 15%) underOLS. Standard error estimates would be unbiased. This, as in the former case, means that itwould be more difficult to obtain statistically significant estimates for the parameters inEquations 6a, 8a, 21 and 22. Thus OLS could be effectively used to estimate Equations 6a,8a, 21 and 22.Malinvaud’s conclusions regarding small sample properties of estimates ofautoregressive models are, however, contingent upon the existence of non-autocorrelated errorterms. OLS would not be appropriate if the error terms in Equations 6a, 8a, 21 and 22 wereautocorrelated. What test could be applied in order to determine if the errors in Equations 6a,8a, 21 and 22 were autocorrelated?When a lagged independent variable is used as a dependent variable, the standardDurbin-Watson test (using Ordinary Least Squares) would give a misleading indication of theextent of that autocorrelation. It could even show no autocorrelation exists when in fact itdoes exist. Normally, equations which do contain a lagged independent variable as a141dependent variable could be estimated by OLS and the extent of autocorrelation could bedetermined through a relatively new test, Durbin’s h. But Durbin’s h is a large sample test.Its properties have not yet been determined for small samples (Ostrom, 1978c).The autocorrelation test best suited to the conditions under which we must operateunder the micro approach to mathematical arms race analysis is the Run Test. The Run Testcan be applied against both autoregressive and non-autoregressive models and its conclusionsare valid under sample sizes as small as nine. The Run Test for autocorrelation is applied asfollows (Gujurati (1978). First arrange the signs (+ or-) of the estimated errors for the modelin question in a consecutive sequence. It must be that n = N1 + N2 where n is the sample size,N1 is the total number of positive signs and N2 is the total number of negative signs. Definea run as an uninterrupted sequence of one symbol (+ or-) from the consecutive sequence ofall plus or minus signs. The test for randomness of errors is done by asking if the observednumber of runs for a given n is too high, too low, or about right as compared to the numberof runs expected in a sthctly random sequence of n observations. Too many runs suggestsnegative autocorrelation and too few suggests positive auto correlation. If N1 or N2 is smallerthan ten, special tables (see Gujurati (1978: 440-441)) give critical values of the runsexpected in a strictly random sequence of n observations. If N1 or N2 are large, then thenumber of runs is distributed normally, and therefore, the Z-test can be used to test if thenumber of runs observed is statistically significant.142How likely is it that estimates of Equations 6a and 8a and hence Equations 21 and 22would exhibit autocorrelation? The errors in autoregressive models such as the adaptiveexpectations model or the Koyck model can be theoretically shown to have a high likelihoodof being autocorrelated (Gujurati (1978: 266-276). It can also be demonstrated that the errorsin the partial adjustment model, which forms the GSR Equations 6a, 8a, 21 and 22, do in facttheoretically satisfy the OLS assumption of non-autocorrelation. Hibbs (1974) shows that OLSestimates obtained for a partial adjustment model will be consistent, but biased in smallsamples. This is consistent with Malinvaud (1970).What happens if, nevertheless, it turns out that the errors in any of Equations 6a, 8a,21 and 22 are autocorrelated? Normally, that is when n is large, when the error term in anautoregressive Equation is found to be autocorrelated, one must reestimate the equation usinga technique such as IV-pseudo GLS. This procedure would give more efficient estimates thanOLS (Hibbs (1974)). But the small sample properties of the IV-pseudo GLS method have notyet been determined (Smith (1980). Here, then, is where the greatest weakness of the microapproach to mathematical arms race analysis lies. If the errors in any of Equations 6a, 8a, 21and 22 are found to be autocorrelated, and if one then reestimated those equations using IV-pseudo GLS, the resulting estimates would have unknown properties. The utility of suchestimates would be quite limited.Thus, by following the procedures set out in this section, my GSR model and its143component equations can be estimated. With those estimates in hand, one could, in a givenarms race context, determine which of propositions 1-4 hold.In this chapter, I have laid out the conditions under which the form of arms interactionbetween two military rivals can materialize as action-reaction interaction, and with the rivalsbeing no less hostile toward each other, when it can materialize as a non-interaction outcomeor an asymmetric outcome. I have also set out propositions concerning the validity of theRational Expectations formation in arms competitions. More formally, there are four mainpropositions concerning the arms race behavior of states which derive from my model. TheyarePROPOSITION 1.0: If States K and X do in fact form their expectations of each other’sarming behavior in accordance with the Rational Expectations hypothesis, then the errors inState K and X’s arming behavior, as modelled by Equations 21a and 22a, should be stronglycorrelated. Mathematically corr(YL, ZJ 0.PROPOSITION 2.0: If, for any given arms race dyad, estimates of Equations 21 and 22 showan asymmetric arms race, that outcome will be due the conditions summarized by either S= 1, 0 < 5’ <1 and B1, B’1 do not equal zero or 5’ = 1, 0 < S < 1 and B1, B’1 do not equalzero.144PROPOSITION 3.0: If, for any given arms race dyad, estimates of Equations 21 and 22 showan action-reaction arms race, that outcome will be due the conditions summarized by 0 <S,S’<1 and B1, B‘do not equal zero.PROPOSITION 4.0: If, for any given arms race dyad, estimates of Equations 21 and 22 usingdata showing the aggregate military strength of each of the dyad members show a non-interaction outcome, that outcome will be due the conditions summarized by S = S’ = 0 andB1 and B’1 do not equal zero.In the next chapter, I will estimate the parameters of my GSR model and its componentequations for the US-Soviet nuclear warhead race. More specifically, I will test my modelagainst data on SLBM and the ICBM based warhead deployment levels, and against data onUS and on Soviet aggregate strategic nuclear warhead deployment levels. With the estimatesof my GSR model in hand, I will then make a determination as to which of Propositions 1-4hold in which cases.145CHAPTER VI: AN EMPIRICALTEST OF THE GSR MODEL:THE US-SOVIET NUCLEARARMS RACEWere the United States and the Soviet Union engaged in an arms race? A great dealof time and energy has been expended over the years in a search for quantitative evidencesuggesting that the US and Soviets are (or now were) so engaged. The effort, thus far, hasbeen to little avail. Most quantitative analyses of the US-Soviet arms competition show acomplete absence of interaction. Was it really the case that the US and Soviet Union acquiredarms without regard to what the other was acquiring? Or is it that we have yet to develop amethodology which can uncover the interactive component of US-Soviet arms acquisitions?How one approaches a study of this competition determines, in large measure, whatone will find. One can either take a macro approach or a micro approach. The formerapproach, most common among arms race researchers, takes the US-Soviet arms competitionas a competition of total armed might versus total armed, in the tradition of Richardson. Theunit of analysis under the macro perspective is most commonly aggregate militaryexpenditures. The micro approach, developed in this dissertation, sees the larger US-Sovietmilitary competition as made up of a series of subraces, the object of each subrace being aparticular weapons system, deployed by one side, and individual weapons system with a cross146purpose, deployed by the other, and designed to counter the former’s political and militaryeffect. Under this perspective, two rivals can engage each other in several different races, orsubraces, the object of each being a particular weapons system and a corresponding weaponssystem with a cross purpose. A picture of an overall military competition can then sketchedfrom an analysis of each of those subraces.There are sound theoretical and methodological reasons, which I have set outelsewhere in this dissertation, for approaching the US-Soviet military competition via the- micro approach, as opposed to the macro approach. In my study of the US-Soviet arms race,I will adopt the micro approach. The specific micro-level questions I will attempt to answerare, Were the US and Soviets engaged in individual action-reaction arms races over SLBMwarhead deployments and ICBM warhead deployments? Were they engaged in an arms raceover total strategic nuclear warhead deployments? In effect, mine will be a study of the USSoviet nuclear arms race. The micro approach will yield the intuitively appealing conclusionthat the US and Soviets were indeed engaged in an action-reaction nuclear arms race.1. A GENERAL OVERVIEW OF TRENDS IN THE US-SOVIET NUCLEAR ARMSRACEThe US-Soviet nuclear arms race began, for all intents and purposes, on 6 June 1945when the US had tested the world’s first atomic bomb. In August of that same year, the US147dropped a 15-20 kiloton bomb on the Japanese city of Hiroshima. In 1949, the Soviet Uniondetonated its first atomic bomb. By this time, the US and Soviets were bitter political,economic and military rivals. Each came to see the other as a threat to its very existence. Inthis section, I will provide an overview of the basic trends and developments of what becamethe potentially most dangerous arms race in history, the US-Soviet nuclear arms race(Freedman, 1983, 1986; Holloway, 1985; Gray, 1976; MccGwire, 1987a, 1987b, 1991;Koenig, 1982; SIPRI, 1971-90; Military Balance, 1969-1990; Sagan, 1989; Trachtenberg,1988-89; Garthoff, 1990; Catudal, 1988).In the early 1950s, the only means that the US and USSR possessed for deliveringnuclear payloads was the longrange bomber. As arms races go, the US-Soviet bomber racewas really not a race at all. In the very early 1950s, the Soviets were able to field twodifferent intercontinental bombers, the M4 Bison, with a range of 7000 miles and the Tu95Bear, with a range of 7800. The Bison proved unequal to its appointed task and waseventually relegated to the role of in-flight tanker-refueler (Koenig, 1982). The Bear proveda much better aircraft and by 1980 some 113 Bears remained in service in the Soviet Airforceas longrange bombers. Neither plane was, however, produced in any real quantity and thusthe Soviet bomber threat was considered small in the US. The US, on the other hand, hadbuilt a large force of B52s. The Soviets, in contrast to the Americans, were thus obliged toconstruct a formidable air defence system. Soviet air defence rested on a missile defencenetwork and on some 2600 interceptor aircraft (Koenig, 1982).148By 1960, ICBM technology had improved to the point where both sides had begunto deploy large numbers of longTange nuclear tipped missiles. The ICBM quickly began tosupplant the long range bomber as the principal means for delivering nuclear payloads(Koenig, 1983). ICBMs were more accurate than the longrange bomber and more difficultto intercept. The first generation US and Soviet ICBMs were primitive, by today’s standards.They were inaccurate and each could only carry one warhead. The USSR first deployed thehugh SS-6 and the US deployed the Atlas.It was not long, however, before the US had developed MIRV technology whichallowed it to place more than one warhead inside each missile. Guidance systems alsoimproved making ICBMs more accurate. Eventually, the Soviets also developed improvedguidance technology and MIRV technology. In consequence, both American and SovietICBMs became vulnerable to a first strike by the late 1960s. Both sides, thus, made an effortto develop anti-ballistic missile defence systems. Both the US and Soviets eventually putoperational systems into place, but today, only the Soviet system remains operational.Another way both sides secured their second strike forces was to step up SLBMdeployments. Indeed, the US had begun to place a strong emphasis on SLBM basing in theearly 1960s. The SLBM basing mode meshed well with US nuclear doctrine. By the mid1960s, US doctrine called for nuclear strikes against the USSR only in retaliation for a Sovietstrike on the US. SLBMs were secure from Soviet preemptive attack and thus could be used149either to deter an initial first strike by the Soviets or to retaliate for a first strike by theSoviets. The Soviets, on the other hand, did not begin to emphasize SLBM deployments intheir strategic calculations until the early 1970s. MIRV and improved guidance technologywas making their ICBM force vulnerable to a US first strike. This was no small concern forthe Soviets since the bulk of their nuclear forces were ICBM based. Also, Soviet militarydoctrine had, in the mid-late 1960s, changed. Initial Soviet plans called for a first-strikeagainst US nuclear forces if nuclear war looked imminent. The Soviets now began to see thevirtue in shifting to a second strike posture. This would require the development of largeSLBM force and, accordingly, Soviet SLBM deployments began to increase substantiallythroughout the 1970s and 1980s.Originally, launchers (ICBMs, SLBMs, and bombers) were the focus of the US-Sovietnuclear competition. In 1972, SALT I came into effect. It was significant because it placedlimits on the number of launchers each side could deploy. It was also significant for what itdid not limit. It placed no controls on the number of warheads each side could deploy. From1972 onward, the US-Soviet race thus shifted from a race over launchers to a race overwarheads (Keating, 1985). Soon, START will cut back the number of strategic nuclearwarheads from 11,000 to 7,000 on the Soviet side and from 12,000 to 9,000 on the US side.1502. THEORETICAL BASIS FOR A US-SOVIET NUCLEAR ARMS RACEWere the US and the Soviets engaged in an action-reaction arms competition? KeyUS defence officials have stated that the US-Soviet military rivalry must, logically, beconducted in an action-reaction mode, particularly when it comes to nuclear weapons.Enthoven and Smith (197 1:176-177) who were two of McNamara’s key assistants during histime as US Secretary of Defense wroteIt is important to understand [... the] interaction of opposing strategic forcesand its relation to the strategic force planning process. If the overridingobjective of our strategic nuclear forces is to deter a first-strike against us, theUnited States must have a second-strike capability .... This capability todestroy him after absorbing his surprise attack must be a virtual certainty, andclearly evident to the enemy. This is the foundation of the U.S. deterrentstrategy. Consequently, as long as deterrence remains the priority objective, theUnited States must be prepared to offset any Soviet effort to reduce theeffectiveness of our assured destruction capability below the level we considernecessary.At the same time, however, if deterrence is also the Soviets’ objective (as theavailable evidence has consistently and strongly suggested), we would expectthem to react in much the same way to any effort on our part to reduce theeffectiveness of their deterrent (or assured destruction) capability against us.And we would also expect them, in their planning, to view our strategicoffensive forces as a potential first-strike threat (just as we do theirs) andprovide for second-strike capability. In other words, any attempt on our partto reduce damage to our society would put pressure on the Soviets to strivefor an offsetting improvement in their assured destruction forces, and viceversa. Each step by either side, however sensible or precautionary, would elicita precautionary response from the other side. This ‘action-reaction’phenomenon is central to all strategic force planning issues as well as to anytheory of an arms race.151The logic that obliges the US and Soviet Union to engage each other in an action-reactionnuclear arms race is compelling. Action-reaction arms accumulation is an essential means tomaintaining the balance of terror between the US and USSR. What empirical evidence hasbeen adduced to verify the existence of a nuclear arms race between the US and SovietUnion?3. PAST QUANTITATIVE STUDIES OF THE US-SOVIET NUCLEAR ARMS RACELittle quantitative empirical evidence of a nuclear arms race between the US andSoviet has been found. Kugler, Organski and Fox (1980), for example, argued that if the USand Soviet Union were set upon maintaining a balance of terror between them, then each sideshould increase its nuclear capabilities in response and in proportion to increases in thenuclear capabilities of the other in order to maintain that balance. In other words, the US andSoviet Union should be engaged in an action-reaction nuclear arms race. The purpose of theirstudy was to find empirical evidence to support this proposition. They found none. Theywrote:According to our data, then, the presence of nuclear arms race, far fromconstituting a given of international politics, proves to be a chimera. We havetried again and again to test for the presence of arms competition or armsracing and we have failed to find anything each time. It is obvious that the USand USSR are building arms, but are not doing so, as they allege, because theyare racing or competing with one another [p. 128).152McGuire (1976) has, similarly, found little evidence to support the view that the US andSoviet Union are engaged in a nuclear arms race. Why could neither Kugler et al (1980) norMcGuire (1976) find any evidence indicating the existence of a US-Soviet nuclear arms race?Kugler et al’s and McGuire’s failure to find any empirical evidence to support theview that the US and USSR are engaged in an action reaction nuclear arms race is due moreto their particular approach to the problem than it is to the genuine absence of such acompetition. In their study, Kugler et al estimated the parameters of Richardson’s modelusing data showing annual US and Soviet expenditures on all strategic nuclear forces,including, ICBM, submarine and bomber forces (given in US dollars). They considered therace, then, to be one of total strategic nuclear capability versus total strategic nuclearcapability. McGuire took a slightly different approach. Instead of using expenditure data, heused nuclear weapons inventory levels to indicate US and Soviet strategic capability. Forexample, he tested Richardson’s model against data on the total annual megatonage in the USand Soviet strategic arsenals. He also tested Richardson’s model against data on the totalstrategic warhead counts of the US and Soviet Union. Why did these studies not revealevidence that the US and Soviet Union are engaged in a nuclear arms race? Indeed, if the USand Soviets are engaged in a nuclear arms race, over what, specifically, are they racing?1534. A NEW APPROACH TO THE STUDY OF THE US-SOVIET NUCLEAR ARMSRACEAs discussed in more detail elsewhere in this thesis, McCubbins (1983) argues thatrival nations do not, as Richardson (1960a) suggested, engage each other in a single race over“total armed might.” McCubbins argues that the US and Soviet Union compete with eachother, in an action-reaction mode, when it comes to specific, individual weapons systems withcross-purposes, e.g., one side deploys long range bombers, the other side will counter withjet interceptors. Theoretically, two rivals could engage each other in a number of differentraces over different sets of weapons and corresponding counter-weapons systems. At anygiven time, some of these sub-races, as we may call them, may be heating up while othersare cooling down. As argued elsewhere in this thesis, looking at the US-Soviet rivalry as acompetition of total armed might versus total armed might, then, can result in any evidenceof action-reaction interaction being masked.McCubbins’ empirical analysis, indeed, showed that the US-Soviet conventionalweapons competition occurs over individual weapons systems with cross-purposes and thatthe competition is action-reaction in nature. Yet McCubbins’ study was limited in its scope.His analysis of the US-Soviet arms race was confined to the competition over conventionalweapons. He might have done well to consider applying his analysis to the US-Soviet nucleararms race as well.154Both the US and Soviet Union do try to counter the nuclear force deployments of theother and thereby maintain a balance of terror. For example, each side has attempted tocounter the other’s strategic nuclear missile deployments with anti-ballistic missile systems(Rathjens, 1969). This is an example of a race over individual weapons systems with cross-purposes. A less obvious example would be the attempt by the superpowers in the early1960s to out deploy each other in ICBMs. An ICBM-ICBM race might seem like a race overindividual weapons systems with identical purposes. But, if, in this case, the policy goal ofeach side was to prevent the other from gaining a first strike capability, and hence avoidnuclear war, then one way to achieve that goal would have been for each side to prevent theother from gaining numerical superiority in ICBMs.A further point about that ICBM race is that it did not last long. By the 1970s, boththe US and Soviets had begun to focus more resources into SLBM basing. Accordingly, itbegins to become clear why Kugler et al and McGuire were unable to discern any realevidence that the US and Soyiet Union are engaged in a nuclear arms race. The total weaponscounts for the US and Soviet Union that McGuire used contain information on three differentweapons systems: ICBMs, SLBMs and long range bombers. Clearly, while the ICBM racewas heating up, the bomber race was dying down. When taken together, the peaks and valleysin the ICBM, the SLBM and the bomber race cancel each other out. This may also explainthe outcome of Kugler et al’s study. Kugler et al used US and Soviet data which reflectedtotal expenditures on ICBM, submarine and bomber based nuclear forces by each side.155From this discussion, it is clear that the conventional approach to the quantitativeanalysis of the US-Soviet nuclear arms race must be amended. In my study of the US-Sovietnuclear arms race, I will follow the McCubbins’ data approach. In particular, as Snow (1981)argues, the US and USSR consider each side’s strategic nuclear warhead count to be a goodindicator of each side’s strategic capability. Missile counts can be misleading since somemissiles contain more than one warhead and it is the warhead which does the actual damagein a war. The unit of analysis in my study, then, will be annual US-Soviet strategic nuclearwarhead deployments. More specifically, following the historical trend of the US-Sovietnuclear arms race, laid out above, I will conduct three separate tests of my GSR model.Specifically, I will test my GSR model against data on annual US and Soviet deploymentsof SLBM warheads from 1972 to 1987 and against data on US-Soviet annual ICBM warheaddeployments from 1960 to 1971 for evidence of action-reaction interaction. Setting the startand end points of each of these races to match the historical record should yield valid results(see Lucier, 1979). Finally, I will test my GSR model against data on total annual US-Sovietstrategic nuclear warhead counts from 1967 to 1984. The reason for this test is that it willestablish whether or not the US and Soviets did react to each other’s total strategic capability.The data set time span for this study is arbitrarily set. If the US and Soviet Union did not soreact, then any sample drawn between 1949 and 1991 should show this to be the case. Fromthe estimates obtained in each of these tests, I will then make a determination as to whichof the arms race Propositions 1-4, derived in the last chapter, hold, and in which cases theydo hold.1566. THE SLBM WARHEAD RACEOf all the strategic nuclear warheads distributed across the US and Soviet thads, noneare more invulnerable to a surprise first-strike than those based onboard the submarines ofthe US and Soviet navies. Neither side has yet been able to develop a device, such as asatellite, which is capable of scanning vast areas beneath the surface of the seas and locatingenemy submarines. Each side does have systems to help narrow down the location of enemysubmarines, but these systems are effective only when the general location of a submarineis already known (Handler, 1987). Because the seas are so vast and because the SLBMswhich the submarines carry have a long range, a submarine can hide virtually anywhere andremain undetected while awaiting launch instructions.The US was the first to develop SLBM technology. By 1960, it had some 32 singlewarhead SLBMs, the Polaris Al, deployed at sea (Military Balance, 1969-70: 55). It was notuntil 1963 that the Soviets were able to field their first SLBM. From the beginning, the UShad placed a high priority on developing and deploying SLBM systems. The US wascommitted to a doctrine of engaging in nuclear war only in response to a first-strike by theSoviet Union. SLBM basing mode meshed well with the US strategic doctrine of secondstrike because submarines are invulnerable to surprise attack and destruction. Thus by basingwarheads onboard submarines, the US would always have a force which, in the event of aSoviet first-strike, could be used to retaliate against Soviet cities. The bulk (50% as of 1984)157of the US strategic warhead inventory is currently based on SLBMS. The Soviets, on theother hand, had from the beginning of the nuclear race, placed a high priority on thedevelopment of landbased ICBM systems. For one thing, it lagged behind the US insophisticated SLBM and SLBM basing technologies. In emphasizing landbased ICBMdevelopment, the Soviets were drawing on the greatest strength, the enormous size of theSoviet Union. Much of the Soviet landmass is uninhabited. Large numbers of ICBMs could,then, be safely deployed in these areas, away from population centers. The US, then, wouldhave to commit large numbers of missiles to target those systems (Snow, 1980: 147).By the 1970s, several factors had come together to channel Soviet efforts intoincreasing its SLBM warhead deployments: the development of MIRV technology, the SALTI arms control agreement, changes in Soviet military doctrine. One of the most significanttechnological developments to occur in the US-Soviet nuclear arms race was the development,first by the US and then by the Soviets, of MIRV technology. MIRV technology allowed eachside to place more than one nuclear warhead inside each of its ballistic missiles. Eachwarhead could be individually and independently targeted. This development was to put atrisk the Soviet Union’s landbased ICBM force as it made it theoretically easier for the U.S.to destroy those ICBMs while still in their silos. This was no small concern for the Sovietssince the bulk of their strategic nuclear capability was in their landbased ICBMs. Somethinghad to be done. As Scoville (1972: 37) writes:158By the late 1960s it must have been obvious to military planners in theU.S.S.R. that their land-based ICBM’s would become increasingly vulnerableto the U.S. MIRV’s, which were then under development and which had beenpublicly justified as providing an improved counterforce capability. TheRussian deterrent needed shoring up with a more effective SLBM force, whosevalue had been demonstrated by the U.S.Indeed, from 1972-87, Soviet SLBM warhead deployments rose from 458 to 3408 (NuclearNotebook, May, 1988).Second, in 1972, the Strategic Arms Limitation Treaty (SALT) I came into effect.SALT I placed ceilings on the number of strategic delivery vehicles that the US and SovietUnion could possess. The term delivery vehicle applies to longrange bombers, and ICBM andSLBMs, excluding the warheads they each carry. Each side was assigned strict maximumlauncher deployment levels in each category. The US took the opportunity afforded it duringthe SALT negotiations to influence Soviet thinking and Soviet action. The US, as alreadymentioned, placed a great deal of emphasis on SLBM development and deployment. BecauseSLBMs were invulnerable to preemptive attack, the US saw them as a stabilizing element inthe overall East-West nuclear game. The US thus wanted the Soviets to also emphasizeSLBM basing in their strategic calculations. The Americans were able to get the Soviets toagree to a “one-way mixing clause” in SALT I. Each side would be permitted to dismantleICBMs and replace them with SLBMs, but not vice-versa, while keeping total launcherceilings constant (Snow, 1980: 98).159Thirdly, Soviet military planners began in the late 1960s to rethink Soviet militarydoctrine (MccGwire, 1987, 1991). Originally, Soviet military planners had expected that warwith the United States would ultimately turn nuclear and would become a decisive showdownbetween the capitalist and the communist systems. The Soviet Union intended to emerge thevictor in this contest. In order to limit the damage that the USSR would have to sustain insuch a war, Soviet planners adopted a strategy of striking American nuclear (particularlyICBM) forces first in order to blunt the weight of an American nuclear attack. But by the late1960s, Soviet planners began to think that war with the United States might not necessarilyinvolve large scale strategic nuclear strikes and counter strikes. Indeed a conventional warwith the West need not escalate beyond the conventional level. Strategic nuclear war, andhence the devastation of the Soviet Union, could be avoided. Certainly, in the event of war,the US would have to be ejected from Europe, but this could be done by conventional means.In order to deter the US from launching nuclear strikes against the Soviet Union itself whenfaced with the defeat of its forces in Europe, the Soviet Union had to have a secure secondstrike capability. Because of MIRV technology, however, its main nuclear forces, landbasedICBMs, were becoming vulnerable to an American first strike. Soviet efforts to develop amobile land based ICBM were not bearing fruit and so it was decided that SLBMdeployments should be stepped up in order to secure the Soviet Union’s second strikecapability. The Soviet Union thus began expending vast resources on its navy throughout the1970s and into the 1980s. By the 1980s, the Soviets began to reduce this emphasis on SLBMdevelopment and deployment. Relations with the West were easing, and significant new160strides were being made in mobile landbased ICBM technology. Mobile ICBMs are muchmore difficult to hit than fixed based ICBMs. Currently, the Sovietshave deployed the railbased SS-24 and the roadmobile SS-25.These factors, the development of MIRV technology, SALT I, and a new Sovietmilitary doctrine were critical in setting in motion a US-Soviet SLBM warhead deploymentcompetition spanning the early 1970s to the late 1980s. It was these factors, principally,critically, which determined the course and dynamics of that competition. Because it is thedestructive capacity of the second strike forces of each side which deter the other fromlaunching a first strike, it would be reasonable to expect that each side would be trying tomaintain a second strike force comparable to that of the other. Parameter estimates for myGSR model suggest that the US and Soviet Union have, in fact, been engaged in an action-reaction competition over SLBM warhead deployments over the years 1972-1987. I began byestimating the simplified versions of Equations 21 and 22, Equations 23 and 24.K = L0 + L1X + L2K + Y (USA) (23)= 2023.0 + 0.158X + 0.550K1(432.6) (0.11) (0.130)** * **R2 = 0.9132, n = 15, ** = sig. at 0.05 and * = sig. at 0.10. (Note Equation 23/21 wasestimated using OLS. Run Test results indicated no autocorrelation at a 0.05 level ofsignificance: see Table 6.1 for test results).161TABLE 6.1: Autocorrelation test results (Run Test). See page 141 for an explanation of thistest. Figures contained in this table show the observed number of runs, the number of positiveerrors, and the number of negative errors, in that order, for each indicated equation and foreach indicated race.SLBM ICBM TOTALWarheads Warheads Warheads1972-87 1960-7 1 1967-84Equation la 6,6,9 8,4,7 7,8,9Equation 2a 7,6,9 6,4,7 7,8,9Equation 6a 6,6,9- 8,4,7 7,8,9Equation 8a 7,6,9 6,4,7 7,8,9Equation 23 7,7,8 8,4,7 7,8,9Equation 24 9,7,8 8,6,5 7,8,9162X = L’0 + L’1K + L’2XL1 + Z (USSR) (24)= -399.9 + 0.201ç, + 0.80X(323.4) (0,097) (0.087)** **R2 = 0.9775, n = 15 (Note Equation 22/24 was estimated using OLS. Run Test resultsindicated no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).My estimates of the GSR coefficients in Equations 23/2 1 and 24/22 show that over the period1972 to 1987, the US and Soviet Union have been engaged in an action-reaction arms raceover SLBM warhead deployments. The US GSR grievance, defence and stock adjustmentterms are all significant at 0.05 or 0.10. Of those parameter estimates which are significant,only the US GSR defence term is significant at 0.10. Less weight must, accordingly, beattributed to this estimate compared to the others. Specifically, my estimates show that USSLBM warhead deployments at time t depend upon 0.158 times Soviet SLBM warheaddeployments at time t-1 and 0.55 times US SLBM warhead deployments at time t-1 plus aGSR grievance term of 2023. The Soviet GSR defence and stock adjustment terms aresignificant at 0.05. Soviet SLBM warhead deployments at time t, similarly, depend upon 0.20times US SLBM warhead deployments at time t-1 and 0.80 times Soviet SLBM warheaddeployments at time t-l. The Soviet GSR grievance term is not statistically significant at 0.10.The finding that the US and USSR were engaged in an action-reaction race over163SLBM warhead deployments (1972-87) is not surprising. Indeed it is important. It suggeststhat the logical conditions for the maintenance of strategic deterrence were being met.The mechanism behind the action-reaction competition of the US and Soviet Unionwith respect to SLBM warhead deployments can be explained thusly. Consider first, that boththe US and the Soviet Union see their sea based nuclear forces as second-strike, or deterrentforces. The theoretical link between action-reaction arms racing and deterrence, as set outby Kugler et al (1980: 108) is as follows:It should be noted that in the context of deterrence theory it is not necessaryto assume that both sides in nuclear arms races must make the same amountof effort or have the same level of capabilities. One must assume, however,that each of the contestants will allocate substantial portions of the resourcesscheduled to be used in the improvement of nuclear capabilities in directresponse to the other’s allocations. Hence one must compete and even racewith one’s opponent. And the race continues even after both contestants reacha second strike capability. One must always keep in mind that theinvulnerability of the defendant’s deterrent depends on the power of theaggressor’s initial attack.Two points must, accordingly, be made. The first is that both the US and Soviets have, withtheir SLBM forces, each achieved a second sthke capability. However, each side has madea point, and this is particularly true of the Americans, of developing the means and themethods for destroying the other’s SLBM forces in the event of war. The US Navy has longhad a policy of pursuing and destroying, in the event of war, Soviet nuclear missile carryingsubmarines (Handler, 1987). Never was that policy made more explicit than in 1983 by US164Navy Secretary Lehman. In this context it would, then, be prudent for each side to try tomaintain an SLBM force proportional to that of the other at all times. What that proportionwould be would depend on how capable each side thought the other was of destroying itsSLBM forces and on how each side evaluated the technological span between its own SLBMsand those of its adversary. In this regard, much can be learned by referring to the estimatesof Equations la, 2a, 6a and 8a.What are the values of the parameters, B0, B1 and S, the US Richardsonian grievance,defence and stock adjustment parameters and B’0, B’1 and S’, the Soviet Richardsoniangrievance, defence and stock adjustment parameters? Equations 6a and 8a, reproducedimmediately below.K = SB0 + SB1XL + (1- S)K1 (USA) (6a)= 2177.0 + 0.226X ÷ 0.479K1(439.9) (0.131) (0.143)** * **R2 = 0.9200, n = 15. (Note: Equa.tion 6a was estimated using OLS. Run Test resultsindicated no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).165X = S’B’0 + S’B’1KL + (1 - S’)X1 (USSR) (8a)= -958.2 + 0.322K, + 0.77X1(501.8) (0.13) (0.08)** **R2 = 0.9794, n = 15. (Note: Equation 8a was estimated using OLS. Run Test resultsindicated no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).The estimate of Equations 6a reveals that the US stock adjustment coefficent is S = 1 - 0.479= 0.521 and the estimate of Equation 8a reveals that the Soviet stock adjustment coefficientS’ = 1 - 0.77 = 0.23. Within the context of the SLBM warhead race, then, the US has morethan double the political, economic and institutional capacity, compared to the USSR, toadjust its actual deployment level at time t to a level where a desired balance between its ownSLBM warhead levels and those it expected the Soviet Union was going to deploy at timet would obtain. This finding is not unreasonable given the US’s stronger economic-industrialbase and the longstanding US doctrinal preference for the SLBM basing model. But since 0<S < 1, US could not fully adjust its actual SLBM warhead levels at time t to their desireddeployment levels at any t, nor could the Soviet Union. From Equations 6a and 8a, we canalso calculate, respectively, the US Richardsonian grievance and defence parameters, B0 andB1 and the Soviet Richardsonian grievance and defence parameters B’0 and B’1. FromEquation 6aB0 = SBQ/S = 2177.0/0.521 = 4178.5166B1 = SB1/S = 0.226/0.521 = 0.43From Equation 8aB’0— S’B’dS’ — -9582/023 — -416608B’1 = S’B’1/S’ = 0.322/0.23 = 1.4Before I will attempt to analyze the meaning of the values obtained for the parameters B0 andB1, B’0 and B’1 from Equations 6a and 8a, I will verify their accuracy by estimatingEquations la and 2a.K*t= B0 + B1X (USA) (la)= 4183.9 + 0.43X,(237.7) (0.102)** **R2 = 0.5830, n = 15. (Note: Equation la was estimated using OLS. Run Test resultsindicated no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).167= B’0 + B’IKL (USSR) (2a)= -4285 + 1.43K(1257) (0.25)** **R2 = 0.7115, n = 15. (Note: Equation 2a was estimated using OLS. Run Test resultsindicated no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).The estimates of the Equations la and 2a coincide with the values obtained for B1 and B’1from Equations 6a and 8a. In Equation la, B1 = 0.43 and from Equation 2a, B’1 = 1.43.Clearly, the Soviet Richardsonian defence coefficient, the defence context being the US-Soviet SLBM warhead race, is much larger than the US Richardsonian defence coefficient.This result is consistent with the Soviet Union’s reappraisal of its military doctrine in the late1960s (MccGwire, 1987, 1991). It reflects the heavy emphasis placed by the Soviets on thedevelopment of a sea based deterrent in the 1970s, as discussed above, as a hedge againstsome American technological break through which might render their ICBM impotent. Incontrast, the US has long recognized the great disparities between US and Soviet technologyand as such has felt secure with a numerically larger Soviet SLBM force. Indeed, underSALT I, the US agreed to give the Soviets a 30% higher ceiling on its SLBM deploymentlevels than it took on for itself. Nevertheless, as of 1987, the US had 5632 SLBM warheadsdeployed while the Soviet had only 3408 deployed. The Soviets have always been behind inthis race.168The fact that the Soviet Richardsonian grievance terms, in Equation 8a and Equation2a, were negative, and that the Soviet Union’s GSR grievance term in Equation 22/24 waszero is worthy of consideration. The negative values of the Soviet Richardsonian grievanceterms, in Equations 8a and 2a, may well be indicative of some internal constraint on theSoviet Union’s ability to match American armament (SLBM warhead) levels, a constraintwhich is not reflected in the value of its stock adjustment coefficient S’. It acts to reduce the“match.” Whatever this constraint may be, however, its impact is neutralized when the SovietUnion explicitly takes into account the US’s SLBM warhead deployment program asindicated by the zero value of the Soviet Union’s GSR grievance term. When negative, theconstraint may well reflect the impact of Soviet doves on armament policy in the SovietUnion. The fact that it is ultimately neutralized may reflect the impact of Soviet hawks.One of the most interesting aspects of the US-Soviet SLBM warhead race is itsequilibrium properties. Equations 29 and 30 show the equilibrium SLBM deployment levelsfor the US and USSR.L0(1 - L’2) + L1’0K, (USA) (29)(1- L2)(1 - L’2) - L’12023(1 - 0.8) + (0.158)(0)IreL—(1 - 0.55)(1 - 0.8) - (0.158)(0.20)169= 6928.08- L) + L’10x, =- (USSR) (30)(1 - L2)(1 - L’2)- L’1= 5)+0.207(2032(1 - 0.55)(1- 0.80) - 0.207(0.158)6928.08Equilibrium in the US-Soviet SLBM warhead race, (K1,X1), then, occurs at the point(6928.08, 6928.08). That is, US and Soviet SLBM warhead deployments should converge to6928.08 SLBM warheads each. The equilibrium point (6928.08, 6928.08) in the US-SovietSLBM warhead race is, moreover, stable as indicated by the slopes of Equations 31 and 32which give the equilibrium demarcation curves for the US and Soviet Union, respectively.L0 (L2-1)XI,1 = K1 (USA) (31)L1 L1170X1 = -12803.7 + 2.8K1L’1X1 = + K (USSR) (32)(1 - L’2) (1— L’2)X11 = 1.0KGraph 6.1 shows the plots of Equations 31 and 32. The plot suggests that the US-SovietSLBM nuclear warhead race (1972-87) was stable. The robustness of this conclusion can bedemonstrated by adding the estimate obtained for L1, the US GSR defence term to itsestimated standard error, the value obtained for L2, the US GSR economic term, to itsestimated standard error, L’1, the Soviet GSR defence term to its estimated standard error andby adding L’2, the Soviet GSR economic term to its estimated standard error and thenrecalculating Equations 31 and 32. If the SLBM race is still found to be stable, then theoriginal conclusion that the race is stable is strengthened. An additional robustness test couldbe performed by subtracting, rather than adding, the estimated standard errors for the GSRparameters from the corresponding GSR parameter estimates and then recalculating Equations31 and 32. Here, too, if the resulting calculations show that the US-Soviet SLBM warheadrace is stable, then the original conclusion regarding the stable nature of the race isstrengthened. In the latter instance, the data continue to show that the US-Soviet SLBMGRAPH 6.1: Stable equilibrium in the US-Soviet SLBM warhead race, 1972-876928K,171xt-1K,-Ic, = 0xt - x.1 = 06928172warhead race was stable. A stable node obtains. In the former instance, the recalculation ofEquations 31 and 32 shows a saddle point. That is, the evidence, one way or another, is notconclusive. The original finding, then, that the US-Soviet SLBM warhead race (1972-87) wasstable seems to be robust.Substantively, it must be emphasized that Equations 31 and 32 were plotted on thebasis of the values obtained for B1 and S, the US defence and stock adjustment parameters,and B’1 and 5’, the Soviet defence and stock adjustment parameters. The interplay of thesevalues dictates that the Soviet Union would consider any point where it and the US have anequal number of SLBM warheads to be an equilibrium point. Because the Soviet Union’sstock adjustment coefficient is so small, it could not surpass the US in SLBM warheaddeployments as its high defence coefficient suggests it would like to. At best, Graph 6.1indicates, the Soviets would be satisfied to match the US SLBM warhead for SLBM warhead.The US in contrast will consider any point where it has 0.35 SLBM warheads to every 1.0Soviet SLBM warhead to be an equilibrium point.6. THE ICBM WARHEAD RACEThe perfection of Intercontinental Ballistic Missile technology, such as it was in thelate 1950s early 1960s, marked a turning point military history (Koenig, 1982). Tipped witha nuclear warhead, the ICBM provided its masters with the potential to do what had never173been done before: obliterate an enemy’s society without ever having to engage his militaryforces in the field.In the beginning (1960) ICBM technology was relatively primitive. The first SovietICBM, the SS-6, was enormous. It had no less than 32 engines all of which had to be firedin concert if it was to get off the ground. The American Altas was not much smaller. TheAtlas had to be lifted out of its silo and fuelled before it could be fired. Each carried a single,highly inaccurate warhead in the megaton range. The Atlas had a CEP of about 1800 metersand the SS-6 had a CEP of about 2500. By today’s standards, these missiles were crude.Given the fact that they were inaccurate, their only real utility was city busting.ICBM technology, however, advanced rapidly. Solid fuels were developed which wasto reduce the time between the order to launch and the launch to minutes. Guidancetechnology improved making warheads more accurate. The second generation US ICBM, theTitan II, had a CEP of 1500 meters. The third generation Minutemen had CEPs of 350meters. The newer Soviet SS-9s and SS-lls had a CEP of about 1300m. Secondly, atechnique was developed which allowed first the US and then the Soviet Union to pack morethan one warhead inside each side each missile. Later, the Americans developed a techniquecalled MIRVing which allowed them to independently target multiple warheads. Why werethese developments significant?174While both sides have encased their land based ICBMs in concrete and steel silos,they are not completely secure. In order to destroy a silo with a nuclear blast, the attackingwarhead must land almost nearly on top of it in order to exert the maximum overheadpressure thereby shattering it and the missile within it. Warheads must, accordingly, be highlyaccurate and it may take several, detonated in concert, to actually destroy a silo. The effectof improved guidance technology and MIRV technology, then, was to make each side’sICBM forces became vulnerable to attack and destruction in a first-strike. At present, USsilos are designed to withstand a maximum over head blast pressure of about 2000 psi.Concrete silos could not be made much stronger. The technical limit on concrete’s ability towithstand overhead blast pressure is about 3000 psi (Davis and Schilling, 1973). In a nuclearwar, ICBMs, because of their vulnerability, would be the first nuclear weapons launched byan aggressor and they would be the first of the victim’s nuclear weapons to be destroyed.From the Soviet point of view, these developments were at once both good and bad.Soviet military doctrine called for preemptive strikes against US nuclear forces (ICBMs) ifnuclear war appeared imminent. However, since the US also possessed these sametechnologies (and in each case had them first), Soviet ICBM forces were also vulnerable toan American preemptive-strike. The Soviets had invested more heavily in their ICBM forcesthan they had in SLBM or long range bomber forces. Indeed the bulk of Soviet nuclear poweris ICBM based. In order to reduce the actual and perceived probability that the Americanscould successfully destroy their ICBM forces on the ground, the Soviets developed an anti-175ballistic missile defence system, still employed to this day, and began to step up their ICBMdeployments, basing them throughout their vast country, thereby complicating American warcalculations by giving them more targets to hit. The Soviets ought reasonably to have tiedtheir ICBM deployments to US ICBM deployments.The Americans were faced with the same sorts of problems. Their ICBM forces mighteventually also become vulnerable to a Soviet first strike. The Americans were to mount threeresponses. One was to defend their ICBM bases with anti-ballistic missile defence systems.An operational system was actually put into place, the Safeguard system, but was shut downin the 1970s. A second American response was to step up their ICBM deployments. A thirdAmerican response to the growing vulnerability of their ICBM force was to begin basingmore warheads at sea where they would be safe from a Soviet first strike. The US, in fact,had a preference for SLBM basing. Submarines are extremely difficult to detect and henceextremely difficult to destroy. Eventually, both sides would look to the sea as the safestbasing mode for their nuclear forces. By the early 1970s, both the US and the Soviet Unionwere to step up the rate of SLBM deployment.There is today a resurgence of interest in the ICBM. ICBM technology continued toprogress through the deployment stasis of the 1970s and 80s. The US has recently deployedthe 100 new MX missiles in refurbished (superhardened) Minute Man silos. The MX carriesten warheads and is said to have a CEP of lOOm which would make it the most accurate176ICBM ever developed. The Soviets have finally developed mobile ICBM systems. They haverecently deployed the railmobile SS-24 and the roadmobile SS-25. The SS-24 is MIRVed,carrying 7 to 10 warheads of 550kt each. Because these systems are mobile they will beextremely difficult for the Americans to target and destroy in the event of war.To what extent did the US and Soviet Union tie their ICBM warhead deploymentcalculations in the 1960s to what each thought the other was going to deploy? The estimatesof Equations 23/21 and 24/22 for US and Soviet ICBM warhead deployments over the years1972-1987 are given immediately below.K, = L0 + L1X + L2K1 + Y, (USA) (23)= 206.18 + 0.0347X1+ 0.843K..1(80.37) (0.120) (0.149)** **R2 = 0.9 102, n = 12 (Note Equation 23/21 was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).X = L’0 + L’1KL + L’2XL1 + Z (USSR) (24)= -10.70 + 0.254K1 + 0.937X1(72.61) (0.134) (0.109)** **R2 = 0.9676, n = 12 (Note Equation 24/22 was estimated using OLS. Run Test results177indicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).What do the estimates of Equations 23/2 1 and 24/22 show?The estimates of Equations 23/2 1 and 24/22, respectively, for US and Soviet ICBM warheaddeployments over the years 1960-7 1 show an asymmetric arms race. The US GSR grievanceand stock adjustment terms only are significant at 0.05. US ICBM warhead deployments attime t depend only upon 0.84 times US ICBM warhead deployments at time t-1 plus agrievance of 206.18. In contrast, the USSR GSR defence and stock adjustment terms aresignificant at 0.10 and 0.05 respectively. Soviet ICBM warhead deployments at time t dependupon 0.25 times US ICBM warhead deployments at time t-1 and 0.93 times Soviet ICBMwarhead deployments at time t. In this case, the USSR is racing over ICBM warheaddeployments with the US, while the US’s ICBM warhead deployments are self-driven. Muchcan be learned about the nature of the US-Soviet ICBM competition from the estimates ofEquations 6a and 8a.K = SB0 + SB1X + (1 - S)K1 (USA) (6a)= 204.4 + 0.0240X + 0.848K1(80.6) (0.122) (0.169)** **R2 = 0.9097, n = 12. (Note: Equation 6a was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).178X, = S’B’ + S’B’1K + (1- S’)X1 (USSR) (8a)= -31.805 + O.225K + 0.969X1(97.487) (0.152) (0.113)* **R2 = 0.9633, n = 12. (Note: Equation 8a was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).From the estimates of Equations 6a and 8a, we see that the US stock adjustment coefficientS = 1 - 0.84 = 0.16 and the Soviet stock adjustment coefficient S’ = 1 - 0.96 = 0.04. Boththe US and Soviet stock adjustment coefficients are very small. From the estimates ofEquations la and 2a, we obtain values for the US and Soviet Richardsonian defence andgrievance terms. The regression results were as follows.K*t= B0 + B1X, (USA) (la)= 1295.7 + 0.1825X(345.6) (0.416)**R2 = 0.0209, n = 12 (Note: Equation la was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).179X, = B’0 + B’1K (USSR) (2a)= -867.4 + 5.85K,(2134) (2.40)**R2 = 0.3977, n = 12 (Note: Equation 2a was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).The estimates of Equations la and 2a show that the US Richardsonian defence coefficient,the defence context being the US-Soviet ICBM warhead race, is not significantly differentfrom zero. The US Richardsoniari grievance terms is, on the other hand, significant at 0.05and has a value of 1295.7. In contrast, the Soviet Union’s Richardsonian defence term issignificant at 0.05 and has a value of 5.85. It Richardsonian grievance terms is not, however,significantly different from zero.In order to get a more meaningful interpretation of these parameter estimatcs it wouldbe useful to first summarize the fact that for the US B0 is non-zero, B1 is zero and S liesbetween zero and one. For the USSR, B’0 is zero, B’1 is non-zero and S’ lies between zeroand one. When these values are plugged into Equations 21 and 22, the GSR model, thefollowing result obtains:K=SB0+(1- )K. (21b)180X, = (S’B’SB) + S’B’1(1- S)KL1 + (1 - S’)X1 ÷ (22b)Equations 21b and 22b do show an asymmetric arms race. Here State X, in this case, theSoviet Union, does take into account State K’s, the US’s, defence policy parameters whencalculating its ICBM warhead deployment requirements for time t. Equation 22b contains B0and S, the US’s Richardsonian grievance and stock adjustment terms. Equation 21b shows,in contrast, that the US did not take account of any of the Soviet Union’s defence policyparameters when it calculated its ICBM warhead deployment requirements for time t.The underlying structure of this race, as suggested by Equations 21b and 22b, is notinconsistent with historical, technological and doctrinal developments in the US and USSRin the 1960s. By 1960, the US had not only developed and deployed ICBM systems it haddeveloped and deployed SLBM systems. The Soviet Union, in contrast, was not able to begindeploying SLBMs until 1963 (Military Balance, 1969-70). ICBMs were the Soviet’s mainnuclear force and during the 1960s, Soviet military doctrine called for preemptive nuclearstrikes against US landbased nuclear forces if it appeared nuclear war was imminent. Thiswould mean that the Soviets would have to take account of the size of the US ICBM forcewhen determining their own ICBM warhead deployment requirements, as Equation 22bsuggests that they have. The US, on the other hand, operating under a second strike doctrine,had a preference for SLBMs. It would be less important for the US to maintain an ICBMforce numerically large enough to survive a first strike by Soviet ICBM forces. A second181strike mission could easily be carried out by the US’s SLBM force. In this sense, the UScould maintain its deterrent without tying their ICBM warhead deployment calculations toexpected Soviet ICBM warhead deployments. This is clearly supported by the constructionof Equation 21b. US ICBM warhead deployments were internally driven. That is not to say,however, that the US was oblivious to Soviet ICBM warhead deployments as the US did havea non-zero Richardsonian grievance term, B0.Equations 29 and 30 show the equilibrium ICBM deployment levels for the US andUSSR.L0(1 - L’2) + L1’0K, = (USA) (29)(1 - L2)(1 - L’2) - L’1=(1 - 0.84)(1- 0.93) - (0.25)(0)= 1288.39L’0(1-L2) + L’10X, = (USSR) (30)(1 - L2)(1 - L’2) - L’1182x,(1 - 0.84)(1- 0.93) - 0.25(0)= 4602.23Equilibrium in the US-Soviet ICBM warhead race, (K1, then, occurs at the point(1288.39, 4602.23). The point (1288.39, 4602.23) is, moreover, stable. This can be shownwith Equations 31 and 32 give the equilibrium demarcation curves for the US and SovietUnion, respectively.L0 (L2-1)X1=(USA) (31)L1 L1X = ooK1L’0 L’1X = + K1 (USSR) (32)(1-L’2) (1-L’2)X1 = 3.57KGraph 6.2 shows the plots of Equations 31 and 32. One can conclude from this plot that theGRAPH 6.2: Stable equilibrium in the US-Soviet ICBM warhead race, 1960-7 14602‘ci183xt-i Ixt - x1 = 0K - K1 01288184US-Soviet ICBM warhead race (1960-71) was stable. That this conclusion is robust issuggested by the fact that if one were to add the estimated standard errors of the GSRparameter estimates to the corresponding GSR parameter estimates and then, using that data,recalculate Equations 31 and 32, the result would still indicate that the US-Soviet ICBMwarhead race was stable. The same conclusion would still hold if one were to subtract theestimated standard errors of the GSR parameter estimates from the corresponding GSRparameter estimates.Substantively, Graph 6.2 suggests is that the US would be satisfied with 1288 ICBMwarheads deployed irrespective of the number deployed by the Soviet Union. The Soviets,in contrast, wanted numerical superiority, on the order of 3.57 to 1, as indicated by theslope of Equation 32. The ratio 3.57 to one may reflect Soviet calculations as to what theydeemed they needed to have a first strike capability vis-a-vis the US ICBM force. The USposition is not an unreasonable one given that it had a substantial warhead force kept onboardits submarines which the Soviets could not destroy and which could be used in retaliation fora Soviet strike on their ICBMs.7. TOTAL STRATEGIC WARHEAD COUNTSThe findings recorded above that the US and Soviet Union were engaged in anasymmetric arms race over ICBM warhead deployments 1960-71 and an action reaction arms185race over SLBM warhead deployments over the years 1972-87 lend a great deal of supportfor the validity of the micro approach to mathematical anns race analysis. States do, thesefindings suggest, compete over individual weapons systems with cross-purposes. It would,now, be useful to conduct a test of total US and Soviet strategic nuclear warhead levels(ICBM, SLBM and bomber based warhead counts) for evidence of interactive deployment.A finding of action-reaction interaction in US-Soviet aggregate strategic nuclear warheaddeployments would undermine the validity of the micro approach to mathematical arms raceanalysis and a finding of non-interaction would add further support for the approach.I tested my GSR model against a data set comprised of aggregate US and Sovietwarhead deployments (the sums of ICBM, SLBM and Bomber warheads for each country)running from the period 1967 to 1984. The time span for this study, 1967-84, is arbitrary. Ifthe US and Soviet Union do compete over aggregate strategic nuclear warhead deployments,then any sample drawn from period 1945-9 1 should show interaction.The figures contained in this data set are official US estimates, compiled by SIPRI.It is important to note that over the years, the US has amended its published count of US andSoviet aggregate strategic nuclear warheads for the year 1976. In 1976, it listed 8900warheads for the US and 3500 warheads for the Soviet Union. Later, in 1981, the US relistedthe 1976 count as 8400 for the US and 3300 for the Soviet Union. Which set of valuesshould be used? Ostrom (1978b) and Ostrom and Marra (1986) make the point that arms race186models should be tested using the data that the actors in question would have had at theirdisposal at the time that they were making their armament decisions. This is an importantpoint because, in general, my model suggests that the US would base a 1977 weaponsdeployment decision on what it and the Soviet Union had deployed in 1976. In the specificcase at hand, in 1977, the US would have been working with the figures 8900 warheads forthe US and 3500 warheads for the Soviet Union. The amended figures of 8400 for the USand 3300 for the Soviet Union were not available in 1977. Thus in testing my GSR model,I will use the figures of 8900 warheads for the US and 3500 warheads for the Soviet Union.I estimated the simplified versions of Equations 21 and 22, Equations 23 and 24.K = L0 + L1X + L2K + Y (USA) (23)= 577.91- 0.018X1 + 0.971K1(499.2) (0.08) (0.097)**R2 = 0.9629, n = 17, ** = significant at 0.05, * = significant at 0.10. (Note: Equation23/2 1 was estimated using OLS. Run Test results indicate no autocorrelation at a 0.05 levelof significance: see Table 6.1 for test results).X = L’0 + L’1K, + L’2XL1 + Z (USSR) (24)= -263.37 + 0.117K + 0.967X1(489.5) (0.095) (0.0826)**187R2 = 0.9706, n = 17 (Note: Equation 24/22 was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).Accordingly, the estimates of the GSR parameters in Equations 23/2 1 and 24/22 suggest anon-interaction outcome in the US-Soviet aggregate warhead race. In Equation 23, only theUS GSR stock adjustment term is significant, at 0.05. US aggregate warhead deployments attime period t are a function of US aggregate warhead deployments at time t- 1 only. Similarly,in Equation 24, only the US’s GSR stock adjustment coefficient is significant, at 0.05. Sovietaggregate warhead deployments at time period t-1 depend only on Soviet aggregate warheaddeployments at time t- 1.The estimates of Equations 6a and 8a tend to conform well with the estimates ofEquations 23 and 24. Estimates of Equations 6a and 8a also show non-interaction.K = SB0 + SBIXL + (1 - S)K1 (USA) (6a)= 548.47 - 0.02X + 0.98 1KL(503) (0.082) (0.101)**R2 = 0.9641, n = 17 (Note: Equation 6a was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).188X S’B’0 + S’B’1KL + (1- S’)X1 (USSR) (8a)= -196.76 + 0.093Kg + 0.987X1(521.01) (0.093) (0.079)**R2 = 0.9696, n = 17 (Note: Equation 8a was estimated using OLS. Run Test resultsindicate no autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).Estimates of Equations 6a and 8a suggest that both the US and Soviet Union have extremelysmall stock adjustment coefficients, the defence context being aggregate strategic nuclearwarhead deployments. The US stock adjustment coefficient is S = 0.019 and the Sovietcoefficient is S’ = 0.0 13. As suggested in Chapter 5, a non-interaction outcome can occur ifthe rivals in question have extremely small (if not zero) stock adjustment terms or if therivals both have zero Richardsonian defence coefficients or both. We can determine whichis the case by estimating Equations la and 2a.The estimates of Equations la and 2a are given immediately below.K=B0+B1X (USA) (la)= 29082 - 1.37X(11867) (2.37)**R2 = 0.0218, n= 17. Equation la was estimated using OLS. Run Test results indicateno autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).189X = B’0 + B’1KL (USSR) (2a)= -15350 + 7.2KL(34066) (4.34)*R2 = 0.1567, n = 17. Equation 2a was estimated using OLS. Run Test results indicateno autocorrelation at a 0.05 level of significance: see Table 6.1 for test results).The estimate of Equation la suggests that the US has a zero Richardsonian defencecoefficient and a non-zero Richardsonian grievance term. The estimate of Equation 2a, on theother hand, suggests that the Soviet Union has a non-zero Richardsonian defence term anda zero grievance term. What these findings suggests is that the US does not set strategicnuclear policy on the basis of total capability. Total warhead counts do, in contrast, seem toenter into the Soviet strategic calculus although their extremely small adjustment coefficientprevents an interactive response on their part to total US strategic nuclear warheaddeployments. In effect, the US seems to address the total strategic nuclear problem on aweapons systems by weapons systems basis as suggested by estimates of the US-SovietSLBM and ICBM warhead races.Estimates of Equations la, 2a, 6a, 8a, 23 and 24 suggest that the US and Soviet Unionwere not engaged in an action-reaction competition over total strategic warhead deployments.The extremely small stock adjustment coefficients obtained for each side may suggest aninability or disinclination on the part of the US and Soviet Union to react to each other’s total190strategic strength as a single package with a single response. Indeed, the historical recordshows that the preference for one nuclear weapons basing mode over another in the US andSoviet Union rises and falls, in part, in concert with technological change. For example, by1960, US bomber forces lost standing as ICBM technology developed. And ICBM forces loststanding as SLBM technology came to fruition. It may be that the total strategic count doesnot constitute a degree of weapons system individuation sufficient to constitute a basis foraction-reaction interaction between the US and Soviet Union.In the next section, I will, on the basis of estimates obtained for the US-Soviet SLBM,ICBM, and aggregate strategic nuclear warhead competitions, attempt to determine which ofPropositions 1-4, derived in the last chapter, concerning the arming behavior of states, hold.8. PROPOSITIONS 1-4.Did the US form its expectations of its future SLBM and ICBM nuclear warhead deploymentrequirements and its expectations of the Soviet Union’s future SLBM and ICBM warheaddeployment behavior in accordance with the Rational Expectations hypothesis? Did Sovietbehavior also accord with the Rational Expectations hypothesis? These questions can beanswered by referring to the value corr(Y, the correlation between the US and theSoviet Union’s GSR error terms, for each of the specified races, as per Propositions 1 and1.1, derived in the previous chapter.191Corr(Y, 4) for the US-Soviet SLBM race (1972-87) race is 0.3245. This correlationis not statistically significant at even 0.10. That is, the errors in the US-Soviet SLBMwarhead deployment race are not contemporaneously correlated. However, a systematicevaluation of the correlation between all possible combinations of Y and Z and laggedvalues of Y and 4 up to t-2 reveals that corr(Y, Z) = -0.414. This correlation is significantat 0.10. Corr(YL, corr(Y1,ZJ, corr(Y24), corr(Y.14) and corr(YL2,Zt2) were foundto not be statistically significant.Corr(Y, Z) = 0.0733 in the US-Soviet ICBM warhead race (1960-71). Thiscorrelation is not significant at 0.10. But, again, a systematic evaluation of the correlationbetween all possible combinations Y and Z and lagged values of Y and Z up to t-2 doesreveal a significant correlation: corr(YL1,4) = -0.48 which is significant at 0.10. Corr(Y2,4), corr(Y, ZL1) corr(Y, Z2), corr(Y1,Z1) and corr(Y2,Z) are not significant.Substantively, these results suggest that inefficiencies exist in US and Soviet reactionsto newly received information regarding innovations in the other’s SLBM and ICBM warheaddeployment behavior. The non-zero value of Corr(Y, Z2), from the SLBM warheaddeployment race, suggests that the US needed two years to respond to an innovation in SovietSLBM warhead deployment behavior. The Soviets, on the other hand, seemed to have neededmore than two years to respond to an innovation in American SLBM warhead deploymentbehavior. The negative value of corr(YL, 42). furthermore, suggests that when some shock192caused the Soviet Union to lower its SLBM warhead deployments for a given year below apreviously planned level, the US responded two years later by setting its SLBM warheaddeployment level above what it otherwise had planned for that year. In short, whatever theSoviet the action, the US reaction was to do the opposite.The non-zero value of corr(Y1,ZJ for the US-Soviet ICBM warhead competitionsuggests that the Soviet Union required one year to respond to innovations in US ICBMdeployment behavior. The US required more than two years to respond to innovations inSoviet ICBM warhead deployment behavior. The negative value of corr(YL1,Z) suggests thatUS and Soviet behavior was opposite.In short, these results suggest that Proposition 1.1 holds. The US and Soviet Uniondid form their expectations of their own armament requirements and of the other’s armingbehavior in accordance with the Rational Expectations hypothesis, but bureaucratic andtechnological inefficiencies existed in each side’s ability to transform newly receivedinformation regarding innovations in the other’s arming behavior into changes in each side’sown behavior.It must be stated at this point that such inefficiencies in information transformationcould also be indicative of an adaptive expectations formation process, as discussed inChapter 5. The existence of a non-contemporaneous cross correlation in State K and X’s193prediction errors is only a necessary, but not sufficient, condition for Rational Expectations.What is more certain is that because these inefficiencies do exist, the US-Soviet nuclear armsrace is not driven by prediction errors alone.It was, accordingly, revealed that the US and Soviet Union were engaged in an action-reaction arms race over SLBM warhead deployments (1972-87). In this instance, the US andSoviet Richardsonian defence terms were both non-zero and that their stock adjustment termsboth lay between zero and one. Accordingly, in the case of the US-Soviet SLBM warheadrace, Proposition 3.0 holds.The US-Soviet ICBM warhead race is an interesting case. Estimates of Equations 21and 22 showed the US and Soviet Union to be engaged in an asymmetric arms race overICBM warhead deployments. The USSR’s Richardsonian defence term was non-zero and itsstock adjustment term lay between zero and one. The US’s Richardsonian defence term, onthe other hand, was zero and its stock adjustment term fell between zero and one. ThusProposition 2.1, as set out in the previous chapter, does hold in the case of the US-SovietICBM warhead race (1960-71).Finally, an analysis of US-Soviet aggregate strategic nuclear warhead deployments(bombers, ICBMs and SLBMs) data (1967-84) showed no interaction. The US Richardsoniandefence coefficient was zero and its stock adjustment coefficient was so small as to be nearly194zero. The Soviet Richardsonian defence coefficient was non-zero, and its stock adjustmentcoefficient was even small than the US coefficient. These conditions are those set out inProposition 4.2 for the occurrence of non-interaction.Estimates of my GSR model for the US-Soviet SLBM, ICBM and aggregate strategicnuclear warhead competitions are summarized in Table 6.2.9. SUMMARYThe findings in this chapter reverse much of the previous quantitative work onthe US-Soviet nuclear arms race. For the most part, the existence of a US-Soviet nuclear armsrace could not be confirmed through quantitative analysis. This result has always beenpuzzling given the theoretical connection between the maintenance of nuclear deterrence andnuclear arms racing. Tests of my GSR model, on the other hand, suggest that the US andSoviet Union were engaged in an action reaction arms race over SLBM warhead deploymentsover the years 1972-87, and an asymmetric arms race over ICBM warhead deployments from1960-71. Tests of Equations 21 and 22 for US and Soviet aggregate warhead deployments,in contrast, showed no interaction. In the next chapter, I will attempt to assess why my studywas more successful than previous attempts to uncover the interactive components of the USSoviet nuclear competition.195TABLE 6.2: GSR and Richardsonian parameter estimates for the US-Soviet nuclear warheadcompetition. (Note: parameters significant at 0.05 are marked **• Those which are significantat 0.10 are marked *)SLBM ICBM TotalWarheads Warheads Warheads1972-87 1960-7 1 1967-84GSR GrievanceTermUSA 2023 201.6 577.9(432.68) (80.37) (499)** **USSR -399 -10.7 -263.37(323.49) (72.61) (489)GSR DefenceTermUSA 0.158 0.03 -0.018(0.1) (0.12) (0.0842)*USSR 0.20 0.25 0.1172(0.097) (0.134) (0.09)** **GSR StockAdjustment TermUSA 0.550 0.843 0.9717(0.130) (0.149) (0.09)** ** **USSR 0.80 0.937 0.9677(0.087) (0.109) (0.08)** ** **TABLE 6.2 continued196TABLE 6.2 (Continued): GSR and Richardsonian parameter estimates for the US-Sovietnuclear warhead competition. (Note: Parameters which are significant at 0.05 are marked **•Those which are significant at 0.10 are marked *.)SLBM ICBM TotalWarheads Warheads Warheads1972-87 1960-71 1967-84RichardsonianGrievance TermUSA 4183 1295 29082(237.72) (345.6) (11867)** ** **USSR -4285 -867 -15350(1257) (2134.5) (34066)**RichardsonianDefence TermUSA 0.43 0.18 -1.3(0.102) (0.416) (2.37)**USSR 1.43 5.85 7.26(0.25) (2.40) (4.34)** ** *Stock AdjustmentCoefficientUSA 0.521 0.16 0.019USSR 0.23 0.04 0.013Form of Action Asymmetric NoInteraction Reaction Reaction Reaction197My data, furthermore, suggest that the SLBM race was stable with equilibriumoccurring at 6928 SLBM warheads for the US and 6928 SLBM warheads for the SovietUnion. The ICBM warhead competition, too, was found to have been stable with anequilibrium point of 1288 warheads for the US and 4602 for the Soviet Union.Error analysis tentatively suggests, finally, that the US and Soviet Union formedexpectations of each other’s future nuclear weapons arming behavior in accordance with theRational Expectations hypothesis. But each side’s reactions were non-contemporaneous tostimulation from the other. Reaction time may have been slowed due to bureaucratic interiaor to the time needed to develop new weapons systems. My error analysis, nevertheless,suggests a two-way linkage existed between nuclear weapons deployment policy in the US andnuclear weapons deployment policy in the USSR. One would expect such a linkage to haveexisted if mutual nuclear deterrence was to exist and be maintained.In the next and final chapter, I will summarize the overall achievements of myresearch effort and raise a number of questions, suggested by my research, for futureconsideration.198CHAPTER VII: SUMMARYIn this, the final chapter of my dissertation, I will assess the overall contributions ofmy research effort to mathematical anns race research in general and to the quantitative studyof the US-Soviet arms race in particular. In this dissertation, I rejected the traditional macroapproach to arms race analysis. I specified and then applied a new approach, which I havetermed the micro approach. Did I uncover anything new about the nature of the US-Sovietarms race with this approach? Could we have learned as much using the macro approach?To summarize, the micro approach to arms race analysis revolves around the idea thatstates engaged in an arms race, in particular the US and Soviet Union, compete on a weaponssystem versus a cross-purpose weapons system basis. My GSR model, reproducedimmediately below, was formulated on that basis and thus constitutes an appropriateanalytical framework for an analysis of the US-Soviet arms race.(SB0 + SB1’ ’0) SB1(1- S’) (1- S)K= + X1+ K1÷Y, (21)1-SBS’B’ 1-SBS’B’ 1-SBS’B’199(S’B’0 + S’B’10) S’B’1(1- S) (1- S’)X =- +-K + -X + Z, (22)1-SBS’B’ 1-SBS’B’ 1-SBS’B’Other formulations, yet to be derived, may be as suitable. Finally, time frames for anyweapons system, cross-purpose weapons system analysis must be set to accord with thehistorical period in which those systems were dominant in US-Soviet military calculations.The macro approach, in contrast; is based on the view that states engaged in an armsrace compete on the basis of total armed might versus total armed might. A state’s annualaggregate military expenditures are, generally, thought to constitute a good indicator of itstotal anned might. Models, then, are most often tested against such data. Analytical timeframes are generally set to cover as lengthy a period as possible. No reasons are normallygiven for so setting analytical time frames, but one can surmise that it is done in order toachieve statistical consistency in arms race parameter estimations.1. COMPARING THE OLD AND THE NEWAs I stated at the outset of the previous chapter, how one approaches the study of theUS-Soviet arms race determines, in large measure, what one will find.200My application of the micro approach to the US-Soviet nuclear arms has yieldedstrong evidence that the US and Soviets were engaged in an action-reaction arms race overSLBM warhead deployments over the years 1972-87. This finding stands in marked contrastto finding reported by Kugler et al (1980) in their detailed macro study of the US-Sovietnuclear arms race. They could find no quantitative evidence of action-reaction in US-Sovietaggregate expenditures on strategic nuclear forces over the years 1952-76. Kugler et al (1980:128), accordingly, stated thatThe absence of competition or races between the two countries [the US and- USSR] leads to the tling finding that the local conditis for deterrenceare absence, and by inference to the conclusion that, mutual deterrence is nottaking place.They further stateIn the absence of competition, furious nuclear arms stockpiling is not easy toevaluate. While it may not be appropriate to think of the process whereby thetwo countries acquire strategic weapons as ‘neurotic’, it is the term thatnevertheless comes to mind. ‘Neurotic’, then, may have to do [p. 131].My findings, in contrast to Kugler et al (1980), permit the intuitively appealing conclusionthat US-Soviet nuclear arms acquisitions, at least US-Soviet SLBM warhead acquisitions,were systematically driven, specifically, action-reaction driven. The logical conditions for the201maintenance of strategic deterrence, one may accordingly surmise, were in fact being met.Who’s findings should one accept: mine or Kugler et als? The answer to this questionrests, in part, on the theoretical validity of the macro approach as compared to the microapproach to arms race analysis. Secondly, one must also weigh the findings generated by themicro approach against the findings generated by the macro approach. Which best accordwith past experience, informed intuition and established theory?The macro approach is based, to reiterate, on the assumption that states engaged inan arms race compete on the basis of total anned might versus total armed might. In fact, thehistorical record is replete with examples of the US and Soviets each justifying theiracquisitions on a weapons system, cross-purpose weapons system basis. ICBM deployments,for example, were originally justified on this basis.Secondly, it was reasonable to find that the US and Soviets were engaged in action-reaction arms race over SLBM warheads in the period 1972-87 and it was reasonable to findthat they were engaged in an asymmetric race over ICBM warhead deployments over theperiod 1960-71. The period 1972-87 was one where both sides, at the same time, placed astrong emphasis on having second strike reserve forces. Given the vulnerability of longrangebombers to interception and given the vulnerability of the ICBM to preemptive attack, the202SLBM basing mode provided the best protection for each side’s nuclear warheads.In theworst case, where each side’s bomber and ICBM forces were decimated in a series ofcounter-force, counter-counter force strikes, then the side which had the most effective, if notthe largest, SLBM warhead force would be in a position to dictate terms to the other. Action-reaction interaction, in this context, is reasonable. Similarly, one should expect to find thatthe USSR was racing against the US over ICBM warhead deployments in 1960-7 1, and theAmericans racing themselves. Soviet doctrine, over that period, called for preemptive strikesagainst US ICBM forces in the event that nuclear war appeared imminent. In order to for theSoviet Union to be able to effect this mission, Soviet ICBM forces would have to be kept insome mathematical proportion to US ICBM forces. (For most of the period 1960-7 1, both USand Soviet ICBMs carried only one warhead each. Thus the missile count and the warheadcount, for most of that period, were one and the same.) The US in the period 1960-7 1, incontrast, focused its efforts on developing a secure second-strike force. They did not thinkthe ICBM basing mode to be particularly secure. Instead, the US invested heavily in SLBMsin the period 1960-7 1. It is, finally, interesting to note that tests of my GSR model againstUS-Soviet aggregate warhead deployments showed no interaction, that is, no race. Indeed,a macro analysis of the US-Soviet nuclear arms race masks any notion of an action-reactionSLBM warhead race (1972-87) and an asymmetric ICBM warhead race (1960-7 1).My contribution to the field of mathematical arms race research, then, is a case for203reorienting away from macro analysis to the micro approach to mathematical arms raceanalysis which I have specified in this dissertation.2. QUESTIONS FOR FUTURE ARMS RACE RESEARCHTwo sets of questions emerge from my theoretical and empirical arms race analysis.The first set of questions concerns the role of Rational Expectations formation in the US-Soviet arms competition. In their studies of the impact of Rational Expectations formationon the US-Soviet aims competition, Williams and McGinnis (1988) and McGinnis andWilliams (1989) concluded that the non-interaction and asymmetric reactions puzzles werenot puzzles at all. Under Rational Expectations, they argued, the US and Soviet Union wouldnot react in the current time period to each other’s previous period military expenditures.Instead, the US and the Soviet Union would, in the current period, react only to innovations,deviations, from the other’s planned level of military expenditure. Only if the US and SovietUnion were inefficient in their collection and use of information regarding innovations in theother’s behavior would past period expenditures play a role in determining current periodexpenditures. Williams and McGinnis thought that such inefficiencies would be negligible,however.On the basis of their findings, Williams and McGinnis (1988: 992) suggested thatarms race research turn away from the attempt to204verify empirically the mere existence of reaction [in the US-Soviet armscompetition] to focus instead on a broader set of research questions concerningthe consequences of the intimate linkage between domestic policy processesof rival states.I concur with Williams and McGinnis that we should now focus attention on theconsequences of the linkages between US and Soviet arms decision making processes. Inparticular, we should begin to focus on questions such as, What specific information did theUS and USSR draw on when forming their expectations of each other’s future arming goalsand strategies. To what extent did each side factor information used by the other in itsarmament calculations into its own calculations? For example, public opinion, as Williams -and McGinnis note, has an impact on US military expenditure levels. To what extent did theSoviets take that opinion into account when formulating their estimates of future US arminggoals and strategies?In contrast to Williams and McGinnis, however, I suggest, based on my researcheffort, that efforts to uncover the existence of action-reaction interaction in US-Soviet armsacquisitions be stepped. The search, then, should focus on action-reaction in the deploymentof individual weapons systems with cross-purposes. McCubbins (1983) began this processwith an analysis of the US-Soviet conventional arms competition. I carried on the search withan analysis of the US-Soviet strategic nuclear arms competition. Tests should now beconducted to determine if US-Soviet short range nuclear weapons acquisitions follow the205weapons system-cross purpose weapons system deployment pattern established at theconventional and strategic nuclear levels. From such efforts, we would have an invaluablebase from which to assess the overall nature and dynamics of the US-Soviet militarycompetition, arguably, the most important military competition in human history. There are,however, certain issues which must be resolved before we could set about sketching a pictureof the overall nature and dynamics of the US-Soviet military competition.Under the macro perspective where arms races are seen as competitions of total armedmight versus total armed might, the meaning of the term “arms race equilibrium” is straightforward. “Arms race equilibrium” refers to overall equilibrium in the military forces of thecontending parties. But when arms races between rival states are viewed as occurring overindividual weapons systems with cross purposes, and when, theoretically, any given pair ofrivals could thus engage each other in several distinct system level arms races, the meaningof the term arms race equilibrium becomes less clear. Theoretically, each individual systemlevel competition can have its own distinctive dynamics and its own distinctive equilibriumpoint. We have seen this to be the case with respect to the US-Soviet SLBM warhead andthe US-Soviet ICBM warhead races. Can we, then, speak of overall arms race stability orinstability if, for any given pair of rivals, we find that some individual system level races arestable while others are unstable. How do the dynamics of one individual system levelcompetition between two rivals affect other individual level competitions between the samerivals? How do these questions impinge on the theory that unstable arms races lead206competing states inevitably to war? These are extremely difficult questions, which must beanswered before a sketch of the overall US-Soviet arms competition, based on system levelanalyses, can be drawn. Their importance merits them a position at the top of the arms raceresearch agenda.3. THE MICRO APPROACH TO MATHEMATICAL ARMS RACE ANALYSIS ANDTHE EN]) OF THE COLD WAR.With the end of the US-Soviet Cold-War, one might question the continued relevanceof the micro approach to mathematical arms race analysis as a tool of international analysis.It is true that East-West relations have become much friendlier in recent years. It is true thatEastern Europe, set free by Moscow, is now in the process of democratizing itself. It is alsotrue that East and West have recently signed a major European conventional reduction treatyand major strategic arms reduction agreement.But the US-Soviet military, political and economic rivalry was not an isolated event.Rivalry, aggressive intentions and aggression itself have been the hallmarks of internationalinteraction since the time of Alexander. The US-Soviet rivalry was a particularly important,and dangerous, historical example in a long line of cases.207Even though the Cold War is over, the US-Soviet rivalry will stand as an excellentexample of a broader theoretical problem in international relations, the problem of rivalry,and arms racing. It will continue to be the focus of much scholarly debate and research. Themicro approach to mathematical arms race analysis could serve as a useful analytical devicein that research. And, unfortunately, so long as the world remains divided into nation-states,the potential for some new rivalry, and hence, a new arms race, to arise will not diminish.208BIBLIOGRAPHYAbelson, R.P. (1963) “A ‘Derivation’ of Richardson’s Equations.” JOURNAL OFCONFLICT RESOLUTION 7:13-15.Abolfathi, F. (1975) “Obstacles to the Quantitative Analysis of Public Expenditure Policies:The Case of Military Spending,” mimeo. CACI, Inc., Federal: Washington, D.C.Abolfathi, F. 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Woodbury (1986) “U.S. Defense Spending, Electoral Cycles, and SovietAmerican Relations.” JOURNAL OF CONFLICT RESOLUTION 30:445-68.242APPENDIX A: US-Soviet NuclearArms Control AgreementsHOT LINE AGREEMENT (1963): This agreement allowed for the establishment of directradio and telegraph links between Washington and Moscow to ensure direct communicationbetween heads of government in times of extreme crisis.HOTLINE MODERNIZATION AGREEMENT (1971): This agreement allows for theestablishment of a satellite link between Moscow and Washington to be used in times ofextreme crisis.ACCiDENTS MEASURES AGREEMENT (1971): Requires the signatories to take steps toimprove safeguards against the accidental or unauthorized use of nuclear weapons.ANTI-BALLISTIC MISSILE TREATY (1972): This treaty limits the deployment of antiballistic missile defence systems to two sites for each signatory. A 1974 Protocol furtherlimited the deployment of ABM systems to one site for each signatory.STRATEGIC ARMS LIMITATION TREATY (SALT) I INTERIM AGREEMENT (1972):This treaty imposes ceilings (maximum deployment levels) on each side’s strategic nuclearforces. More specifically, it places ceilings on the numbers of strategic launchers (deliveryvehicles) that each side can deploy across its triad. (Note: Under SALT I, the USSR wasgiven higher launcher ceilings than the US. Richard Nixon allowed this arguing the US hada substantial technological lead over the Soviet Union and thus the Soviet Union should beallowed to have numerically larger nuclear forces. In 1972, the principal technologicaladvantage that the US had over the Soviet Union was its MIRV technology (the ability toplace more than one nuclear warhead inside a launch vehicle.)PREVENTION OF NUCLEAR WAR AGREEMENT (1973): This agreement commits thesignatories to consultation in the event that there is a danger of nuclear war.STRATEGIC ARMS LIMITATION TREATY (SALT) 11(1979): (Note: The treaty cameabout due mainly to US disaffection with SALT I. The USSR had quickly developed thetechnology to MIRV its missiles and by the late 1970s some Americans were arguing thatthe USSR had a substantial advantage over the US. The USSR had larger a larger deliveryvehicle force than the US due to SALT I and that force was being MIRVed.) SALT II thusaimed at establishing an overall equivalence between US and Soviet nuclear forces. SALTII was never ratified by the US Senate.243APPENDIX A (CONT)THRESHOLD TEST BAN TREATY (1974): This treaty prohibits the signatories fromconducting underground nuclear test explosions where the yield exceeds 150 kilotons.PEACEFUL NUCLEAR EXPLOSIONS TREATY (1974): This treaty bans “groupexplosions” of nuclear weapons when the aggregate yield would exceed 1500 kilotons. (Note:Such tests are useful for determining the force configuration and yield required to destroyICBM silos. Both sides agree that attacking ICBM silos would only be done in a first-strikesituation.) The PEACEFUL NUCLEAR EXPLOSIONS TREATY also provided for on-siteinspection where nuclear test detonations would have a yield in excess of 150 kilotons.INTERMEDIATE NUCLEAR FORCES (INF) TREATY (1987): This treaty committed theUS and USSR to remove and destroy all of their intermediate range ballistic missiles fromthe European theatre, the US Tomahawk Cruise and Pershing II Missiles and the Soviet SS20s. INF also provided for intrusive verification procedures to ensure compliance with theterms of the treaty.STRATEGIC ARMS REDUCTION TREATY (START) (signed in Moscow on 31 July1991): This treaty requires the US and Soviets to reduce their current strategic nuclearwarhead deployments levels. The Soviets will reduce their strategic nuclear warhead inventoryfrom 11,000 to 7,000 and the US will reduce from 12,000 to 9,000. (Note: While SALT I didplace ceilings on the number of launchers each side could deploy, it did not place any sortof limits on warhead deployments. MIRV technology was to allow each side to substantiallyincrease its warhead count while staying within the launcher ceiling imposed by SALT I.)244APPENDIX B: Multilateral NuclearArms Control AgreementsANTARCTIC TREATY (1959): This treaty prohibits signatory states from using the Antarcticfor military purposes. In particular, it forbids nuclear testing in the Antarctic. (30 states havesigned this treaty.)PARTIAL TEST BAN TREATY (1963): This treaty bans nuclear testing in the atmosphere,outerspace, and underwater. (111 states)OUTER SPACE TREATY (1967): Under this treaty, signatories are prohibited from basingnuclear weapons in earth orbit or outerspace. (81 states)LATIN AMERICAN NUCLEAR FREE ZONE TREATY (1967): This treaty prohibitssignatory states from testing, possessing, or deploying nuclear weapons in Latin America. Itfurther requires signatories to strict safeguards on any nuclear facilities they may possess. (25states)NON-PROLIFERATION TREATY (1968): This treaty bans the US, China, the USSR, Franceand Britain from transferring nuclear weapons and nuclear weapons technology to non-nuclearstates. It also commits the five nuclear powers to seek to end the nuclear arms race. (129states)SEABED TREATY (1971): This treaty forbids the deployment of nuclear weapons on theseabed beyond a 12-mile costal limit. (74 states)SOUTH PACIFIC NUCLEAR FREE ZONE TREATY (1985): In this case, signatories areprohibited from testing, manufacturing, acquiring, or stationing nuclear weapons in the SouthPacific area. The SOUTH PACIFIC NUCLEAR FREE ZONE TREATY further requests thatthe five major nuclear states sign a protocol banning the use, the threat of use, and the testingof nuclear weapons. (8 states)245APPENDIX C: Annual US-Soviet SLBMNuclear Warhead Deployments1972-1987(Source: BULLETIN OF THE ATOMIC SCIENTISTS May, 1988)YEAR US USSR1972 2384 4581973 3536 5561974 3824 6881975 3968 8281976 4688 9541977 4832 15031978 5120 19701979 5088 21051980 4896 21981981 4976 27141982 4992 27621983 5152 27501984 5536 29341985 5760 31601986 5632 31761987 5632 3408246APPENDIX D: Annual US-Soviet ICBMNuclear Warhead Deployments1960- 197 1(Source: Military Balance, 1969-70, SIPRIYearbook, 1976, 1981)YEAR US USSR1960 18 351961 63 501962 294 751963 424 1001964 834 2001965 854 2621966 934 3381967 1054 7221968 1054 9021969 1054 11981970 1074 14981971 1274 1527247APPENDIX E: Annual Aggregate US-Soviet StrategicNuclear Warhead Deployments1967-1984(Source: SIPRI Yearbook 1976, 1981, 1982, 1983, 1984)YEAR US USSR1967 4500 10001968 4200 11001969 4200 13501970 4000 18001971 4600 21001972 5700 25001973 6784 22001974 7650 25001975 8500 25001976 8900 35001977 8500 40001978 9000 45001979 9200 50001980 9200 60001981 9000 70001982 9540 88021983 9681 87811984 9665 8880248APPENDIX F: Alternative testsof the GSR modelIn Chapter 6, the GSR model and its component equations were tested against dataon US-Soviet strategic nuclear warhead deployments. Equations 21/23, 22/24, 6a, 8a, la, and2a were estimated from data on US-Soviet SLBM warhead deployments, 1972-87, from dataon US-Soviet ICBM warhead deployments, 1960-71 and from data on US-Soviet totalstrategic nucl’ar warhead deployments, 1967-84. For each competition, the GSR model andits component equations were estimated as single equation models. Arguably, however, eachof the equation pairs 21/23 and 22/24, 8a and 6a, and la and 2a constitutes a simultaneousequation system and must be estimated as such. Because the errors in each pair of equationsare theoretically cross correlated, estimation by the method of Seemingly UnrelatedRegression would be appropriate.In this appendix, I will present estimates of the GSR model and its componentequations under the assumption that each of the equation pairs, Equations 6a and 8a, andEquations 1 a and 2a, constitutes a simultaneous system. It will not be necessary to reestimatethe system comprised of Equations 2 1/23 and 22/24. Because Equations 2 1/23 and 22/24contain the same independent variables, the method of Seemingly Unrelated Regression wouldyield results identical to those obtained when Equations 2 1/23 and 22/24 are estimatedseparately.The estimates obtained under the method of Seemingly Unrelated Regression forEquations 6a and 8a and Equations la and 2a for the US-Soviet SLBM warhead race, 1972-87, for the US-Soviet ICBM warhead competition 1960-7 1 and for the US-Soviet totalstrategic nuclear warhead competition were not much different from those presented inChapter 6 when each equation was estimated as a single equation model. In fact, the resultspresented below are completely consistent with those presented in Chapter 6. Thus, noalteration is necessary in the substantive analysis presented in Chapter 6.2491. THE US-SOVIET SLBM WARHEAD RACE, 1972-87 (SEEMINGLY UNRELATEDREGRESSION RESULTS)K = SB0 + SB1X + (1- S)K1 (USA) (6a)= 2292.5 + 0.27X, + 0.43K.1(431) (0.129) (0.14)** ** **R2 = 0.9192, n = 15. ** indicates significant at 0.05 and * indicates significant at 0.01.(Run Test results indicate no autocorrelation at a 0.05 level of significance: 6 Runs, 6 positiveand 9 negative)X = S’B’0 + S’B’K + (1 - S’)X1 (USSR) (8a)= -1287.8 + 0.41K, + 0.72X1(492) (0.128) (0.085)** ** **R2 = 0.9787, n = 15 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 7 Runs, 7 positive and 8 negative)2501. SLBM WARHEAD REGRESSIONS (CONT)K*t= B0 + B1X (USA) (la)= 4045.3 + 0.48X,(215) (0.09)** **R2 = 0.6573, n = 15 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 6 Runs, 6 positive and 9 negative)X = B’0 + B’1KL (USSR) (2a)= -4746.8 + 1.5Kg(1034) (0.20)** **R2 = 0.7719, n = 15 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 7 Runs, 7 positive and 8 negative)2512. THE US-SOVIET ICBM WARHEAD RACE 1960-7 1 (SEEMINGLY UNRELATEDREGRESSION RESULTS)K = SB0 + SB1X + (1 - S)K,1 (USA) (6a)= 209.1 + 0.06X + 0.80KL1(80.2) (0.12) (0.16)** **R2 = 0.9087, n = 11 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 8 Runs, 5 positive and 6 negative)X = S’B’0 + S’B’K + (1 - 5)X1 (USSR) (8a)= -68.6 + 0.29K + 0.92X1(96.7) (0.14) (0.11)** **R2 = 0.9623, n = 11 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 6 Runs, 5 positive and 6 negative)2522. ICBM WARHEAD REGRESSIONS (CONT)K*= B0 + B1X (USA) (la)= 1055.8 + 0.33X,(273) (0.32)**- R2 = 0.0599-, n 11, (Run Test results indicate no autocorrelation at a 0.05 level of-significance: 8 Runs, 5 positive and 6 negative)X = B’0 + B’1KL (USSR) (2a)= -940 + 3.8Kg(1050) (1.17)**R2 = 0.4620, n = 11 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 6 Runs, 5 positive and 6 negative)2533. THE US-SOVIET AGGREGATE STRATEGIC NUCLEAR WARHEADCOMPETITION, 1967-84 (SEEMINGLY UNRELATED REGRESSIONRESULTS)K = SB0 + SB1XL + (1- S)KL1 (USA) (6a)= 593.6 - 0.11X, + 0.96KL1(502) (0.08) (0.10)**R2 = 0.95 18, n = 17 (Run Test results indicate no autocorrelation at a 0,05 level ofsignificance: 7 Runs, 8 positive and 9 negative)X = S’B’0 + S’B’1K + (1 - S’)X1 (USSR) (8a)= -304 + 0.115K, + 0.972X1(502) (0.09) (0.07)**= 0.9695, n = 17 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 7 Runs, 8 positive and 9 negative)2543. US-SOVIET AGGREGATE STRATEGIC WARHEAD REGRESSIONS (CONT)K*t= B0 + B1X (USA) (la)= 17460- 0.30K(6605) (1.3)**R2 = 0.0072, n = 17 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 7 Runs, 8 positive and 9 negative)X, = B’0 + B’1K (USSR) (2a)= -11255 + 4.18Kg(15738) (2.0)**R2 = 0.1942, n = 17 (Run Test results indicate no autocorrelation at a 0.05 level ofsignificance: 7 Runs, 8 positive and 9 negative)

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