Essays in Group Decision-MakingbySHASIKANTA SINGH NANDEIBAMB.A., The University of Madras, 1984M.A., The University of Delhi, 1986A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Economics)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Shasikanta Singh Nandeibam, 1993In 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.(Signature)Department of FLO 1\) 0 I'v\ 1 C The University of British ColumbiaVancouver, CanadaDate I S^"Tu"-.12- 19 9 S DE-6 (2/88)AbstractThe thesis comprises of two essays. Although the two essays deal with somewhat differentsituations and use different approaches, both of them essentially examine the problem ofmaking decisions that affect some group of individuals.The first essay is on moral hazard and looks at the principal's problem in a principal-agent(s) free-rider problem in which, unlike most existing work, the principal is not precludedfrom participating in the production process. Furthermore, there are no uncertainties, butmoral hazard is caused by joint production which renders the action of each individual in theproduction process unobservable. A multi-stage extensive game in which only the principalcan propose the output sharing rule determines both the set of individuals who actuallyparticipate in the joint production process and the output sharing rule.The main conclusion we draw in the first essay is that, when designing the optimal outputsharing rule, the principal need not look for any output sharing rule more sophisticated thanthe linear or piecewise linear rules we frequently observe. We also characterize the conditionunder which the principal chooses to take part in production, and conclude that the issueof mitigation of moral hazard and sustainability of efficiency crucially hinges on whetherthe principal actually participates in production or not. More concretely, we show thatiimoral hazard dissipates completely whenever the principal does not participate in production,however, even then she does not achieve as much welfare as in the First Best situation if herbest option in the First Best situation is to take part in production.The second essay is in stochastic social choice theory. In a paper published in 1986 inEconometrica, Pattanaik and Peleg formulated stochastic analogues for each of Arrow's ax-ioms and concluded that the stochastic social choice functions that satisfy their axioms areessentially randon dictatorships when individuals have strict preferences. More precisely,there is a unique weight associated with each individual such that the vector of these individ-ual weights has the properties of a probability distribution over the set of individuals, and,given any preference profile and any feasible set, the probability that a feasible alternativeis chosen is equal to the sum of the weights of those individuals who have this alternative astheir best feasible alternative.We extend the analysis of Pattanaik and Peleg by allowing individuals to have weak pref-erences. As in their paper, it turns out that the probabilistic versions of Arrow's conditionsimply that there are individual weights. However, now, given a preference profile and a fea-sible set, we partition the society so that any two individuals from different elements of thepartition have no common best feasible alternatives, but the set of best feasible alternativesof each individual in an element of the partition overlaps with that of some other individual inthe same element. Using this partition, it is shown that the only restriction on the stochasticsocial choice function is that the sum of the weights of all individuals belonging to the sameelement in the partition is equal to the probability that some alternative which is best in thefeasible set for one of these individuals is chosen. When everyone has unique best feasiblealternatives, the rules characterized here reduce to those of Pattanaik and Peleg.iiiCONTENTSAbstract^ iiAcknowledgementIntroduction^ 1Chapter 1: A Free-Rider Problem with a Free-Riding Principal^71.1: Introduction^ 71.2: Production and Preferences^ 121.3: First Best^ 201.4: Optimal Outcomes^ 231.5: Limited Liability 371.6: Uniqueness of SPE Payoffs^ 421.7: An Example^ 451.8: Nonquasilinear Utilities^ 47Chapter 2: The Power Structure for Stochastic Social ChoiceFunctions with an Unrestricted Preference Domain^482.1: Introduction^ 482.2: Notation and Definitions^ 542.3: Coalitional Weights 572.4: The Structure of SSCFs^ 67Conclusion^ 82References 85Appendix A^ 88Appendix B 98ivAcknowledgementI would like to thank my thesis supervisor Dr. John Weymark, and my thesis supervisorycommittee members, Dr. Kenneth Hendricks and Dr. Guofu Tan, for their many helpfulcomments and suggestions. I have also benefited from the comments of Dr. Charles Blackorby,Dr. John Cragg, Dr. David Donaldson, Dr. Mukesh Eswaran, Dr. Ashok Kotwal, Dr.William Schorm, and the students in the economics Ph.D. seminar class (Econ 640, 1990-92)in the Department of Economics at the University of British Columbia.As a Ph.D. student, I received financial assistance in the form of University GraduateFellowships from the University of British Columbia for the 1988-92 academic years. I wouldalso like to thank the Whiteley family for the Albert Whiteley Memorial Fellowship I receivedfor 1988-89.IntroductionGiven the ever-increasing diversity of human wants and the limited resources at thedisposal of any one individual, we frequently observe two or more individuals pooling theirresources in some activity whose outcome affects all of them. Examples of such arrangementsrange from the more specific case of a group of individuals taking part in a team productionprocess whose outcome affects all of them to the more general case of a society as a groupof individuals trying to achieve certain social objectives that affect every member of thesociety. In such situations, individuals in the group often tend to have different preferencesover the set of possible outcomes leading to conflicting incentives. This makes it interestingand important to analyse the thorny issue of making decisions that affect the outcome. Thetwo essays in this thesis look at this issue in two very different situations using very differentapproaches.The first essay deals with moral hazard in team production. Whenever a group of indi-viduals jointly participate in some production process, by observing only the final output,it is difficult to determine the input contribution of each individual to this output. For ex-ample, when two individuals jointly load heavy cargo into a truck, by observing the totalweight loaded, it is not possible to find out the marginal productivity of each individual. The1difficulty in determining the individual contributions in a joint production process gives eachindividual a free-riding incentive, thereby, creating a moral hazard problem. This makes itimportant to analyse the issue of designing reward schemes for the individuals participatingin a joint production process, because different reward schemes will affect the free-ridingincentives of the individuals differently.We consider a situation in which each member of a group of individuals can choose to takepart in a deterministic joint production process. However, although there are no uncertaintiesin production, the nonobservability and/or nonverifiability of the actions in the productionprocess due to joint production gives each participating individual a free-riding incentive.We analyse the problem of output sharing in this setup using a principal-agent(s) frameworkwhich differs from traditional principal-agent problems in one important respect. In ourproblem, one member of the group acts as the principal and the rest act as agents. Thismeans that, unlike the principal in most existing principal-agent models, our principal is aresidual claimant who is not precluded from participating in the joint production process inwhich she has a free-riding incentive just like the agents. Being the residual claimant, theprincipal's problem is to come up with an output sharing rule which depends only on theobservable final output. This output sharing rule is designed to determine those individualsin the group who will decide to take part in the production process, and also, to inducethem to take actions that will maximize the benefit of the principal. The special positionof the individual in the group who acts as the principal may stem from her ownership ofcertain tangible inputs required in the joint production process like technology, equipment,capital. license to operate the business, etc. Such principal-agent(s) arrangements are not souncommon among self-owned and operated businesses as their operation frequently involve2the services of individuals other than the owner. In these businesses, possibly in addition tosome other activities, the owner tends to perform the task of supervision and coordinationof the activities of all the individuals which often becomes an essential part of the operation.In the free-rider problem described above, we primarily focus attention on the structureof the optimal output sharing rule the principal comes up with. This as well as the impor-tant issue of mitigation of moral hazard and sustainability of efficiency crucially depends onwhether the principal decides to take part in the production process, in which case she is afree-riding residual claimant, or not to take part in the production process, in which case sheis still a residual claimant but not a free-rider. So we first characterize the condition whichdetermines the principal's participation decision. Our main conclusion is that the optimaloutput sharing rule the principal designs does not have to be any more complicated thanthe linear or piecewise linear rules we often observe. Regarding mitigation of moral hazard,we conclude that the problem of moral hazard can be dissipated completely only when theprincipal decides not to take part in the production process. This is not surprising, becausethe role of the principal in this case is more or less the same as that of the outsider in Holm-strom [23] who administers "budget-breaking" incentive schemes. We also conclude that theprincipal can achieve the same amount of welfare as in the First Best situation, which isthe hypothetical situation in which the actions are observable and the output sharing ruleis a function of these actions, provided her best option in the First Best situation is not totake part in the production process. As a result, even when it is best for the principal tocompletely mitigate the moral hazard problem by not taking part in the production process,she will still suffer welfare loss if her best option in the First Best situation requires herparticipation in the production process.3The second essay in this thesis deals with social choice theory. Social choice theory isan area of socio-politico-economic theory which is concerned with the study of collective de-cision rules that aggregate individual preferences over social alternatives to arrive at socialpreference relations or socially chosen alternatives. The primary objective of this area is toformulate systems of axioms that seem "reasonable" according to some ethical, social, polit-ical, or economic criteria and characterize collective decision rules that satisfy these systemsof axioms. One of the most important result in social choice theory is the ImpossibilityTheorem of Arrow [2]. This theorem states that there does not exist a nondictatorial socialwelfare function that satisfies a certain reasonable system of axioms formulated by Arrow.Since the seminal work of Arrow, numerous attempts have been made, without muchsuccess, to escape his impossibility result by modifying his system of axioms in various ways.This has led to a new direction of research involving stochastic collective decision rules. Theserules are more general than their deterministic counterparts as they aggregate individualpreferences over social alternatives to arrive at lotteries over social preferences or lotteriesover feasible social alternatives. Compared to deterministic collective decision rules, whatmakes stochastic collective decision rules attractive is their ability to incorporate certainelements of fairness and reasonable compromise For example, it seems fair and reasonableto toss a coin to decide who gets an indivisible good to be given to one of two individualswho want it.In 1986 Pattanaik and Peleg published a paper on stochastic social choice functions inEconometrica. Stochastic social choice functions are collective decision rules that map eachcombination of a social preference profile and a feasible set of social alternatives to a socialchoice lottery over the feasible set. Pattanaik and Peleg [29] formulated stochastic analogues4for each of Arrow's [2] axioms (namely. Independence, Pareto and Collective rationality) andcharacterized the stochastic social choice functions that satisfy their axioms when individ-uals have strict preferences. These stochastic social choice functions are essentially randomdictatorships. More precisely, there is a unique weight associated with each individual andthe vector of individual weights is like a probability distribution over the set of individuals inthe society. Given any combination of a social preference profile and a feasible set of socialalternatives, the social probability of choosing a feasible alternative is equal to the sum ofthe weights of those individuals who have this alternative as their most preferred feasiblealternative.In the second essay, we extend the analysis in Pattanaik and Peleg [29] by permittingindividuals to have weak preference orders over the social alternatives (i.e. we allow individ-uals to have indifference between alternatives). The key to the result of Pattanaik and Peleg[29] is that, when individuals have strict preferences, they have unique best alternatives inthe feasible set, and hence, different individuals cannot have overlapping sets of best feasiblealternatives that are not the same. However, when individuals have indifference betweenalternatives, they need not have unique best feasible alternatives and it is possible for differ-ent individuals to have overlapping, but nonidentical, sets of best feasible alternatives. Thismakes the extension to the larger preference domain technically demanding. Nonetheless,as in Pattanaik and Peleg [29], the stochastic versions of Arrow's [2] axioms still imply thatindividuals have unique weights. But now, given a social preference profile and a feasibleset of social alternatives, we partition the society into coalitions so that any two individualsfrom different coalitions have no common best feasible alternative, but the set of best feasiblealternatives of each individual in a coalition overlaps with that of some other member of the5same coalition. We then show that the only restriction on the stochastic social choice functionis that the sum of the weights of individuals belonging to the same coalition in the partitionjust mentioned gives the probability that some alternative which is best in the feasible set forone of these individuals is chosen. This restriction is a generalization of the restriction derivedin Pattanaik and Peleg [29] because it reduces to their's whenever everyone has unique bestfeasible alternatives.6Chapter 1A Free-Rider Problem with a Free-Riding Principal1.1 IntroductionOne of the most prominent features of existing work on moral hazard in the principal-agentframework is the separation of ownership from labour. In most previous work (e.g. Grossmanand Hart [18], Harris and Raviv [19], Hart and Holmstrom [20], Holmstrom [22], Rees [31],Ross [32], Shavell [36]) only the agent takes part in production but her action is unobservableand unverifiable because of uncertainty in the production process, and the principal is thereonly as a passive residual claimant (because she does not take part in production). So, asthe residual claimant, the problem of the principal is to design, before production begins,a payment schedule for the agent that depends only on the observable final output. Thispayment schedule is designed to induce the agent to choose a level of action which willmaximize the expected benefit of the principal (from the residual) subject to the incentiveand individual rationality constraints.7However, we often observe organizations in which there is a residual claimant who hiresthe services of other individuals in the production process. designs the output sharing rule,and also, unlike the principal in most principal-agent models, takes part in the produc-tion process. In these organizations, as in most principal-agent relationships, the residualclaimant role of the individual who has the right to design the output sharing rule usuallystems from her ownership of tangible essential inputs like technology, equipment, capital oreven the license to operate the business. Also, the participation of the residual claimant inthe production process is often because of her comparative advantage in certain input that isspecific to the production technology. Quite a sizable proportion of the so called "self ownedand operated" businesses fall under this category, because the self employed owner in suchbusinesses often tend to hire the services of other individuals as well. Such organizations arealso quite prevalent in the "small-scale and cottage industries" sector of most less developedcountries, where a single individual (or household), because of her ability to make the nec-essary investments or her ability to acquire the credits for the necessary investments or evenher ability to influence the bureaucracy (as is often required) to secure the proper businesslicense, may often start a business that employs herself and others. Also, even though mostexisting work use the landlord-tenant relationship in the agrarian economies of the less devel-oped countries as a good example of a principal-agent relationship, as Eswaran and Kotwal[14] pointed out, often the landlord not only designs the rule for sharing the crop with thetenant but also makes farm management decisions that are not observable to the tenant.So there is enough evidence to suggest the coexistence of the two kinds of principal-agent(s) relationship, namely. those in which the principal is only a residual claimant, andthose in which the principal is not only a residual claimant but also an active participant in the8production process. Then the following question comes to mind immediately. Contrary to thecommon assumption that the principal's nonparticipation (or participation) is exogenouslydetermined, is it possible that the principal is not precluded from participating in productionex ante in a sizable number of cases, but whether she does participate or not is a choicewhich she makes? For instance, as Eswaran and Kotwal [14] pointed out, this questionoften has an affirmative answer in the case of landlord-tenant relationships, because, ratherthan being an absentee landlord, the landlord can often choose to participate in farming, forexample, by making farm management decisions. Once the principal's participation is notprecluded, the condition which determines her participation decision will depend on thingslike her opportunity cost, the degree of complementarity between her action and the actionsof the agents, the opportunity costs of the agents, the relative efficiency of any agent whoseaction can substitute the principal's action. etc. Thus, using a fairly general model with somestandard assumptions, we want to characterize the condition which determines the principal'sparticipation decision.Apart from the principal's participation decision, an equally important question is, whatdo the optimal output sharing rules look like? We have the following answer to this question.If the principal chooses not to participate, there is an optimal output sharing rule in whicheach participating agent (we allow the possibility of two or more agents) has a linear paymentfunction with a slope of one. On the otherhand, if the principal's best option is to partici-pate, there is an optimal output sharing rule in which each participating agent has a linearpayment function with a slope which lies strictly between zero and one, and moreover, theresidual function of the principal is also linear with a slope which lies strictly between zeroand one. Although we have quasilinear utility functions, our result on linearity of optimal9output sharing rules depends not on this but on another important feature of our modelwhich has to do with the cause of moral hazard. As we want to concentrate on the moralhazard caused by joint production, in contrast to the single-agent case (which precludes theprincipal's participation) where moral hazard is caused by the presence of uncertainty in pro-duction, there are no uncertainties in our production process and moral hazard is a pure jointproduction phenomenon. Thus, unlike the case with uncertainty where incentive constraintsimpose conditions on the behaviour of the output sharing rule throughout the support of thedistribution of output, because of the absence of uncertainty, incentive constraints imposeonly a local condition on the behaviour of the output sharing rule around the equilibriumoutput level. This leaves sufficient degrees of freedom to choose the behaviour of the outputsharing rule elsewhere. So, in contrast to the case where there is uncertainty in production,qualitatively, our result on linearity of optimal output sharing rules will hold even if theutility functions are no longer quasilinear provided they are still concave and there are nouncertainties in production.The amount of freedom provided by the absence of uncertainty on the behaviour of optimaloutput sharing rules away from a local neighbourhood of the equilibrium output level alsohas another important implication for the case where individuals have limited liabilities.Even when individuals have limited liabilities, there is still sufficient freedom to modify thelinear optimal output sharing rules in such a way that we get piecewise linear optimal outputsharing rules that satisfy the limited liability constraints. This will not be always possible ifthere are uncertainties in the production process.We also show that, if the principal does not participate in production, then she can-not do any better even if the actions of the participating agents were made observable. So10moral hazard is completely mitigated whenever the principal acts only as a residual claimant.This is because of the fact that, if the principal does not participate in production, then, asin Holmstrom [23], her role is just like the role of the outsider who administers "budget-breaking- incentive schemes. It must be pointed out that, like the result on the linearityof optimal output sharing rules, this result on complete mitigation of moral hazard in thecase of nonparticipation by the principal does not depend on the quasilinearity of the utilityfunctions. As long as the utility functions are concave and the production process is deter-ministic, moral hazard can always be completely mitigated in the case of nonparticipation bythe principal. This is in sharp contrast to the case where there is uncertainty in productionand risk sharing.On the otherhand, if the principal does participate in production, then she cannot com-pletely mitigate the moral hazard problem, and hence, she can do better if the actions wereobservable. This is because of the fact that there is an inherent conflict between the principal'srole as the residual claimant and her incentive to shirk in the production process.Thus, the answer to another important question, "Can the principal sustain efficiency?",depends crucially on whether she has to participate in the production process in the fullinformation case (which is the hypothetical situation where all actions are observable) to getthe maximum utility. In particular. if the principal does not have to participate in productionin the full information case to get the maximum utility, then and only then can she sustainefficiency when actions are unobservable.In the next section, we describe a simple deterministic joint production process. Thereare two or more individuals. one of whom is the principal and the rest are agents, who maytake part in the joint production process. but their actions in the production process are11unobservable. This section also describes the preferences of the individuals and the SecondBest Game. The Second Best Game is a multi-stage extensive game played by the principaland all the agents to determine the set of individuals who will take part in production alongwith the output sharing rule they will follow. It is worth noting that the moral hazardproblem in the actual production process is similar to the moral hazard problem in teamsconsidered in section 2 of Holmstrom [23]. Hence, as mentioned later on, our efficiency resultcan also be derived using his results. However, unlike here, Holmstrom [23] focuses attentionon the issue of mitigating moral hazard in a joint production process and does not look atthe problem faced by a principal who can actively participate in the production process'.The third section looks at the First Best situation and describes the appropriate notionof efficiency. We derive some optimal outcomes of the Second Best Game that involve linearoutput sharing rules in the fourth section. This section also looks at the issue of mitigationof moral hazard and sustainability of efficiency. In the fifth section, we show that there areno significant changes in our results when there is limited liability. The uniqueness of thesubgame perfect equilibrium utility tuple in the Second Best Game, which does depend onthe quasilinearity of the utilities, is established in the sixth section. The seventh sectionillustrates most of our findings in a simple example. Why our results are robust to moregeneral utility functions is briefly discussed at the end of the chapter.1.2 Production and PreferencesThere are m (> 2) individuals who can participate in a joint production process. Wheneveran individual, indexed i, participates in the production process, she takes an unobservable'Although the outsider who administers the "budget-breaking" incentive scheme in Holmstrom's proposedsolution to the moral hazard problem is often interpreted as the principal. this outsider, unlike the principalin this chapter, is precluded from taking part in production.12and/or unverifiable action ai E Ai = R± . For each individual i, ci : Ai R+ is thecost function that specifies the cost she incurs from her action when she participates in theproduction process. All inputs other than the actions of the individuals are assumed to beobservable, and hence, suppressed in the specification of the model. In the production process,the actions of the individuals determine a joint monetary outcome. This production process,assumed to be deterministic, is represented by a function, f : A —4 W.* , where A = jIm 1 Ai.For each individual i, her preference relation over money-action pairs is represented bya quasilinear utility function, Ui x Ai --+ n, which is of the form Ui (pi , ai) = pi — ci(ai)for any (pi, ai) E x Ai . Because we consider the case in which it is possible for individualsto get negative payments, note that the utility function Ui is defined even for pairs withnegative amounts of money. Later on, we discuss how the results are qualitatively affected ifwe abandon the quasilinear form of the utility functions.We use the following standard notation: A_i is the Cartesian product of Al over all jnot equal to i; a = (ai, ..., am ) E A; a_i = (ai,^ai_i, ai+ ,^am ) E A_i; and a = (a_i, ai).Throughout, we assume that the production function f , the cost functions ci , and the util-ity functions Ui are common knowledge. In addition, we maintain the following assumptionsabout the functions ci and f :Al. For each i, ci is continuously differentiable, strictly increasing and strictly convex onAi; ci(0) = 0; c'i (0)= 0 and lim,„,c'i (ai)= oc, where cz denotes the derivative of ci.A2. f is continuously differentiable, strictly increasing and concave on A; f (0, ..., 0) = 0;for each i and each a_ i E A_i , limai _o fi(a-i. ai) > 0 and lima i —oo fi(a-i. ai) < Do, where fiis the partial derivative of f with respect to the action of individual i .Assumption Al is standard and needs no explanation. In assumption A2, the smooth-13ness, monotonicity and curvature properties of the production function, and the requirementthat the output be zero when everyone takes zero action are standard. Also, the limitingbehaviour of the marginal product of any individual's action when it approaches infinity isstandard. However, our stipulation in assumption A2 about the limiting behaviour of themarginal product of an individual's action when it approaches zero is not so standard, be-cause it says that the marginal product of an individual's action in a neighbourhood of zerois positive even when every other individual takes zero action. This means that, no matterwhat the actions of the other individuals are, the total output is strictly increasing in theaction of each individual. Hence, nobody is essential for production, as total output is equalto zero only when every individual takes zero action. This particular property of the produc-tion function along with the quasilinearity of the utilities are exploited in the derivation ofthe results on uniqueness of equilibrium.We treat individual m as the residual claimant in the following sense - (i) like the principalin the standard agency models, individual in is the only one who can design and proposeoutput sharing rules; and (ii) whether she takes part in production or not, individual malways keeps that part of the output left after making payments to the other participatingindividuals2 . Unlike individual in, each of the first m - 1 individuals receives a paymentonly if she takes part in production. Thus, throughout the remainder of this chapter, we callindividual in - the principal, and the first in - 1 individuals -- the agents. However, it mustbe noted that, unlike the principal in standard agency models, individual in can choose toparticipate in the production process.2 1t is implicitly assumed that this residual claimant role of individual rn is a consequence of reasonsexogenous to the specification of the model. As mentioned in the introduction, one such good reason may beindividual In owning certain tangible inputs like technology, equipment, capital, etc.14If the actions taken by the agents in the production process are unobservable and/orunverifiable, then the principal cannot make the payment to any agent depend on that agent'saction. Hence. the payment to an agent for participating in the production process can dependonly on the observable and/or verifiable total output of the production process. Thus, ingeneral, a payment function for an agent is a real valued function defined on 94, the set ofall possible output levels. However, in this chapter we shall require the payment functions tosatisfy a regularity condition. This condition essentially says that the curve of the paymentfunction of each agent has only a finite number of jumps and kinks on any bounded range ofoutput levels. Thus, the payment function of each agent must be drawn from the set(i) s is piecewise continuous on every (qL ,q 11 ) C R+ ;:^---> ll^(ii) if s is continuous on (qL , qH ) C 94, then it ispiecewise continuously differentiable on (qL qH).Although a technical restriction, requiring the payment functions of the agents be drawnfrom the class S seems quite reasonable, because most of the observed payment functions injoint production processes belong to this class of payment functions.Let .M be the set of all subsets of {1...., m}. So each member of .A/1 is a subset of theset of m individuals. We call each member of a team, and denote them by T, T , T, etc.Given a team T and an individual i, we denote the set of all individuals in T other than i byT_i; i.e. T_i = T — {i}4 . Thus, T = T_ i, if and only if i T.Given T E M, let AT be the Cartesian product of Ai over all i in T, and aT = (ai)ieTE AT . Also, given any T E M. let f : AT —+ R+ be the restriction of f to AT in thefollowing sense — for each aT E AT . fIT(aT) = f (a`), where a' = (a'1 , a'n1 ) E A is such that3 Recall that piecewise continuity on an interval means continuity everywhere on it except for a finitenumber of discontinuities of the first kind.4 Given any two sets D and E, we use the convention of letting D E = {d E D : d E}.15= ai if i E T and di = 0 if i 0 T. For each T E M and each i E T, we use filT to denotethe partial derivative of f IT with respect to the action of individual i.As production can take place with the participation of any subset of individuals, it isclear that, when only the members of some T E M take part in production, the relevantproduction process is f IT.Given any T E ,A4, if only the members of T take part in production, then an outputsharing rule for T is a tuple of payment functions, (si)iET_ where s, E S is the paymentfunction of agent i E 7„. Obviously, for each output level q E whether the principalbelongs to T or not, she gets the residual q — si(q)•Each individual has an outside option which she can exercise instead of taking part inthe production process. The utility of individual i from her outside option is equal to u > 0.So, when agent i exercises her outside option, she automatically gets zero payment from theprincipal and her utility is equal to ui. However, as the principal is the residual claimant,she still gets her residual in addition to u rn even when she exercises her outside option.Because the action taken by any individual in the production process is neither observablenor verifiable, once the team which will take part in production along with the output sharingrule become common knowledge, the members of the team actually play a noncooperativegame in the production process conditional on the common knowledge output sharing rule.Suppose it becomes common knowledge that a team T will take part in production and theoutput sharing rule will be (si)JET_„,• Then the strategies and payoffs of the players in theensuing noncooperative game of production (NGP), which we denote by {T, (si)j E T_,, }, areas follows: (i) a strategy of player j E T is an action ai E Aj; and (ii) when the actions takenby the players in T are aT E AT , the payoff of player j E T is si(f IT(aT)) — cj( ) if j is an16agent (i.e. if j^2) and fIT(aT) —^(f IT(aT)) — cm (^if j is the principal (i.e. ifj^7-n).Given any NGP {T, (si)T_,„, such that 7n T, aT E AT is a Nash equilibrium of thisNGP if and only ifai E argmax,, EA, [si(f IT(aT_ „ a i )) — ci (di)] V i E T.Similarly, given any NGP {T, ( )T_ Ta } such that m E T, aT E AT is a Nash equilibrium ofthis NGP if and only ifai E argmaxa: eAt [si(f^ci(a'i)] V i E^andan, E argmaxa:n EAm. [f^, aim ) — E^— cm (arn )] •We denote the set of all Nash equilibria of each NGP {T, (si^} by NE({T, (s i )For each team in M. it is clear that there are lots of possible output sharing rules. Thismeans that, as production can take place with the participation of any one team in M, thereare lots of possible team and output sharing rule combinations according to which productioncan take place. So we need a procedure that determines a single team and output sharingrule combination according to which actual production takes place.Suppose the principal deals secretly with different subgroups of agents and ultimatelyarrives at a single team T and a corresponding output sharing rule (S i ) ET_ m according towhich actual production takes place. Then it is very likely that the output sharing rule(si) ET_„, is not common knowledge among the agents in T. So the strategic behaviour ofeach agent in T in the production process depends on her belief about the output sharing rule,her beliefs about the beliefs of the others in T and so on. Clearly, the beliefs of each agent in Tdepend on all the information that she has, for example, her own payment function and may17be that of some of the other agents in T. However, given all the information that is availableto an agent, the manner in which she uses them to form her beliefs is quite complicated tomodel and well beyond our scope. Therefore, we assume that the principal does not dealsecretly with any subgroup of agents and whatever she proposes becomes common knowledgeimmediately.As we assume that no one can be forced to participate in production, even though theprincipal, as the residual claimant, is the only individual who can propose any team andoutput sharing rule combination, an agreement must be reached on a single team and out-put sharing rule combination according to which actual production takes place. However,handling the complex strategic issues involved when individuals are allowed to collude withone another in trying to reach an agreement on a single team and output sharing rule com-bination are again beyond our scope. Moreover, it is seldom easy to justify the credibility ofcommitment of any member of a coalition to the coalition. Therefore, we assume that theindividuals behave noncooperatively when trying to reach an agreement on a single team andoutput sharing rule combination.Thus, we use a very simple multi-stage procedure to determine a single team and out-put sharing rule combination according to which production takes place. This multi-stageprocedure, which we call the Second Best Game (SBG), is described as follows:Stage I: In the first stage. the principal announces a NGP which becomes commonknowledge immediately.Stage II: In the second stage, each agent who is a player of the NGP announced bythe principal must announce whether she agrees to play this NGP or not. These announce-ments by the agents are made sequentially. so that, the announcement of an agent becomes18common knowledge before the announcement of any subsequent agent. If any agent who is aplayer of the NGP announced by the principal announces a disagreement, then the procedureterminates at this point and everyone exercise their respective outside options. Of course,those individuals who are not players of the NGP announced by the principal automaticallyexercise their outside options5 .Stage III: This stage is reached only if every agent who is a player of the NGP an-nounced by the principal announced an agreement in the second stage. Once the third stageis reached, the NGP announced by the principal is played in the production process.Obviously, if the third stage is not reached, then the payoff of each individual i in theSBG is her outside option utility, ui. However, if the third stage is reached, then the payoffsin the SBG are given as follows: (i) each agent who is not a player of the NGP played inthe third stage gets her outside option utility; (ii) each agent who is a player of the NGPplayed in the third stage gets her payoff from this NGP; (iii) if the principal is a player ofthe NGP played in the third stage, then she gets her payoff from this NGP; and (iv) if theprincipal is not a player of the NGP played in the third stage, then she gets u rn (her outsideoption utility) plus the output left after distributing the payments to the players of the NGPplayed in the third stage.Given T E .M and an output sharing rule (si)JET„ such that si E S for each i E T_,„there is no gaurantee that the NGP {T. (si)i E T_„,} has a Nash equilibrium. However, as wewant to focus only on subgame perfect equilibria of the SBG, we cannot allow the principalto propose NGPs that do not have any Nash equilibrium. Hence, we impose the restrictionthat the NGP announced by the principal in the first stage of the SBG be drawn from the'The assumption of sequential move in the second stage of the SBG is required only for the uniqueness ofthe subgame perfect equilibrium payoff vector discussed in section 1.6.19set{g= {T,(si)JET_,, }(i) T E M; (ii) si ESViE T_ m ; and(iii) N ENT, (si)jET_„,}) 0 01.3 First BestTo understand the efficiency properties of the model it is necessary to know the meaningof efficiency in the present context. So, as in most standard moral hazard models, we lookat the First Best (FB) situation in order to find an appropriate notion of efficiency for ourmodel.The FB situation is the hypothetical situation in which the action of each individual inthe production process is observable, and hence, the principal pays each agent who takes partin production according to her action. So, in the FB situation, the principal can dictate theaction of each agent in the following sense — if the principal wants an agent to participatein production and take a particular level of action, then she can solicit the desired actionvoluntarily from the agent with a sufficient payment for that action and zero payment for anyother action. Therefore, when only the members of a team T E M take part in production inthe FB situation, the principal chooses a tuple of payment-action pairs for all the agents in Tand an action for herself if she belongs to T to maximize her utility subject to the conditionthat each agent in T gets at least as much utility as from her outside option. We denote thismaximum utility of the principal by W (T).Suppose T E .M is the set of individuals who take part in production in the FB situation.Then in T means that the principal exercises her outside option and also gets the residual.So, if in E' T. then W(T) is given by20MaXTEM W( T) I in 0 71 otherwise.umum__{ if W(T) < u rn V TEM such that M, T) p i — ci(ai) > ui V i E T;W(T) = u, + max(aT,PT)f (aT Pd (ii) aT E AT ; andJET(iii) PT = (Pi)iET EOn the otherhand, the principal can belong to T only if she does not exercise her outsideoption. Hence, if T is such that rn E T, then W(T) is defined by(i) pi — ci(ai) > ui V EW(T) = max(aT,p7-__ rnfIT(aT) — Epi — cm(arot (ii) aT E AT ; and(iii) PT^= (pi)JET_,, E RIT- rnIn the definition of W(T) in either case, the constraints, pi > ui V i E T_„2 , ensure thateach participating agent is no worse off than exercising her outside option. These constraintsare obviously necessary for the participating agents to be willing participants. Also, we mustpoint out in passing that W(T) is well defined for any T E M, because, using assumptionsAl and A2, it can be easily shown that the maximization problem in the definition of W(T)in either case has a solution.Suppose the principal chooses to exercise her outside option in the FB situation. Clearly,if W(T) < um for every T E M such that in T, then the highest utility she can get is u m .On the other hand, if there exists T E M such that in T and W(T) > um , then the highestutility she can get is the maximum of W(T) over all T E M such that in 0 T. Thus, if theprincipal chooses to exercise her outside option in the FB situation, then the highest utilityshe can get, which we denote by urn _, is given byNow, suppose the principal chooses to participate in the production process instead of6 Given any set D, we follow the usual convention of denoting the number of elements in D by IDS.21exercising her outside option in the FB situation. Then the highest utility she can get, whichwe denote by urn+ , is given byuF + = max {W (T) 771 Ern TEMTherefore, the highest utility the principal can get in the FB situation, which we denoteby um, is given by21 ?Fn. = max{u rn _ , urn+ }Clearly, the principal cannot gain anything from the joint production process if urn =So, throughout this chapter we assume that urn >When the action taken by each individual in the production process is unobservable andunverifiable, the principal and the m — 1 agents play the SBG. In the SBG the equilibriumconcept we use is subgame perfect equilibrium (SPE). Therefore, if the SBG has a SPE atwhich the utility of the principal is at least as large as umF, then we shall say that the principalcan sustain efficiency.It is straightforward to see that the principal maximizes her utility in the FB situation onlywhen every participating individual takes positive action and the utility of each participatingagent is pushed down to her outside option utility. Then, because of the quasilinear utilityfunctions and assumptions Al and A2, when the principal maximizes her utility in the FBsituation, the marginal product equals the marginal cost for every participating individual.Thus, we have the following two lemmas.Lemma 1.1 : Suppose assumptions Al and A2 hold. If T E M and (pT,aT) E^x ATare such that m T, pi — ci(ai) > ui V i E T and u m + f iT(aT) EiETPi = W(T). then:(i) Pi — (ai) = ui V i E T; (ii) ai >0 V i E T; and (iii) f ilT(ar) — c(ai) =0 Vi E T.22Proof. See Appendix A.Lemma 1.2 : Suppose assumptions Al and A2 hold. If T E M and (pT_„,, aT) E^xAT are such that m E T, pi — ci(ai) > ni V i E^and f IT(aT) EiET n, Pi — cm (a,n ) =W(T), then: (i) pi—^= ui V i E T--01; (ii) ai >OViE T; and (iii) filT(aT)—^i) = 0V i E T.Proof. See Appendix A.1.4 Optimal OutcomesAn outcome of the SBG consists of a NGP which is played in the third stage and the tupleof actions taken by the players in this NGP if the third stage is reached. So the set of allpossible outcomes of the SBG conditional on reaching the third stage is= {({T, (8. )i€T_„, aT)1^(.9i)jET„.„} E g and aT E AT } .Using the standard notion of SPE, it is easy to check that, if ({T, (si)j eT__„,}, aT) E C2 isa SPE outcome of the SBG, then aT is a Nash equilibrium of {T,(si)j ET_ m }. Also, as eachagent i has an outside option which gives ui utility, it is quite obvious that, at any SPEoutcome ({T,( 8 i)iET,_„, aT) E St, the utility of each agent i E T must be at least as large asui. Thus, the set of all outcomes in C2 that can arise as SPE outcomes of the SBG must bea subset of the following set:{({T, (si)iET_ m }, aT) E C21(i) aT E N E({T,(si)iET,}); and(ii) si(TIT(aT)) -- ci(ai)^ui V i EGiven any outcome ({T.(si)iET.-„,.} , aT) E 12, let 7(({T. (si)iET__,,, aT)) denote the utilityof the principal at this outcome; i.e.23fIT(aT) — EiET_, si(fIT(aT))— e rn (am) if 711 E T74({T.(si)iET_,.},aT))um + flT(aT) — EiET si(PT(aT))^if m T.Among all the outcomes in S2, we shall refer to any outcome at which the utility of theprincipal is the highest as an Optimal Outcome (00). Formally, 00s are defined as follows:Definition 1.1: ({T,(si)JET.„}, aT) E C2 is an 00 if and on ly if: (a) ({T, (si)iET_.}, aT) ES2; and (b) ((IT, (s .)JET_J,aT)) 7(({T, (sz)iET_„,},aT)) V (IT, E O. Wedenote the set of all 00s by Q*.Clearly, it is meaningless to analyze the SBG if it does not have a SPE outcome at whichthe utility of the principal is higher than um . So we make the following assumption:A3. There exists ({T,(si)ieT_^aT) E O such that 7MT, (sdiET_„,}, aT)) > um .Intuitively, it makes sense for the principal to consider a NGP only if, at the very least, sheexpects a positive action from every player. So it is highly improbable to expect the principalto choose a NGP only to realize a Nash equilibrium at which some player takes zero action.Also, from a practical point of view, we seldom notice joint production processes in whichsome participants remain inactive in production. Thus, we want to avoid the possibility ofhaving an 00 of the SBG at which some participating individual remains inactive. So weshall explicitly assume that the following is true:A4. There is a ({T, (gi) iep_ m }, aT ) EO such that Cti>OViEP, and 74({T,^m }, at ))> ir(({T, (si). E T_„, }, aT)) V ({T, (si)JET_,,},aT) E O such that ai = 0 for some j E T.Because of the deterministic production process. when only one individual takes part inthe production process, her action can be exactly verified from the observed total output,and hence, there is no moral hazard problem. This means that, if there is an 00 with aNGP which has only one player (whether an agent or the principal), then we need not go any24further. Therefore, for the subsequent analysis to be meaningful, it is necessary to make thefollowing assumption:A5. If ({T, (si)J E T_,,, }, aT) E^, then 171> 2.Let ({T, ( i^aT) E Q. Suppose every individual in T believes that the action tupleag, will result whenever the NGP {T, (si)j E T_,,,} is played. Then, as the utility of each agentin^at the outcome ({T, ( si)701.„ aT) is at least as large as her outside option utility,using a straightforward backward induction logic, we know that any agent in^can do nobetter than agree to the NGP {T, (si)j ET_,,,, } whenever it is announced by the principal. Sothe principal can announce the NGP IT, (sdiET_,„1 and get 71 - ( ({T, (si)j ET_„., }, aT)) amountof utility. But, as any SPE outcome in et must belong to el, we also know that the principal'sutility in any SPE outcome must be no larger than her utility in any 00. Therefore, it isobvious that ({T, (\si)?eT„, }, aT) must be a SPE outcome. Thus, the following claim holds.Claim 1.1 : Every outcome in Q* can be supported as a SPE outcome of the SBG.Because of Claim 1.1, we can now focus our attention on the set Q*. In particular, weshall show that there are outcomes in et* that involve very simple linear output sharing rules.At this stage we do not know anything about the principal's participation or nonpartici-pation in the production process at any of the 00s in Cr. So let us begin by looking for thebest outcomes (from the principal's perspective) in et_ n, C e2, which is the set of all outcomesin et where the principal does not participate in production; i.e.O—m^{({T, (si)iET_„,},aT) E f21 m T} .Let TF be a team in M such that in TF , and W(TF) maxrEm{W(T)1 m T}.Also, let (pC , F ) E RI TF I x ATF be payment and action tuples such that pr — ci(ar)25^> u, V i E T. and^+ fiTF(aCF) E2ETFpr^= W(TF ). So the team TF along withthe payment and action tuples (pTF F . a TF ) give the highest utility to the principal if she doesnot participate in production in the FB situation. Clearly, W(TF ) > 7(({T, (8/)?ET }. aT))V ({T, (sz)zETI,aT) E f2_„. Then, according to assumptions A3 and A5, W(TF ) > umand ITF I > 2 are necessary for any outcome in (2_,-,„ to be an 00. Thus, unless otherwisementioned, it must be understood that we are only looking at the case in which W(TF ) >and ITF I > 2.Because of Lemma 1.1, we know that the following are true:pr — ci (ar) =^V E T;^ (1.1)^urn + fiTF (aCF )— E pr = w(TF) ; and (1.2)i ET1^filT(aCF) — c(ar) = 0 V i E T.^ (1.3)So we seek to construct an output sharing rule which will induce the agents in TF to takethe actions 4,F and also pay pr to each agent i E TF at the output level fITF (4F ). Now,(1.3) says that any output sharing rule which induces the actions aTFF and is smooth in aneighbourhood of the output level fITF(aTFF ) must only have payment functions that haveunit slopes around a neighbourhood of the output level fITF(aFrE ). However, because of thedeterministic nature of the production function f, (1.3) does not say anything about how theoutput sharing rule should behave away from a neighbourhood of the output level fITF (aTFF ).This gives sufficient freedom that allows us to construct a desired output sharing rule whichis linear.Suppose the agents in TF play a NGP in which the payment to each agent i E TF is equalto the total output plus the constant ui + ci(aF) — fiTF(aTFF ) for every level of output. Then26the quasilinear utility functions, the strict convexity of the cost functions and the concavityof the production function imply that a tuple of actions for the agents in TF is a Nashequilibrium if the marginal product (which is the same as the marginal benefit) is equal tothe marginal cost for every agent in TF . However, we already know from (1.3) that themarginal product is equal to the marginal cost for each agent in TF at aTFF . Therefore, aTFFis a Nash equilibrium. Also, it is easy to check that the utilities of each agent i E TF and theprincipal at aTF F are ui and W(TF ), respectively. Furthermore, because of the quasilinearutility functions and the monotonicity, curvature and limiting marginal properties of the costfunctions and the production function, aTF F is in fact the unique Nash equilibrium.For each i E TF , let kr be the constant such thatkr^ui+ ei (ar)_ fiTF (4F) V i E TF .^ (1.4)Now, for each agent i E TF , define the payment function sr as follows:sr (q)^kr + q VqEW+.^ (1.5)Then, more formally, we have the following proposition.Proposition 1.1 : If assumptions Al and A2 hold, then: (i) ({TF , (sr) iETr} , 4,F ) E-m ; (ii) sr ( fITF (aS: F ))—ci(ar ) ui V i E TF ; (iii) um+ fITF — sr ( f iTr(4 F ))W(TF); and (iv) N E({TF , (sr) jETF}) = faC F 1 .Proof. See Appendix A.The intuition behind Proposition 1.1 is as follows. As the principal is the residualclaimant, when the actions are not observable and only the individuals in TF take partin production, the role of the principal here and the role of the outsider who administers a27budget-breaking incentive scheme in Holmstrom [23] are the same. So, whenever the totaloutput deviates from fiTF (aTF F ), although the principal does not know the agent(s) whoseaction(s) caused this deviation, she can find the entire team of agents TF at fault. Hence, theprincipal can punish everyone in TF for any deviation in the total output from f ITF (4 .p).In particular, the principal can make each agent i E TF fully responsible for any devia-tion in the total output from f ITF (aC,F ) by paying her the total output plus the constantui + c,(ar) — fiTF F ) for every level of output.Next, let us look at the other subset of the set O, denoted by 52 +7,, which contains allthose outcomes from (2 where the principal participates in production; i.e.{({T, aT) E E .When the principal is the only player, there is only one NGP, namely, the one in whichthe principal keeps the entire output for herself. Although it is obvious that the principal canget W({m}) in this NGP, because of assumption A5, there cannot be any 00 which involvesthis NGP. So we only need to pay attention to those outcomes in 52 + ,,,, that have at leastone agent participating in production along with the principal. Also, as we are interested in00s, assumption A4 allows us to ignore those outcomes in 52 + , at which some participantin the production process takes zero action.Thus, among all the outcomes in 52.0„ that have two or more individuals participating inproduction with every one of them taking a positive action. we are interested only on thosethat are best from the principal's perspective. Therefore. as an intermediate step, for eachT E M such that in E T and IT! > 2, we need to look at the following maximization problem:28max^{,f IT( aT) —^Si(f (aT)) cm(arn)]^ (PT)oT 4 8 i) JET__,7subject to:ai > 0 V i E T; and^ (C1)({T,(82)zeT„,},aT) E O+m•^ (C2)As discussed above, constraint (C1) requires a positive action for every individual in T.Constraint (C2) obviously follows from the fact that SPE outcomes have to be in the set O.Clearly, constraint (C2) of problem (PT) involves maximization problems of the playersin T. So we use a standard method, commonly known as the first order approach, to solveproblem (PT). As the first order approach uses only the necessary conditions of the opti-mization problems involved in the constraint, sometimes the solution(s) obtained by usingthis approach may not be solution(s) of the original problem. However, we need not worryabout such a possibility in the present case, because the solutions we derive by using the firstorder approach are indeed solutions of problem (PT).Appendix A proves a technical lemma that enables us to use the first order approach.Given T E A4 such that in E T and 17'1 > 2, if a NGP and an action tuple corresponding toT satisfy (C1) and (C2), then this lemma asserts that the curve of the payment function ofeach agent in T is smooth at the output level corresponding to the given action tuple.Lemma 1.3 : Suppose assumptions Al and A2 are satisfied, and T E M is such thatE T and DTI > 2. If ({T,(MieT__„,},^satisfies (Cl) and (C2), then si is differentiableat fIT(aT) for each i EProof See Appendix A.29Suppose ({T, (8i)iot m },ay') E S2 -1- „t is such that IT! > 2 and ai > 0 V i E T. Becauseof the quasilinear utility functions, it is clear that the marginal benefit of each individualin T is equal to her marginal cost at aT. Also, because of Lemma 1.3, we know that, foreach agent i E T_,,, her marginal benefit at aT can be written as the product of the slopeof her payment function si at f IT(aT) and her marginal product at aT. Hence, for eachagent i E T_„2 , the slope of si at fIT(aT) must be equal to the ratio of her marginal costand marginal product at aT. As the principal's residual for any level of output q E 94 isq — E ieT_ m s,(q), Lemma 1.3 also implies that the marginal benefit of the principal at aT isequal to her marginal product at aT times one minus the sum of the slopes of the paymentfunctions of all the agents in T at fIT(aT). Therefore, one minus the sum of the slopes ofthe payment functions of all the agents in T at fIT(aT) must be equal to the ratio of theprincipal's marginal cost and marginal product at aT. But we already know that the slopeof the payment function of agent i E T, at f IT(aT) is equal to the ratio of her marginalcost and marginal product at aT. So the marginal cost to marginal product ratios of all theindividuals in T at aT must add up to one. More formally, we have the following lemma.Lemma 1.4 : Suppose assumptions Al and A2 are satisfied, and T E M such that m E Tand ITS > 2. if ({T,(si)jET_„,},aT) satisfies (C1) and (C2), then EiETVi(ai)1 filT(aT)] = 1.Proof See Appendix A.Given T E M such that m E T and ITS > 2, suppose we replace (C1) and (C2) by themarginal condition in Lemma 1.4. Also, for each i E T--m, suppose we replace si(fIT(aT))by ui ci(ai) in the objective function of problem (PT). Then we get the following newmaximization problem:30max [f (aT) — EiETci(ai) EiET-aT EATsubject to:EETV,(a?)1.1.11T(aT)] = 1 .^ (C3)Note that the actions of the individuals in T are no longer required to be positive inproblem (4). Also, unlike problem (PT), the only choice variables in problem (4) are theactions of the individuals in T. The intuitive logic behind the transformation of problem (PT),which involves payment functions, into problem (4), which does not involve any paymentfunction, is as follows. Because of the deterministic nature of the production function f ,constraints (Cl) and (C2) only tell how the output sharing rule (8,), E7-_,,, must behave arounda neighbourhood of the output level f IT(aT) and not elsewhere. Then, as the payments tothe agents and the principal's residual has to add up to the total output, this local conditionimplied by (C1) and (C2) translates into (C3) and eliminates the payment functions.Lemma 1.5 : Suppose assumptions Al and A2 hold, and T E M such that 772 E T and171 > 2. Then problem (4) has a solution.Proof. See Appendix A.Given any T E M such that 772 E T and ITS > 2, the value of the objective function ofproblem (4) at a solution is denoted by V(T). So, if aT is a solution of problem (4), thenf 1T(aT) — E iET ci(ai) — E iET_ m u i = V(T)•Remember that, for any combination of (si)iE T„ and aT which is feasible for problem(PT), the individual rationality conditions of the agents in T, si( f IT( T)) — ci(a i ) > 14 V i ET_,„ are included in constraint (C2). So the value of the objective function of problem (PT)at (aT.(sniET„,) cannot be greater than the value of the objective function of problem (4)31at ay.. But, because of Lemma 1.4, the action tuple at any feasible point of problem (PT) isalso feasible for problem (/=',). Therefore, we cannot find a feasible point of problem (PT) atwhich the value of its objective function is greater than the value of the objective function ofproblem (/=',) at a solution. Thus we have the following lemmaLemma 1.6 : Suppose assumptions Al and A2 are satisfied, and T E M such that m ET and IT1 > 2. If (aT, (si)JET_,„) is feasible for problem (PT), then V(T) > f IT (aT) —EJET_,, si(fIT(aT)) — cm (am)•Proof See Appendix A.Let T* maximize V(T) over all T E M such that in E T and IT > 2; i.e.T* E argmaxTEm {V (T) m E T and 171 > 2} .The existence of T* follows from Lemma 1.5 and the finiteness of the number of teams thathave the principal and at least one agent. Also, let 4. E AT* be a solution of problem (4.).Then we haveE iET .[ci(a7)/fi lr(d7;.)] = 1; and (1.6)fir(4.) — EJET.ci(ai) EJET.,n ui = V(T*). (1.7)Thus, we want an output sharing rule which will induce the action tuple 4. and pay ui+ ci(a7) to each agent i E T*,,, at the output level f1T*(4*). However, as mentioned above,because of the absence of uncertainty in the production process, we know that we have somefreedom in choosing the behaviour of the desired output sharing rule away from the outputlevel f1T*(aT**). Below, we show that this freedom is indeed enough for us to construct alinear output sharing rule.32For each i E T* n„ let the two constants 77 and k7 be as follows:-Y7 =^(a7) /LIT* (4. ); and^ (1.8)k:` = ui^ci(a7) —^f^(4.). (1.9)Then, for each i E T* in , let s be the linear payment function whose slope is -y7 and interceptis k7; i.e.87(q) =^± -y7 q V qE 94.^ (1.10)Clearly, for each i E T*^the slope of 87, 77, is nonnegative and equal to zero only if0, which we have not yet ruled out. Also, the intercepts of the payment functions in(s si`)iET* ra 7 ( k7)iET*, are set in such a way that, if (87)t ET. m is the output sharing rule and* is the action tuple taken by the individuals in T*, then the utilities of agent i E T*, andthe principal are ui and V (T*), respectively. So Lemma 1.6 implies that (4*, (87)t ET* m ) isindeed a solution of problem (PT*) if feasible.Suppose agent i E T* m is paid according to s7 and the actions of the other individualsin T* are fixed at a*T. . Then, because of her quasilinear utility function, the utility of agenti as a function only of her own action at E Ai can be separated into the benefit function,+y7 f , at), and the cost function, ci(ai). We know that the cost function is strictlyincreasing. However, the benefit function is just the constant k7 if 77 is equal to zero, andstrictly increasing if 77 is positive. So it is obvious that, if 71 = 0, which can happen onlyif a 0. then the best action for agent i is a 0). On the otherhand, if a7 > 0, then 77is positive and the benefit function is strictly increasing. but there is a trade-off between theincrease in the benefit and the increase in the cost as the action of agent i increases. Then itmakes sense for agent i to choose a7 if a7 > 0. because her marginal benefit is equal to her33marginal cost at a7.When the agents in T* are paid according to the output sharing rule (s7)i ET. m , theresidual function of the principal, —^+ (1 -E iET . m -)1')q, is also linear in the outputq E 94. Because of (1.6) and (1.8), the slope of the residual function, (1 —^-y7), isequal to c'n,(a7,,)/fn, IT" (^). Then, using a similar intuition as in the case of the agents, wecan say that a7,., is the best action for the principal if the output sharing rule is (s7)iET. ,,, andthe actions of the agents in T* are fixed at 4.. Therefore, we have our next proposition.mProposition 1.2 : If assumptions A1-A5 are satisfied, and V (T*) > W(TF ), then: (i)(4., (s7)i ET. m ) is a solution of problem (PT*); (ii) s7 (f 1T*(4.)) — ci(a7) = n• V i E(iii) fIT*(a'7%) —^( f IT* (4.)) — cm (a,* ) = V (T*) ; and (iv) N E ({T* =Proof See Appendix A.The condition, V(T*) > W(TF ), plays a crucial role in Proposition 1.2. Whenever itholds, because of assumptions A4 and A5, the principal's utility at an 00 cannot exceedV (T*). So, once we establish ({T*. (87) ET*„, }, 4*) E O and 74({T*, ( 87)iET*,,}, a )) =V (T*), ai > 0 for every i E T* follows from assumption A4.Also, note that 4,, is the unique Nash equilibrium of {T*. (87)j ET* m } according to (iv)in Proposition 1.2. This result follows from the quasilinearity of the utility functions, theconcavity of the production function and the strict convexity of the cost functions.The most obvious but important message of Proposition 1.2 is that, if it is better for theprincipal to participate in production. then she need not look for any output sharing rulethat is more sophisticated than those in the class of simple linear output sharing rules.34In contrast to the output sharing rule in Proposition 1.1, there is pure sharing in theoutput sharing rule in Proposition 1.2 in the sense that — every participating individual getsa constant (which may be negative) plus a positive proportion of the total output. This canbe roughly interpreted as follows. If the principal is better off participating in production,then the principal can only get worse off with an output sharing rule which punishes only aparticular proper subset of the set of participating individuals for every deviation in the totaloutput from the optimal output.If the principal is better off participating in production, then, because of (iii) of Lemma1.2, the only way she can get the FB utility is if the actions taken in the production process aresuch that the marginal product is equal to the marginal cost for each participating individual.But Proposition 1.2 says that this cannot happen, because the slopes of the payment functionsand the residual function of the principal in the output sharing rule in Proposition 1.2 areall less than one. Thus, another important implication of Proposition 1.2 is that, if it isbetter for the principal to participate in production, then she cannot completely mitigatethe moral hazard problem; i.e. V(T*) < W(T*). This highlights the presence of an inherentconflict between two things, namely, the principal's role as a residual claimant and her roleas a free-rider whenever she participates in production.(iv) of Proposition 1.2 also has an important implication. As we shall demonstrate later,this result along with the quasilinearity of the utility functions can be used to show theuniqueness of the SPE utility tuple.Now, what we originally set out to show, namely, there is some 00 in C2* in which theoutput sharing rule is linear. is a rather obvious corollary of Propositions 1.1 and 1.2.35Corollary 1.1 : Suppose assumptions Al-A5 are satisfied. If W(TF) > V(T*), then(ITF,(sni„0 ,4F) E S2 . If V(P) > W(TF ), then (IT*, (sniET_%, 1. E Q.From our derivations up to this point, we can naturally draw the following conclusionsabout the principal's participation decision: (i) if W(TF ) > V(T*), then the principal willchoose not to participate in production; (ii) if V(T*)>W(TF), then the principal will chooseto participate in production; and (iii) if W(TF) = V(T*), then the principal may or may notchoose to participate in production.Let us now look at the principal's ability to sustain efficiency. Obviously, the only waythe principal can sustain efficiency is if her utility at every 00 is equal to urn . This meansthat the principal can sustain efficiency if W(TF) = u n2F , because we know that in this casethe principal's utility at any 00 is equal to W(TF). So the question remains, what if 24> W(TF )? If urn > W(TF ), then it is clear that the only way the principal can sustainefficiency is if urn V(T*). We know that 4.„ is at least as large as W(T*). But we alsoargued above that W(T*) is greater than V(T*). Therefore, urn > V(T*) always holds. Sothe principal cannot sustain efficiency if urn > W(TF ). Thus, we can claim the following.Claim 1.2 : Suppose assumptions A 1 -A5 hold. Then the principal can sustain efficiencyif and only if ur,F, = w (TF ).As the principal does not belong to TF , the condition uniF W(TF ) means that the principaldoes not participate in production in the FB situation to obtain the maximum utility. SoClaim 1.2 can also be put in a slightly different way as follows. To sustain efficiency it isnecessary and sufficient that the principal obtain the maximum utility in the FB situationwithout her participation in production.36As the principal plays the role of a residual claimant, if she takes part in production,we get a joint production process to which Theorem 1 of Holmstrom [23] is applicable. Onthe otherhand, if the principal does not take part in production, she is just like the outsiderin Holmstrom's solution to the moral hazard problem, who administers "budget-breaking"incentive schemes. Therefore, Claim 1.2 can also be viewed as an implication of the resultsin Holmstrom [23].The intuition behind Claim 1.2 is as follows. When the actions are not observable, everyindividual who takes part in production (including the principal) has an incentive to freeride in the production process. So, when only the members of some team T E M take partin production, to get W(T), the FB situation utility, the principal must design an outputsharing rule which has sufficient punishments for everyone in T for any deviation of thetotal output from the FB situation output level corresponding to W(T). But, whenever theprincipal punishes every agent in T, as the residual claimant she can only reward herself,which means that the principal cannot punish everyone in T if she herself is a member of T.So, whenever the principal takes part in production along with a group of agent(s), there isbound to be an inherent conflict between her residual claimant role and her incentive to freeride in the production process. Therefore, when the actions are not observable and only themembers of some team T take part in production, the principal can get W(T) only if shedoes not belong to T.1.5 Limited LiabilitySo far we have allowed output sharing rules that can award sufficiently large negative pay-ments to some agents or the principal. However, such output sharing rules may no longer37be feasible if individuals have limited liabilities. Thus, in this section, we look at the casein which individuals do have limited liabilities. In particular, we impose an extreme formof limited liability constraint, namely, no one, including the principal, can commit to anyamount of negative payment. So the NGP announced by the principal in the first stage ofthe SBG must be drawn from the following set:{T, } E g i{ (i) si(q) ..> OVqE 94 and each i E T_ ni ; andg+ .(in q — EiET_,,, si(q) >0Vq . ER,4_The question we ask is, is there any 00 ({T, (si)iET_,,I, aT) E C2* such that {T, (si)iET_,}belongs to g+? The answer is yes. In fact, we modify the linear output sharing rule inthe appropriate 00 derived in the previous section in such a way that the limited liabilityconstraint is met, and the modified output sharing rule along with the original action tupleremains an 00. This modification is carried out in such a way that the payment functionsof the agents (and the residual function of the principal if V(T") > W (TF )) are continuous,piecewise linear and nondecreasing.The reason that allows us to perform our modifications is quite obvious. Suppose W(TF)> V(T*) (V(T*) > W(TF)). Then, as long as we keep the output sharing rule (sn iETF((s7)j ET..) intact on an appropriate range of output around fITF(4,) (fIT*(4.)), theabsence of uncertainty in the production process provides enough freedom that allows us tochange (sn iET , ((s7 )J E T.,,,) quite arbitrarily elsewhere such that the action tuple aTF F (ai,.)is still induced.Let us first consider the case W(TF ) > V(T"). Then it can be easily checked that, foreach i E TF , kr < 0 and the payment function sr awards negative payments to agent ionly at output levels below _kr. Also, the principal's residual, q — E iETF sr (q), becomes38negative only beyond a certain output level greater than f ITF (aTFF )• So, for each i E TF ,we can modify sr in such a way that there are no changes between the output levels _krand fITF (aTFF ), but the payment is fixed at zero for output levels below _kr and at kr +fITF(4,,) for output levels above f 1TF (aTF F ). More formally, for each i E TF , we define thepayment function gr as follows:0s iF (q)kF + f ITF (4F)if 0 < q < _krif _kr < q < fITF(4,)if q > f ITF (aCF).It is obvious that the payment function .§-f is continuous, piecewise linear, and nondecreasing.Also, it can be easily verified that the NGP {TF ,(e)i,} belongs to c+ .Clearly, for each i E TF , the payment according to gr can exceed the payment accordingto sr only at output levels below _kr. However, the payment according to gr for any outputlevel below _kr, which is fixed at zero, is no larger than ui. Also, for each i E TF , gr andsr award the same payment at the output level f ITF(aTF ). Then, as we already know that(sn iETF induces arF , (gn iETF must also induce aF F and hence, we have the followingTproposition.Proposition 1.3 : Suppose assumptions A1-A5 are satisfied. If W(TF) > V (T*), then({TF^E * .Proof See Appendix A.Next, consider the other case V (T") > W (TF ). Partition the set of agents^into thethree subsets. 7; , 73 and T",. such that i E T+ if and only if k7 > 0, i E T6' if and only if= 0, and i E T* if and only if k7 < 0. By relabeling the agents if necessary, without loss ofgenerality, we let 17 be the first IT; I agents, T6 be the 1116 1 agents after 177; I, and 7'"_ be the39177.11 agents after ITV + IT ^i.e. 77 = {1, ...,I771} if 711 0 0, T6' = {1T1_" I+ 1, ...,^U T6'1}if T6 0 0, and 7-7 = {177 U T61+ 1,^mil if T1." 00.For each i E^, let §7 be the payment function that pays zero wherever si pays anonpositive amount and the same as s7 everywhere else. Formally, for each i E T' , as thecritical output level at which s7 starts paying nonnegative amounts is —k7/77, we have=^°^if 0 < q < —k7 177^(1.12)s;' (q) if q > —k7 1-y7.For each i E TP, let4(q) = 4(q) V q E^ (1.13)Suppose 77 is nonempty. Then agent 1 belongs to T. Now, if agent 1 is paid q —EiET6' UT* (q) for every q E n+ , then it is obvious that there is an output level below whichshe does not get as much as in sI but above which she gets more than in sT. So, let -4' be theunique output level such that sl(e ) ql — EiET. .uT 4(4-1 )0 ^• Then the payment function 4is defined as follows:q — EiETo*UT* "qi (q) if 0 < q < -41§i(q) =^ (1.14)8'1(0^if q >Following a similar procedure as above, for any i E^— {1} , qi is iteratively defined asthe unique critical output level such that s7 (q)^— EJ ET,NT^—^(qi). Then,for each i E^— {1}, the payment function g7 is iteratively defined as follows:q — EJET0-uT. g:;(q) —^(q) if 0 < q < qi4 ( 0 = (1.15)s7 ( q)^if q > 41 .Note that. if T+* has more than one agent, then there is an asymmetry in the behavioursof .51 at output levels on or below 4-1 ands7 at output levels on or below qz, where i E 77 is40distinct from 1. For each i E 111 distinct from 1, s pays zero at every output level lower thanq4-1 , the critical output level of the agent just before i. On the otherhand, if is nonempty,then sl pays zero only at zero output level. For each i E T4I distinct from 1, when the finaloutput is on the interval W-1 , pays agent i the output that is left after paying eachagent h before her according to s';', and each agent j E T6 UT* according to g;!. So, if T11 hasmore than one agent, for each i E T11 distinct from 1, the behaviour of 4 between 4-4-1 andis similar to that of WI between 0 and e.It can be easily verified that, for each i E^4 is continuous, piecewise linear, andnondecreasing. Furthermore, the principal's residual, q —^(q), is continuous, piece-wise linear, and nondecreasing. Our construction also ensures that the NGP {T*belongs to g+ .Clearly, for each i E T. U Tp, the curve of 4 always lies on or below that of 4 . Onthe otherhand, for each i E T*, wherever the curve of si lies above that of s7 its value isequal to zero, and hence, no larger than the utility of agent i at 4., ui. Also, the curves ofthe principal's residual in JET* ET* m(re)^and ( )^are such that, if there are output levels,at which the former lies above the later, then the value of the former is equal to zero atthose output levels. Then, because (s7)iET* induces 4. , ) JET. should also induce 4..Hence, we have the following proposition.Proposition 1.4 : Suppose assumptions Al-A5 are satisfied. If V(7*) > W (TF), then({T* , (4) j ET.}, ai%) E^.Proof. See Appendix A.41Thus, according to Propositions 1.3 and 1.4, except for the fact that the principal may haveto look for slightly more sophisticated output sharing rules than those of the linear variety(namely, piecewise linear rules), there are no other significant changes when individuals havelimited liabilities. This result, as we have pointed out all along, is a consequence of thedeterministic production process. In contrast, when the production process is no longerdeterministic, often, there is not enough freedom to modify the unlimited liability optimaloutput sharing rule to a limited liability optimal output sharing rule. Hence, imposing limitedliability condition often reduces the principal's optimal utility when there are uncertaintiesin the production process.1.6 Uniqueness of SPE PayoffsTo begin with, we must point out that the result presented in this section relies on thequasilinearity of the utility functions, and hence, may not hold for the more general utilityfunctions that are concave, but not necessarily quasilinear.Our objective is to show that the utility tuple remains the same in every SPE outcomeof the SBG. Precisely, we show that at any SPE outcome the utility of each agent i is ui andthe utility of the principal is her utility from any 00.Suppose ({T, (si)i G T_„, }, aT) E Sr, but the utility of some agent j E T_,, si ( f IT(aT)) —ci(aj), is greater than ui. Now, if we keep the payment function of every other agent intactand give agent j an € > 0 less for every level of output, where e is such that si(f IT(aT)) —— ci( ) > ui, then, because of the quasilinear utility functions, aT is still a Nash equilibriumat which every agent in 7t,„ (including agent j) get at least their outside option utility andthe principal's utility has increased by e. But this means that there is an outcome in O at42which the principal's utility is higher than at an outcome in Q*, which is not possible. So wecan claim the following.Claim 1.3 : If assumptions A1-A5 hold, and ({T.(8 . )jET_ Th }• ciT) E Q*, then si(fIT(aT))— ci(ai) = u i, V i ESuppose W(TF ) > V(T*) (V(T*) > W(TF)). Then, as assumption A3 implies W(TF )> u rn > 0 (V(T*) >^> 0), let e > 0 be such that W(TF ) — 1TF le > u rn (V(T*) —Tri > urn). Consider the NGP {TF , (sp)iETF} ({7--,(sr)ieT,„,}), which is obtainedfrom {TF, (8r),ETF} ({T* ?*) iET* ) by paying each agent in TF c more for ev-ery level of output. Then, because of the quasilinear utility functions and Proposition 1.1(Proposition 1.2), it obviously follows that: (i) each agent i E TF (E T* rn ) gets u, c atthe outcome ( ITF, ( sr, ) , ET F L aC,F ) (({T* (4 1 )20-1%„ } , 4*)); (ii) the principal gets W(TF )— !Vic (V(T*) — iT'!_ rn ic) at the outcome ({TF. (sNiET,}, 4-,F) (({T* • ( 8 7 E)i T* ,n } aT*));and (iii) aTF (4.) is the unique Nash equilibrium of {TF , (sNiET, } ({T* ,( 87 6 )iET*,,})•In the case of limited liabilities, we can exploit the freedom provided by the deterministicproduction process to modify the above mentioned NGPs in such a way that the limitedliability condition is met without loosing any of the conclusions drawn. Thus. Appendix Bproves the following lemmasLemma 1.7 : Suppose assumptions A1-A5 hold, and W(TF ) > V(T*). Then, for eache > 0 such that W (TF ) — ITF > u,n , there exists {TF , (kJ ' ) jeTF} E c+ such that: (i) aTFis its unique Nash equilibrium; and (ii) the utilities of the principal and each agent i E TF ataTF are W(TF ) — 1TF le and ui e, respectively.Proof. See Appendix B.43Lemma 1.8 : Suppose assumptions A1-A5 hold, and V (T*) > W(TF). Then, for each> 0 such that V (T*) — 171",,Ic > u, and k7 +c<OViE T.: (the existence of such an cis assured by the fact that k7 <OViE TI), there exists {T* ,(k7 6 )./€7*„} E g+ such that:(i) 4 is its unique Nash equilibrium; and (ii) the utilities of the principal and each agenti E T*,,, at 4. are V (Ts) — le and ui + e, respectively.Proof. See Appendix B.Now, suppose there is a SPE outcome which does not belong to Q*, and the principal'sutility at this outcome is it,. Clearly,^is less than the principal's utility at an 00. Then,because of Lemmas 1.7 and 1.8, we can find some outcome, say ({T, (s i ) , aT) , suchthat: (i) {T, (s,), ET_,, } E ; (ii) aT is the unique Nash equilibrium of {T, (iii)T, is nonempty and the utility of each agent in T_,, at aT is more than her outside optionutility; and (iv) the principal's utility at aT is more than lint. But then, using a simplebackward induction logic, it is easy to see that all the agents in T_, will always agree tothe NGP {T, (s i ), ET_„, } whenever the principal announces it in the first stage of the SBG.So there cannot be any SPE outcome at which the principal's utility is nn,, and hence, ouroriginal supposition must be false. Therefore, there are no SPE outcomes outside Q*. Theuniqueness of the SPE utility tuple is then an immediate consequence of Claims 1.1 and 1.3.Claim 1.4 : Suppose assumptions Al-A5 hold. Then, whether there is limited liability ornot, we have the following: (i) if W (TF ) > V (T*), then at any SPE the utility of theprincipal is W(TF ) and the utility of each agent i is ui ; and (ii) if V(7*) > W (TF ), then atany SPE the utility of the principal is V (7') and the utility of each agent i is ui.441.7 An ExampleConsider a situation with three individuals (in, = 3). So individuals 1 and 2 are the agents,and individual 3 is the principal. The joint production process and the cost functions aregiven by: (i) f (a) = 2(1/6 + al) 1 / 2 (1/6 + K2a2 K3a3) 1 /2 — 1/3 V a = (ai, a2, a3) E 9VF ,where K2 and K3 are constants to be specified; and (ii) ci (a z ) = 4/2 V a, E 94, i = 1,2,3.All individuals have the same outside option utility, which is equal to 1/3; i.e. u, = 1/3,i = 1,2,3. The constants K2 and K3 can be interpreted as parameters that express therelative efficiency between the action of agent 2 and the action of the principal. We look atthree different scenarios corresponding to different values of the efficiency parameters K2 andK3.Case 1: K3 1, and K2 > 0 but sufficiently close to zero.In this case, it can be easily verified that W({1}) = W({3}) = 5/24 and W({1,3}) = 2/3.We can also find K2 small enough such that W({2}) < 5/24, W({1, 2}) < 5/24, W({2,3})< 5/24, and W({1, 2, 3}) < 5/12. Then it is easy to see that the principal must take partin production along with agent 1 to get the maximum utility in the FB situation; i.e.= W(11,31) = 2/3. So the principal cannot sustain efficiency in this case. Clearly, TF ={1} and W(TF ) = 5/24. Also, straightforward maximization shows that V({1,3}) = 5/12,and hence. T' = {1,3}. Thus, W(TF) < V (T*) in this case. Therefore, only agent 1 andthe principal participates at an 00. The optimal actions are (al, a5) = (1/2, 1/2), and theoptimal linear and piecewise linear payment functions for agent 1 are given by:s7(q) =^—1/24 + (1/2)y V y E 94; and0 if q <1112(q) =—1/24 + (1/2)q if q > 1/12.45Case 2: K2 = 1, and K3 > 0 but sufficiently close to zero.Here, W({1}) = W({2}) = 5/24 and W(11.21) = 2/3. We can also find K3 small enoughsuch that W({3}) < 5/24. W({1,3}) < 5/24, W(12,31) < 5/24, and W(11,2,31) < 2/3.Then it is obvious that only agents 1 and 2 must take part in production for the principal toget the maximum utility in the FB situation; i.e. ?IC W({1, 2}) = 2/3. So TF = {1, 2}, andW(TF) = uC. Therefore, the principal can sustain efficiency in this case. Clearly, whateverbe the T* , we have W(TF) > V (T* ). Hence, at an 00. only the two agents participatein production and moral hazard is completely mitigated. It can be easily verified that theoptimal actions are (cif , = (1,1). Thus, the optimal linear and piecewise linear paymentfunctions for the two agents are given by:sf (q) = (q) = —7/6 +q V q E W+; and0 if q < 7/6gr(q) = (q) = —7/6 + q if 7/6 < q < 25/6 if q > 2.Case 3: K2 = 1, and K3 > 1 but sufficiently close to one.As in case 2 above, W({1})^W({2}) = 5/24 and W({1,2}) = 2/3. Now, we can find K3close enough to one such that W({3}) < 2/3, W({1, 3}) > 2/3, W({2, 3}) < 1/3, W({1, 2, 3})< W({1, 3}, and V(11,2,31) < V({1, 3}) < 2/3. Then it is obvious that W({1,3}) > W (T)V T C {1,2,3}. So only agent 1 and the principal must take part in production for theprincipal to get the maximum utility in the FB situation: i.e. u ;3 W({1. 3}) > 2/3. It isalso easy to see that TF 11,21, and T* = {1, 31. This means that uC > W(TF ) > V (T*).Therefore, at an 00. as in case 2, only the two agents participate in production and moralhazard is completely mitigated. Moreover, the optimal actions and the linear and piecewise46linear optimal payment functions of the two agents remain the same as in case 2. However,unlike case 2. the principal can no longer sustain efficiency as her best option in the FBsituation requires her participation in production.1.8 Nonquasilinear UtilitiesFor each agent i, when she participates in production and takes an action a, and receives apayment p i , suppose her utility is given by U,(p„ a,), where U, is concave and satisfies all theother standard assumptions. Similarly, when the principal participates in production andtakes an action a, and receives a residual r, suppose her utility is given by Um (r, am ), whereUm is concave and satisfies all the other standard assumptions. On the other hand, when theprincipal does not participate in production but exercises her outside option and receives aresidual r, suppose her utility is given by (1,-,,(r), where Om is concave and satisfies all theother standard assumptions.With the above specified utilities, the logic about the deterministic production processleaving sufficient room that allows the optimal output sharing rules to behave quite arbitrarilyaway from the optimal output level is still applicable. Therefore, although it is slightly moredemanding technically, we can derive analogues of conditions (1.1)-(1.3) that do not dependon any output sharing rules. Also, for any T such that m E T and IT! > 2, we can eliminatethe output sharing rule from the appropriate analogue of problem (PT) and transform it intothe appropriate analogue of problem (4). Thus, except for the section on the uniquenessof SPE utility tuple (namely. section 1.6). which relies heavily on the quasilinearity of theutility functions, the analysis in the rest of this chapter can be repeated with the more generalutility functions without any qualitative changes in the results.47Chapter 2The Power Structure for Stochastic Social ChoiceFunctions with an Unrestricted Preference Domain2.1 IntroductionDepending on the social choice rule used to aggregate individual preferences on social alter-natives, there are two standard paradigms in deterministic social choice theory. In the first,commonly known as the Arrovian or social preference framework, the social choice rule takesthe form of a social welfare function which maps the set of all possible profiles of individualorderings of the social alternatives into the set of social preferences on them. In the secondparadigm, commonly known as the social choice framework, for each possible combinationof a profile of individual orderings of the social alternatives and a feasible set (which is asubset of the universal set of social alternatives), a social choice function specifies a set ofalternatives from the feasible set as the socially chosen alternatives. Arrow's [2] impossibilitytheorem can be formulated in either framework. Since Arrow's seminal work, many attemptshave been made, without much success, to escape his impossibility result by modifying the48conditions which Arrow required the social welfare function or the social choice function tosatisfy, including weakening of the basic collective rationality requirement. The numerousimpossibility results in the deterministic social choice literature bear testimony to the robust-ness of Arrow's impossibility result in the deterministic case. These negative results are partlyresponsible for a line of research which considers probabilistic social choice rules to aggregateindividual preferences on social alternatives. Probabilistic social choice rules are more generalthan their deterministic counterparts and increase the possibility for satisfactory aggregationof individual preferences. As in the deterministic case, the probabilistic social choice rulesthat are considered in the probabilistic social choice literature can be broadly classified intotwo categories, namely, those that map each social preference profile to a lottery over socialpreferences, and those that map each combination of a social preference profile and a feasibleset to a lottery over the feasible set. We refer to these kinds of rules as stochastic socialwelfare functions and stochastic social choice functions, respectively.Apart from opening up the possibility of escaping the Arrow-type impossibility results, anequally important and attractive aspect of the probabilistic framework is the scope it providesfor incorporating certain notions of fairness and reasonable compromise into the collectivedecision-making process. For example, consider the situation of two seriously injured accidentvictims who each must have a pint of blood to survive, but there is only one pint of bloodavailable and each individual wants to have it. In this situation of conflict of preferences,flipping a coin to determine the actual recipient of the single available pint of blood seemsto incorporate a certain element of fairness and reasonable compromise which is lacking indeterministic social choice rules.The Arrow conditions in the deterministic framework imply that the social choice pro-49cedure satisfies a neutrality property which plays a key role in the dictatorship theorem,namely, if any group of individuals is decisive over some pair of social alternatives, then thatgroup must be decisive over every pair of social alternatives. A probabilistic analogue ofneutrality is satisfied in the probabilistic framework when the Arrow conditions are appropri-ately translated into their probabilistic counterparts. However, in this framework, neutralityis a more appealing principle because of the randomness present in probabilistic social choicerules. For example, a random dictatorship satisfies neutrality while avoiding many of theundesirable features of a deterministic dictatorship.A major concern of probabilistic social choice theory is to characterize the properties ofthe power structures that can arise in this framework when the appropriate probabilistic ana-logues of the axiom systems used in the various impossibility theorems in the deterministicframework are adopted. Loosely speaking, by power structure we mean the distribution ofthe degree of influence in the social decision process that different groups of individuals mayhave. In the case of stochastic social welfare functions, this line of investigation was initiatedby Barbera and Sonnenschein [9] and subsequently pursued by Bandyopadhyay, Deb andPattanaik [5], Heiner and Pattanaik [21], and McLennan [26]. In this literature, the power ofa coalition to determine the social choice probabilities in pairwise comparisons are inducedby the probabilities assigned to the social preferences. However, unlike the deterministicframework, where the distribution of coalitional power in nonbinary choice situations is com-pletely determined by the distribution of coalitional power for binary comparisions, it is notat all clear if the results in these papers can be used to derive restrictions on the distributionof coalitional power for nonbinary choice. This is a serious weakness if we maintain thatfrom a social action perspective the significance of the probabilities assigned to the social50preferences lies in the social choice probabilities they induce over each possible feasible set ofsocial alternatives.Among the literature which considers stochastic social choice functions, to our knowl-edge, Barbera and Valenciano [10], and Pattanaik and Peleg [29] are the only articles thatsystematically investigate the distribution of coalitional power 7 . However, as Barbera andValenciano [10] implicitly consider only those feasible sets that contain exactly two socialalternatives, their work also suffers from the same weakness mentioned in the previous para-graph. Pattanaik and Peleg [29], which is henceforth referred to as PP, consider feasible setswith arbitrary numbers of alternatives. For the axioms considered by them, they were ableto characterize the distribution of coalitional power even when the feasible set contains morethan two social alternatives.PP considered stochastic social choice functions that satisfy three conditions they calledindependence of irrelevant alternatives, ex-post Pareto optimality and regularit . Indepen-dence of irrelevant alternatives requires that, given a feasible set of alternatives B, the sociallottery over B must be the same for any two profiles in winch the individual preferencesrestricted to B are the same. Ex-post Pareto optimality says that, given a feasible set ofalternatives B and a pair of alternatives x and y in B, if everyone prefers x to y, then thesocial probability of choosing y from B must be zero. Regularity requires that, given a profileof individual preferences and a feasible set of alternatives B, the social choice probability as-signed to each alternative in B must not increase from its original value when the individualMost of the other works in this literature (Barbera [6.7], Fishburn [15], Fishburn and Gehrlein [16], andIntriligator [24] to mention a few) focus attention on the properties of particular stochastic social choicefunctions.8 lndependence of irrelevant alternatives and ex-post Pareto optimality are respectively the probabilisticcounterparts of Arrow's [2] Independence and Pareto conditions. Regularity is a natural probabilistic versionof a rationality condition in the deterministic framework due to Chernoff [12] known as condition a.51preferences remain the same but the feasible set is expanded by adding more alternatives toB. When the universal set of social alternatives contains at least four elements and individualpreference orderings are strict (i.e. each individual can only have a linear ordering over theuniversal set of alternatives), PP showed that a stochastic social choice function is essentiallya random dictatorship if it satisfies their three conditions. More precisely, they first derived aunique weight for each individual such that the vector formed by these individual weights hasthe properties of a probability distribution over the set of all individuals. Then they showedthat, given any feasible set B distinct from the universal set and any preference profile, theprobability of society's choosing an alternative x from B is equal to the sum of the weights ofthose individuals who have x as their best (or greatest) alternative in B. When the numberof alternatives in the universal set exceeds the number of individuals in the society by at leasttwo, their result extends to the case in which the feasible set is the universal set itself.The assumption that individuals can only have strict preference orderings plays an im-portant role in the result of PP. However, as the authors themselves pointed out, this is arather restrictive assumption. Therefore, the next logical step is to permit individuals tohave indifference between alternatives and ask whether there are any natural extensions ofthe results of PP with this expanded individual preference domain. This particular line ofinvestigation is the subject matter of the current chapter.To understand the difficulty that arises when the individual preference domain is expandedto permit indifference between alternatives, consider a feasible set of alternatives B and apreference profile R. If x and y are two alternatives in B and the individual preferences in Rare strict, then, as there is only one best alternative in B for each individual, the intersection ofthe set of individuals who have x as their best alternative with the set of individuals who have52y as their best alternative is empty. This fact is crucial for the weighted random dictatorshipresult of PP as it allows a specific partitioning of the society in which each member coalitionconsists of all those individuals who have the same best alternative in B. However, sucha partition of the society may no longer exist if we permit the individual preferences in Rto have indifference between alternatives. Unlike the case with strict individual preferences,when the individual preferences in It are not necessarily strict, as each individual may havemore than one best alternative in B, it is possible for two individuals to have best sets in Bthat intersect but are not equal to each other. Thus, when the assumption of strict individualpreferences is dropped, we must deal with the possibility that, given any two alternatives xand y in B, the two sets of individuals whose members respectively have x and y as a bestalternative in B may intersect with each other.Suppose there are at least four elements in the universal set of alternatives and thestochastic social choice function satisfies the three conditions of PP, namely, independence ofirrelevant alternatives, ex-post Pareto optimality and regularity. Also, suppose B is a propersubset of the universal set of alternatives and It is a social profile of preference orderings.Let us partition the society in such a way that each coalition of individuals S in the partitionsatisfies: (i) if individual i belongs to S and individual j does not belong to 5, then theirbest sets in B according to their preferences in R do not have any common alternative; and(ii) if individual i belongs to S and there are at least two individuals in S, then at least onealternative which belongs to individual i's best set in B according to her preference in it alsobelongs to the best set in B according to the preference in R of some other individual in S.This way of partitioning the society is in some sense a generalization of the one describedbefore for the case of strict individual preferences. because it yields the same partition of the53society as before whenever the individual preferences in R are strict. We show the following:(i) as in PP, there is a unique nonnegative weight for each coalition of individuals in thesociety; and (ii) when B is the feasible set of alternatives and R is the social preferenceprofile, for each coalition S in the partition just described, the sum of the social probabilitiesassigned to all the alternatives, each of which is a member of the best set in B according tothe preference in R. of at least someone in S, is equal to the weight of the coalition S. If thesocial preference profile R is such that at least two alternatives do not belong to any of thebest sets in the universal set according to the preferences in R, or the number of alternativesin the universal set exceeds the number of individuals in the society by at least two and thesocial preference profile R is such that at least one alternative does not belong to any of thebest sets in the universal set according to the preferences in R, then the feasible set B doesnot have to be a proper subset of the universal set for our result to hold. These results reduceto the results of PP whenever everyone has unique best feasible alternatives. So our resultsare generalizations of those in PP.In the next section, we introduce some prerequisite notation and definitions. The uniquenonnegative weight of each coalition of individuals in the society is derived in section 2.3.The results, which we have briefly outlined above, are presented in section 2.4.2.2 Notation and DefinitionsThe universal set of alternatives, denoted by X. has 1 alternatives with oo > 1> 2. Let X bethe set of all nonempty subsets of X (i.e. X = 2x — {0}) 9 . An ordering on X is a reflexive,complete and transitive binary relation on X. We denote by R. the set of all orderings on X.9 As in Chapter 1, given any two sets D and E, we use the notation D—E={rED:r0 E} Similarly,given any set D. we use IDI to denote the number of elements of D.54Let N^{1, ..., n} be the set of all individuals in the society, where oc > n > 2. Also,let K be the set of all nonempty subsets of N (i.e. ,Y = 2 N — 01). Then, as any subset ofindividuals in the society is a coalition, K is the set of all possible coalitions in the society.We denote the coalitions in Ai by S, .Given any S E K, RS denotes the ISI-fold Cartesian product of R. We use the term pref-erence profile for the members of RN and denote them by R, R , ft, . Given a preferenceprofile R E RN , the ith coordinate of R, which we denote by Ri, represents the preferencerelation on X of individual i in the preference profile R. As usual, for each possible order-ing Ri E R of individual i, Pi and Ii denote the asymmetric and symmetric parts ofrespectively.Definition 2.1: A stochastic social choice function (SSCF) is a function F : X x X x RN^92which satisfies: (i) F(x,B,R)> 0 V (x,B,R) E X xXx RN ; and(ii) ExE , F(x,B,R) ExEx F(x,B,R) = 1 V (B, R) E X x RN .We denote the set of all SSCFs by .7'. Given a SSCF F E .F, a feasible set of alternativesB E X, a preference profile It E RN and an alternative x E X, we interpret F(x, B, R)as the probability of x being chosen by the society when the feasible set is B and society'spreference profile is R. Thus, for each feasible set and each preference profile, a SSCF alwaysassigns zero probability to any nonfeasible alternative.Definition 2.2: For each (B, R) E X x RN , x E B is weakly Pareto optimal (WPO) withrespect to (B. R) if and only if there does not exist y E B such that yPix V i E N.Given any (B. R) E X x RN, we denote the set of all WPO alternatives with respect to(B,R) by WPAR(B,R); i.e.55WPAR(B, R) =^E B: x is WPO with respect to (B, R)}.Given a SSCF F, let POS(F, B, R) be the set of all alternatives that are assigned positiveprobabilities by F when the feasible set is B E X and the preference profile is R E RN ; i.e.POS(F,B,R)^E B: F(x,B,R) > 01.Definition 2.3: A SSCF F is weakly Paretian ex-post (WP) if and only ifPOS(F,B,R) C WPAR(B,R) V (B,R) E X x RN .Let B E X. For each i E N and each Ri E /Z, we denote the restriction of Ri to B by RiPB.Similarly, for each R E RN , we denote the restriction of R to B by RIB = (R1IB, IB).Also, we denote the set of all possible orderings on B by RIB and the n-fold Cartesianproduct of RIB by TZNIB. Thus, for any R E RN , it is obvious that RiI.B E V i E N,and RIB E "R, N IB.The following definitions of independence of irrelevant alternatives (IIA) and binary in-dependence of irrelevant alternatives (BIIA), which are the appropriate counterparts of thosein the deterministic framework, are as given in PP.Definition 2.4: A SSCF F satisfies IIA if and only if V B E X and V R, R. E RN[RIB =RIB] z [F(x,B,R) = F(x,B,R.)V x EDefinition 2.5: A SSCF F satisfies BIIA if and only if V B E X, with IBI = 2, and VR,R E RN[RIB = 14,1B]^[F(x B, R) = F(x. B, ft) V x E56The final condition we want to impose on a SSCF is regularity. This condition is a naturalextension to the current framework of condition a of Chernoff [12], which is a minimumconsistency condition for rationalizability of choice functions. For a more detailed discussionof this regularity condition, the interested reader is referred to PP.Definition 2.6: A SSCF F is regular if and only if V B, B E X and V R E RN[x E B C ij] [F(x, B, R) > F(x. B, R)].2.3 Coalitional WeightsIn this section, corresponding to each SSCF F E .T which satisfies certain conditions, wederive a unique nonnegative number for each coalition of individuals S E Al called the weightof coalition S according to the SSCF F. Given any partition of the society, the sum of theseweights over all coalitions in this partition is equal to one. Although we carry out our analysisin a slightly different manner, these weights are essentially the same as those in PP.Given a SSCF F which satisfies certain conditions and a coalition S E A1, we establishthat, for any two pairs of alternatives (x, y) and (z, w), and for any pair of preference profilesR and it, if everyone in S strictly prefer x to y and z to w according to their preferences inR and it, respectively, and everyone outside S strictly prefer y to x and w to z accordingto their preferences in R and R, respectively, then the social probability of choosing x when{x, y} is the feasible set and R is the preference profile must be the same as the socialprobability of choosing z when {z, w} is the feasible set and R is the preference profile. It isthis social probability which is the weight of the coalition S according to the SSCF F. Thus,to some extent, the weight of a coalition S according to a SSCF F indicates the power that57coalition S has to influence the social choice probabilities. These weights play a key role inthe characterization of the coalitional power structure.We begin with a lemma, which is the counterpart of Lemma 4.1 in PP. Given a feasible setB and a preference profile R, this lemma completely characterizes the condition under whichthe probability assigned to each alternative in B by a regular SSCF F remains unchangedwhen the feasible set is expanded to B D B.Lemma 2.1 : Suppose F E 1. is regular and B, B E X with B C E. Then, for any R E RN ,POS(F,E,R) C B if and only if F(x,B,R) = F(x,E,R) V xE B.Proof. Let B, 1'3 E X with B C and R E R,N(Necessity): Suppose Pos(F,E,R) C B. Then we have1^E F(x,E,R) =^E^F(x,B,R) < E F(x, /3,R) < 1.xEf3^xEpos(F,n,R) xEBThus, ExEB F(x, B, R) = 1 = ExEB F(x, B, R). But, by regularity, F(x, B, R) > F(x, B, R)V x E B. Hence, F(x, B, R) = F(x, B, R) V x E B.(Sufficiency): Suppose F(x,B,R) F(x,E,R) V x E B. Then 1 Ex€B F(x, B, R)= ExEB F(x, B, R). Clearly, POS(F,E,R) C B for any B c 13 such that ExEb F(x, B, R)= 1. Hence, POS(F, B, R) C B.The next two lemmas consider the neutrality features of the stochastic social choice func-tion.Lemma 2.2 : Let F be a regular SSCF that satisfies WP and IIA , and let x, y. z E X bethree distinct alternatives. If S E Ai and R, R E R -v are such that xPiy and xl3iz V i E S,and yPix and zPix ViEN— S. then F(x, {x,y},R) = F(x. fx,z1,1i).58Proof. Let x, y, z E X be distinct. Suppose S E N and R, R E 7?.. /v are such that x Piy andxPiz V i E 8, and yPix and zPix Vi E N — S. Because of WP, the lemma is trivially true ifS N. So we suppose S NSuppose F (x , {x, y}, R) > F (x , {x, z} , R.). Consider R E 7Z,N such that xPiyPizViESand yPizPix Vi E N — S. Clearly, WP implies that F(z, {x, y, z}, R) = 0. Then, because ofLemma 2.1, we haveF(x, {x, y, z},^= F(x,{x,y},11).^ (2.1)Also, because of IIA, we have^F(x,{x,y},R) = F(x, {x, y}, ft);^and (2.2)^F(x, {x, z}, R) = F(x, {x, z},it). (2.3)Then we get^F(x, {x, y}, R) > F(x, {x, z}, R)^[by supposition]^F (x , {x, z} , it)^[by (2.3)]^> F(x, {x, y, z},it)^[by regularity]^F(x, {x, y}, it)^[by (2.1)]^F(x, {x, y}. R)^[by (2.2)],which is not possible. Hence, it must be the case that F(x, {x, y}, R) < F(x, {x,By a similar argument. we can also conclude that F(x. {x. y}. R) > F(x, {x, z}, R).^Therefore. F(x. {x. y}. R) = F(x, {x, z},^II59Lemma 2.3 : Let F be a regular SSCF that satisfies WP and IIA, and let x, y, z E X bethree distinct alternatives. If S E .AT and R, It E 7Z-v are such that xPiy and zi'iy i E S,and yPix and yPiz ViEN—S. then F(x, {x,y},R) = F(z, {y,z},f1).Proof Let x, y, z E X be distinct. Suppose S E Al and R. ft E 1Z,N are such that xPiy andzPiy V i E S, and yPix and yPiz ViEN— S.Using Lemma 2.2, we get F(y,{x,y},1t) = F(y. {y, z}, ft), which implies that1— F(y.fx,y1,1t) = 1— F(y,{y,z},ft).Hence, F(x, {x, y}, R) = F(z, {y,z},ft).^IIGiven a SSCF F, define the correspondence al, : Al. ---4—> 1? as follows: for any S E .Ai,/a = F(x, {x, y}, It) for some distinct x, y E X and ItaF(S) = a E R.* :^ .ETA such that xPiy ViES and yPixViEN—SAs a convention, we use aF(0) = {0} F E .F. Clearly, aF(S) is nonempty for any F E .Tand any S C N (including S = 0). If a SSCF F satisfies WP, then it is also easy to checkthataF(N) = {1}.^ (2.4)Thus, given any F E^that satisfies WP, the following follows from the definition of thecorrespondence aFfor any S C N :^[a E aF(S)] <=> [(1 —^E aF(N — S)]•^ (2.5)Now, given a SSCF F and a coalition S E Al. consider any pair of distinct alternativesx, y E X and a preference profile R E 7Z. N such that xPiy V i E S and yPix ViEN— S.As long as F(x.{x,y},R) = F(x,{x.y}.1i) for every ft E R, -\ that is identical to R when60restricted to {x, y}, which is the case if F satisfies IIA, the social probability of choosing x fromthe feasible set {x, y} when R is the preference profile. F(x, {x, y}, R), can be interpreted asthe probability for coalition S to be decisive for x over y in the SSCF F. As a consequence, ifaF(S) contains a single nonnegative number, then the probability for coalition S to be decisivefor some alternative over another alternative is invariant to the pair of distinct alternativesconsidered. Further, this probability is equal to the single number in aF(S). Proposition 2.1demonstrates that aF(S) is a singleton under our assumptions.Proposition 2.1 : Suppose 1 > 3. If F E .7' is a regular SSCF that satisfies WP and IIA,then laF(S)I = 1 V SC N.Proof Clearly, la F ( 0 )1 = la F (N )1 = 1 .Suppose S E .Ar such that S 0 N. Let et, 6e. E aF(S), ft, ft E RN and x,y,z,w E X besuch that xPiy and zPiw V i E 8, yPix and wPiz Vi E N— 5, =^and= F(z, {z,w},ii). Then the proposition is true if & = ee holds in each of the followingexhaustive list of possibilities: (a) x = z and y w; (b) x z and y = w; (c) x = w andy z; (d) y = z and x w; (e) x = z and y = w; (f) x = w and y = z; and (g) x, y, z, ware distinct.(a) x = z and y w: Then, using Lemma 2.2, it can be easily checked that & = a.(b) x z and y = w: Then & = et. readily follows from Lemma 2.3.(c) x = w and y z: Consider R E RN such that zPiy ViES and yPiz ViEN— S.Obviously, Lemma 2.2 implies that ee=^F(z, {z. y}, R). Then, using Lemma2.3, we get & = F(x,{x,y},14,) = F(z, {z, y}. R) = et.(d) y = z and x w: An argument similar to the one used in (c) shows that & =61(e) x = z and y = w: Then, because of IIA, (ft =(f) x = w and y = z: As 1 > 3, consider v E X distinct from x and y. Also, letR', R" E RN be such that vPly V i E S, yPiv Vi E N— S, v/311x V i E S and xPI'vV i E N— S. Now, using Lemma 2.3, a = F(x, {x, y}, it)^F(v, {v, y}, IC) and a =F(y, Ix, yl,^= F(v, {v, x}, R"). Then Lemma 2.2 implies that 'a =(g) x, y, z, w are distinct: Let R E R N be such that xPw Vi ES and wPix V i E N—S.So, using Lemma 2.3, we get a = F(z,{z,w},ft) F(x, {x,,w},ft). Then & = a" followsfrom Lemma 2.2.Hence, jaF(S)I = 1 V S C N. IIThe significance of Proposition 2.1 lies in the fact that, if aF(S) is a singleton for everycoalition S E Al- , then the single number in aF(S) is a good indicator of the power of coalitionS E , and hence, there is a possibility of using aF to derive restrictions on the structure ofcoalitional power. It is worth noting that Proposition 2.1 also confirms that the neutralityfeature implied by the Arrow conditions in the deterministic framework, mentioned earlier,holds in the current framework, because the conclusion of Proposition 2.1 can be viewed asa probabilistic version of this neutrality condition.Henceforth, whenever there are at least three social alternatives and F E .F is a regularSSCF that satisfies WP and IIA, we shall treat aF as a function; i.e. for each S C N, aF(S)is a nonnegative real number rather than a set containing a single nonnegative real number.We have the following straightforward corollary to Proposition 2.1.Corollary 2.1 : Suppose 1 > 3. If F E^is a regular SSCF that satisfies WP and IIA,then aF(S) + aF(N — S) =1VSCN.62Proof Follows from (2.5) and Proposition 2.1.When there are at least three alternatives in the universal set X, and a SSCF F satisfiesregularity, WP and IIA, Proposition 2.1 opens up the possibility of characterizing the coali-tional power structure under F in terms of the function a F. However, to be able to do so it isessential that the function aF be additive, so that, given any partition of the society, the sumof aF(S) over all coalitions S in the partition is equal to the power of the grand coalition N,which is equal to one. Thus, the objective which remains to be accomplished in this sectionis to show that this additivity property for aF is satisfied. To achieve this objective we needto introduce some prerequisite notation and prove a preliminary proposition.Given B E X, i E N and Ri E R,, letGi(RilB)^{x E B : xRiy V y E B}; andt-1^ t-1xEB—Gt(RilB) =^U Gi(RilB) : x Riy VyE B— UGT(RilB)^Vt>2.t--.--1 t.--1Thus, Gi(Ri IB) is the best set in B according to Ri , G2(Ri1/3) is the second best set in Baccording to Ri, and so on.Given B E X and R E RN , letOM RIB)^0 and/3(S, RIB) =^Gi(RilB)^V S E .iESSo 0(5, RIB) is the set of all alternatives that belong to the best set in B according to thepreference in the profile R of at least one individual in the coalition S.Given x E B E X and R E R N , let L(x, B, R) be the set of all individuals who have xin their best sets in B according to their preferences in the profile R. More formally, given63xEBEXandRERN ,L(x, B,R) = {i, EN: x E Gi(Ri1B)}.Consider a SSCF F which satisfies regularity, WP and IIA. and consider any feasibleproper subset B of the universal set X. Suppose x is an alternative in B and It is a preferenceprofile such that x E G1(R3 IB) for some individual j, and for each individual i, either x isthe unique member of the best set Gi (R, 1B) or x G1(.11, 1B); i.e. Gi(R,IB) = {x} for eachi E L(x, B, R). Then the following proposition, which is in the spirit of Claim 4.7 of PP,shows that we can find some feasible set {z, w} and a preference profile it that satisfy zi'iwfor each i E L(x, B. R) and wi3iz for each i E N — L(x, B, R) such that the social probabilityof choosing x from B, F(x, B, R), is at least as large as that of choosing z from {z, w},F(z,{z,w},14.).Proposition 2.2 : Let 1 > 2, and suppose F E .7" is a regular SSCF that satisfies WP andIIA. If x E B E X and R E RN are such that B X and (L(x , B, R), RIB) = {x}, thenF(x, B, R) > a for some a E aF(L(x, B ,R)) •Proof. Suppose x E B E X and R ERN are such that B 0 X and [3(L(x, B. R), RIB)= {x}. Then, as L(x, B, R) = N and WP imply F(x, B, R) = 1 and aF(N) {1}, theproposition is trivially true if L(x, B, R) = N.Suppose L(x, B, R) 0 N. Let y E X —B and B = BU{y}. Also. let ft E RN be such thatRIB = RIB, G2( 11i1E) = {y} V i E L(x. B, R) and Gi(-1?‘ ilf3)^{y} V i E N — L(x, B, R).By WP, F(z, B , R) = 0 V z E f3 — {x,y}. Then, by Lemma 2.1, F(x, B, R) = F(x, {x, y }, ft)E aF(L(x, B ,R)). Regularity and IIA also imply F(x, B, R) < F(x,^= F(x, B, R).Hence, F(x, B, R) E a F(L(x. B, R)) and F(x, kit) < F(x,B,R). II64Note that the conclusion in Proposition 2.2 can be strengthened if 1 > 3. In lightof Proposition 2.1, when 1 > 3, the conclusion of Proposition 2.2 becomes F(x, B, R) >aF(L(x, B, R)).We are now ready to formally state our desired result in the form of Proposition 2.3,which is similar to Claim 4 8 of PP. This proposition shows that, if there are at least threesocial alternatives and F E T is a regular SSCF that satisfies WP and IIA, then aF is asubadditive function, which is additive whenever there are four or more social alternatives.Proposition 2.3 : Let F E^be a regular SSCF that satisfies WP and IIA,and supposeE Al are disjoint. If 1 > 3, then aF(S)+aF(S) > aF(SUS) . If 1 > 4, then aF(S)+aF( ,-)= aF(S UProof Let S, S E Al be disjoint.Suppose 1 > 3. Clearly, the proposition is trivially true if S U S = N. So we supposeS U S N. Let x, y, z E X be distinct. Also, consider R E RN such that xPiyPiz V i E S,yPizPix V i E S, and zPixPiy Vi E N — (S U S). Then we have the following:aF(S) + aF(S) F(x, {x, z}, R) + F(y, {x, y}. R)> F(x, {x, y, z}. R) + F(y, , {x, y, z} ,R)^[by regularity]1 — F(z, {x, y, z} ,R)> 1 — F(z, {y,z},R)^ [by regularity]1 — aF(N — (S U ,§))= aF(S U^ [by Corollary 2.1].Now, suppose 1 > 4. Then we have {x, y, z} X • L(x, {x, y, z}, R) = S, L(y, {x. y, z}, R)65= S, L(z, {x, y, z} ,R) = N — (SUS), 13(S,RI{x,y, z}) = {x}, [3(T,RI{x,y, z}) = {y} and/3(N — (S U^z}) = {z} . So, using Proposition 2.2 and regularity, we getF(x, {x, y, z}, R)^aF(S) = F(x. {x, z}, R) > F( ,{x,y, z}, R).Hence, F(x, {x, y, z}, R) = aF(S). Then, using a similar argument, we can also concludethat F(y, { x,^= aF(S) and F(z, {x, y, z} ,R) = aF(N — (S U ,§)). Therefore,aF(S) + aF(S) = F(x, {x, y, z} ,R) + F(y, {x, y, z}, R)1 — F(z,{x,y, z} ,R)1 — cep(N — (S UaF(S U^ [by Corollary 2.1].^IIThus, given an universal set with at least four social alternatives and any regular SSCFF that satisfies WP and IIA, it follows from Proposition 2.3 that aF(S) = E E S aF ({i} )V S E Al. Further, if {St}L i is a partition of the societyl°, then aF(St ) is a nonnegativereal number for each coalition St in the partition {St}L, and the sum of aF(S t ) over allcoalitions St in the partition {St}it=i is equal to one. So, if the universal set has at least foursocial alternatives and F is a regular SSCF that satisfies WP and IIA, for each coalition ofindividuals S E Ar, we refer to aF(S) as the weight of the coalition S according to F. It isworth pointing out that the preference profile R used in the proof of Proposition 2.3 is thewell known Condorcet paradox profile, which also figures in the proof of Arrow's impossibilitytheorem.We conclude this section with the following corollary, which implies the weighted randomdictatorship result of PP (Theorem 4.11).1° So {S' }i =1 is such that S' CNVt=1.....,i. S 1 U U S t = N and s'' n sf " = 0 for 1 < < t" <66Corollary 2.2 : Suppose 1 > 4. Let F E .F be a regular SSCF that satisfies WP and IIA.If B E X and R E R,N are such that B X and IG (RilB)I = 1 V i E N, then F(x, B ,R)= aF(L(x, B, R)) V x E B.Proof. Suppose B E X and R ERN are such that B X and IGi(RilB)1 = 1 V i E N.Then it is obvious that P(L(x, B, R), RIB) = {x} V x E p(N,RIB). So, using Propositions2.1 and 2.2, we getF(x, B ,R) > aF(L(x, B, R))^V x E )3(N, RIB).^ (2.6)Clearly, {L(x, B ,R) }a.€13(N,RIB) is a partition of N. Therefore, Proposition 2.3 implies thatE^ceF(L(x,B,R)) = aF(N) = 1.xo(N,Ris)Then, because of (2.6) and the last equation, we get1 ?_^E^F(x, B, R)^E^aF(L(x,B,R))^1.^(2.7)xo(N.RIB) rEo(N,R1B)Now, (2.6) in conjunction with (2.7) implyF(x, B, R) = aF(L(x, B, R))^V x E /3(N. RIB).^ (2.8)Then, using (2.7) and (2.8), it is easy to check thatF(x. B, R) = 0 = aF (0) = aF(L(x, B, R))^V x E B — /9(N, RIB).2.4 The Structure of SSCFsSuppose there are at least four alternatives in the universal set X. Let B be a feasibleproper subset of X, and let R be a preference profile. If the individual preferences inR are strict, then {L(x, B, R) },.0(N,R IB) defines a partition of the society that satisfies67O(L(x, B. R), RIB) = {x} for each x E 13(N, RIB). Using this partition, one can derive theweighted random dictatorship result in PP from the structure of coalitional power for two-element feasible sets. But such a partition of the society may no longer exist once the individ-ual preferences in R are not restricted to be strict. However, when individual preferences inR are not necessarily strict, using two key properties of the partition {L(x, B,R)}, E0( N,RIB)when individual preferences in R are strict, we can generate a partition of the society whichcan be used to derive the structure of coalitional power for nonbinary choice from the restric-tions for two-element feasible sets derived in the previous section. These two key propertiesof the partition {L(x, B,R)},E0 ( N ,RI B) when individual preferences in R are strict are: (i)13(L(x, B, R), RIB) n [3(N — L(x, B, R), RIB) = 0 for every x E I3(N ,RIB); and (ii) for eachx E I3(N, RIB), if L(x, B, R) has more than one member and i E L(x, B, R), then there issome other member j E L(x, B, R) such that Gi(R,IB) fl Gi(R) IB) 0 0.Thus, given X , B and R as above, let S E Al be a coalition in the partition of the societywhich satisfies the above mentioned two properties; i.e. S is such that: (i) 0(8, RIB) fl/3(N — S, RIB) = 0; and (ii) if 1S1 > 2 and i E S, then there is j E S — {i} such thatGi(R,IB) fl Gi(Ri IB) 0. Hence, there is nothing in common between the best sets in Bof any two individuals according to their preferences in the profile R if only one of thembelongs to the coalition S. Also, if the coalition S has more than one member, then everyindividual in S has some best feasible alternative according to her preference in R which isalso a best feasible alternative according to the preference in R of some other individual inS. Our main result shows that given any regular SSCF F that satisfies WP and IIA, for Ssatisfying the above properties, the social probability of choosing one of the alternatives fromthe set 0(S, RIB) when B is the feasible set and R is the social preference profile is equal to68the weight of the coalition S according to F; i.e. L',.0(s ,RiB) F(x, B, R) = aF(S).The steps we use to prove our result are as follows: (i) we first establish our result when1 = 4; (ii) the next crucial step shows that, for any feasible proper subset of X and anypreference profile, the social probability of choosing an alternative which is a best feasiblealternative for no one in the society is zero; and (iii) finally, using an induction argument, weprove our result for the more general 1 > 4 case.Given any B E X and any R E 7Z,N , let P(B, R) be the unique partition of N such that il :(P1) if S, S E P(B, R) and S S, then 13(S.RIB)C) /3(S, RIB) = 0; and(P2) if i E SE P(B, R) and IS! > 2, then Gi(R,IB) fl Gi(Ri IB) 0 for some j E S — {i}.Thus, P(B, R) is the partition of the society which satisfies the two important propertiesdiscussed above.The first proposition in this section is our result on the structure of coalitional power,which we stated above, for the case 1 = 4. As with most impossibility results, the proofexploits the fact that, given any feasible proper subset B of the universal set X and apreference profile R, there are sufficient degrees of freedom to choose another preferenceprofile R such that both R and R are the same on B and R also has certain desirableproperties on some other subsets of X. The new profile R is then used to derive somerestrictions on the social choice probabilities for alternatives, corresponding to some feasiblesets distinct from B. Then, by invoking IIA and regularity, which is a collective rationalitycondition, these restictions are shown to imply the desired restictions when B is the feasibleset and R is the preference profile.'It can be easily checked that P(B, R) is uniquely determined by (P1) and (P2).69Proposition 2.4 : Suppose 1 = 4. Let F E be a regular SSCF that satisfies WP and IIA.If B E X but B 0 X , and R E -R,N , then 7xE.3(S.R1B) F(x, B. R) = a F(S) VS EP (B , R).z-,Proof Let B E X but B 0 X, and R E R.N . Then we have the following exhaustive list ofpossibilities: (a) 10(5, RIB)! = 1V S EP(B,R); (b) IBI = 2 and I/3(S, RIB)I = 2 for someS e P(B,R); (c) = 3, INN, RIB)I = 2 and 1,3(S. RIB)I = 2 for some S E P(B,R): and(d) IBI = 3 and /3(N, RIB) B.(a) 10(S, RIB)! = 1 V S E P(B,R): Corollary 2.2 immediately implies the propositionin this case.(b) IBI = 2 and INS, RIB) I = 2 for some S E P(B,R): Then it is clear that P(B, R){N}. So the proposition holds in this case.(c) IBI = 3, 10(N,RIB)I = 2 and 10(S, RIB)! = 2 for some S E P(B,R): Clearly,P(B, R) = {N} in this case. Let /3(N, RIB) = {x, y}, B — /3(N, RIB) = {z} and X — B ={w}. Also, let {Ni, N2, N3} be the partition of N such that Gi(RilB) = {x, y} V i E N1,Gi(RilB) {x} V E N2 and Gi(RilB) = {Y} V E N3 . If N2 = 0 or N3 = 0 it canbe easily checked that z W PAR(B,R), and hence, the proposition obviously follows fromWP. So let us suppose N2 0 0 and N3 0 0. Now, consider R E "RN such that RIB = RIB,G2(RilX)={w}Vi EN1 UN2 i and vFw if v E B andi E N3. Clearly, as xi'iw V i E N,w WPAR(X,it ). So WP implies F(w,X.1i) = 0. Let {y,z,w}. Then it can beeasily verified that Gi(.hill3) = {y} V i E N1 U N3 and Gi(i,j.ti) = {w} V i E N2. Thus,we have L(y. E. U L(w, i3, R) = N, /3(L(y, b,k),R.,113) = {y} and /3(L(w, ={w } . Hence, using Corollaries 2.1 and 2.2, we getF(y, B. R) + F(w.I3,^aF(L(y.B,A)) + aF(L(w,E,i)) = 1,70which implies F(z,B,ft) = 0. Then, because of regularity, F(z,X,ft) = 0. But we alsoknow that F(w, X, ft) = 0. So we have F(x,X,ft) + F(y, X, ft) = 1. Then, using regularityonce more, we get F(x,B,ft) + F(y, B, R) = 1. Therefore, as RIB = RIB , IIA impliesF(x,B,R) + F(y, B, R) 1. Hence, EvE,f3(N,RIB) F(v, B, R) = 1 = aF(N).(d) 1BI = 3 and /3(N, RIB) B: If P(B, R) {N}, then the proposition is obvious.So we suppose P(B, R) {N}. As the proposition is true in case of possibility (a), we onlyneed to consider the case where 10(S,RIB)I = 2 for some S E P(B, R). Then it can beeasily checked that P(B, R) {S, N — S}, = 2 and I/3(N — S, RIB) = 1. So let/3(5, RIB) = {x, y} and )3(N — S,RIB) {z} . Then, because of Corollary 2.1, it is sufficientto show that F(z, B, R) = aF(N — S). However, as Proposition 2.2 implies F(z,B,R) >aF(N — S), the proof is complete if we show that F(z, B, R) < aF(N — S). Now, as before,let {w} = X — B. Also, let R E RN be such that RIB = RIB, G2(kIX) = {w} V i E Nsuch that Gi(R,IB) fl {x} (i, and xijiw and y/3iw V i E N such that Gi(R,I.B) fl {x} = 0.Clearly, as xPiw V i E N, w W PAR(X,A). So WP implies F(w, X, R) = 0. Next, let B ={y, z, w}. Then it can be verified that IG1(// i 1E)I = 1 V i E N. and L(z, /3, ft) N — S. So,using regularity and Corollary 2.2, we get F(z, X ,ft) < F(z. = aF(N — S). Therefore,as we already know that F(w, X, = 0, Corollary 2.1 impliesF(x, X, ft) + F(y, X, R) = 1 — F(z, X, ft) > 1 — aF(N — S) = aF(S).Then, because of regularity. we get F(x,B,ft) + F(y, B ,ft) > aF(S). So, as RIB = RIB,IIA implies F(x. B, R) F(Y,B,It) > aF(S). Hence, using Corollary 2.1 once more, we getF(z, B, R) < 1 — aF(S) = aF(N — S).This completes the proof of Proposition 2.4.^II71Given any F E .F that satisfies IIA and any B E X, let FIB : B x (2B —^x RN 113be the restriction of F to the set of all nonempty subsets of B, 2B — {O}; i.e. for anyE 2 B - {O} and any itIB EFI B (x, E,itIB) F(x, B, R) VxEB and any R E R,N such that 14.1/3 = RIB.It can be easily checked that, if F E T satisfies regularity, WP and IIA on X x X x RN, thenFIB also satisfies regularity, WP and IIA on B x (2B — {0}) x R, N AB for any B E X. Thisobservation is important for the proof of our next proposition and the main result.The final result required for the proof of the main result is Proposition 2.5. As statedabove, it shows that an alternative from a feasible proper subset of the universal set mustbe assigned zero social choice probability if it is not a best feasible alternative for anyone.The proof uses a simple induction logic which shows that, if the proposition is true whenthe universal set has t > 4 alternatives, then it is also true when the number of alternativesin the universal set increases by one to t 1. Depending on whether the feasible set B hastwo or more fewer alternatives, or exactly one fewer alternative than the universal set, theproof has two parts. In the first case, we consider the restriction of the SSCF to a supersetof B with exactly one alternative less than in the universal set and exploit the observationmade above about the properties it inherits from the original SSCF. In the later case, as inthe proof of Proposition 2.4, we rely on the amount of freedom available to choose a newpreference profile with certain desirable properties, and then appeal to IIA and regularity.Proposition 2.5 : Suppose 1 > 4. Let F E ,F be a regular SSCF that satisfies WP and IIA.If B E X but B X, andR E RN , then F(x, B, R) =0VxE B-0(N,111B).Proof. Because of Proposition 2.4. the proposition is obviously true for 1 = 4. So, if we show72that the proposition holds for 1 = t +1 whenever it holds for 1 = t > 4 (t < oo), then, usingan induction argument, the proof of the proposition is complete. Thus, we only need to showthat the proposition also holds for 1 = t + 1 if it holds for 1= t > 4 (t < Do).Suppose the proposition holds for 1 = t > 4 (t < oc). Consider X such that IX I = 1= t +1. Let B E X but B 0 X, and R E R. Then we look at two possibilities: (a)IBI < t — 1; and (b) t.(a) IBI < t — 1: Clearly, there exists B° E X such that IB°I = t and B E 2B° — {0}.Now, consider FIB°, the restriction of F to 2B ° — {(4}. As the proposition is true for 1= t, itmust be the case that FIB° (v, B,RIB°) = 0 V v E B — d(N. RIB). Hence, F(v, B, R) = 0 Vv E B — 0(N, RIB).(b) IBI = t: Let x* E B — [3(N, RIB). Also let {w} = X — B. Then we need to showF(x*,B,R) = 0. If IGi(RiI/3)I = 1 V i E N, then, because of Corollary 2.2, the propositionis true. So we suppose Pi (RilB)I > 2 for some j E N. Then we distinguish between twopossibilities: (bl) IGi(Ril.B)1 > 3 for some j E N; and (b2) IGi(RilB)1 <2ViEN andIG1(RhIB)I = 2 for some h E N.(bl) IGi(RilB)1 > 3 for some j E N: Let x y E Gi(R3 1B) be distinct, and let I)= (B — {x*,^U {w}, where i E Gi(RAB) is different from i and y. Also, define 13'X - 041. Now, consider it E RN such that itIB = RIB. G2(i?',IX) = {w} if i E Nis such that 0,0 fl Gi(RilB)^0, and vPiw Vv E B if i E N is such that 041 flGi(RilB) = 0. Then it is easily verified that LEI = t — 1, IBI = t — 1, w E B - p(N, RIB)and x* E B - /3(N. RIB). So, using the conclusion drawn in possibility (a) above, weget F(w, B, R) = 0 and F(x*,.1j,lit) = 0. Then regularity implies F(w, X, R) = 0 andF(x*. X, R) = 0. Therefore. E i , Ex _ {^F(v, X, R) = 1. But, clearly, X — {x*, w} =73B — lel CB C X. So, using regularity, we get E t,„_ { ,,. } F(v,B,it) = 1. Then, asRIB = RIB, E vEB---{ x.} F(v B R) = 1 follows from IIA. Hence, F(x*, B, R) = 0.,^, (b2)^(R,IB)1 < 2 V i EN and Pi (Rh IB)I = 2 for some h E N: Here, we look attwo mutually exclusive cases: (b2.1) I13(N ,RIB)I < 2; and (b2.2) AN,RIB)1> 2.(b2.1) I/3(N, RIB)I < 2: Let B = 13(N ,RIB) U lx*I. Clearly, IBI < 3 < t — 1 and{x*} = B - /3(N, RIB). So, using the conclusion drawn in possibility (a) above, we haveF(x*,B,R) = 0. Therefore, F(x*,B,R) = 0 follows from regularity.(b2.2) IP(N, RIB)I > 2: Let -0, = Gi(RhIB) and z E [3(N, RIB) — 0,0. Also, let= x -{x*, and B = X-0,0. Now, consider ft E RN such that RIB =RIB G2 (Ili IX)= {w} if i E N is such that 0, fl Gi(RilB) 0, and vi'm Vv E B if i E N is such that{ (RilB) = 0. Then it is easily verified that 113I = t = w E - 13 (N, f1 1 1-3 )and x* E B — ,13(N,RIB). So we can use the conclusion drawn in possibility (a) once moreand get F(w, B , R) = 0 and F(x*, B, R) = 0. Then regularity implies F(w, X, R) = 0 andF(x* , X, R) = 0. Therefore, E t,Ex_ {x*,„, } F(v, X, R) = 1. However, X — {x*, w} = B — {x*}C B C X. So, using regularity once more, we have E, EB_ Ix * I F(v, B, R) = 1. Then, asRIB = RIB , 7vEB—{x*} F(v, B, R) = 1 follows from IIA. Hence, F(x* , B, R) = 0.—Thus, we have shown that the proposition holds for 1 = t 1 if it holds for 1 = t > 4(t < oc). This completes the proof of Proposition 2.5.^IIWe are now ready to present our characterization of the structure of coalitional powerwhen the universal set has four or more alternatives. This theorem states that the distributionof power under a SSCF F. which satisfies regularity, WP and IIA, is such that, given a feasibleproper subset B of the universal set and a preference profile R, for each coalition S in the74partition P(B, R), the social probability of choosing an alternative that is a best feasiblealternative for some member of S is equal to the weight of the coalition S according to theSSCF F. The proof uses an overall strategy which closely resembles that of the proof ofProposition 2.5; i.e. it is based on induction and has two parts, one for B with two or morefewer alternatives than in X and another for B with exactly one fewer alternative than in X.Theorem: Suppose 1 > 4. Let F E^be a regular SSCF that satisfies WP and IIA. IfB E X but B 0 X , and R E RN , then ExE,3(s,RIB) F(x, B, R) = a F(S) V S E P(B, R).Proof. We already proved the theorem for 1 = 4 in Proposition 2.4. So, if we prove thetheorem for 1 = t +1 whenever it holds for 1 = t > 4 (t < oo), an induction argumentcompletes the proof of the theorem for any finite 1> 4. Thus, it is sufficient to show that thetheorem also holds for 1 = t +1 if it holds for 1 = t > 4 (t < oo).Suppose the theorem holds for 1 = t > 4 (t < oo). Consider X such that 12(1 = 1 = t +1.Let B E X but B 0 X, and R E RN . Then we have two possibilities: (a) < t — 1; and(b) IBI = t.(a) IB1^t — 1: Clearly, there exists B° E X such that IB°1 = t and B E 2 B° — 91.Then, as the theorem is true for 1 = t, it must be the case thatE^FIB. (v, B, RIB')^afb3„(S) VSEP(B ,R1B°).vel3(s,R1B)However, it can be easily checked that a F I B ,, (S) = aF(S) VS C N and P(B,R113°) =P(B,R). Therefore, 7 vE3(s,R4B) F(v,B,R) = aF(S) V S E P(B,R).(b) IBS = t: Depending on the cardinality of ,13(N, RIB), we have two mutually exclusivecases to look at: (b1) I/3(N, RIB)! < t: and (b2) 10(N. RIB)! = t.(bl) 1,13(N,R1B)1 < t: In this case it is clear that P(B,R) = P(O(N,RIB),R), and75/3(S, RIB) -= ,(3(S,R1/3(N ,RIB)) V S E P(B ,R). Then, using 1,3(N, RIB) I < t and theconclusion drawn in possibility (a) above, we haveE^F(v,p(N,R1B), R) = aF(S)^V S E P(B, R).^(2.9)vo(s,RIB)Because of Proposition 2.5, PO S(F, B, R) C^RIB). So, using Lemma 2.1, we also haveF(v, , /3(N, RIB),^= F(v, B, R)^VvEfl(N, RIB).^ (2.10)Then it is obvious from (2.9) and (2.10) that 7^F( v, B,^= aF(S) V S E P(B, R).(b2) 10(N, RIB)I = t (i.e. 0(N ,RIB) = B): Here, we suppose IP(B, R) I > 2, because,if P(B, R) = {N}, then Evo(N,R1B) F(v, B, R) = 1 = a F(N). Pick any ,§ E P(B, R). Let{w} = X — B, B = x - 0(& RIB), and B = X - 13(N — S, RIB). Now, consider R. E RNsuch that RIB = RIB, G2(-hi IX) = {w} if i E N — S , and vi',w Vv E B if i E k Then itis obvious that B o X and w E B - )3(N,EIE). So Proposition 2.5 implies F(w, = 0.Therefore, using regularity, we getF(w, X ,A) = 0.^ (2.11)It can also be easily verified that^X, L(w, B ,ft) N — S, and 0(N — S,EIB) = {w} .Then Propositions 2.1 and 2.2 imply F(w,E,it)> ap(N - ,§). So, because of Corollary 2.1,EvEb_ { ,,, } F(v, B, it.) _< aF(S) must be true. But it can be easily verified that E - {w} =Therefore, EvE ,_ .RIB) F(v, B . ft)^aF(S). Then regularity impliesE^F(v,x,it) < aF(S).^ (2.12)vE3(.§,RIB)Now, using (2.11), (2.12) and B - {w} = 0(,§,R113), we have EueE F(v, X, it) < aF(S).Then, because of Corollary 2.1 and X — B = [3(N — S,RIB), A LE3(N-&R1B) F(v, X, ft) ?_76aF(N — must hold. Therefore, Evo(N_ s;,RIB) F(v,^> aF(N — Th must also be truebecause of regularity. However, as RIB = RIB, IIA implies F(v,B,A) = F(v,B,R) V v E B.Thus,E^F(v, B, R) > aF(N —^ (2.13)vE,(3(N-.§,11.1/3)We know that 4§, RIB) = B — /3(N —^So EvE," ,RIB) F(v, B, R) aF(S) followsfrom (2.13) and Corollary 2.1. But S was arbitrarily chosen from P(B, R). Therefore, wecan conclude thatE^F(v, B. R) < ap(S)^V S E P(B, R).^ (2.14)vE0(S,RIB)As P(B, R) is a partition of N and aF(N) = 1, Esep ( B,R ) aF(S) = 1 easily follows fromProposition 2.3. Then, using (2.14), we get1 = E F(v,B,R)^E^> F(v,B,R) < E aF(S) = 1.vEB^SEP(B,R) vE0(S ,RIB)^S EP(B ,R)Hence, the inequality in (2.14) cannot be strict for any S E P(B, R).Thus, we have shown that the theorem holds for 1 = t +1 if it holds for 1 = t > 4 (t < oo).This completes the proof of the theorem.^IIGiven any S E .N', let= {F e^Exo(s,RIB) F(x, B. R) = 1 V (B,R) E X x RI .Ys can be interpreted as the class of all SSCFs that give oligarchic power to the coalition Sin the following sense - given any feasible set B and any preference profile R, an alternativewhich is not best in B for anyone in the coalition S according to their preferences in R haszero probability of being the socially chosen feasible alternative. Clearly, for each i E N,77is the appropriate stochastic counterpart to the class of deterministic social choice functionsin which individual i is a weak dictator 12 .The next two straightforward corollaries of our main result essentially show that, underthe conditions of the Theorem, a SSCF can be interpreted as a weighted random oligarchyor as a weighted random dictatorship. As noted earlier, the weighted random dictatorshipresult of PP (Theorem 4.11) is implied by our Corollary 2.2. It can also be viewed as a specialcase of Corollary 2.4 given below.Corollary 2.3 : Suppose 1 > 4, and F E F is a regular SSCF that satisfies WP and IIA.Let B E X but B X, and R E RN . If {ds}sEP(B.R) is a set of SSCFs such that ds E .Tfor each S E P(B, R) thenE F(x,B,R) =^E^E aF(s)ds (x,B,R) V S E P(B.R).xE,3(g,RIB) seP(B,R)Proof. Suppose B E X but B X, and R E RN . Let -ids IsEP(B,R) be such that ds E •FS'for each S E P (B , R). Also, let S E P(B ,R). Then we haveE^E aF(s)ds(x,B,R).^aF(S)^E ds( ,B,R) . (2.15)^xE13(g.RIB) SEP(B,R)SEP(B,R)^rE3(g,RIB)By definition,^RIB) n 13 ( , RIB) = 0 and ETE/3( ,,,RIB) d,Ox,B,R) = 1 V S E P(B, R) —So ExE , R1B) (1,.§(x,B,R) = 0 V S E P(B,R) — {S}. Hence, (2.15) can be rewrittenasE^E aF (s ) ds (x,B,R ) = crF(:5)^E^,/,(x,B,R) aF(:5'). (2.16)xE3(§,R1B) SEP(B,R)^ TE3(g.RIB)However, because of the Theorem, (2.16) is all that we need to show.12 When we restrict individual preference relations to the set of all linear orderings on X, .F{ i ) is equivalentto the class of decision schemes in which individual i is a dictator, as defined in PP.78Corollary 2.4 : Suppose 1 > 4, and F E T is a regular SSCF that satisfies WP and IIA.Let B E X but B 0 X, and R E R,N . If {di }i E N is a set of SSCFs such that di E Tfo. foreach i E N, thenE F(x,B,R) =^>2^E aF({i})di(x,B,R) V S E P(B,R).xE/3(S,R111)^ xE23(S,RIB) iE NProof. Suppose B E X but B 0 X, and R E RN . Let Idili E N be such that di E .Ffil foreach i E N. Also, let S E P(B, R). Then we haveE^E aFeciDdi(x,B,R) = >2 [aF({i})^>2^di (x, B,R).^(2.17)xEi3(S,RIB) iEN^ iEN^xE3(S,RIB)But it is obvious that ExE/3(s,RIB) di (x, B, R)^1 V i E S and Ex s,RIB) di , B, R) = 0ViEN— S. So (2.17) can be rewritten asE^E aF({i})di(x,B,R)^aF({i}) = aF(S),^ (2.18)xe*S ,R1B) iEN^ iESwhere the last equality follows from Proposition 2.3. However, because of the theorem, (2.18)^is sufficient for the corollary to hold.^IISuppose one of the following two conditions are satisfied in addition to those specified inour Theorem: (i) the preference profile R is such that there are at least two alternatives thatare not best in the universal set X for anyone according to their preferences in R; or (ii) thenumber of alternatives in the universal set exceeds the number of individuals in the societyby at least two, and the preference profile R is such that there is at least one alternativewhich is not best in the universal set X for anyone according to their preferences in R. Thena simple consequence of regularity is that the conclusion of Proposition 2.5 remains valideven when the feasible set B is the universal set X. Therefore, when one of the above two79additional conditions is satisfied, Lemma 2.1 allows us to consider the universal set as thefeasible set in our Theorem. This extended result is formally stated as our last proposition.Proposition 2.6 : Suppose 1 > 4. Let F E .F be a regular SSCF that satisfies WP and IIA,and let R ERN . If (i) I X — [3(N ,RIX)I > 2, or (ii) 1 > n + 2 and IX — )3(N , RIX)! > 1,then 7L..0.0(s,Rix) F(x, X, R) = aF(S) V S E P(X, R).Proof: Suppose R E RAT is such that: (i) IX — /3(N, RIX) > 2, or (ii) 1 > n + 2 andIX —0(N, RIX)I > 1. We first show that F(x, X, R) =0VxEX —0(N, RIX) in both cases.(i) IX— /3(N,RIX)I _> 2: Let z E X —13(N, RIX) and B = 13(N,RIX)U{z}. Clearly, {z}= B — 13(N, RIB). By Proposition 2.5, we then have F(z, B, R) = 0. Therefore, regularityimplies F(z,X,R) = 0.(ii) 1 > n + 2 and IX — /3(N. RIX) I > 1: As IX — /3(N, RIX)I > 2 has already beenconsidered, we only look at IX — (3(N,RIX)I = 1. Suppose, for each x E )3(N, RIX), thereexists i E N such that Gi(R,IX) = {x}. Then n > 10(N, RIX)I. which implies 1 > n + 2 >n+1 > If3(N • RIX)I +1 = /, an impossibility. Hence. there is an alternative in 0(N, RIX), sayw, such that {w} Gi(R7 IX) for every i E N. Let X — 13(N.RIX) = {y} and 13 X - {w}.Then y E B — /3(N, RIB). So F(y, B, R) = 0 follows from Proposition 2.5. Therefore, byregularity, F(y, X, R) = 0.Thus, the following holds in any case:POS(F. X, R) C /3(N, RIX) C X.^ (2.19)Then, because of Lemma 2.1. (2.19) gives usF(x. /3(N, RIX), R) = F(x, X. R)^V x E 13(N,RIX).^ (2.20)80Obviously, P (13(N ,RiX), R) = P(X, R) and /3(S, RI/3(N. RIX)) = 13(S, RIX) V S E P(X, R).Hence, (2.20) in conjunction with the Theorem implyE F(x, X. R) =^E^F( ,O(/‘L RIX), = aF(S) V S E P(X, R).x€0(s,R1x) TE3(S,RI,3(N,RIX))In Proposition 2.6, the two additional conditions are independent of each other, becauseneither necessarily implies the other. However, when individual preference orderings arerestricted to be strict, as 1> n+2 implies IX-0(N, RIX)I > 2, the second condition is strongerthan the first. Thus, although PP's additional condition for their extended result (Theorem4.14) is 1 > n + 2, our Proposition 2.6 shows that their result is actually a consequence of aweaker condition on the preference profile, namely, IX — 0(N,RIX)1> 2.Needless to say, using Proposition 2.6, it is easy to show that Corollaries 2.3 and 2.4 alsohold when we consider the universal set X as the feasible set provided at least one of the pairof additional conditions specified in Proposition 2.6 is satisfied.In this chapter, although we derived restrictions on the distribution of coalitional powerfor SSCFs, it is clear that the results presented here do not fully characterize the structure ofcoalitional power under SSCFs. So how close to a complete characterization are our results?We offer the following simple answer to this question. The examples in PP can be extendedconsistently to our expanded preference domain in a straightforward manner to show that ourTheorem may no longer hold when any one of its conditions is violated, and also, Proposition2.6 may not be true when both additional conditions specified in it are dropped. Thus,in a way, the results of this chapter can be conceived as representing an almost completecharacterization of the structure of SSCFs in our framework.81ConclusionIn the first chapter, we looked at a simple moral hazard problem in a principal-agent(s)framework. However, unlike most existing work, our principal was not precluded from activeparticipation in the production process. Also, unlike the single agent case, there was nouncertainty and the moral hazard problem was caused by joint production. A simple multi-stage extensive game, the SBG, determined the set of individuals who actually took part inproduction along with the output sharing rule they followed.Although the principal was not precluded from participation in the production process,whether it was optimal for her to participate or not depended on the values of W(TF ) andV (T*). In particular, it was best for her to participate only if V(T*) > W(TF ).Whenever the principal did not participate in production, moral hazard was completelymitigated although there was potential for moral hazard if two or more agents participatedin production. On the otherhand, it was impossible to mitigate the moral hazard problemcompletely if the principal participated in production along with at least one agent. Thesefindings, as we argued, depend on the deterministic production process and not on the quasi-linear utility functions. In contrast, unless there is risk neutrality, moral hazard cannot becompletely mitigated in most principal-agent moral hazard problems with uncertainty.82From the above remarks we can also draw some other interesting conclusions. Firstly,although the principal could completely mitigate moral hazard by not participating in pro-duction, it is quite conceivable that she could be better off introducing moral hazard byparticipating in production. Secondly, even if it was optimal for the principal to completelymitigate moral hazard by not participating in production, she might still be worse off thanat her best option when actions are observable, because her best option when actions areobservable might require her participation in the production process.Except when agents are risk neutral, in most standard principal-agent moral hazardmodels with nondeterministic production processes, it is the norm rather than the exceptionthat the principal has to look for output sharing rules that are much more sophisticatedthan linear or piecewise linear output sharing rules. However, although we did not presentit formally for the more general concave utility case, we showed that the principal need notlook any further than the class of linear output sharing rules (piecewise linear output sharingrules in case of limited liability) if the production process is deterministic.Also, in principal-agent moral hazard models with nondeterministic production processes,the results that are obtained without limited liability may change significantly when thereis limited liability, for example, the principal's optimal utility often decreases when limitedliability is imposed. But we showed that the deterministic production process made most ofour results robust to the introduction of limited liability.In Chapter 2, we explored the issue of distribution of coalitional power when aggregatingthe preferences of the individuals in a society in terms of stochastic social choice functions. Inparticular, we provided a natural extension of the results in Pattanaik and Peleg [29] to thecase in which individuals are permitted to have indifference between alternatives. When there83are at least four elements in the universal set of social alternatives, and the stochastic socialchoice function is regular, weakly Paretian ex-post and satisfies independence of irrelevantalternatives, as in Pattanaik and Peleg [29], there is a unique nonnegative weight associatedwith each coalition of individuals in the society. For each social preference profile, whenthe above mentioned conditions hold and the feasible set is a proper subset of the universalof social alternatives, we showed that the society could be partitioned into coalitions ofindividuals in such a way that the sum of the social probabilities of all the alternatives in theunion of the best sets of the members of each coalition is equal to the weight of the coalition.When the universal set of alternatives itself is the feasible set, we showed that our result stillholds provided the social preference profile is such that there are at least two alternatives thatare not best in the universal set for anyone, or the number of alternatives in the universal setexceeds the number of individuals in the society by at least two and the preference profile issuch that there is at least one alternative which is not best in the universal set for anyone.84References[1] Alchian A. and Demsetz H. 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Then let € > 0 be such that pi — c— ci(ai) >and define 13T E IT as^piViET—{j} and fij = pi — c. Clearly, (15T, aT) is suchthat pi — ci(ai) > ui V i E T and u rn + f IT(aT) EiET > W(T), a contradiction to thedefinition of W(T). Hence, (i) of Lemma 1.1 holds.(i) of Lemma 1.1 implies that u rn + f 1T(aT) EiET q(ai) EiET ui = w(T)• Now,suppose there exists aT E AT such that um + f IT(aT) — EiET^— EiET ui > um. +fIT(aT)— EiET ci (ai) EiET ui. Let 13± E pIT I be such that Pi = ui +^V i E T. Then itis obvious that (fiT, aT) satisfies Pi — ci (ai) = ui V i E T and 'um -F.PT(aT) EiET pi > W(T),which contradicts the definition of W(T). Thus, aT solves the problemmax [u rn + f IT(4) EiET^EiET 1-td•&T EATBecause of the limiting properties of the derivatives of ci and f given in assumptions Al andA2, the above maximization problem can only have interior solutions. So (ii) of Lemma 1.1must hold. Also, as an interior solution of the above maximization problem, al, must satisfythe first order conditions, filT(aT) — =0V i E T, which are exactly the conditions in88(iii) of Lemma 1.1.^IIProof of Lemma 1.2: Similar to the proof of Lemma 1.1. 11Proof of Proposition 1.1: (ii) and (iii) of Proposition 1.1 readily follow from (1.4) and(1.5). Also, it is obvious from (1.5) that sr ESViE TF . So, if we show (iv) of Proposition1.1, then we have also shown (i) of Proposition 1.1.The strict convexity of ci. the concavity of f and (1.5) imply that sr (f ITF (aTF)) — ci(ai)is concave in aTF E ATF for each i E TF . Thus, because of (1.5), if EcTF E ATF satisfiesfilTF(aTF) — i) = 0 V i E TF, then aTF E N E({TF , (sr) JETF}) But we already knowfrom Lemma 1.1 that filTF (4F ) — c'i (ar) =0ViE TF . So aTF E N E({TF , (sr),JETF}).Using (1.5) and the limiting properties of the derivatives of ci and f given in assumptionsAl and A2, it is quite obvious that, at any Nash equilibrium of {TF , F),,s, iieTF}, the actionsof all the agents in TF are positive. Thus, in fact, any aTF E ATF is a Nash equilibrium of{TF , (sniETF} if and only if filTF (aTF) — ci (ai )=0ViE TF.Now, consider the following maximization problem:aTF EATFmax [f 1TF (aTF) — iETF ci(ai)]• (A.1)The concavity of f and the strict convexity of ci imply that the objective function of themaximization problem in (A.1) is strictly concave in aTF E ATF . Also, because of the limitingproperties of the derivatives of ci and f given in assumptions Al and A2, the problem in(A.1) can only have interior solutions. Thus, aTF E ATF is a solution of the problem in (A.1)if and only if it satisfies filTF (a.TF)— =0ViE TF, which are the first order conditions.But this immediately implies that aTF E A TE is a Nash equilibrium of {TF, (sf ) . ETF } if andonly if it is also a solution of the problem in (A.1). However, the problem in (A.1) can89have at the most one solution, because we already know that its objective function is strictlyconvave. Therefore, {TF, (sn iETF I can have at the most one Nash equilibrium. Hence,because al,T, E NE({TF .(sr) jETF}), (iv) of Proposition 1.1 must hold as well.Proof of Lemma 1.3: Let T E M , with m E T and IT1 2. Suppose (IT, (s )jET_,,,/,aT)satisfies (C1) and (C2). Let i be any member of T_m . Then the proof of Lemma 1.3 iscompleted in two steps. In the first step we show that s i is continuous at fIT(aT). Thecontinuity of s at f IT (aT) for every j E T_m is then used in the second step to show that Siis differentiable at f IT (aT).Step 1:Because of (i) in the definition of 8, we know that both limbio si(f IT(aT) + b) andlimbo s,(fiT(aT) b) exists (where we use the convention of letting b T 0 and b 0 de-note b approaching 0 through negative and positive values, respectively). Then the proof ofcontinuity of s i at f IT (aT) (> 0) is complete if we show thatlimTo si(f1T(aT) + b) = limsz(fIT(aT) +^= si(fIT(aT))•b10 (A.2)Clearly, constraints (C1) and (C2) imply that, if e E hJ2 and ai + e E Ai, then si(fIT(aT_ i ,ai+e)) — ei(ai + e) < si(f (aT)) ei(ai), and hence, neither of limeio[si(f 1T(aT_i , ai + e)) —ci(ai + e)] or limelo[si(f IT(aT_,, ai + e)) — ci(ai + e)] can be greater than si(f IT(aT)) ci(ai)•Then, because of the continuity of ci , we have111118 i(f1T(aT_,,ai^e)) 5_ Si(fiT(aT)),el0111118 i(fIT(aT_i,ai +^Si(fiT(aT))•and90However, because of the continuity and monotonicity of f we also havelim si(f IT(aT_ i , ai + e))^limsi(fIT(aT) + b),ei0 bi0limsi(fIT(aT_ i , ai +^=et0Thus, the following must be true:limbIO si(f IT(aT) + b) <^IT(aT)),^and^(A.3)lim si(f IT (aT) +^si( f IT (aT)). (A.4)(C1) and (C2) also imply that, if e E J22 and a m + e E A rn , then f IT (aT_ ni , am, + e) —EjET_„, si(fIT(aT_,,,^+ e)) — cm (am +e) f IT(aT) — EjET_, j(f (aT)) Crn (am), andhence, neither of lime -toff IT (aT_ n, ,^— EET_+ e)^si(f IT(aT_m, am + e)) —^(am, + e)] orlimeto[f IT(aT_,,,, am + e) — EjET_,, S j f IT am + e)) — c rri (a, + e)] can be greater thanfiT(aT) — EjET„, sj (fIT(aT )) — cm (am ). Then, because of the continuity of c m and f, wehavelimeio^, am + e)) ?_ >j ET_m si (f IT(aT)), andE j E T- m limeto si (f IT (aT„, a m + e))^EJET_ Tn sj(f 1T(aT ))•However, for each j E T_„2 , because of the continuity and monotonicity of f , we also haveliM j f IT^am + e)) = limsj(fIT(aT)+ b),eTO^ b/0lim sj (fIT(aT, arn + e)) = lim si(f IT( T) + b).b10Hence, the following must hold:EjET„^si(fIT(aT)+ b)^si(fIT(aT)), and^(A.5)E^limbo si(f IT(aT) + b)^EjET_„ si(f IT(aT)). (A.6)andlim si(ilT(aT) + b).b10and91Now, (A.2) readily follows from (A.3)-(A.6). Therefore, si is continuous at f IT (aT) •Step 2 :Let q° = f IT (aT) . Then, as si is continuous at q° > 0, by (i) in the definition of S. thereexists e > 0 sufficiently small such that q° — e > 0 and si is continuous on (q° — e, q° + e).By (ii) in the definition of S, we can find S > 0 such that S < e and si is continuouslydifferentiable on the two intervals (q° — 6. q°) and (q° q° + S). Denote by s the derivativeof si wherever it exists. Then (ii) in the definition of S also ensures the existence of bothlimbto s'i (q° + b) and limbo s'i (q° + b). Therefore, to prove that si is differentiable at q°, it issufficient to show thatlim Wi (q° + b) = lim s'i (q° + b).brO^b10(A.7)As f and ci are continuously differentiable, (C1), (C2) and the continuous differentiabilityof si, on (q° — S, q°) and (q° , q° + 15) immediately imply thatlim di g IT (aTL i , aiCr0ihnsi(fIT(aT , aiet0lei0iM(1 — EjET_„, si (fiT(aT_,„„ am +eihn.(1 — EjET, Si (fIT(aT_ rn .^+ e0.0+ e))f (aT)e)) f (aT)))) frnIT(aT) —))) finIT (aT) ———cm (am )em (a n ,)> 0,< 0,> 0, and< 0.Given (C1), for each j E T_,, the continuity and monotonicity of f also imply thatlim 'f (07_ 3 , a^e))^lim W. ; ( f (aTi m am e))eiO lim^f IT (a7-_, aj^e)) = lim (.f (aT^e)) =el0So, by using (A.12) and (A.13) in (A.8)-(A.11), we getc/ (ai)lim s^ii (q°^b) > ^ > lim (q°^b).bi- () f^(aT)^btOlimbio si,(q° b), and (A.12)b10^(q° + b).^(A.13)and^(A.14)92s', (q° + b)] < ^-RH^<^f„,IT(aT) - 813(q° +^(A.15)Now. (A.7) easily follows from (A.14) and (A.15). Hence, s i is differentiable at q°.Proof of Lemma 1.4: Let T E M. with m E T and iTi > 2. Suppose ({T, (83 ) JET_„z},aT)satisfies (C1) and (C2). Then, because of Lemma 1.3, (C1) and (C2) immediately imply thats',(fIT(aT)) fIIT(aT) - eijaz) = 0 V i E T_,„ and^(A.16)— E jET_ sji (f IT(aT)))f.171 (aT) — cm(arn) = 0.^ (A.17)Also, because of the monotonicity of the cost functions and the production function, and(C1), it is obvious that, for each i E T, c'i (ai)> 0 and filT(aT) > 0. Hence, by some simplealgebraic manipulations of (A.16) and (A.17), we get E iET [c'i (a i )Ifi lT(aT )] = 1. IIProof of Lemma 1.5: Let T E M be such that m E T and Ill > 2. The limiting propertiesof the derivatives of the cost functions and the production function in assumptions Al andA2 imply limilaT ii-÷0EiETVi(ai)/filT(aT)] = 0 and Ern liaTii _ oc E iET [ei (ai)/filT(aT)1 = Do,where 11.11 is the standard Eucleadian norm. Then, because of the continuous differentiabilityand the monotonicity and curvature properties of the cost functions and the productionfunction in assumptions Al and A2, it is easily verified that the feasible set of problem (P-,)is nonempty and compact. As the cost functions and the production function are continuouslydifferentiable, it is obvious that the objective function of problem (/',) is continuous. Hence,the theorem of Weierstrass implies the existence of a solution to problem (/1),).Proof of Lemma 1.6: Let T E M, with Til E T and 171 ?_ 2. Suppose ({T, (s )JET_ rn },aT)satisfies (Cl) and (C2). Then Lemma 1.4 implies that aT is feasible for problem (/:'„). Obvi-ously, because of Lemma 1.5, V(T) is well defined. Hence, V(T) > fIT(aT) - E, ET (al) -93ui. But (C2) implies s j( f IT (aT)) — ci(aj) > fl V j E T—m . Therefore, f IT(aT) —E iET ci(ai) — EiEr—^fIT(aT) EjET_,, Sj(fiT(aT)) — (am ). Thus, it must be thecase that V(T) > f IT (aT) EjET_„, sj(f IT(aT)) — ern (ain)•^IIProof of Proposition 1.2: Suppose assumptions A1-A5 are satisfied and V(T*) > W(TF).(1.7), (1.9) and (1.10) immediately imply (ii) and (iii) of Proposition 1.2. It is obviousfrom (1.10) that s7 ESViE T*,,. So, if we show that a7 >OViE T* and (iv) of Proposition1.2 holds, then, because of Lemma 1.6, the proof of (i) of Proposition 1.2 is also complete.Clearly, (1.6), (1.8) and the monotonicity of the cost functions and the production functionimply that 77 >OVi E Ti` „, and (1—E jET* .ra 71) > 0. Then, because of the strict convexity ofthe cost functions, the concavity of the production function and (1.10), s7 (f IT* (aT.))— ci(ai)is concave in E AT* for each i E T* 9.„, and f IT* (aT*) — s; (f IT* (aT*)) — cm,(a rn ) isconcave in aT. E AT* . Thus, because of (1.10), if^E AT* satisfies -y7 filT*(EtT.)—^= 0V i E^and ( 1 — j ET* m 7;)frnIT* (aT*)^(am) = 0, then aT* E N E({T* , (sP ET* „.,}) •Hence, (1.6) and (1.8) imply 4* E N E({T* (sMET* „,}) •Thus, it immediately follows that ({T* (sj )j ET* n, }, a ) E e2. But suppose ah = 0 forsome h E T*. Then assumptions A4 and A5, and V(T*) > W(TF ) imply that there is some({T, (8 j) ET_,} , aT) E O+n, such that 7(({T, aT)) > V (T*) , > 0 V j E T, andITI > 2. But then, because of Lemma 1.6. we get 71 - (({T, (si)^aT)) > V(T*) .> V(T)> MT. (si)jeT_,n 1, aT)), which is impossible. Therefore, a7 > 0 V i E T*.Having shown that a7 > 0 V i E T*, we can immediately conclude that1 >^>0 V i E^m , and 1 > (1 — J E T* m. 77) >^ (A.18)So, using (1.10), (A.18) and the limiting properties of the derivatives of the cost functions94and the production function given in assumptions Al and A2, it is quite obvious that atany Nash equilibrium of {T*, (spj ET. m } the actions of all the individuals in TX are positive.Thus, in fact any aT* E AT* is a Nash equilibrium of IT* ( 8 ';)jET* n, if and only if '%IT* (aT*) V i E T_* and (1 — 6) = 4, (a^fm IT* (aT*) •-Consider the following maximization problem:max [f IT* (aTaT* EAT*) EJETL%„ Icd (ai)/73 — (cm (am)/(1 — E jET ,, m 11))].^(A.19)The concavity of the production function, the strict convexity of the cost functions and (A.18)imply that the objective function of the maximization problem in (A.19) is strictly concavein E AT* . Also, because of (A.18) and the limiting properties of the derivatives of thecost functions and the production function given in assumptions Al and A2, the problemin (A.19) can only have interior solutions. Thus, eii , * E AT* is a solution of the problem in(A.19) if and only if it satisfies '' = ,)/filT*(etT*) V i E T__* in and (1 EjET*,„, 7) =c'n,(ano f m ir" (aT*), which are the first order conditions. But this immediately implies thataT. E AT* is a Nash equilibrium of {T*, ( ) ET* m } if and only if it is also a solution of theproblem in (A.19). However, the problem in (A.19) can have at the most one solution, becausewe already know that its objective function is strictly convave. Therefore, {T*, (s'AET* 7„ }can have at the most one Nash equilibrium. Hence, as 4* E NE(IT* 1( 8DIET*„,}), (iv) ofProposition 1.2 must hold.Proof of Proposition 1.3: Using (1.11), and (ii) and (iii) of Proposition 1.1, it is quiteobvious that(fITF (aTFF )) — ci(ar) = ui V i E TF, and^(A.20)u rn f 1TF (aTF) — E T F)) = W (71 )^ (A.21)ET F95Then, given W(TF ) > V (T'), it is sufficient to show that aTF E NE({TF , (e)iETF}).For each i E TF , as ci is strictly increasing and ci(0) = 0, (1.11), (A.20) and ui > 0imply gr(fITF(aTFF,ai)) — ci(ai) < gr(f ITF (al; F )) — ci(ar ) for any ai E Ai such that 0 <fITF (aTFF ,ai) 5_ -kr.For each i E TF , using (ii) and (iv) of Proposition 1.1, (1.11) and (A.20), it can beverified that sfft ITF( a. TFF ai)) — egai)_)) ci(ar) for any ai E A • such that-hr < fITF(aTF,,ai) < f ITF(aTF).For each i E TF , as both ci and f are strictly increasing in ai E Ai, it is obvious that,for any ai E Ai , f ITF (aTF , ai) f ITF (all; F ) if and only if ci(ai) > ci(ar). Hence, for eachi E TF , (1.11) implies that gr(fiTr(aFF ' ai)) — ci(ai)T_i^< gr(fITF(aTF)) — ci(ar) for anyai E Ai such that f 1TF (aTF ai) > IITF (4F)•Thus, we have shown that, for each i E TF , ( f 1TF (aF a -)) — ci(a 2 ) < (fITF (aF )) —^TF^ TFci(ar) for any ai E Ai. Therefore, aTF E NE({TF,(e)i,}).Proof of Proposition 1.4: Suppose assumptions Al-A5 are satisfied and V(T*) > W(TF ).Because of (ii) and (iii) of Proposition 1.2, (1.12)-(1.15) readily imply that.§-7(fIT*(a'4,.)) — ci(a7) = ui V i E T* 7,, and^(A.22)f IT* (aT* — E 4(f IT* (aT* .)) — cm (am*) = V (T*).^ (A.23)jET*mSo it is sufficient to show that aT. E N E^,^) iET1',1) •By (1.13) and (iv) of Proposition 1.2, it is obvious that, for each i E T4`, ,§-7(f IT' (4. , ai))— ci(ai) < g7(fIT*(4.)) — ci(a7) V ai E A.For each i E^as ci is strictly increasing and ci(0) = 0, (1.12), (A.22) and ui > 0imply .7(f17—(4.,a i )) — ci(ai) <^(f^— ci(a7) for any ai E Ai such that 0 <96fIT*(di-,* , ai) <For each i E T*, using (iv) of Proposition 1.2 and (1.12), it is easily verified thatf^ai)) — ci(ai) 5_ g7(f^)) — ci(a7) for any ai E Ai such that f IT*^ai) >—k7 /77.It can be checked in (1.14) and (1.15) that, for each i E T. , g7(q) < (q) V q E^. So, foreach i E 77, (ii) and (iv) of Proposition 1.2 and (A.22) imply that ,§-7(f IT*(4:% , ai)) — ci(ai)si (f IT* (4.)) — ci(a7) V ai E Ai.Clearly V(T*) >^> 0. Now, suppose 1771 > 0. Then it is easily verified from(1.12)-(1.15) that q — EJET . m .73:7! (q) = 0 V q E [0AFIT20] and q —^(q) < q —EjETi, „, 8.7(0 b q > q174=1 . Then, as cm is strictly increasing and cm (0) = 0, (A.23) andV (T*) > 0 imply that f IT*(aT* am ) —E^f IT* (4. , am )) — cm (am ) < f IT*IT* (4.)) — cm (en ) V a, E A m such that 0 < f^, a m ) < qIT+ I . Also, by(iii) and (iv) of Proposition 1.2 and (A.23), f^(4 !m , am ) — EiEr m^ant)) —cm (am ) 5_ fIT*(asi,,,) EJET.! g.:;(fIT*(4*)) — cm(am) V a, E Am such that f^am)> q,174:1 .Next, suppose ITV = 0. Then it is clear from (1.12) and (1.13) that q — EjETi n, g3(q)8 '; (q) V q E 14. So, by (iii) and (iv) of Proposition 1.2 and (A.23), f IT* (a',1%! , am ) —EjETL.„, f m, am )) — cm (am ) f IT* (4*)• §;( f IT* (4*)) — (a,*n ) V am EA m .Thus, we have shown that. for each i E T* , s ( f IT* (4 ! . ai)) — ci(ai) :§-7(f IT* (4*)) —ci(a7) V ai E Ai • and f IT* (41. m . am ) — Ei^g;*(f^(q,„ . am )) — cn (am ) < f IT*^—•jEri%n g';(f IT* (a7%)) — c ri (a m* ) V a m E A n,. Hence, a7-,* E NE({T*^) ?Er: „,})^II97Appendix BWe begin this appendix with a preliminary lemma which is required in the constructionof the output sharing rule used to prove Lemma 1.7.Lemma B : Suppose assumptions Al and A2 are satisfied, and {TF ,(sn iETF} is suchthat sr (q) = qIITFI V q E 94 and each i E TF . Then there exists of > 0 such that of <f ITF (aTF) V aTF E NE({TF ,(sr) iETF}).Proof of Lemma B: Suppose {TF , 82 ) iETF } is as given in the statement of the lemmaThen the limiting properties of the derivatives of ci and f given in assumptions Al and A2imply that, if aTF E ATF is a Nash equilibrium of {TF, (sr) ieTF}, then ai > 0 V i E TF,and hence, f 1TF (aTF) > 0. So Lemma B must hold. 11Suppose W(TF ) > V(T"). Then it is obvious that W(TF ) > ur,, > 0. Now, pick any e> 0 such that W(TF) — ITF Ic > um . Define sr'(q) = sr (q) + e V q E 94 and each i E TF.Using Proposition 1.1 and the quasilinearity of the utility functions, it can be easily checkedthat^srE(fr(aTF))_ ei(ar)^VzET ,•^F^+ fITF(aTFF)— E sr f (f ,,,F (aTFE))^_)^1 74- 1 E,^andiETF(B.1)(B.2)98NE({TF ,(sNiETF})^140 .^ (B.3)Next, let of > 0 be such that of < mil-4E7'F fuil + c, of < fITF (4F ), and of < oF , whereof is as given in Lemma B. Also, let of < Of < fITF (4F). Furthermore, let ac > fiTF (4, )be such that bc — EzETF srE ( 5f ) > 0. The existence of bE is gauranteed by W(TF ) — ITFic> um and (B.2). Then, for each i E TF , define the payment function ,4r€ as follows:qIITFI if 0 < q < of0^if of < q <(B.4)s e (q)^if of < q < O f0^if q>It can be easily checked that the output sharing rule (.;rf) jETF always awards nonnegativepayments to every agent in TF , and also, the principal's residual is always nonnegative.Proof of Lemma 1.7: Suppose W(TF ) > V(T*). Let c > 0 be such that W(TF ) — ITF IE> u rn . Then it is obvious from (B.1), (B.2) and (B.4) that(f ITF (4F )) — ci (ar) = ui^ViE TF , and^(B.5)um + fITF (afF )—^..r,(fITF(aC,F)) = W(T F )—ITF Ic.^(B.6)iETFSo it is sufficient to show that aTF is the unique Nash equilibrium of {TF , cerf, ETF } .As of < ui eViE TF, using (B.1) and (B.3)-(B.5), we can develop an argumentsimilar to the one in the proof of Proposition 1.3 and show that c4; fi. is a Nash equilibrium of{TF, (,§T) ieTF }. Thus, it remains to be shown that {TF, (er')JETF} does not have any Nashequilibrium other than aTF(B.4) and the limiting properties of the derivatives of ci and f given in assumptionsAl and A2 imply that, if aTF E NE({TF ,(srf), ETF}) and 0 < fITF (aTF) < of, thenC (q)99f (aTF ) IITF — ei (ai) = 0 V i E TF. But it is obvious that, if aTF E ATF satisfiesf iiTF (aTF)I1TF I — ei (ai) = 0 V i E TF, then, because of the concavity of f and the strictconvexity of ci, aTF is a Nash equilibrium of {TF, (sR ETF}, which is as given in Lemma B.Hence, there does not exist aTF E NE({TF ,(.§n iETF}) such that 0 < f ITF (aTF) < of.Suppose aTF E N E({TF (sff)iET,}) is such that of < f 1TF (aTF) < of or f ITF (aTF) >5'. Then there exists j E TF such that ai > 0, and hence, cj(ai) > 0. So .§r(f ITF (aTF))— ci(ai) = 0 — ci(ai) < 0 < q' f( fITF (aTF ,0)) — ci(0), where the last inequality followsfrom the fact that ,§TE always pays a nonnegative amount to agent j and ci (0) = 0. But thiscontradicts our supposition that aTF E N E({TF , 9iETF}). Hence, there does not existaTF E N E({TF (sFE),ETF}) such that of < f ITF (aTF) < Of or f ITF (aTF) > of.From (1.5) and (B.4), it is clear that, if aTF E N E({TF , (,;r, ),ETF }) and of < f 1TF (aTF)< a', then f,ITF (aTF) — ,(a,) = 0 V i E TF . But we also know from the proof of (iv) ofProposition 1.1 that, if aTF E ATF and f i iTF (aTF) — 0 V i E TF, then aTF = aC7F •So, if aTF E NE({TF , (sFe),ETF}) and of < f ITF (aTF) < of, then aTF = 4F •So we have shown that {TF, (SFE),ETF} has only one possible candidate for a Nash equi-librium, namely, aTF . Thus, as we already know that c4; E N ENTF (,4ri)zETF}), the proofof Lemma 1.7 is complete.Suppose V (T*) > W(TF ). Then V(T*) > u.172 > 0. Now, pick any € > 0 such that V(T*)— IT1` ra lE > u rn and k7^< 0 V i E^and define s7f(q) = si (q) + e V q E R+ and eachi E T*^Using Proposition 1.2 and the quasilinearity of the utility functions, it is obviousthat.5.7f(fIT'(aT'.)) — ci e^= ui +6 V i^ (B.7)100f IT*^— E .97 , (fIT*(6.)) — cm (em) = V (7' ) — !r in k, and^(B.8)i ET mN E({T* (S7 c ) iErl,„})^ (B.9)Let o' > 0 be such that -yIo*E < ui + e V i E 71',„ and (1 — E iGT:m 72) 0*E < V (T*) —IT* ni le. Also, for each i E T*, let IT be the smallest positive integer such that —(k;" + ON'< /1o*E. Then, for each i E 771, define the payment function '&7' as follows:-4`{q — (I — 1)o*E] if (/ — 1)o*c < q < I o' for / = 1, ..., /I4f(q) =87(0 if q >^.(B.10)Given any nonnegative integer I and any i E T* , let0 if I = 0 2.9*E^if 0 < / <A(/) =7o*E — 87€(.4o*E)^if /if / > /T.Then, for each nonnegative integer I, let AE(/)_ 4_,JET* k(/), and AE(/) = E1_0 AE(/). Also,let /EH be the largest positive integer such that A (If") > 0.Now, if — EiET. m (k'; €) > 0, then, for each i E T6' U n , define the payment function,§7f as follows:[^ k. _LE \ AE(/)^-y;" q if /o*E < q < (I + 1) o*E ,ETwhere I = 0,^— 1^(B.11)s7c(q)^ if q >^IOn the otherhand, if — L-4iET.„ (kT + 6 )^LET< 0, then let , be the smallest positive integersuch that EiET. (k ^e) I (1 — iETc, 7t) < In% o' . Now, given any nonnegative integer I,define(q) =101if I = 0if 0 < / <AErn (I) =( 1 -^77)0' - (q - L2ET" m 87f (i- i3O')) if I0 if / >Then, for each nonnegative integer I, let AE(/) = AE(/) + A;(/), and AE(/) = E1 =0 AE(/).Also, let /EH be the largest positive integer such that AE(PH) > 0.Thus, if — EiET*„,(k7 + E) < 0, then, for each i E T6 U 1;`, replace the definition of^in(B.11) by the following: [^^VETOUq( +E^)+ q,;.";^) =-_if /o*€ < q < (I + 1)o* ,where I = 0, ..., /EH — 1if q >(B.12)It can be verified that the output sharing rule (,§7E)j E T. m , as defined above, always awardsnonnegative payments to every agent in T* „2 , and also, the principal's residual is alwaysnonnegative.Proof of Lemma 1.8: Suppose V (T*) > W (TF ). Let c > 0 be such that V (T* ) — IT* „2 1€ >um and k7 +€<OViE T. Then, using (B.7), (B.8) and (B.10)-(B.12), it can be checkedthat4`(f171*(4.)) - ci(a7) = + € V i E and (B.13)f IT* (a;-,) — E ( f IT* (4.)) — cm (am) = V (T*) — IT_* „de. (B.14)iET*„,So it is sufficient to show that 4 is the unique Nash equilibrium of {T*, (.§7E)j E T. m }.For each i E T*, if q E [0, ./.,o* E ), then (B.10) implies that ,47f (q) < ui + c. So. for each i ET*, as ci is strictly increasing and ci(0) = 0, (B.10) and (B.13) imply that .§7' ( f IT' (al,. ai))— ci(ai) < (f 1T"(4.)) — ci(a7) V ai E Ai such that 0 < f (4. ai) < I, o"f.102Also, it is clear from (B.9) and (B.10) that, for each i E^4E(f IT*^ai)) — ci(ai).§7 6 (f^— c (a7) V ai E Ai such that f IT*(4 .ai)^o'.For each i E Tp U77, using (B.11) or (B.12) depending on whether — EiEr: m (k; + f)is nonnegative or not, it can be checked that 4' (q) < s7f(q) V q E 94. Thus, for each i ET(7 U7+, because of (B.7), (B.9) and (B.13), .47E (fIT*(4.. ai)) — ci(ai) < :sr (f (4*)) —ci(a7) V ai E Ai.It can be verified that, if — E3 ET_*_,n (k; + e) > 0, then (B.10) and (B.11) imply q —,§7(q)^q^EJETi%„ sr(q) b q E 94. Thus, because of (B.8), (B.9) and (B.14),if — EiEr:m (k; + e) 0, then f^am,1 — 7^( f^am)) — ern (am) <f IT* (4.) — EJETL' m '79:;e (f IT*(4*)) cm (4) V^E Am .It can also be verified that, if — Ej E T . (k; E) < 0, then (B.10) and (B.12) imply q —EJET! rn :4.7(q) < V(T*) — IT%„je q E [0,/,E,,,,p *€ ). So, if —7 jET:'„,(k; E) < 0, then, by themonotonicity of cm , cm (0) = 0 and (B.14), f IT* (a'7" m , am ) — EjET_*_ rn :97 (f IT* (ar. , am ))— cm (am) fIT*(4*) EjET* (f IT* (4*)) — cm (4) V am E Am such that 0 <f^am) < .4n o*f.Once again, it can be verified that, if — E.IET*„, (k; +e) < 0, then (B.10) and (B.12) implyq^EjET* m `;';€ (q) = q^JET*„, .97(q) V q^I;n o*f. So, if — 4-a jET*,,,, (k;^E) < 0, then,because of (B.9), f IT* (4* m , am) — EJET*„, !; E (f iT* (ai-* am)) — cm (am) C f^(4,0 —EjET * ru^( f IT* (4*)) — cm ( ) V am E Am such that f IT*(4* m . am ) >Thus, we have shown that a7, E NE({T* ,(.§7)JET*„,})•For each i E T' „„ it is clear from the definitions in (B.10)-(B.12) that :s7E is piecewiselinear and has a slope of -K` (> 0) on each linear piece. Similarly, (B.10)-(B.12) also implythat the principal's residual, q E,ET.„, is piecewise linear in q E 94 and has a103slope of 1 — jET*,,^( > 0) on each linear piece. Therefore, the limiting properties of the<--, derivatives of the cost functions and the production function in assumptions Al and A2immediately imply that, if aT* E N E({T* (‘'3' € ) T* m }), then a, >0Vi E T*.As one can easily check, the proof of Lemma 1.3 uses only the Nash equilibrium conditionin constraint (C2). So a similar argument to the one used in the proof of Lemma 1.3 showsthat, if aT* E N E({T* (g7) JET* „,}) then glf is differentiable at f IT* (aT*) V i E TIK.n . Hence,if aT* E N E ({T* , (_. 1*€) 3 ET* m }), then (B.10)-(B.12) imply that -y:` = I f i lT* (aT*) V i ET* Tri and 1 — E ET* „, 7* = cim (a rn ) I^(aT*). However, we know from the proof of (iv) of3 Proposition 1.2 that aT* E AT satisfies 77 = (a,) 1 fi lT* (aT*) V i^and 1 —^T*in=^(am )/f m IT* ( aT* ) if and only if aT* = 4*. Therefore, as we have already shown that4* E N E ({T* , (g7) JET* }), the proof of Lemma 1.8 is complete. 1 1104
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Essays in group decision-making Nandeibam, Shasikanta S. 1993
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Title | Essays in group decision-making |
Creator |
Nandeibam, Shasikanta S. |
Date Issued | 1993 |
Description | The thesis comprises of two essays. Although the two essays deal with somewhat different situations and use different approaches, both of them essentially examine the problem of making decisions that affect some group of individuals. The first essay is on moral hazard and looks at the principal's problem in a principal-agent(s) free-rider problem in which, unlike most existing work, the principal is not precluded from participating in the production process. Furthermore, there are no uncertainties, but moral hazard is caused by joint production which renders the action of each individual in the production process unobservable. A multi-stage extensive game in which only the principal can propose the output sharing rule determines both the set of individuals who actually participate in the joint production process and the output sharing rule. The main conclusion we draw in the first essay is that, when designing the optimal output sharing rule, the principal need not look for any output sharing rule more sophisticated than the linear or piecewise linear rules we frequently observe. We also characterize the condition under which the principal chooses to take part in production, and conclude that the issue of mitigation of moral hazard and sustainability of efficiency crucially hinges on whether the principal actually participates in production or not. More concretely, we show that moral hazard dissipates completely whenever the principal does not participate in production, however, even then she does not achieve as much welfare as in the First Best situation if her best option in the First Best situation is to take part in production. The second essay is in stochastic social choice theory. In a paper published in 1986 in Econometrica, Pattanaik and Peleg formulated stochastic analogues for each of Arrow's axioms and concluded that the stochastic social choice functions that satisfy their axioms are essentially randon dictatorships when individuals have strict preferences. More precisely, there is a unique weight associated with each individual such that the vector of these individual weights has the properties of a probability distribution over the set of individuals, and, given any preference profile and any feasible set, the probability that a feasible alternative is chosen is equal to the sum of the weights of those individuals who have this alternative as their best feasible alternative. We extend the analysis of Pattanaik and Peleg by allowing individuals to have weak preferences. As in their paper, it turns out that the probabilistic versions of Arrow's condition simply that there are individual weights. However, now, given a preference profile and a feasible set, we partition the society so that any two individuals from different elements of the partition have no common best feasible alternatives, but the set of best feasible alternatives of each individual in an element of the partition overlaps with that of some other individual in the same element. Using this partition, it is shown that the only restriction on the stochastic social choice function is that the sum of the weights of all individuals belonging to the same element in the partition is equal to the probability that some alternative which is best in the feasible set for one of these individuals is chosen. When everyone has unique best feasible alternatives, the rules characterized here reduce to those of Pattanaik and Peleg. |
Extent | 4406039 bytes |
Subject |
Decision-making, Group. |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0086434 |
URI | http://hdl.handle.net/2429/1749 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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