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The relevance of game theory in its application to decision making in competitive business situations Ng, Kim Seah 1968

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THE RELEVANCE. OF GAME THEORY IN ITS APPLICATION TO DECISION MAKING IN COMPETITIVE BUSINESS SITUATIONS by KIM SEAH NG 3. E... (Plonsv) U n i v e r s i t y of Malaya, 1965* A THESIS SUBMITTED IN PARTIAL FULFILMENT 0] THE REQUIREMENT FOR THE DEGREE 0? MASTER OF BUSINESS ADMINISTRATION In the F a c u l t y of. Commerce and Business A d m i n i s t r a t i o n Me accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY. OF BRITISH COLUMBIA J u l y , . 1968. In presenting this thesis in part ial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for reference and Study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by hils representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Coronerce and Business A d m i n i s t r a t i o n . The University of Br i t ish Columbia Vancouver 8, Canada Da te T u l y 22. ? 1 9 6 8 ABSTRACT I t i s known t h a t i n s i t u a t i o n s where we have to make d e c i s i o n s , and i n which the.outcomes depend on opponents whose a c t i o n s we have no c o n t r o l , c u r r e n t q u a n t i t a t i v e t e c h -niques employed are inadequate. These are d e p i c t e d i n the complex c o m p e t i t i v e environment o f to-day's - b u s i n e s s w o r l d . The technique t h a t proposes t o overcome t h i s problem • i s the methodology of game theory.- Game theory i s not a new i n v e n t i o n but has been i n t r o d u c e d t o economists and. mathema-t i c i a n s w i t h the p u b l i c a t i o n of von Neumann and H o r g e n s t e r n 1 s book "Theory of Games and Economic Beh a v i o r " . S i n c e then i t has remained r e l a t i v e l y obscured as a t o o l f o r a i d i n g manage-ment d e c i s i o n making p a r t l y because i t has become more mathe-m a t i c a l i n nature, and, as a r e s u l t of the v a s t amounts of computation i n v o l v e d f o r any reasonable s i a e problem. Hov/ever, w i t h the advent of the h i g h speed e l e c t r o n i c computer the p o s s i b i l i t y of p r a c t i c a l a p p l i c a t i o n to l a r g e s c a l e business problems i s w i t h i n s i g h t . E f f o r t s are t h e r e -f o r e seen to undergo r e o r i e n t a t i o n . The r e c e n t development of a c o m p e t i t i v e d e c i s i o n model at S h e l l Development Company i n C a l i f o r n i a i s a step forward i n t h i s d i r e c t i o n . This t h e s i s examines v a r i o u s aspects of game- theory t h a t would a p p r o p r i a t e l y l e a d to areas of f r u i t f u l p r a c t i c a l a p p l i c a t i o n s . A s e r i e s of examples were d i s c u s s e d to demon-lv s t r a t e the a n a l y t i c a l process of game theory. The f e a t u r e s o f the model t h a t the S h e l l r e s e a r c h group set up were analysed and d i s c u s s e d f o r p o s s i b l e a p p l i c a t i o n s t o i n d u s t r i a l s i t u a -t i o n s . Our g e n e r a l c o n c l u s i o n i s t h a t , although the S h e l l model does i n d i c a t e to us the s o r t of r e s u l t s t h a t a game theo-r e t i c , model c o u l d p r o v i d e and to l e a v e us w i t h a c l e a r e r un-de r s t a n d i n g of the problem, i t has not answered the q u e s t i o n of how we should proceed t o use these r e s u l t s . TABLE 0? CONTENTS CHAPTER ' ' PAGE I . INTRODUCTION '. . 1 Trends i n Business Research . . . . . . . . 1 Examples of A p p l i c a t i o n of Game Theory 7 Purpose and Scope of T h e s i s Ik-I I . PERSPECTIVES IN THE THEORY 0? GAMES 17 I n t r o d u c t i o n . . 17 Theory of S o l u t i o n of 2-Person Constant Sum Games 2k-T h e o r i e s of S o l u t i o n of n-Person Games 31 Game T h e o r e t i c P o i n t s : T h e i r I m p l i c a t i o n s t o Decision'Mailing' . . . . k-7 I I I . GAMES AS DECISION TOOLS IN DYNAMIC SETTINGS. . 63 I n t r o d u c t i o n 63 Continuous V a r i a b l e S t r a t e g y and Payoff Space 69 Dynamics of Game S i t u a t i o n s . . . . . . . . . ' 7 7 Games of Economic S u r v i v a l 87 IV. STRATEGY AND STRUCTURE IN THE OIL AND GAS INDUSTRY: AN APPLICATION 07 THE SHELL MODEL 96 Overview of -the Indu s t r y 96 S t r a t e g i c Decision's i n the O i l Ind u s t r y . . : 98 The S h e l l Model 102 v i CHAPTER PAGE The 3. C. G a s o l i n e Market S t r u c t u r e . . . . I l l A p p l i c a b i l i t y of the S h e l l Model 115 .V SUMMARY AND CONCLUSIONS 120' BIBLIOGRAPHY 125 APPENDIX 131 LIST OF TABLES TABLE PAGE (2-1) Payoff M a t r i x f o r 2-Person Constant Sum Game .27 (2-2) P r i s o n e r s ' Dilemma Game 33 (2-3) M a t r i x Game: P-, 1 s Choice Known to P p 58 LIST OF FIGURES IGURE PAGE 2-1) Pareto Optimal and N e g o t i a t i o n Sets . . . . . . . 39 2-2) L i n e a r T r a n s f o r m a t i o n of Convex U t i l i t y F u n c t i o n 39 2-3) Comparison of Nash's and Other " F a i r D i v i s o n " C r i t e r i a **1 2-4) Game T h e o r e t i c P o i n t s on Payoff Space 45 2-5) E f f e c t of C o n s t r a i n t s on P o s i t i o n of E q u i l i b r i u m P o i n t s 50 2-6) E f f e c t of Small Changes i n Investment oh P o s i t i o n of E q u i l i b r i u m P o i n t s 52 2-7) Cross S t r a t e g i e s When One F i r m Uses Co o p e r a t i v e S t r a t e g y 55 2-8) Cross S t r a t e g i e s When One F i r m Uses E q u i l i b r i u m S t r a t e g y . 55 2-9) Game Tree R e p r e s e n t a t i o n of Table (2-2) 58 2-10) Game Tree R e p r e s e n t a t i o n of Table (2-3) 58 2- 11) A Simple Dynamic Duopoly Game 62 3- D 2-Person Game i n V a r i a b l e Space 72 3-2) 2-Ferson Game i n Payoff Space 72 3-3) Market Trend With Share D i s t r i b u t i o n R e l a t i o n s h i p 80 3- 1:-) G r a p h i c a l R e p r e s e n t a t i o n of a Game of Economic S u r v i v a l . . . . ' 91 4- 1) STRATCOH: S t r u c t u r e of Model 104 4-2) • S i m p l i f i e d G r a p h i c a l R e p r e s e n t a t i o n of B u l k A l g o r i t h m . 109 ACiaroWLEDGHEHT Much of the i d e a s generated i n the t h e s i s were l a r g e -l y due t o the many s e s s i o n s of s t i m u l a t i n g d i s c u s s i o n s w i t h P r o f e s s o r J . S w i r l e s . I t was through him "that the s u b j e c t o f the t h e s i s was brought to my a t t e n t i o n . I wish t o o f f e r my most s i n c e r e thanks to him f o r the i n t e r e s t and encouraging words he had shown i n my work as my t h e s i s s u p e r v i s o r . CHAPTER I INTRODUCTION• Trends i n Business Research The expansion of b u s i n e s s o r g a n i z a t i o n s to m u l t i -n a t i o n a l , m u l t i - p r o d u c t and. m u l t i - l e v e l dimensions i n an en-vironment of e v e r - i n c r e a s i n g c o m p l e x i t i e s c a l l s f o r more s c i e n t i f i c a l l y based r a t h e r than " r u l e s of thumb" or " i n -t u i t i v e " d e c i s i o n s . T h i s t r e n d to s y s t e m a t i c methodologies has sparked o f f the growth and development of q u a n t i t a t i v e techniques of o p e r a t i o n s r e s e a r c h and management s c i e n c e , the t o o l - k i t of tomorrow's manager. F u r t h e r impetus has been giv e n to t h i s advancement w i t h the advent of the high-speed e l e c t r o n i c computer. The impact of such developments and the s u c c e s s f u l a p p l i c a t i o n t o bu s i n e s s a c t i v i t y i s summed up by H i l l i e r and Lieberman: "... some of the problems have been s o l v e d by p a r t i c u l a r techniques of opera-t i o n s r e s e a r c h . L i n e a r programming has been' used s u c c e s s f u l l y i n the s o l u t i o n of problems concerned with assignment of personnel, b l e n d i n g of m a t e r i a l s , d i s t r i -b u t i o n and. t r a n s p o r t a t i o n , and investment p o r t f o l i o s . Dynamic programming has been suc-c e s s f u l l y a p p l i e d to such areas as p l a n -ning a d v e r t i s i n g e x p e n d i t u r e s , d i s t r i b u t i n g s a l e s e f f o r t , and p r o d u c t i o n s c h e d u l i n g . Queueing theory has had a p p l i c a t i o n s i n s o l v i n g problems concerned with- t r a f f i c c o n g e s t i o n , s e r v i c i n g machines s u b j e c t e d to breakdown, determining the l e v e l of a s e r v i c e f o r c e , a i r t r a f f i c s c h e d u l i n g , 2 d e s i g n of dams, p r o d u c t i o n s c h e d u l i n g , and h o s p i t a l o p e r a t i o n . Other techniques of o p e r a t i o n s r e s e a r c h , such as i n v e n t o r y theory, game theory, and s i m u l a t i o n , a l s o have been s u c c e s s f u l l y a p p l i e d i n a v a r i e t y of c o n t e x t . " 1 Yet, i n the midst of a l l t h i s progress we can s t i l l f i n d ^ r e s i s t a n c e to change. • Change i s something t h a t cannot be achieved o v e r n i g h t , but i s a g r a d u a l p r o c e s s . However, we cannot a t t r i b u t e t h i s r e s i s t a n c e t o . i n a d e q u a c i e s i n coping w i t h such development. T h i s i s perhaps due to apathy and t o the attitu.de._of some of the managers b e l o n g i n g to the s c h o o l which s u b s c r i b e s to the b e l i e f t h a t management i s an a r t , not a s c i e n c e . In the past and. c u r r e n t l y , the emphasis on ma-nagement s c i e n c e development has been i n the areas of o p t i -m i z a t i o n t e c h n i q u e s . The l o g i c of the "optimum'7 s o l u t i o n and the e l a b o r a t e mathematical procedures f o r s o l v i n g , i t are Indeed beyond dispute.. However, i n s i t u a t i o n s where s t o c h a s -t i c s i m u l a t i o n and the g e n e r a t i o n of s t a t i s t i c a l d i s t r i b u t i o n f u n c t i o n s are no l o n g e r a v a i l a b l e and we can no l o n g e r des-c r i b e the v a r i a b l e s as random, we have to l o o k f o r other s o l u t i o n t e c h n i q u e s . The s i t u a t i o n we have i n mind i s t h a t where d e c i s i o n s have to be made i n a. complex c o m p e t i t i v e environment. T h i s i s one area where the s t o c h a s t i c techniques of d e c i s i o n t h e o r y , 1. H i l l i e r , F.S., Liebernman, G.J., " I n t r o d u c t i o n to  Operations Research," H-olden Bay Inc., -Sen F r a n c i s c o 1967 P. 7. 3 s i m u l a t i o n i n i n d u s t r i a l dynamics, r i s k anc venture a n a l y s i s , both l i n e a r and n o n - l i n e a r o p t i m i z a t i o n techniques and o p t i -misation, of s t o c h a s t i c s i t u a t i o n s are inadequate and i n c a -pable of g i v i n g s a t i s f a c t o r y s o l u t i o n To i l l u s t r a t e t h i s i n a very elementary way, l e t us c o n s i d e r the f o l l o w i n g s i t u a t i o n : Three o i l companies, A, B and C marketing g a s o l i n e i n a g i v e n area have r e s p e c t i v e l y 500, 300 and 200 s e r v i c e s t a -t i o n s t o serve as o u t l e t s f o r t h e i r p r o d u c t s . Through r e s e a r c h and f o r e c a s t i n g f a c i l i t i e s i t i s known to each t h a t the volume of the g a s o l i n e market w i l l i n c r e a s e by approximately 10 $ each year f o r the next ten y e a r s . B e s i d e s , the r a t e of de-t e r i o r a t i o n of s e r v i c e s t a t i o n s measured as a percentage of the existing- t o t a l can be c a l c u l a t e d from h i s t o r i c a l r e c o r d s or operations r e s e a r c h s t u d i e s . Faced with t h i s s i t u a t i o n what would f o r example be the a p p r o p r i a t e s i z e of the marketing investment of com-pany B viewed over the next ten 2/ears? A f i r s t r e a c t i o n c o u l d be to m a i n t a i n the s t a t u s quo, t h a t i s , to work f o r the same share i n the expanding market by i n v e s t i n g i n a p r o p o r t i o n ' of 5: 3' 2. A l t e r n a t i v e l y company B would, l i k e ' t o know what would ham;en i f i t makes investment such as to •'- * lu i n c l u d e a s i z a b l e amount, say 100 s t a t i o n s which i n c l u d e both the new market o p p o r t u n i t i e s and d e t e r i o r a t i o n of the o l d e r s t a t i o n s . Would company A and C l e t such a move go by uncha-l l e n g e d and allow B to b i t e i n t o t h e i r share of the new market? R e t a l i a t i o n f o l l o w e d by r e t a l i a t i o n c o u l d ensue with nobody g a i n i n g a n y t h i n g . The answer t o seek i s t h e r e f o r e an o p t i -mum Xg i n view of the competitive environment i n which the f i r m i s o p e r a t i n g . Game theory may overcome t h i s decision-making problem. In essence, game theory i s , as Shubik d e s c r i b e s : "...a method f o r the study o f decision-making i n s i t u a t i o n s of c o n f l i c t . I t d e a l s w i t h pro-blems i n which the i n d i v i d u a l decision-maker i s not i n complete c o n t r o l of the f a c t o r s i n -f l u e n c i n g the outcome.... The essence of a game problem i s t h a t i t i n v o l v e s i n d i v i d u a l s w i t h d i f f e r e n t g o a l s or o b j e c t i v e s whose f a t e s are i n t e r l o c k e d . . . . . The problem of game theory i s more d i f f i c u l t than t h a t of simple maximization. The i n d i -v i d u a l has to work out how to achieve as much as p o s s i b l e , t a k i n g i n t o account t h a t there are others whose goals are d i f f e r e n t and whose actions have an e f f e c t on a l l . A d e c i s i o n -maker i n a game f a c e s a cross-purposes maxi-m i z a t i o n problem. He must pl a n f o r an optimal r e t u r n , t a k i n g i n t o account the p o s s i b l e ac-t i o n s of h i s opponents." 2 Much of the r e s e a r c h done i n game theory has been main-l y concerned w i t h the mathematical aspects of f o r m u l a t i o n and s o l u t i o n s i n c e the i n c e p t i o n of the monumental work of von 2. Shubik, M a r t i n , "The Uses of Game Theory i n Management  S c i e n c e , " Management Sc i e n c e , Volume 2, Wo.l, October 1955. p. 1,0. 5 Neumann and Morgenstern,^ but r e l a t i v e l y l e s s on i t s a p p l i c -a b i l i t y t o p r a c t i c a l decision-making s i t u a t i o n s . One of the most o u t s t a n d i n g i s the r e s e a r c h done by S h e l l i n C a l i -f o r n i a . ^ ' The o f f s p r i n g of t h i s r e s e a r c h i s a body of b e l i e f t h a t t h i s technique w i l l soon be a • s i g n i f i c a n t a i d to b u s i n e s s d e c i s i o n making i n s t e a d o f 'remaining a mathematician's d e l i g h t f u l pastime. To see where and when the me t h o d o l o g i c a l t o o l s of game theory as provided by gaming and s i m u l a t i o n , w i t h the advancement of computer t e c h n o l o g y , a s s i s t i n the e f f e c t i v e a n a l y s i s of b u s i n e s s and other economic s i t u a t i o n s , we t u r n to some assessments of P r o f e s s o r M a r t i n Shubik. "The methodologies we have d i s c u s s e d are new . .• The b r i n g i n g i n of r a d i c a l l y new approaches and techniques u s u a l l y take a s i z a b l e f r a c t i o n of a g e n e r a t i o n .... I t i s not unreasonable however, t o expect t h a t w i t h i n twenty years many l a r g e f i r m s and s e c t i o n s of the government w i l l have d e t a i l e d s i m u l a t i o n s of d i f f e r e n t aspects of the e n v i -3. von Neumann, J . , and Oskar Morgenstern. "Theory of Games  and Economic Behavior." P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n (1947)• 'Mathematical r e s e a r c h i n t o the theory of games i n c l u -des: Kuhn, H., and Tucker, A.W., (eds.) " C o n t r i b u t i o n s to the Theory of Games." V o l . I and I I : Annals of Mathematical S t u d i e s - 2 A and 26. F r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n 1950, 1953. Various j o u r n a l a r t i c l e s of the S o c i e t y of I n d u s t r i a l and A p p l i e d Mathematics and. p u b l i c a t i o n s of Rand C o r p o r a t i o n . • L. Hughes, R.R., and. Ornea, J.C. "Decision-Making i n Competi-t i v e S i t u a t i o n s . " Proceedings World Petroleum Congress, Mexico C i t y , mimeograph copy. 6 ronment i n which they operate. I f they do so they w i l l be i n a p o s i t i o n t o explore po-l i c y a l t e r n a t i v e s arid, i n some cases, w i l l a l s o f i n d i t worthwhile to use t h e i r simula-t i o n s to provide the e n v i r o n m e n t ' f o r o p e r a t i o -n a l games i n the same way as m i l i t a r y opera-t i o n a l games are used today. There i s no r o y a l road to t h i s s t a t e of a f f a i r s , nor w i l l i t n e c e s s a r i l y h e r a l d the m i l l e n i u n i . A new methodology i s becoming a v a i l a b l e a t the appro-p r i a t e time, when the i n c r e a s e i n speed of t e c h n o l o g i c a l change, j o i n e d w i t h the s i z e of p o p u l a t i o n and the complexity of modern s o c i e -t y , makes i t i m p e r a t i v e f o r us t o be i n a po-s i t i o n to examine and i n t e g r a t e models of economic f i n e s t r u c t u r e The methodological t o o l s provided by gaming and s i m u l a t i o n are making i t f e a s i b l e to uncover and examine i n an o r g a n i z e d manner much of the important f i n e s t r u c t u r e of the f i r m s and markets i n which f i r m s operate. Me-thodology and techniques alone do not provide answers and c u r e - a l l s . However, the progress t o date and the c o s t s of the progress i n ga-ming and s i m u l a t i o n i n d i c a t e t h a t they can be of immediate and d i r e c t use i n b u s i n e s s ope-r a t i o n s , t h a t they are b e g i n n i n g to supply data and e n l a r g e the knowledge of both the e m p i r i c i s t and economic t h e o r i s t s , and t h a t they are b e g i n n i n g to provide a base f o r the c o n s t r u c t i o n of a i d s to guide i n the framing of p o l i c y . " 5 I t i s r e c o g n i z a b l e t h a t i n the next two- or t h r e e deca-des, the r e v o l u t i o n i n b u s i n e s s t h i n k i n g and the e n v i -ronment i n which b u s i n e s s operates w i l l have changed so much, the s u r v i v a l of a f i r m i n a competitive world depends. 5 . Shubik, Hart i n , " S i m u l a t i o n end Gaming;: T h e i r Value to the Study of P r i c i n g and Other Market V a r i a b l e s , " i n "iiodel of Markets," e d i t e d by A l f r e d K. Oxenfelclt, Columbia U n i -v e r s i t y Press, New York, 1 9 6 3 , pp. 307-336". 7 e n t i r e l y on the s t r a t e g i c d e c i s i o n s which i t s e x e c u t i v e s make. In the process,one of the most c h a l l e n g i n g i n t e l l e c -t u a l f r o n t i e r s of b u s i n e s s r e s e a r c h l i e s i n the dynamics of bus i n e s s o r g a n i z a t i o n and d e c i s i o n making i n a. h i g h l y com-p e t i t i v e enviroiiment. I t would thus appear t i m e l y f o r management t o i n v e s t i g a t e i n the s t r a t e g i c area of d e c i s i o n making u s i n g game theory once computational b a r r i e r s are overcome and the problematic areas of the theo r y ' a r e r e s o l -ved. Of these a r e a s , the most important are 'those .of measurement and i n t e r c o m p a r a b i l i t y o f u t i l i t i e s . . '. Examples of A p p l i c a t i o n of Game Theory The Canadian pulp and paper i n d u s t r y , and, f o r t h a t matter the B r i t i s h Columbia pulp and paper i n d u s t r y as we were t o l d i n the e d i t o r i a l of "The Pro v i n c e " , "has been s e -r i o u s l y weakened .by massive i n c r e a s e of o v e r - c a p a c i t y " . These were the remarks of Mr. R. II. Fowler, P r e s i d e n t of the Cana-6 d i a n Pulp and Paper A s s o c i a t i o n . Mr.. Fowler had. c o r r e c t l y . i n d i c a t e d t h i s t o t h e ' " s p i r i t of c o m p e t i t i o n " . However, he vent ..on.to p o i n t out t h a t these c o m p e t i t i v e f o r c e s were '•'vigorously a i d e d and abe t t e d by i n t e r p r o v i n c i a l c o m p e t i t i o n which f i n d s i t s r o o t s i n f e d e r a l p o l i c i e s to support p r o v i n -6. "The Pr o v i n c e " , e d i t o r i a l "Competition and the Dino-saurs ," February 6, 196S. 8 c i a l e f f o r t to expand." In a statement t o the e d i t o r of "The P r o v i n c e " Dr. J.W. Sut h e r l a n d has t h i s t o say: "...Under the f r e e e n t e r p r i s e system, c a p a c i t y i n c r e a s e s i n a n t i c i p a t i o n of, or i n response t o , i n c r e a s e i n demand. I f the r e has been an in c r e a s e i n c a p a c i t y i n the pulp and paper i n d u s t r y we hope i t i s because a t some p o i n t producers f e l t t h a t there would be an i n c r e a s e i n f u t u r e demand. Current o v e r - c a p a c i t y merely i n d i c a t e s t h a t the f o r e c a s t o f produ-cers were i n e r r o r one cannot presume t h a t the government's aims and o b j e c t i v e s are i d e n t i c a l w i t h those of i n d u s t r y . I n d u s t r y must compete i n the market f o r economic advantage. Government on the other hand, are p r i m a r i l y concerned with p o l i t i c a l advantage. They i n v e s t i n the benevolent h a s i s of r e a l i z i n g advantages f o r the i n d u s t r i a l sector."* 5" B e a r i n g what has been s a i d , we can now take a c l o s e r l o o k a t the B r i t i s h Columbia pulp and paper i n d u s t r y . The s i t u a t i o n as i t i s , s p e l l s en urgent need f o r a realignment of market and' p r o d u c t i o n s t r a t e g i e s . Vvhat a i l s the companies i n the comp e t i t i o n as shown by t h e i r past'performance of a steady d e c l i n e i n p r o f i t s over t h e y ears 1965-1967 d e s p i t e 9 i n c r e a s e d s a l e s c a l l s f o r s p e c i a l d i a g n o s i s . I t i s here t h a t the new methodology of game theory would seem a p p r o p r i a t e , though not as a panacea f o r the s t r a t e g i c i l l s , but could 7. I b i d . £ . 3uther 1 and, J .V/., ''Exn'orts Suffer- f o r Lack of Free  E n t e r p r i s e " i n a l e t t e r t o the e d i t o r of "The Pr o v i n c e " , February 12, 1968. 9. "Pulp and Paoer I>;"agasine of Canada." (August 1967). host companies r e p o r t e d l o s s e s . Motably a l l the companies have a t t r i b u t e d l o s s e s t o p r i c e i m m o b i l i t y and r i s e i n wages. Other so u r c e s : Company r e p o r t s 196-5-1967. 9 improve a company's o v e r a l l p o s i t i o n . . Granted-that there may be a s h o r t , world market demand., and t h a t at the moment the pulp and paper i n d u s t r y i s i n the doldrums,a company as a p a r t y i n the c o m p e t i t i o n must not overlook c e r t a i n a s p e c t s of the markets. Competitive p r e s -sures have been b u i l d i n g up w i t h the e n t r y of new f i r m s and w i t h e x t e n s i v e i n c r e a s e s ' i n e x i s t i n g c a p a c i t i e s . . I n the i n d u s t r i a l survey of B r i t i s h Columbia pulp- and paper i n d u s t r y , p u b l i s h e d by B'. G. H y d r o ^ t h e r e were 17 m i l l s operated by 10 companies. T a t l i i s date we have 19 m i l l s , a n d s t i l l more m i l l s are to be c o n s t r u c t e d or are under construction"!" 1 Have these investments been undertaken w i t h t o t a l d i s r e g a r d f o r c o m p e titive p r e s s u r e s and f o r the over-supply p r e v a i l i n g i n the pulp and paper markets? There i s c o n s i d e r a b l e s i m i l a r i t y of t h i s i n d u s t r y , r e g i o n a l as w e l l as n a t i o n a l , w i t h the o f t - q u o t e d and f a m i -l i a r examples of o l i g o p o l i s t i c i n d u s t r i e s i n s t e e l , auto-mobiles, o i l s and o t h e r s . V/e can thus p o r t r a y the B r i t i s h Columbia pulp and paper market as a problem i n o l i g o p o l i s -t i c c o m p e t i t i o n . The l e a d i n g competitor i s HaeMl11an B l o e d e l L i m i t e d 10. "The Pulu and Paper Ind u s t r y of B r i t i s h Columbia." I n d u s t r i a l Development Department, B.C. Hydro and Power A u t h o r i t y . . October 1966. 2nd. E d i t i o n . 11. "Province of B r i t i s h Columbia Budget Speech" by V/.A.C. Bennett, Premier and M i n i s t e r of Finance, February i 9 6 0 , p. 25. 10 w i t h a number of other competitors who are e i t h e r s u b s i d i a -r i e s or a s s o c i a t e s of major companies i n the U.S., B r i t a i n and Sweden or formed from groups of l o c a l independent com-panies. Here the entrepreneur o f a f i r m can no l o n g e r a c t a c c o r d i n g to well-known laws of maximization of economic t h e o r y . He becomes a competitor whose success i s determined not only by .his. own d e c i s i o n but by o t h e r entrepreneurs who he i s o p e r a t i n g a g a i n s t , i n a s i t u a t i o n t h a t a f f o r d s them the o p p o r t u n i t y to e x e r c i s e t h e i r f r e e w i l l i n a manner t h a t w i l l v i t a l l y i n f l u e n c e him. J o i n t ventures are t y p i c a l i n any new investment because of t h e i r immense s i z e . A s i d e from t e l l i n g where the investment.should stop, the techniques o f " c a -p i t a l budgeting t h e o r y are t o t a l l y l a c k i n g : i n ' a l l these c o n s i d e -r a t i o n s . Each f i r m t r i e s t o outdo the others with l a r g e r m i l l s . Although t h i s i s i n l i n e with economies of l a r g e -s c a l e , there i s a. p o i n t where l e s s f a v o r a b l e o p e r a t i o n s may r e s u l t upon b r i n g i n g i n the e f f e c t s of competition", t h r e a t s , c o u n t e r - t h r e a t s , e n t r y and the l i k e . In p l a n n i n g i t s moves,a f i r m should c o n s i d e r how i t s competitors w i l l r e a c t to them. T h i s , i n e f f e c t i s the crux of the problem. F o r some time l a t e l y a s e r i e s of shutdowns were i n s t i -12 t u t e d by a number of f i r m s , and yet 'che c o n s t r u c t i o n of m i l l s i s s t i l l g oing on. I t would be i n t e r e s t i n g t o know what 12. 196?). "Pulp and Faper Magazine of Canada" (June and J u l y 11 the bases f o r such s t r a t e g i e s were. N e v e r t h e l e s s , i n the midst of a l l t h i s c o m p e t i t i o n and o u t c r y some f i r m s were able to improve t h e i r share of the markets and i n c r e a s e e a r n i n g s . V/hat p o s s i b l e e x p l a n a t i o n can we o f f e r ? In the f a c e of i n f l a t e d l a b o u r c o s t s and c o m p e t i t i o n , r e g i o n a l , n a t i o n a l and i n t e r n a t i o n a l , what s t r a t e g i c choice should a f i r m whose investment runs t o m i l l i o n s make? To advocate the use of game t h e o r e t i c methods we must a l s o note the l i m i t a t i o n s of the mechanics of the so-l u t i o n . Investment d e c i s i o n s w i l l i n v o l v e both buyers and s e l l e r s market and a f?irm making a d e c i s i o n w i l l have to 13 c o n s i d e r both o l i g o p o l i s t i c and o l i g o p s o n i s t i c c o m p e t i t i o n . A l l these f a c t o r s can and -will make a complete game th e o r e -t i c model extremely unwieldy but i f we accept the f a c t t h a t computational problems can be s o l v e d i n the near f u t u r e by l a r g e - s c a l e computing machine technology we can f o r e s e e immense a p p l i c a b i l i t y of game t h e o r y . We have thus f a r encountered a problem i n i n d u s t r y where game .theory c o u l d be o p e r a t i v e . As f a r as the l i t e r a t u r e on the s u b j e c t goes, the areas i n which game theory i s a p p l i c -able have, over the y e a r s , been c o n s i d e r a b l y extended. 1 3 . For a f i r m the v a r i a b l e s i n an o l i g o p o l i s t i c market may be p r i c e , d i f f e r e n t i a t i o n ' of product, e t c . . In an o l i -g o p s o n i s t i c market we have i n t e r e s t on l o a n s , l a b o r , wages, t r a n s p o r t a t i o n and m a t e r i a l c o s t s , e t c . . 12 Shubik, 14 Friedman 1^ and Charnes and Cooper,"1" have shown how game theory can be used i n the a l l o c a t i o n of a d v e r t i s i n g ex-pen d i t u r e s . Game t h e o r e t i c models f o r c o m p e t i t i o n between two r e f i n e r i e s w i t h f i x e d demand and othe r problems i n c l u d -i n g c a p i t a l budgeting i n the o i l i n d u s t r y were d i s c u s s e d by G.H. S y m o n d s 1 7 ' l S and E.G. B e n n i o n . 1 9 V/ith i n an o r g a n i s a t i o n a l system there are competing u n i t s . S u c c e s s f u l d e c e n t r a l i z a t i o n w i t h r e s p e c t t o a s e t of d e c i s i o n s w i t h i n a system, means t h a t the independent a c t i o n s of the i n d i v i d u a l s i n c o n t r o l of the sub-systems 14. Op . c i t . p. 47-/4-6 15. Friedman, Lawerence, "Game-Theory Models i n the A l l o c - a t i o n of A d v e r t i s i n g E x p e n d i t u r e s O p e r a t i o n s Research Volume VI, No. 5, September-October 195&, pp*. 699-709. A l s o l o c a t e d In "Mathematical Models and Methods i n Ma r k e t i n g " by F.M. Bass, R.D. B u z z e l l and ot h e r s , R i c h a r d D.- Ir w i n , Inc., 1961, op.220-244. 16. Charnes, A. and Cooper, V,r.W., "A C o n s t r a i n e d Game Fo r -mulation of A d v e r t i s i n g S t r a t e g i e s , " Econornetrica, 22, October 1954. . 17• Symonds, G.H., " A p p l i c a t i o n s t o I n d u s t r i a l Problems, I n c l u d i n g S c h e d u l i n g and T e c h n o l o g i c a l Research," Econome-t r i c a , 22, October 1954, p. 526*7 16. , " L i n e a r Programming: The S o l u t i o n o f Re f i n e r y Problems," Esso Standard O i l Company, Mew York, 1955 Chapter 5. 19• Bennion, E.G., " C a p i t a l Budgeting and Game Theory", Harvard Business Review, Volume 34, No. 6 (November-December) 1956, PP. 115-123. 13 a c h i e v e the same outcomes as a s i n g l e d e c i s i o n maker mak ing a l l the d e c i s i o n s . The p o s t u l a t i o n o f a method o f r e s o l v i n g t h i s c o n f l i c t o f i n t e r e s t f o r e a s i e r d e c e n t r a l i z e d d e c i s i o n - . mak ing as a s o l u t i o n o f an n - p e r s o n game was examined by 20 S h u b i k . A n o t h e r a r e a where game t h e o r y has drawn a t t e n t i o n i s 21 i n t he dynamics o f l sbour-managernent b a r g a i n i n g . The i n t e r -a c t i o n s o f t he p a r t i e s c o n c e r n e d r e q u i r e an a n a l y s i s o f t he p r o c e s s o f c o n c e s s i o n and the d e t e r m i n a t i o n o f the p o i n t where agreement would, be r e a c h e d on the b a s i s o f some a r b i -22 t r a t i o n schemes . S h u b i k has shown how game t h e o r e t i c methods c o u l d be u s e d i n a c c o u n t i n g i n the a s s i g n m e n t o f j o i n t c o s t s and the c o n s t r u c t i o n o f an i n c e n t i v e s y s t e m f o r d e c e n t r a l i z e d c o n t r o l . Such a t t e m p t s t o a p p l y the t e c h n i q u e t o v a r i o u s 2 0 . S h u b i k , M a r t i n , "Games, D e c i s i o n s and I n d u s t r i a l O r g a -n i z a t i o n , " Management S c i e n c e , Volume 6 , J u l y I 9 6 0 , p p . 4 5 5 -4 7 4 . The paper d i s c u s s e d t h e a p p l i c a t i o n and i n f l u e n c e o f game theor j r f r o m 2 - p e r s o n t o n - p e r s o n and e s p e c i a l l y o r i e n t e d t o w a r d p o s s i b l e a p p l i c a t i o n s i n i n d u s t r y . I n t h i s c o n n e c t i o n i t i s w o r t h w h i l e t o m e n t i o n t h a t t h e r e a r e many s o l u t i o n c o n c e p t s t o n - p e r s o n games a c c o r d i n g t o L u c e and R a i f f a ( R e f . F o o t n o t e 8, C h a n t e r I i ) . However i n a r e c e n t m a t h e m a t i c a l e x p o s i t i o n J . B . Rosen ( R e f . F o o t n o t e 7, C h a p t e r I I I ) has shown the e x i s t e n c e o f a u n i q u e e q u i l i b r i u m p o i n t under c e r -t a i n c o n s t r a i n e d , c o n d i t i o n s . 2 1 . C r o s s , J . C . "A Theo ry o f t h e B a r g a i n i n g P r o c e s s , " A m e r i c a n Economic R e v i e w , Volume 5 5 , 1 9 6 5 , p p . 6 7 - 9 4 . 2 2 . S h u b i k , M a r t i n , " I n c e n t i v e s , D e c e n t r a l i z e d C o n t r o l ; The Ass ignmen t o f J o i n t C o s t s and I n t e r n a l P r i c i n g , " Mana-gement S c i e n c e , A p r i l 1962 , p p . 3 2 5 - 3 4 3 . Ik management d i s c i p l i n e s go to show the wide scope and v e r s a t i -l i t y of i t s applications. Game theory has also made inroads into other d i s c i p l i -nary studies, namely national and i n t e r n a t i o n a l p o l i t i c s and psycnology. Purpose and Scope of Thesis I t is. becoming more evident that the a b i l i t y to deal e f f e c t i v e l y with various facets of competition, within and without the industry, ultimately determines whether an enter-pr i s e stands or f a l l s . A major objective of the thesis i s to draw attention to a body of theory, namely game.theory, which has drawn considerable i n t e r e s t i n recent years, and which i n the course of tin e may become an indispensable t o o l which ma-nagement can r e l y on i n seeking optimum decisions i n competi-t i v e s i t u a t i o n s . For t h i s reason, an attempt i s made at br-inging together the main body of game theory i n an a p p l i c a -t i o n oriented context, rather than the rigorous matnematical treatment that the subject i s currently receiving since the 25 von i'ieuraann and i-Ior gens tern c l a s s i c . 23 • A book which, gives the general d i s c i p l i n a r y areas where game theory has made progress i s "Game Taeopy-^ivi..Belated Approaches to S o c i a l Behavior", John Wiley ITSons ~ l h c . , New York, I9S 1:-, by Martin Shubik. The Journal of C o n f l i c t Re-solution contains a number of a r t i c l e s which deals with va-rious aspects of game theory. 2L:-. This i s es p e c i a l l y done i n "Strategy and Market Struc-ture" , John-'Wiley (2nd E d i t i o n ) , I960, by Martin Shubik. 25• op. c i t . 15 Some areas of management where the theory has found use and c o u l d p o t e n t i a l l y be used are observed. On the assumption t h a t advances i n computer technology are such t h a t v a s t amounts of d a t a and computation can e v e n t u a l l y be handled we then pro-' pose to see how game theory can e l u c i d a t e and c o n t r i b u t e t o s t r a t e g i c d e c i s i o n making i n the o i l i n d u s t r y * She mathemati-c a l model t h a t we examine i n t h i s c o n n e c t i o n i s the S h e l l mo-d e l . A t t e n t i o n i s drawn t o the a p p l i c a b i l i t y of the model from the standpoint of the r e s u l t s t h a t c o u l d be produced, and. the p o s s i b l e use t h a t could, be made of such r e s u l t s so t h a t more l i g h t w i l l be shed on d e c i s i o n making i n c o m p e t i t i v e s i t u a -t i o n s . Having o u t l i n e d the aim of the study, Chapter I I i s de-signed to b r i n g out the fundamental aspects of game th e o r y . We s t a r t o f f v/ith simple 2 - p e r s o n constant sum games. The more d i f f i c u l t t h e o r i e s of s o l u t i o n of nonconstant sum games are i n t r o d u c e d . Wherever a p p l i c a t i o n s of these t h e o r i e s have made important c o n t r i b u t i o n s , mention i s made of them. I m p l i c a -t i o n s ' of game t h e o r e t i c r e s u l t s - to our d i s c u s s i o n , e s p e c i a l l y 2 6 those from Hughes and Ornea's paper are h i g h l i g h t e d . E x t e n -s i v e games, a framework f o r moving to dynamics, form the l a s t s e c t i o n of Chapter I I . In Chapter I I I the focus i s on continuous v a r i a b l e and 2 6 . op. c i t . 16 p a y o f f spaces. .This forms the b a s i s f o r examining c o n t i n u -ous games. The d i r e c t i o n pursued i s , as the t i t l e suggests, the dynamics of game s i t u a t i o n s . To t i e up a l l t h a t has been s a i d on the essence of game theory we d i s c u s s the s t r u c t u r a l f e a t u r e s of the S h e l l model i n Chapter IV. An attempt i s made a t c o n s i d e r i n g the a p p l i c a b i l i t y of such a model t o an i n d u s t r i a l s i t u a t i o n cha-r a c t e r i z e d by the B r i t i s h Columbia o i l i n d u s t r y . The S h e l l model has been s u c c e s s f u l i n p r o v i d i n g r e -s u l t s of problems a r i s i n g from c o m p e t i t i v e s i t u a t i o n s . In view of t h i s , Chapter V sums up the survey w i t h a note on the re l e v a n c e of game the o r y . CHAPTER I I PERSPECTIVES IN THE THEORY OF GAMES 1. I n t r o d u c t i o n The v i r t u e s of mathematics are indeed many and v a r i e d . I t s use as a powerful device to make p r e c i s e concepts i n s o c i a l and economic s c i e n c e s has l e d to the f o r m u l a t i o n o f v a r i o u s t h e o r i e s of s o c i a l and economic b e h a v i o r . For, "Mathematical a n a l y s i s i s as e x t e n s i v e as nature i t s e l f , i t d e f i n e s a l l p e r c e p t i b l e r e -l a t i o n s , measures time, spaces, f o r c e s , tem-peratures I t s c h i e f a t t r i b u t e i s c l e a r -ness, i t has no marks t o confused n o t i o n s . I t b r i n g s t o g e t h e r phenomena the most d i v e r s e , .and d i s c o v e r s the hidden a n a l o g i e s which u n i t e them.... I t seems to be a f a c u l t y of the human mind d e s t i n e d to supplement the shortness of l i f e and the i m p e r f e c t i o n of the senses." — F o u r i e r , " A n a l y t i c a l Theory of Heat""^ However, we are ever i n danger of t r e a t i n g mathema-t i c s i n a bu s i n e s s r e s e a r c h paper f o r the sake of mathematics i t s e l f . T h i s has caused us to l o s e s i g h t of what we were i n i t i a l l y a f t e r , namely to make concepts p r e c i s e , c l e a r e r and u s e f u l . Mathematics, as a language i n s o c i a l and economic s c i e n c e s of which b u s i n e s s i s a p a r t , should be used on the 1. Quoted i n Herbert A. Simon, "Models of Man", John Wiley and Sons, Inc, New York, 1961, (2nd. L d i t i o r j p . l . 18 b a s i s of u t i l i t a r i a n r a t h e r than a e s t h e t i c grounds. The l a r g e number.of mathematical essays i n management s c i e n c e d e a l i n g w i t h game theory l o o k mainly at i n t e r e s t i n g , mathematical q u e s t i o n s . As d e c i s i o n , makers, our i n t e r e s t i n mathematical techniques i s not t h e - e l a b o r a t e proofs,, but r a t h e r t h e i r a p p l i -c a b i l i t y t o p r a c t i c a l . s i t u a t i o n s . On the oth e r hand opponents of the r o l e of mathemati-c a l models i n b u s i n e s s have claimed t h a t these are too s t e r i l e and r e s t r i c t e d , and, t h a t i n the o v e r s i m p l i f i c a t i o n and abs-t r a c t i o n of the r e p r e s e n t a t i o n of the phenomena, the s u b t l e -t i e s and v i t a l content of the s u b j e c t matter ere o f t e n des-t r o y e d . However, even with h i g h l y s i m p l i f i e d assumptions, f r e q u e n t l y the problems posed are mathematically unmanage-a b l e . True, the language of mathematics i s poor i n a d j e c t i v e s . But, i n s p i t e of t h i s drawback, a good mathematical model can d i s p l a y - a n a l y s i s o f the type which enables us to f o l l o w through a cha i n of re a s o n i n g not p o s s i b l e i n a v e r b a l des-c r i p t i o n of business- phenomena f u r n i s h e d w i t h r i c h a d j e c t i v e s . We have t o admit t h a t mathematics does serve us as a v e h i c l e f o r the study of the p r o p e r t i e s of the system. To see t h i s i m p l i c a t i o n we t u r n t o Shubik. "We need i n s i g h t s and breadth of view. I t i s necessary to couple t h i s w i t h l o g i c a l c l a r i t y and a n a l y t i c a l a b i l i t y . The type of mathema-t i c a l t h i n k i n g e x e m p l i f i e d by the theory of ga-. mes pr o v i d e s a u s e f u l methodology such t h a t with care a t l e a s t some aspects of the e s s e n t i a l f e a t u r e s of important problems can be examined."2 At the micro-economic l e v e l of the f i r m and market, i n s t i t u t i o n a l w r i t i n g s have been f a r too broadly d e s c r i p -t i v e and of use only i n conveying a general p i c t u r e of a f i r m or i n d u s t r y , not as an a i d t o decision-making f o r the operator of a f i r m . In the study of the competitive e n v i -ronment, we are t o l d ' t h a t , "The theory of o l i g o p o l y has long, been one of the most u n s a t i s f a c t o r y areas of economic theory. On the one hand, there have been a. s e r i e s of mathematical- models of markets which have defended on s i m p l i f i c a t i o n s so d r a s t i c as to render them of h i g h l y l i m i t e d value f o r any purpose. On the other hand, man]/ of the v e r b a l d e s c r i p t i o n s have tended to be h i s t o r i c a l anecdotal i n nature, pro-v i d i n g l i t t l e p o s s i b i l i t y f o r g e n e r a l i z a t i o n . , the o l i g o p o l i s t i c form i s domi-nant. The elegant and simple t h e o r i e s of pure competition and of monopoly <:>o not provide s u f f i c i e n t i n s i g h t i n t o the major market mechanisms of market s t r u c t u r e . Among the most g l a r i n g omissions from economic a n a l y s i s has been the l a c k of e x p l i c i t a t t e n t i o n to important d i s t i n g u i -shing f e a t u r e s between d i f f e r e n t competitive s i t u a t i o n s , such as type of d i s t r i b u t i o n and r e t a i l i n g system w i t h i n which a manufacturing f i r m operates..."3 2. Shubik, Martin,- '-Game Theory and Related Approaches  to S o c i a l Behavior", op . c i t . , p.4. 3 . Shubik,. M a r t i n , " Simulation and the Theory of the Firm", i n " C o n t r i b u t i o n to S c i e n t i f i c Research i n Manage-ment", The proceedings of the S c i e n t i f i c Program-Following the D e d i c a t i o n of the Western Data Processing Center, Gra-duate School of Business A d m i n i s t r a t i o n , U n i v e r s i t y of C a l i f o r n i a , Los Angeles, Jan. 2 9 - 3 0 , 1959, P . 7 0 - 7 1 . What we need thus i s a methodology which would p r e s -c r i b e g u i d e l i n e s f o r the p o s i t i v e treatment of d i f f e r e n t market forms. T h i s approach would of n e c e s s i t y be d i f f e r e n t from the patchwork theory t h a t has surrounded the economics of o l i g o p o l y . I t i s g e n e r a l l y ; b e l i e v e d ' . here t h a t the • methods of game theory c o u l d h e l p us f i n d answers t o p r o -blems a r i s i n g " from'competitive s i t u a t i o n s , , thereby' irnpro-... v i n g our d e c i s i o n malting c a p a b i l i t i e s . H o p e f u l l y , new i n -s i g h t s and f u r t h e r l i g h t would be shed on the.nature of cross-purposes maximization problems concerning competi-t i o n and. c o l l u s i o n as we probe deeper i n t o our a n a l y s i s . Before we begin to d i s c u s s the v a r i o u s t h e o r i e s of s o l u t i o n of games i t might be a p p r o p r i a t e to i n t r o d u c e the b a s i c concepts t h a t are used to c h a r a c t e r i z e a game s i t u a -t i o n . • Some of the important assumptions t h a t are i m p l i c i t or e x p l i c i t i n the f o r m u l a t i o n process w i l l be o u t l i n e d . !<e d e s c r i b e a c o m p e t i t i v e s i t u a t i o n as a game i n v o l -v i n g p l a y e r s or i n d i v i d u a l autonomous decision-making u n i t s , each of which has a s e t of s t r a t e g i e s which s p e c i f i e s cour-ses of a c t i o n f o r every p o s s i b l e s e t of moves of h i s oppo-nents. The r u l e s of the game s p e c i f y the v a r i a b l e s t h a t each p l a y e r c o n t r o l s , the i n f o r m a t i o n c o n d i t i o n s , and oth e r r e l e v a n t aspects of the environment p r i o r to the choice of an a c t i o n from a s e t of a l t e r n a t i v e s . The p a y o f f s are the values assigned to the outcome of the p l a y of the game. These elements form the b u i l d i n g b l o c k s c f game t h e o r y . 21 A h i g h l y complicated model of r e a l i t y w i l l mean the examination of a multitude of d e t a i l s , and, i t i s very l i -k e l y t h a t the purpose of the i n v e s t i g a t i o n would be l o s t i n the maze. By making an a b s t r a c t i o n w i t h s i m p l i f y i n g assumptions we can d i s c o v e r some of the non-obvious .pro-p e r t i e s of the r e a l w o r l d . A game theory model of a r e a l • world c o n f l i c t s i t u a t i o n has these o b j e c t i v e s i n mind when we attempt to put the l a t t e r down .on paper. I t i s not r e a -l i s t i c at the ou t s e t to. enumerate a l l the assumptions t h a t are a s s o c i a t e d with game theory as d e t a i l s and problems w i l l a r i s e as we proceed w i t h our a n a l y s i s . What we propose to do i s to reco g n i z e some of the more important ones and p o i n t out those t h a t we w i l l encounter as-we move a l o n g . True, some of the assumptions r e s t on shaky grounds, but we w i l l have something t o say about them i n a l a t e r c h a p t e r . The r u l e s of beh a v i o r i n game theory p e r t a i n t o those of the gen e r a l d e c i s i o n theory p o s t u l a t e s of which one of the most comprehensive i s found i n the Savage Axioms.' By r a t i o n a l b e h a v i o r of the d e c i s i o n maker, the assumption A. Savage, Leonard, J . , "The Foundations of S t a t i s t i c s " , John Wiley and Sons, Inc., Hew York", I S 0 4 . The book con-t a i n s new concepts i n the modern s u b j e c t i v i s t i c theory of d e c i s i o n making. In the main, the p o s t u l a t e s of d e c i s i o n theory and that' of game the o r y have very much i n common e.g. a. preference o r d e r i n g of outcomes, i n a d m i s s i b i l i t y of dominated a c t i o n s , maximization of expected u t i l i t y , e t c . , except t h a t i n game theory the assignment of p r o b a b i l i t i e s Is c o n d i t i o n a l on the r e s u l t s . 22 does not imply value judgement,but, that given h i s d e s i r e s as known, h i s choice of a c t i o n s should be c o n s i s t e n t w i t h h i s d e s i r e s . However, we know that as human beings we are not i n f a l l i b l e and thus i m p e r f e c t i o n s are u s u a l l y present. This assumption of i n d i v i d u a l behavior i s c r u c i a l to the whole superstructure of game theory and i t must be s t r e s s e d t h a t there should be no confusion about t h i s . The term r a t i o n a l i s , u n f o r t u n a t e l y f a r from being p r e c i s e and while i t assumes i n d i v i d u a l s maximizing something, i t i n a way p i c t u r e s the i n d i v i d u a l as a co n s e r v a t i v e , p e s s i m i s t i c being. The opponent i s assumed to be going a l l out to render him as much damage as p o s s i b l e and i s i n t e l l i g e n t and f u l l y unders-tands how t o take advantage of any s i t u a t i o n . To supplement our r a t i o n a l i t y p o s t u l a t e we t u r n to the concept of a u t i l i t y f u n c t i o n which describes the de-c i s i o n maker's preferences, and. the problem i n game theory becomes one of maximizing expected u t i l i t y . The measur-a b i l i t y of u t i l i t y i s an age old. controversy and we s h a l l 5 not dwell on i t here. To be able to a s s i g n a numerical 5. Some experiments have been performed on the messura-b i l i t y of u t i l i t y as an aid. to decision-making. V-hile there are many the ones that stand out are: Grayson, C. Jackson, "Decisions Under U n c e r t a i n t y , D r i l l i n g  D e cisions by O i l and Gas Operators", Boston, Harvard Univer-s i t y , D i v i s i o n of Research, Graduate School of Business A d m i n i s t r a t i o n , I960. Qwalm, R.0. " U t i l i t y Theory-Insights i n t o Risk Taking", Harvard Business Review, Rovember-December, 196"6~7 pp.123- • 126. Green, P.E. "Risk A t t i t u d e s and Chemical Investment  Decisions","Chemical Engineering Progress, January 1963, pp. 35-40. 23 value to the element i n the game mat r i x , to be d e s c r i b e d i n the next s e c t i o n , we have t o assume t h a t the p a y o f f s from an a c t i o n under a l l p o s s i b l e circumstances are n u m e r i c a l l y measurable. Game theory which von Neumann and Morgenstern^ deve-lo p e d i s a normative t h e o r y . The concept of s o l u t i o n i s p l a u s i b l y a s e t of r u l e s f o r each p a r t i c i p a n t which t e l l s them how to behave i n every s i t u a t i o n t h a t may c o n c e i v a b l y a r i s e . , F u r t h e r , as mentioned above, i t i s assumed we are p l a y i n g an adversary who i n a l l r e s p e c t s i s r a t i o n a l too, and, whose i n t e r e s t s are d i a m e t r i c a l l y opposed to ours. The q u e s t i o n we might ask i s , i f our opponents d i s p l a y i r r a t i o n a -l i t y does the s o l u t i o n s t i l l h o l d . I f we assume the p l a y e r s are s i m i l a r l y motivated the problem does not a r i s e . However, i n the event of facing, an i r r a t i o n a l opponent under complete i n f o r m a t i o n the normative theory would not n e c e s s a r i l y be the b e s t to f o l l o w as we c o u l d e x p l o i t our opponent's weakness. In r e a l i s t i c s i t u a t i o n s the assumption of complete i n f o r m a t i o n must give way to ignorance, u n c e r t a i n t y and i n d e t e r m i n a t e eco-7 nomic s i t u a t i o n s r e q u i r i n g more complex game a n a l y s i s . To be able to c o n s t r u c t the p a y o f f m a t r i x i n the d i s -c r e t e choice games and p a y o f f space i n continuous v a r i a b l e 6 . op . c i t . "Theory of Games and Economic Behavior." . 7 . Shubik,. M a r t i n , "Information, Risk, Ignorance and  Indeterminacy", Q u a r t e r l y J o u r n a l of Economics, Volume 6S. (1954), pp. 6 2 9 - 6 4 0 . 2h space games we must have a l l the necessary information. In-order to characterize a game i t must be assumed that we can specify a l l the variables which control the possible outcomes and a l l the values they may assume under different circums-tances. Leaving the controversial issues for discussion later we shall in the following sections turn our attention to the formulation of games.. 2. Theor^' of Solution of 2-person Constant Sum Games. This class of 2-person games is by far the most exten-sively developed and. familiar part of game theory. ' In general they f a l l into two categories, namely games where both players have f i n i t e strategies and those where at least one of the players has an in f i n i t e set of pure strategies. The f i r s t of these are known as.matrix games or games in norma- lized, form while the second type includes games of search and duels. The solution concept of the la t t e r class of ga-mes, having value in military work and weapon defence system, is the same as the former but the mathematical d i f f i c u l t i e s £. 2-p.-rson games have played a central role in the theory of games mainly because economic theories of linear program-ming and activity analysis receive much of their impetus from the interrelationship with 2-person games. For an elaborate enumeration of the historical and. mathematical prominence of 2-person- theory, one preferred to Luce, R.D. and Raiffa, H., "Games and Decisions", John V/iley and Sons, Inc., New York, (1957) PP. 56-57. are g r e a t e r . In the d i s c u s s i o n t h a t f o l l o w s we w i l l be concerned with matrix games. A f u r t h e r type of 2-person games c a l l e d e x t e n s i v e games w i l l be i n t r o d u c e d i n a l a t e r s e c t i o n as we move away from the p r i m a r i l y s t a t i c nature, of normalized games to the dynamic f e a t u r e s i n h e -r e n t i n e x t e n s i v e games f o r m u l a t i o n . I f 3-^  and S 2 are the set of s t r a t e g i e s f o r ' p l a y e r 1 ( h e n c e f o r t h c a l l e d P-^ } and p l a y e r 2 ( h e n c e f o r t h c a l l e d P^) r e s p e c t i v e l y , every p a i r of s t r a t e g i e s s^e S-^  and s . € S 2 w i l l have two p a y o f f f u n c t i o n s R , ( s . , s . ) and R„(s.,s.) a s s o c i a t e d w i t h i t . These r e p r e s e n t amounts going r e s p e c t -i v e l y to P-j_ and Po when they choose the s t r a t e g i e s s^ and By d e f i n i t i o n i n a constant sum game the a l g e b r a i c sum of the pay o f f s equals some co n s t a n t . ( 2 - 1 ) R - L U ^ S J ) * R 2 ( s i , S j ) = C In the l i m i t we have a zero sum game where C = 0 . In othe r words we have a s t r i c t l y c o m p e t i t i v e game, meaning what i s gained by- one of the p l a y e r s i s l o s t by the o t h e r . There i s thus no m o t i v a t i o n f o r c o l l u s i o n i n 2-person zero sum games. ( 2 - 2 ) R 1 ( s i , s J . ) r - R 2 ( S j L , s J 26 As the name i m p l i e s we can r e p r e s e n t the p a y o f f s t o P-j i n the form of a matrix as shown i n t a b l e (2-1). The b e h a v i o r i s t i c assumptions concerning the method of p l a y of zero-sum games i s g i v e n by the minimax and maxi-min p r i n c i p l e s . • i n the above game, i f he assumes the pos-t u l a t e s o u t l i n e d e a r l i e r , w i l l achieve h i s optimal s t r a t e g y by choosing i such t h a t max min R-^  ( , s ^ ) S i m i l a r l y P2 s choice of j must be such as to provide him a t l e a s t (2-3) max j (2-4) max i The value of the game i s thus determined and s i n c e a 9 s o l u t i o n e x i s t s we have an e q u i l i b r i u m p a i r of s t r a t e g i e s . The game i s s a i d to possess a s a d d l e - p o i n t and i s c a l l e d a s t r i c t l y ' determined game. A mixed s t r a t e g y must be employed i n cases where pure s t r a t e g i e s do nob y i e l d s a d d l e -p o i n t s o l u t i o n s . By a mixed s t r a t e g y we mean f o r P-, an 9. F u r t h e r d i s c u s s i o n of the mathematical concepts end proofs of s o l u t i o n s of zero-sum 2-person games can be found i n McKinsey, J . " I n t r o d u c t i o n to the Theory of Games". The Rand S e r i e s , McGraw-Hill Book Co., Inc.,. New York, 1952. Chapter 1 and 2. min Rgts^jS.) = max min -R-. (s.j_,s-) i J j i 3 or min R - ^ s ^ j S j ) = min max R - J _ ( S J L , S | ) 3 3 *^ p£ s Pure S t r a t e g i e s s . n ' R 1 ( s 1 , s 1 ) R 1 ( s 1 , s 2 ) R 1 ( s 1 , s . ) R l ^ s l , 3 n ^ •H &0 • CD • P * g • - p . CO CQ R L ( S 2 » S 1 ) m R l ( 8 2 ' 8 2 ) R l ( s 2 ' S j } R l ( s 2 ' S n } R l ( s i ' S l } R l ( s i ' S 2 } ' R l { s i > S i ) H l ( s i ' s n ) R (s ,s ) • R (s , s ) R , (s , s ) R,'(s ,s ) l m l 1 m 2 I m j x m n Table (2-1): Payoff M a t r i x f o r 2-Person Constant Sum Game 28 ordered M-tuple / T~ xi J s a t i s f y i n g the c o n d i t i o n s (2-5) X i > 0 (1-1,2 m 1 x ±- 1 , i = l t h a t i s , the p r o b a b i l i t y with which chooses s t r a t e g i e s i to m. S i m i l a r l y f o r P 2 we have an ordered n - t u p l e r?\*X2* • • i y n _7 s a t i s f y i n g (2-6) y. > o ( j = l , 2 , ,n) n I f P-^  employs mixed s t r a t e g y X." /~x^,Xp, >xm__7 anc* P 2 employs mixed s t r a t e g y Y- Lr~y1>Yz> »YN _7 then the mathematical e x p e c t a t i o n of P-j i s n m (2-7) E (X,Y) = 2 I R 1 { s i t s . ) X j L y . j= l i = l J J I f i t happens t h a t f o r some X i n and some Y i n 3^ vie have, (2-8) E (X,Y") « E ( A V ' ) < E(X | Y ) f o r a l l X i n S-j and a l l Y i n S 2 then X and Y. are the op-t i m a l mixed s t r a t e g i e s f o r P]_ and P 2 r e s p e c t i v e l y . E(}'!,'Y') i s c a l l e d the value of the game to 1\, X and Y b e i n g the 2 9 s o l u t i o n t o the game or as McKihsey c a l l s , a s t r a t e g i c s a d d l e - p o i n t ^ Thus, f o r games without pure e q u i l i b r i u m s t r a t e g i e s we are l e d to a mixed s t r a t e g y e q u i l i b r i u m . As a- matter of f a c t , .the pure s t r a t e g y e q u i l i b r i u m p a i r i s a s p e c i a l case of the g e n e r a l • mixed s t r a t e g y e q u i l i b r i u m s o l u t i o n w i t h the p r o b a b i l i t y of one of s e l e c t i n g the maximin s t r a t e g y ( f o r P^). In t h i s sense the theory o f 2-person s t r i c t l y c o m p e t i t i v e game should be adequately d e s c r i b e d by the theory of mixed s t r a t e g y e q u i l i b r i u m p a i r s . To pursue the n o t i o n of mixed s t r a t e g y f u r t h e r we note t h a t ma thematic all}'- the mixed s t r a t e g y s o l u t i o n i s i r r e p r o a c h -a b l e , but the problem i s i n i n t e r p r e t i n g i t . In p r a c t i c e we would q u e s t i o n s e l e c t i n g a s t r a t e g y on some randomized proce-dure. I t might be argued t h a t mixed s t r a t e g i e s are d e s i r a b l e i n order to w i t h h o l d from our opponents knowledge of the pure s t r a t e g y we w i l l use to a v o i d them e x p l o i t i n g us. Whenever we r e f e r t o chance device i n a s s i g n i n g p r o b a b i l i t i e s to v a r i o u s c h o i c e s , we are i n t u i t i v e l y having i n our minds games which w i l l be pla y e d many times i n s u c c e s s i o n . Business d e c i s i o n s are u s u a l l y of "one-shot" n a t u r e . Once we have committed o u r s e l v e s to a c a p i t a l expenditure the f l e x i b i l i t y of change 10. I b i d . Chanter 2. The chanter i n c l u d e s mathematical proofs o f the ex p r e s s i o n (2-7 i and the minimax theorem. on the whole i s minimal. There are, however, cases where mixed s t r a t e g i e s are o p e r a t i v e , and, i n f a c t , q u i t e u s e f u l as i n open market o p e r a t i o n s and currency d e v a l u a t i o n where i t i s used to minimize the chance of s p e c u l a t o r s t a k i n g advantage of i n f o r m a t i o n l e a k s . N e v e r t h e l e s s we are drawn to t h i s s t r i k i n g statement by Luce and R a i f f a . "A s t r a t e g y which i s good i n the t o t a l context of the c o n f l i c t of i n t e r e s t may appear to be poor i n a l i m i t e d c o n t e x t . In e v a l u a t i n g s t r a t e g i e s t h i s d i s t i n c t i o n between context i s , , of course, important, but i t i s o f t e n d i f f i c u l t t o maintain, when c o n s i d e r i n g p a r t i c u l a r cases . . . . U n f o r t u n a t e l y , the s t r a t e g i s t i s o f t e n e v a l u a t e d i n terms of the outcome of the adopted choice r a t h e r than i n terms of i t s s t r a t e g i c d e s i r a b i l i t y i n the whole r i s k y s i t u a t i o n . " 1 1 .It must be pointed out t h a t i n . , the d i s c u s s i o n so f a r , " the f o r m u l a t i o n of 2-person zero-sum or s t r i c t l y c o m p e t i t i v e games g l i d e s over the method of s o l u t i o n . We have not g i v e n i t any r i g o r o u s mathematical treatment. Indeed s e v e r a l a r t i c l e s and bocks have been w r i t t e n t h a t are con-cerned with the mathematical d e t a i l s of s t r i c t l y compe-t i t i v e games and we have g i v e n l i t t l e c o n s i d e r a t i o n to t h i s aspect. T.re have f o r reasons e x p l a i n e d i n Chanter I t h a t the purpose here i s to determine a p p l i c a t i o n s - f o r game theory, not 11. OP . c i t . p. 76. 31 g e t t i n g involved, i n mathematical i n t r i c a c i e s . However, a number of methods are employed, i n the s o l u t i o n of. 2-person zero-sum games. These i n c l u d e t r i a l and e r r o r , simplex 1? method and the l i k e . " As f a r as a p p l i c a t i o n s go, there are few s i t u a t i o n s or o r g a n i z a t i o n s t h a t can be c h a r a c t e r i z e d s u c c e s s f u l l y by means of 2-person constant sum games. One may t h i n k of a s i t u a t i o n where two fi r m s are i n v o l v e d i n an. a d v e r t i s i n g competition f o r a market th a t i s already saturated, so t h a t 13 what i s l o s t by one i s gained by the other. As f a r as d i r e c t a p p l i c a t i o n i s concerned there appears according t o Shubik^^'ve'ry few i n the areas of economics, s o c i a l , or indus-t r i a l o r g a n i z a t i o n but have been a p p l i e d . i n m i l i t a r y problems. 3• Theories of S o l u t i o n of n-Person Games The moment we move away from constant.sum 2-person games the a n a l y s i s becomes involved- E x t r a game t h e o r e t i c considera-t i o n s such as "bargaining psychologies of the p l a y e r s " and " i n t e r p e r s o n a l comparison of u t i l i t y " w i l l - a d d to . the com-p l e x i t i e s and modify the s o l u t i o n . By n-person games we 12. For a d e t a i l of the methods, see Appendix 6 i n Luce and R a i f f a , I b i d . 13. See Charnes end Cooper, OP . c i t . I/:.. Op . c i t . "Games, Decisions and I n d u s t r i a l O r g a n i z a t i o n . " 32 include variable sum games for n>2 and constant sum games for n>2. As the t i t l e suggests there i s no single theory of n-person games but a number of them."'*'' We shall however be concerned v/ith von Neumann-Morgenstern and Nash"*"^  theories. Broadly, the theories f a l l within two main dis-tinctions, namely cooperative and non-cooperative theories, with r e a l i s t i c cases f a l l i n g in between these polar extremes. In cooperative theory we are dealing with cases where there i s a tendency to joint maximization through cooperation or collusion, open preplay communication and side payments. The latter type examines situations where there i s no or limited communication, no side payment and independent individual action. Before oroceeding to give the definition of a non-cooperative 2-person nonconstant sum game i t i s proposed to present a classical example of this class of games. The purpose i s to bring out some of the interrelationships that exist between the strategies and objectives of several decision makers. In particular, the concept of equilibrium would be introduced. Let us turn to the famed "Prisoners' Dilemma Game" attributed to A.W. Tucker. Two suspects taken separately 15. Luce and Eaiffs. OP . c i t . Chapter 6 gives a c r i t i c a l analysis of the important theories. 16. Hash, J.F., "The Bargaining Problem", Econcmetrica, Volume 18, (April 1950), pp." 155-162'and "Two-Person Co-operative Game s ", E c onome t r i c a , Vol unie 21, (January 1953) pp. 128-140. into custody are confronted with the problem of either con-fessing and turning state evidence or not confessing. Both know that i f they do not confess they w i l l escape with re-latively light sentences. However, i f one confesses and the other does not, the non-confessor w i l l be faced with heavy punishment. Finally i f both confess they w i l l receive char-ges that are heavy but not as heavy as the most severe one. The game matrix i s drawn with figures respresenting the prisoners' payoffs in table (2-2) Prisoner 2 Not Confess b, Confess bo Mot " Confess a-^  Prisoner 1 (-1,-1) (-10,0) Confess a^ (0,-lC) (-6,-fi) Table ( 2 - 2 ) : 1 7 Prisoners' Nil emma Game Since the str^ tegy a2 domina oes a-^  and strate.gy b 2 17. Note that the f i r s t numeral within the brackets accord ing with .convention refers to nayoff goirirr to row nlaver, i.e Prisoner 1, while the second numeral refers to payoff to columnplayer, i.e. Prisoner 2. 3h dominates b-^ , i t w i l l be observed that by applying the cri t e -rion of.individual rationality both prisoners w i l l be moti-vated to adopt the position of confessing. The strategies a 2. and b 2 are maximin strategies and the outcome (-8,-8) is said to be an equilibrium outcome. The outcome (-1,-1) i n -18 volving strategies a and.b i s termed by Luce and Raiffa as a cuasi-equilibrium outcome and occurs only in a repeated game which eventually settles at ( - £ , - 8 ) , caused by one of the players "defecting" or "double-crossing". One point of note i s that although the payoff (-8,-8) is in equilibrium, i t is not a jointly desirable one. Many 2x2 games of this nature have been developed and a complete classification of 2x2 games -with outcomes on an ordinal scale can be found in an article by Rapoport and Guyer. J It is not proposed to go into a discussion of these, as, although recognizing the value of such games in bringing, out the essential nature of conflict and cooperation and enabling anyone interested in thinking logically about compe-ti t i v e situations, they are too restrictive for most analysis of industrial situations. It must be pointed out that even 18. op . c i t . p. 98. 19. Rapoport, Anatol and Cuyer, Melvin, "A Taxonomy of 2x2 Games", General Systems: Yearbook for the Society of General Systems- Re search, Volume XI- (1966). op. 2C3-21Z..." 35 in a 2x2 game we do not necessarily have unique stable e q u i l i -brium as the above case, but conditions such as strongly stable or Pareto equilibrium, stable and unstable e q u i l i -brium and other properties such as non-conflict, threat-vul-20 nerable, essential , inessential and the like* Although we began to define the concept of solution of non-cooperative nonconstant sum games for 2 players i t can be generalized for any number of players using mixed strate-21 gies. Using the notations set out in formulating 2-person zero sum games we have for a pair of strategies (s-pS^) to be in equilibrium the conditions that (2-9) ^ ( ^ , § 2 ) > R 1 ( s 1 , s 2 ) for s ] *S 1 ^2(2-) ,^2) ^ R/pls-pSg) for S2 eS2 This amounts to saying that given that P 2 selects his strategy "~2' ^ l w i l l , select his strategy s-]_ which maximizes his payoff against P 2 ' s choice. The same applies to ? 2 . Thus for equilibrium the strategy pair (s-j , s 2) must be jo int ly ad-missible. In general i f ^ is mixed strategy to player i (denoted by P i ) then the payoff function to P^ is defined by 20. Ibid. We shall have more to say about Pareto e q u i l i -brium later in the section. 21. Nash, J . F . , has shown in his paper ''Non-cooperative  Games" Annals of Mathematics, 54, (September] 1951. pp.286-296, . that ' a mixed strategy equilibrium solution ex-ists i n non-cooperative games with f in i te sets of pure strategies. 36 R j _ ( ^ , » ^ n ^* t * l e n - t u p l e of mixed s t r a t e g i e s ^ l ' ^ 2 ' ' ^ S ^ e n o t e d $ w e C G n l e t t h e e x p r e s s i o n l ^ ) to stand f o r ( ^ , ^ ^ • > ? „ ) • F o r $ to be i n e q u i l i b r i u m we must s a t i s f y (2-10) R. ($) max R ± ( ^ , a l l *?. meaning t h a t each p l a y e r maximizes h i s expected p a y o f f i f the s t r a t e g i e s of the o t h e r s are h e l d f i x e d . T h i s d e f i n i t i o n of e q u i l i b r i u m i s e s s e n t i a l l y s t a t i c and does not r e f l e c t the p s y c h o l o g i c a l f a c t o r s . o f non-cooperative games. We w i l l con-s i d e r the dynamics of a c t i o n and r e a c t i o n , i n t e r p l a y and motion-in the n e x t ' c h a p t e r . A d e c i s i o n to cooperate a r i s e s because i n general, there i s much more to g a i n by c o o p e r a t i n g than not - c o o p e r a t i n g . In s p i t e of gains from j o i n t a c t i o n , i n t e r e s t s are u s u a l l y opposed when we come to d i v i d i n g the p r o f i t s . It. i s here the c o o p e r a t i v e t h e o r i e s p l a y t h e i r r o l e s i n d e c i d i n g what i s a " f a i r " d i v i s i o n . In von Neumann and Morgensterntheory we are i n t r o d u c e d to the n o t i o n of the c h a r a c t e r i s t i c f u n c t i o n of a game, a f u n c -t i o n which a s s i g n s a value to every subset of p l a y e r s . I f we n _ have n- p l a y e r s the c h a r a c t e r i s t i c f u n c t i o n has 2 v a l u e s . For example, i f we have a s i t u a t i o n i n v o l v i n g 3 p l a y e r s , we have 37 where $ i s the empty s e t i n v o l v i n g no p l a y e r = z1 *i2) = z2 *(3) = z 3 ^(1,2) = z 4 \*(l , 3 ) = z 5 ^(2,3) = z 6 *(1,2,3) = z y where 1,2,3 r e p r e s e n t the p l a y e r s and Zj_ f o r i = 1,2, ,7 are the pa y o f f s to t h e . v a r i o u s c o a l i t i o n s . The c h a r a c t e r i s t i c f u n c t i o n has a f u r t h e r p r o p e r t y t h a t g i v e n a p a r t i t i o n of the p l a y e r s i n t o two subsets ^ and T, then (2-11) A M S U T) > ^ ( S ) + t f(T) when S O T = / The c o n d i t i o n given i n (2-11) i s fundamental i n ex p r e s s -i n g von Neumann and Morgenstern s o l u t i o n concept of the s e t of imput a t i o n s . By imputation i t i s meant t h a t i f - ( re p r e s e n t s a s e t of values i n a 2-person s i t u a t i o n such t h a t (2-12) * 1 + ^ = ^(1,2) cC > >H1) <*2 > ^(2) then i s an imp u t a t i o n . The s o l u t i o n to t h i s game c o n s i s t s of a s e t of imputa-t i o n s which dominates a l l others but do not dominate each other. T h i s i s e q u i v a l e n t to say i n g t h a t the s o l u t i o n l i e s 38 tie on the Pareto surface. ' Beyond that the solution does not narrow the f i e l d of choice as much as we would l i k e . Pictorially the outcomes of the game can be represen-ted by the convex region R shown in figure (2-1). The undomi-nated set or the Pareto optimal set l i e s on the section formed by the extreme points a,b,c and d. If (u,v) represents . the maximin values of the game then the boundary of R between the dashed vertical and horizontal through (u,v) as represented by lbcrn, is the negotiation set of the game. This i s what von Neumann and Morgenstern terms the cooperative solution. 23 The Mash solution for 2-person cooperative games i s based on four basic assumptions. They are; (1) there i s no interpersonal comparison of u t i l i t i e s or in other words we can pick a natural position of status quo, (2) the solutions are Pareto optimal, (3) the solution i s independent of i r r e -levant alternatives and (/..) the symmetry nature of "fairness" i s specified. ¥e w i l l not dwell on their limitations and question their validity as the opportunity w i l l arise again when \»:e discuss the dynamics of the bargaining process. VJe should'however recognize that the formulation has advanced 22. A division of proceeds is said to be Paretc optimal i f no individual can increase his welfare by departing from i t without at least one other individual suffering a. decrease in his welfare. This icea is similar to the contact curve of Edgeworth indifference solution in 2-dimensionai economics. 23. Nash, J.F., "The Bargaining Froblern" and "'Two-person ' Cooporative Games", op . c i t . 3 9 i i t P l a y e r 1 s u t i l i t y F i g u r e (2-^1). Pareto Optimal and Ne g o t i a t i o n . Sets P'2 s U t i l i t y (V) F i g u r e ( 2 - 2 ) . L i n e a r T'rensforrnation of Convex U t i l i t y F u n c t i o n our understanding o f c o n f l i c t s i t u a t i o n s by i n t r o d u c i n g the r o l e of t h r e a t s . Thus i f we accept the Mash assumptions the a n a l y s i s proceeds as f o l l o w s : Given two u t i l i t y f u n c t i o n s U and V of two p l a y e r s P-^  and F 2 we can d e s c r i b e a convex r e g i o n ^ r e p r e s e n t i n g . a l l p o s s i b l e p rospects of p a y o f f s . T h i s i s shown i n f i g u r e (2-2). Amongst these t h e r e i s one s i t u a t i o n i n which no trade ensues or the s t a t u s quo p o s i t i o n where the p l a y e r s a c t n o n - c o o p e r a t i v e l y . We denote t h i s by ( u 0 , v Q ) . By a l i n e a r t r a n s f o r m a t i o n of u t i l i t y we can always reduce t h i s to the ( 0 , 0 ) p o i n t . The Nash s o l u t i o n i s the d e t e r m i n a t i o n of a unique 25 p o i n t '• based on h i s assumptions of the game re p r e s e n t e d by the r e g i o n S and the s t a t u s quo p o i n t (u , v Q ) . We can denote such a game by / ~ 3 , ( u 0 , v 0 ) J . T h i s p o i n t must of course be on the Pareto s u r f a c e because no p o i n t s w i l l be c o n s i d e r e d i f both p l a y e r s can j o i n t l y improve t h e i r p o s i t i o n by choos-i n g another prospect. Hence i f we t r a n s f o r m ( u 0 , v Q ) i n t o ( 0 , 0 ) the r e s u l t a n t game d e s c r i b e d by the r e g i o n above and to the r i g h t of s t a t u s quo p o i n t can be c o n s i d e r e d t o be independent of . t h e l a r g e game S (assumption 8) . Nash s o l u t i o n i s then the maximum of the product of a l l (u',v') of the new game /V , ( 0 , 0 ) J , i . e . 2.1,. The r e g i o n i s convex because f o r any two prospects w i t h d i f f e r e n t u t i l i t y p a i r s there w i l l always be oth e r prospects whose u t i l i t y p a i r l i e s on the s t r a i g h t l i n e connecting"them. I f the r e g i o n r e p r e s e n t s f i n i t e s e t of t r a d e s , i . e . commodities are not i n f i n i t e l y d i v i s i b l e we have a convex polygon. 2 5 . Convexity o f S or S ensures one maximal element. hi F i n d u;|. v.., > u v f o r a l l (u,v ) b e l o n g i n g t o S G e o m e t r i c a l l y the s o l u t i o n i s g i v e n by the p o i n t of tangency of the Pareto s u r f a c e and the e q u i l a t e r a l h y p e r b o l a UV-constant, shown i n f i g u r e (2-3). One i s able to compare Nash's c r i t e r i o n of " f a i r d i v i s i o n " w i t h one i n which the u t i l i t i e s U and V are made equal and the other which i s the 26 maximization of the sura of u t i l i t i e s U and V. The p o i n t s , a, b and c i n f i g u r e (2-3) r e p r e s e n t these r e s p e c t i v e l y . F i g u r e (2-3): Comparison of Nash's and Other " F a i r D i v i s i o n " C r i t e r i a 26. U=f(V) U+V=f(V)+V ^ L i = f (V)+1=0 dv .'.f (V)=-l i . e . the p o i n t of j o i n t maximun on the Pareto s u r f a c e has a slope of -1. Hash's axioms can be g e n e r a l i z e d f o r n - p l a y e r s . The problem becomes, (2-13) max TTui f o r i ^ ^ , ,n The problem.however i s i n f i n d i n g the s t a t u s quo p o s i t i o n because u n l i k e 2-person games there i s a p o s s i b i -l i t y of forming c o a l i t i o n s t r u c t u r e s . We s h a l l d e a l b r i e f l y here with the c o n c e p t of op-t i m a l t h r e a t s t r a t e g i e s w i t h and without s i d e payments. A " f a i r d i v i s i o n " of p r o f i t s as a measure of pay o f f can only be determined a f t e r a n a l y s i n g the optimal t h r e a t s t r a t e g i e s e q u i v a l e n t t o the s t a t u s quo p o i n t outlined, above. Two p l a y e r s i n a c o o p e r a t i v e game, having the o b j e c t of making; the outcome to themselves as f a v o u r a b l e as p o s s i -bles w i l l p i c k t h r e a t s t r a t e g i e s t h a t guarantee the l a r g e s t amount of damage to t h e i r opponent. In doing so i t i s i n e v i t a b l e t h a t the t h r e a t e n e r w i l l have to bear a c e r t a i n c o s t . I f we assume as Nash does t h a t both p l a y e r s have the same e v a l u a t i o n f o r outcomes then the optimal t h r e a t s t r a -t e g i e s w i l l be f o r the p l a y e r s to (2-li.) min max (R,-R 0) = max min (R.-R;,) 1 2 1 * 2 1 1 <• where R^  and R^ are the payoffs t o P-^  and P 2. What t h i s means i s f o r P^.to maximize (R^-R 2) and P 2 t o minimize t h i s q u a n t i •4-3 I t i s f o r t h i s reason that the t h r e a t curve i s sometimes known as the maximin surface ( a l s o c a l l e d Pareto minimal s u r f a c e ) . The p a i r of t h r e a t , s t r a t e g i e s f o r games where players c o l l u d e and side payments are permitted w i l l be d i -f f e r e n t from t h a t i n which no side payment i s permitted. When the players have i d e n t i c a l cost s t r u c t u r e they c o i n -c i d e . Where side payment i s permitted the poin t of " f a i r d i v i s i o n " can be determined g e o m e t r i c a l l y by a 45° l i n e drawn from the t h r e a t p o i n t t o the j o i n t maximun l i n e , the c o n s t r u c t i o n of which was demonstrated i n f i g u r e ( 2 - 3 ) . The determination of no side payment optimal t h r e a t s t r a t e g i e s and the Nash s o l u t i o n i s s l i g h t l y more c o m p l i c a t e d . ^ We w i l l however proceed w i t h the general approach of o b t a i n i n g the Fareto optimal and optimal t h r e a t surface which i s fundamentally more important and forms the b a s i s f o r c a l c u -l a t i n g optimal t h r e a t and Nash e q u i l i b r i u m p o i n t s . A r a t i o n a l p l a y e r 1 (P^ ) w i l l attempt to max min R,.(s. ,s .) • s i - V P l a y e r 2 (P 2) w i l l make every e f f o r t to keep t h i s minimal. This c o n d i t i o n alone i s i n s u f f i c i e n t to define an outcome. I f we assume th a t P 2 i s r e s t r i c t e d to making a f i x e d p r o f i t 27- ohubik,''Strategy and Market Appendix C, pp. 360-361.. . S t r u c t u r e " , op . c i t . R->(sj,)=C, then f o r the p r o f i t t o be Pareto o p t i m a l , ? R 1 ( s s.) ^ R ^ s . , 5 , ) (2-15) dR. (s,, s .) = — ^ r — ^ - d s ± + SrT~— d a , = O S i m i l a r l y , (2-16) ^ R 2 U i ' S . i ' A ? » 2 ( » i - . * , i ) d s i + 9 s ds..= 0 The above two equations are summarized i n the J a c o b i a n (2-17) ? R , (s. ,s .) 9 S j L ?R, ( s . , s . ) 1 1 . 1 ^ R 2 ( s i ' a j ) 3 s , = 0 The e x p r e s s i o n (2-17) i s s a t i s f i e d by 2 curves, the Pareto optimal and the optimal t h r e a t c u r v e s . The t y p i c a l shape of the curves and the l o c a t i o n of the game t h e o r e t i c p o i n t s f o r the 2-person case d i s c u s s e d above is.shown i n f i g u r e (2-4). We have shown i n g e n e r a l how Nash theory c o u l d be g e n e r a l i z e d f o r n-person non-cooperative games / " e q u a t i o n (2-10) J and co o p e r a t i v e games /""equation (2-13) _7« I t was noted then t h a t we have t o e x p l i c i t l y i n c o r p o r a t e c o a l i t i o n s t r u c t u r e s i n our a n a l y s i s , g i v e n t h a t c o o p e r a t i o n i s allowed I f X denotes the p a r t i t i o n of the p l a y e r s i n t o c o a l i t i o n , the Nash e q u i l i b r i u m i s r i g h t f u l l y d e s c r i b e d by, >5-Payoff to P, • Figure ( 2 - 4 ) : Game Theoretic Points on Payoff Space n < ; 2 , > s n > , * j where ( > c o n s t i t u t e , as defined e a r l i e r the n-tuple of mixed s t r a t e g i e s i n e q u i l i b r i u m . I f X - T {1} , { 2 } , { n } J the c o a l i t i o n s t r u c t u r e i s s a i d to. have no n o n - t r i v s l c o a l i -t i o n s and thus we have the c o n d i t i o n of non-cooperation. I n cooperative games the s p e c i f i c a t i o n of f i s important and 2 S . Source: Chapter 5 . Shubik, 1 'Strategy and Market St r u c t u r e h6 the e q u i l i b r i u m c o n d i t i o n w i l l depend on i t . Between the cases where.we have p e r f e c t c o o p e r a t i o n and non-cooperation there i s a host of p o s s i b i l i t i e s . I t i s observed i n . p r a c -t i c e t h a t the c o a l i t i o n s t r u c t u r e has c e r t a i n l i m i t a t i o n s p l a c e d on i t s contemplated, changes. T h i s i s s p e c i f i e d i n a r u l e of a d m i s s i b l e c o a l i t i o n changes summarized i n the 29 f u n c t i o n Y'(T). 7 These r u l e s may be thought of as s o c i o -30 l o g i c a l , c o sts o r o t h e r r e s t r i c t i o n , l i k e n e d t o the bound-ary . c o n d i t i o n s of c l a s s i c a l mathematical, p h y s i c s problems. In g e n e r a l one i s l e d i n a c o o p e r a t i v e game to search f o r a p a i r / " { ^ U ^ ) , £ ( T 2 ) , , ^ j ) , T_7, where t M T - p T g , ,T - t) and ^ ( T ^ ) denotes a t y p i c a l c o r r e l a t e d mixed s t r a t e g y j o i n t l y chosen by-the p l a y e r s i n the c o a l i t i o n s t r u c t u r e T\. I f i t i s i n e q u i l i b r i u m then f o r a t l e a s t one p l a y e r j i n M4 V( t ) i . e . K i s a p o s s i b l e c o a l i t i o n change i n ; t , R j i f ^ V , t ( T 2 ) , . . . . , £ ( T n ) J > R . £ X ( ^ A ^ ) J : T^(I-i) i s the.mixed s t r a t e g y of the c o a l i t i o n formed by the remaining p l a y e r s not i n K, chosen such as t o minimize the payof f t o j . T h i s , of c o u r s e , i s a c o n s e r v a t i v e d e f i n i t i o n of e q u i l i b r i u m of an n-person c o o p e r a t i v e game. 29. A d i s c u s s i o n of the f u n c t i o n V( TT) i s g i v e n i n Luce end Ra.iffa, OP . c i t . S e c t i o n 7.6 of Chapter 7 and Chapter 10. * 30. For example, i n the c o a l i t i o n s t r u c t u r e *~-= / " *{. 1} , {^ } » {*0 J formed by p l a y e r s 1,2,3 we might impose the c o n d i t i o n t h a t the a d m i s s i b l e c o a l i t i o n s are y( X ) 1 \ , {2} , {3} , {1,2} , (2,3} , {1,3} J . k7 Although there are few direct'applications of the theories of n-person nonconstant sum games to operating problems in industry, the potentialities of this part of game theory is great. Host of the d i f f i c u l t i e s are com-putational in nature, but, with the rapidity at which computer technology is advancing this w i l l eventually be .overcome. Ive have already mentioned one breakthrough in 31 the use of these t h e o r i e s in the o i l industry. Charnes and Cooper-^ had made a study using an n-person game model with Nash equilibria in association with the Chicago Area Transportation Study to simulate patterns of t r a f f i c flow when the huge size of the problem made ordinary cut-and-tried simulations prohibitive for electronic computer runs. 33 -In other aspects Shubik" has -written about the possible use of game theory in joint costs allocation for decentra-lized, decision making. 4• Game Theoretic Points: Their Implications to Decision  Making. The above discussion has mainly centred around the 3 1 . Hughes and Ornea, op . c i t . 32. Charnes, A. and Cooper, VI. W. "Extremal Principles  for Simulating Traffic Flows Over a. Network of City Streets" Proceedings of the National Academy of Sciences, Volume 44:2 February 1 9 5 P P - 201-204. 33-. Shubik, Martin, "Incentives, Decentralized Control, The Assignment of Joint Costs end Internal Pricing," Loc.cit. concept of e q u i l i b r i u m and t h r e a t s t r a t e g i e s under v a r i o u s context i n a s t a t i c s e t t i n g . I t i s thus a p p r o p r i a t e here t o examine q u a l i t a t i v e l y i n g r e a t e r d e t a i l s what i m p l i c a - t i o n s , b a s i c a l l y these concepts have i n d e c i s i o n making i n  comp e t i t i v e s i t u a t i o n s . The word e q u i l i b r i u m i n a sense conveys a f e e l i n g of a s t a t e of balance or i m m o b i l i t y . In a r r i v i n g a t the e q u i l i b r i u m ( i n the non-cooperative s e n s e ) , we s t a t e d t h a t any u n i l a t e r a l movement by a p l a y e r away from i t w i l l make him l e s s w e l l o f f than he was a t the e q u i l i b r i u m . I t i s reasonable t o expect t h a t none of the p l a y e r s w i l l be mo-t i v a t e d to s h i f t . But, i t i s c e r t a i n l y p o s s i b l e f o r anyone p l a y e r to move away from the e q u i l i b r i u m ( a g a i n s t h i s immediate i n t e r e s t ) i n order t o motivate another, t o s h i f t ( i n p u r s u i t of immediate 'gains or t a k i n g any consequence of the s h i f t ) so as to o b t a i n f o r the i n i t i a t o r of the s h i f t a l a r g e r p a y o f f e v e n t u a l l y than he gets at the e q u i -l i b r i u m . An example would be the p r i c e c u t t i n g move by a f i n a n c i a l l y l a r g e r f i r m t o d r i v e out the weaker ones so t h a t i t w i l l be l e f t w i t h l a r g e r shares a t the end. One c o u l d induce another p l a y e r to s h i f t s t r a t e g y i f the other p l a y e r sees t h a t i t i s to h i s advantage to s h i f t r a t h e r than t o s u f f e r the consequences of the f i r s t p l a y e r s h i f t i n g . In another s i t u a t i o n , the s h i f t i n g ; of the s t r a t e g y of a. p l a y e r may f o r c e a second p l a y e r to s h i f t a l s o because of the 9^ advantage of the l a t t e r i n doing so. In a competitive d e c i s i o n s i t u a t i o n we can broadly i d e n t i f y f o u r general s t r a t e g i e s - e q u i l i b r i u m , c o o p e r a t i v e , t h r e a t and defence. We can, as was done i n the s e c t i o n s before this,determine these points arid p l o t them out g r a -p h i c a l l y i n payoff space. I f i t i s a one d e c i s i o n v a r i a -b l e problem i n v o l v i n g two p l a y e r s , which i s very u n l i k e l y i n p r a c t i c a l s i t u a t i o n s we could a l s o draw out the c o n f l i c t s i t u a t i o n on a 2-dimensional v a r i a b l e plane. Host game theory a n a l y s i s i n t e x t and papers end here l e a v i n g the •decision maker who wovuld e v e n t u a l l y use the technique to decide f o r h i m s e l f ' . a host of unanswered questions. Let us p i c t u r e a s i t u a t i o n i n which two competing fi r m s are faced w i t h manufacturing and marketing investment d e c i s i o n s . \Te could on the assumption of p e r f e c t informa-t i o n construct the payoff f u n c t i o n p r o f i l e s r e p r e s e n t i n g the c o n f l i c t s i t u a t i o n i n f i g u r e ( 2 - 5 ) . One question that comes i n t o the f o r e i s the changes that are l i k e l y to occur on the o o s i t i o n of the e q u i l i b r i u m point (S/E) given changes i n the r e s t r i c t i o n s that may be Placed on the a c t i o n s of the d e c i -s i o n maker. For example, an obvious one would be that investment cannot be i n f i n i t e , g i v e n l i m i t a t i o n s of f i n a n -cing;, shareholders s a n c t i o n s , e t c . 34-c i t . An exception i s the paper by Hughes and Ornea, op . 50 Pareto S u r f a c e i '—. i E 1 i - I . E ' x - ^ 2 IE \ a l E / E \ s. 1 \ A S 2(T/D) \ S 1(D/T) E.. ,E ' E_, E,, are the e a u i l i b r i u m p o i n t s r e s u l t i n g 1' 3 4 • from v a r i o u s r e s t r i c t i o n s p l a c e d on the f i r m s . Figure' (2-5): E f f e c t of C o n s t r a i n t s on  P o s i t i o n of E q u i l i b r i u m P o i n t s In the f i g u r e (2-5) any a c t i o n t h a t would, cause the movement of the e q u i l i b r i u m p o i n t t o the l e f t of the v e r t i c a l through E/E towards the Pareto s u r f a c e would mean gains t o p l a y e r 2. T h i s a p p l i e s t o any movement of the e q u i l i b r i u m p o i n t above the h o r i z o n t a l through E/E f o r p l a y e r 1. A movement i n the south-west d i r e c t i o n from E/E i s unwel-35. I b i d . Adapted from r e s u l t s i n Hughes and Crnea's ivlodel., 51 corned to both as t h i s means t h a t both w i l l have t o throw i n l a r g e investment w i t h r e l a t i v e l y l i t t l e g a i n s . An uncoordinated a c t i o n would l e a d to t h i s mess. The' oppo-s i t e t r a v e r s e of the e q u i l i b r i u m p o i n t w i l l only come about through c o o p e r a t i o n , and, as we have p o i n t e d out e a r l i e r , the u l t i m a t e p o i n t of d i v i s i o n w i l l depend on the p l a y e r s . Just as much as we are i n t e r e s t e d i n post optimal a n a l y s i s i n l i n e a r programming and o t h e r mathematical t e c h -niques we must not l i m i t our i n t e r e s t i n game the o r y a t the e q u i l i b r i u m s o l u t i o n s . . As d e c i s i o n makers we would l i k e t o know how s e n s i t i v e the r e s u l t s of the a n a l y s i s are t o s m a l l changes i n the v a r i a b l e s of the problem. These may be due to i n a c c u r a c i e s i n the f o r e c a s t s and other c o n t i n g e n c i e s and, t h a t f o r a l l vie know we might i n f a c t not be op e r a t -i n g a t where we thought we are or ought t o be. F u r t h e r i t would be i n t e r e s t i n g to i n v e s t i g a t e the movement of the e q u i l i b r i u m p o s i t i o n s when the firms- as competitors e i t h e r s i n g l y or both s i m u l t a n e o u s l y undertake s m a l l changes i n v a r i o u s forms of investment. I t might b e . p o s s i b l e t h a t we f i n d i t advantageous i n making a c e r t a i n k i n d of investment w h i l e i n o t h ers we would only sow the seeds of our f i n a n c i a l r u i n . F i g u r e ( 2 - 6 ) i s a t y p i c a l example of such an a n a l y s i s and i t w i l l be noted t h a t there are c e r t a i n investments, i n t h i s p a r t i c u l a r case marketing investments which are generally/ c o s t l i e r ' t o f i r m 2 w h i l e manufacturing investments are ^ R 2 ^ = Incres.se i n marketing investment K = Increase i n manufacturing investment 1,2,E = P l a y e r 1,2 or both r e s p e c t i v e l y F i g u r e (2-6): E f f e c t of Small Changes i n Investment on P o s i t i o n of E q u i l i b r i u m Points c o s t l i e r to f i r m 1. I t a l s o appears that f i r m i could w e l l improve i t s market share by adopting an aggressive marketing p o l i c y . Such a n a l y s i s would be an i n v a l u a b l e a i d to mana-gement d e c i s i o n making as they form g u i d e l i n e s to s t r a t e g y 36. Source: Hughes and Ornea, I b i d . 53 s e l e c t i o n and. p o l i c y f o r m u l a t i o n . T h i s would be a f a r cry from j u s t o f f e r i n g management an e q u i l i b r i u m p o i n t and. ex-p e c t i n g r a t i o n a l maximin be h a v i o r to respond without i n v e s -t i g a t i n g any u n d e r l y i n g f l e x i b i l i t y i n the r e s u l t s o f f e r e d . Turning to t h r e a t and defence s t r a t e g i e s we note t h a t they are r e c i p r o c a l i n - n a t u r e , meaning t h a t when one of the p l a y e r s t h r e a t e n s the oth e r i s a c t u a l l y employing a d e f e n s i v e s t r a t e g y and v i c e v e r s a . To be sure, a t h r e a t once c a r r i e d out i s no more a t h r e a t . I t i s only e f f e c t i v e a t the bar -gaining, or n e g o t i a t i n g t a b l e i f i t s use i s kept at bay, t o be unleashed anytime the outcomes are not f a v o u r a b l e .to the p a r t y concerned. We w i l l have the o c c a s i o n to view the dynamics of the process of b a r g a i n i n g i n the next chapter where we w i l l a l s o i n t r o d u c e the concept of c o n c e s s i o n . The t h r e a t p o i n t S-^  i n f i g u r e (2-5) i s the maximun of. the p a y o f f w i t h r e s p e c t t o f i r m 1's s t r a t e g y and. minimum t o f i r m 2's s t r a t e g y . In e f f e c t f i r m 1 i s defending a g a i n s t any t h r e a t from f i r m 2 . S i m i l a r l y S 2 i s the maximin p o i n t of payoff f u n c t i o n R 2 f o r f i r m 2 . There i s , of course, a l i m i t at which f i r m 1 and 2 can e x e r c i s e t h e i r t h r e a t s t r a -tegy. T h i s i s t i e d i n with the r e s t r i c t i o n placed on e q u i l i b r i u m p o i n t c a l c u l a t i o n s i n the e a r l i e r p a r t of the s e c t i o n . F i n a l l y p o i n t .S^  d e s c r i b e s the threat-de'fence " J s t r a t e g i e s d i s c u s s e d i n s e c t i o n 3 equation ( 2 - 1 4 ) , where f i r m 1 t r i e s to maximize R n-R„ and f i r m 2 attempts to m i n i -mize t h i s q u a n t i t y . In going through the t h e o r e t i c a l e x e r c i s e of game th e o r y we have i m p l i c i t l y assumed a n a t u r a l tendency f o r the p l a y e r s to c o o r d i n a t e t h e i r c h o i c e of s t r a t e g i e s . The e q u i l i b r i u m p o i n t was a r r i v e d a t w i t h both p l a y e r s choos-i n g t h e i r e q u i l i b r i u m s t r a t e g y . L i k e w i s e , the Pareto so-l u t i o n r e s u l t s when both p l a y e r s are drawn by v a r i o u s reasons, i n c l u d i n g the prospect of j o i n t maximal p a y o f f to cooperate. I t seems thus t h a t . c o n s i d e r a t i o n have not been made f o r s i t u a t i o n s where i t i s q u i t e p o s s i b l e f o r independent uncoordinated a c t i o n s on the p a r t of the p l a -yers.. What would then be the outcome i s a q u e s t i o n t h a t b r i n g s us to the i n t e r e s t i n g area of c r o s s - s t r a t e g i e s . 37 T h i s was done.in hughes and Ornea's paper and f i g u r e (2-7) and f i g u r e (2-8 ' ) are some of the r e s u l t s . I t ap-pears t h a t w i t h c o o p e r a t i o n both p l a y e r s have p a y o f f s on the Pareto s u r f a c e . I f one of the pla^^ers had i n i t i a l l y chosen a. c o o p e r a t i v e s t r a t e g y , a defence or e q u i l i b r i u m s t r a t e g y taken by the other w i l l improve the l a t t e r T s pay-o f f . With a threat- s t r a t e g y both f i r m s s u f f e r reduced p a y o f f s . T h i s i s shown i n f i g u r e ( 2 - 7 ) . F i g u r e ( 2 - 8 ) i s the case where the r e f e r e n c e s t r a t e g y i s the e q u i l i b r i u m s t r a t e g y . In t h i s case, too, i n c r e a s e d p a y o f f goes to the n l a y e r s e l e c t i n g a. c o o p e r a t i v e s t r a t e g y while a defence s t r a t e g y makes l i t t l e d i f f e r e n c e to the r e s u l t s . Again 37. I b i d . 55 \ \ \ E/C \ Fare to Surface n v \ \ " ^- *"* \ \ T/C \)t C/D \ V C/E \ . C/T \ \ s \ R. • F i g u r e ( 2 - 7 ) : Cross S t r a t e g i e s Uhen One F i r m Uses Cooperative S t r a t e g y 56" threat' s t r a t e g i e s are seen to run again s t mutual i n t e r e s t . -I f the above were a "one-shot" game then we would expect the r e s u l t s expounded. In the case where the game i s repeatable we would not expect the r e s u l t to remain to the advantage of one p l a y e r f o r l o n g , as,.by then the good Samaritan w i l l have l e a r n t t h a t h i s good w i l l s are being abused and we can expect the game to degenerate t o a v i c i o u s end or to some more e q u i t a b l e r e s u l t s . 5. Extensive Games In our cursory survey thus f a r we were drawn on va-r i o u s occasions to gl o s s over the dynamic aspects of game theory. P a r t l y to set the foundations f o r the m a t e r i a l i n the next chapter which deals w i t h t h i s area of game theory, and, as an a l t e r n a t i v e f o r m u l a t i o n of a game i n normalized form, the extensive form of a game-will be described here. 'The matrix gams i n t a b l e (2-2) can be represented by a game tree shown.in f i g u r e (2-9). Player P^ s t a r t s at p^, the vertex w i t h the name of the p l a y e r , making h i s choice from two a l t e r n a t i v e s . Since we are c o n s i d e r i n g a case where P 0 has.no knowledge of P-t ' s choice, s i m i l a r to . 3 6 . The tendencies f o r p l a y e r s • t o behave i n such a manner were found i n experiments by R u s s e l l L. Ackoff, David V:. Conrath and. N i g e l Howard i n t h e i r work, "A Model Study of the E s c a l a t i o n and D e - e s c a l a t i o n of  C o n f l i c t " , Management science Center - U n i v e r s i t y of Pennsylvania (196?). the case where they make a s i n g l e move si m u l t a n e o u s l y , the v e r t i c e s ? 2 are encl o s e d by a s i n g l e curve, I n d i c a t i n g t h a t the group of v e r t i c e s belong t o the same i n f o r m a t i o n s e t . In the case where P 2 i s allowed to make h i s move a f t e r P^ has done and r e v e a l e d h i s c h o i c e , the v e r t i c e s P^ are c i r -c l e d s e p a r a t e l y producing two i n f o r m a t i o n s e t s , i n d i c a t i n g t h a t P 2 can d i s t i n g u i s h P^'s c h o i c e . We thus have the con-d i t i o n of p e r f e c t i n f o r m a t i o n i n r e s p e c t of P^/s choice f o r P,,. T h i s i s shown i n f i g u r e (2-10) and the e q u i v a l e n t normalized game i s shown i n t a b l e (2-3). When both p l a y e r s have made t h e i r c h o i c e , the game l e a d s to one o f the f o u r p a y o f f s through a path s t a r t i n g from P-^  and ending at the payo f f , g i v i n g a complete d e s c r i p t i o n of the p l a y o f the game. Often d u r i n g the pl a y of a game chance moves as d i s -t i n g u i s h e d from ;oersonsl moves of the p l a y e r s are i n v o l v e d . To t h i s which we a s s i g n P Q we must a s s o c i a t e a p r o b a b i l i t y d i s t r i b u t i o n on the a l t e r n a t i v e s . N o t i c e t h a t the v e r t i -ces i n the same i n f o r m a t i o n s e t c o n t a i n the same number of a l t e r n a t i v e s (otherwise the v e r t i c e s can be d i s t i n g u i s h e d ) . Proceeding thus the anatomy of any game can be d e s c r i b e d , no matter how complicated the moves or i n f o r m a t i o n c o n d i -t i o n s may be. More g e n e r a l l y we have an n-person game i n e x t e n s i v e form, d e s c r i b e d mathematically as f o l l o w s : ( - 1 , - 1 ) ( - 1 0 , 0 ) ( 0 , - 1 0 ) ( - £ , - £ ) F i g u r e ( 2 - 9 ) : Game Tree R e p r e s e n t a t i o n of Table ( 2 - 2 ) 1 / . \ v 2 / V 1 / \ 2 ( - 1 , - 1 ) ( - 1 0 , 0 ) ( 0 , - 1 0 ) (-8,-8) F i g u r e ( 2 - 1 0 ) : Game Ti -ee R e p r e s e n t a t i o n of Table ( 2 - 3 ) ( 1 , 1 ; 2 , 1 ) . ( 1 , 1 ; 2 , 2 ) ( l , 2 ; 2 , l ) ( 1 , 2 ; 2 , 2 ) ( - 1 , - 1 ) ( - 1 , - 1 ) ( - 1 0 , 0 ) ( - 1 0 , 0 ) ( 0 , - 1 0 ) (-e,-g) ( 0 , - 1 0 ) (-8,-8) Table ( 2 - 3 ) ; M a t r i x Game: P ^ s Choice Knownjfco P 2 59 1) A p a r t i t i o n of the moves i n t o n + 1 index s e t P^P^... ...,P n c o n s i s t i n g of chance and pe r s o n a l moves. T h i s i s known as the p l a y e r p a r t i t i o n . 2) For each chance move wit h j a l t e r n a t i v e s , a p r o b a b i -l i t y d i s t r i b u t i o n , which a s s i g n s a p o s i t i v e p r o b a b i -l i t y t o each a l t e r n a t i v e s . 3) F o r each p l a y vv, a p a y o f f f u n c t i o n R(W) =/^ ~R-j_ (W)......, R n (w) _7. 4) A p a r t i t i o n of moves i n t o s e t s U so t h a t each U i s cont a i n e d i n some P. and A. f o r some i and j , and no U c o n t a i n s two moves b e l o n g i n g to the same p l a y . T h i s p a r t i t i o n i s c a l l e d the i n f o r m a t i o n p a r t i t i o n . ^ The f i r s t three of these s p e c i f i c a t i o n s are s e l f - e x -p l a n a t o r y . In (4), what i s meant i s t h a t the s e t of a l l moves (a move i s the s e l e c t i o n o f one among a s e t of a l t e r -n a t i v e s at a choice p o i n t i n the game) are p a r t i t i o n e d i n t o s e t s A] , A 2 , ,-Aj, where Aj c o n t a i n s those moves with j a l t e r n a t i v e s . Thus each U must belong to some P^ and A. f o r some i and j . The game proceeds as a continuous path w i t h no branching, or, i n other words one cannot d i s -t i n g u i s h the v e r t i c e s i n the same i n f o r m a t i o n s e t and makes a choice as i f fa c e d w i t h one move. 39. Shubik, "Stra t e g y and Karket S t r u c t u r e " , op . c i t p.190 We can t h i n k of a pure s t r a t e g y i n an ext e n s i v e game of p l a y e r i w i t h q d i f f e r e n t i n f o r m a t i o n s e t s as the s e l e c t i o n of a q-tuple of i n t e g e r s (y^ . •. • >y ^, • •••>Yq) such t h a t the number y k r e p r e s e n t s the branch chosen . i n the k t h i n f o r m a t i o n s e t , where y^ i s p o s i t i v e i n t e g e r ts j , the number of a l t e r n a t i v e s i n the kth i n -f o r m a t i o n s e t . The mixed s t r a t e g y c o u l d be regarded as a "random" s e l e c t i o n from a number of f i x e d plans, the pure s t r a t e g i e s } b y the s t r a t e g i s t or. the c o r p o r a t i o n d i -r e c t o r s or o f f i c e r s . . I n s p i t e of the dynamic overtones of the e x t e n s i v e game model presented above we have not a c t u a l l y achieved a dynamic model of a market. The above game i s l i m i t e d to a f i n i t e l e n g t h w i t h p a y o f f s at the end. In r e a l l i f e economic s i t u a t i o n s payments are made c o n t i n u o u s l y a n d the game has no d e f i n i t e t e r m i n a t i o n p o i n t . U n l i k e the one p e r i o d game, i n an i n f i n i t e game a t any pie.";/ d u r i n g a sub-game, the knowledge t h a t the supergame i s expected to l a s t f o r subsequent p e r i o d s , no matter how small the pro-b a b i l i t y , alone makes i t p o s s i b l e to c a r r y out r e p r i s a l s i n the event of d e f e c t i o n or d o u b l e - c r o s s i n g . The whole course of s t r a t e g i e s may have to be changed under such circumstances where the degree of f i x e d investment, f l e -x i b i l i t y i n a d v e r t i s i n g , r e s e a r c h or p r i c i n g p o l i c y a l l of which a f f e c t s the m a n e u v e r a b i l i t y of the f i r m s and t h e i r a b i l i t y ; t o enforce s t a b i l i t y i n the market. For a mathematical d e s c r i p t i o n o f an n-person i n -f i n i t e game "P ,• we have, wi t h the f i r s t and second con-d i t i o n s s i m i l a r to t h a t f o r the f i n i t e game the remaining as f o l l o w s : 3) A p a r t i t i o n ^ of the moves into- i n f o r m a t i o n s e t s U 4 F o r each path of p l a y up to the t . t h move h^fi (the s e t of a l l p l a y s up t o the t . t h move], a pay-o f f f u n c t i o n R t ( h t ) = / T R l t ( h t ) , R 2 t ( h t ) , , R n t ( h t ) J 5) For any p l a y e r i , the sum £ R i t ( h t ) . i s bounded.*1'0 t=o F i g u r e (2-11) r e p r e s e n t s f o r s i m p l i f i c a t i o n a duopoly dynamic game. During, the f i r s t p e r i o d p l a y e r s P-^  and P^ make simultaneous d e c i s i o n s , a t which p o i n t p a y o f f s as •a. r e s u l t of these d e c i s i o n s and d e s c r i b e d by the f u n c -t i o n R-j.(h-j_) are made. The game proceeds through subseo/aent p e r i o d s d u r i n g which f u r t h e r p a y o f f s are made. One of the c r i t e r i a . f o r a m u l t i - p e r i o d d e c i s i o n i s the maximization of a present value f u n c t i o n . Through a d i s c o u n t r a t e which could-be a. f u n c t i o n of time, the di s c o u n t e d f u t u r e income stream s e t s a bound to the sum of p a y o f f s , s a t i s f y i n g , con-d i t i o n ( 5 ) . In the event of one of the plajrers dropping out 40. I b i d . p. 198 62 F i g u r e (2-11): A simple Dynamic Duopoly Game the remaining p l a y e r i s l e f t to e x e r c i s e m o n o p o l i s t i c con-t r o l from then on. A l l the above f o r m a l i z a t i o n of the i n f i n i t e game t r e e g i v e s only a complete d e s c r i p t i o n of what can p o s s i b l y happen. The game tree- per se does not p r e s c r i b e any p a t t e r n of beha-v i o r but r e p r e s e n t s a method whereby the dynamic f e a t u r e s of an i n t e r a c t i o n a l competitive environment can be examined i n t o t o . We s h a l l l e a v e the d i s c u s s i o n of s o l v i n g dynamic ga-mes to the next chapter. CHAPTER I I I • GAMES AS DECISION TOOLS IN DYNAMIC SETTINGS 1. I n t r o d u c t i o n . There has been' a tendency i n the development of tech niques i n management s c i e n c e and o p e r a t i o n s r e s e a r c h t o b u i l d from simple s t a t i c models to those i n v o l v i n g complex dynamic a n a l y s i s . T h e o r i e s and models proceeding . from elementary e x p o s i t o r y forms of s i n g l e - p e r i o d s i t u a t i o n s are extended to cases v/ith a touch of pseudo-dynamic f l a -vor., leading, e v e n t u a l l y to all-encompassing dynamic models where the time element p l a y s a major r o l e . The•path which game theory has t r a c e d f o l l o w s a very s i m i l a r course with one e x c e p t i o n , t h a t i s , the emphasis and swing to the con-s i d e r a t i o n of the dynamic aspect of the theory s t i l l l e a v e s many gaps to be f i l l e d . In t h i s chapter, we s h a l l examine how d i f f e r e n t f o r m u l a t i o n s can be made to c o n s t r u c t models of dynamic games s i t u a t i o n s . Notably the work of Shubik"'" and Hughes and Ornea which we have c o n s i s t e n t l y emphasized at the outset w i l l be i n v o l v e d . , • 1. OP . c i t . " S t r a t e g y and Market S t r u c t u r e " . 2. op . c i t . As we have seen in our encounter with the essence of game theory in the previous chapter, formulation becomes increasingly complicated when we leave 2-person analysis to the general case of multiperson situations. We could there7 fore expect the mathematics of dynamic game situations for the general n-person case to be even more complex. In many of the situations we w i l l be discussing in the following pages, we are of necessity drawn to make some simplifying assumptions to bypass the d i f f i c u l t i e s of mathematical for-mulations, rather than to be bogged down i n our effort to deal effectively with real world situations. On the other hand, we should not. be necessarily satisfied' with the tools pre-sently available. The seemingly dynamic overtones of a game i n . i t s normalized form played many times i s inadequate representa-tion of actual situations. When we conjecture a competi-tive situation as a one period model we have .. evidently l e f t aside such considerations as the learning processes and other' psychological attitudes as well as. adaptive cha-racteristics of the players. The combined interplay of these forces resulting from internal (endogenous to the players), and external (environment) sources w i l l have bear-ing on the payoff functions. Kence, i t would be reasonable to assert that the payoff functions to the players could not remain unchanged as the game progresses over many plays. In the main, the e x p e c t a t i o n s of the p l a y e r s are l i k e l y to e x h i b i t ephemeral b e h a v i o r . As a. consequence, i n order to make game theory t r u e l y a normative theory vie are bes i e g e d by a need to formulate s u i t a b l e c r i t e r i a f o r a c h i e v i n g op-t i m a l i t y In our d e c i s i o n s . The answer i s i n some way pro-vid e d by the processes d e a l t w i t h i n dynamic programming. Le a v i n g the s i n g l e - s t a g e or - p e r i o d game we are i n v a -r i a b l y drawn i n t o a c o n s i d e r a t i o n of the way i n which the q u a n t i t y of reso u r c e s change i n a m u l t i - s t a g e p r o c e s s . I t i s obvious t h a t as time goes on the amount of re s o u r c e s c r e d i t e d to each p l a y e r through which they are f u r t h e r able t o o b t a i n o r develop w i l l be f u n c t i o n a l l y r e l a t e d t o one another and to time. What t h i s means to the s t r a t e g i c a l -t e r n a t i v e s a v a i l a b l e to the p l a y e r s i s that they w i l l not be l i m i t e d to any same number of a l t e r n a t i v e s i n any one sub-game p e r i o d . I t i s thus a p p r o p r i a t e to view the a l t e r n a t i v e s a v a i l a b l e as a continuous f u n c t i o n r a t h e r than d i s c r e t e value v a r i a b l e s . The dynamics are not r e s t r i c t e d t o c o n t r o l l a b l e v a r i a b l e s i n t r i n s i c to the p l a y e r s but to environmental f o r -ces, e.g. changes i n t a s t e over time, adding to the comple-x i t y of the choice s i t u a t i o n . A l l these, as was. remarked much e a r l i e r i n the t h e s i s , can make the model extremely un-3. Bellman, R i c h a r d , "Dynamic Programming" P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , 1 9 5 7 . 66 wield.y and, i t would be wise t o pause and examine what or d e r s of magnitude i n the f e a t u r e s of the model we are e x p e c t i n g b e f o r e we pass the p o i n t of d i m i n i s h i n g r e t u r n s . Imagine the immensity of the computer program i f we have to enumerate a l l the a l t e r n a t i v e s p e r t a i n i n g to the v a r i o u s stages, what-' more, the combinations of such I T h i s i s c l e a r l y an i m p o s s i -b l e t ask and such exhaustive examination i s out of the ques-t i o n . F o r t u n a t e l y most o f these are i r r e l e v a n t and do not war-r a n t a t t e n t i o n and work i n h e u r i s t i c programming and a r t i f i c i a l i n t e l l i g e n c e ^ have taken a s t e p i n the d i r e c t i o n of f o r m a l i z -in g v a r i o u s r u l e s of thumb i n attempting to p o r t r a y the search processes i n d e c i s i o n making. In the context of dynamic games v:e are n a t u r a l l y i n t e -r e s t e d i n the a p p l i c a b i l i t y of the concepts o f e q u i l i b r i u m and o t h e r game t h e o r e t i c p o i n t s to i t s s o l u t i o n . T h e i r im-portance cannot be underplayed even as a r e s u l t of the hazy p i c t u r e we have presented above, f o r , they form the b a s i s from which the normative s t r u c t u r e of game the o r y i s b u i l t . How would we then view the s i n g l e p e r i o d e q u i l i b r i u m s o l u t i o n which was so e m p h a t i c a l l y d i s p l a y e d i n the preceding chapter? In the words of Bellman, "These m u l t i - s t a g e game may be c o n s i d e r e d not only to c o n s t i t u t e an e x t e n t i o n of the s i n g l e - s t a g e t h e o r v . but i n manv-wavs they ij.. I'.'insky, K. "Stens Toward A r t i f i c i a l I n t e l l i g e n c e " , Pro-ceedings of the I i lE, January 196.1, pp. £ - 3 C 67 may be considered to be more fundamental. The s i n g l e - s t a g e game may be conceived of as a steady-stage v e r s i o n of an o r i g i n a l dynamic process, namely the mult i - s t a g e process."5 We might also'make some remarks i n t h i s regard about the type of e q u i l i b r i u m s o l u t i o n t h a t can p o s s i b l y a r i s e i n dynamic games. However, i t must be r e a l i z e d , f i r s t t hat no matter what the goals a c o r p o r a t i o n or i n d i v i d u a l professes they are i n some ways r e l a t e d to the problem of s u r v i v a l , f o r unless a f i r m can s u r v i v e , i t i s p o i n t l e s s to t h i n k of maximizing p r o f i t s and growth. With t h i s view i t appears t h a t when we apply the non-cooperative concept to a game of economic s u r v i v a l , almost any s t a t e y i e l d i n g every imputation of wealth can be enforced, as an e q u i l i b r i u m s t a t e by c a r r y i n g a s u f f i c i e n t l y v i o l e n t t h r e a t . For, i n a game t h a t i s played, only once, as i n the case of the p r i s o n e r s ' dilemma game we would expect the Nash non-cooperetive e q u i l i b r i u m to r e s u l t , w i t h the players s e l e c t i n g t h e i r second s t r a t e g y . Even i f the game was to be played many times over a f i n i t e l e n g t h of time, by a process of "backward, i n d u c t i o n " t h i s e q u i l i b r i u m would p e r s i s t because of the' f e a r hovering i n the game before the l a s t of the opponent d e f e c t i n g from the j o i n t maximal s o l u t i o n . With the game over, r e p r i s a l s cannot be c a r r i e d out. No such l a c k of opportunity e x i s t s i n an i n f i n i t e game. 5. op . c i t . p. 283. The temporary gains of the d o u b l e - c r o s s e r may c o s t him more i f h i s opponent c a r r i e s - out a s t i n g i n g r e p r i s a l . F u r t h e r once a d e f e c t i o n i s made , i n a world where s u s p i c i o n f i n d s permanent r o o t s i n the minds of people, much e f f o r t i s r e -q u i r e d t o erase t h i s and to r e b u i l d the c o n f i d e n c e t h a t 'was s h a t t e r e d . Thus, i t may even be p o s s i b l e as the above c o n s i d e r a -t i o n shows, to have a j o i n t maximum enfo r c e d n o n - c o o p e r a t i -v e l y i f we c o n s i d e r economic s u r v i v a l as a mainstream f o r c e i n the g o a l s of a firm.- I t i s t h e r e f o r e d i f f i c u l t from t h i s s t a n d p o i n t to draw a c l e a r cut d i s t i n c t i o n between coopera-t i v e and non-cooperative behavior i n a dynamic s i t u a t i o n . In t h i s regard the s t r i c t dichotomy between these two forms of behavior must g i v e way to v a r i o u s measures of c o o p e r a t i o n . Of t h i s , we have mentioned a t the s t a r t of our preamble i n t o the d i f f e r e n t t h e o r i e s of s o l u t i o n of games i n chapter I I . However, Shubik b e l i e v e s t h a t , "The key to the examination of the p o s s i b l e e q u i l i b r i u m s t a t e s r e s t s w i t h our a b i l i t y t o £ develop a c a l c u l u s of p l a u s i b i l i t y f o r t h r e a t s . " With t h i s note we s h a l l i n the f o l l o w i n g s e c t i o n s de-termine how the. processes of game theory ere employed i n 6. op . c i t . "Game Theory and R e l a t e d Approaches to  S o c i a l B e h a v i o r . " p. 69• d e c i s i o n making i n dynamic s e t t i n g s . V/hen, i n f a c t a n a l y t i c a l f o r m u l a t i o n of d e s c r i p t i v e or i n t r i c a t e d q u a n t i f i a b l e know-ledge becomes exce e d i n g l y d i f f i c u l t we s h a l l have to r e s o r t to s i m u l a t i o n techniques'. S i m u l a t i o n models w i t h the a i d of e l e c t r o n i c computers have considerably- extended our a b i l i t y to d e a l with a wide range of the s t i l l l a r g e l y d e s c r i p t i v e and nebulous areas of management d e c i s i o n making. 2. Continuous V a r i a b l e S t r a t e g y and Payoff Space One of the prime purposes of c o n s t r u c t i n g p a y o f f ma-t r i c e s i n 2-person games i s to show the e s s e n t i a l nature of c o n f l i c t and why i t a r i s e s i n s i t u a t i o n s where i n t e r e s t s are opposed. In s e t t i n g up m a t r i c e s i n v o l v i n g a few s t r a t e g i e s the o b j e c t i s to demonstrate c e r t a i n b a s i c concepts- on which game theory i s b u i l t . The r e a l world c o u l d be very simple to d e a l w i t h i f i n f a c t such are what we encounter i n the b u s i n e s s environment, which we know to be h i g h l y complex. Not d i s c o u n t i n g t h a t we can have d i s c r e t e choice s i t u a t i o n s but s u r e l y c o n t a i n i n g much more s t r a t e g i e s than what we have shown, i t i s i n t e r e s t i n g to note t h a t i n i n d u s t r i a l l i f e we w i l l have to d e a l with choice of s t r a t e g i c v a r i a b l e s des-c r i b e d by continuous f u n c t i o n s . T h i s f a c t i s f o r example observed i n investment i n marketing or other f a c i l i t i e s , where over a p e r i o d of time the s i z e of such investment can be v a r i e d c o n t i n u o u s ! v . 7 0 To f a c i l i t a t e our d i s c u s s i o n of the concept of c o n t i -nuous v a r i a b l e snd p a y o f f f u n c t i o n s we s h a l l c o n s i d e r a sim-p l e example of a 2-person s i t u a t i o n where each p l a y e r i s con-cerned with one d e c i s i o n v a r i a b l e . By i n c r e a s i n g the number of v a r i a b l e s or p l a y e r s or both we w i l l only complicate our a n a l y s i s with an i n c r e a s e i n the d i m e n s i o n a l i t y of the pro-blem, making i t d i f f i c u l t t o v i s u a l i z e on a 2-dimensional plane. C o n s t r u c t i n g the g r a p h i c a l r e l a t i o n s h i p i s a f i r s t s t e p i n attempting to show how the path to an e q u i l i b r i u m i s f o l l o w e d on the continuum of s t r a t e g i c c h o i c e . F i g u r e (3-1) i s such a r e p r e s e n t a t i o n . The q u a n t i t i e s w i t h s u b s c r i p t 1 and 2 are those a s s o c i a t e d w i t h p l a y e r s 1 and 2 r e s p e c t i v e l y . X i s the d e c i s i o n v a r i a b l e and Rs the p a y o f f s to the p l a y e r s . The d o t t e d and l i g h t contours are i s o - p r o f i t or p a y o f f l i n e s I !! 11! 111! such t h a t Rj_ *> R i > Rj_ > Rj_ f o r i = 1 and. 2. T h i s g r a -p h i c a l r e p r e s e n t a t i o n i s e q u i v a l e n t to the Edgeworth i n d i f -f e r e n c e a n a l y s i s f o r a duopoly market i n economic t e x t . The l o c i of the p l a y e r s maxima, are shown as dark unbroken and broken l i n e s w i t h and M2. the monopoly p r o f i t s . At t h e i r i n t e r s e c t i o n we have the non-cooperative e q u i l i b r i u m , the p o i n t denoted by E/E. In a c o o p e r a t i v e game the p l a y e r s would attempt to achieve j o i n t maximization on the Pareto s u r f a c e . N a t u r a l l y the set of s o l u t i o n s , the n e g o t i a t i o n s e t i n the sense of von Neumann and. L'iorgenstern, on t h i s s u r f a c e must be such t h a t each p l a y e r i n d i v i d u a l l y must 71 obtain more than they each alone could have obtained at the e q u i l i b r i u m p o i n t E/E. The s i g n i f i c a n c e of the D/T and T/D, the defence and t h r e a t p o i n t s were mentioned i n S e c t i o n 4 of the preceding chapter. F o r a 2-person case, as the number of v a r i a b l e s i n -creases, the i n c r e a s e . i n d i m e n s i o n a l i t y of the problem makes i t impossible to represent t h i s g r a p h i c a l l y on a v a r i a b l e space. • Representation on payoff space could overcome t h i s . The equivalent of f i g u r e ( 3 - 1 ) p l o t t e d i n payoff space i s shown i n f i g u r e (3-2). In general the payoff to a player i i n an n-person s i t u a t i o n i s r e l a t e d by a continuous func- . t i o n R^ having a f i n i t e range over h i s d e c i s i o n v a r i a b l e s Xi,yj_,Zj_, and so on. The f u n c t i o n f o r p l a y e r i i s given by R^j/ -(x-^,y^,z^,...),....., ( x ^ . , y ^ , z ^ , . . . ) , . . . . ( ^ I Y ^ Z N > • • • • )_7 and l i m i t e d to a f i n i t e range because i n p r a c t i c e we would not expect the p l a y e r s to have i n f i n i t e resources. Other r e s t r i c t i o n s out of p r a c t i c a l c o n s i d e r a t i o n s t h a t would place c o n s t r a i n t s on the a c t i o n s of the p l a y e r s i n c l u d e cases where one or more playe r s c o n t r o l access to a f i x e d amount of phy-s i c a l resources, l e g a l matters concerning the operation of the business e n t e r p r i s e s and o t h e r s . The e x i s t e n c e and uniqueness of s o l u t i o n of such a game where the c o n s t r a i n t s and payoff f u n c t i o n s to each p l a y e r depends on the s t r a t e g y of every player, r e q u i r e s complex 'mathematical a n a l y s i s . F i g u r e ( 3 - 2 ) : 2 -Person Game i n P a y o f f onace 73 However, J.B. Rosen has shown the c o n d i t i o n to h o l d under requirements of a p p r o p r i a t e c o n c a v i t y i n the payo f f f u n c -t i o n s . For a simple e x p o s i t i o n of the type of f o r m u l a t i o n i n v o l v i n g continuous f u n c t i o n s of dynamic games we s h a l l o u t l i n e the approach u s i n g the technique of dynamic program-ed ming., ' To r e l i e v e us of undue mathematical, and conc e p t u a l d i f f i c u l t i e s , 2 - p e r s o n games w i l l be examined. 'In the case where the game i s zero-sum the e x p e c t a t i o n of p l a y e r 1 (P]_) when he chooses h i s s t r a t e g i e s from a c o n t i -nuum r a t h e r than a. d i s c r e t e s e t of moves i s d e f i n e d by E(F,G), o where F i s the d i s t r i b u t i o n f u n c t i o n ' from which Pn chooses •X. h i s s t r a t e g i c v a r i a b l e x d e f i n e d i n the i n t e r v a l /~0,I_7« G i s the d i s t r i b u t i o n f u n c t i o n from which p l a y e r 2 ( P 2 ) chooses h i s s t r a t e g i c v a r i a b l e y d e f i n e d i n the i n t e r v a l £~Q,lj• 1 p Then E(F,G") i s g i v e n by the S t i e l t j e s i n t e g r a l 7 . . Rosen, J.B., "E x i s t e n c e and Uniqueness of E q u i l i b r i u m  P o i n t s f o r Concave N-Person Games." Econometrica, volume 3 3 » No. 3 , J u l y 1 9 6 5 , Pp. 5 2 0 - 5 3 4 . " 8 . Bellman, op . c i t . , Dreyfus, Stewart, "Dynamic Program-ming" , i n "Progress i n Operations Research", Volume I . e d i t e d by Russel L. A c k o f f . , John Wiley and.Sons, New York 1 9 6 1 . pp. 2 1 1 - 2 4 2 . 9 - The d i s c u s s i o n of d i s t r i b u t i o n f u n c t i o n s as a p p l i e d to game theory i s found i n Chapter 8" of KcKinsey " I n t r o d u c t i o n t o the Theory of Games", op . c i t . In t h i s case F. i s the cumulative d i s t r i b u t i o n f u n c t i o n governing the frequency with which p l a y e r 1 chooses x. 1 0 . T h i s i s a continuous e q u i v a l e n t of the d i s c r e t e f o r -mulation given i n equation ( 2 - 7 ) of Chapter I I . 7h ( 3 - D E(F,G)=f f R(x,y)dF(x)dG(y) J o J o If R(x.,y) the payoff function i s jointly continuous in x and y in the closed unit square then i t can be shown'''"'" that (3-2) f 1 r 1 R( X }y) dF(x)dG(y) = f 1 f 1 R(x,y)dG(y)dF(x) J o J o J o J o and hence the condition (3-3) max min E(F,G) = min max E(F,G) F G G F Taking the analysis one step further we put res t r i c -tions on the resources of the players represented at some stage of the dynamic game by m-dimensional vectors x and y. For a zero-sum case where the survivor of the game benefits v/ith the ruin of his opponent, -we c a l l this a game of sur- vival . In the more general case v;e have a non-zero sum dynamic game, a game of economic survival which we shall leave to a later section. Let us for the moment side step from our object of formalizing an in f i n i t e process because of conceptual d i f f i c u l t i e s and consider an M-state nrocess. of each stage by a vector u and by a vector v such that 0 ^  u ^ rx,C v ^  y, where the inequalities hold component 2kes an allocation of his resources at the beginning 11. McKinsey, op .cit.. pp. 3.86-193 • 75 wise. This i s equivalent to a strategy i n the subgame under consideration. As a consecuence of t h i s action P, receives 1 payoff of Pi(u }v;x,y), a scalar function and P 2 by d e f i n i t i o n the negative of t h i s quantity. With these payoffs the resour-ces of P-^  i s transformed from x into T(x,y;u,v) and P 2 from y 12 into S(x,y;u,v). After N periods the t o t a l return to P^ i s (3 -4 ) RN = RN (u, u x , u 2,..., u K_ ]_, v, v x , v 2,..., v N _ x ; x, y) =R(u,v) + R(u 1,v 1) -i- ... + R(ujf_i»VN-1 ^  I f R(u,v;x,y) i s a. continuous function i n u and v f o r a l l f i n i t e values of u,v,x and y and T(x,y;u,v) and. S(x,y;u,v) are also continuous over f i n i t e values of the vector variables then the value of the N-stage game i s given by ( 3 - 5 ) VN=E(F,G)= max min f[\ R^F (u, u±, u ^ u ^ ) dG(v }v 1,v 2,...,v K_ 1) _7 = max min £~ 7 G F where F and G are d i s t r i b u t i o n functions over regions of quite complicated forms defined by the i n e q u a l i t i e s 12. These values should r i g h t l y be adjusted to i t s present value through a disccunt rate function. 7 6 (3-6) 0 JS u -S x, 0 s v $ y, 0 * « T, 0 v «s S, 0 * V l « ' T H - 1 ' ° S V - - V l The q u a n t i t i e s T and S depend upon x;y,u and v; T and depends upon x,y,u,v,u-^ and v-^  , and so on. On applying the p r i n c i p l e of o p t i m a l i t y and f u n c t i o n a l 13 equations of dynamic programming where the sequence of func-t i o n s i s defined as (3-7) f (x,v)=V • f o r N=l,2 .. N • N we obtai n the f o l l o w i n g recurrence r e l a t i o n s ' ^ (3-£) f-^ixyy)- max min £~\ JR (U , v;x,y )dF(u)dG(v) _7 F G 0 s u c x 0 -S v ^  y = min max £~ J G F f (x,y)= max min / " J J / f R ( u , v ; x , y ) f (T,S)_7dF(u)dG(v)_7 ' ' F G 0 e £ U sj X . ' 0 •* v « y . - min max £~ J 1 3 . For proof of a p p l i c a b i l i t y of t h i s p r i n c i p l e to game processes see Bellman, OP . c i t . pp. 291-292. 1 4 . The existence and uniqueness of s o l u t i o n of t h i s approach i s found i n Bellman, I b i d . 77 With t h i s background we t u r n t o the general case of an . i n f i n i t e game i n which we c o n s i d e r the f u n c t i o n f(>:,y;k), the value to P-^  of the i n f i n i t e game b e g i n n i n g a t the k t h stage or subgame where P-^  and P 2 possess r e s o u r c e s x and y at t h i s stage and both employ optimal s t r a t e g i e s . The f u n c -t i o n a l equations d e s c r i b i n g the subsequent p e r i o d s are giv e n i n the re c u r r e n c e r e l a t i o n . (3-9) f ( x , y ; k ) = max min / " J J / " R ( u , v ; x } y ; k ) f (T, ,S k;k+ 1) 7 F • G . 0 * u a? x 0 « v <s y dF(u)dG(v) J - min max £~ J G • F . F u r t h e r c o m p l i c a t i o n s c o u l d be added t o the model but t h i s would take us away from the purpose of t h i s s e c t i o n which i s - to show the change i n f o r m u l a t i o n when we c o n s i d e r con-ti n u o u s r a t h e r than d i s c r e t e f u n c t i o n s . 3. Dynamics of Game S i t u a t i o n s With the background of game t h e o r e t i c r e a s o n i n g deve-loped thus f a r we s h a l l i n v e s t i g a t e some s i t u a t i o n s e x h i b i t -i n g dynamic game c h a r a c t e r i s t i c s . There are b a s i c a l l y two approaches i n the c o n s i d e r a t i o n of dynamic o p e r a t i o n a l systems, naraelj' continuous or d i s c r e t e time sequences."'-^ We u s u a l l y 15. Morse, P h i l i p Ii., ''Dynamics of O p e r a t i o n a l Systems", i n A c k o f f s "Progress i n Operations Research", op . c i t . Chapter 6. 78 t h i n k of the f i r s t c ategory as those where the s t a t e s of the system changes as a continuous f u n c t i o n of time. D e c i s i o n c r i t e r i a c o u l d be based on a comparison of the times f o r the systems to reach a predetermined s t a t e or other measures t h a t i n v o l v e c o n s i d e r a t i o n of time as a continuous v a r i a b l e . The case of d i s c r e t e time systems are e x e m p l i f i e d by the p e r i o d by p e r i o d games we observed, e a r l i e r and which we w i l l d i s c u s s as games of economic s u r v i v a l i n the remaining s e c t i o n of the chapter. For-the present we w i l l be concerned w i t h continuous time systems. The f i r s t s i t u a t i o n we s h a l l a nalyse i s the d e c i s i o n t o . p e n e t r a t e markets e x h i b i t i n g v a r i o u s growth p a t t e r n s . " ^ A number of. complex factors, are i n v o l v e d , of which the c a p a b i l i t y of the company, o b j e c t i v e s , time, markets, compe-t i t i o n , c o s t s , investments and experience are a few. At best we s h a l l c o n s i d e r some of these because the i n t e r r e l a t i o n s h i p of these f a c t o r s can be e x c e e d i n g l y complicated. The e n v i r o n -ment p i c t u r e d i s a market composed of many products i n which companies are competing f o r a share. To s i m p l i f y our e x p o s i -t i o n we s h a l l c o n s i d e r the company p l a y i n g a g a i n s t a u n i f i e d 16. The a n a l y s i s i s based on the i d e a s of the model i n R.W. A l l e n ' s " F a c t o r s I n f l u e n c i n g Market P e n e t r a t i o n " , Management Science, S e r i e s A, Volume 13, September 1966, pp. 22-43> but modified f o r the i n c l u s i o n of e x p l i c i t game t h e o r e t i c reason-i n g and method of f o r m u l a t i o n . 79 opponent, namely " c o m p e t i t i o n " i n the same product market. Thus i f , 1 7 S = market share of company at time t . C = market share of c o m p e t i t i o n a t time t . S' = r a t e of change of market share of company a t time t . C*= r a t e of change i n market share of c o m p e t i t i o n a t time t . . «*= a payoff f a c t o r s 1 to the company which i s a f u n c -t i o n * o f the market investment s t r a t e g i e s of the com-pany and c o m p e t i t i o n . The p a y o f f expectations' chan-ges w i t h time through experience gained ( f o l l o w i n g the e x p o n e n t i a l l e a r n i n g curve theory) i n the pro-c e s s . That i s «*-f (x,y;t) where x and y are the s t r a t e g i e s of the company and c o m p e t i t i o n r e s p e c -t i v e l y . .p = a pa y o f f f a c t o r < 1 to c o m p e t i t i o n s i m i l a r l y - de-f i n e d as f o r oL . Thus 0=g(x,y;t)l£ "V = a l o s s r a t e f a c t o r on the company's shares ( b u s i -ness l o s s e s in' the form of l o s i n g c o n t r a c t s , e t c . . ) S = as d e f i n e d f o r V but on the c o m p e t i t i o n ' s s n a r e s . H s = the a v a i l a b l e market a t time t . Mg=K-C-S A p i c t o r i a l r e p r e s e n t a t i o n of the market i n t e r a c t i o n i s shown i n f i g u r e (3-3)• We have the r a t e of change of market- shares 17. A l l the v a r i a b l e s d i s c u s s e d can be reduced to the common y a r d s t i c k of d o l l a r u n i t s . 18. For a constant or f i x e d a v a i l a b l e market <*.+. 0 = 1 at the steady s t a t e e q u i l i b r i u m , i . e . we have a con s t a n t -sum game. market d o l l a r s > a v a i l a b l e market > competition's share company's share time F i g u r e ( 3 - 3 ) : Market Trend with Share D i s t r i b u t i o n R e l a t i o n s h i p of a company and co m p e t i t i o n g i v e n by, (3-9) . < p = 6 t J V L-ys dt c-S u b s t i t u t i n g f o r K,. i n e c u a t i o n (3-9) and (3-10) we have cl -(3-11) S 1 = ot(K-S-C)- VS (3-12) C = ^ ( k - S - C ) - « T G 81 D i f f e r e n t i a t i n g (3-11) w i t h r e s p e c t to time we o b t a i n (3-13) s " = * ( i ^ _ s f - c ! ) - y s 1 R e p l a c i n g f o r 0* and C and r e a r r a n g i n g we are l e f t w i t h the 2nd order l i n e a r d i f f e r e n t i a l e q u a t i o n (3 -H) s"+ + + r + &)s'+ +(M -;-y£)s= *.(M' +6K) The equation can be s o l v e d f o r v a r i o u s market c o n d i t i o n s ( f o r example, constant, l i n e a r or t r a n s c e n d e n t a l ) and the competi-t i v e behavior of the p a r t i c i p a n t s as d e s c r i b e d by t h e i r i n t e r -l o c k i n g s t r a t e g y spaces. We are l e f t w ith f i n d i n g a c r i t e r i o n f o r the d e c i s i o n to penetrate the market. Two measures c o u l d be thought of, (1) the time to reach a p o s i t i v e g a i n G given by (3-15) G = S - K where K i s the t o t a l c o s t i n v o l v i n g the i n i t i a l investment, r e c u r r i n g investment and o p e r a t i n g c o s t , a l l the c o s t s and gains being reduced to present v a l u e s , and (2) the g a i n over investment r a t i o s . These two c r i t e r i a are l i k e n e d to the net present values and b e n e f i t c o s t r a t i o s of c a p i t a l budgeting t h e o r y . The r e s u l t s should i n d i c a t e t o us what investment plans or s t r a t e g i e s to f o l l o w i n order to reap most from a market p e n e t r a t i o n . We do r e c o g n i z e t h a t s u b j e c t i v e f a c t o r s w i l l i n f l u e n c e the r e s u l t s but u n t i l we are a b l e to q u a n t i f y them, they r e s t mainly i n i n d i v i d u a l judgement and can be 82 accounted f o r i n the a n a l y s i s by p u t t i n g more or l e s s severe c o n s t r a i n t s on the problem or changing the v a l u e s of the constant terms. Le t us now t u r n t o another process, the b a r g a i n i n g process, where economists have t r a d i t i o n a l l y l e f t to psy-c h o l o g i s t s the problem of p r o v i d i n g a s p e c i f i c s o l u t i o n r a t h e r than d e f i n i n g a d e l i m i t e d area as i n the case of von Neumann and Morgenstern n e g o t i a t i o n s e t . T h i s i s thought to depend on the " b a r g a i n i n g a b i l i t i e s " of the p a r t i e s concerned. We have seen how the Hash Axiomatic approach i s o l a t e s f o r us a p o i n t on the Pareto optimal s u r f a c e t h a t i s s a t i s f i e d by maximizing the product of the p l a y e r s u t i l i t i e s measured from the p o i n t of t o t a l disagreement. J.G. Cross had pointed out two short-comings of Nash's the o r y i n t h a t , " . . . F i r s t , i t o f f e r s no a n a l y s i s of the dynamic process of disagreement-concession-agreement t h a t c o n s t i t u t e s the very essence of the b a r g a i n i n g p r o c e s s . We are g i v e n • only a s o l u t i o n . c r i t e r i o n with no i n s i g h t i n t o i t s r a i s o n d'etre Second, acceptance of the d e s c r i p t i v e i n t e r p r e t a t i o n of the Nash model would imply acceptance of the c o n c l u s i o n t h a t a l l the i n f o r m a t i o n which i s necessary f o r the a n a l y s i s i s c o n t a i n e d i n the s e t of p o s s i b l e u t i l i t y - p a y o f f c o m b i n a t i o n . " 1 9 1 9 . Cross, John G,'"A Theory of the B a r g a i n i n g Process" op . c i t . p. 6 9 . 8 In what f o l l o w s we w i l l i n c o r p o r a t e the above obser-2 0 v a t i o n s i n a labour-management b a r g a i n i n g model. Cross makes two d i s t i n c t i v e cases i n the b a r g a i n i n g process, the pure b l u f f i n g and the pure i n t r a n s i g e n t cases, but r e c o g -n i z e s t h a t both are present i n v a r y i n g magnitudes i n a c t u a l s i t u a t i o n s . The a n a l y s i s of b a r g a i n i n g and n e g o t i a t i o n s i n v o l v i n g b l u f f i n g i s s t i l l i n i t s q u a l i t a t i v e stages and 21 remains to .be formulated mathematically. Our a t t e n t i o n however, w i l l be. d i r e c t e d at the i n t r a n s i g e n t case where genuine disagreement a r i s e s . For a 2 - p e r s o n s i t u a t i o n disagreement r e s u l t s because (3-16) q -i- c > M " 1 • ' 2 where q^ and'q^ are the q u a n t i t i e s of a commodity demanded b y ' p l a y e r 1 (p..) and p l a y e r 2 ( F 9 ) , having an a v a i l a b l e 2 0 . Ibid..pp. 7 0 - 7 1 . 2 1 . An e n l i g h t e n i n g a r t i c l e b r i n g i n g i n m i s r e p r e s e n t a t i o n of preference ( b l u f f i n g ) i n a b a r g a i n i n g and n e g o t i a t i o n i n t e r a c t i o n and the p o s s i b l e change i n u t i l i t y e v a l u a t i o n as the ne g o t i a t i o n proceeds i s " N e g o t i a t i o n : A Device f o r M o d i f y i n g  U t i l i t i e s " by F.C. I k l e i n c o l l a b o r a t i o n w i t h M. L e i t e s , Con-f l i c t Resolution,Volume VI, No. 1 , pp. 1 9 - 2 8 . A l s o found i n Shubik's "Game Theory and R e l a t e d Approaches to S o c i a l Be-h a v i o r " , op . c i t . pp. 2 4 3 - 2 5 7 . 8»+ 22 amount M. In b u s i n e s s , time has a monetary c o s t brought out i n the d i s c o u n t i n g f u n c t i o n and u t i l i t y i n t h a t our 23 d i s p o s i t i o n and value of an agreement changes through time. I f i n i t i a l l y expects P^ to concede a t some r a t e r ^ and P^ expects a c o n c e s s i o n r a t e r ^ from P-j then the expected time f o r . P j t o r e c e i v e q± i s + q 2 - Mj/r 2 = r The v a l u e of the u t i l i t y of agreement t o P^ i s the d i f f e r e n c e between the present value a t d i s c o u n t r a t e a of the u t i l i t y of q and the c o s t s which i s assumed to be f i x e d and r e c u r r i n g through time. Hence, X - a t e dt o (3-17) \ =. f t q j e ~ a t - f C •'o = f (q. )e " a t + h e a a To determine the value of q^ which maximizes we d e r i v e the f i r s t and second d e r i v a t i v e s of with r e s p e c t to q^ and pi a r r i v e a t two c o n d i t i o n s : • 22. The q u a n t i t y M need not be a f i x e d q u a n t i t y . I f we are considering; labour-management wage n e g o t i a t i o n , d i s a g r e e -ment a r i s e s whenever Q] > Q2 where Q]_ i s the union wage demand. U n l i k e the case where the two p l a y e r s i n d i v i d u a l l y s t r i v e s f o r the g r e a t e s t q^ p l a y e r 2 (management) i n the labour-management n e g o t i a t i o n p r e f e r s a low Qg. Hence t o a v o i d s i g n c o n f u s i o n i n the f o r m u l a t i o n we w i l l r e t a i n the d e f i n i t i o n q, -j- q 2 > M f o r disagreement. 23. I k l e and L e i t e s , op", c i t . 24. T h i s i s done i n the Appendix 85 (3-18) / ~ f ( q n ) + C  4 j a , ^ ( q ) a r2 1 t! (3-19) f (q-,) As the play progresses each p a r t y acquires i n f o r m a t i o n and modifies t h e i r e x p e c t a t i o n s , v/e can c h a r a c t e r i z e the l e a r n i n g process by the f o l l o w i n g c o n d i t i o n s : d r 0 .: . dq« > 0 I f - — > r dt a t ^ dr~ d q ? (3-20) = 0 i f - J J S = r 2 d r 9 dop — - < 0 ' i f — < r 9 dt dt meaning that i f concedes f a s t e r than i s expected P^ w i l l increase h i s e s t i m a t e of P 2's concession r a t e : i f F 2 concedes "iiist as r a o i d l v as i s expected P, w i l l r e t a i n h i s o r i g i n a l " • - 1 . estimate and so on. In f a c t P^'s expectations change d i -r e c t l y i n p r o p o r t i o n with the discrepancy between h i s expecta-t i o n s and'F^'s concession r a t e , that i s , (3-21). df? ^ n where fv = ' > 0 -c a t a i(-r„-qJ 2 -2' L2 " dt To determine -q-j , the concession rate of P-^  we d i f f e r e n t i a t e 8 6 equation ( 3 - 1 8 ) w i t h r e s p e c t to time. We get, ( 3 - 2 2 ) d r 2 11 d t a From equation ( 3 - 1 9 ) end ( 3 - 2 2 ) we see t h a t v a r i e s i n the c same d i r e c t i o n as r _ . What t h i s means i s t h a t i f P d i s c o v e r s t h a t P i s conceding at a r a t e h i g h e r than h i s e x p e c t a t i o n s , he w i l l , i n c r e a s e h i s demand q^, i n l i n e with the l e a r n i n g theory c o n d i t i o n s above. S i m i l a r e x p r e s s i o n s can be a r r i v e d f o r P^ , the s u b s c r i p t s being c o r r e s p o n d i n g l y r e p l a c e d and a d i f f e r e n t d i s c o u n t r a t e , b put i n place of a. met r i c c o n d i t i o n s , w i t h a p p r o p r i a t e t r a n s f o r m a t i o n of the o r i g i n of the u t i l i t y f u n c t i o n s i s a p a r t i c u l a r case of the ge n e r a l model. By assuming s u i t a b l e u t i l i t y and l e a r n i n g f u n c t i o n s and a p p l y i n g the general model we can determine equations d e s c r i b i n g the p l a y e r s demand as a f u n c t i o n of • time and the time taken to reach agreement. T h i s r e q u i r e s the s o l u t i o n of complicated d i f f e r e n t i a l equations which are be s t l e f t to r e s e a r c h o r i e n t e d t o comoutional methods and numerical s o l u t i o n s . I t can be shown t h a t the Nash s o l u t i o n under sym-2 5 . 2 6 . Refer t o Appendix Cross, John G, op .. c i t . pp. £ 1 - S A . 87 4. Games of Economic S u r v i v a l Vie s h a l l p o r t r a y i n t h i s s e c t i o n the r e l a t i o n s h i p between the market and f i n a n c i a l a spects of a f i r m through the f o r m u l a t i o n of a. game of economic s u r v i v a l as developed 27 by Shubik. ' I t was mentioned i n the i n t r o d u c t i o n t o t h i s chapter t h a t s u r v i v a l per se co u l d be a go a l as co u l d the maximization of the di s c o u n t e d value of the d i v i d e n d payment. To c h a r a c t e r i z e the dynamic p l a y we d i s t i n g u i s h the f o r t u n e s of the f i r m as d i v i d e d i n t o a corpora t e and a withdrawal account. The ga i n s and l o s s e s of the f i r m are r e f l e c t e d i n the former w h i l e the d e c i s i o n to make payments i n t o h i s w i t h -drawal account r e s t s w i t h the d e c i s i o n maker and h i s a s s e s s -ment of h i s chance of s u r v i v i n g . One r e s t r i c t i o n , however, i s that once payments are made to the withdrawal account they can no l o n g e r be a v a i l a b l e f o r use i n the corporat e account. To some extent t h i s i s u n r e a l i s t i c because c a p i t a l i s normally a v a i l a b l e i n the sense t h a t owners can s h e l l but r e s e r v e s f o r reinvestment or l o a n s c o u l d be f l o a t e d . But, t o reduce the complexity•of the problem and to b r i n g out e s s e n t i a l r e l a t i o n -ships we have to forgo t h i s added c o m p l i c a t i o n . 2 ? . Shubik, "Str a t e g y and Market S t r u c t u r e " , op . c i t . , and Shubik and Thomson, G.L., "Game of Economic S u r v i v a l " Naval Research L o g i s t i c s Q u a r t e r l y , Volume 6, 1 9 5 9 , po.. I I I - 1 2 3 . 88 Before we venture to f o r m a l i z e the case f o r n-person l e t us make a b r i e f e x c u r s i o n i n t o 2-person games of economic s u r v i v a l . In each p e r i o d p r i o r t o the r u i n o f one of them we can I d e n t i f y f o r each p l a y e r a market and a f i n a n c i a l move. The market a c t i o n determines the p a y o f f while the f i n a n c i a l move decides the payment t o the withdrawal.account. A 2-person game of economic s u r v i v a l can thus be c h a r a c t e r i z e d by: 1) a pay o f f f u n c t i o n Rj_>£ i n continuous v a r i a b l e space at time t f o r p l a y e r i = l , 2 . 2) a d i s c o u n t r a t e t where the s u b s c r i p t s are s i m i -l a r l y d e f i n e d . ' 3) A^ and A 2 are r u i n payments, B j and B 2 the r u i n con-d i t i o n s -such t h a t A < B^ ; A^ < B^ .. 4) C - | t , J c 2 ' t t - Q e corporate accounts and W - j , t ; W o , t the withdrawal accounts to the p l a y e r s denoted by the s u b s c r i p t s 1 and 2 a t time t . 5) a f i r m w i l l be r u i n e d i n time t i f i t s a s s e t s which we have condensed i n t o the f u n c t i o n s we termed corporate accounts f a l l below the r u i n c o n d i t i o n s . That i s , C i > t < B ± . 6) R j _ , t , the value of the income t o the s u r v i v o r i as scon a h i s opponent i s r u i n e d . In any p e r i o d t the cor p o r a t e account t o the i t h p l a y e r i s given by, (3-23) C i > t * C i , t _ 1 + R i J t - ^ ± , t * I f the f i r m i never l i q u i d a t e s the long-run d i s c o u n t e d payoff 89 (3-24) D ± = I w i > t f . t = 0 1 and i f i t l i q u i d a t e s at the end of t p e r i o d s . t = 0 i • In a game of economic s u r v i v a l and i n g e n e r a l , games having i n f i n i t e d u r a t i o n , the t h r e a t of r e p r i s a l s induces e q u i l i b r i a which do not e x i s t i n s i n g l e p e r i o d games. A p l a y e r can employ a v i o l e n t t h r e a t i n an e f f o r t to ma i n t a i n c e r t a i n s t a t u s QUO. I t i s thus p o s s i b l e to enforce non-coo p e r a t i v e e q u i l i b r i u m a t the j o i n t maximum. Much of the a b i l i t y t o c a r r y out e f f e c t i v e t h r e a t s l i e s i n the f l e x i -b i l i t y of the f i r m s , a f a c t o r founded on t h e i r c o s t s and f i -n a n c i a l structures-. Assuming t h a t each p l a y e r adopts a s t a t i o n a r y s t r a t e g y , one i n which the market and f i n a n c i a l moves are repeated through time, we can d e f i n e a s t a t i o n a r y  s t a t e s o l u t i o n as the set of s t a t i o n a r y outcomes, which the p l a y e r s can enforce by a'pair, o f t h r e a t s t r a t e g i e s . We denote a s t a t i o n a r y s t r a t e g y f o r a p l a y e r i by s.,. i S . ,,, h i s s et of p a r t i c u l a r s t r a t e g i e s . There i s a note of un-r e a l i s m i n our assumption, but i t b r i n g s out the a s s e r t i o n that- i n i n f i n i t e games our n o t i o n of e q u i l i b r i u m becomes nebulous, and the more so the more paranoid.are the p l a y e r s e s p e c i a l l y when we c o n s i d e r t h e i r s u i c i d a l attempts to c o n t r o l t h e i r opponents. Asymmetry i n the a s s e t s of the p l a y e r s are important • c o n s i d e r a t i o n s i n a game of economic s u r v i v a l . L e g a l and other e t h i c a l q u e s t i o n s s e t up c o n s t r a i n t s on the a c t i o n s o f the p l a y e r s . L e a v i n g these a s i d e a f i n a n c i a l l y s t r o n g e r p l a y e r w i l l not accept a long-run s t a t i o n a r y e q u i l i b r i u m i f i t i s p o s s i b l e to d r i v e out the opponent and.recoup more than enough t o equal the s t a t i o n a r y s t a t e income. The e q u i v a l e n t steady s t a t e p a y o f f a t which he i s i n d i f f e r e n t i s determined by s o l v i n g the f o l l o w i n g e q u a t i o n : (3-26) Dl - f = I R i . t f * I R ^ t f " t=0 t=0 t=t-l C l e a r l y we cannot d i s c o u n t the f a c t t h a t the opponent j ^ i can r e t a l i a t e other than adopting a d e f e n s i v e s t r a t e g y . Therefore IL i s b e t t e r d e f i n e d as: min max j). s j ' t s i » t 1 As we noted, l e g a l and s o c i a l r e s t r a i n t s w i l l modify somewhat the g o a l s of the f i n a n c i a l l y s t r o n g e r f i r m s . The f o r e g o i n g argument i s brought out more c l e a r l y i n a g r a p h i c a l a n a l y s i s i n a one d e c i s i o n v a r i a b l e problem. The ©C l i n e s i n f i g u r e (3-4) represent, the e q u i v a l e n t steady s t a t e payoff Kj_ t h a t p l a y e r 1 can enf o r c e as o u t l i n e d i n equa t i o n (3-26). ots wit h i n c r e a s i n g number of (') r e p r e s e n t lower i n i t i a l a s s e t s of p l a y e r 1. L n and L ? l i n e s are the l i q u i d a t i o n v a l u e s , the r e t u r n s from investment at l o n g run r a t e ' oi" r e t u r n of p l a y e r s 1 and 2 r e s p e c t i v e l y ( f o r o l a y e r 1 the number of ( T) i n the Ls correspond to the a s s e t l e v e l s i n the «Cs). At cC we note t h a t no p o s s i b l e e q u i l i b r i u m i s a t t r a c t i v e to o l a y e r 1 except the r u i n of h i s opponent. I f p l a y e r 1 s t a r t s cut w i t h a s s e t s i n the ot" p o s i t i o n then the p o s s i b l e steady s t a t e e q u i l i b r i a i s l i m i t e d to the r e g i o n 28. Adapted from Shubik, " S t r a t e g y and Market S t r u c t u r e " , op . c i t . P . 239. 92 on the Pareto curve marked o f f by the l i n e s and L^. S i m i l a r e x p l a n a t i o n can be made f o r the oc"' and L 2 l i n e s , only t h a t i n t h i s case the a s s e t s of p l a y e r 1 are such t h a t he can b a r e l y r u i n p l a y e r 2. We observe t h a t as the a s s e t s of the p l a y e r s become more symmetrical the s e t of p o s s i b l e e q u i l i b r i a becomes i n c r e a s i n g l y l a r g e r . A . f a c t o r which makes the e q u i l i b r i u m concept i n a dynamic game more confounding i s the r o l e played by chance. An otherwise s t a b l e s t a t i o n a r y s t a t e w i l l unexpectedly be destroyed as chance f l u c t u a t i o n s i n the p a y o f f s cause mis-f o r t u n e s to run against'one of the p l a y e r s . We might t h i n k of such a s i t u a t i o n as a f a i l u r e i n an a d v e r t i s i n g campaign or an even more d r a s t i c event. The values a s s i g n e d to a pa y o f f matrix element i s an' expected v a l u e over some proba-b i l i t y d i s t r i b u t i o n . As a r e s u l t of c e r t a i n market and f i -n a n c i a l p o l i c i e s , there i s a p r o b a b i l i t y t h a t the f i r m w i l l be r u i n e d . T h i s has been c o n s i d e r e d by Shubik and Thompson as a random walk problem. ' Other f a c t o r s t h a t c o n t r i b u t e , to the shaping of comp e t i t i o n are the s t r u c t u r e s of the markets d i s t i n g u i s h e d by t h e i r manufacturing, d i s t r i b u t i o n and r e t a i l i n g systems and the type, of products t r a d e d . The combined a c t i o n i n -v o l v i n g the mu l t i t u d e of d e c i s i o n v a r i a b l e s , f o r example, 29. Shubik, I b i d . pp. 247-249 and Shubik and Thompson, op . c i t . 9 3 p r i c e , a d v e r t i s i n g , p r o d u c t i o n and i n v e n t o r y s c h e d u l i n g , e t c . , w i l l expand c o n s i d e r a b l y the formal apparatus we have e r e c t e d ahove. Computational d i f f i c u l t i e s w i l l c a l l f o r experimenta-t i o n by s i m u l a t i o n w i t h h i g h speed computers. When we expand our a n a l y s i s t o s i t u a t i o n s i n v o l v i n g more than two p l a y e r s the n o t i o n of t h r e a t becomes even more awkward to handle. Measures s i m i l a r to the \£ f u n c t i o n f o r c o a l i t i o n s t r u c t u r e s c o u l d be employed t o d e s c r i b e the i n t e r -a c t i o n . The assumption i s made t h a t we can d i v i d e an n - p l a -y e r i n t e r a c t i o n i n t o 3 s e t s c o n t a i n i n g k, r and n-k-r p l a y e r s . The (kj[) - ( r j ) s t a b l e market e q u i l i b r i u m " ^ s p e c i f i e s t h a t the j o i n t a c t i o n by the r p l a y e r s i n the r s e t cannot y i e l d them more than m a i n t a i n i n g the steady s t a t e on the assump-t i o n t h a t each of the k p l a y e r s i n the k s e t i s committed to a t h r e a t s t r a t e g y w h i l e those of the plaA^ers i n the n-k-r use t h e i r stead}' s t a t e s t r a t e g y . The problem i s i n d e f i n i n g these d i f f e r e n t s e t s and j u s t as i n the y s t a b i l i t y f u n c t i o n economic and extra-economic i n f o r m a t i o n w i l l be r e q u i r e d to l i m i t the c o m b i n a t o r i a l p o s s i b i l i t i e s . In the case of a market dominated by a. s i n g l e f i r m , i t can be d e s c r i b e d by (1^) - (r) . s t a b i l i t y . (1^) stands f o r the dominant f i r m , (r) stands f o r any s e t of r p l a y e r s not i n c l u d i n g the domi-nant f i r m , and, s i n c e we do not need to d i s t i n g u i s h between 3 0 . Shubik,. I b i d . Chapter 1 1 . 9h d i f f e r e n t s e t s of r p l a y e r s we do not use a s u b s c r i p t w i t h ( r ) . For complete domination we have (1^) - ( n - l ) s t a b i l i t y . T h e ' c o n d i t i o n f o r the general case of (k. ) - ( r . ) s t a b i l i t y can be expressed as f o l l o w s : — •" k * * ' k (3-27) max R-, (S,T V) - R (S",T i) f o r j ^ i ' • . VJ which means th a t the p l a y e r j i n s e t r i s maximizing h i s p a y o f f u s i n g h i s optimal s t r a t e g y a g a i n s t the t h r e a t s t r a t e g i e s of a l l the p l a y e r s i n the k s e t d e s c r i b e d by the v e c t o r T ^ i w h i l e the r e s t n-k-r p l a y e r s use t h e i r s t a t i o n a r y s t r a t e g i e s r e p r e -sented by the v e c t o r . S. The r e a s o n i n g can be extended t o other p l a y e r s i n r . F o r e q u i l i b r i u m the index of ( r ^ ) - (k^) s t a b i l i t y must be d e f i n e d with the r o l e s i n the e x p r e s s i o n ( 3 - 2 7 ) r e v e r s e d . The s t a b i l i t y c r i t e r i a t h a t we examined i n the g e n e r a l case i s h e l p f u l only i n showing us what we can i n f e r from an e q u i l i b r i u m s t a t e . No- e x p l i c i t c o n s i d e r a t i o n of the dynamic path to e q u i l i b r i u m or from one e q u i l i b r i u m t o another i s g i v e n except i n the 2 - p e r s o n cases we have gi v e n i n s e c t i o n 3 and the e a r l i e r p a r t of t h i s s e c t i o n . ^ As was suggested a t the beginning of the chapter we note t h a t our concept of s o l u t i o n depends on the understanding, of t h r e a t s t r a t e g i e s , , 3 1 . The path to e q u i l i b r i u m f o r a 2 - p e r s o n case i s d i s - . cussed, by means of g r a p h i c a l a n a ^ s i s by Shubik and Thompson, op . c i t . D P . 1 2 0 - 1 2 3 Indeed the t h r e a t components of a t h r e a t s t r a t e g y can be almost any f u n c t i o n and can i n v o l v e very complex b e h a v i o r . I t i s hoped t h a t some l i g h t would be thrown on the nature of these c o m p l e x i t i e s through gaming and s i m u l a t i o n ex-periments. CHAPTER IV STRATEGY AND STRUCTURE IN THE OIL AND GAS INDUSTRY: AN APPLICATION CF THE SHELL MODEL.1 1. Overview of the Industry: A major goal of t h i s study i s related to the problem of how game theory methods can be used to elucidate and con-tribute to strategic decision making. Factors a f f e c t i n g gaso-l i n e sales i n the medium range future of the North American o i l and gas industry are areas o f f e r i n g opportunities f o r such invest i g a t i o n . The economic background of the industry had given r i s e to strongly oriented o l i g o p o l i s t i c competition requiring the use of sophisticated decision tools'. "Fortune" provides an informative l i s t i n g of o i l companies. The vast number of competitors i n the business reaching f o r shares of the market makes i t of considerable urgency f o r a competing firm to search f o r p o l i c i e s aimed not only f o r survival or status quo p o s i t i o n but f o r expand-ing p r o f i t a b l e ventures. • By f a r and large we can perceive the petroleum industry as comprising three stages i n the v e r t i c a l structure, namel}/ crude producing,- r e f i n i n g and marketing, exploration and 1. ' Hughes and Ornea, op . c i t . 2. "Fortune", June 1967, pp. 196-213. For the Canadian scene, re f e r to "survey of O i l s 1968", publication of the Financial.Post. 97 d r i l l i n g being o r d i n a r i l y conceived w i t h i n crude p r o d u c t i o n functions.-^ I nterspaced between these d i v i s i o n s and the consuming p u b l i c at one end are c o m p e t i t i v e markets. I f there were no i n t e g r a t i o n the crude and products would pass through the s u c c e s s i v e markets to the. consumer market. Conversely i f every f i r m were f u l l y i n t e g r a t e d and complete-l y balanced there would be no market, beyond those over which the f i n a l products pass ' i n t o the hands of the u l t i m a t e consumer. However, although there i s f u l l i n t e g r a t i o n , no major^ company i s completely balanced and most independents are n o n - i n t e g r a t e d or p a r t i a l l y integrated.'^ F o r the f u l l y i n t e g r a t e d concerns, sometimes more crude may be produced f o r them to r e f i n e w h ile a t other times they may r e f i n e more crude than they produce .. The i n t e r m e d i a t e markets provide b u f f e r s f o r such imbalance. In the product market i t i s p o s s i b l e t h a t they may market more than they r e f i n e or v i c e v e r s a . The u n p r e d i c t a b l e occurrence and e x h a u s t i b l e nature of crude d e p o s i t s have made i t o f t e n times economical f o r a company to s e l l the crude i t produces f o r other r e f i n e r i e s , 3. Hamilton, D.C. "Competition i n O i l ' 7 , Harvard U n i v e r s i t y P r e s s . Cambridge (1958), p. 4« 4. A d i s t i n g u i s h i n g f a c t o r between a major and an indepen-dent marketer i s the customary 2 cent p r i c e d i f f e r e n t i a l a l l o w -ed to the l a t t e r on the market f r o n t . 5. Hamilton, D.C, op . c i t . 98 while i t may f i n d more to i t s advantage to buy crude from sources l o c a t e d near i t s r e f i n e r i e s . The i n t e r a c t i o n of these f o r c e s as determined by the p o l i c i e s of the v a r i o u s companies gives r i s e to c o m p e t i t i o n or c o l l u s i o n which we can analyse by examining the i n t e r l o c k i n g s t r a t e g y spaces of the p a r t i e s t o c o n f l i c t or c o - o p e r a t i o n . 2 . S t r a t e g i c D e c i s i o n s i n the O i l I n d u s t r y The f o r e g o i n g d i s c u s s i o n of the g e n e r a l s t r u c t u r e of the o i l i n d u s t r y p r o v i d e s a b a s i s f o r examining the t3>-pes.of d e c i -s i o n s t h a t an o i l company must c o n s i d e r . B r o a d l y , i n l i n e w i t h the f e a t u r e s of the i n d u s t r y , three areas of s t r a t e g i c d e c i s i o n s can be i d e n t i f i e d . These are concerned with marketing, r e f i n i n g , and e x p l o r a t i o n , a f a c t noted i n our d i s s e c t i o n of the i n d u s t r y i n t o d i f f e r e n t l e v e l s - . In the model to be d i s c u s s e d we are mainly i n t e r e s t e d i n marketing d e c i s i o n s , l a r g e l y because i t i s here t h a t c o m p e t i t i o n i s keenest.. Importance of optimum d e c i s i o n making i n the o t h e r l e v e l s of the i n d u s t r y are not to be underplayed. In f a c t the proper approach i s to adopt a systems viewpoint i n or der to s t r i v e f o r o v e r a l l o p t i m a l i t y . Being an area where there i s tremendous growth, c o n t i -nual change i n c h a r a c t e r , l o c a t i o n and o p e r a t i o n of f a c i l i t i e s the consummer market i s c l o s e l y . w a t c h e d by r i v a l f i r m s . In 6 . I b i d . p. 7» no other sphere i s i t e a s i e r f o r op e r a t o r s t o e n t e r or e x i t . Excess c a p a c i t y i s recognized to predominate among r e t a i l s e r v i c e s t a t i o n s and i s g e n e r a l l y the cause of depressed e a r n i n g s . Marketing, investment d e c i s i o n s thus determine t o a l a r g e extent the success of a f i r m . These investments . both f o r new o p p o r t u n i t i e s and to develop, modernize•and expand e x i s t i n g investments i n s e r v i c e s t a t i o n s and o t h e r f a c i l i t i e s run t o s u b s t a n t i a l p r o p o r t i o n s of the o v e r a l l investment f i g u r e s of o i l f i r m s . To quote some comparative f i g u r e s , i n 1965 I m p e r i a l - O i l i n v e s t e d 21.6 m i l l i o n d o l l a r s i n marketing, which i s 25.2 percent of t o t a l c a p i t a l ex-p e n d i t u r e . For the same p e r i o d S h e l l i n v e s t e d 12.1 m i l l i o n d o l l a r s , which i s 28.6 percent of c a p i t a l e x p e n d i t u r e . P r i c e s t a b i l i z a t i o n i n the l a s t few years has produc-ed a general s h i f t i n marketing s t r a t e g i e s t o non - p r i c e and •other forms of c o m p e t i t i o n . T h i s has opened a new arena f o r implementing marketing investment d e c i s i o n s . A l s o i t i s p r e d i c t e d , though not an immediate i s s u e i n our present de-c i s i o n problem, t h a t trends i n shopping h a b i t s w i l l g r e a t l y change the manner of marketing g a s o l i n e . S e r v i c e s t a t i o n s w i l l g r a d u a l l y become p a r t of r e t a i l complexes as-powerful 7. I m p e r i a l O i l L i m i t e d , Annual Report 1965. 8. S h e l l Canada L i m i t e d , Annual Report 1965. 100 competition from g i a n t r e t a i l s t o r e s and g e n e r a l merchandise o chains become an important f a c t o r i n g a s o l i n e m a r k e t i n g . J R e f i n i n g has been the most f u l l y developed branch of the i n d u s t r y and where independent e n t r y f i n d s staunchest b a r r i e r s l a r g e l y because of ' h i g h e r investment o u t l a y s . R e f i n i n g processes are e s s e n t i a l l y the same and only w i t h g r e a t e f f o r t through r e d u c t i o n , i n c o s t s can a f i r m expand i t s m a r g i n by f r a c t i o n s of a cent. Thus the areas of s t r a -t e g i c concern to a r e f i n e r are p r i c e and a v a i l a b i l i t y of crude, p r i c e and a v a i l a b i l i t y of p i p e l i n e and o t h e r means of t r a n s p o r t a t i o n , problem of access to markets and techno-l o g i c a l development."^ In e x p l o r a t i o n and d r i l l i n g independents have u n t i l r e c e n t l y played prominent r o l e s . ' The majors w i t h a b i l i t y at m a r s h a l l i n g the ever i n c r e a s i n g c a p i t a l r e q u i r e d f o r expen-d i t u r e i n such a c t i v i t i e s have taken over l a r g e p r o p o r t i o n s of these a c t i v i t i e s . Competition i n the o i l i n d u s t r y i s l i n k e d to the' e r r a t i c f l u c t u a t i o n s and annual a d d i t i o n s to c a p a c i t y and new o i l f i e l d d i s c o v e r i e s . Competitive s t r a t e g i e s are ne-9. "Canadian Petroleum", October 1967, p. 17. 10. de Chazeau, H.G., and Kahn, A.S., " I n t e g r a t i o n and  Competition i n Petroleum I n d u s t r y " , Y a l e U n i v e r s i t y Press, New Haven (1959). 1 0 1 c e s s a r y i n the face of u n c e r t a i n t y a r i s i n g from l a c k of con- . t r o l of raw materials., t r a n s p o r t a t i o n , p r o c e s s i n g and market-i n g . In the p u r s u i t of s e c u r i t y i t i s observed t h a t o i l f i r m s have over the years proceeded on an e x t e n s i v e h o r i z o n t a l and v e r t i c a l program of i n t e g r a t i o n . For, as de Chazeau and Kahn remarked, "Competitive s t r a t e g i e s have many f a c e t s , e s p e c i a l l y i n an i n d u s t r y as dynamic as t h i s one. I n t e g r a t i o n i t s e l f has been one o f these f a c e t s . Investment, inn o -v a t i o n and non-price c o m p e t i t i o n are o t h e r s " . H At the present time, t h i s i s however the s u b j e c t of much con t r o v e r s y e s p e c i a l l y i n view of p u b l i c p o l i c i e s d i r e c t -ed a t the o r g a n i z a t i o n of the o i l i n d u s t r y and i t s tendency • to m o n o p o l i s t i c c o n t r o l . F i n a l l y , .though i t may seem t h a t the d i r e c t i o n i n which the g a s o l i n e market i s heading s i g n a l s f o r new s t r a t e g i c moves, "Competition f o r g a s o l i n e s a l e s i n the l a s t 2 years has s t r a n g e l y brought very few r a d i -c a l changes i n marketing p a t t e r n s a c r o s s Canada and the U.S... Markets are growing but major companies appear t o h o l d tena-c i o u s l y onto t h e i r shares of the. b u s i n e s s . . . The r e g i o n a l c h a r a c t e r of the market p r e v a i l s ( i n the U.S.) d e s p i t e e f f o r t s of some majors to expand n a t i o n a l l y . " 12 11. I b i d . p. 375. 12. "The O i l and Gas J o u r n a l " , February 1967, Volume 65, No. 6, p. 56. 1.02 O i l companies are f a c e d w i t h other problems of a co'mpe -t i t i v e n a t u r e • o u t s i d e the c o n f i n e s of the immediate i n d u s t r y . Trends i n i n t e r - f u e l c o m p e t i t i o n have loomed l a r g e of r e c e n t 1 years . ^ F u r t h e r m o r e , s e r i o u s a i r . p o l l u t i o n problems have taken changes i n p u b l i c p o l i c i e s and c o n t r o l and a c c e l e -r a t e d the advent of oth e r forms of energy. The output, com-p o s i t i o n of the i n d u s t r y w i l l then have to take d r a s t i c changes t o meet w i t h t i m e s . ^ 3. The .Shell Model 1 $ Although we may have k i n d l e d an i n t e r e s t i n the model developed a t S h e l l by f r e q u e n t l y r e f e r r i n g to i t at v a r i o u s stages i n the preceding chapters, we have not made any s i g -n i f i c a n t examination of the composition c f i t s b a s i c s t r u c -t u r e . In t h i s s e c t i o n We s h a l l attempt to o u t l i n e i t s main 1 3 . I b i d . p. 5 2 . 1A. Success by B r i t i s h Petroleum to produce e d i b l e p r o t e i n s at t h e i r French r e f i n e r y goes to show the gradual change i n t y p i c a l r e f i n e r y product balances, "Canadian Petroleum News," June, 1 9 5 7 , p. 1 2 " . . . . ~ 1 5 . P a r t of t h i s s e c t i o n , e s p e c i a l l y the a n a l y t i c a l d e t a i l s i s the r e s u l t of p r i v a t e communication with P r o f e s s o r J . Swir-l e s , a member of the team r e s p o n s i b l e f o r s e t t i n g up the model. 103 f e a t u r e s , paving a way f o r p o s s i b l e a p p l i c a t i o n t o an i n d u s -t r i a l s i t u a t i o n c h a r a c t e r i z e d by the B r i t i s h Columbia o i l and gas i n d u s t r y . But, i t must be a s s e r t e d t h a t we s h a l l not i n  any way r e s o r t t o d e t a i l e d computation f o r o b t a i n i n g game theo-r e t i c p o i n t s nor to concern o u r s e l v e s s o l e l y w i t h the v a l i d i t y  o f the mode of c h a r a c t e r i z a t i o n . The S h e l l model i s an  attempt at i n t e g r a t i n g what we have d i s c u s s e d i n the f o r e - going c h a p t e r s . I t r e p r e s e n t s one of the few e f f o r t s a t  endeavouring to c o n s t r u c t a model of a complex c o m p e t i t i v e  s i t u a t i o n i n order to provide meaningful r e s u l t s to a i d  competent management i n making d e c i s i o n s . Hence, taken t h a t the model does provide us w i t h a method of c a l c u l a t i n g these p o i n t s our purpose i s to r a i s e i s s u e s as to the manner we proceed to u t i l i z e these r e s u l t s , a q u e s t i o n t h a t has not been r e s o l v e d f u l l y . The S h e l l model code named STRATCOH was i n t e n d e d t o l o o k a t a m u l t i p e r s o n non-zero sum game i n continuous v a r i a b l e space. In t h e i r paper Hughes and Ornea"""^ c o n s i d e r e d the case of a duopoly market s i t u a t i o n o n l y . F i g u r e (4-1) i n g e n e r a l shows the s t r u c t u r e of the model from the viewpoint of a s i n g l e company. A b a s i c c o n s i d e r a t i o n i n the model i s the e x c l u s i o n of e x p l o r a t i o n and p r o d u c t i o n a c t i v i t i e s . The 16. op . c i t . 10h Net Return to Shareholders ~1 Other D i v i s i o n s !and Ventures Stocki Issues! t SDividends F i n a n c i a l C o n t r o l Banks and Bondholders T LPG, Avgas, Kerosene, F u e l O i l , A s p h a l t , e t c Manufacturing Bulk G a s o l i n e Market Marketing 1 S a l e s Purchases M a t e r i a l Flows Cash Flows Motor G a s o l i n e F i g u r e (4-1): STRATCOM; S t r u c t u r e of Model p r i c e of crude oil-was f i x e d f o r t h i s ' r e a s o n and f l u c t u a t i o n s i n i t s a v a i l a b i l i t y were not c o n s i d e r e d . Other v e n t u r e s , e.g. chemical companies which c o n t r i b u t e to t h e ' f i n a n c i a l success . ' ' 17 of ah o i l company were not c o n s i d e r e d as p a r t of the model. • G a s o l i n e was the prime product w i t h r e g a r d to decision..mak-ing'.'Other r e f i n e r y products were assumed to be s o l d a t f i x e d p r i c e s i r r e s p e c t i v e of the p r o d u c t i o n l e v e l of t h e - r e -17. Hughes and Ornea. 105 A s e t of competitors which may compose of s i n g l e or groups of companies are chosen. F o r the i n i t i a l s t a t e s , the c u r r e n t t o t a l debt, investment i n manufacturing f a c i l i t i e s (including; f a c i l i t i e s f o r marketing byproducts and h a n d l i n g the bulk g a s o l i n e market) and investment i n r e t a i l g a s o l i n e marketing f a c i l i t i e s of each competitor are taken. The per-formance of the competitors over a number of time p e r i o d s are c a l c u l a t e d . A l t o g e t h e r three time p e r i o d s i n a ten year h o r i z o n are c o n s i d e r e d . D e c i s i o n s i n the f i r s t p e r i o d apply to the f i r s t and. second y e a r s , those i n the second p e r i o d run from the t h i r d to the f i f t h year, v/hile the t h i r d p e r i o d concern the remainder, of the t e n y e a r s . In each time p e r i o d the s e t of d e c i s i o n s i n v o l v e investments i n r e -f i n e r i e s and marketing, debt, r e t a i l p r i c e of g a s o l i n e and the buying and s e l l i n g of .bulk g a s o l i n e . To.enumerate they a r e : 1) Increase i n debt (AB) 2) Hew investment i n manufacturing ( AK) . 3) New investment i n marketing ( AM) Z|.) Tankwagon p r i c e f o r r e t a i l s a l e s (P) 5) F r i c e margin f o r s e l l i n g g a s o l i n e i n bul k market (Q ) 6) F r i c e margin f o r buying g a s o l i n e i n b u l k market (Q~) The e f f e c t of the f i r s t t h r e e d e c i s i o n s i s f e l t t h r -oughout the subsequent p e r i o d s . The l a t t e r t hree a f f e c t the c u r r e n t time p e r i o d only and these p r i c e s are assumed cons-t a n t over the e n t i r e p e r i o d concerned, not too r e a l i s t i c f o r implementation at f i r s t s i g h t . T h i s can be taken to be 1.06 an average over the e n t i r e p e r i o d and i s reasonable a p p r o x i -mation to an o v e r a l l s t r a t e g y . In a c t u a l p r a c t i c e day-to-day c o n s i d e r a t i o n of the balance of r e f i n e r y runs, i n v e n t o r y po-s i t i o n and i n d i v i d u a l bulk s a l e s and purchases govern the bulk p r i c e margin. A market i n i t i a l l y h aving a s a l e s volume of V(0) and p r e d i c t e d t o grow a t a uniform r a t e such t h a t at time t . (4-1) V(t) = V C 0 ) ( l + - o c t ) where c i i s a constant, i s taken as. the s t a r t i n g p o i n t . I f , M.(t) = the marketing investment a t time t of 1 p l a y e r i K. ( t ) = the manufacturing investment a t time t 1 of p l a y e r i Bj.(t) = the debt, p o s i t i o n at time t of p l a y e r i t = 0 f o r the v a r i a b l e s above r e p r e s e n t s i n i t i a l c o n d i t i o n then t o t a l marketing investment at time t i s (4-2) M(t) -- I M ± ( t ) ' • i and s i m i l a r l y f o r manufacturing investment (4-3) K(t) = \ K t ( t ) M^(t) and X ^ ( t ) the marketing and manufacturing investment of player i a t time t i s g i v e n by (4-4) M j ( t ) = aM(t-l) +.AM(t-l) 107 (4-5) K ±(t) * bK(t-l) -:- AK(t-l) where a and b are parameters, their values being determined by the deterioration of investment functions. I f i t i s assumed that the deterioration i s 10 percent for manufac-turing and marketing f a c i l i t i e s then a ="b = 0.9. With these investment levels the supply at time t, V^(t) of competitor, i to meet the market demand i s determined by the 18 following equation. (4.-6) V t ) _Mi(t) -v(tT " M(t) • t i v t j ' m - u -where P^(t) i s the tankwagon price for r e t a i l sales of compe-t i t o r i and P(t) i s the weighted average price given by, ."(4-7) P(t) = I M.(t)F.(t) M(t) The bulk prices are determined by an algorithm"deviced for the model. A simple interpretation of the Eulk Algorithm for a 2-person case is shown graphically in figure (4-2). The horizontal scale represents the volume of sales at time t. This is divided between the players, V-^  going to player 1 and to player 2, these values having been calculated from the variables of foregoing equation (4-6). At prices Q-j_' o n t^ l e upper broken curve player 1 sells bulk gasoline and buys at 18. The relationship is based on an empirical studv done at Shell. 108 p r i c e s Q£ on the lower dashed curve. The same goes f o r p l a y e r 2, s u b s c r i p t 1 being r e p l a c e d w i t h 2. In the g i v e n r e p r e s e n t a -t i o n p l a y e r 1 buys, g a s o l i n e t o f u l f i l l h i s market demands. I t must be r e a l i z e d t h a t the a c t u a l program i s much more complicated than these -• when, we c o n s i d e r more than' 2 p l a y e r s . The above have been a s i m p l i f i e d a b s t r a c t i o n to h i g h -l i g h t the important f e a t u r e s . Proceeding , . wi t h g i v e n v a l u e s f o r d e c i s i o n s and i n i t i a l s t a t e s , the STRATCOM pro-gram computes year by year f o r each p l a y e r (1) the i n v e s t -ment l e v e l s ( ^ ( t t l ) = % ( t + l ) - ¥i±[t) and S K j U + l ) = K ^ t + l ) - K - J t ) ) , (2) s a l e s (V i ( t}) and p r o d u c t i o n l e v e l s ( C j ) i a n d (3) the f i n a n c i a l balance i n c l u d i n g new l o a n s (6Bj_(t)). L i m i t s are imposed on the model f o r economic reasons and c a l c u l a t i o n a l c o n s i d e r a t i o n s i n order to achieve r e a l i s m . • The r e t a i l g a s o l i n e price' i s l i m i t e d to P i ( t ) *c Constant. f o r otherwise a c o o p e r a t i v e s t r a t e g y ' by a l l the p l a y e r s w i l l place no bounds on the p r i c e . In r e a l i t y consumers'' r e a c t i o n , l e g a l and e t h i c a l r e s t r i c t i o n s , t h r e a t of e n t r y set the l i m i t s . F o r a s i m i l a r reason bulk g a s o l i n e p r i c e s are kept at l e v e l s comparable to a c t u a l s i t u a t i o n s . The i n t e r e s t , r a t e i s based on the p o l i c y o f s e t t i n g a debt c a p a c i t y of 109 the company. A l i n e a r r e l a t i o n s h i p between debt a n d . t o t a l c a p i t a l investment i s assumed. For the model, (4 -8 ) E I (t) ^ k ( K i ( t ) -J- K ^ ( t ) ) where k. i s a con s t a n t . F i n a l l y , the bond i n t e r e s t r a t e r ^ i s taken t o be a f u n c t i o n of the debt - c a p i t a l r a t i o . Thus., (4 -9 ) is = f ( B i ( t ) ) K ^ t ) + M i ( t ) For each s e t of d e c i s i o n s the r e s u l t i n g f i n a n c i a l ba-l a n c e s can he compared, the c r i t e r i o n being t o maximize the Bulk P r i c e - x = volume g a s o l i n e t r a n s f e r r e d AY / / / / — • — _ — ^^^^ X v 2(t) V ( t ) B. Gasoline- s a l e s volume. Fi g u r e ( 4 - 2 ) : S i m p l i f i e d G r a p h i c a l R e p r e s e n t a t i o n of Bulk A l g o r i t h m 110 t o t a l d i s c o u n t e d cash flows over the s p e c i f i e d time h o r i z o n ( i n t h i s case 10 y e a r s ) . In each y e a r the cash flow e q u a t i o n f o r i t h p l a y e r i s gi v e n by, (4-10) Zt= P i ( t ) V i ( t ) + A B ^ t ) - A M ^ t + l ) - A K - ^ t+ l ) - r ^ U ) - r C ± ( t ) +xQ~ + ( p r o f i t from s a l e of by products) where C j ( t ) = the crude p r o d u c t i o n V - crude o i l c o s t •+ Q i ' = p r i c e margin f o r s e l l i n g b u l k Q^ "" = p r i c e margin f o r buying bulk x s the bulk bought or s o l d as determined by the Bulk A l g o r i t h m From the y e a r l y cash flow the t o t a l d i s c o u n t e d cash flow (R) i s determined, a d i s c o u n t r a t e f r e l a t e d t o the c o s t of c a p i t a l b e i n g used. Thus w i t h the I n i t i a l parameters known, the v a l u e s of the d e c i s i o n v a r i a b l e s chosen, f o r the i t h p l a y e r the t o t a l d i s c o u n t e d cash flow can be r e p r e s e n t e d f o r an n - p l a y e r s i t u a t i o n by a f u n c t i o n , ( 4 - l D . R i Z " M 1 ( 0 ) , K 1 ( 0 ) , B 1 ( 0 ) , . . . , f / L L ( 0 ) , K i ( 0 ) , B i ( 0 ) M n ( 0 ) , K n ( 0 ) , B n ( 0 ) , . . . , Al^, A K j / Q j A E 1 , P 1 > . . . , AB ,P 7 n ' n -I f f o r each p l a y e r we can summarize the i n i t i a l s t a t e s and d e c i s i o n v a r i a b l e s by a v e c t o r X as was done i n chapter I I I , s e c t i o n '2.., then Of-11), becomes, ^j-(Xj ,X,>, ... ,X^,... ,X^) Fo r a 2 p l a y e r s i t u a t i o n P l a y e r l ' s p a y o f f becomes Rj(Xj,X2) . . P l a y e r 2's payoff becomes R 2 ( X ,X 2) T O The A l t e r n a t e Play A l g o r i t h m ' i n the program searches f o r an e q u i l i b r i u m p o i n t f o l l o w i n g the steps o u t l i n e d i n e q u a t i o n ( 2 - 9 ) of chapter I I . Having obtained the game t h e o r e t i c p o i n t s i t would be i n t e r e s t i n g t o analyse t h e i r s i g n i f i c a n c e and u s e f u l n e s s i n d e c i s i o n making. Before e n t e r i n g ' i n t o such a d i s c u s s i o n the B.C. g a s o l i n e market w i l l be examined so as to pr o v i d e the necessary background. L. The B.C. Gas o l i n e Market S t r u c t u r e The' petroleum market i n B r i t i s h Columbia i s dominated by seven major f i r m s , two of which are r e a l l y the marketing 1 9 . F o r d e t a i l s of t h i s , one is r e f erred to the paper "A H e u r i s t i c A l g o r i t h m f o r C a l c u l a t i n g E q u i l i b r i u m P o i n t s i n  Competitive S i t u s t i o n s " , by P r o f e s s o r J . S w i r l e s presented a t the Canadian Operations Research S o c i e t y Annual Meeting i n Toronto, fMay 1 0 , 1 9 6 S . 112 s u b s i d i a r i e s of two of the majors. In e f f e c t the number of separate e n t i t i e s engaged i n the market i s f i v e . Minor f i r m s i n B.C. have r e c e n t l y taken on a primary importance i n the 2 0 major m e t r o p o l i t a n a r e a s . B.C. i s a t t r i b u t e d w i t h a l a r g e r number and v a r i e t y (both i n s i z e and l o c a t i o n ) o f r e f i n i n g f a c i l i t i e s than would be expected i n a market of i t s s i z e . As a r e s u l t i t has r e f i n i n g c a p a c i t y i n excess of i t s im-mediate needs. In. a d d i t i o n r e f i n e r y outputs from A l b e r t a and 2 1 the U.S. have a l s o e n t e r e d the market. S i x r e f i n e r i e s w i t h v a r y i n g c a p a c i t i e s serve the market. D e s p i t e the advantage of economies of l a r g e s c a l e , h i g h t r a n s p o r t a t i o n c o s t s f o r both crude petroleum and r e f i n e d products.and l i m i t e d s i z e .of p a r t i c u l a r markets make-, i t . f a v o r a b l e t o operate s m a l l e r r e f i n e r i e s with h i g h e r o p e r a t i n g c o s t s . C o n s i d e r a t i o n s of these kinds can'be o p t i m i z e d by an a n a l y s i s u s i n g l i n e a r programming techniques. The S h e l l model d e s c r i b e d p r o v i d e s only the s i z e of the manufacturing investment t o be undertaken. Since 1 9 5 5 the shares of major brands i n the r e t a i l oo market have seen a d e c l i n e . " These changes have taken 2 0 . "An A n a l y s i s of Competition and P r i c e Behavior i n the  B r i t i s h Columbia Petroleum I n d u s t r y , " prepared, f o r I m p e r i a l O i l L i m i t e d , Toronto, O n t a r i o , Canada, by S t a n f o r d Research I n s t i t u t e , Menlo Park, C a l i f o r n i a , May 1 9 6 4 , p. V I I - 3 5 . Most of the o b s e r v a t i o n s i n t h i s s e c t i o n are d e r i v e d from t h i s study. 2 1 . I b i d - p. V I I - 3 4 . 2 2 . I b i d . p. V I I - 1 4 . 113 p l a c e amidst a p e r i o d of g e n e r a l expansion. I n f l u x of minor and independent brands have accounted l a r g e l y f o r t h i s . During t h i s p e r i o d the independents have i n v e s t e d i n 1 8 . 6 $ of the 23 t o t a l number of s e r v i c e s t a t i o n s b u i l t . The e n t i r e p r o v i n c e of B. C. may not be r e p r e s e n t a t i v e f o r the whole Canadian or U. S. markets, i t s r e g i o n a l c h a r a c -t e r i s s u f f i c i e n t l y d i f f e r e n t t o r e q u i r e separate a n a l y s i s . E f f e c t i v e g a s o l i n e r e t a i l i n g t o combat e f f e c t i v e com-p e t i t i o n i s most accentuated around the h i g h l y po]3ulated areas of Vancouver and Vancouver I s l a n d . S t u d i e s and da t a have shown t h a t i n c r e a s e i n o u t l e t s i n these h i g h volume mar-kets have not matched the g e n e r a l p a t t e r n of i n c r e a s i n g s a l e s . T h i s time l a g c o u l d be regarded as an area worth e x p l o r i n g and e x p l o i t i n g by o i l companies l o o k i n g f o r p r o f i t a b l e ven-t u r e s and by r e s e a r c h e r s seeking problems f o r a p p l i c a t i o n of ope r a t i o n s r e s e a r c h t e c h n i q u e s . The v a r i a t i o n s i n market shares among competing f i r m s between r e g i o n s i s -an o b s e r v a t i o n worth n o t i n g . I m p e r i a l O i l L i m i t e d w i t h the h i g h e s t p r o v i n c e wide average market share •does not command the same percentage i n the major po-pu l a t e d areas as i t does i n the Northern and Southern I n t e r -i o r of 3. C. Major companies have experienced s u b s t a n t i a l de-2 3 . I b i d . p. V l l - l ' r . l i f -e l i n e s i n the c i t y areas. By c o n t r a s t minor brands wi t h a lower p r o v i n c e wide average have gained c o n s i d e r a b l e prominence i n the m e t r o p o l i t a n markets around Vancouver and Vancouver I s l a n d . T h i s d i v e r s i t y and i n s t a b i l i t y i n r a n k i n g of v a r i o u s brands among r e g i o n s over time i s f u r t h e r evidence o f the need to cope w i t h market v a r i e t y and to c o n s t r u c t e f f e c t i v e a n a l y s i s of the impact of s t r u c t u r a l changes on company p o l i c i e s . P r i o r t o 1955 "the majors accounted f o r a l l the o u t l e t s i n B. C. By 19&3 marketing o u t l e t s f o r minor and independent brands have become s i g n i f i c a n t . The t h r e a t of e n t r y of these independent brands and t h e i r impact on the g a s o l i n e r e t a i l i n g s t r a t e g i e s of the majors are areas o f worthwhile i n v e s t i g a t i o n and r e s e a r c h f o r students of game theory* 25' i n the S t a n f o r d Research I n s t i t u t e Study ' the s m a l l f i r m s are found to experience r a p i d i n c r e a s e s i n market shares a g a i n s t the d e c l i n e s of t h e i r major c o u n t e r p a r t s . T h i s i s a p p a r e n t l y i n agreement w i t h the s i t u a t i o n Hughes and Ornea c o n s t r u c t e d i n t h e i r paper to demonstrate game t h e o r e t i c c a l c u l a t i o n s . Here e x t e n s i v e marketing investment i s not a t t r a c t i v e to l a r g e s i z e f i r m s . The s t r u c t u r e of the petroleum i n d u s t r y which we have 2h. I b i d . p. VII -20 25- I b i d . p. VII -3 1 : -26. op. c i t . j u s t o u t l i n e d c o u l d provide a s u i t a b l e - framework f o r an a n a l y s i s of com p e t i t i v e choice and the development of e f f e c t i v e marketing p o l i c i e s of a f i r m . 4. A p p l i c a b i l i t y of the S h e l l Model The model, based on the framework discussed, i n the previous s e c t i o n we l i k e to see run on the B.C. g a s o l i n e market and p o s s i b l y other i n d u s t r i a l s i t u a t i o n s should be able t o handle cases i n v o l v i n g m u l t i p e r s o n s i t u a t i o n s . The f i v e m a j o r s i n c l u d i n g the two marketing s u b s i d i a r i e s and the group of independents c o u l d be c h a r a c t e r i z e d as a 6-p l a y e r game. The e x t e n t of i n t e g r a t i o n and r e t a i l p r i c e o f g a s o l i n e have been c i t e d as d i s t i n g u i s h i n g f a c t o r s between majors and independents. Assuming t h a t the model c o n s t r u c t e d i s f e a s i b l e i n terms of producing c a l c u l a b l e r e s u l t s , and th a t f o r the out put'we are given a s e t of e q u i l i b r i u m s t r a t e g i e s f o r each p l a y e r , the purpose then as keynoted i n the t h e s i s i s t o draw a t t e n t i o n . • to what p o s s i b l e use can be made of these r e s u l t s . To v a r i o u s extents we have i l l u s t r a t e d the type of a n a l y s i s t h a t was done i n Hughes and Ornea's paper. At t h i s p o i n t we would l i k e t o r e i t e r a t e our b e l i e f t h a t i n the near f u t u r e the approach p r o v i d e d by game t h e o r e t i c a n a l y s i s to e l a b o r a t e p r a c t i c a l problems concerning corapeti t i o n w i l l surpass s o p h i s t i c a t e d techniques c u r r e n t l y i n use However, the answers to the questions of when and how we 1 1 6 s h o u l d , , t a k e - o f f ' a f t e r r e s u l t s p r o v i d e d by a model s i m i l a r to t h a t done at S h e l l have been o b t a i n e d remain to be determined. Le t us r e t u r n to the simple example i n Chapter I and s t r e t c h i t to b r i n g out t h i s p o i n t . Assume the g a s o l i n e market share d i s t r i b u t i o n t h a t p r e v a i l s i n B.C. i s such that- the l e a d i n g major (Impe r i a l O i l L i m i t e d ) ' h o l d s . 3 0 p e r c e n t of the t o t a l , the r e s t of the majors hold..! 1 5 percent each w h i l e 'the independent group holds' the remaining 1 0 p e r c e n t . Our coneern then becomes one .of viewing the market from the p o i n t of a f i r m i n the major c l a s s but excluding: t h a t of the l e a d i n g major. Given t h a t the r e l e v a n t data have been ob t a i n e d , a com-puter s i m u l a t i o n on a model s i m i l a r to the S h e l l model but modified to s u i t • c o n d i t i o n s and e x p l i c i t p o l i c i e s of the com-pany i s conducted. I f f e a s i b l e r e s u l t s from the computer run are produced, should we then adopt them as p o l i c i e s or s t r a -t e g i c g u i d e l i n e s , or should we-employ s t r a t e g i e s o t h e r than the recommendations of the e q u i l i b r i u m s t r a t e g i e s ? In s h o r t how do we use these r e s u l t s . We have t o some l e n g t h d i s c u s s e d the i m p l i c a t i o n s of game t h e o r e t i c p o i n t s t o d e c i s i o n making i n S e c t i o n A, Chapter I I . Some of these concepts c o u l d be the b a s i s f o r formulating, our i n i t i a l t a k e - o f f , but they are by no means the a l l - i n c l u s i v e answers. Even then fundamentally we are s t i l l f a c e d with a s e r i e s of u n r e s o l v e d q u e s t i o n s as to what a c t i o n s we should u l t i m a t e l y choose i n order to be 117 most w e l l o f f i n the medium and long.run f u t u r e . Of course we could e x p l o r e p o s s i b l e immediate g a i n s . Coming back to the example, saippose t h a t the market has e t present 1000 s e r v i c e s t a t i o n s and i s p r e d i c t e d t h a t the volume of g a s o l i n e consumption i s l i k e l y to grow i n the next 2 years by 20 percent. I t i s noted t h a t each. year, the e x i s t i n g s t a t i o n s s u f f e r a 10 percent d e t e r i o r a t i o n r a t e . T h i s means t h a t c u r r e n t investment would not only have t o take i n t o c o n s i d e r a t i o n new market o p p o r t u n i t i e s but a l s o the r e b u i l d i n g of o l d e r investments. The estimated number of s e r v i c e s t a t i o n s r e q u i r e d t o f i l l t h i s new volume i s taken to respond p r o p o r t i o n a l l y t o the i n c r e a s e , t h a t i s to 1200 s t a t i o n s . Under these circumstances, the q u e s t i o n any mana-gement would l i k e to see answered i s :"Given t h a t a computer run on the r e l e v a n t data recommends an investment of 30 s t a t i o n s , should management accept these r e s u l t s or should, they take a l t e r n a t i v e a c t i o n s based on the p i c t u r e of the r e s u l t i n g s i t u a t i o n which the output of the computer program prov i d e s ? " Some i d e a of the courses of a c t i o n to l o o k f o r c o u l d be approached by examining the i n f l u e n c e of t h r e a t p o i n t s and Fareto s u r f a c e on the d e c i s i o n . To achieve outcomes on the Fareto s u r f a c e we observed i n Chapter I I t h a t some form of co o p e r a t i o n or c o l l u s i o n may have to be .obtained. Threat s t r a t e g i e s may i n a l l e v e n t u a l i t y keep everyone on the s t a t u s 1 1 8 quo or i f carried, out w i l l r e s u l t ' in., l o s s e s o r l e s s gains on the p a r t of the c o m p e t i t o r s . There may be some o p p o r t u n i t i e s f o r a l i t t l e edging i n on the shares of o t h e r c o m p e t i t o r s . I t i s here t h a t we hope to d i s c o v e r , through the methodology of game theory..the b e s t l i n e of p e n e t r a t i o n or defence. In a simple one d e c i s i o n v a r i a b l e of a one p e r i o d cons-t a n t sum model the-normative r u l e of game-theory p r e s c r i b e s the maximin s o l u t i o n . As the model becomes non-constant sum and more complex w i t h a d d i t i o n a l v a r i a b l e s and extending over many per i o d s t h i s r u l e does not apply and w i l l not n e c e s s a r i l y y i e l d e q u i l i b r i u m s o l u t i o n s . Thus i t appears t h a t the qiies-t i o n of how we can use the r e s u l t s p rovided by a model such as t h a t done at S h e l l remains to be r e s o l v e d . The t h e s i s ; i n i t s e f f o r t t o f i n d a p p l i c a b i l i t y f o r the use of game theory i n the development of c o m p e t i t i v e models,has reached such an impasse. I t i s t h e r e f o r e hoped t h a t f u r t h e r work i n t h i s d i r e c t i o n would be c a r r i e d out. Such c o n t r i b u t i o n would be most welcomed by p r a g m a t i s t s . 0nl3>- thus can the technique of game theory methods f i n d a p l a c e i n management d e c i s i o n a i d s as o t h e r s such as l i n e a r programming have c u r r e n t l y found. A few g e n e r a l remarks c o u l d be made about the S h e l l model.-Indeed i t can be a t t r i b u t e d as a s i g n i f i c a n t c o n t r i b u t i o n i n the f i e l d of o p e r a t i o n s r e s e a r c h and management s c i e n c e . Kuch have been s a i d and views have been made r e g u l a r l y i n l e a r n e d j o u r n a l s r e g a r d i n g p o s s i b l e use of game the o r y to p r a c t i c a l 119 business s i t u a t i o n s but very l i t t l e have a c t u a l l y been done. The main areas seemed to be those i n psychology, where simple m a t r i x .games have been designed to t e s t human b e h a v i o r . The S h e l l model c o u l d be'regarded as one of the f i r s t attempts at modeling c o m p e t i t i v e b u s i n e s s s i t u a t i o n s . I t r e p r e s e n t s a cornerstone from which more s o p h i s t i c a t e d and more r e a l i s t i c models can be c o n s t r u c t e d . • However, as i t now stands, i t i s not without shortcomings. Host of these have been r e c o g n i z e d by Hughes and Ornea. These are a s s o c i a t e d w i t h the d e t e r m i -n i s t i c nature of 3TRATC0H. More r e a l i s t i c models should take i n t o account u n c e r t a i n t i e s which c o u l d as a f i r s t approxima-t i o n be taken care of i n an e l a b o r a t e s e n s i t i v i t y a n a l y s i s . C l e a r l y the model has not reached a stage of refinement f o r f u l l - p r o o f a p p l i c a t i o n t o r e a l s i t u a t i o n s . The i n a d e q u a c i e s as demonstrated e a r l i e r and i t s s i m p l i c i t y and d e t e r m i n i s t i c c h a r a c t e r are not r e a l l y l i m i t s t o i t s e v e n t u a l f r u i t f u l use i n dynamic c o m p e t i t i v e s i t u a t i o n s because the model can be e n l a r g e d to take i n t o account such f a c t o r s . What i t l e a v e s us w i t h , i s the s o r t of r e s u l t s and. comparisons which might be made wit h a more ac c u r a t e model. 120 CHAPTER V SUMMARY AND CONCLUSIONS While i t i s r e c o g n i z e d t h a t i n the past progress i n the use of game theory f o r b u s i n e s s a p p l i c a t i o n s has been r e l a -t i v e l y i n s i g n i f i c a n t , i t s p o s i t i o n as a management t o o l f o r the e f f e c t i v e a n a l y s i s of c o m p e t i t i v e b u s i n e s s s i t u a t i o n s i n the f u t u r e w i l l no., doubt improve. T h i s has been l a r g e l y due to advances i n h i g h speed computer technology. The i n c r e a s -i n g complexity of the b u s i n e s s environment i s demanding a t o t a l l y new way of examining the f o r c e s at p l a y w i t h i n and without the f i r m . E f f o r t s have been expended i n showing the nature of such s i t u a t i o n s i n Chapter I where the g e n e r a l con-c l u s i o n reached i s t h a t there i s a need f o r a new and e f f e c t -i v e t o o l to s o l v e the type of s t r a t e g i c problems which modern management f a c e s . In Chapter I I the d i r e c t i o n pursued i s one mainly of e x p o s i t i o n . Though many books and papers have d e a l t i n some ways w i t h v a r i o u s aspects of game theory, they are e i t h e r too r i g o r o u s l y mathematical i n nature or they l a c k of s u f f i c i e n t depths to be u s e f u l f o r a person seeking o p e r a t i o n a l p r i n c i -p l e s . . An e x c e p t i o n , however, i s the book " S t r a t e g y and Mar-ket S t r u c t u r e " by M a r t i n Shubik ~. T h i s cnapter i n e f f e c t . 1... Op. c i t . 121 attempts at g i v i n g a b i r d ' s eye view of the v a r i o u s aspects and concepts of game- theory and e s p e c i a l l y to h i g h l i g h t impor-t a n t r e l a t i o n s h i p s . I t i s o r i e n t e d from a p r a c t i c a l p o i n t of view and examples from the next two chapters draw c o n s i d e r a -b l y from the p r i n c i p l e s enunciated. I n p a r t i c u l a r , we saw how game t h e o r e t i c p o i n t s i n f l u e n c e d s t r a t e g i c d e c i s i o n making. To extend an understanding of game theory beyond simple con-cepts, and i n view of a d e s i r e t o f i n d a p p l i c a t i o n s , vie have demonstrated the kinds of r e s u l t s t h a t c o u l d be o b t a i n e d from 2 an experiment of the type done at S h e l l That t h e r e i s a complete and c o r r e c t 2-person zero' or constant sum game theory i s the g e n e r a l concensus. Indeed t h i s t heory i s so s o l i d l y founded t h a t i t i s d i f f i c u l t t o con-c e i v e t h a t i t c o u l d be i n c o r r e c t . On the other hand as we have shown t h i s i s not the case w i t h non-zero sum and n-person games. T h i s , we have seen i s r e l a t e d to the problem o f c o a l i -t i o n f o r m a t i o n . But, w i t h the i n c r e a s i n g c a p a b i l i t i e s o f com-puters such c o m b i n a t o r i a l problems c o u l d e v e n t u a l l y . b e o v er-come. Furthermore our answer to a problem i s o n l y as a c c u r a t e as the data themselves, so much so we f i n d t h a t i t i s o f t e n necessary i n d e a l i n g w i t h the s u b j e c t i v e environment to make s i m p l i f y i n g assumptions. A number of c r i t i c a l comments have been d i r e c t e d at 2. Op. c i t . 122 the r e l e v a n c e of game theo r y . David W. M i l l e r ° had a t t r i b u -t e d the shortcomings of the theory to b a s i c m e t h o d o l o g i c a l con-s i d e r a t i o n s r e l a t i n g t o the measurement of u t i l i t i e s , and some p a r a d o x i c a l d i f f i c u l t i e s w i t h the. concept of r a t i o n a l i t y . ' How-ever, he d i d r e c o g n i z e . t h a t , i t might be assumed t h a t f o r many k i n d s of c o m p e t i t i v e s i t u a t i o n s , d o l l a r amounts, i n one form or another, r e p r e s e n t s s a t i s f a c t o r y measures of p a y o f f . " e B e s i d e s , Hughes and Ornea^ have p o i n t e d out t h a t game theory per se assumes n o t h i n g . The r a t i o n a l i t y i s s u e thus does not a f f e c t our r e s u l t s i n any a p p r e c i a b l e way. What game t h e o r e t i c methods go out to f i n d i s the best a c t i o n for, us regardless, of what cur opponents do. The o b j e c t of going .through the a n a l y s i s of dynamic games i n Chapter I I I i s p r i m a r i l y to b r i n g out c o n c e p t u a l d i f -f i c u l t i e s i n proceeding from s i n g l e p e r i o d to continuous games. The a p p l i c a t i o n of the processes of dynamic programming methods to a game s i t u a t i o n was d i s c u s s e d . D i s c r e t e v a r i a b l e f u n c t i o n s give way to continuous v a r i a b l e f m i c t i o n s as we probe deeper i n t o the dynamics of game s i t u a t i o n s . . Two examples of i n d u s t r i -a l s i t u a t i o n s were taken to show the use of games as d e c i s i o n 3. M i l l e r , David W . "TJie_ilelevance of Game Theory" ? i n "Model of Markets" e d i t e d by A l f r e d O x e n f e l d t , Columbia u n i v e r s i t y Press, New York, 1963, pp. 26'5~306. h» • I b i d . p. Loc. c i t . 123 t o o l s i n dynamic s e t t i n g s . In g e n e r a l i t was found t h a t the computation can be s u b s t a n t i a l and i n v o l v i n g . The l a s t s e c t -i o n of the chapter shows how an on-going market s i t u a t i o n can be c h a r a c t e r i z e d as a game of economic s u r v i v a l . Leaving a-s i d e s o c i a l and other r e s t r a i n t s , f i n a n c i a l s t r e n g t h , as ex-pected? i s a major f a c t o r i n determining who the s u r v i v o r s would be. The a n a l y s i s a l s o brought out the e f f e c t s of t h r e a t s t r a t e g i e s on e q u i l i b r i u m p o i n t s o l u t i o n s 5 w h i c h .cover an i n -c r e a s i n g l y l a r g e r r e g i o n of the Pareto s u r f a c e as the f i n a n -c i a l a s s e t s of the p l a y e r s or competitors become more symmetri-c a l . Our i n t e r e s t subsequently turned to the a p p l i c a b i l i t y of game theory to the o i l i n d u s t r y w i t h s p e c i a l i m p l i c a t i o n s to the s i t u a t i o n i n B r i t i s h Columbia. The o v e r a l l s t r u c t u r e of the i n d u s t r y and the kin d s of s t r a t e g i c d e c i s i o n problems t h a t a r i s e at the v a r i o u s l e v e l s of the i n d u s t r y were examined. One of the most c r u c i a l we noted i s i n the area of marketing i n v e s t -ment d e c i s i o n s . I t thus appears t h a t the s i t u a t i o n c a l l s , f o r the s o r t of a n a l y s i s p r o v i d e d by game theory. In t h i s con-n e c t i o n , the b a s i c approach t h a t c o u l d be a p p l i e d i s STRATCOH, the competitive d e c i s i o n model developed at S h e l l 0 . Some of the e s s e n t i a l f e a t u r e s of the model were d i s -cussed so as to i n d i c a t e i t s r e l e v a n c e i n such a p r a c t i c a l s i t u -6. Op. c i t . 1 2 * + a t i o n . As to i t s a p p l i c a b i l i t y and p o t e n t i a l i t y f o r other i n -d u s t r i a l s i t u a t i o n s we noted t h a t f u r t h e r m o d i f i c a t i o n s are r e -qu i r e d before any i d e a of implementation could s u c c e s s f u l l y ma-t e r i a l i z e . Vie are e s p e c i a l l y concerned here w i t h the q u e s t i o n of how the r e s u l t s p r o v i d e d by a model such as t h i s should be used. N e v e r t h e l e s s , i t i s g e n e r a l l y agreed t h a t the S h e l l model r e p r e s e n t s one of the f i r s t few co n c r e t e attempts a t p r o -v i d i n g the s o r t of r e s u l t s t h a t are meaningful to management contemplating game theory f o r complex c o m p e t i t i v e d e c i s i o n p r o -blems. I t must be s a i d t h a t even the ardent c r i t i c of game theory, D. W. M i l l e r would agree to t h i s by h i s statement t h a t , ' "... A theory may very w e l l deepen understand-i n g by r a i s i n g q u e s t i o n s even i f the answer o f f e r e d by the theo r y ? a r e f o r one or another reason, u n s a t i s f a c t o r y . " 7 F i n a l l y , i t must be mentioned t h a t w h i l e the t h e s i s has not c o n t r i b u t e d .to b a s i c r e s e a r c h other than the cormnents on some of the foremost l i t e r a t u r e d e a l i n g w i t h p r a c t i c a l ap-p l i c a t i o n s of game, theory, i t i s hopjed, i t lias, at l e a s t aroused an i n t e r e s t and opened up new e f f o r t s i n seeking use f o r game t h e o r e t i c methods. 7 . Op. c i t . p. 2 6 7 BIBLIOGRAPHY BIBLIOGRAPHY A c k o f f , R. L., e t . a l . A Model Study of the E s c a l a t i o n and De-e s c a l a t i o n of C o n f l i c t . Management Sc i e n c e Center, U n i -v e r s i t y of P e n n s y l v a n i a ( 1 9 5 7 ) • A l l e n , A. W. F a c t o r s I n f l u e n c i n g Market P e n e t r a t i o n . 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Feb-ru a r y 6 , 1 9 6 8 . The P r o v i n c e . S u t h e r l a n d , J . V/. Exports S u f f e r s f o r Lack of Free E n t e r p r i s e . , February 1 2 , 1 9 6 * 8 " . von Neumann, J . and Morgenstern, Oscar. Theory of Games and Economic Be h a v i o r . P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n • I W . " 131 APPENDIX 1). % = f ( q ; L ) e " a t + ^ e " a T _ C i a a -=-^=6 f ' q 1 + . f q -a e ^ - - ; - - ^ - a ) e Given t h a t (q •+ q 2 -M)/i" 2 = r f'(q ) - f ( ^ ^ : 0 1 1 r 2 2 or / T f ( q i ) + ^ _7 ^ = f , ^ l } To s a t i s f y maximum c o n d i t i o n s 4 < 0 J l - a t - r 2 _ a t ^ 2 -- a f ^ H - a J e ( « ) - a f l ^ J e | - | J ± . • q i G-, ae I r — r 1 - h e ——•« 1 3^1 1 * q f T h i s reduces to f " ( q ) - . i - f ( q 3 ) - 0 1 1 ? J-2 r„ - a < 0 Rearranging, r 2 f ' ( q 1 ) = a f ( q ] L) + D i f f e r e n t i a t i n g with r e s p e c t to time t Brxting j ^ - = qx and ^ = f2 T h i s reduces to a f t e r r e a r r a n g i n g t o , 9i = - 2 " r a - a 

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