Does game theory offer “new” mathematical images of economic reality? GIORGIO ISRAEL Dipartimento di Matematica Università di Roma “La Sapienza” P.le A. Moro, 2 00185 – Roma (Italy) [email protected]

Modern developments in game theory, in the early 20th century, have no special links with economic theory. The contributions by Ernst Zermelo and Emile Borel, for instance, were related above all to traditional parlour games. It is to John von Neumann that we owe the axiomatic foundation of the theory and its close linkage with economics. Indeed, the mention made in von Neumann’s first paper providing proof of the fundamental minimax theorem (J. Von Neumann 1928) refers directly to the economic equilibrium theory: «… any event – given the external conditions and the participants in the situation (provided that the latter are acting on their own free will) – may be regarded as a game of strategy if one looks at the effects it has on the participants […] this is the principal problem of classical economics: how is the absolute selfish homo œconomicus going to act under given external circumstances». Nevertheless this is the only reference by von Neumann to Walras’ theory that is not openly critical. As early as 1928 and subsequently, on numerous other occasions, he displayed a polemical attitude to general economic equilibrium theory (see R. J. Leonard 1992, 1995). Morgenstern further emphasized this point of view, going as far, in the ‘seventies, as to speak of the advisability of actually dispensing with the term “equilibrium” (O. Morgenstern 1973, see also G. Israel, forthcoming b). Von Neumann’s and Morgenstern’s critique of the neoclassical theory of equilibrium may be summed up as follows: (a) This theory is based on a form of mechanistic reductionism while, in order to be original and effective, any formalized (i.e. mathematized) economic theory must be based on an autonomous conceptualization and not simply imported from mechanics or physics. (b) One manifestation of this dreary reductionism is the adoption of the concept of

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equilibrium, the approach in terms of equations and the use of the methods of classical maximization: these must be replaced by the concepts of fixed point, minimax, disequation, and an “interactive” form of optimization, as it were. (c) Game theory may represent the conceptual core of a new and original form of the mathematization of social and economic processes. However, to cut all remaining ties with the theory of equilibrium, it is necessary to consider as central the cooperative point of view since, according to von Neumann, it makes «more social sense», rather than the non cooperative one which is closer to the microeconomic approach. This statement was made by von Neumann to Shubik, as reported by the latter, precisely in a controversial reference to Nash’s approach in which priority is given to the non cooperative approach. Furthermore, the approach of Theory of Games and Economic Behaviour (J. von Neumann, O. Morgenstern 1944) overshadows the concepts of minimax with respect to the centrality of the concepts of imputation and stable set. As Morgenstern observed later (O. Morgenstern 1973), the notion of stability that emerges in connection with imputations and stable sets «has nothing to do with the usual equilibria of physics […] indeed the present notion differs so profoundly with the usual ideas of stability and equilibrium that one would prefer to avoid even the use of the words. But no better ones have yet been found». (d) It is unwise to nourish too many illusions concerning the possibility of making a satisfactory descriptive analysis of a social and economic system because of the extreme complexity of the factors involved. On this point von Neumann goes as far as to adopt an almost holistic attitude when he asserts, in a letter to Harold Kuhn written on 14 April 1953, that «I think that nothing smaller than a complete social system will give a reasonable “empirical” picture». This view entails a preference for the normative approach, which is particularly apparent in the interpretation given by von Neumann and Morgenstern to the concept of “mixed strategy” (see L. Dell’Aglio 1995). Indeed they rarely appeal to the frequentist approach according to which the mixed strategy represents an average behaviour with respect to a large number of games and the randomized choice corresponds to the distribution of the frequencies of the individual strategies that are available to each player. They never utilize the psychologistic approach, which was preferred by Borel, in which the mixed strategy represents the description of the internal process leading a player to his final choice and in which the assignment of a probability to

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a strategy is a symbolic representation of the psychological mechanism involved in the final choice. They rather prefer the rationalist approach in which, if a player plays regularly, the opponent can guess his intentions, so that the choice of a mixed strategy corresponds to an attempt to protect oneself by probabilizing one’s behaviour and thus the choice of a set of probabilities is a “rational” choice. In other words, von Neumann is not attempting to describe human psychology. He is pursuing the more limited and specific aim of axiomatizing as many of its aspects as possible, as this is the only available scientific method for objectivizing them: «There are of course many – and most important – aspects of psychology which we have never touched upon, but the fact remains that a primarily psychological group of phenomena has been axiomatized» (J. von Neumann, O. Morgenstern 1944). The gap between this approach and that of social atomism is quite evident. Von Neumann and Morgenstern have sharply deviated from any attempt to construct a theory of individual rationality, aiming rather, as Robert Leonard (R. J. Leonard 1992, 1995) pointed out, at the construction of a mathematics of society intended as a whole. In order to explain this diffidence towards what is now called methodological individualism it is necessary to consider the roots of von Neumann’s social conceptions which, probably under the influence of the political events in which he was involved, were increasingly influenced by a fundamental distrust of the individual’s rational behaviour. Unlike the theory of economic equilibrium, which is based on a definition of the agent’s rational behaviour, he aimed rather at the determination of “wise” and “reasonable” conditions capable of establishing a comparatively rational social order. Taking these various aspects into account, together with the centrality the axiomatic approach had for von Neumann, we observe, en passant, that the reconstruction made by Mirowski (P. Mirowski 1992, 2002) of the significance of von Neumann’s contribution to economics is nothing more than a literary invention (see G. Israel, forthcoming). Among the various points mentioned above, (b) is the central one as it has represented a source of misunderstanding among mathematicians and economists concerning the actual import of the game theoretic approach. The difference of opinion between von Neumann and Samuelson is significant in this regard. The former, in a letter dated 8 October 1947 to Morgenstern, accused Samuelson of using primitive mathematics, at most acceptable at the time of Newton, and of having «murky ideas about

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stability», adding that «even in 30 years he won't absorb game theory». The latter, also quite recently (P. A. Samuelson 1989), accused von Neumann of having attempted a bluff with his theory about a new mathematics for economics allegedly based on game theory, and claimed that economics had never had any need for optimization processes other than those of Newton, nor would it ever. From a general standpoint, Samuelson was quite wrong and his attitude only confirmed von Neumann’s opinion that he would understand nothing about game theory even after 30 years. The difference between the competitive approach and the game theoretic approach is encapsulated in Osborne and Rubinstein’s effective statement: «To clarify the nature of game theory, we contrast it with the theory of competitive equilibrium that is used in economics. Game theoretic reasoning takes into account the attempts by each decision-maker to obtain, prior to making his decision, information about the other players’ behavior, while competitive reasoning assumes that each agent is interested only in some environmental parameters (such as prices), even though these parameters are determined by the actions of all agents. To illustrate the difference between the theories, consider an environment in which the level of some activity (like fishing) of each agent depends on the level of pollution, which in turn depends on the levels of the agents’ activities. In a competitive analysis of this situation we look for a level of pollution consistent with the actions that the agents take when each of them regards this level as given. By contrast, in a game theoretic analysis of the situation we require that each agent’s action be optimal given the agent’s expectation of the pollution created by the combination of his action and all the other agents’ actions» (M. J. Osborne, A. Rubinstein 1994). Also in the modern version based on Nash’s concept of equilibrium, this difference remains as, although the player’s behaviour in Nash’s equilibrium and Walras’ agent are similar, as both are parametric behaviours (H. Moulin 1981), for the former the parameters are not prices but the others’ strategies, which is a radical difference, precisely in the sense illustrated by Osborne and Rubinstein’s above remark. However, even though Samuelson was formally wrong and did not understand the true nature of the problem, from the point of view of subsequent historical developments things by no means went in the direction predicted and hoped for by von Neumann and Morgenstern.

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The first reversal with respect to this predicted and hoped-for direction came about as a result of Gerard Debreu’s work. Even though he acknowledged von Neumann and Morgenstern’s «outstanding influence which freed mathematical economics from the traditions of differential calculus and compromises with logic» (G. Debreu 1959), his Theory of Value represents a skilful reconversion of results obtained in the field of game theory for the purpose of making a reappraisal in axiomatic terms of Walras’ theory and of providing proof of one of its central results – the equilibrium existence theorem. This reconversion was achieved by “pruning” all the results utilizable for this purpose of all references to game theory. It is no coincidence that the results obtained by von Neumann in different contexts become utilizable thanks to Nash’s theorem. As Defalvard pointed out «the existence of Nash’s equilibrium refers to the same conditions of Walrasian equilibrium as the Arrow-Debreu model, as those given by Brouwer’s fixed point theorem and as Kakutani for his extensions to correspondences» (Defalvard 2001). Consequently, an analysis more strongly focused on Nash’s equilibria overshadows the dynamic aspects in favour of the fixed points of this dynamics. There is no doubt that von Neumann (more than Morgenstern) was relatively uninterested in the dynamic aspects, not because he considered them insignificant, but because he considered it preferable to put off considering them to a later stage. In his opinion, the mathematization of a complex subject such as economics would require much time and patience and to tackle the dynamic problem too precipitously would not lead to any worthwhile results. However, as far as the concept of equilibrium was concerned, von Neumann could not fail to realize that an approach à la Nash ultimately enhanced the centrality of Walras’ theory. It was again Defalvard who correctly pointed out that «while the formal kinship between the two concepts of equilibrium [of Nash and Arrow-Debreu] had already been noted en passant, it had never been subjected to any in-depth scrutiny». Any such scrutiny would probably reveal how Debreu’s use of the results of game theory, and in particular Nash’s, was instrumental in reproposing a classical microeconomic interpretation and of sterilizing or at least of marginalizing the role of game theory in the formalization of economics which should instead be considered central in the view of von Neumann and Morgenstern. Attention should be focused on the fact that the operation performed through Debreu’s Theory of Value (even more than through Arrow and Debreu’s model) corresponds to a programme designed to reinstate the centrality of methodological

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individualism, that is, precisely the point of view that von Neumann and Morgenstern’s research programme tended radically to challenge. Another aspect of this programmatic clash is the emphasis laid by von Neumann and Morgenstern on the primacy of the cooperative approach. In their view, the possibility of cooperation arises as soon as the number of players exceeds 2 and so the players form coalitions: «Interdependence has, of course, been recognized, but even where neoclassical economics of the Walras-Pareto type tried to describe this interdependence, the attempt failed because there was no rigorous method to account for interaction which is evident especially when the number of agents is small, as in oligopoly (few sellers). Instead large numbers of participants were introduced (under the misnomer of “free competition”) such that asymptotically none had any perceptible influence on any other participant and consequently not on the outcome, each merely facing fixed conditions. Thus the individual’s alleged task was only to maximize his profit or utility rather than to account for the activities of the “others”. Instead of solving the empirically given economic problem, it was disputed away; but reality does not disappear. In international politics there are clearly never more than a few states, in parliaments a few parties, in military operations a few armies, divisions, ships, etc. So effective decision units tend to remain small. The interaction of decisions remains more obvious and a rigorous theory is wanting». (O. Morgenstern 1973). Research took a quite different path from that predicted by von Neumann and Morgenstern and, also as a result of the decline of the cooperative approach, gave a fresh lease of life to methodological individualism. That research went in this direction was explicitly acknowledged by Nash. In an interview given to Robert Leonard he declared he had followed a different «individualistic» approach from that of von Neumann, who, in his view, had a more «European» outlook. Therefore, Nash had correctly identified a divergence in the research programmes and not only a branching off of different formal approaches, as many tend to think today, reducing the question to a mere fact of modeling or mathematical convenience (G. Giraud 2000). The mathematical aspect no doubt plays an important role but only to the extent to which it is possible to provide a conclusive and fully general proof that every model expressed in cooperative terms may be related back to an equivalent model expressed in non cooperative terms. If such a result were to be proved true – we repeat, in a fully general way and not only for specific cases, as is still

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the case today - von Neumann and Morgenstern’s programme could be considered to have failed, at least as far as the claim to the primacy of the cooperative approach is concerned. The utility of game theory in economics would therefore be limited to proposing an optimization approach different from that of Walras’ theory. Nevertheless, it would not be correct to present von Neumann and Morgenstern’s approach as being dictated by a series of prejudices. This is what Ken Binmore does when he accuses von Neumann and Morgenstern of having themselves placed obstacles in the path of their research (K. Binmore 1992). By so doing, Binmore apparently fails to realize that a research programme represents precisely a series of objectives, the deliberate choosing of which limits the direction of movement. For Binmore, who sees nothing wrong with adopting the methodological individualism approach, a programme like von Neumann and Morgenstern’s is legitimately unacceptable. However, this does not mean that the programme itself is absurd. Von Neumann and Morgenstern could likewise have considered the programme deemed feasible by Binmore as an ad hoc and meaningless way of proceeding. In von Neumann and Morgenstern’s intentions, game theory had the role of providing new and more plausible images of economic reality. In their view, this role could be allowed to emerge only by demolishing the central role of the microeconomic methodological individualism approach. As it turns out, their research programme has been surreptitiously set aside rather than confuted. What is unsatisfactory about the way this happened is the vague boundary that separates issues of formal effectiveness from those related to the research programme: although no one can deny the effectiveness of Nash’s approach, it is completely unsatisfactory that a positive assessment can be influenced by an inextinguishable a priori attachment to the mythology of Walrasian equilibrium. Even if it were indeed true that game theory offers no substantial advantage to economic analysis, it would be necessary to arrive at a conclusion of this kind without being influenced by Walrasian equilibrium mythology. This becomes all the more necessary when the mediocrity of the results obtained in the framework of the general equilibrium model is taken into account. It has indeed been conclusively demonstrated that it is impossible to make any significant progress in the direction of dynamic analysis. It has been pointed out (H. Defalvard 2001) that many have fallen and continue to fall «into the equilibrium trap, which consists in believing that it contains the process, while it

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actually leads to its being discarded». Conversely, the impression is that if game theory is to make any useful contribution to economic analysis, it can do so to only the extent in which it allows us not only to make use of optimization processes that are conceptually richer than the classic ones, but also to rid ourselves of the concept of equilibrium and of the entire mechanistic apparatus imported from mathematical physics: in other words, if it can contribute to the introduction of a holistic approach and the abandoning of the customary inadequate reductionist schematas. In concluding this short note, I should like to mention that, in this climate in which so many hasty and superficial formalizations, each of which claims to say the final word, are being pursued, the most valid teaching we can draw from von Neumann and Morgenstern is expressed in the warning that «the decisive break which came in physics in the seventeenth century […] was backed by several millennia of systematic, scientific, astronomical observation, culminating in an observer of unparalleled caliber, Tycho de Brahe. Nothing of this sort has occurred in economic science. It would be absurd in physics to expect Kepler and Newton without Tycho, - and there is no reason to hope for an easier development in economics». (J. von Neumann, O. Morgenstern 1944).

References K. Binmore 1992, Fun and Games: A Text on Game Theory, Lexington, Ma., D. C. Heath & co. G. Debreu 1959, Theory of Value. An Axiomatic Analysis of Economic Equilibrium, New Haven, Yale University Press. R. Defalvard 2001, “Le hors d’équilibre dans la théorie des jeux: une affaire de croyances”, in L’économie hors de l’équilibre (coordonné par J. Cartelier, R. Frydman), Paris, Economica, pp. 41-62. L. Dell’Aglio 1995, “Divergences in the history of mathematics: Borel, von Neumann and the genesis of Game Theory”, Rivista di Storia della Scienza, s. II, 3, No. 2, pp. 1-46. G. Giraud, La théorie des jeux, Paris, Flammarion, 2000. B. Ingrao, G. Israel 1985, “General Economic Equilibrium Theory. A History of Ineffectual Paradigmatic Shifts, I, II, Fundamenta Scientiae, 6, Nos. 1-2, pp. 1-45, 89-125. B. Ingrao, G. Israel 1987, La mano invisibile. L'equilibrio economico nella storia della scienza, Roma-Bari, 2 3 Laterza Editori, 1996 , 1999 ; English version: The Invisible Hand. Economic Equilibrium in the History 2 of Science, Cambridge, Mass., M.I.T. Press, 1990, 2000 . G. Israel 1993, “The Two Paths of the Mathematization of the Social and Economic Sciences. The Decline of the "Mathématique Sociale" and the Beginnings of Mathematical Economics at the Turn of the Eighteenth Century”, Physis, 30, No. 1, pp. 27-78. G. Israel, A. Millan Gasca 1995, Il mondo come gioco matematico. John von Neumann, scienziato del Novecento, Roma, La Nuova Italia Scientifica; Spanish transl.: El mundo come un juego matemático. John von Neumann, un científico del siglo XX, Madrid, Nivola, 2001. G. Israel 1995, La mathématisation du réel. Essai sur la modélisation mathématique, Paris, Editions du Seuil; Italian transl: La visione matematica della realtà, Introduzione ai temi e alla storia della 2 3 modellistica matematica, Roma-Bari, Laterza, 199&, 1997 , 2003 .

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G. Israel 2001, “La matematizzazione dell’economia: aspetti storici ed epistemologici”, in Matematica e cultura 2001 (M. Emmer ed.), Milano, Springer Verlag Italia, pp. 67-80. G. Israel 2002, “The Two faces of Mathematical Modelling: Objectivism vs. Subjectivism, Simplicity vs. Complexity”, in The Application of Mathematics to the Sciences of Nature. Critical Moments and Aspects (P. Cerrai, P. Freguglia, C. Pellegrini, eds.), New York, Kluwer Academic/Plenum Publishers, pp. 233-244. G. Israel forthcoming a, “Al di là del mondo inanimato: la storia travagliata della matematizzazione dei fenomeni economici e sociali”, Bollettino dell’Unione Matematica Italiana, sez. B. G. Israel forthcoming b, “Teoria dei giochi ed economia matematica secondo von Neumann e Morgenstern”, Bollettino dell’Unione Matematica Italiana, sez. A. R. J. Leonard 1992, "Creating a Context for Game Theory", in Weintraub E. R. (ed.), Toward a History of Game Theory, Durham-London, Duke University Press, pp. 29-76. R. J. Leonard 1995, “From Parlor Games to Social Science: Von Neumann, Morgenstern, and the Creation of Game Theory, 1928-1944”, Journal of Economic Literature, 33, no. 2, pp. 730-761. P. Mirowski 1992, "What Were von Neumann and Morgenstern Trying to Accomplish", in Weintraub E. R. (ed.), Toward a History of Game Theory, Durham-London, Duke University Press, pp. 113-147. P. Mirowski 2002, Machine Dreams. Economics Becomes a Cyborg Science, Cambridge, Cambridge University Press. O. Morgenstern 1973, “Game Theory”, Dictionary of the History of Ideas, II, New York, Ch. Scribner’s & sons, pp. 263-275. H. Moulin 1981, Théorie des jeux pour l’économie et la politique, Paris, Hermann. J. von Neumann 1928, “Zur Theorie der Gesellschaftsspiele”, Mathematische Annalen, 100, pp. 295-320. J. von Neumann, O. Morgenstern 1944, Theory of Games and Economic Behaviour, Princeton, Princeton 2 University Press, 1947 . M. J. Osborne, A. Rubinstein 1994, A Course in Game Theory, Cambridge, Mass., The MIT Press. P. A. Samuelson 1989, "Revisionist View of von Neumann's Growth Mode", in M. Dore, S. Chakravarty, R. Goodwin. John von Neumann and Modern Economics, Oxford, Clarendon Press, pp. 100-122.

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