Eshet 2010, Amsterdam, March 25-28

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“The Holy Grail is a solution concept which, for every game, picks out one and only one combination of strategies as the solution. The Holy Grail of game theory is the uniqueness of game solution. This objective was not pursued by Nash, who was interested only to prove that all games have a solution (the existence problem). This objective was pursued by Shapley before and Harsanyi after, who find an unique solution to cooperative games and then by Harsanyi-Selten (1988) with equilibrium refinements.”

Sudgen (2001)

Von Neumann and Morgenstern assessed the multiplicity of solutions “not something to run from but rather to embrace” (Schotter 1992)

Nash before - and Harsanyi after – aimed at finding a unique solution to cooperative and to non-cooperative games through the axiom of symmetry

Shubik was a key actor in moving economic applications of game theory from TGEB towards Nash-Harsanyi’s approach

Game theory was moulded such as “to make it accessible as a research vehicle only to mathematicians” (Luce and Raiffa 1957)

interdisciplinary vs. economic-oriented method neutral tool vs. supportive/critical of neoclassical

economics cooperative vs. non-cooperative equilibrium-based vs. disequilibrium analysis solution multiplicity vs. uniqueness

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All these issues were already patently pointed out in the way vision of the co-founder of game theory, Oskar Morgenstern, was received and then abandoned by his student Martin Shubik

Shubik took the first steps on the path leading the application of game theory

from cooperative to non-cooperative approach from disequilibrium to equilibrium analysis from solutions multiplicity to uniqueness

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The inadequacy of the assumption of maximizing behaviour, whose removal undermined the metaphysics of neoclassical economics

No straightforward principle of social rationality in games as constrained maximization is for individual decision-making

Game theory as the foundation of a social and interdisciplinary theory of strategic interaction based on “live” variables and disequilibrium analysis (not on Hicks’ “dead” variables and perfect foresight)

To solve a game did not mean to determine how opposing maximizing choices could be balanced by means of assumptions on others’ behaviour

Solutions are multiple and often indeterminate as the outcome of the process of coalition formation

The word equilibrium is practically expunged from TGEB

Cooperative games were offered as a new foundation for a social science such as economics

“All these considerations illustrate once more what a complexity of theoretical forms must be expected in social theory. Our static analysis alone necessitated the creation of a conceptual and formal mechanism which is very different from anything used, for instance, in mathematical physics. Thus the conventional view of a solution as a uniquely defined number or aggregate of numbers was seen to be too narrow for our purposes, in spite of its success in other fields. The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominate mathematical physics.” (von Neumann and Morgenstern 1947, p. 45)

“It is my belief that von Neumann was even more committed than Morgenstern to the idea of a solution as a set of imputations. He felt that it was premature to consider solutions which picked out a single point and he did not like noncooperative equilibrium solution. In a personal conversation with von Neumann (on the train from New York) to Princeton in 1952) I recall suggesting that I thought that Nash’s non-cooperative equilibrium solution theory might be of considerable value in applications to economics. He indicated that he did not particularly like the Nash solution and that a cooperative theory made more social sense. Albert Tucker, in a personal conversation, informed me that in his conversations with von Neumann, he had displayed somewhat the same attitude to the single point solution, the value, proposed by Lloyd Shapley.” (Shubik 1992)

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“We give two independent derivations of our solution to the two-person cooperative game. In the first, the cooperative game is reduced to a non-cooperative game. To do this, one makes the players’ steps of negotiation in the cooperative game become moves in non-cooperative model. (…) The second approach is by the axiomatic method. One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other” (Nash 1953, p. 129)

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What is a “rational” prediction for Nash in playing a game? “Starting from the principle that a rational prediction should be unique, that the players should be able to deduce and make use of it, and that such knowledge on the part of each player of what to expect the others to do should not lead him to act out of conformity with the prediction, one is led to the concept of a solution defined before.” [Nash Ph.D.Th. 1952]

The axiomatic method is described as follows: “One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely.” (Nash 1953)

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Nash (1950) and (1953) give the axiomatic foundation of bargaining theory: 1) Independence of utility scales’ calibration, 2) Maximization of individual utility, 3) Pareto efficiency, 4) Independence of Irrelevant Alternatives, 5) Symmetry

The symmetry axiom: The bargaining solution does not depend on who is labelled player I and player II. If the players’ labels are reversed, each will still get the same payoff

The axiom “expresses equality of bargaining skill” (Nash 1950)  

“The symmetry axiom, Axiom IV, says that the only significant (in determining the value of the game) differences between the players are those which are included in the mathematical description of the game, which includes their different sets of strategies and utility functions. One may think of Axiom IV as requiring the players to be intelligent and rational beings. But we think it is a mistake to regard this as expressing ‘equal bargaining ability’ of the players, in spite of a statement to this effect in ‘The Bargaining Problem’. With people who are sufficiently intelligent and rational there should not be any question of ‘bargaining ability’, a term which suggests something like skill in duping the other fellow.” (pp. 137-8)

 

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“The bargaining parties will follow identical (symmetric) rules of behaviour (whether because they follow the same principles of rational behaviour or because they are subject to the same psychological laws).”

“This postulate is here equivalent to the assumption which we would express on the common-sense level by saying that each party will make a concession at a given stage of the negotiations if and only if he thinks he has at least as much “reason” as his opponent has to yield ground at that point.”

Harsanyi (1956)

Harsanyi supports Nash’s model because it

determines a unique solution to the bargaining game.

The symmetry axiom makes “possible to define a unique rational solution in terms of the directly relevant independent variables alone. Consequently, rational players can have no reasonable ground for making their bargaining strategies dependent on other variables intrinsically irrelevant for utility maximization.” (Harsanyi 1961, p. 190)

To emphasize the early achievement of unique solutions as primary goal

To introduce various devices designed to suppress strategic interaction in the basic assumptions of their models.

The symmetry assumption permit each player to make fully accurate predictions of the (probable) choices of the others

The consequence of symmetry is to give players perfect information along all dimensions, even though perfect information is not introduced as an explicit assumption.

1949 (Fall) Arrival to Princeton1952 GT applications to business cycle Econometrica1952 GT to information theory JPE1953 (June) Ph.D. discussion1953 GT to duopoly (with Nash-Mayberry) Econometrica1954 GT to information theory QJE1955 (August) Center for Advanced Study in Behavioral

Sciences (CASBS), Stanford1956 GT to management science Management Science1959 “Strategy and Market Structure”1959 Edgeworth market games (equivalence between

Edgeworth’s contract curve and the core)

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1953 Ph.D. Thesis was largely representative of Morgenstern’s view

Cooperative games are more useful than non-cooperative games for economics

Solution are settled by the “standards of society” based on some form of cooperation (explicit or implicit)

The core of Shubik’s thesis is the analysis of disequilibrium processes with no perfect foresight and non-homogeneous beliefs (Morgenstern 1935)

“The nature of a non-cooperative equilibrium point as a static phenomenon which has an objective existence of its own, but which may never be attained by the individuals in the economy is stressed. The possibility that many economic situations may not inherently possess any equilibrium state is discussed.” (Shubik’s PhD Thesis 1953 p. 151)

“How can the player maximize his own ends? He is not forced with a simple maximization problem of the variety that appears so often in mechanics. He must make his way as best he is able in the face of the countervailing power. An examination of his action must result in an analysis of coalitions, the possibility of collusion, the analysis of threats” (Shubik 1954 p. 7)

A Comparison of Treatments of the Duopoly Situation, discusses various duopoly analyses of Cournot, Edgeworth and others from the perspective of game theory.

There is no mention of Nash equilibrium: Nash's work on non-cooperation does not even appear in the bibliography.

Cournot solution is very briefly described and the main reference is Fellner's (1949) book on oligopoly, where the concept of equilibrium in Cournot is dismantled

“In these two early papers, therefore, Shubik displays a certain scepticism towards the Nash equilibrium, on the grounds that the hypothesis of non-cooperation is of questionable relevance, and towards the Cournot solution.” (Leonard 1995)

This shows that the connection between Cournot and Nash was not perceived instantly by Shubik, but involved a reinterpretation of the work of Cournot from Fellner's critique to the implausibility of the reaction curve dynamics to the static interpretation in which the dynamics are suppressed, and the equilibrium point preserved.

In 1955 Shubik moved from Princeton to Stanford

Shubik acknowledged that when he was writing “Edgeworth Market Games” in 1955, “except for some assistance by Howard Raiffa, I had no one to check my abominal [sic] mathematics, and it appeared with several false theorems.” (Shubik diaries)

Just like his mentor Morgenstern, Shubik’s mathematics were limited, a restriction that would not only hinder the influence of “Edgeworth Market Games”

The paper relates a neoclassical economic concept, the contract curve, to a game theoretic concept, the core

Proof was largely incomplete (convergence to Walrasian general equilibirum was shown by Debreu and Scarf 1963)

This notwithstanding, “From that time on, economics has remained by far the largest area of application of game theory (…) core theory was extensively developed and applied to market economies.” (Aumann 1985)

Edgeworth market games do not converge to a single point as the number of players increases to infinity, but under certain conditions, the core of these games does approach a single imputation in the limit.

Shubik noted that this “single imputation has economic meaning either in the theory of monopoly or the theory of pure competition.“

In one particular class of games, the core approaches a single imputation where a single player will act as a perfectly discriminating monopolist

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While the Von Neumann-Morgenstern stable set contains numerous imputations, the core delineates the true solution

As the number of players became large, the core shrunk to one point that is the competitive equilibrium

The core is an effective solution concept for market games because as Shubik discovered if the competitive equilibrium point exists, then it must be in the core.

In this economy there is only one imputation that is not dominated

Cooperative games are restricted to the case of collusive behaviour

Nash bargaining solution picks out the efficient solution in bilateral monopoly (vs. von Neumann Morgenstern’s solution)

Emphasis on standards of society is replaced with the static analysis of threats and bargaining process as in Nash’s bargaining model

Nash equilibrium is defined as the only reasonable solution in oligopolistic models (strictly enforceable)

Dynamic processes are described in terms of the static adjustment process to equilibrium

The outcome of the Cournot duopoly model is interpreted as a Nash equilibrium, i.e. a static fixed point

This interpretation of Cournot “marks therefore the true landmark for the application of game-theoretic,static equilibrium theory to economic analysis” (Giocoli 2004)

Shubik’s definition of external symmetry

Games are based on the assumption that players are alike for everything is not explicitly included in the formal description of the game

This assumption is the corollary of the axiom of symmetry postulated by Nash and Harsanyi

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TGEB “in general does not yield determinate solutions for two-person nonzero-sum games and for n-person games” (Harsanyi 1976)

Many economists, and especially applied economists, are troubled that games had “too many equilibria and no way to choose, Kreps (1990)

If considerations of rationality only narrowed the range of possible equilibria in game-theoretic models without providing unique, determinate results and economists were even more disturbed by the variety of solution concepts proposed in cooperative game theory.

Non cooperative game theory received increasing attention in the 1970s for this reasons which differentiate it from cooperative game theory

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The central question of orthodox pre-war microtheory is how market equilibrium actually attained and it has been shunted aside ever since the Formalist Revolution of the 1950s. In general equilibrium theory, the question of whether it exists at all dominated the issue of convergence to equilibrium so successfully as to swallow it up entirely (Blaug 2005)

Conditions guaranteeing the uniqueness of solutions are crucial to their applications in comparative statcis models based on equilibrium state. “Lacking conditions that guarantee uniqueness, economists must resort to considerations of historical conditions and dynamic stability, which greatly complicate the analysis”

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In the 1950s Shubik changed his view from that intellectual indebted to his advisor Oskar Morgenstern to that promoting the next-to-come mainstream view of the application of game theory to economics.

An explanation is given by the necessity of making game theory acceptable for economists. In this way game theory was acknowledged as fully integrated in the body of economics but it weakens its empirical meaning.