A New Mathematics of Society

Morgenstern’s introductory chapter of Theory of Games and Economic Behavior (von Neumann and Morgenstern 1944 [1947]), as the book was finally titled, is the volume’s most accessible and most widely read part.

The most prominent extant treatment of the field, general equilibrium theory, is, in the authors’ opinion, merely the imitative grafting of physical science methods onto social science. The importation of rational mechanics brings with it assumptions about the underlying continuity of change, whereas the social domain likely requires attention to discretely separate structures and discontinuities, and thus recourse to mathematics of a different kind. General equilibrium theory has also failed to account for the properly interactive nature of social behaviour, particularly that which is manifest in situations involving “small” numbers of agents, be they involved in the exchange of goods or in the distribution of gains through the formation of social and political groups. Throughout the book, von Neumann’s preference for “modern”, discrete mathematics (that is, set theory and combinatorics) over the “outdated” differential and integral calculus, is evident.

Chapter 2 of Theory of Games and Economic Behavior (von Neumann and Morgenstern 1944 [1947]) lays out the mathematical structure of a game, introducing the concepts of sets and partitions and describing the game axiomatically in these terms. The whole is presented as a piece of modern mathematics, in the spirit of Hilbert: although the axioms are stimulated by the common sense features of games, the latter are soon allowed to recede into the background and the theory pursued in a spirit of relative abstraction. While the mathematics is being followed through, the empirical is held at arm’s length and everyday terms are introduced in inverted commas. Hence, “class”, “discrimination”, “exploitation” and so on. Only during periodic returns to the heuris­tics is the vocabulary of the everyday re-invoked, and the “common sense” meaning of the results discussed. The minimax theorem is proved in the next chapter, using, not von Neumann’s earlier proof, but a modification of the elementary 1938 proof by Borel’s student Jean Ville, based on the theory of convex sets. From here on, chapter upon chapter, von Neumann systematically goes through the zero-sum game for three, four and more players, exploring the combinatorial possibilities for coalition-formation and compensations (side payments). Each game is described in terms of its characteristic function, which shows the maximal payoff available to each possible coalition of the game, assuming that the coalition plays minimax against its complement and that utility is transferable between players. In chapter 9, von Neumann introduces the concept of strategic equivalence to show how the move from the zero-sum restriction to a con­stant sum retains the basic features of the game, thus allowing it to be solved by the same means.

As already mentioned, the central theoretical contribution of the Theory of Games is the stable set, von Neumann’s solution to coalitional games. It is a “complicated com­binatorial catalogue”, indicating the minimum each participant can get if he behaves rationally. He may, of course, get more if the others behave “irrationally”, that is, make mistakes. Were the solution to consist of a single imputation - a unique vector of the amounts to be received by each player - then the “structure of the society under consid­eration would be extremely simple: there would exist an absolute state of equilibrium in which the quantitative share of every participant would be precisely determined” (von Neumann and Morgenstern 1944 [1947]: 34). However, such a unique solution does not generally exist - a given society can be organized in various ways - so the notion needs to be broadened. The solution is thus a set of possible imputations.

Any particular alliance describes only one particular consideration which enters the minds of the participants when they plan their behaviour. Even if a particular alliance is ultimately formed, the division of the proceeds between the allies will be decisively influenced by the other alliances which each one might alternatively have entered... It is, indeed, this whole which is the really significant entity, more so than its constituent imputations. Even if one of these is actually applied, i.e., if one particular alliance is actually formed, the others are present in a “virtual” existence: Although they have not materialized, they have contributed essentially to shaping and determining the actual reality. (Ibid.: 36)

In an n-person game, therefore:

[a] solution should be a system of imputations possessing in its entirety some kind of balance and stability the nature of which we shall try to determine. We emphasize that this stability - whatever it may turn out to be - will be a property of the system as a whole and not of the single imputations of which it is composed.

This stability is based on the notion of “dominance”. One imputation, x, is said to domi­nate another, y, “when there exists a group of participants each one of which prefers his individual situation in x to that in y, and who are convinced that they are able, as a group - i.e. as an alliance - to enforce their preferences” (von Neumann and Morgenstern 1944 [1947]: 38). Since the demurring coalition may be different in each case, the dominance relation is not a transitive ordering. Von Neumann defines the solution to an n-person game as a set of imputations, S, with the following characteristics: (1) no imputation y contained in S is dominated by an imputation x contained in S; (2) every y not contained in S, is dominated by some x contained in S.

A solution is thus a set of possible imputations, stable in so far as none of its member imputations dominates each other and every non-member imputation outside the set is dominated by at least one member. Not only is a solution comprised of possibly many imputations, linked by these stability criteria: a given game may have many solutions. To take a simple example, consider the fixed-sum game in which a “pie” of value 1 has to be divided among three people. It has the following four solutions:

Here, not only are there multiple solutions, but three of them actually admit an infinite number of possible imputations. Note also that the observation of a given imputation, such as (1/2, 1/2, 0), says nothing about which solution obtains, as that imputation could occur in any of the four solutions possible.

The question of which solution will obtain in a given situation, the authors say, can be broached only by considering “standards of behaviour”, the various rules, customs or institutions governing social organization at the time. These are extra-game consid­erations, not contained in the information provided by the characteristic function.

Let the physical basis of a social economy be given, - or to take a broader view of the matter, of a society. According to all tradition and experience human beings have a characteristic way of adjusting themselves to such a background. This consists of not setting up one rigid system of apportionment, i.e. of imputation, but rather a variety of alternatives, which will probably express some general principles but nevertheless differ among themselves in many particular respects. This system of imputations describes the “established order of society” or “accepted standard of behaviour”. (Ibid.: 41)

Thus, in the above game, in solution 2, player 3 is held to an amount, c, that may be as small as zero, or as high as 1/2. The actual value of c would reflect the social norms governing that player’s social standing. Depending on tradition, the marginal member might be completely exploited or might not. As von Neumann and Morgenstern write: “A theory which is consistent at this point cannot fail to give a precise account of the entire interplay of economic interest, influence and power” (ibid.: 43).

When one considers the social and political backdrop against which von Neumann developed this theory, the Theory of Games, with its emphasis on stability and its perva­sive reference to norms, discrimination and power, appears as an attempt not simply to replace general equilibrium theory, but to achieve a mathematical description of social organization, much more broadly defined. And, after the publication of the book, von Neumann continued to speak of game theory in these terms. In 1953, when Harold Kuhn wrote to him, asking what he thought of the experimental games then being conducted at the RAND Corporation, von Neumann replied:

I think that nothing smaller than a complete social system will give a reasonable “empiri­cal” picture [of the stable set solution].

In 1955, at a Princeton conference on game theory, when the young John Nash objected to the great multiplicity of solutions to cooperative games, von Neumann replied that “this result was not surprising in view of the correspondingly enormous variety of observed stable social structures; many differing conventions can endure, existing today for no better reason than that they were here yesterday” (in Wolfe 1955: 25).