von Neumann & Morgenstern

von Neumann & Morgenstern Theory Of Games And Economic Behavior, 1944

This introduces game theory, and shows that economies can be  viewed as games, and hence have interesting strategic aspects. For example, there may be multiple stable economic equilibria for the same ‘real’ resources. Prior to this economics had often been seen as analogous to thermodynamics in Physics, where behaviors tend to a unique solution. This is qualitatively wrong: such methods can (at best) be used to consider behaviors within an equilibrium: they cannot be used to consider stability issues. Thus one needs to distinguish between cyclical booms and busts, which may be a symptom of equilibrium-seeking, and ‘catastrophe’s’ which mark the end of an equilibrium and hence ‘new economics’.

Here we comment on the theory of games from an economic perspective, although it is also of great interest independently of economics.

The Problem of Rational Behavior

Much of the book uses the conventional notions of probability and utility as a technical device, and von Neumann went on to use entropy in a similar way. Some have supposed that such usages validate the use of these concepts more widely, but this is to misinterpret the book. If we – hypothetically – thought that an economy was simple and stable in the short-run, and hence describable in terms of utility etc, then the book shows that in the long-run one can get instability, such that notions of utility maximization etc are not adequate. It is the conclusion that matters: there is no attempt to prove that even in the short-run utility maximization is meaningful.

Why do we buy insurance? One might suppose that we consider the expected return on it and our ‘attitude to risk’. But in this book all we do is consider the immediate worry we have from not having insurance and the reassurance we would immediately gain from buying some. One might expect that in a stable situation the increase in satisfaction would reflect the ‘real’ benefit from having it, but it is not necessary to suppose that the purchaser actually buys on this basis. Nor is it supposed that people are ‘rational’, or even that rationality (outside of an equilibrium situation) has any meaning.

2.1.2. The individual who attempts to obtain these respective maxima [of utility] is also said to act “rationally.” But it may safely be stated that there exists, at present, no satisfactory treatment of the question of rational behavior. … The chief reason for this lies, no doubt, in the failure to develop and apply suitable mathematical methods to the problem; this would have revealed that the maximum problem which is supposed to correspond to the notion of rationality is not at all formulated in an unambiguous way. Indeed, a more exhaustive analysis … reveals that the significant relationships are much more complicated than the popular and the” philosophical” use of the word “rational” indicates.

Thus, whereas the popular view may be that utility maximization is a mathematical concept, it is actually one whose limitations are exposed by the appropriate use of mathematics.

2.2.3. Consider now a participant in a social exchange economy. … This is certainly no maximum problem, but a peculiar and dis-concerting mixture of several conflicting maximum problems. Every participant is guided by another principle and neither determines all variables which affect his interest.

This kind of problem is nowhere dealt with in classical mathematics. We emphasize at the risk of being pedantic that this is no conditional maximum problem, no problem of the calculus of variations, of functional analysis, etc.

Section 66 ‘Generalization of the concept of utility’ discusses the problem of reforming the concept of utility (which will no longer be just a number) so that it can represent economic problems more generally, but does not give a solution.

(Wikipedia says that this book led to people becoming interested in Ramsey’s common-sense approach, despite the above criticisms.)

Methodology

The book is critical of the unprincipled application of theories developed to support Physics.

[2.2.3] One would be mistaken to believe that it can be obviated … by a mere recourse to the devices of the theory of probability. Every participant can determine the variables which describe his own actions but not those of the others. Nevertheless those “alien” variables cannot, from his point of view, be described by statistical assumptions. This is because the others are guided, just as he himself, by rational principles whatever that may mean and no modus procedendi can be correct which does not attempt to understand those principles and the interactions of the conflicting interests of all participants.

2.4.2. In all fairness to the traditional point of view this much ought to be said: It is a well known phenomenon in many branches of the exact and physical sciences that very great numbers are often easier to handle than those of medium size. … This is, of course, due to the excellent possibility of applying the laws of statistics and probabilities in the first case. This analogy, however, is far from perfect for our problem. For the social exchange economy i.e. for the equivalent “games of strategy” … only after the theory for moderate numbers of participants has been satisfactorily developed will it be possible to decide whether extremely great numbers of participants simplify the situation. … The current assertions concerning free competition appear to be very valuable surmises and inspiring anticipations of results. But they are not results and it is scientifically unsound to treat them as such as long as the conditions which we mentioned above are not satisfied.

2.4.3. A really fundamental reopening of this subject is the more desirable because it is neither certain nor probable that a mere increase in the number of participants will always lead [in fine] to the conditions of free competition. The classical definitions of free competition all involve further postulates besides the greatness of that number. E.g., it is clear that if certain great groups of participants will for any reason whatsoever act together, then the great number of participants may not become effective; the decisive exchanges may take place directly between large “coalitions,” few in number, and not between individuals, many in number, acting independently. Our subsequent discussion of “games of strategy” will show that the role and size of “coalitions” is decisive throughout the entire subject. Consequently the above difficulty though not new still remains the crucial problem. Any satisfactory theory of the “limiting transition” from small numbers of participants to large numbers will have to explain under what circumstances such big coalitions will or will not be formed i.e. when the large numbers of participants will become effective and lead to a more or less free competition. Which of these alternatives is likely to arise will depend on the physical data of the situation. Answering this question is, we think, the real challenge to any theory of free competition.

The Notion of Utility

The classical notion of utility is derived, and many have subsequently taken this to justify its widespread use without further consideration. This may seem in flat contradiction to the above discussion. The resolution is that conventional utility is shown to depend on a complete set of preferences (3.6.1), which presumes precise probabilities. More broadly, these precise probabilities can be used as a gauge (3.3). It is argued that the assumption of utilities is no more restrictive than the then conventional assumption of indifference curves, but not that this assumption is always valid. In 3.7.1 it is noted that behavioral values, such as a ‘utility of gambling’ are excluded by the assumptions.

We have practically defined numerical utility to be that thing for which the calculus of mathematical expectations is legitimate. (3.7.1) [This has a footnote noting that Bernoulli’s notion of utility as log payoff satisfies these axioms.]

Solutions and standards of behavior

They consider a 3-person zero-sum game as the simplest example that brings out some key considerations.

[4.3.2] [A]ny two players who combine and cooperate against a third can thereby secure an advantage. The problem is how this advantage should be distributed among the two partners in this
combination. Any such scheme of imputation will have to take into account that any two partners can combine; i.e. while any one combination is in the process of formation, each partner must consider the fact that his prospective ally could break away and join the third participant.

Common sense suggests that one cannot expect any theoretical statement as to which alliance will be formed, but only information concerning how the partners in a possible combination must divide the spoils in order to avoid the contingency that any one of them deserts to form a combination with the third player.

[4.3.3] It is clear that in the above three-person game no single imputation from the solution is in itself anything like a solution. Any particular alliance describes only one particular consideration which enters the minds of the participants when they plan their behavior. 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. Thus only the three alliances and their imputations together form a rational whole which determines all of its details and possesses a stability of its own. 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.

[4.6.2] [The] solutions correspond to such “standards of behavior’ as have an inner stability: once they are generally accepted they overrule everything else and no part of them can be overruled within the limits of the accepted standards. This is clearly how things are in actual social organizations, and it emphasizes the perfect appropriateness of the circular character of our conditions in 4.5.3.

4.6.3.  … [G]iven the same physical background different “established orders of society” or “accepted standards of behavior” can be built, all possessing those characteristics of inner stability which we have discussed. Since this concept of stability is … operative only under the hypothesis of general acceptance of the standard in question these different standards may perfectly well be in contradiction with each other.”

Strategies

[11.1.1] Let us return to the course of an actual play TT of the game F.

Imagine now that each player … instead of making each decision as the necessity for it arises, makes up his mind in advance for all possible contingencies; i.e. that [each player] begins to play with a complete plan … which specifies what choices he will make in every possible situation, for every possible actual information which he may possess at that moment in conformity with the pattern of information which the rules of the game provide for him for that case. We call such a plan a strategy.

Observe that if we require each player to start the game with a complete plan of this kind, i.e. with a strategy, we by no means restrict his freedom of action. In particular, we do not thereby force him to make decisions on the basis of less information than there would be available for him in each
practical instance in an actual play. This is because the strategy is supposed to specify every particular decision only as a function of just that amount of actual information which would be available for this purpose in an actual play. The only extra burden our assumption puts on the player
is the intellectual one to be prepared with a rule of behavior for all eventualities, although he is to go through one play only. But this is an innocuous assumption within the confines of a mathematical analysis.

[11.2] … The player’s judgement must be directed solely by [the] “mathematical expectation” … .

To be clear, this notion of a strategy only applies to playing a specific game, such as making the most money from a particular economic situation. It does not concern the broader type of strategy, that considers how the rules of the game may change, for example through changes in coalitions. In this sense it is ‘pragmatic’. Von Neumann and Morgenstern nowhere argue that such pragmatic behavior is in any sense ‘good’, and it would seem rather stupid if the current game seems likely to end.

Comment

The book makes scant reference to other authors, but the findings on stability mirror those of Keynes, and the implications for uncertainty mirror those which Keynes, Knight and (later) Ellsberg emphasise.

As an example, suppose that an outcome, u, has utility ‘u’, and that one has the choice of either u conditional on some event E, denoted u|E, or u conditional on not E, denoted u|¬E. If E is assigned a definite probability, p, and we choose u|E or u|¬E on the basis of a fair coin, then it follows from the utility axioms that the result is worth ‘u/2’, as usual. Moreover, one of u|E or u|¬E must be worth at least ‘u/2’. Hence a free choice between u|E and u|¬E must be worth at least ‘u/2’, so that preferring anything with lesser probability would be irrational, as in Ellsberg.

A conventional belief is that even if one cannot assign some definite probability to E, it is still irrational to prefer anything with utility less than ‘u/2’ to a choice between u|E and u|¬E. But while this is often attributed to von Neumann, it is not a formal consequence. Indeed, if E is an event that is informed by a the decision then one potentially has a game, for which the effective (minimax) value of the choice is 0.

Following Boole and Jack Good,  we can apply von Neumann’s approach more widely by adopting the notation u(A|B:C) to denote the utility of A, conditional upon B, that arises when one makes extra assumptions C in order to yield definite probabilities and utilities. The axioms will then apply for a fixed context, C, such that one needs additional axioms if one is to make judgements across different contexts. This leaves room for different interpretations and extensions to the theory.

Dave Marsay

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