Nash’s Equilibrium points

John F. Nash Equilibrium points in n-person games PNAS Jan. 1 1950, Vol. 36 (1).

One may define a concept of an n-person game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the pay-off functions are the expectations of the players, thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.

One such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expectation for its player against the n − 1 strategies of the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point

[T]here is an equilibrium point.

In the two-person zero-sum case the “main theorem”2 and the existence of an equilibrium point are equivalent. In this case any two equilibrium points lead to the same expectations for the players, but this need not occur in general.

Technical Comments

As with any mathematics, but particularly that expressed in ‘plain language’, one has to be careful what how one interprets the terms.

Imagine saving for retirement. It makes a difference whether one considers the payoff to be the size of the fund or (following Bernouilli) the logarithm of the fund. In a managed fund the fund manager might well consider their bonus whereas an investor might consider either their ‘pot’ or its logarithm. Thus one the notion of ‘expected payoff’ is not straightforward.

The notion of ‘expectation’ is also problematic. There may be an important distinction between what a player thinks will happen and some more ‘objective’ probabilities. With this in mind we can only conclude that under the conditions of the theorem there is some strategy that all players consider yields their highest expectation, given the strategies of the other players. We cannot say that the strategies are not actually abysmal.

There is also a subtlety here. In game theory one often has to consider ‘extended form games’, and the conclusion do not consider specific acts but the long-run strategies. What we have from Nash is that there is always a ‘self-countering’ strategy. But as we actually play a game we tend to learn things that are not just about the strategies of other players, but may be about the payoff function. This may change our expectations and hence the ‘equilibrium’.

Nash’s final remark is important, but perhaps obscure. Perhaps the first thing to note is that there may be multiple equilibria. Typically it is a good idea to form coalitions in multi-player games. Mainstream game theory as usually understood only covers what happens next. But typically there can be multiple viable coalitions, giving rise to different expectations and equilibria. Even without formal coalitions there may be social effects. For example, in an economic depression the equilibrium may be for everyone to be bearish, whereas in a period of sustained growth it may be an equilibrium to be bullish. The challenge is to get from one to the other. From Nash we see that we need to modify expectations and payoffs, but not how.

Nash is assuming a fixed game, with given ‘rules’. It is always open to ue to ‘raise the level of the game’, for example to game around the rules. Game theory is useful at this higher level, so long as their is sufficient of a defined context to be able to identify the game and its rules. In practice one often ends up playing a lower level game as if it were the only game, but keeping an eye on the higher-level game in case conditions uner which the lower-level description applies may be violated. For example, one might be investing bullishly (along with everyone else), but looking at for evidence that it might be wise to run to safety.

Nash’s result  doesn’t address the stability of the equilibria. Typically exploration and learning have a cost, so changing strategy will typically have a cost, and hence there is a ‘payoff from stability’. But the equilibria for the game which ignores this payoff may also be equilibria for the actual game. But this needs to be checked in particular cases. In zero-sum games, for example, it may be that ‘shaking the tree’ stresses one’s ability to learn, but distresses your opponents more.

Implications

A naïve reading of Nash would be that we should expect equilibria in all kinds of systems, and be looking for external explanations for manifest instabilities. This would be quite wrong. But many systems can be regarded as games whose rules stay reasonably fixed over the short-run. For example in social systems most people will benefit from conserving the status quo, at least in the short-run, so while Nash’s theorem doesn’t exactly apply, it seems a reasonable interpretation – in the short-run. It is also suggestive of what one might do if one doesn’t like the status quo: look for some preferable, achievable, alternative equilibrium, and even suggests how to get there: change the payoff function, e.g. by forming alliances.

 

Dave Marsay

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