Tuesday 11 June 2024

The problem with Nash's 1950 paper

There was a time when people thought that the universe must obey whatever stupid logic or mathematics cretinous pedants came up while sitting comfortably in their armchairs. Then, it turned out that denser objects don't fall faster and gravity bends light waves and, by the Madam Wu experiment, 'incongruent counterparts' don't exist, and the Bell inequality, not Von Neumann's theorem, is correct. 

In the field of Econ, Arrow's theorem showed that math is useless unless you use words with a degree of precision. A Dictator is a guy who kills or incarcerates his opponents. He isn't a guy who always happens to vote for the winning candidate. 

Social Choice, like individual choice, is something done by a Brouwerian 'creative subject' and thus only capturable by a choice sequence which may be wholly 'lawless'. The same thing can be said of the word 'strategy' in game theory. It isn't an option forced on you- like a gangster giving you the option to hand over your money or get a bullet in the head- but an approach you can take to getting what you want- which itself is something you can decide to alter as the situation evolves. 

If you have taken a lot of trouble to rig a game, you may be able to use mathematics to prove to yourself that the game is indeed rigged. This, is what is established by one of most cited Econ papers of all time- viz Nash's 'Equilibrium points in n-person games' from 1950. It relies on Kakutani, and therefore, Brouwer's fixed point theorems and applies only to endomorphic, compact, convex continuous functions. The problem here is that there is no way of telling whether something we believe to be a function is actually a function. This is because the 'intension' which the function is supposed to fulfil may not have a definite or decidable 'extension'. 

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

this is an 'intension'. Is the 'extension' knowable? By arbitrary stipulation- sure. There is some exogenous Game-master who fixes the pay-off matrix in advance. But, in that case, Nash is offering a proof of something which has been arbitrarily imposed. In other words, he is merely showing that, as Game-master, he can rig things a particular way. But that isn't very interesting. I have a proof that all cats are dogs. This is because I define 'dog' as cat. But this merely proves I can utter any arbitrary shite.  

Suppose we have to play a game rigged in advance by a Game-master. We may choose, either in a coordinated or uncoordinated manner, to give different 'psychic' or other pay-outs to different outcomes and thus frustrate the Game-master. Here, the 'intension' of 'pure strategy' has a different 'extension' from that which the Game-master stipulated, and thus his calculations are confounded. 

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,

which are formed according to a rule Nash imposes 

thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.

So, pure strategy just means 'arbitrarily imposed option'. It isn't a strategy at all.  

Any n-tuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the players.

It is the matrix arbitrarily imposed by the game-master on the basis of options permitted to each player. 

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.

It is the outcome the game has been rigged to produce.  

The correspondence of each n-tuple with its set of countering n-tuples gives a one-to-many mapping of the product space into itself.

Thus making it endomorphic but if players can assign their own private values to outcomes, this ceases to be the case. 

From the definition of countering we see that the set of countering points of a point is convex.

Only if no player can introduce a private valuation to outcomes 

By using the continuity of the pay-off functions

which has been arbitrarily imposed 

we see that the graph of the mapping is closed...

by arbitrary stipulation.  

Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani’s theorem 1 that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point.

Why do people engage in zero-sum games they are bound to lose? The answer is that there can be a psychic or reputational benefit in taking part in the game. Otherwise the equilibrium point would be for only equally well matched players to ever indulge in that particular type of game.  

A superior solution concept to Nash is Aumann's correlated equilbria. Add in Brouwer type choice sequences as 'strategies' and you can still have fixed points though they may not be computable or accessible in any way. Still, they can drive dynamics. 

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