Friday, December 12, 2014

First Formulation of Folk Theorem and Indeterminacy in Game Theory

Initial and Chaotic Learning in Rock-Paper-Scissors

Consider a game, as games are defined in game theory. And consider some strategy for some player in some game. The folk theorem states, roughly, that any strategy can be justified as a solution for a game by considering an infinitely repeated game. (An amusing corollary might be stated as saying that competition is the same as monopoly, if you do the math right.) The following seems to me to state the folk theorem (abstracting from the distinction between Nash equilibria and Von Neumann and Morgenstern's solution concept):

"21.2.3. If our theory were applied as a statistical analysis of a long series of plays of the same game - and not as the analysis of one isolated play - an alternative interpretation would suggest itself. We should then view agreements and all forms of cooperation as establishing themselves by repetition in such a long series of plays.

It would not be impossible to derive a mechanism of enforcement from the player's desire to maintain his record and to be able to rely on the on the record of his partner. However, we prefer to view our theory as applying to an individual play. But these considerations, nevertheless, possess a certain signiificance in a virtual sense. The situation is similar to the one we encountered in the analysis of the (mixed) strategies of a zero-sum two-person game. The reader should apply the discussions of 17.3 mutatis mutandis to the present situation." -- John Von Neumann and Oscar Morgenstern (1953) p. 254.

I have heard it claimed that economic theory has developed such that any moderately informed graduate student can now provide you with a model that yields any conclusion that you like. The folk theorem, as I understand it, is not even the most threatening finding for the ability of game theory to yield determinate conclusions.

Consider an iterated game before an equilibrium, under some definition or another, has been achieved. The players are trying to learn each others' strategies. Even a simple game, such as Rock-Scissors-Paper, can yield chaotic dynamics (Sato, Akiyama, and Farmer 2002; Galla and Farmer 2013). An equilibrium might never be established, for it is worthwhile for some players to deliberately choose "irrational" moves so as to ensure that other players do not achieve equilibrium, instead of a result that benefits the supposedly irrational player (Foster and Young 2012). (I hope I found this reference from reading Yanis Varoufakis, who, in one paper in one of his books, makes this point with the centipede game.) Apparently, this irrationality does not disappear by moving towards a more meta-theoretic level. And one player, who understands the evolutionary behavior of the other player in a Prisoner's Dilemma, can manipulate the other player to result in a asymmetric result - that is, a case where the non-evolutionary player extorts the player following a mindless evolutionary strategy (Press and Dyson 2012, Stewart and Plotkin 2012).


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