Determinator - More General Games - Games of Imperfect Information

Games of Imperfect Information

In any interesting game with imperfect information, a winning strategy will be a mixed strategy: that is, it will give some probability of differing responses to the same situation. If both players' optimal strategies are mixed strategies then the outcome of the game cannot be certainly determinant (as it can for pure strategies, since these are deterministic). But the probability distribution of outcomes to opposing mixed strategies can be calculated. A game that requires mixed strategies is defined as determined if a strategy exists that yields a minimum expected value (over possible counter-strategies) that exceeds a given value. Against this definition, all finite two player zero-sum games are clearly determined. However, the determinacy of infinite games of imperfect information (Blackwell games) is less clear.

In 1969 David Blackwell proved that some "infinite games with imperfect information" (now called "Blackwell games") are determined, and in 1998 Donald A. Martin proved that ordinary (perfect-information game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass. This, combined with the Borel determinacy theorem of Martin, implies that all Blackwell games with Borel payoff functions are determined. Martin conjectured that ordinary determinacy and Blackwell determinacy for infinite games are equivalent in a strong sense (i.e. that Blackwell determinacy for a boldface pointclass in turn implies ordinary determinacy for that pointclass), but as of 2010, it has not been proven that Blackwell determinacy implies perfect-information-game determinacy.

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