Blackwell Games

Abstract

Blackwell games are infinite games of imperfect information. The two players simultaneously make their moves, and are then informed of each other's moves. Payoff is determined by a Borel measurable function $f$ on the set of possible resulting sequences of moves. A standard result in Game Theory is that finite games of this type are determined. Blackwell proved that infinite games are determined, but only for the cases where the payoff function is the indicator function of an open or $G_\delta$ set. For games of perfect information, determinacy has been proven for games of arbitrary Borel complexity. In this paper I prove the determinacy of Blackwell games over a $G_{\delta\sigma}$ set, in a manner similar to Davis' proof of determinacy of games of $G_{\delta\sigma}$ complexity of perfect information. There is also extensive literature about the consequences of assuming AD, the axiom that _all_ such games of perfect information are determined. In the final section of this paper I formulate an analogous axiom for games of imperfect information, and explore some of the consequences of this axiom.