Infinite games with imperfect information

Abstract:

We consider an infinite, two person zero sum game played as follows: On the $n$th move, players $A,B$ select privately from fixed finite sets, ${A_n},{B_n}$, the result of their selections being made known before the next selection is made. After an infinite number of selections, a point in the associated sequence space, $\Omega$, is produced upon which $B$ pays $A$ an amount determined by a payoff function defined on $\Omega$. In this paper we extend a result of Blackwell and show that if the payoff function is the indicator function of a set in the Boolean algebra generated by the ${G_\delta }$’s (with respect to a natural topology on $\Omega$) then the game in question has a value.