Infinite Linear Programming in Games with Partial Information

An area of considerable recent research interest has involved the extension and modification of the basic model for two-person zero-sum game theory. One particular type of extension found in the literature involves the introduction of risk and uncertainty into the model by allowing the m × n payoff matrix A = (a_ij) to be a discrete random matrix that can assume a finite set of values. This paper considers both one- and two-person games and investigates the situation in which A is a discrete random matrix that can assume a countably infinite set of values {A(k)}_k=1^∞. We assume that the players possess certain partial information about P, the distribution of A, in which case the game problems for players 1 and 2 can be reduced to programming equivalents. We prove minimax theorems for both semi-infinite and infinite games, and give some properties of optimal mixed strategies. The paper also develops some extensions of a theorem due to Caratheodory.