Chance-Constrained Games with Partially Controllable Strategies

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Abstract
This paper is the second of a series directed at the investigation of relations of chance-constrained programming to problems in game theory. The model introduced and analyzed herein is a two-person game model with zero-sum payoff matrix in which the strategies selected by the players do not in themselves determine the payoffs, but in which random perturbations with known distributions modify the strategy of each player before actual implementation of the strategies. A major hypothesis is that, while the strategy perturbations may be random vectors with known distributions, the selection of strategies (or more accurately strategy policies) is to be made before any observations of the random variables are made so that the strategies chosen are to be “zero-order” decision functions of the random perturbations, in the customary terminology of chance-constrained programming. The strategy selected by each player is to be chosen to extremize that payoff which the player can be assured with at least some (a priori specified) probability. A key result shows that the deterministic equivalents for these problems yield a deterministic two-person game that is not zero-sum.

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