Stochastic Programs as Nonzero Sum Games

Abstract

A linear programming (LP) model for determining an optimal decision vector x, which satisfies the following system

$$\max \text{ z={c}x} $$

(1.1)

$$\text{subject to x }\!\!\varepsilon\!\!\text{ R, R:}\left\{ \text{x Ax }\le \text{ b, x }\ge \text{ 0} \right\} $$

(1.2)

may be related to a two-person game-theoretic formulation in two ways. In the first case, which is based on the saddle point property of the optimal solution of the LP problem, the two players have an identical strategy space, say $p' = \left( {\bar{x}',\bar{y}',\bar{v}} \right) = q'$ and the payoff matrix is B:

$$B:\left[ {\begin{array}{*{20}{c}} 0 & { - A'} & c \\ A & 0 & { - b} \\ { - c'} & {b'} & 0 \\ \end{array} } \right]$$

(2.1)