Two Person Zero-Sum Game with Two Sets of Strategic Variables

Abstract

We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by [Formula: see text] and [Formula: see text] for players A and B. The payoff function of Player A is [Formula: see text]. Then, the payoff function of Player B is [Formula: see text]. [Formula: see text] is upper semi-continuous and quasi-concave on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]), upper semi-continuous and quasi-concave on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]), lower semi-continuous and quasi-convex on [Formula: see text] for each [Formula: see text] (or each [Formula: see text], and lower semi-continuous and quasi-convex on [Formula: see text] for each [Formula: see text] (or each [Formula: see text]). We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome. (1) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (2) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (3) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]). (4) Players A and B choose [Formula: see text] and [Formula: see text] (competition by [Formula: see text]).