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

We consider a two-person zero-sum game with two sets of strategic variables which are related by invertible functions. They are denoted by (sA,sB) ∈ (SA,SB) and (tA,tB) ∈ (TA,TB) for players A and B. The payoff function of Player A is uA. Then, the payoff function of Player B is − uA. uA is upper semi-continuous and quasi-concave on SA for each sB ∈ SB (or each tB ∈ TB), upper semi-continuous and quasi-concave on TA for each tB ∈ TB (or each sB ∈ SB), lower semi-continuous and quasi-convex on SB for each sA ∈ SA (or each tA ∈ TA), and lower semi-continuous and quasi-convex on TB for each tA ∈ TA (or each sA ∈ SA).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 sA and sB (competition by (sA,sB)).(2) Players A and B choose tA and tB (competition by (tA,tB)).(3) Players A and B choose tA and sB (competition by (tA,sB)).(4) Players A and B choose sA and tB (competition by (sA,tB)).