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) and (tA, tB) for players A and B. We will show that the following four patterns of competition are equivalent, that is, they yield the same outcome. 1. Player A and B choose sA and sB (competition by (sA, sB)). 2. Player A and B choose tA and tB (competition by (tA, tB)). 3. Player A and B choose tA and sB (competition by (tA, sB)). 4. Player A and B choose sA and tB (competition by (sA, tB)).