Symmetric multi-person zero-sum game with two sets of strategic variables

We consider a symmetric multi-person zero-sum game with two sets of alternative strategic variables which are related by invertible functions. They are denoted by (s1, s2, ., sn) and (t1, t2, ., tn) for players 1, 2, ., n. The number of players is larger than two. We consider a symmetric game in the sense that all players have the same payoff functions. We do not postulate differentiability of the payoff functions of players. We will show that the following patterns of competition, 1) all players choose si, 2) all players choose ti and 3) m players choose ti, i=1, ., m and n-m players choose sj, j=m+1, ., n where 1<=m<=n-1, are equivalent, that is, they yield the same outcome. However, in an asymmetric zero-sum game with more than two players the equivalence does not hold.