Optimal strategies for symmetric matrix games with partitions

We introduce three variants of a symmetric matrix game corresponding to three ways of comparing two partitions of a fixed integer (sigma) into a fixed number (n) of parts, In the random variable interpretation of the game, each variant depends on the choice of a copula that binds the marginal uniform cumulative distribution functions (cdf) into the bivariate cdf. The three copulas considered are the product copula T-P and the two extreme copulas, i.e. the minimum Copula T-M and the Lukasiewicz copula T-L. The associated games are denoted as the (n, sigma)(P), (n, sigma)(M)and (n, sigma)(L) games. In the present paper, we characterize the optimal strategies of the (n, sigma)(M) and (n, sigma)(L) games and compare them to the optimal strategies of the (n, sigma)(P) games. It turns out that the characterization of the optimal strategies is completely different for each game variant