Optimal Strategies for Symmetric Matrix Games with Partitions

Abstract

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 $T_{\bf P}$ and the two extreme copulas, i.e. the minimum copula $T_{\bf M}$ and the Łukasiewicz copula $T_{\bf L}$. The associated games are denoted as the $(n,\sigma)_{\bf P}$, $(n,\sigma)_{\bf M}$ and $(n,\sigma)_{\bf L}$ games. In the present paper, we characterize the optimal strategies of the $(n,\sigma)_{\bf M}$ and $(n,\sigma)_{\bf L}$ games and compare them to the optimal strategies of the $(n,\sigma)_{\bf P}$ games. It turns out that the characterization of the optimal strategies is completely different for each game variant.