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.