Symmetric Games Revisited

Based on the works of Nash(1951), Peleg et al. (1999) and Alos-Ferrer and Kuzmics (2013), we investigate several of the most natural classes of symmetric games. We mainly distinguish among three types of symmetric games, which we name ordinary symmetric games, renaming symmetric games, and name-irrelevant symmetric games, in an order of increasing broadness. The second class of symmetric games is new. Making these distinctions is meaningful even for some elementary two by two games, e.g., Battle of Sexes is renaming symmetric but not ordinary symmetric, and Matching Pennies is name-irrelevant symmetric but not renaming symmetric.Some nice properties are preserved when ordinary symmetric games are extended to renaming symmetric games; e.g., when each player has two strategies, they are both exact potential games, which is in general not true for name-irrelevant symmetric games.We investigate these classes of games through exploring the corresponding symmetry groups of normal form games, which turn out to have rich mathematical structures that are of game theoretical interests. We develop the coveringness idea of Peleg et al. (1999) and adapt their results to characterize these symmetry groups. We extend the ordinally symmetric games of Osborne and Rubinstein (1994) from two to n players, and show that ordinally symmetric games with two strategies are ordinal potential games. We also define more classes of symmetric games through pairwise symmetries and discuss computational issues.