Symmetric games revisited

We 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 and has bite even on 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. When each player has two strategies, both ordinary and renaming symmetric games are exact potential games, a property not possessed by their name-irrelevant counterparts. We establish a characterization of the renaming symmetric game through the player symmetry group.