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We introduce several methods of decomposition for two player normal formgames. Viewing the set of all games as a vector space, we exhibit explicitorthonormal bases for the subspaces of potential games, zero-sum games, andtheir orthogonal complements which we call anti-potential games andanti-zero-sum games, respectively. Perhaps surprisingly, every anti-potentialgame comes either from the Rock-Paper-Scissors type games (in the case ofsymmetric games) or from the Matching Pennies type games (in the case ofasymmetric games). Using these decompositions, we prove old (and some new)cycle criteria for potential and zero-sum games (as orthogonality relationsbetween subspaces). We illustrate the usefulness of our decomposition by (a)analyzing the generalized Rock-Paper-Scissors game, (b) completelycharacterizing the set of all null-stable games, (c) providing a large class ofstrict stable games, (d) relating the game decomposition to the decompositionof vector fields for the replicator equations, (e) constructing Lyapunovfunctions for some replicator dynamics, and (f) constructing Zeeman games-games with an interior asymptotically stable Nash equilibrium and a purestrategy ESS.

Decompositions of two player games: potential, zero-sum, and stable games