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In this paper we introduce a novel flow representation for finite games instrategic form. This representation allows us to develop a canonical direct sumdecomposition of an arbitrary game into three components, which we refer to asthe potential, harmonic and nonstrategic components. We analyze natural classesof games that are induced by this decomposition, and in particular, focus ongames with no harmonic component and games with no potential component. We showthat the first class corresponds to the well-known potential games. We refer tothe second class of games as harmonic games, and study the structural andequilibrium properties of this new class of games. Intuitively, the potentialcomponent of a game captures interactions that can equivalently be representedas a common interest game, while the harmonic part represents the conflictsbetween the interests of the players. We make this intuition precise, bystudying the properties of these two classes, and show that indeed they havequite distinct and remarkable characteristics. For instance, while finitepotential games always have pure Nash equilibria, harmonic games genericallynever do. Moreover, we show that the nonstrategic component does not affect theequilibria of a game, but plays a fundamental role in their efficiencyproperties, thus decoupling the location of equilibria and their payoff-relatedproperties. Exploiting the properties of the decomposition framework, we obtainexplicit expressions for the projections of games onto the subspaces ofpotential and harmonic games. This enables an extension of the properties ofpotential and harmonic games to "nearby" games. We exemplify this point byshowing that the set of approximate equilibria of an arbitrary game can becharacterized through the equilibria of its projection onto the set ofpotential games.

Flows and Decompositions of Games: Harmonic and Potential Games