k1 x k2 x .... x kn games are uniquely decomposed into their strategic and behavioral parts. The strategic part contains all information needed to determine a specified strategic outcome (e.g. Nash, Quantal Response Equilibria (QRE); the behavioral portion is what motivates cooperative approaches such as tit-for-tat, side payments, etc. the decomposition reduces complexity; e.g., the space of 2x2 games is reduced from R^8 to points in a three-dimensional cube. Applications include characterizing all 2x2 games, finding which Nash (or QRE) structures accompany certain classes of k1 x ... x kn games, and how results concerning two and three player tit-for-tat strategies differ.