In evolutionary game theory we characterize the pair interaction by the payoff matrix.

We start the general analysis of such interactions with the symmetric two-strategy games, because in this case a potential can always be derived, which guarantees that these systems are equivalent to Ising-type models and, using a proper dynamics, the results of statistical physics become applicable.

We demonstrate that if the payoff matrix is cast in a vectorial form, we can choose a set of basis vectors/games that allows us to identify a part of the payoff that depends on ourselves, a part that is due to the partner, and a coordinative component of the game. In the case of more than two strategies (or the indistinguishability of the partners) there might be a deviation from potential games due to the presence of a cyclic game (like the rock-paper-scissor game), which, together with the symmetries present, has characteristic consequences in spatial models.