Orthogonal elementary interactions for bimatrix games

In symmetric matrix games, the interaction is defined through a single payoff matrix that can be decomposed into elementary interactions of four types representing games with self- and cross-dependent payoffs, coordination-type interactions, and rock–paper–scissors-like cyclic dominance. Here, this analysis is extended to a similar anatomy of bimatrix games given by two matrices. The respective self- and cross-dependent components describe extended versions of donation games expanding the range of social dilemmas. The most attractive classification utilizes the fact that games can be separated into the sum of a fraternal and a zero-sum part whose payoff matrices can then be further divided into symmetric and antisymmetric terms. This approach revealed two types of interaction not present in symmetric games: directed anticoordination components share some features with both partnership games and rock–paper–scissors-like cyclic dominance; the combinations of matching pennies components prevent the existence of a potential, which precludes detailed balance with the Boltzmann distribution under Glauber-type dynamics. In another departure from symmetric games, bimatrix games may admit a non-Hermitian potential matrix, which could possibly give rise to thermodynamic behaviors not found in classical spin models. Some curiosities of the directed anticoordination interaction are illustrated by simulations when the players are located on a square lattice.