Four classes of interactions for evolutionary games

The symmetric four-strategy games are decomposed into a linear combination of 16 basis games represented by orthogonal matrices. Among these basis games four classes can be distinguished as it is already found for the three-strategy games. The games with self-dependent (cross-dependent) payoffs are characterized by matrices consisting of uniform rows (columns). Six of 16 basis games describe coordination-type interactions among the strategy pairs and three basis games span the parameter space of the cyclic components that are analogous to the rock-paper-scissors games. In the absence of cyclic components the game is a potential game and the potential matrix is evaluated. The main features of the four classes of games are discussed separately and we illustrate some characteristic strategy distributions on a square lattice in the low noise limit if logit rule controls the strategy evolution. Analysis of the general properties indicates similar types of interactions at larger number of strategies for the symmetric matrix games.

Figure 1

Dynamical graph with the independent and relevant loops indicated by dashed (red), dotted (blue), and dashed-dotted (green) circles. Along the main diagonal the solid (yellow) circles refer to symmetric $2\times 2$ games.

Figure 2

Typical $K$-dependence of strategy frequencies for evolutionary games on the square lattice where the pair interactions are composed of ${\mathbf{f}}^{\left(pq\right)}$ with strength chosen at random. The MC data are denoted by open squares, diamonds, circles, and pluses for the strategies 1, 2, 3, and 4, respectively.

Figure 3

Two snapshots illustrate strategy distributions during the domain growing process on a square lattice at $K\simeq K_c/2$ if the interaction is described by the four-state Potts model (left) and by one of its relatives (right) obtained by exchanging the third and fourth rows of the payoff matrix.

Figure 4

Strategy frequencies as a function of noise $K$ for evolutionary games with pair interactions ${\mathbf{f}}^{\left(12\right)}$ on the square lattice. The MC data are denoted by the same symbols used in Fig. 2.

Figure 5

Monte Carlo data for the strategy frequencies versus noise on the square lattice when the pair interaction is defined as $\mathbf{A}={\mathbf{f}}^{\left(12\right)}+{\mathbf{f}}^{\left(23\right)}+{\mathbf{f}}^{\left(13\right)}$. Symbols agree with those used in Fig. 2 (here the circles and diamonds coincide).

Figure 6

The three cyclic basis games, Eqs. (40, 41, 42), are defined by adjacency matrices of the directed graphs $a$, $b$, and $c$, respectively.