Fourier decomposition of payoff matrix for symmetric three-strategy games

In spatial evolutionary games the payoff matrices are used to describe pair interactions among neighboring players located on a lattice. Now we introduce a way how the payoff matrices can be built up as a sum of payoff components reflecting basic symmetries. For the two-strategy games this decomposition reproduces interactions characteristic to the Ising model. For the three-strategy symmetric games the Fourier components can be classified into four types representing games with self-dependent and cross-dependent payoffs, variants of three-strategy coordinations, and the rock-scissors-paper (RSP) game. In the absence of the RSP component the game is a potential game. The resultant potential matrix has been evaluated. The general features of these systems are analyzed when the game is expressed by the linear combinations of these components.

Figure 1

Average frequencies of strategies are illustrated by solid lines as a function $K$ for the Potts model in the absence of external field $[\mathbf{A}=\mathbf{g}(8)\text{]}$. Dashed lines show the same quantities if $\mathbf{A}=\mathbf{g}\left(8\right)+\varepsilon \mathbf{g}\left(4\right)$ for $\varepsilon =0.01$.

Figure 2

Typical snapshot of self-organizing strategy distributions on a square lattice for a three-strategy model where the pair interaction is described by $\mathbf{A}=\mathbf{g}\left(8\right)+\beta \left(9\right)\mathbf{g}\left(9\right)$ for $\beta \left(9\right)=0.1$ at a low noise level $\left(K=0.3\right)$.

Figure 3

Snapshots illustrate typical strategy distributions during the domain growing process on a square lattice at a low noise level if the interaction is described by $\mathbf{g}\left(8\right)$ (left) and $\mathbf{g}\left(6\right)$ (right). For both cases the finite system evolves into a mono-domain state dominated by one of the three ordered strategy arrangements.

Figure 4

Games defined by Eq. (35) are located along the dashed line in the two-dimensional subspace of parameters. Closed circles represent games related closely to the attractive Potts model. Open circles illustrate models equivalent to the repulsive three-state Potts model. Open squares show six models that can be transformed into $\mathbf{g}\left(6\right)$ by the permutation of the strategy labels.

Figure 5

Average strategy frequencies in the sublattices as a function of $K$ for the interactions defined by Eq. (35). Solid, dashed, and dotted lines show the MC results for $\phi =0,\pi /6$, and $\pi /4$, respectively.

Figure 6

Typical snapshots illustrating strategy distributions during the domain growing process at a low noise level if the interactions are given by (left) $\mathbf{g}\left(7\right)$ and (right) $-\mathbf{g}\left(7\right)$.

Figure 7

Log-log plot of ${\gamma }_1$ and ${\gamma }_2$ versus $K_c-K$ for $\phi =0$ [crosses, here ${\gamma }_1={\gamma }_2$, and $K_c=0.9952\left(1\right)\text{]}$, $\pi /6$ [open and closed circles, for $K_c=0.58966\left(2\right)\text{]}$, and $\pi /4$ [open and closed boxes, for $K_c=0.4467\left(1\right)\text{]}$. The dashed line refers to the theoretical prediction (with a slope of $1/9)$ for the Potts model.