Payoff components and their effects in a spatial three-strategy evolutionary social dilemma

We study a three-strategy spatial evolutionary prisoner's dilemma game with imitation and logit update rules. Players can follow the always-cooperating, always-defecting or the win-stay-lose-shift (WSLS) strategies and gain their payoff from games with their direct neighbors on a square lattice. The friendliness parameter of the WSLS strategy—characterizing its cooperation probability in the first round—tunes the cyclic component of the game determining whether the game can be characterized by a potential. We measured and calculated the phase diagrams of the system for a wide range of parameters. When the game is a potential game and the logit rule is applied, the theoretically predicted phase diagram agrees very well with the simulation results. Surprisingly, this phase diagram can be accurate even in the nonpotential case if there are only two surviving strategies in the stationary state; this result harmonizes with the fact that all $2\times 2$ games are potential games. For the imitation dynamics, we found that the effects of spatiality combined with the presence of two cooperative strategies are so strong that they suppress even substantial changes in the payoff matrix, thus the phase diagrams are independent of the cyclic component's intensity. At the same time, this type of strategy update mechanism supports the formation of cooperative clusters that results in a cooperative society in a wider parameter range compared to the logit dynamics.

Figure 1

Phase diagram determined by the maximal value of the potential matrix (that exists for $w=1/2$). Black labels in the regions refer to the preferred strategy. In the AllC+AllD phase, AllC and AllD players occupy the square lattice according to a chessboard or anti-chessboard-like arrangement. Gray labels and lines show the different social dilemma parameter regions, the prisoner's dilemma (PD), the hawk-dove game (HD), the staghunt game (SH), and the harmony game (HG).

Figure 2

Strategy frequencies versus $K$ with logit rule at $T=2$ and $S=-0.5$ for $w=1/2$. The accuracy of MC results is comparable to the line thickness.

Figure 3

Coincidence between the strategy frequencies for two consecutive runs when $K$ is increased gradually during the simulations at $S=-1$ and $T=1$ for $w=1/2$. In the first run, the system is started from homogeneous AllD state; the average strategy frequencies are indicated by lines. In the second run, all players use the WSLS strategy initially; strategy frequencies are illustrated by symbols.

Figure 4

Strategy frequencies versus $K$ for two consecutive series of runs at $S=-0.5$ and $T=1.5$ for $w=1/2$. The notations of strategy frequencies are the same as in Fig. 3.

Figure 5

Strategy frequencies in the $T\text{−}S$ plane when the logit strategy update rule is applied for $w=1/2$ and $K=0.1$. Blue (very dark gray) surface indicates the average frequency of WSLS players, red (dark gray) surface denotes the average frequency of the AllD strategy. AllC players [green (light gray)] appear only in the hawk-dove region, forming twofold degenerated sublattice ordered structures together with the AllD players.

Figure 6

Comparison of strategy frequencies in the $T\text{−}S$ plane when the logit strategy update rule is applied for (a) $w=0.1$ and (b) $w=0.9$ at $K=0.1$. The color code for the strategies is the same as in Fig. 5.

Figure 7

Frequency of the WSLS strategy as a function of $T$ at $S=-0.5$ for different noise levels when imitation update rule is applied.

Figure 8

Stationary strategy frequencies in the $T\text{−}S$ plane for any $w$ parameter value when imitation rule is applied. The color code for the strategies is the same as in Fig. 5.