Opportunistic migration in spatial evolutionary games

We study evolutionary games in a spatial diluted grid environment in which agents strategically interact locally but can also opportunistically move to other positions within a given migration radius. Using the imitation of the best rule for strategy revision, it is shown that cooperation may evolve and be stable in the Prisoner's Dilemma game space for several migration distances but only for small game interaction radius while the Stag Hunt class of games become fully cooperative. We also show that only a few trials are needed for cooperation to evolve, i.e., searching costs are not an issue. When the stochastic Fermi strategy update protocol is used cooperation cannot evolve in the Prisoner's Dilemma if the selection intensity is high in spite of opportunistic migration. However, when imitation becomes more random, fully or partially cooperative states are reached in all games for all migration distances tested and for short to intermediate interaction radii.

Figure 1

(Left image) The games phase space (H = Harmony, HD = Hawk-Dove, PD = Prisoner's Dilemma, and SH = Stag Hunt) as a function of $S,T$ ($R=1$, $P=0$). (Right image) Replicator dynamics stable states [20, 22] with $50\%$ cooperators and defectors initially in a well-mixed population for comparison purposes. Lighter tones stand for more cooperation. Values in parentheses next to each quadrant indicate average cooperation in the corresponding game space.

Figure 2

Average cooperation levels with opportunistic migration and IB rule as a function of $R_g$ and $R_m$. (Left) $N_{\mathrm{test}}=20$; (right) $N_{\mathrm{test}}=1$. The size of the population is 1000 players and the density $\rho$ is $0.5$. In all cases the initial fraction of cooperators is $0.5$ randomly distributed among the occupied grid points.

Figure 3

Illustration of the effect of the playing radius $R_g$ on the payoff of individuals. (Left image) $R_g=1.5$; (right image) $R_g=3$. The drawings refer to a locally full grid and are intended for illustrative purposes only (see text).

Figure 4

Average convergence time $T_c$ with IB rule as a function of $N_{\mathrm{test}}$ for $R_g=1.5$ and $R_m=1.5,3,5$. (Left image) $S=-0.5$, $T=0.5$. (Right image) $S=-0.1$, $T=1.1$. The time to convergence is defined as the number of simulation steps needed for the number of cooperators $N_c$ or defectors $N_d$ to be smaller than $0.1N$. Times of convergence are averaged over 500 independent runs.

Figure 5

Average cooperation levels with opportunistic migration and the Fermi rule as a function of $R_g$ and $R_m$. From left to right $\beta =1.0,0.1,0.01,0.001$. The density $\rho$ is $0.5$ and $N_{\mathrm{test}}=20$. The size of the population is 1000 players. In all cases the initial fraction of cooperators is $0.5$ randomly distributed among the population.

Figure 6

Time evolution for the case of a PD with $T=1.5$, $S=-0.5$, $R=1$, $P=0$. Here $\beta =0.01$, $R_g=1.5$, $R_m=3$. The density $\rho =0.5$ and $N_{\mathrm{test}}=20$. There are 1000 players and the strategies are initially attributed uniformly at random in a 50:50 proportion.