Stochastic win-stay-lose-shift strategy with dynamic aspirations in evolutionary social dilemmas

In times of plenty expectations rise, just as in times of crisis they fall. This can be mathematically described as a win-stay-lose-shift strategy with dynamic aspiration levels, where individuals aspire to be as wealthy as their average neighbor. Here we investigate this model in the realm of evolutionary social dilemmas on the square lattice and scale-free networks. By using the master equation and Monte Carlo simulations, we find that cooperators coexist with defectors in the whole phase diagram, even at high temptations to defect. We study the microscopic mechanism that is responsible for the striking persistence of cooperative behavior and find that cooperation spreads through second-order neighbors, rather than by means of network reciprocity that dominates in imitation-based models. For the square lattice the master equation can be solved analytically in the large temperature limit of the Fermi function, while for other cases the resulting differential equations must be solved numerically. Either way, we find good qualitative agreement with the Monte Carlo simulation results. Our analysis also reveals that the evolutionary outcomes are to a large degree independent of the network topology, including the number of neighbors that are considered for payoff determination on lattices, which further corroborates the local character of the microscopic dynamics. Unlike large-scale spatial patterns that typically emerge due to network reciprocity, here local checkerboard-like patterns remain virtually unaffected by differences in the macroscopic properties of the interaction network.

Figure 1

Equilibrium cooperation level in dependence on $T$ in the WSLS model, as obtained with Monte Carlo simulations and the integration of the master equation (see legend). “Monte Carlo whole lattice” refers to the aspiration being equal to the average payoff of the whole lattice rather than just the nearest neighbors, thus representing results for the two limiting cases concerning the interaction range of individual players. There are small quantitative differences between the presented curves. But more importantly, we see that the inflection point of the curves and the minimum level of cooperation are both very similar in all cases, thus pointing to a good qualitative agreement and a deeper mechanism promoting this effect.

Figure 2

Time evolution of the average cooperation level in the mean-field approximation. The graph shows two sets of curves, each obtained with different initial conditions. The system always reaches a stable state, independent of the initial condition for each value of $T$.

Figure 3

Fraction of cooperator as a function of $T$, as obtained with Monte Carlo simulations. We used the weak prisoner's dilemma with $S=0$ and $k=0.1$. The behavior of WSLS is different from the imitation dynamics, especially above $T=1$, where here we have a nonzero cooperation level. Note also how the WSLS has a smooth drop in $T\simeq 0.6$.

Figure 4

Phase diagram depicting the fraction of cooperators (color bar), depending on $S$ and $T$ for the two different models (WSLS and imitation). In the imitation model (top), as expected, there is full cooperation in the harmony game quadrant, some coexistence in the SD quadrant, a sharp division in the SH quadrant, while most of the PD quadrant is dominated by defection, with cooperators surviving only near the weak PD regime, with $T_c\approx 1.04$. For the WSLS model (bottom) the cooperation is widespread everywhere, although lower than for imitation in the HG quadrant. We can see that the model is efficient in maintaining the coexistence of strategies even for high values of $T$.

Figure 5

Typical snapshots from the imitation (top) and WSLS (bottom) models, cooperators are dark blue and defector light red. We can see different spatial organization patterns that spontaneously emerge. We depict a typical PD game where $\rho \approx 0.25$ and $S=-0.01$ for both models. The temptation level is imitation ($T=1.023$) and WSLS ($T=1.1$).

Figure 6

Cooperator (dark blue) surrounded by defectors (light red) in the deterministic version of the WSLS model. Although the focal site changes to defection, its second neighbors become cooperators due to the lower payoff when compared to the first neighbors.

Figure 7

Fraction of cooperation as a function of $T$ in two different topologies: square lattice and scale-free network with absolute and normalized payoff. We see for the imitation model (top) that the scale-free network with absolute payoff highly enhances cooperation, while the normalized payoff dampens this boost to a substantial degree. The lowest cooperation is obtained on the square lattice. For the WSLS (bottom) the different topologies have little effect on the evolution of cooperation.