Aspiring to the fittest and promotion of cooperation in the prisoner's dilemma game

Aspiring to the fittest and promotion of cooperation in the prisoner’s dilemma game

Zhen Wang and Matjaž Perc

Phys. Rev. E 82, 021115 – Published 13 August 2010

Abstract

Strategy changes are an essential part of evolutionary games. Here, we introduce a simple rule that, depending on the value of a single parameter $w$ , influences the selection of players that are considered as potential sources of the new strategy. For positive $w$ players with high payoffs will be considered more likely, while for negative $w$ the opposite holds. Setting $w$ equal to zero returns the frequently adopted random selection of the opponent. We find that increasing the probability of adopting the strategy from the fittest player within reach, i.e., setting $w$ positive, promotes the evolution of cooperation. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process and for different interaction networks. Since the evolution to widespread defection is tightly associated with cooperators having a lower fitness than defectors, the fact that positive values of $w$ facilitate cooperation is quite surprising. We show that the results can be explained by means of a negative feedback effect that increases the vulnerability of defectors although initially increasing their survivability. Moreover, we demonstrate that the introduction of $w$ effectively alters the interaction network and thus also the impact of uncertainty by strategy adoptions on the evolution of cooperation.

Received 25 May 2010

DOI:https://doi.org/10.1103/PhysRevE.82.021115

Authors & Affiliations

Zhen Wang¹ and Matjaž Perc²

¹School of Physics, Nankai University, Tianjin 300071, China
²Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

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Vol. 82, Iss. 2 — August 2010

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Images

Figure 1
(Color online) Characteristic snapshots of cooperators [red (dark gray in black-white print)] and defectors [light blue (light gray in black-white print)] for different values of the selection parameter $w$ . From top left to bottom right $w = - 0.2$ , 0, 0.2, 0.5, 1.0, and 4.0, respectively. All panels depict results obtained for $r = 0.022$ and $K = 0.1$ on a $100 \times 100$ square lattice.Reuse & Permissions
Figure 2
(Color online) Top panel: frequency of cooperators $ρ_{C}$ in dependence on the cost-to-benefit ratio $r$ for different values of the selection parameter $w$ . From left to right $w = - 0.2$ , 0, 0.5, 1.0, 2.0, and 4.0, respectively. Note that negative values of $w$ impair the evolution of cooperation, while $w > 0$ moves the survivability of cooperators toward larger values or $r$ . Bottom panel: critical threshold values of the cost-to-benefit ratio $r = r_{c}$ , marking the transition to the pure $D$ phase (extinction of cooperators), in dependence on the selection parameter $w$ . Note that $r_{c}$ converges in both the negative and the positive limits of $w$ . In particular, $r_{c} \to 0$ for negative and $r_{c} \to 0.35$ for positive values of $w$ . Depicted results were obtained for $K = 0.1$ (both panels).Reuse & Permissions
Figure 3
(Color online) Frequency of cooperators $ρ_{C}$ in dependence on the cost-to-benefit ratio $r$ for different values of the selection parameter $w$ for the random regular graph (RRG) and the scale-free (SF) network. From left to right $w = - 0.2$ , 0, and 1.0 for the random regular graph, and $w = - 0.2$ , 0, and 1.0 for the scale-free network, respectively. Note that these results are in qualitative agreement with those obtained on the square lattice in that negative values of $w$ impair the evolution of cooperation, while $w > 0$ moves the survivability of cooperators toward larger values or $r$ . Depicted results were obtained for $K = 0.1$ .Reuse & Permissions
Figure 4
(Color online) Time courses depicting the evolution of cooperation for $w = 0$ (solid black line), $w = 1.0$ (dashed gray line), $w = 2.0$ (dotted blue line), and $w = 4.0$ (dashed-dotted orange line). Note that while for $w = 0$ cooperators die out, for $w > 0$ they recover from what appears to become an even faster extinction to eventually rise to near dominance. Notably, the stronger the initial temporary downfall, the better the recovery (see also the inset). All time courses were obtained as averages over 20 independent realizations for $r = 0.03$ and $K = 0.1$ on a $400 \times 400$ square lattice. Note that the horizontal axis is logarithmic and that values of $ρ_{C}$ were recorded also in between full iteration steps to ensure a proper resolution.Reuse & Permissions
Figure 5
(Color online) Full $r − K$ phase diagram for $w = 0$ (top panel) and $w = 2.0$ (bottom panel), obtained via systematic simulations of the prisoner’s dilemma game on the square lattice. Dashed blue and dotted red lines mark the border between stationary pure $C$ and $D$ phases and the mixed $C + D$ phase, respectively. In agreement with previous works [57, 38], it can be observed that for $w = 0$ (top panel) there exists an intermediate uncertainty in the strategy adoption process (an intermediate value of $K$ ) for which the survivability of cooperators is optimal, i.e., $r_{c}$ is maximal. Conversely, while the borderline separating the pure $C$ and the mixed $C + D$ phases for the $w = 2.0$ case (bottom panel) exhibits a qualitatively identical outlay as for the $w = 0$ case, the $D \leftrightarrow C + D$ transition is qualitatively different. Note that in the bottom panel there exists an intermediate value of $K$ for which $r_{c}$ is minimal rather than maximal, while toward the large $K$ limit $r_{c}$ increases, saturating only for $K > 4$ (not shown).Reuse & Permissions

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