Memory-based snowdrift game on networks

Wen-Xu Wang, Jie Ren, Guanrong Chen, and Bing-Hong Wang

Phys. Rev. E 74, 056113 – Published 28 November 2006

Abstract

We present a memory-based snowdrift game (MBSG) taking place on networks. We found that, when a lattice is taken to be the underlying structure, the transition of spatial patterns at some critical values of the payoff parameter is observable for both four- and eight-neighbor lattices. The transition points as well as the styles of spatial patterns can be explained by local stability analysis. In sharp contrast to previously reported results, cooperation is promoted by the spatial structure in the MBSG. Interestingly, we found that the frequency of cooperation of the MBSG on a scale-free network peaks at a specific value of the payoff parameter. This phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation. The memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale-free networks. Our work may shed some new light on the study of evolutionary games over networks.

Received 16 June 2006

DOI:https://doi.org/10.1103/PhysRevE.74.056113

Authors & Affiliations

Wen-Xu Wang^1,2, Jie Ren², Guanrong Chen^1,*, and Bing-Hong Wang²

¹Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
²Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

^*Electronic address: gchen@ee.cityu.edu.hk

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Vol. 74, Iss. 5 — November 2006

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Images

Figure 1
(Color online). The frequency of cooperation $f_{C}$ as a function of the payoff parameter $r$ for two-dimensional (a) four-neighbor and (b) eight-neighbor lattices, respectively. $(▵)$ , $(엯)$ , and $(◻)$ are for $M = 2$ , $7$ , and $30$ , respectively. Each data point is obtained by averaging over $40$ different initial states and $f_{C}$ for each simulation is obtained by averaging from MC time step $t = 5000$ to $t = 10 000$ , where the system has reached a steady state. The top inset of (a) is $f_{C}$ as a function of memory length $M$ for two different cooperation levels. The bottom inset of (a) is a time series of $f_{C}$ for $r = 0.4$ in the case of $M = 1$ . Since for $M = 1$ , $f_{C}$ as a function of $t$ displays a big oscillation, we do not compute the $f_{C}$ over a period of MC time steps. The inset of (b) is $f_{C}$ depending on $M$ for four cooperation levels in the range of $0 < r < 0.5$ . The network size is $N = 10 000$ .Reuse & Permissions
Figure 2
Typical spatial patterns in two distinct payoff parameter ranges (a) $0 < r < 0.25$ , (b) $0.25 < r < 0.5$ . The C is in black and the D is in white. A $50 \times 50$ portion of the full $100 \times 100$ lattice with four neighbors is illustrated. (c) and (d) are the relevant stable local patterns of (a) and (b). $W_{C}$ and $W_{D}$ are the payoffs of the center individual A by choosing C and D with fixing strategies of neighbors for each local pattern. $r = 0.25$ in (c) and $0.5$ in (d).Reuse & Permissions
Figure 3
Typical spatial patterns in two distinct payoff parameter ranges (a) $0.25 < r < 0.375$ ; (b) $0.375 < r < 0.5$ . The color coding is the same as Fig. 2. A $50 \times 50$ portion of the full $100 \times 100$ lattice with eight neighbors is illustrated.Reuse & Permissions
Figure 4
Typical patterns for the time step $t = 8001$ and $t = 8002$ in the case of memory length $M = 1$ , $r = 0.4$ . $C$ is in black and $D$ is in white. A $50 \times 50$ portion of the full $100 \times 100$ lattice with four neighbors is illustrated.Reuse & Permissions
Figure 5
(Color online) $f_{C}$ as a function of $r$ in BA networks with (a1) average degree $⟨ k ⟩ = 4$ and (a2) $⟨ k ⟩ = 8$ for different $M$ . A time series of $f_{C}$ for $M = 1$ is shown in the inset of (a2). (b1) and (b2) are $f_{C}$ as a function of $M$ in the case of $⟨ k ⟩ = 4$ and $⟨ k ⟩ = 8$ for a special range of $r$ . (c1) and (c2) are average degrees $⟨ k_{s} ⟩$ of $C$ and $D$ players depending on $r$ in the case of $M = 7$ for $⟨ k ⟩ = 4$ and $⟨ k ⟩ = 8$ , respectively. The network size is $10 000$ . Each data point is obtained by averaging over $30$ different network realizations with $20$ different initial state of each realization. $f_{C}$ for each simulation is obtained by averaging from MC time step $t = 5000$ to $t = 10 000$ , where the system has reached a steady state.Reuse & Permissions
Figure 6
Distributions of strategies in BA networks. Cooperators and defectors are denoted by gray bars and black bars, respectively. Each bar adds up to a total fraction of $1$ per degree, the gray and black fractions being directly proportional to the relative percentage of the respective strategy for each degree of connectivity $k$ . (a) is for the case of $⟨ k ⟩ = 4$ with $r = 0.1$ and (b) is for the case of $⟨ k ⟩ = 4$ with $r = 0.49$ , at which $f_{C}$ peaks. (c) shows the case of $⟨ k ⟩ = 8$ with $r = 0.05$ and (d) displays the case of $⟨ k ⟩ = 8$ with $r = 0.16$ , which corresponds to the maximum value of $f_{C}$ . All the simulations are obtained for network size $N = 1000$ in order to make figures clearly visible.Reuse & Permissions

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