Spatial prisoner’s dilemma game with volunteering in Newman-Watts small-world networks

Abstract

A modified spatial prisoner's dilemma game with voluntary participation in Newman-Watts small-world networks is studied. Some reasonable ingredients are introduced to the game evolutionary dynamics: each agent in the network is a pure strategist and can only take one of three strategies (cooperator, defector, and loner); its strategical transformation is associated with both the number of strategical states and the magnitude of average profits, which are adopted and acquired by its coplayers in the previous round of play; a stochastic strategy mutation is applied when it gets into the trouble of local commons that the agent and its neighbors are in the same state and get the same average payoffs. In the case of very low temptation to defect, it is found that agents are willing to participate in the game in typical small-world region and intensive collective oscillations arise in more random region.PACS numbers: 02.50. Le, 87.23.Kg, 87.23.Ge, 89.75.Hc There has been a long history of studying complex behaviors qualitatively of biological, ecological, social and economic systems using special game models. After the prisoner's dilemma game (PDG) was first applied by Neumann and Morgenstern [1] to study economic behavior, great developments have been made by a lot of subsequent studies. Recently, more and more attentions have been focused on the applications of the PDG in the fields of biology [2], economy [3], ecology [4], and other domains [5]. Game theory and evolutionary theory provide a powerful metaphor for simulating the interactions of individuals in these systems [6].Most realistic systems can be regarded as composing of a large number of individuals with simple local interactions. For example, human beings are limited in territory and interact more frequently with their neighbors than those far away. Therefore, the spatial structure may greatly affect their activities. Since Axelrod [7] suggested ideas of the PDG on a lattice, spatial prisoner's dilemma games (SPDG) have been extensively explored in various kinds of network models in the past few years, including regular lattices [8,9,10], random regular graphs [11], random networks with fixed mean degree distribution [12], small-world networks [13,14,15] and realworld acquaintance networks [16], etc. In the general SPDG, each agent can take one of two strategies (or states): cooperator (C) and defector (D). There are four possible combinations: (C, C), (C, D), (D, C) and (D, D), which get payoffs (r, r), (s, t), (t, s), and (p, p), respectively. The parameters satisfy the conditions t > r > p > s and 2r > t + s, so that lead to a so-called dilemma situation where mutual trust and cooperation is beneficial in a long perspective but egoism and guile can produce big short-term profit. Agents update their states by imitating the strategy of the wealthiest among their neighborhoods in subsequent plays. The system is easy to get into an absorbing state: all agents are D for large values of t, which is known as the tragedy of the commons [17].Recently, Szabó ...

Separation of cyclic and starlike hierarchical dominance in evolutionary matrix games

“…For imitation type dynamical rules [3,4] the multiagent voluntary prisoner's dilemma on a square lattice exhibited a self-organizing pattern with rotating spiral arms [22,25,26]. In well-mixed populations [21,27] and also on some small-world networks [28][29][30] this interaction yielded global oscillations in the strategy frequencies. The "tragedy of the commons" is avoided in all these systems because the cyclic component supports the survival of all three strategies.…”

Evolutionary games on graphs

“…Similar results were reported by Aktipis (2004) who considered the effect of contingent movement of cooperators: the players walked away once a defection occurred in the previous step. In some sense, this "win-stay, lose-move" strategy is analogous to voluntary participation which also enhance cooperation (Saijo and Yamato, 1999;Hauert et al, 2002;Szabó and Hauert, 2002a;Wu et al, 2005a) as detailed in the subsequent section.…”

Networking effects on cooperation in evolutionary snowdrift game

“…Mathematicians, biologists, and physicists alike have found the phenomena of emergence of cooperative behavior fascinating. Since the ground-breaking work on repeated or iterated games based on the Prisoner's Dilemma (PD) game by Axelrod [1,2], there has been a continuous effort on exploring the determining factors on possible cooperative behavior in evolutionary games based on the PD game and its variations [3,4,5,6,7,8], with a recent emphasis on the effects of spatial structures such as regular lattices [5,6,9,10,11] and networks [8,12,13,14,15,16,17]. Remarkably, it was found that cooperation can be induced in a repeated PD game by cleverly designed strategies.…”