复杂网络中不对称种群博弈的合作演化研究

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"该研究论文探讨了复杂网络上不对称种群博弈中合作演化的机制。通过在不同复杂网络上构建不对称种群博弈模型，并借助Matlab进行模拟，研究发现合作演化的关键因素包括SD型玩家的成本效益比(r2)，PD型玩家的数量(nPD)，以及复杂网络的平均度(<k>)。模拟结果揭示了多个现象：当r2较小(小于0.6)且nPD固定时，无论是规则网络还是BA无标度网络都不利于合作，平均度越小，游戏中的合作程度越低。然而，当r2较大(大于0.6)时，在给定的nPD条件下，规则网络和BA无标度网络都促进了合作，而且平均度越小，合作程度反而越高。" 本文详细分析了复杂网络环境中的合作演化问题，重点在于分析了三个关键因素对合作演化的影响。首先，成本效益比(r2)是决定SD型玩家是否倾向于合作的关键。当r2较低时，意味着合作的成本相对收益较高，这会抑制合作的形成和发展。而当r2增加到一定程度，合作变得更有利可图，从而促进合作行为的出现。其次，PD型玩家的数量(nPD)也起着重要作用。nPD决定了与SD型玩家互动的对手数量，更多的PD型玩家可能使得合作策略难以维持，因为PD型玩家通常会选择自私的行为。然而，研究并未具体说明nPD如何影响合作，这可能是未来研究的一个方向。再者，复杂网络的平均度(<k>)反映了网络中节点的连接密度。在r2较小的情况下，网络的紧密连接性并不能促进合作，反而可能导致竞争性的策略占据优势。而在r2较大的情况下，更稀疏的网络结构反而有助于合作的扩散，这可能是由于在低连接度的网络中，合作策略能够更好地抵御非合作策略的侵蚀。关键词涉及"演化博弈"、"复杂网络"和"合作行为"，暗示了该研究深入探讨了这些领域的交叉点。通过理解这些因素如何相互作用，可以为设计更有效的社会经济系统、优化网络结构以及推动集体行动提供理论指导。此外，这些发现对于理解和预测现实世界中如资源分配、环境保护等多方面的问题的合作动态具有重要意义。

Study on Evolution of Cooperation by an Asymmetric Population

Game on Complex Networks

JIANG Yanmei, TIAN Yuping

School of Automation, Southeast University, Nanjing, 210096,P. R. China

E-mail: jiangyanmei0619@126.com, yptian@seu.edu.cn

Abstract: This paper studies on the evolution of cooperation by an asymmetric population game on complex networks. By

designing an asymmetric population game on different complex networks and with the help of Matlab simulation, we ﬁnd that

the cooperation evolution of this population game depends on three important factors: cost-to-beneﬁt of SD-type players (r

the number of PD-type players (n

), and the average degrees of complex networks (<k>). The simulation results reveal

several facts. For small r

< 0.6) and the given n

, neither regular networks nor BA scale-free networks are in favor of

cooperation. And the smaller the average degree is, the less cooperation in the game becomes. However, for large r

> 0.6)

and the given n

, both regular networks and BA scale-free networks are conducive to cooperation. Further more, the smaller

average degree is, the more cooperation becomes.

Key Words: Evolution of Cooperation, Asymmetric Population Game, Complex Networks

1 Introduction

Cooperation plays an important role in the evolution of

species. The cooperation phenomena exist widely in the na-

ture [1-3]. Evolutionary game theory [4-5] has become a

powerful tool to investigate theses phenomena, and research

on cooperation mechanisms becomes more and more sig-

niﬁcant. Besides, two simple games, the Prisoner’s Dilem-

ma game (PDG) [6] and the snowdrift game (SDG) [7]

have been used widely in theoretical and experimental stud-

ies. Based on the achievement in the past, Nowak pointed

out ﬁve mechanisms which are conducive to the evolution

of cooperation [8]: kin selection, direct reciprocity, indirec-

t reciprocity, group selection, and network reciprocity. The

effect of spatial structures on cooperation evolution in the

population games has been studied for a long time. In the

literatures [9-10], Nowak et al. studied Prisoner’s Dilem-

ma game on regular lattices where every agent has different

number of the neighbors; their research showed that the spa-

tial structures of regular networks can promote cooperation

in the population game. Meanwhile spatial structures can

also do a contrary effect on the snowdrift game. The result-

s in literature [11] manifested that spatial structures reduce

the proportion of cooperators for a wide range of parameters.

But fortunately, scale-free networks provide both PDGand

SDG a unifying framework for the emergence of coopera-

tion [12].

However, the studies mentioned above are all based on

symmetric population game, in which two players are cho-

sen to play a game, their payoff matrixes are identical. As

we all know, players often obtain different beneﬁts even they

adopt the same strategy in a game, the asymmetric of payoff

exists commonly in practice [13]. And such a game can be

established as an asymmetric game. In the literature [14], the

authors researched the evolution of cooperation in an asym-

metric well-mixed population game. When the agents are

either PD-type agents or SD-type agents, the proportion of

This work is supported by National Natural Science Foundation (NNS-

F) of China under Grant 61273110 and 61573105, and the Research

Fund for the Doctoral Program of Higher Education of China under grant

20130092130002.

cooperators is determined by two factors: the numbers of

PD-type players and the parameters in SD-type players’ pay-

off matrixes. When the number of PD-type players is less

than a threshold value, equilibrium of the well-mixed popu-

lation game achieves 1 − r

. Nevertheless, population game

on complex networks is more common in the nature. It is

an urgent task to study on the evolution of cooperation by an

asymmetric population game on complex networks.

Inspired by the literature [14], we study on the evolution

of cooperation by an asymmetric population game on the

complex networks. By using Matlab simulation, the propor-

tion of cooperators in the population game on different av-

erage degrees of different complex networks is studied. The

simulations reveal that the cooperation evolution of a struc-

tured population game is determined by three important fac-

tors: the number of PD-type players (n

), cost-to-beneﬁt

of SD-type players (r

), and the average degrees of complex

networks (<k>). With increasing of the ﬁrst two factors,

cooperation is hindered in general. However, the last factor

can promote the evolution of cooperation in the population

game under some certain conditions. Concretely speaking,

for the given n

it is more favorable to cooperation in the

population game if the cost-to-beneﬁt of SD-type players is

smaller. For given r

and <k>, with the PD-type player-

s’ number growth, cooperation in the population is hampered

by n

after a certain threshold. Besides, for the given n

, neither regular networks nor BA scale-free networks are in

favor of cooperation with small r

. And the smaller the aver-

age degree is, the less cooperation in the game becomes. But

for the given n

, both regular networks and BA scale-free

networks are conducive to cooperation with large r

. What

is more, the smaller average degree is, the more cooperation

becomes.

The rest of the paper is organized as follows. The sys-

tem model is put forward in the section 2. The experimental

design and the experimental results are presented Section 3.

And concluding remarks are stated in the Section 4.

2 The Model

Generally speaking, a structured population evolutionary

game consists of the player set, strategy set, payoff matrix,

Proceedings of the 35th Chinese Control Conference

Jul

27-29, 2016, Chen

du, China

10339

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