The replicator equation on graphs
The replicator equation on graphs
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DOI：
10.1016/j.jtbi.2006.06.004
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发表时间：
2006-11-07
影响因子：
2
通讯作者：
Nowak, Martin A.
中科院分区：
文献类型：
--
作者：
Ohtsuki, Hisashi;Nowak, Martin A.
我们研究图上的进化博弈。每个玩家都由图中的一个顶点表示。边缘表示谁遇到了谁。玩家可以使用n种策略中的任何一种。玩家从与所有近邻的互动中获得回报。我们考虑了三种不同的更新规则，称为“出生-死亡”、“死亡-出生”和“模仿”。在我们的模型中，第四个更新规则“成对比较”被证明等同于生灭更新。我们用对近似来描述k度正则图上的进化博弈动力学。在弱选择的极限下，我们可以得到一个描述图上每种策略的平均频率如何随时间变化的微分方程。值得注意的是，这个方程是一个带有转换后的收益矩阵的复制者方程。因此，将游戏从混合良好的群体(完整图)转移到规则图上只会导致收益矩阵的转换。新的收益矩阵是原收益矩阵与另一个描述战略局部竞争的矩阵之和。我们讨论了我们的理论在四个具体例子中的应用，它们是囚徒困境、雪-漂流博弈、协调博弈和石头-剪刀布博弈。(C)2006爱思唯尔有限公司。保留所有权利。
We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called 'birth-death', 'death-birth' and 'imitation'. A fourth update rule, 'pairwise comparison', is shown to be equivalent to birth-death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner's Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors Paper game. (c) 2006 Elsevier Ltd. All rights reserved.