کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4499357 | 1319027 | 2006 | 12 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: The replicator equation on graphs The replicator equation on graphs](/preview/png/4499357.png)
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.
Journal: Journal of Theoretical Biology - Volume 243, Issue 1, 7 November 2006, Pages 86–97