The Replicator Equation on Graphs

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The Replicator Equation on Graphs

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Title: The Replicator Equation on Graphs
Author: Ohtsuki, Hisashi; Nowak, Martin A.

Note: Order does not necessarily reflect citation order of authors.

Citation: Ohtsuki Hisashi, Martin A. Nowak. 2006. The replicator equation on graphs. Journal of Theoretical Biology 243(1): 86-97.
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Abstract: 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.
Published Version: doi:10.1016/j.jtbi.2006.06.004
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at
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  • FAS Scholarly Articles [8087]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University

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