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
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