The Evolution of Cooperation in a Lattice-Structured Population Under Two Different Updating Rules

Abstract

The evolution of cooperation is studied in a lattice-structured habitat under two different updating rules: the score-dependent viability model (viability model) and the score-dependent fertility model (fertility model). Each individual in each lattice site is assigned to Tit-for-Tat (TFT) or All Defect (AD). Each individual plays the iterated prisoner’s dilemma game with its nearest neighbors, and obtains the total payoff. In the viability model, its total payoff determines its mortality. After the death of an individual, the site is replaced with a copy of a randomly chosen neighbor. In the fertility model, an individual dies randomly, one of the neighbors is chosen proportionally to its total payoff, and the vacant site is replaced with a copy of the neighbor. The model is analyzed by invasion probability analysis, pair-edge method, mean-field approximation, pair approximation, and computer simulations. Results are: (1) In both updating rules, TFT players come to form tight clusters, and then rare TFT can invade and spread in a population dominated by AD, unlike in the completely mixing model where AD is always an ESS. The dynamics in a one-dimensional lattice are predicted by all the techniques except mean-field approximation. (2) The fertility model is more favorable for TFT than the viability model because spatial structure not only facilitates cooperation but also inhibits cooperation in the viability model because of the advantage of being spiteful by killing neighbors.