The Evolution of Cooperation in a Lattice-Structured Population

Abstract

The evolution of cooperation among unrelated individuals is studied in a lattice-structured habitat, where individuals interact locally only with their neighbors. The initial population includes Tit-for-Tat (abbreviated as TFT, indicating a cooperative strategy) and All Defect (AD, a selfish strategy) distributed randomly over the lattice points. Each individual plays the iterated Prisoner's Dilemma game with its nearest neighbors, and its total pay-off determines its instantaneous mortality. After the death of an individual, the site is replaced immediately by a copy of a randomly chosen neighbor. Mathematical analyses based on mean-field approximation, pair approximation, and computer simulation are applied. Models on one and two-dimensional regular square lattices are examined and compared with the complete mixing model. Results are: (1) In the one-dimensional model, TFT players come to form tight clusters. As the probability of iteration w increases, TFTs become more likely to spread. The condition for TFT to increase is predicted accurately by pair approximation but not by mean-field approximation. (2) If w is sufficiently large, TFT can invade and spread in an AD population, which is impossible in the complete mixing model where AD is always ESS. This is also confirmed by the invasion probability analysis. (3) The two-dimensional lattice model behaves somewhat in between the one-dimensional model and the complete mixing model. (4) The spatial structure modifies the condition for the evolution of cooperation in two different ways: it facilitates the evolution of cooperation due to spontaneously formed positive correlation between neighbors, but it also inhibits cooperation because of the advantage of being spiteful by killing neighbors and then replacing them.

A symmetry of fixation times in evoultionary dynamics

“…Our result holds for games on cycles (Nakamaru et al, 1997(Nakamaru et al, , 1998Nakamaru & Iwasa, 2005). Furthermore, our result also applies to a wide range of imitation processes of interest to economists (Ellison, 1993;Binmore & Samuelson, 1997;Maruta, 2002), when the detailed balance condition holds.…”

“…In fact, our results can also be applied to the study of evolutionary game dynamics on graphs, of which there is a great deal of current interest (Nakamaru et al, 1997(Nakamaru et al, , 1998Nakamaru & Iwasa, 2005;Lieberman et al, 2005;Ohtsuki et al, 2006). In the case of a cycle graph with N nodes, the game dynamics starting from a single mutant can be described by the Moran process.…”

Evolutionary game dynamics with non-uniform interaction rates

“…There is much current interest to study evolutionary game dynamics on graphs, which also leads to non-uniform interaction rates. [Ellison, 1993, Nakamaru et al, 1997& 1998, Epstein, 1998, Abramson & Kuperman, 2001, Ebel & Bornholdt, 2002, Szabo & Vukov, 2004, Ifti & et al, 2004, Nakamaru & Iwasa, 2005, Lieberman et al, 2005 The fundamental Lotka-Volterra equation of ecology is equivalent to the replicator equation of evolutionary game theory [Hofbauer & Sigmund, 2003].…”

Does mobility decrease cooperation?

“…These spatial games, where the interactions are localized and non random, have been studied and extended in many ways (see, for example, Refs. [1,2,4,9,10,15,19,22,23,24,25,29,30,31,34,37,38,40,41,44,45,46,47,49,52]). Once the population is spatially structured, a natural question concerns the effects of mobility that, along with other important biological factors, is often neglected [28]: is it possible to evolve and sustain cooperation in a population of mobile agents, where retaliation can be avoided by moving away from the former partner?…”