Elsevier

Journal of Theoretical Biology

Volume 231, Issue 2, 21 November 2004, Pages 175-179
Journal of Theoretical Biology

Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay

https://doi.org/10.1016/j.jtbi.2004.06.012Get rights and content

Abstract

We construct two models of discrete-time replicator dynamics with time delay. In the social-type model, players imitate opponents taking into account average payoffs of games played some units of time ago. In the biological-type model, new players are born from parents who played in the past. We consider two-player games with two strategies and a unique mixed evolutionarily stable strategy. We show that in the first type of dynamics, it is asymptotically stable for small time delays and becomes unstable for big ones when the population oscillates around its stationary state. In the second type of dynamics, however, evolutionarily stable strategy is asymptotically stable for any size of a time delay.

Introduction

The evolution of populations can be often described within game-theoretic models. The pioneer of such approach, John Maynard Smith, introduced the fundamental notion of an evolutionarily stable strategy (Maynard Smith, 1974, Maynard Smith, 1982). If everybody plays such a strategy, then the small number of mutants playing a different strategy is eliminated from the population. The dynamical interpretation of the evolutionarily stable strategy was later provided in (Taylor and Jonker, 1978, Hofbauer et al., 1979, Zeeman, 1981). A system of differential or difference equations, the so-called replicator equations, were proposed. They describe the time-evolution of frequencies of strategies. It is known that any evolutionarily stable strategy is an asymptotically stable stationary point of such dynamics (Hofbauer and Sigmund, 1988; Weibull, 1997). In fact, in two-player games with two strategies, a state of population is evolutionarily stable if and only if it is asymptotically stable.

Recently, Tao and Wang (1997) investigated the effect of a time delay on the stability of interior stationary points of the replicator dynamics. They considered a two-player game with two strategies and a unique asymptotically stable interior stationary point. They proposed a certain form of a time-delay differential replicator equation. They showed that the mixed evolutionarily stable strategy is stable if a time delay is small. For sufficiently large delays it becomes unstable.

Here, we discuss discrete-time replicator dynamics. We introduce a time delay in two different ways. In the so-called social model, we assume that players imitate a strategy with the higher average payoff, taking into account time-delayed information. When the limit of zero time interval is taken, one obtains equations discussed by Tao and Wang (1997). In the so-called biological model, we assume that the number of players born in a given time is proportional to payoffs received by their parents in a certain moment in the past. We analyze the stability of the interior stationary point in our discrete time-delay replicator dynamics. In the first model, we prove results analogous to those of Tao and Wang (1997). However, in our second model, we show the stability of the interior point for any value of the time delay. We prove our results in an elementary way, that is without any reference to the theory of time-delay equations.

In Section 2, we present our discrete-time replicator dynamics with two types of time delay. In Section 3, we discuss a model with a social-type time delay and in Section 4, with a biological-type time delay. Discussion follows in Section 5. Proofs are given in Appendix A and B.

Section snippets

Replicator dynamics

We assume that our population is haploid, that is the offspring have identical phenotypic strategies as their parents (we assume that there are two different pure strategies; denote them by A and B). In discrete moments of time, individuals compete in pairwise contests and the outcome is given by the following payoff matrix: U=ABAabBcdwhere the ij entry, i,j=A,B, is the payoff of the first (row) player when he plays the strategy i and the second (column) player plays the strategy j. We assume

Social-type time delay

Here, we assume that individuals at time t replicate due to average payoffs obtained by their strategies at time t-τ for some delay τ>0 (see also a discussion after (3.6)).

We propose the following equations:pi(t+ε)=(1-ε)pi(t)+εpi(t)Ui(t-τ)i=A,B.Then for the total number of players we getp(t+ε)=(1-ε)p(t)+εp(t)U¯0(t-τ),where U¯0(t-τ)=x(t)UA(t-τ)+(1-x(t))UB(t-τ).

We divide (3.1) by (3.2) and obtain an equation for the frequency of the strategy A,x(t+ε)-x(t)=εx(t)[UA(t-τ)-U¯0(t-τ)]1-ε+εU¯0(t-τ)and

Biological-type time delay

Here, we assume that individuals born at time t-τ are able to take part in contests when they become mature at time t or equivalently they are born τ units of time after their parents played and received payoffs. We propose the following equations:pi(t+ε)=(1-ε)pi(t)+εpi(t-τ)Ui(t-τ),i=A,B.Then the equation for the total number of players readsp(t+ε)=(1-ε)p(t)+εp(t)x(t)pA(t-τ)pA(t)UA(t-τ)+(1-x(t))pB(t-τ)pB(t)UB(t-τ).We divide (4.1) by (4.2) and obtain an equation for the frequency of the first

Discussion

We proposed and analyzed certain forms of discrete-time replicator dynamics with a time delay. We introduced the time delay on the level of the number of individuals playing different strategies and not on the level of their relative frequencies. We showed that the stability of a mixed evolutionarily stable strategy depends on the particular form of the time delay.

It would be interesting to analyze time-delay replicator models taking into account the limited capacity of an environment (for

Acknowledgements

JM would like to thank the Polish Committee for Scientific Research for a financial support under the grant KBN 5 P03A 025 20.

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