Evolutionary dynamics of the prisoner's dilemma with expellers

Abstract

Expulsion refers to the widespread behavior of expelling intruders from the owners' territories, which has not been considered in current models on the evolutionary dynamics of cooperation so far. In the context of prisoner's dilemma, we present a simple game-theoretical model of expulsion in which punishing cooperators (i.e. expellers) are able to banish defectors from their own neighborhoods. In the mean-field limit, our theoretical analysis of prisoner's dilemma with expellers shows that the increment of either vacant sites ratio or time scale parameter between pairwise interaction process and strategy updating process can slow down evolutionary speed though defection is the only stable fixed point anyway. In more realistic spatial settings, we provide both analytical and numerical results for the limiting case where pairwise interaction dynamics proceeds much faster than strategy updating dynamics. Using the extended pair approximation methods and Monte Carlo simulations, we show that the introduction of expellers not only promotes coevolution of expulsion and cooperation by means of both direct and indirect domain competition but also opens the gate to rich dynamical behaviors even if expulsion is costly. Phase diagrams reveal the occurrence of frozen as well as dynamical stationary states, between which continuous or discontinuous phase transition may happen. For intermediate ranges, we investigate numerically the coupled interplay between pairwise interaction dynamics and strategy updating dynamics, and show that the validity of main results for the limiting case can be extended to this general case. Interestingly, there exists an optimal value of time scale parameter that results in the maximum fraction of altruistic players, which resembles the coherence resonance phenomenon in dynamical systems. Our results may provide insights into understanding coevolutionary dynamics of expulsive and cooperative behavior in social dilemma situations. the one-short prisoner's dilemma. However, if both players choose defection then their individual income is lower than that obtained for mutual cooperation when maximum overall payoff is shared equally. Meanwhile, we observe that cooperation is ubiquitous in nature as well as human society [6][7][8]. The evolutionary puzzle here arising is how natural selection can lead to cooperation [9].Until now, five fundamental rules, i.e. kin selection [10, 11], direct reciprocity [12], indirect reciprocity [13], network reciprocity [14-16] and group selection [17,18], are reported to be able to enforce the evolution of cooperation in different contexts [19]. Of the five mechanisms for the evolution of cooperation, considerable attention from physics community has been paid to network reciprocity [20][21][22]. In evolutionary games on graphs, players occupy the vertices of a graph. The edges determine who interacts and competes with whom. Following the pioneering work of Nowak and May [14], evolutionary prisoner's dilemma has been explored on various spatial networks [23-25] and di...

Robust cooperation against mutations via costly expulsion

“…In this letter, we aim to study the evolution of robust cooperation against mutations by comparing the mutationselection dynamics of the spatial public goods game with costly expulsion [19,20] and defection as the two competing strategies with that of the traditionally spatial public goods game. In the spatial public goods game with costly expulsion, we intentionally leave out cooperators who do not pay a cost to expel defectors from their neighborhoods so as to avoid the second-order free-riding problem [21][22][23], and thus to be able to concentrate solely on the effectiveness of costly expulsion against mutations.…”