Exit Option Induced by Win-Stay-Lose-Leave Rule Provides Another Route to Solve the Social Dilemma in Structured Populations

Abstract

Theoretical have shown that cooperation can be promoted by the so-called reciprocity, where cooperation can coexist with defection via the compact cooperative clusters. However, this mechanism has one assumption interpreted as players have no chance to exit from the game even the situation is extremely bad, which is in sharp contrast with real-life situations. Here, we relax this assumption in the spatial prisoner's dilemma game by giving players the right to choose one of their state between active (participate in the game) and inactive (exit from the game) in each round of the game determined by win-stay-lose-leave rule. This new rule attracts a winner whose payoff is larger than the average payoff of its neighbors to stay in its current state, thus retains its current advantage to its neighbors. Conversely, a loser is pushed to leave from its current state and thus increase its chance to obtain a higher payoff. Specifically, we incorporate exit costs into consideration by assuming anyone who decides to exit from the game must pay a cost $\gamma$. Extensively numerical simulation shows that for the requirement of exit cost is neither too high or too small, a full cooperation plateau is achieved, where cooperation evolves with the support of enhanced network reciprocity. In fact, inactive players can only exist at the boundary of cooperative clusters during its evolution dynamics, which create a crucial buffer area for the endangered cooperators. Finally, the joint effect of this protective film composed of inactive players and cooperative clusters forming the foundation of enhanced network reciprocity. Our findings may provide new insight into network reciprocity.