Probability Punishment in Spatial Optional Public Goods Games

In Optional Public Goods Games, players can choose among three strategies in each round: cooperate, defect, or abstain (loners). The option to abstain adds complexity to the game, but free-riding (defection) threatens the stability of cooperation. To address this issue, incentive mechanisms such as decentralized (peer punishment) and centralized (pool punishment) sanctions are used. Following previous methodologies, we introduce a peer punishment system that adjusts payoffs: cooperators who punish incur a cost alpha for each defector they sanction, while punished defectors see their benefit reduced by an amount beta imposed by each cooperator who sanctions them, where alpha is less than beta. The decision to punish is determined randomly using a Bernoulli variable, which selects a group of cooperators to impose sanctions. Cooperators not selected as punishers receive their normal payoff. Payoffs are defined based on the number of cooperators, defectors, and loners, the interest rate, and the contribution to the game. This approach aims to study how the introduction of a probabilistic punishment mechanism influences cooperative behavior and the stability of strategies in the game.