Emergence of Cooperation and Commitment in Optional Prisoner's Dilemma

Commitment is a well-established mechanism for fostering cooperation in human society and multi-agent systems. However, existing research has predominantly focused on the commitment that neglects the freedom of players to abstain from an interaction, limiting their applicability to many real-world scenarios where participation is often voluntary. In this paper, we present a two-stage game model to investigate the evolution of commitment-based behaviours and cooperation within the framework of the optional Prisoner's Dilemma game. In the pre-game stage, players decide whether to accept a mutual commitment. Once in the game, they choose among cooperation, defection, or exiting, depending on the formation of a pre-game commitment. We find that optional participation boosts commitment acceptance but fails to foster cooperation, leading instead to widespread exit behaviour. To address this, we then introduce and compare two institutional incentive approaches: i) a strict one (STRICT-COM) that rewards only committed players who cooperate in the game, and ii) a flexible one (FLEXIBLE-COM) that rewards any committed players who do not defect in the game. The results reveal that, while the strict approach is demonstrably better for promoting cooperation as the flexible rule creates a loophole for an opportunistic exit after committing, the flexible rule offers an efficient alternative for enhancing social welfare when such opportunistic behaviour results in a high gain. This study highlights the limitations of relying solely on voluntary participation and commitment to resolving social dilemmas, emphasising the importance of well-designed institutional incentives to promote cooperation and social welfare effectively.