Emergence of cooperation and commitment in optional Prisoner’s Dilemma

• Modelling commitment in a two-stage optional Prisoner’s Dilemma game. • Optionality boosts commitment but fails to sustain cooperation, leading to exit. • A strict rule boosts cooperation by closing an ‘opportunistic exit’ loophole. • Flexible rules may yield higher social welfare when exit options are beneficial. • Guides the institutional design to maximise both cooperation and social welfare. Commitment is a well-established mechanism for fostering cooperation in human society and multi-agent systems, while most research on commitment-based cooperation assumes compulsory participation. This overlooks many real-world scenarios where individuals are free to abstain from an interaction. We address this gap by proposing a two-stage evolutionary game model that integrates commitment with the optional Prisoner’s Dilemma. In our model, players first decide whether to commit and then choose to cooperate, defect, or exit. We find that optional participation boosts commitment acceptance but fails to foster cooperation, leading instead to widespread exit behaviour. To address this, we compare two institutional incentives: a strict rule rewarding only committed cooperators, and a flexible rule rewarding any committed non-defector. Results show that the strict rule is better for promoting cooperation, as the flexible rule creates a loophole for opportunistic exits. However, the flexible rule can yield higher social welfare when exit options are highly beneficial. Our study highlights the need for well-designed incentives in voluntary settings to effectively promote both cooperation and social welfare.