In the classic snowdrift game, the choice is binary: cooperate or defect. Takesue adds a third option --- the individual solution, a safe fallback that guarantees a constant payoff regardless of what neighbors do. Under myopic best-response dynamics on a square lattice, this exit option restructures the evolutionary landscape entirely. Monte Carlo simulations produce three distinct equilibria depending on parameters: dominance of the individual solution, coexistence of cooperators and defectors, or coexistence of all three strategies. Neighborhood size plays a dual role. Smaller neighborhoods promote cooperation when the individual solution is expensive, but they simultaneously demand greater benefits from cooperation to sustain it when the individual solution is cheap. This duality is absent under imitation dynamics, where smaller neighborhoods uniformly favor cooperation. Bounded rationality changes not just the equilibrium but the topology of the parameter space.
The structural finding is that the individual solution is not a neutral addition to the strategy space --- it is a cooperation inhibitor whose strength depends on the decision rule. Under imitation dynamics, agents copy successful neighbors, so a mediocre individual solution cannot spread against a locally dominant cooperative cluster. Under myopic best response, agents calculate their optimal action given current neighbors, and the guaranteed payoff of the individual solution acts as an outside option that siphons agents away from cooperation whenever the local cooperative surplus is thin. The transition between all-strategy coexistence and cooperator-defector coexistence maps onto local strategy configurations that exist on a two-dimensional lattice but not on a one-dimensional chain, confirming that the spatial structure co-determines the outcome.
An exit option reshapes the game even when nobody takes it. Its presence changes the threshold at which cooperation is rational, because every agent now evaluates cooperation not against defection alone but against a guaranteed alternative. The general principle extends beyond game theory: any system of conditional participation --- markets, alliances, collaborations --- is altered by the availability of an unconditional outside option, and the alteration depends on whether agents optimize locally or imitate globally. The decision rule is not a detail of the dynamics. It is a parameter as fundamental as the payoff matrix.
(arXiv:2603.18128)