The Effect of Punishment and Reward on Cooperation in a Prisoners' Dilemma Game

This paper characterizes how different incentive instruments shape cooperation in a repeated Prisoner`s Dilemma with a continuum of players. A simple tit-for-tat strategy competes against unconditional defection, and the long-run outcome is summarized by a tipping-point share of cooperators, above which cooperation spreads and below which defection prevails. Closed-form expressions for this tipping-point are derived as a function of four payoff classes: targeted punishment of defectors, general punishment applied to all deviations, and two symmetric reward instruments. The formula implies sharply diminishing returns to targeted punishment so that increasing the penalty lowers the temptation payoff and reduces the cooperation threshold but can only drive the threshold asymptotically toward zero, so a positive mass of defectors persists even under extreme sanctions. By contrast, sufficiently strong general incentives (taxes, subsidies, or reputational payoffs) that shift the entire payoff profile can increase cooperation and make it self-enforcing. The framework nests several standard strategy refinements (evil and generous tit-for-tat, Win-Stay-Lose-Shift, heterogeneous horizons, and imperfect monitoring) and derives the corresponding shifts in the cooperation threshold. A numerical calibration illustrates these comparative statics and links them to policy environments where unilateral incentives conflict with collective welfare, including climate agreements, online platform governance, and systemic financial regulation. The results suggest that durable cooperation is rarely secured by a single heavyweight mechanism, but rather that robust outcomes emerge when policy simultaneously tilts payoffs away from unilateral defection and extends the effective horizon over which cooperative gains are realized, combining moderate targeted measures with broader, general instruments.