Preplay Losing Contracts: Inducing Strong Nash Equilibrium in the $n$-player Prisoner's Dilemma

Abstract

In strategic games such as the prisoner's dilemma, allowing players to make binding offers of utility transfers before play has been shown to alter incentives and potentially support cooperative outcomes. These preplay exchange mechanisms reshape payoffs by transferring utility while being contingent on actions; however, they typically require side payments that can reduce individual benefits relative to joint cooperation. In this paper, we extend the analysis to a finite $n$-player prisoner's dilemma with ordered strategy sets, defined such that any restriction of strategies by any subset of players still yields a prisoner's dilemma. To achieve a robust cooperative outcome that resists group deviations, we introduce a novel class of mechanisms: $\textit{losing contracts}$. Unlike transfer-based preplay mechanisms, losing contracts require players to irrevocably reduce their own utility if they defect, thereby aligning individual incentives with cooperation without inter-player payments. With appropriately chosen loss amounts, losing contracts induce joint cooperation as the unique strong Nash equilibrium in the modified game and in every restricted game within it, ensuring that cooperative incentives persist even under possible external constraints on strategy sets. We show that our contracts can be constructively defined, reducing the preplay stage to a simple and binary decision for each player: whether to sign the contract or not. Furthermore, if the losing contract is only executed when all players sign, signing is a strictly dominant strategy for all. Finally, we extend these results to certain public goods games.