Maximizing Social Welfare with Side Payments

We examine normal-form games in which players may \emph{pre-commit} to outcome-contingent transfers before choosing their actions. In the one-shot version of this model, Jackson and Wilkie showed that side contracting can backfire: even a game with a Pareto-optimal Nash equilibrium can devolve into inefficient equilibria once unbounded, simultaneous commitments are allowed. The root cause is a prisoner's dilemma effect, where each player can exploit her commitment power to reshape the equilibrium in her favor, harming overall welfare. To circumvent this problem we introduce a \emph{staged-commitment} protocol. Players may pledge transfers only in small, capped increments over multiple rounds, and the phase continues only with unanimous consent. We prove that, starting from any finite game $\Gamma$ with a non-degenerate Nash equilibrium $\vec{\sigma}$, this protocol implements every welfare-maximizing payoff profile that \emph{strictly} Pareto-improves $\vec{\sigma}$. Thus, gradual and bounded commitments restore the full efficiency potential of side payments while avoiding the inefficiencies identified by Jackson and Wilkie.