Risk Sharing Among Many: Implementing a Subgame Perfect and Optimal Equilibrium

Can a welfare-maximising risk-sharing rule be implemented in a large, decentralised community? We revisit the price-and-choose (P&C) mechanism of Echenique and N\'u\~nez (2025), in which players post price schedules sequentially and the last mover selects an allocation. P&C implements every Pareto-optimal allocation when the choice set is finite, but realistic risk-sharing problems involve an infinite continuum of feasible allocations. We extend P&C to infinite menus by modelling each allocation as a bounded random vector that redistributes an aggregate loss $X=\sum_i X_i$. We prove that the extended mechanism still implements the allocation that maximises aggregate (monetary) utility, even when players entertain heterogeneous credal sets of finitely additive probabilities (charges) dominated by a reference probability $\mathbb{P}$. Our credal sets are weak$^{\ast}$-compact and are restricted so that utility functionals are uniformly Lipschitz on the feasible set. Finally, we pair P&C with the first-mover auction of Echenique and N\'u\~nez (2025), adapted to our infinite-menu, multiple-prior environment. Under complete information about players' credal sets, the auction equalises the surplus among participants. The result is a decentralised, enforcement-free procedure that achieves both optimal and fair risk sharing under heterogeneous priors.