Marginalism and Stability in Package Allocation Problems and Market Replicas

Abstract

We revisit the classical tension between stability and marginalism in the package allocation model of Milgrom [2007]. Our main finite-market stability results provide multiple necessary and sufficient conditions for the Banzhaf payoff vector to belong to the core. The key condition is the feasibility of an optimal assignment in which each buyer receives one of his most preferred bundles. Further, we show that the Shapley and Banzhaf payoff vectors coincide if and only if the Banzhaf payoff vector is in the core. We then consider replica economies. We construct a market in which, after finitely many replications, neither the Shapley nor the Banzhaf payoff vector belongs to the core of the replicated game. Strikingly, as the number of replicas tends to infinity, none of those payoff vectors converges to the core. Consequently, they do not converge to a competitive-equilibrium payoff vector in the limit. This stands in contrast to Luo et al. [2024], which shows that, in their setting of assignment games with multiple partnership, both payoff vectors converge to a competitive-equilibrium payoff vector as the number of replicas tends to infinity.