Stable and Fair Random Allocations in a Two-Sided Discrete-Concave Market

Abstract

Random allocations are widely used to handle ties and indifferences in two-sided environments. In such environments, commonly used procedures such as random tie-breaking may fail to ensure stability and fairness from an ex ante perspective. We show that when agents have discrete concave (M$^\natural$-concave) valuations, there exists an ex ante stable and fair allocation. To establish this result, we relate our framework to the model of stability introduced by Alkan and Gale. In particular, we show that ex ante stable and fair fractional allocations are exactly characterized as Alkan--Gale stable outcomes under choice functions induced from concave closures together with a symmetric strictly convex tie-breaking rule. We further prove that any ex ante stable fractional allocation can be decomposed into a lottery over stable deterministic allocations, using a generalization of the Birkhoff--von Neumann theorem. Finally, we study a setting that does not rely on cardinal valuations and instead assumes ordinal preferences. Within this ordinal framework, we establish the existence of an ex ante stable and fair fractional allocation. This setting is formulated within the matching-with-contracts framework under matroid constraints. The resulting class includes existing models, such as one-to-many random allocation with responsive choice correspondences, and captures a wide range of applications, including controlled school choice with lotteries.