Stable and Envy-Free Lottery Allocations for Affordable Housing

Abstract

Affordable housing lotteries often enforce a rule--preventing duplicate lottery entries--that makes the model in Hylland and Zeckhauser (1979) (HZ) inapplicable. We revisit HZ so that it accommodates the rule and propose a Tickets algorithm that always achieves an individually stable (IS) allocation. An IS allocation is closely related to an envy-free (EF) one but the two are not the same because an EF allocation may not exist. On the other hand, a strictly envy-free (SEF) allocation is the unique one that is IS and Pareto-optimal (PO). Furthermore, we construct a congestion game along the line of Milchtaich (1996) and find a pure Nash equilibrium (NE) using the Tickets algorithm, which, together with the Top Trading cycles (TTC) algorithm, achieves a pure strong NE (SNE) (Konishi et al. 1997). The SEF allocation induces a unique SNE of the game and the unique competitive equilibrium allocation of the Pseudo market in HZ. The Tickets algorithm always obtains the unique SNE for any order of players. Unlike our algorithm, the New York City affordable housing lottery is neither EF, IS, nor PO.