Thayer Morrill

An alternative characterization of the deferred acceptance algorithm

Kojima and Manea (Econometrica 78(2):633–653, 2010) present two characterizations of when an allocation rule corresponds to the agent-proposing deferred acceptance (DA) algorithm for some substitutable priority rule of the objects being assigned. Building on their results we characterize when an allocation rule is outcome equivalent to the DA algorithm for every substitutable priority rule. In particular, an assignment rule satisfies mutual best (MB) if an agent is always assigned her most preferred object whenever she has the highest priority for it. This mild requirement is a necessary but far from sufficient condition for an assignment rule to be stable. We demonstrate that any allocation mechanism that satisfies MB along with non-wastefulness, population monotonicity and either individually rational monotonicity or weak Maskin monotonicity not only is a stable assignment mechanism but is equivalent to the agent proposing DA algorithm.

The roommates problem revisited

One of the oldest but least understood matching problems is Gale and Shapleys (1962) \roommates problem : is there a stable way to assign 2N students into N roommate pairs? Unlike the classic marriage problem or college admissions problem, there need not exist a stable solution to the roommates problem. However, the traditional notion of stability ignores the key physical constraint that roommates require a room, and it is therefore too restrictive. Recognition of the scarcity of rooms motivates replacing stability with Pareto optimality as the relevant solution concept. This paper proves that a Pareto optimal assignment always exists in the roommates problem, and it provides an efficient algorithm for finding a Pareto improvement starting from any status quo. In this way, the paper reframes a classic matching problem, which previously had no general solution, to become both solvable and economically more meaningful.

Making just school assignments

It is impossible for a mechanism to be strategyproof, Pareto efficient, and eliminate justified envy. However, little is known to what extent a strategyproof and efficient mechanism can limit justified envy. We define an assignment to be unjust if a student i is not assigned to a school a that she prefers to her own assignment, i has higher priority at a than some student j who is assigned to a, and none of the students ranked higher at a than i are dependent on j. We prove that Top Trading Cycles is the unique mechanism that is strategyproof, efficient, and just. This demonstrates that a strictly stronger notion of fairness than justness is either unachievable by a strategyproof and efficient mechanism or it is only achievable by TTC. We extend this characterization to the general case when schools may have arbitrary capacities by introducing the concept of reducibility.

Network formation under negative degree-based externalities

While a relationship in a social or business network should be mutually beneficial, it is ambiguous whether the relationship benefits or harms the rest of the network. This paper focuses on the situation where any new relationship imposes a negative externality on the rest of the network. We model this by assuming an agent’s payoff from a relationship is a decreasing function of the number of relationships the other agent maintains. We solve for the socially efficient and stable networks. While in general the two diverge, we demonstrate that they coincide when agents are able to make transfers to their partners.

Competitive equilibria in school assignment

Top Trading Cycles was originally developed as an elegant method for finding a competitive equilibrium of Shapley and Scarfs housing market. We extend the definition of a competitive equilibrium to the school assignment problem and show that there remains a profound relationship between Top Trading Cycles and a competitive equilibrium. Specifically, in every competitive equilibrium with weakly decreasing prices, the equilibrium assignment is unique and exactly corresponds to the Top Trading Cycles assignment. This provides a new way of interpreting the worth of a students priority at a given school. It also provides a new way of explaining Top Trading Cycles to students and a school board.