A textbook chapter on modeling choice behavior and designing institutional choice functions for matching and market design.

For a model of fractional matching, interpreted as probabilistic matching, together with the allocation of non-negative amounts of money, we show that strategy-proofness, ex post Pareto efficiency of the matching, and a weak version of ex ante individual rationality are incompatible when each agent's utility is a linear function of both their fractional assignment and money. We identify some avenues to escape this impossibility.

We consider a model where each agent has an outside option of privately known value. At a given allocation, we call the set of agents who do not exercise their outside options the "participants". We show that one strategy-proof and individually rational mechanism weakly Pareto-improves another if and only if, at each preference profile, it weakly expands (in terms of set inclusion) the set of participants. Corollaries include: a sufficient condition for a mechanism to be on the Pareto-efficient frontier of strategy-proof mechanisms; uniqueness of strategy-proof Pareto-improvements under true preferences over certain normatively meaningful benchmark allocation rules; and a characterization of the pivotal mechanism.

For combinatorial choice problems, I show that the Irrelevance of Rejected Items condition is equivalent to the Weak Axiom of Revealed Preference (WARP), and is necessary and sufficient for the existence of a complete, reflexive and antisymmetric rationalization of a combinatorial choice function. I also show the equivalence of WARP to path independence and to other classical choice conditions when the choice domain is combinatorial.

We introduce a general concept of layered computation model, of which neural networks are a particular example, and combine tools of topological dynamics and model theory to study asymptotics of such models. We prove that, as the number of layers of a computation grows, the computation reaches a state of "deep equilibrium" which amounts to a single, self-referential layer. After proving the existence of deep equilibria under fairly general hypotheses, we characterize their computability.

In the problem of allocating indivisible objects via lottery, a social planner often knows only agents' ordinal preferences over objects, but not their complete preferences over lotteries. Such an informationally constrained planner cannot distinguish between different utility profiles that induce the same rankings over the objects. In this context, we ask what it means to adjudge an allocation as efficient.
We introduce the concept of unambiguous efficiency, which guarantees no further Pareto improvement regardless of how agents' ordinal preferences extend to lotteries. We compare this concept with the predominant formulation of efficiency in the random allocation literature and explore some structural properties. As an application to mechanism design, we characterize the class of efficient and strategy-proof ordinal mechanisms that satisfy certain regularity conditions.

A rule is pairwise strategy-proof if groups of size one and two never have an incentive to manipulate. When agents have strict preferences over their own outcomes, I show that pairwise strategy-proofness even eliminates incentives for any group of agents to manipulate, therefore implying group strategy-proofness. It is also equivalent to Maskin monotonicity. I obtain the equivalence results assuming preference domains satisfy a richness condition. Decomposing richness into two parts, I explore what brings about the equivalence. The results apply to school choice and matching with contracts, indivisible object allocation, and economies with private or public goods with single-peaked preferences.

We consider a general model of indivisible goods allocation with choice-based priorities, as well as the special case of school choice. Stability is the main normative consideration for such problems. However, depending on the priority structure, it may be incompatible with Pareto-efficiency. We propose a new criterion: an allocation is stable-dominating if it weakly Pareto-improves some stable allocation. We show that if an allocation Pareto-improves on a particular non-wasteful (and therefore stable) allocation, then it matches the same agents to some object, and matches the same number of agents to each object. This is much like the conclusion of the Rural Hospitals Theorem. In fact, we connect the existence of a stable-dominating and strategy-proof rule and the Rural Hospitals Theorem on one hand with the existence of the agent-optimal stable-dominating rule on the other. For the school choice model, we also characterize the weak priority structures that ensure every Pareto-efficient and stable-dominating rule is stable. We also show that if a rule is Pareto-efficient, stable-dominating and strategy-proof, then it is actually stable. We also show an alternative version of this result where we replace Pareto-efficiency with a mild regularity condition.

Many real-life problems involve the matching of talented individuals to institutions such as firms, hospitals, or schools, where these institutions are simply treated as individual agents. In this paper, I study many-to-one matching with contracts that incorporates a theory of choice of institutions, which are aggregate actors, composed of divisions that are enjoined by an institutional governance structure (or mechanism). Conflicts over contracts between divisions of an institution are resolved by the institutional governance structure, whereas conflicts between divisions across institutions are resolved, as is typically the case, by talents' preferences. Noting that hierarchies are a common organizational structure in institutions, I offer an explanation of this fact as an application of the model, where stability is a prerequisite for the persistence of organizational structures. I show that stable market outcomes exist whenever institutional governance is hierarchical and divisions consider contracts to be bilaterally substitutable. In contrast, when governance in institutions is non-hierarchical, stable outcomes may not exist. Since market stability does not provide an impetus for reorganization, the persistence of markets with hierarchical institutions can thus be rationalized. Hierarchies in institutions also have the attractive incentive property that in a take-it-or-leave-it bargaining game with talents making offers to institutions, the choice problem for divisions is straightforward and realized market outcomes are pairwise stable, and stable when divisions have substitutable preferences.

We investigate the impact of a Deferred Retirement Option Plan (DROP) on the age of retirement of employees covered by defined benefit pension plans provided by the City of Philadelphia. We show that the program results in significant and substantial increases in the age of retirement: 1.25 years on average for municipal employees. Employees make use of the program in ways that maximize the expected present value of their pension benefits, with municipal employees entering the program an average 2.1 years before the age at which they would otherwise have retired. Consequently, the program results in a substantial increase in pension cost. We estimate that the program has cost the city around $258 million over the period to 31 December 2009. We construct an indicator of employee quality and find that some classes of high-quality employees are disproportionately likely to delay retirement as a result of the program.

For problems of school choice, I identify conditions on priorities that ensure consistency of: some stable rule; every stable rule; the student-pessimal stable rule.

I examine the relationship between various concepts of stability and the core in choice-based model of matching with contracts, under a range of weakened substitutability conditions. I show the following results. The set of group stable outcomes and the set of core outcomes are identical as long as choice satisfies the irrelevance of rejected contracts (IRC). The set of pairwise stable outcomes is strictly larger than the set of group stable outcomes even when choice satisfies both IRC and bilateral substitutability. These sets are identical with the additional assumption of Pareto separability. I also discuss the role, in preference-based notions of stability, of information that cannot be revealed simply through observations of combinatorial choice.

I study stability of matching in two-sided markets with non-transferable utility, and propose a notion of stability that reflects a requirement that blocks be robust to partial execution. To execute a particular block in which a set of agents stand to gain, agents may have to change multiple independent relationships. The block is only partially executed if some agents do not follow the block's prescription, which may leave some blocking agents worse of than in the original matching. I define a block of a matching to be safe if the gain from participating in the block is robust to partial execution. I show that safe-setwise stable matchings, those that admit no safe blocks, exist for markets where preferences can feature some types of complementarities. In particular, I identify that conditional complementarities that are symmetric do not pose a problem for existence. With many previous notions of stability, some such markets admit no stable matching. For such markets with a decentralized random meeting process, I then show that from any initial matching a safe-setwise stable matching is eventually reached. For centralized matching, I provide a sufficient, and in a sense necessary, condition on preferences for a deferred acceptance procedure to produce a safe-setwise stable matching.