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We study stability of two-sided many-to-one matching in which firms' preferences for workers may exhibit complementarities. Although such preferences are known to jeopardize stability in a finite market, we show that a stable matching exists in a large market with a continuum of workers, provided that each firm's choice is convex and changes continuously as the set of available workers changes. We also study the existence and structure of stable matchings under preferences exhibiting substitutability and indifferences in a large market. Building on these results, we show that an approximately stable matching exists in large finite economies. We extend our framework to ensure a stable matching with desirable incentive and fairness properties in the presence of indifferences in firms' preferences.

Branch selection is a key decision in a cadet's military career. Cadets at USMA can increase their branch priorities at a fraction of slots by extending their service agreement. This real-life matching problem fills an important gap in the market design literature, providing strong empirical legitimacy to a series of elegant theoretical works on matching with contracts. Although priorities fail a key substitutes condition, the agent-optimal stable mechanism is well defined, and in contrast to the current USMA mechanism it is fair, stable, strategy-proof, and respects improvements in cadet priorities. Adoption of this mechanism benefits cadets and the Army. This new application shows that the matching with contracts model is practically relevant beyond traditional domains that satisfy the substitutes condition.

We introduce a two-sided, many-to-one matching with contracts model in which agents with unit demand match to branches that may have multiple slots available to accept contracts. Each slot has its own linear priority order over contracts; a branch chooses contracts by filling its slots sequentially, according to an order of precedence. We demonstrate that in these matching markets with slot-specific priorities, branches' choice functions may not satisfy the substitutability conditions typically crucial for matching with contracts. Despite this complication, we are able to show that stable outcomes exist in the slot-specific priorities framework and can be found by a cumulative offer mechanism that is strategy-proof and respects unambiguous improvements in priority.

The allocation and exchange of discrete resources, such as transplant organs, public housing, dormitory rooms, and many other resources for which agents have single-unit demand, is often conducted via direct mechanisms without monetary transfers. Incentive compatibility and efficiency are primary concerns when designing such mechanisms. We construct the full class of group strategy-proof and Pareto-efficient mechanisms and show that each of them can be implemented by endowing agents with control rights over resources. This new class, which we call trading cycles, contains new mechanisms as well as known mechanisms such as top trading cycles, serial dictatorships, and hierarchical exchange. We illustrate how one can use our construction to show what can and what cannot be achieved in a variety of allocation and exchange problems, and we provide an example in which the new trading-cycles mechanisms are more Lorenz equitable than all previously known mechanisms.

We characterize when a stable and strategy-proof mechanism is guaranteed to exist in the setting of many-to-one matching with contracts. We introduce three novel conditions—observable substitutability, observable size monotonicity, and non-manipulability via contractual terms—and show that when these conditions are satisfied, the cumulative offer mechanism is the unique mechanism that is stable and strategyproof (for workers). Moreover, we show that our three conditions are, in a sense, necessary: if the choice function of some firm fails any of our three conditions, we can construct unit-demand choice functions for the other firms such that no stable and strategy-proof mechanism exists. Thus, our results provide a rationale for the ubiquity of cumulative offer mechanisms in practice.

Distributional constraints are common features in many real matching markets, such as medical residency matching, school admissions, and teacher assignment. We develop a general theory of matching mechanisms under distributional constraints. We identify the necessary and sufficient condition on the constraint structure for the existence of a mechanism that is stable and strategy-proof for the individuals. Our proof exploits a connection between a matching problem under distributional constraints and a matching problem with contracts.

We propose a criterion of approximate incentive compatibility, strategy-proofness in the large (SP-L), and argue that it is a useful second-best to exact strategy-proofness (SP) for market design. Conceptually, SP-L requires that an agent who regards a mechanism’s “prices” as exogenous to her report—be they traditional prices as in an auction mechanism, or price-like statistics in an assignment or matching mechanism—has a dominant strategy to report truthfully. Mathematically, SP-L weakens SP in two ways: (1) truth-telling is required to be approximately optimal (within epsilon in a large enough market) rather than exactly optimal, and (2) incentive compatibility is evaluated ex interim, with respect to all full-support i.i.d. probability distributions of play, rather than ex post with respect to all possible realizations of play. This places SP-L in between the traditional notion of approximate SP, which evaluates incentives to manipulate ex post and as a result is too strong to obtain our main results in support of SP-L, and the traditional notion of approximate Bayes-Nash incentive compatibility, which, like SP-L, evaluates incentives to manipulate ex interim, but which imposes common knowledge and strategic sophistication assumptions that are often viewed as unrealistic.

Matching markets are often fragmented, organized at a small local level. While integration of matching markets may lead to welfare gains by expanding choice, it may also harm some market participants by increasing competition for the same resources. We show that every “good” mechanism fails the monotonicity requirement that no individuals be hurt by integration. Then we provide characterization results that identify conditions under which monotonicity becomes compatible with other desirable properties of matching mechanisms.

In the standard one‐to‐one agent‐object matching model, I consider a central matching authority that publicly announces a strategy‐proof mechanism and then initiates a matching. Following Akbarpour and Li (2020), the authority's commitment to the announced mechanism is limited to mechanisms rendering participants' observations indistinguishable from it. I call an announced mechanism transparent if any deviation from it would be detected. The main findings identify trade‐offs regarding transparency and other desirable properties: under stability or efficiency, strategy‐proof mechanisms are transparent if and only if they are dictatorial. At the same time, the agent‐proposing Deferred Acceptance (DA) mechanism is equivalent to committing to stability, while efficient mechanisms often fail to commit to efficiency. This transparency trade‐off between stability and efficiency persists when strategy‐proofness is guaranteed.

For division problems with single-peaked preferences, we show that all sequential allotment rules, a large subfamily of strategy-proof and efficient rules, are also obviously strategy-proof. Although obvious strategy-proofness is in general more restrictive than strategy-proofness, this is not the case in this setting.