This paper studies a general model of matching with constraints. Observing that a stable matching typically does not exist, we focus on feasible, individually rational, and fair matchings. We characterize such matchings by fixed points of a certain function. Building on this result, we characterize the class of constraints on individual schools under which there exists a student-optimal fair matching, the matching that is the most preferred by every student among those satisfying the three desirable properties. We study the numerical relevance of our theory using data on government-organized daycare allocation.

Many real matching markets are subject to distributional constraints. These constraints often take the form of restrictions on the numbers of agents on one side of the market matched to certain subsets on the other side. Real-life examples include restrictions on regions in medical matching, academic master's programs in graduate admission, and state-financed seats for college admission. Motivated by these markets, we study design of matching mechanisms under distributional constraints. We show that existing matching mechanisms suffer from inefficiency and instability, and propose a mechanism that is better in terms of efficiency, stability, and incentives while respecting the distributional constraints.

A number of labor markets and student placement systems can be modeled as many-to-one matching markets. We analyze the scope for manipulation in many-to-one matching markets under the student-optimal stable mechanism when the number of participants is large. Under some regularity conditions, we show that the fraction of participants with incentives to misrepresent their preferences when others are truthful approaches zero as the market becomes large. With an additional condition, truthful reporting by every participant is an approximate equilibrium under the student-optimal stable mechanism in large markets.

In markets for indivisible resources such as workers and objects, subsidy and taxation for an agent may depend on the set of acquired resources and prices. This paper investigates how such transfer policies interfere with the substitutes condition, which is critical for market equilibrium existence and auction mechanism performance among other important issues. For environments where the condition holds in the absence of policy intervention, we investigate which transfer policies preserve the substitutes condition in various economically meaningful settings, establishing a series of characterisation theorems. For environments where the condition may fail without policy intervention, we examine how to use transfer policies to re-establish it, finding exactly when transfer policies based on scales are effective for that purpose. These results serve to inform policymakers, market designers, and market participants of how transfer policies may impact markets, so more informed decisions can be made.

Accommodating couples has been a long-standing issue in the design of centralized labor market clearinghouses for doctors and psychologists, because couples view pairs of jobs as complements. A stable matching may not exist when couples are present. This article’s main result is that a stable matching exists when there are relatively few couples and preference lists are sufficiently short relative to market size. We also discuss incentives in markets with couples. We relate these theoretical results to the job market for psychologists, in which stable matchings exist for all years of the data, despite the presence of couples.

We give two characterizations of Pareto optimality via “near” weighted utilitarian welfare maximization. One characterization sequentially maximizes utilitarian welfare functions using a finite sequence of nonnegative and eventually positive welfare weights. The other maximizes a utilitarian welfare function with a certain class of positive hyperreal weights. The social welfare ordering represented by these “near” weighted utilitarian welfare criteria is characterized by the standard axioms for weighted utilitarianism under a suitable weakening of the continuity axiom.