Three Essays on Matching with Contracts

Thesis advisor: Tayfun SonmezThis dissertation consists of three theoretical essays. In all essays matching with contracts is a key factor. The first essay tries to explain effects of choosing primitives of the model and irrelevance of rejected contracts condition on some key existence theorems and results in matching with contracts literature. The second essay analyzes the properties of cumulative offer algorithm and presents an application of matching with contracts. It studies the achievability of responsive choices under a constrained setup. The last essay presents a new market design application of program-student matching where affirmative action policies are effective. The first essay develops a hospital-doctor many-to-one matching with contracts model. Doctor preferences over contracts are part of primitive of the model. The other primitive of the model, our first essay suggests, hospital choice functions on sets of contracts. The first essay shows that if choice functions of hospitals are primitives of the model, then existence theorems used in many papers do not hold even when they satisfy strongest conditions. As a remedy, we introduced Irrelevance of Rejected Contracts (IRC) which guarantees stability if it is satisfied along with one substitutes condition. Next, we show the relation between IRC and law of aggregate demand (LAD) conditions. Since LAD is satisfied by many application naturally, many models satisfying LAD and the strongest substitutes conditions are immune to our criticism. On the other hand, many of the new and exiting applications satisfy only weakened substitutes condition. Therefore, assuming IRC explicitly does not only make their proofs accurate and also close the gap between theory and application. The second chapter studies properties of cumulative offer algorithm under weakened substitutes condition. In this part we showed that in many-to-one matching with contracts problems order of proposals of COA does not change the outcome, under bilateral substitutes and IRC conditions. Also, bilateral substitutes and IRC conditions make COA equivalent to generalized deferred acceptance algorithm which produces the outcome in fewer steps. This chapter also presents a new application area of matching with contracts. We used cadet-branch matching problem in USMA. In this application our main objective is, for a given branch, increasing cadet quality without giving up useful properties of allocation mechanism, such as stability and strategy-proofness. The third essay studies a college admission with affirmative action problem. With this application, for the first time in the literature, we presented an affirmative action problem where students need to claim privilege if they want to be subject to affirmative action. We analyzed the current system and showed that current guideline is unfair and causes incentive compatibility issues. Also we showed that it fails to satisfy affirmative action requirements described in affirmative action law. To solve these problems with the current system, we introduced a new choice function which is fair, respects affirmative action requirements and makes student optimal stable allocation stable and incentive compatible when used in conjunction with generalized deferred acceptance algorithm.Thesis (PhD) — Boston College, 2014.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics

Contracts vs. Salaries in Matching

Firms and workers may sign complex contracts that govern many aspects of their interactions. I show that when firms regard contracts as substitutes, bargaining over contracts can be understood as bargaining only over wages. Substitutes is the assumption commonly used to guarantee the existence of stable matchings of workers and firms

Efficiency in Matching Markets with Regional Caps: The Case of the Japan Residency Matching Program

In an attempt to increase the placement of medical residents to rural hospitals, the Japanese government recently introduced "regional caps" which restrict the total number of residents matched within each region of the country. The government modified the deferred acceptance mechanism incorporating the regional caps. This paper shows that the current mechanism may result in avoidable ineffciency and instability and proposes a better mechanism that improves upon it in terms of effciency and stability while meeting the regional caps. More broadly, the paper contributes to the general research agenda of matching and market design to address practical problems.medical residency matching, regional caps, the rural hospital theorem, sta- bility, strategy-proofness, matching with contracts

Dynamic Reserves in Matching Markets

We study a school choice problem under affirmative action policies where authorities reserve a certain fraction of the slots at each school for specific student groups, and where students have preferences not only over the schools they are matched to but also the type of slots they receive. Such reservation policies might cause waste in instances of low demand from some student groups. To propose a solution to this issue, we construct a family of choice functions, dynamic reserves choice functions, for schools that respect within-group fairness and allow the transfer of otherwise vacant slots from low-demand groups to high-demand groups. We propose the cumulative offer mechanism (COM) as an allocation rule where each school uses a dynamic reserves choice function and show that it is stable with respect to schools' choice functions, is strategy-proof, and respects improvements. Furthermore, we show that transferring more of the otherwise vacant slots leads to strategy-proof Pareto improvement under the COM

Strategyproof matching with regional minimum and maximum quotas

This paper considers matching problems with individual/regional minimum/maximum quotas. Although such quotas are relevant in many real-world settings, there is a lack of strategyproof mechanisms that take such quotas into account. We first show that without any restrictions on the regional structure, checking the existence of a feasible matching that satisfies all quotas is NP-complete. Then, assuming that regions have a hierarchical structure (i.e., a tree), we show that checking the existence of a feasible matching can be done in time linear in the number of regions. We develop two strategyproof matching mechanisms based on the Deferred Acceptance mechanism (DA), which we call Priority List based Deferred Acceptance with Regional minimum and maximum Quotas (PLDA-RQ) and Round-robin Selection Deferred Acceptance with Regional minimum and maximum Quotas (RSDA-RQ). When regional quotas are imposed, a stable matching may no longer exist since fairness and nonwastefulness, which compose stability, are incompatible. We show that both mechanisms are fair. As a result, they are inevitably wasteful. We show that the two mechanisms satisfy different versions of nonwastefulness respectively; each is weaker than the original nonwastefulness. Moreover, we compare our mechanisms with an artificial cap mechanism via simulation experiments, which illustrate that they have a clear advantage in terms of nonwastefulness and student welfare

Matching with Couples: a Multidisciplinary Survey

This survey deals with two-sided matching markets where one set of agents (workers/residents) has to be matched with another set of agents (firms/hospitals). We first give a short overview of a selection of classical results. Then, we review recent contributions to a complex and representative case of matching with complementarities, namely matching markets with couples. We discuss contributions from computer scientists, economists, and game theorists.matching; couples; stability; computational complexity; incentive compatibility; restricted domains; large markets

Stable Many-to-Many Matchings with Contracts

We consider several notions of setwise stability for many-to-many matching markets with contracts and provide an analysis of the relations between the resulting sets of stable allocations for general, substitutable, and strongly substitutable preferences. Apart from obtaining "set inclusion results" on all three domains, we introduce weak setwise stability as a new stability concept and prove that for substitutable preferences the set of pairwise stable matchings is nonempty and coincides with the set of weakly setwise stable matchings. For strongly substitutable preferences the set of pairwise stable matchings coincides with the set of setwise stable matchings.Many-to-Many Matching, Matching with Contracts, Pairwise Stability, Setwise Stability.

A Many-to-Many 'Rural Hospital Theorem'

We show that the full version of the so-called 'rural hospital theorem' (Roth, 1986) generalizes to many-to-many matching where agents on both sides of the market have separable and substitutable preferences.matching, many-to-many, stability, rural hospital theorem.