Essays on real life allocation problems

In the first chapter, we introduce a new matching model to mimic inter-college tuition exchange programs for dependents of faculty to attend other colleges tuition-free. Each participating college has to avoid being a net-exporter of students. Programs use decentralized markets making it difficult to achieve balance. We show that stable equilibria discourage net-exporting colleges from exchange. We introduce two-sided top-trading-cycles (2S-TTC) mechanism that is balanced-efficient, student-strategy-proof, and respecting priority bylaws regarding dependent eligibility. Moreover, it encourages exchange, since full participation is dominant strategy for colleges. We prove 2S-TTC is the unique mechanism fulfilling these objectives and introduce new student-strategy-proof mechanisms to achieve other objectives. In the second chapter, we consider a house allocation with existing tenants model in which each transaction is costly for the central authority, a housing office. We compare two widely studied mechanisms, deferred acceptance (DA) and top trading cycles (TTC), based on their costs for the housing offices. A mechanism in which more existing tenants are assigned to their current house is preferred for the housing offices due to the costs of moving. We show that although there is no dominance between the two mechanisms, DA has more desirable features in terms of the cost efficiency for the housing offices. Then we include the welfare of the housing office in the welfare analysis and redefine the Pareto efficiency notion. We show that every fair matching is Pareto efficient. Based on the extended Pareto efficiency definition, the DA mechanism is the unique Pareto efficient, fair, and strategy-proof mechanism. Finally, the third chapter characterizes the top trading cycles mechanism for the school choice problem. Schools may have multiple available seats to be assigned to students. For each school a strict priority ordering of students is determined by the school district. Each student has strict preference over the schools. We first define weaker forms of fairness, consistency and resource monotonicity. We show that the top trading cycles mechanism is the unique Pareto efficient and strategy-proof mechanism that satisfies the weaker forms of fairness, consistency and resource monotonicity. To our knowledge this is the first axiomatic approach to the top trading cycles mechanism in the school choice problem where schools have a capacity greater than one.Economic

When manipulations are harm[less]ful?

We say that a mechanism is harmless if no student can ever misreport his preferences so that he does not hurt but someone else. We consider a large class of rules which includes the Boston, the agent-proposing deferred acceptance, and the school-proposing deferred acceptance mechanisms (sDA). In this large class, the sDA happens to the unique harmless mechanism. We next provide two axiomatic characterizations of the sDA. First, the sDA is the unique stable, non-bossy, and independent of irrelevant student mechanism. The last axiom is a weak variant of consistency. As harmlessness implies non bossiness, the sDA is also the unique stable, harmless, and independent of irrelevant student mechanism

Strategy-proof size improvement: is it possible?

In unit-demand and multi-copy object allocation problems, we say that a mechanism size-wise dominates another mechanism if the latter never allocates more objects than the former does, while the converse is true for some problem. Our main result shows that no individually rational and strategy-proof mechanism size-wise dominates a non-wasteful, truncation-invariant, and extension-responding mechanism. As a corollary of this, the wellknown deferred-acceptance, serial dictatorship, and Boston mechanisms are not size-wise dominated by an individually rational and strategy-proof mechanism. We also show that whenever the number of agents does not exceed the total number of object copies, no group strategy-proof and ecient mechanism, such as top trading cycles mechanism, is size-wise dominated by an individually rational, weakly population-monotonic, and strategy-proof mechanism

Constrained stability in two-sided matching markets

In two-sided matching markets, not every worker-firm (doctor-hospital) pair can match with each other even if they would rather do so due to possible non-poaching contracts among firms or market specific regulations. Motivated by this observation, we introduce a new matching framework and a constrained stability notion, while emphasizing that the usual matching problem and Gale and Shapley, Am Math Mon 69:9–15 (1962)’s stability notion are realized as special cases of our formulation and the constrained stability notion. We first show that some fundamental properties of the stable matchings do not carry over to the constrained stable matchings. The worker-proposing deferred acceptance (DA) mechanism fails to be worker-optimal constrained stable, yet it is the unique constrained stable and strategy-proof mechanism. Lastly, we propose a worker-optimal constrained stable mechanism that also improves the workers’ welfare upon that under DA

Incompatibility between stability and consistency

Stability is a main concern in the school choice problem. However, it does not come for free. The literature shows that stability is incompatible with Pareto efficiency. Nevertheless, it has been ranked over Pareto efficiency by many school districts, and thereof, they are using stable mechanisms. In this note, we reveal another important cost of stability: ‘‘consistency’’, which is a robustness property that requires from a mechanism that whenever some students leave the problem along with their assignments, the remaining students’ assignments do not change after running the mechanism in the smaller problem. Consequently, we show that no stable mechanism is consistent

When preference misreporting is harm[less]ful?

In a school choice problem, we say that a mechanism is harmless if no student can ever misreport his preferences so that he is not hurt but someone else is. We consider two large classes of mechanisms, which include the Boston, the agent-proposing deferred acceptance, and the school-proposing deferred acceptance (sDA) mechanisms. Among all the rules in these two classes, the sDA is the unique harmless mechanism. We next provide two axiomatic characterizations of the sDA. First, the sDA is the unique stable, non-bossy, and "independent of an irrelevant student mechanism". The last axiom requires that the outcome does not depend on the presence of a student who prefers being unassigned to any school. As harmlessness implies non-bossiness, the sDA is also the unique stable, harmless, and independent of an irrelevant student mechanism. To our knowledge, these axiomatizations as well as the well-known Gale and Shapley's (1962), which reveals that the sDA is the student-pessimal stable mechanism, are the only characterizations of the sDA