Coalitions and Cliques in the School Choice Problem

The school choice mechanism design problem focuses on assignment mechanisms matching students to public schools in a given school district. The well-known Gale Shapley Student Optimal Stable Matching Mechanism (SOSM) is the most efficient stable mechanism proposed so far as a solution to this problem. However its inefficiency is well-documented, and recently the Efficiency Adjusted Deferred Acceptance Mechanism (EADAM) was proposed as a remedy for this weakness. In this note we describe two related adjustments to SOSM with the intention to address the same inefficiency issue. In one we create possibly artificial coalitions among students where some students modify their preference profiles in order to improve the outcome for some other students. Our second approach involves trading cliques among students where those involved improve their assignments by waiving some of their priorities. The coalition method yields the EADAM outcome among other Pareto dominations of the SOSM outcome, while the clique method yields all possible Pareto optimal Pareto dominations of SOSM. The clique method furthermore incorporates a natural solution to the problem of breaking possible ties within preference and priority profiles. We discuss the practical implications and limitations of our approach in the final section of the article

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

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

Group Strategyproof Pareto-Stable Marriage with Indifferences via the Generalized Assignment Game

We study the variant of the stable marriage problem in which the preferences of the agents are allowed to include indifferences. We present a mechanism for producing Pareto-stable matchings in stable marriage markets with indifferences that is group strategyproof for one side of the market. Our key technique involves modeling the stable marriage market as a generalized assignment game. We also show that our mechanism can be implemented efficiently. These results can be extended to the college admissions problem with indifferences

Stable Roommate Problem with Diversity Preferences

In the multidimensional stable roommate problem, agents have to be allocated to rooms and have preferences over sets of potential roommates. We study the complexity of finding good allocations of agents to rooms under the assumption that agents have diversity preferences [Bredereck et al., 2019]: each agent belongs to one of the two types (e.g., juniors and seniors, artists and engineers), and agents' preferences over rooms depend solely on the fraction of agents of their own type among their potential roommates. We consider various solution concepts for this setting, such as core and exchange stability, Pareto optimality and envy-freeness. On the negative side, we prove that envy-free, core stable or (strongly) exchange stable outcomes may fail to exist and that the associated decision problems are NP-complete. On the positive side, we show that these problems are in FPT with respect to the room size, which is not the case for the general stable roommate problem. Moreover, for the classic setting with rooms of size two, we present a linear-time algorithm that computes an outcome that is core and exchange stable as well as Pareto optimal. Many of our results for the stable roommate problem extend to the stable marriage problem.Comment: accepted to IJCAI'2

One for all, all for one---von Neumann, Wald, Rawls, and Pareto

Applications of the maximin criterion extend beyond economics to statistics, computer science, politics, and operations research. However, the maximin criterion---be it von Neumann's, Wald's, or Rawls'---draws fierce criticism due to its extremely pessimistic stance. I propose a novel concept, dubbed the optimin criterion, which is based on (Pareto) optimizing the worst-case payoffs of tacit agreements. The optimin criterion generalizes and unifies results in various fields: It not only coincides with (i) Wald's statistical decision-making criterion when Nature is antagonistic, (ii) the core in cooperative games when the core is nonempty, though it exists even if the core is empty, but it also generalizes (iii) Nash equilibrium in $n$-person constant-sum games, (iv) stable matchings in matching models, and (v) competitive equilibrium in the Arrow-Debreu economy. Moreover, every Nash equilibrium satisfies the optimin criterion in an auxiliary game

Re-Reforming the Bostonian System: A Novel Approach to the Schooling Problem

This paper proposes the notion of E-stability to conciliate Pareto efficiency and fairness. We propose the use of a centralized procedure, the Exchanging Places Mechanism. It endows students a position according with the Gale and Shapley students optimal stable matching as tentative allocation and allows the student to trade their positions. We show that the final allocation is E-stable, i.e. efficient, fair and immune to any justifiable objection that students can formulate.School allocation problem, Pareto efficient matching