Top Trading Cycles Research Articles

Affirmative action under common school priorities: The top trading cycles mechanism case

In school choice, there are two types of extensively used affirmative action policies based on majority quotas and minority reserves, respectively. This paper investigates the responsiveness of the top trading cycles school choice mechanism to these two types of affirmative action policies when all schools have the same priority order over students. Under the common school priority (within student type) condition, we show that the top trading cycles mechanism is minimally responsive to the minority reserves type of affirmative action policy, while a stronger majority quotas type of affirmative action policy may yield a Pareto inferior assignment for the minority.

OBVIOUSLY STRATEGY‐PROOF IMPLEMENTATION OF TOP TRADING CYCLES

AbstractAlthough there is a rich theoretical literature extolling the virtues of the top trading cycles (TTCs) mechanism, it is rarely used in practice. Anecdotal evidence suggests that one possible explanation is that TTC is difficult for participants to understand. This article formalizes this intuition by asking whether it is possible to implement TTC in an obviously strategy‐proof (OSP) way. I identify an acyclicity condition that is both necessary and sufficient for OSP implementation of TTC. The condition is unlikely to hold in most applications, which may explain why TTC is rarely used, despite its many appealing theoretical properties.

Justified-envy Minimal Mechanisms in School Choice

We study priority-based matching markets with public and private endowments. We show that efficiency-adjusted deferred acceptance (EADA) is justified-envy minimal in the class of efficient mechanisms, while Top trading cycles (TTC) and other popular mechanisms are not. Our findings highlight EADA as an interesting alternative to TTC in the context of transplantation-organ markets. Restricting attention to strategyproof mechanisms, we show that TTC is justified-envy minimal, strengthening the result of Abdulkadiroglu et al. (2020).

The Crawler: Two Equivalence Results for Object (Re)Allocation Problems When Preferences are Single-Peaked

For object reallocation problems, if preferences are strict but otherwise unrestricted, the Top Trading Cycle rule (TTC) is the leading rule: It is the only rule satisfying efficiency, the endowment lower bound, and strategy-proofness; moreover, TTC coincides with the core. However, on the subdomain of single-peaked preferences, Bade (2019a) defines a new rule, the “crawler”, which also satisfies the first three properties. Our first theorem states that the crawler and a naturally defined “dual” rule are actually the same. Next, for object allocation problems, we define a probabilistic version of the crawler by choosing an endowment profile at random according to a uniform distribution, and applying the original definition. Our second theorem states that this rule is the same as the “random priority rule” which, as proved by Knuth (1996) and Abdulkadiroglu and Sonmez (1998), is equivalent to the “core from random endowments”.

Endowments, Exclusion, and Exchange

We propose a new solution for discrete exchange economies and resource‐allocation problems, the exclusion core. The exclusion core rests upon a foundational idea in the legal understanding of property, the right to exclude others. By reinterpreting endowments as a distribution of exclusion rights, rather than as bundles of goods, our analysis extends to economies with qualified property rights, joint ownership, and social hierarchies. The exclusion core is characterized by a generalized top trading cycle algorithm in a large class of economies, including those featuring private, public, and mixed ownership. It is neither weaker nor stronger than the strong core.

Matching with single-peaked preferences

The crawler is a new efficient, strategyproof, and individually rational mechanism for housing markets with single-peaked preferences. In a housing market each agent is endowed with exactly one house. These houses are ordered – by their size for example – and all agents preferences are single-peaked with respect to that order. The crawler screens agents in order of their houses' sizes, starting with the smallest. The first agent who does not want to move to a larger house is matched with his most preferred house. Agents who currently occupy houses sized between this agent's original and chosen houses “crawl” to the next largest unmatched house. This process is repeated until all agents are matched. The crawler is easier to understand than Gale's top trading cycles and can be extended to allow for indifferences.

Strategy-Proofness of Stochastic Assignment Mechanisms

This paper compares two prominent stochastic assignment mechanisms in the laboratory: Random serial dictatorship (RSD) and top trading cycles with random endowments (TTC). In standard theory, both mechanisms are strategy-proof and Pareto-efficient for the house allocation problem without endowments. In the experiment, RSD outperforms TTC. This can be attributed to more dominant strategy play under RSD. The behavioral theory of obvious strategy-proofness can partly explain this difference in dominant strategy play. Generally, subjects with extremely high and low levels of contingent reasoning play their dominant strategies. These results suggest that one strategy-proof mechanism may outperform another one if individuals are boundedly rational.

Strategy-Proofness of Stochastic Assignment Mechanisms

This paper seeks to identify which algorithm to employ in a situation where goods are distributed to individuals without using money, while treating everyone equally and respecting each individual's preferences. I compare two stochastic assignment mechanisms: Random serial dictatorship (RSD) and top trading cycles with random endowments (TTC). In standard theory, both algorithms are strategy-proof and yield the optimal result. In the experiment, RSD outperforms TTC. This can be attributed to a more dominant strategy play under RSD. Generally, subjects with extremely high and low levels of contingent reasoning play their dominant strategies. These results suggest that one optimal algorithm may outperform another one if individuals are boundedly rational.

Stable and Envy-Free Lottery Allocations for Affordable Housing

Affordable housing lotteries often enforce a rule--preventing duplicate lottery entries--that makes the model in Hylland and Zeckhauser (1979) (HZ) inapplicable. We revisit HZ so that it accommodates the rule and propose a Tickets algorithm that always achieves an individually stable (IS) allocation. An IS allocation is closely related to an envy-free (EF) one but the two are not the same because an EF allocation may not exist. On the other hand, a strictly envy-free (SEF) allocation is the unique one that is IS and Pareto-optimal (PO). Furthermore, we construct a congestion game along the line of Milchtaich (1996) and find a pure Nash equilibrium (NE) using the Tickets algorithm, which, together with the Top Trading cycles (TTC) algorithm, achieves a pure strong NE (SNE) (Konishi et al. 1997). The SEF allocation induces a unique SNE of the game and the unique competitive equilibrium allocation of the Pseudo market in HZ. The Tickets algorithm always obtains the unique SNE for any order of players. Unlike our algorithm, the New York City affordable housing lottery is neither EF, IS, nor PO.

IoT-Enabled Parking Space Sharing and Allocation Mechanisms

This paper is among the first proposing an integrated auction and market design method for the parking space sharing and allocation problem. Drivers (agents) who fail to exchange their own parking spaces can then rent them to the platform. The platform receives private parking spaces from agents and manages some public parking spaces. We first develop the urban parking management cloud platform through Internet of Things. Based on this systemic framework, parking spaces are shared among agents via a price-compatible top trading cycles and chains (PC-TTCCs) mechanism and the platform's parking spaces are reassigned via a one-sided Vickrey-Clarke-Groves (O-VCG) auction. Both the PC-TTCC mechanism with rule e (PC-TTCC [e]) and O-VCG auction are effective in terms of strategy-proofness and (allocative or Pareto) efficiency. In the PC-TTCC [e] mechanism, the platform's payment rule used in private parking space sharing is determined based on historical O-VCG auction prices. Our experimental results further show that the proposed mechanism results in system profitability of 20%-30% and ex post budget balance for the platform.

Implementation of School-Enterprise Two-Sided Matching Based on Pearson and Top Trading Cycle

Due to a large number of graduates and employment information, two-sided matching is increasingly used in employment. However, the general recruitment websites are not rich in student data, so their recommendation will not be very accurate. In order to improve the efficiency of recommendation, a recommendation system based on Pearson and Top Trading Cycle is designed by using the real graduate data of our school. Through the application of Pearson and TTC (Top Trading Cycle), the stable match between the students and the enterprise is realized. We use Pearson similarity algorithm to calculate the similarity between students and sort them from large to small, then we arrange the similarity of order as user preferences in TTC algorithm. Take each of the students as the starting point and the end point, the similarity as the evaluation criteria, and the rest of the students to form a ring. We believe that all the enterprises of the past students in the ring area are the recommended enterprises to be allocated to the students in the ring area, and the students who are to be assigned are recommended students recommended by them.

THE EQUITABLE TOP TRADING CYCLES MECHANISM FOR SCHOOL CHOICE

AbstractA particular adaptation of Gale's top trading cycles (TTC) procedure applied to school choice, the so‐called TTC mechanism, has attracted much attention both in theory and practice due to its superior efficiency and incentive features. We discuss and introduce alternative adaptations of Gale's original procedure that can offer improvements over TTC in terms of equity, along with various other distributional considerations. Instead of giving all the trading power to those students with the highest priority for a school, we argue for the distribution of the trading rights of all slots of each school: This allows them to trade in a thick market where additional constraints can be accommodated by choosing an appropriate pointing rule. We propose a particular mechanism of this kind, the equitable top trading cycle (ETTC) mechanism, which is also Pareto efficient and group strategy‐proof just like TTC and eliminates avoidable justified envy situations. ETTC generates significantly fewer justified envy situations than TTC both in simulations and the lab.

Outcome equivalence in school choice with reciprocal preferences

We show that modifications of the Boston school choice mechanism, the student proposing deferred acceptance algorithm and the top trading cycles algorithm where the colleges’ priorities are adjusted according to students’ preferences in a “first preferences first” manner generate the same outcome. This outcome coincides with the BM outcome under original priorities. As a result, the question which of the three algorithms is chosen becomes immaterial.

Endowments-swapping-proof house allocation

We consider house (re)allocation problems (Shapley and Scarf, 1974) with strict preferences. We are concerned with the possibility that a pair of agents may gain by swapping their endowments before the operation of the chosen rule. A rule is called endowments-swapping-proof if it is immune to this kind of manipulation. Our main result is that the top trading cycles rule is the only rule that satisfies individual rationality, strategy-proofness, and endowments-swapping-proofness.

Preference profiles for efficiency, fairness, and consistency in school choice problems

We study the school choice problem in which a school district assigns school seats to students. There has been a long debate over the three best-known rules for this problem: the deferred acceptance rule (DA), the top-trading cycles rule (TTC), and the immediate acceptance rule (IA). We evaluate these rules by investigating how often they satisfy three central requirements, efficiency, fairness, and consistency. We compare the restricted domains of students’ preferences on which each rule satisfies these requirements. From the containment relations between them, we show that DA performs better than IA, which itself performs better than TTC in terms of efficiency and fairness. If we consider consistency instead, IA performs better than DA, which itself performs better than TTC.

Endowments-Swapping-Proof House Allocation

We consider house (re)allocation problems (Shapley and Scarf, 1974) with strict preferences. We are concerned with the possibility that a pair of agents may gain by swapping their endowments before the operation of the chosen rule. A rule is called endowments-swapping-proof if it is immune to this kind of manipulation. Our main result is that the top trading cycles rule is the only rule that satisfies individual rationality, strategy-proofness, and endowments-swapping-proofness.

LEARNING MATTERS: REAPPRAISING OBJECT ALLOCATION RULES WHEN AGENTS STRATEGICALLY INVESTIGATE

AbstractIndividuals form preferences through search, interviews, discussion, and investigation. In a stylized object allocation model, we characterize the equilibrium learning strategies induced by different allocation rules and trace their welfare consequences. Our analysis reveals that top trading cycles rules dominate serial priority rules under inequality‐averse measures of social welfare.

Teacher Assignment and Minimal Envy Mechanisms

We investigate the teacher assignment problem where one has to assign both newly tenured teachers without an initial assignment and tenured teachers with an initial position. Each school has a priority ordering over the teachers. In this framework, there is no mechanism that is both i) individually rational (IR), i.e., that assigns each teacher to a school that he/she weakly prefers to his/her initial one, and ii) stable, i.e., there are no blocking pairs: teachers and schools that are not assigned together and would prefer to be. A mechanism has minimal envy if there is no other mechanism that always leads to a the set of blocking pairs included in the one of the former mechanism. We show that the modified Deferred Acceptance mechanism (DA*) is a minimal envy mechanism in the set of IR and strategy-proof mechanisms. We also show that, in a one-to-one setting, an extension of the Top Trading Cycle (TTC*) mechanism is also a minimal envy mechanism in the set of IR, strategy-proof and Pareto-efficient mechanisms. These results generalize the results in the school choice literature concerning the standard DA and TTC mechanisms.

Two further impossibility results on responsive affirmative action in school choice

This paper investigates the responsiveness of priority-based affirmative action in school choice. We consider two efficiency-improved mechanisms—Kesten’s EADAM (efficiency-adjusted deferred acceptance mechanism) and DA-TTC mechanism (running deferred acceptance and then running top trading cycles algorithm). We show that neither EADAM nor DA-TTC is minimally responsive to the priority-based affirmative action policy. That is, under EADAM or DA-TTC, there are market situations in which a stronger affirmative action policy may result in a Pareto inferior assignment for the minority students.

Stable two-sided matching of slot allocation in airport collaborative decision making by top trading cycles mechanism

The problem of capacity shortage in some airports needs to be dealt with sustainable solutions including a more efficient use of the existing runway slots at the airports. The Collaborative Decision Making (CDM) is an important approach applied to Air Traffic Management (ATM) to achieve this efficient use of the slots allocation. Using the Matching approach for two-sided markets of Game theory, the Top Trading Cycle CDM (TTC-CDM) algorithm developed in this research is an extension of the CDM approach aggregating the Ground Delay Program (GDP) of the air sector. The paper compared the developed TTC-CDM model to the existing models such as the conventional Compression algorithm in CDM, the Trade Cycle algorithm and the Deferred Acceptance CDM (DA-CDM) model to evaluate the performance of the proposed model. Through a case study, the results show the effective application of TTC-CDM model to slot allocation in ATM and also presents the advantage of considering the preferences of airport managers beside ATC controllers and airlines in the decision processing.