Top Trading Cycles Research Articles

Job Rotation: Core and Mechanism

Yu and Zhang (2020) propose a job rotation model to study rotation schemes that widely exist in real life. In the model agents’ rights to consume own endowments are restricted, but their rights to trade endowments are unrestricted. This poses an interesting contrast with the housing market model of Shapley and Scarf (1974). Recognizing restricted property right in the job rotation model, in this paper we define a solution called the rotation core in the same flavor of the core in the housing market model. It turns out that the rotation core is nonempty, single-valued, and found by a strategy-proof variant of Top Trading Cycle (TTC). This generalizes the well-known result that TTC finds the unique core allocation in the housing market model.

Influence in private-goods allocation

We reinterpret the ‘bossiness’ of a private-goods allocation rule (Satterthwaite and Sonnenschein, 1981) as the ability of an agent to ‘influence’ another’s welfare with no change to her own welfare. In applications where non-bossiness is not possible, we propose simple conditions on (1) which agents may have influence (acyclicity and preservation), and (2) the welfare consequences of influence (positivity and oppositeness). We apply these conditions to three well-known bossy rules: the ‘Vickrey rule’ in single-object auctions (Vickrey, 1961) (acyclic, positive), the ‘doctor-optimal stable rule’ in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and ‘generalised absorbing top-trading cycles (GATTC) rules’ in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies weak group-strategy-proofness (requires acyclicity and either positivity or preservation), weak Maskin monotonicity (acyclicity and positivity) and Pareto-efficiency (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the efficiency-adjusted deferred acceptance mechanism in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.

Joint relay assignment and energy‐efficiency maximization in energy‐harvesting downlink/uplink clustered nonorthogonal multiple‐access networks

AbstractIn this article, the problem of joint relay assignment and energy‐efficiency maximization (J‐RA‐EE‐MAX) in energy‐harvesting downlink (DL) and uplink (UL) clustered nonorthogonal multiple‐access (NOMA) networks is considered. Specifically, the aim is to perform relay assignment to user clusters in the DL and UL directions, while simultaneously maximizing energy efficiency (EE) over each relay via multiobjective optimization, and satisfying users' quality‐of‐service (QoS) constraints. However, problem J‐RA‐EE‐MAX happens to be nonconvex in each link direction, and thus is computationally prohibitive. Alternatively, a low‐complexity solution procedure is devised to solve problem J‐RA‐EE‐MAX in each link direction by: (i) optimally solving the energy‐efficiency maximizing power allocation (EE‐MAX‐PA) for each (user cluster, relay) combination to construct the relays' preference profile, and (ii) performing relay assignment via Gale's top trading cycles (TTC) matching mechanism. In particular, the optimal solution of the EE‐MAX‐PA problem is obtained by transforming it into a concave maximization problem, while the TTC mechanism is executed in linear time‐complexity so as to obtain a stable relay assignment. Simulation results are presented to validate the proposed solution procedure, which is shown to yield comparable user cluster sum‐rate and relay EE to the J‐RA‐EE‐MAX scheme, and superior to other schemes in both link directions, however, at lower computational complexity, while satisfying users' QoS constraints. Finally, this work sheds light on the importance of decoupling relay assignment in the DL/UL directions, which significantly improves user cluster sum‐rate and relay EE in comparison to coupled relay assignment schemes, and thus serves the requirements of 5G networks and beyond.

Influence in Private-Goods Allocation

We reinterpret the `bossiness' of a private-goods allocation rule (Satterthwaite and Sonneschein, 1981) as the ability of an agent to `influence' another's welfare with no change to her own welfare. We propose simple conditions on (1) which agents may have influence (`acyclicity' and `preservation'), and (2) the welfare consequences of influence (`positivity' and `oppositeness'). We apply these conditions to three well-known bossy rules: the `Vickrey rule' in single-object auctions (Vickrey, 1961) (acyclic, positive), the `doctor-optimal stable rule' in matching with contracts (Hatfield and Milgrom, 2005) (acyclic, positive, preserving) and `generalised absorbing top-trading cycles (GATTC) rules' in housing markets with indifferences in preferences (Aziz and Keijzer, 2011) (acyclic, opposite, preserving). Under mild restrictions, we show how the nature of influence under a strategy-proof rule determines whether or not it satisfies `weak group-strategy-proofness' (requires acyclicity and either positivity or preservation), `weak Maskin monotonicity' (acyclicity and positivity) and `Pareto-efficiency' (acyclicity and oppositeness). In addition, we propose an influence-related generalisation of the`efficiency-adjusted deferred acceptance mechanism' in school choice (Kesten, 2010), and characterise influence for strategy-proof GATTC rules in housing markets.

Robust Minimal Instability of the Top Trading Cycles Mechanism

In the context of priority-based allocation of objects, we formulate methods to compare assignments in terms of their stability. We introduce three basic properties that a reasonable stability comparison should satisfy. We show that, for any stability comparison satisfying the three properties, the top trading cycles mechanism is minimally unstable among efficient and strategy-proof mechanisms when objects have unit capacities. Our unifying approach covers basically all natural stability comparisons and establishes the robustness of a recent result by Abdulkadiroglu et al. (2020). When objects have non-unit capacities, we characterize the capacity-priority structures for which our result is preserved.

Structural Estimation of a Model of School Choices: The Boston Mechanism versus Its Alternatives

An important debate centers on what procedure should be used to allocate students across public schools. We contribute to this debate by developing and estimating a model of school choices by households under one of the most pop- ular procedures known as the Boston mechanism (BM). We recover the joint distribution of household preferences and sophistication types using adminis- trative data from Barcelona. Our counterfactual policy analyses show that a change from BM to the Gale-Shapley student deferred acceptance mechanism would create more losers than winners, while a change from BM to the top trading cycles mechanism has the opposite effect.

A market design approach to job rotation

Organizations often rotate employees' jobs. This paper proposes a market design approach to organize job rotation. In our model each employee has occupied a position, and if any employee wants to move to another position, the current occupier must leave the position. This requirement is described by a priority structure in which each employee has the lowest priority for his position and others have the equal priority. It is the “opposite” to the famous housing market priority structure in which every owner has the highest priority for his endowment and others have the equal priority. We adapt Top Trading Cycle to solve our model. Our mechanism is novel in that employees are not allowed to point to their positions during the mechanism and cycles are cleared by backward induction after all of them are generated. Our mechanism is stable, constrained efficient and weakly group strategy-proof.

Top Trading Cycles with Reordering: Improving Priorities in School Choice

Top Trading Cycles with Reordering: Improving Priorities in School Choice

On the Integration of Shapley-Scarf Housing Markets

We study the welfare consequences of merging disjoint Shapley-Scarf housing markets. We obtain tight bounds on the number of agents harmed by integration and on the size of their losses. We show that, in the worst-case scenario, market integration may harm the vast majority of agents, and that the average rank of an agent's house can decrease (asymptotically) by 50% of the length of their preference list. We also obtain average-case results. We exactly compute the expected gains from integration in random markets, where each of the preference profiles is chosen uniformly at random. We show that, on average, market integration benefits all agents, particularly those in smaller markets. Using the expected number of cycles in the top trading cycles algorithm, we bound the expected number of agents harmed by integration. In particular, the expected fraction of agents harmed by integration is less than 50% if each market has the same size and this is below 26 (independent of the number of markets that merge). We conclude by providing a preference domain that ensures that those harmed by market integration are a minority.

The Property Rights Theory of Production Networks

This paper investigates the formation of production and trading networks in an economy with general interdependencies and complex property rights. We argue that the right to exclude, a core tenet of property, grants asset owners local monopoly power that is both amplified and constrained by an economy's endogenous production network. Supply multisourcing and a bias toward shorter supply chains emerge in equilibrium. As a methodological contribution, we generalize the top trading cycles algorithm to a production economy. Applications of the framework include the study of vertical integration and government intervention in supply chains.

Top trading cycles, consistency, and acyclic priorities for house allocation with existing tenants

We study the house allocation with existing tenants model (Abdulkadiroğlu and Sönmez, 1999) and consider rules that allocate houses based on priorities. We introduce a new acyclicity requirement and show that for house allocation with existing tenants a top trading cycles (TTC) rule is consistent if and only if its underlying priority structure satisfies our acyclicity condition. Next we give an alternative description of TTC rules based on ownership-adapted acyclic priorities in terms of two specific rules, YRMH-IGYT (you request my house - I get your turn) and efficient priority rules, that are applied in two steps. Moreover, even if no priority structure is a priori given, we show that a rule is a top trading cycles rule based on ownership-adapted acyclic priorities if and only if it satisfies Pareto-optimality, individual-rationality, strategy-proofness, consistency, and either reallocation-proofness or non-bossiness.

Efficiency and Stability in Large Matching Markets

We study efficient and stable mechanisms in matching markets when the number of agents is large and individuals’ preferences and priorities are drawn randomly. When agents’ preferences are uncorrelated, then both efficiency and stability can be achieved in an asymptotic sense via standard mechanisms such as deferred acceptance and top trading cycles. When agents’ preferences are correlated over objects, however, these mechanisms are either inefficient or unstable, even in an asymptotic sense. We propose a variant of deferred acceptance that is asymptotically efficient, asymptotically stable, and asymptotically incentive compatible. This new mechanism performs well in a counterfactual calibration based on New York City school choice data.

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, social hierarchies, and production. 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.

Optimal Priority-Based Allocation Mechanisms

This paper develops a tractable methodology for designing an optimal priority system for assigning agents to heterogeneous items while accounting for agents' choice behavior. The space of mechanisms being optimized includes deferred acceptance and top trading cycles as special cases. In contrast to previous literature, I treat the inputs to these mechanisms, namely the priority distribution of agents and quotas of items, as parameters to be optimized. The methodology is based on analyzing large market models of one-sided matching using techniques of revenue management, and solving a certain assortment planning problem whose objective is social welfare. I apply the methodology to school choice and show that restricting choices may be beneficial to student welfare. Moreover, I compute optimized choice sets and priorities for elementary school choice in Boston, improving upon the results of Ashlagi and Shi (2015).

On responsiveness of top trading cycles mechanism to priority-based affirmative action

In school choice, priority-based affirmative action favors minority students by means of promoting their priority ranking at schools. This paper investigates the responsiveness of the top trading cycles mechanism to priority-based affirmative action when schools’ priority structure satisfies a certain acyclicity condition. When schools’ priority under the problem without affirmative action satisfies virtual homogeneity condition, a weak responsiveness result is obtained.

On endowments and indivisibility: partial ownership in the Shapley–Scarf model

We introduce a parameterized measure of partial ownership, the $$\alpha $$ -endowment lower bound, appropriate to probabilistic allocation. Strikingly, among all convex combinations of efficient and group strategy-proof rules, only Gale’s Top Trading Cycles is sd efficient and meets a positive $$\alpha $$ -endowment lower bound (Theorem 2); for efficiency, partial ownership must in fact be complete. We also characterize the rules meeting each $$\alpha $$ -endowment lower bound (Theorem 1). For each bound, the family is a semilattice ordered by strength of ownership rights. It includes rules where agents’ partial ownership lower bounds are met exactly, rules conferring stronger ownership rights, and the full endowments of TTC. This illustrates the trade-off between sd efficiency and flexible choice of ownership rights.

A New Solvable Weak Priority Structure

Organizations (especially firms) often rotate employees’ jobs from period to period. We propose a market design approach to organize job rotation in any given period. In our model each employee has occupied a position. Rotation requires that the occupier of a position must leave if any other employee wants to move to the position. This requirement is described by a priority structure in which each employee has the lowest priority for his occupied position and other employees have equal priority. Interestingly, this priority structure can be regarded as the “opposite” to the famous housing market priority structure (Shapley and Scarf, 1974). We adapt the Top Trading Cycle mechanism to solve the model. Our mechanism is innovative in that we preclude employees from pointing to their positions during the procedure of the mechanism and we do not clear cycles immediately when they appear; instead, after all employees are involved in cycles, we clear cycles by backward induction. Our mechanism is stable, constrained efficient and weakly group strategy-proof. Our mechanism (partially) answers an open question of Ehlers and Westkamp (2018) on the solvability of coarse priority structures.

Analysis of Incentive Ratio in Top-Trading-Cycles Algorithms

Analysis of Incentive Ratio in Top-Trading-Cycles Algorithms

Two-Sided Matching via Balanced Exchange

We introduce a new matching model to mimic two-sided exchange programs such as tuition and worker exchanges, in which export-import balances are required for longevity of programs. These exchanges use decentralized markets, making it difficult to achieve this goal. We introduce the two-sided top trading cycles, the unique mechanism that is balanced-efficient, worker-strategy-proof, acceptable, individually rational, and respecting priority bylaws regarding worker eligibility. Moreover, it encourages exchange, because full participation induces a dominant-strategy equilibrium for firms. We extend it to dynamic settings permitting tolerable yearly imbalances and demonstrate that its regular and tolerable versions perform considerably better than models of current practice.

RAM: A collection of mechanisms for (indivisible) resource allocation in oTree

The growing theoretical literature on mechanism design and especially on the allocation of indivisible resources has created the need for experimental testing of mechanisms that evolved from matching theory in order to investigate the mechanisms’ theoretical properties in practice. The collection of mechanisms offered by this package includes Deferred Acceptance, Gale’s Top Trading Cycles and the Boston Mechanism in the context of School Choice Problems and the University of Michigan Bidding System, the Gale–Shapley Pareto-Dominant Market Mechanism, and Random Serial Dictatorship in the context of multi-unit resource allocation. All standalone apps are easy to use, customizable for many purposes, and are implemented in the oTree framework.