Pareto Efficient Mechanism Research Articles

Comparison of the voluntary contribution and Pareto-efficient mechanisms under voluntary participation

Comparison of the voluntary contribution and Pareto-efficient mechanisms under voluntary participation

Reallocation with priorities and minimal envy mechanisms

We investigate the problem of reallocation with priorities where one has to assign objects or positions to individuals. Agents can have an initial ownership over an object. Each object has a priority ordering over the agents. In this framework, there is no mechanism that is both individually rational (IR) and stable, i.e. has no blocking pairs. Given this impossibility, an alternative approach is to compare mechanisms based on the blocking pairs they generate. A mechanism has minimal envy within a set of mechanisms if there is no other mechanism in the set that always leads to a set of blocking pairs included in the one of the former mechanism. Our main result shows that the modified Deferred Acceptance mechanism (Guillen and Kesten in Int Econ Rev 53(3):1027–1046, 2012), is a minimal envy mechanism in the set of IR and strategy-proof mechanisms. We also show that an extension of the Top Trading Cycle (Karakaya et al. in J Econ Theory 184:104948, 2019) mechanism is a minimal envy mechanism in the set of IR, strategy-proof and Pareto-efficient mechanisms. These two results extend the existing ones in school choice.

Truthful and Fair Mechanisms for Matroid-Rank Valuations

We study the problem of allocating indivisible goods among strategic agents. We focus on settings wherein monetary transfers are not available and each agent's private valuation is a submodular function with binary marginals, i.e., the agents' valuations are matroid-rank functions. In this setup, we establish a notable dichotomy between two of the most well-studied fairness notions in discrete fair division; specifically, between envy-freeness up to one good (EF1) and maximin shares (MMS). First, we show that a known Pareto-efficient mechanism is group strategy-proof for finding EF1 allocations, under matroid-rank valuations. The group strategy-proofness guarantee strengthens an existing result that establishes truthfulness (individually for each agent) in the same context. Our result also generalizes prior work from binary additive valuations to the matroid-rank case. Next, we establish that an analogous positive result cannot be achieved for MMS, even when considering truthfulness on an individual level. Specifically, we prove that, for matroid-rank valuations, there does not exist a truthful mechanism that is index oblivious, Pareto efficient, and maximin fair. For establishing our results, we develop a characterization of truthful mechanisms for matroid-rank functions. This characterization in fact holds for a broader class of valuations (specifically, holds for binary XOS functions) and might be of independent interest.

Strategy-proof and Pareto efficient allocation of indivisible goods: general impossibility domains

We consider the problem of allocating several homogeneous indivisible goods when monetary transfers among agents are possible. We study the possibility of constructing strategy-proof and Pareto efficient mechanisms on restricted domains of agents’ valuation profiles. We show that there is no strategy-proof and Pareto efficient mechanism under the weak domain condition that all agents’ sets of possible valuations share at least four common valuations satisfying a certain inequality. Moreover, we prove that this impossibility result is robust to any affine transformation of domains, and we examine this impossibility result when the set of agent’s possible valuations consists of finite integers.

Dynamic assignment without money: Optimality of spot mechanisms

We study a large market model of dynamic matching with no monetary transfers and a continuum of agents. Time is discrete and horizon finite. Agents are in the market from the first date and, at each date, have to be assigned items (or bundles of items). When the social planner can only elicit ordinal preferences of agents over the sequences of items, we prove that, under a mild regularity assumption, incentive compatible and ordinally efficient allocation rules coincide with spot mechanisms. A spot mechanism specifies “virtual prices” for items at each date and, at the beginning of time, for each agent, randomly selects a budget of virtual money according to a (potentially non-uniform) distribution over [0,1]. Then, at each date, the agent is allocated the item of his choice among the affordable ones. Spot mechanisms impose a linear structure on prices and, perhaps surprisingly, our result shows that this linear structure is what is needed when one requires incentive compatibility and ordinal efficiency. When the social planner can elicit cardinal preferences, we prove that, under a similar regularity assumption, incentive compatible and Pareto efficient mechanisms coincide with a class of mechanisms we call Spot Menu of Random Budgets mechanisms. These mechanisms are similar to spot mechanisms except that, at the beginning of the time, each agent must pick a distribution in a menu. This distribution is used to initially draw the agent's budget of virtual money.

A Justified Envy Minimal Variation of the Top Trading Cycles Mechanism

For priority-based allocation of indivisible objects to agents with unit demand, when each object has exactly one copy, Abdulkadiroglu et al. (2019) show that the priority-based top trading cycles mechanism is justified envy minimal, in the sense that no strategy proof and Pareto efficient mechanism has strictly less justified envy than it. It remains open whether there is a unique justified envy minimal mechanism. We propose a strategy proof and Pareto efficient variation of the top trading cycles mechanism and show that it is also justified envy minimal.

The Core of School Choice Problems

We propose a notion of core for school choice problems. We say that a coalition of students is able to enforce a subassignment among them if, given their priorities at schools, other students together cannot exclude any of them from her assignment. An assignment is in the core if no coalition of students can enforce any subassignment among them that Pareto improves their assignments. We show that the core is always nonempty and all assignments in it are Pareto efficient. When the student-proposing deferred acceptance mechanism is a Pareto efficient mechanism, it always produces the unique assignment in the core..

On stable and efficient mechanisms for priority-based allocation problems

For school choice (priority-based allocation) problems, when the priority structure is acyclic, the associated student-proposing deferred acceptance algorithm is Pareto efficient and group strategy-proof (Ergin, 2002). We reveal a hidden iterative removal structure behind such deferred acceptance algorithms. A nonempty set of students is called a top fair set (TFS) if when all students apply to their most preferred schools and all schools accept the best applicants up to their quotas, students in the set are always accepted, regardless of other students' preferences. We provide an elimination process to find the maximal TFS, if any TFS exists. We show that for any priority structure, iterative removal of TFS is equivalent to the associated deferred acceptance algorithm if and only if the latter is a Pareto efficient mechanism.

Essentially stable matchings

We propose a solution to the conflict between fairness and efficiency in one-sided matching markets. A matching is essentially stable if any priority-based claim initiates a chain of reassignments that results in the initial claimant losing the object. We show that an essentially stable and Pareto efficient matching always exists and that Kesten's (2010) EADA mechanism always selects one while other common Pareto efficient mechanisms do not. Additionally, we show that there exists a student-pessimal essentially stable matching and that the Rural Hospital Theorem extends to essential stability. Finally, we analyze the incentive properties of essentially stable mechanisms.

Weighted Matching Markets with Budget Constraints

We investigate markets with a set of students on one side and a set of colleges on the other. A student and college can be linked by a weighted contract that defines the student's wage, while a college's budget for hiring students is limited. Stability is a crucial requirement for matching mechanisms to be applied in the real world. A standard stability requirement is coalitional stability, i.e., no pair of a college and group of students has any incentive to deviate. We find that a coalitionally stable matching is not guaranteed to exist, verifying the coalitional stability for a given matching is coNP-complete, and the problem of finding whether a coalitionally stable matching exists in a given market, is SigmaP2-complete: NPNP-complete. Other negative results also hold when blocking coalitions contain at most two students and one college. Given these computational hardness results, we pursue a weaker stability requirement called pairwise stability, where no pair of a college and single student has an incentive to deviate. Unfortunately, a pairwise stable matching is not guaranteed to exist either. Thus, we consider a restricted market called a typed weighted market, in which students are partitioned into types that induce their possible wages. We then design a strategy-proof and Pareto efficient mechanism that works in polynomial-time for computing a pairwise stable matching in typed weighted markets.

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.

Payoff equivalence of efficient mechanisms in large matching markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We provide a broad set of circumstances under which, as the market grows large, all Pareto efficient mechanisms---including top trading cycles (with an arbitrary ownership structure), serial dictatorship (with an arbitrary serial order), and their randomized variants---produce a distribution of agent utilities that in the limit coincides with the utilitarian upper bound. This implies that Pareto efficient mechanisms are uniformly asymptotically payoff equivalent ``up to the renaming of agents.'' Hence, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

On Stable and Efficient Mechanisms for Priority-Based Allocation Problems

For school choice (priority-based allocation) problems, when the priority structure is acyclic, the associated student-proposing deferred acceptance algorithm is Pareto efficient and group strategy-proof (Ergin, 2002). We reveal a hidden iterative removal structure behind such deferred acceptance algorithms. A nonempty set of students is called a top fair set (TFS) if when all students apply to their most preferred schools and all schools accept the best applicants up to their quotas, students in the set are always accepted, regardless of other students’ preferences. We provide an elimination process to find the maximal TFS, if any TFS exists. We show that for any priority structure, iterative removal of TFS is equivalent to the associated deferred acceptance algorithm if and only if the latter is a Pareto efficient mechanism.

Quadratic voting and the public good: introduction

This introduction to the Public Choice special issue on “quadratic voting (QV) and the public good” provides an opinionated narrative summary of the contents and surveys the broader literature related to QV. QV is a voting rule, proposed by one of us Weyl (Quadratic vote buying. http://goo.gl/8YEO73, 2012), Lalley and Weyl (Quadratic voting. https://papers.ssrn.com/sol3/Papers.cfm?abstract_id=2003531, 2016) building off earlier work by Groves and Ledyard (Econometrica 45(4):783–810 1977a), Hylland and Zeckhauser (A mechanism for selecting public goods when preferences must be elicited, Kennedy School of Government Discussion Paper D, 70, 1980), where individuals buy as many votes as they wish by paying the square of the votes they buy using some currency. An appreciation of the history of research in the field suggests that QV is uniquely practically relevant compared to the other approximately Pareto-efficient mechanisms economists have proposed for collective decisions on public goods. However, it faces a number of sociological and ethical concerns regarding how a political system organized around QV would achieve the efficiency aims stated in abstract theory and whether the pure aggregate income-maximizing definition of efficiency QV optimizes in its simplest form is desirable. The papers in this volume flesh out and formalize these concerns, but also provide important responses in two ways: by suggesting domains where they are unlikely to be applicable (primarily related to survey research of various kinds) and versions of QV (using an artificial currency) that maintain many of QV’s benefits while diffusing the most important critiques. Together this work suggests both a practical path for applying QV in the near-term and a series of research questions that would have to be addressed to broaden its application.

Incentive compatible allocation and exchange of discrete resources

The allocation and exchange of discrete resources, such as transplant organs, public housing, dormitory rooms, and many other resources for which agents have single-unit demand, is often conducted via direct mechanisms without monetary transfers. Incentive compatibility and efficiency are primary concerns when designing such mechanisms. We construct the full class of group strategy-proof and Pareto-efficient mechanisms and show that each of them can be implemented by endowing agents with control rights over resources. This new class, which we call trading cycles, contains new mechanisms as well as known mechanisms such as top trading cycles, serial dictatorships, and hierarchical exchange. We illustrate how one can use our construction to show what can and what cannot be achieved in a variety of allocation and exchange problems, and we provide an example in which the new trading-cycles mechanisms are more Lorenz equitable than all previously known mechanisms.

Payoff Equivalence of Efficient Mechanisms in Large Matching Markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We show that, as the market grows large, all Pareto efficient mechanisms -- including top trading cycles, serial dictatorship, and their randomized variants -- are uniformly asymptotically payoff equivalent "up to the renaming of agents," yielding the utilitarian upper bound in the limit. This result implies that, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

Payoff Equivalence of Efficient Mechanisms in Large Matching Markets

We study Pareto efficient mechanisms in matching markets when the number of agents is large and individual preferences are randomly drawn from a class of distributions, allowing for both common and idiosyncratic shocks. We show that, as the market grows large, all Pareto efficient mechanisms -- including top trading cycles, serial dictatorship, and their randomized variants -- are uniformly asymptotically payoff equivalent “up to the renaming of agents,” yielding the utilitarian upper bound in the limit. This result implies that, when the conditions of our model are met, policy makers need not discriminate among Pareto efficient mechanisms based on the aggregate payoff distribution of participants.

A note on the serial dictatorship with project closures

In this paper, we consider the problem of assigning agents having preferences to projects with capacities and lower quotas. For this problem, Monte and Tumennasan proposed a strategy-proof and Pareto efficient mechanism, called the serial dictatorship with project closures. In this paper, we show that the serial dictatorship with project closures can be extended to a more general setting.

Matching with quorums

In the problem of allocating workers to different projects, where each project needs a minimum number of workers assigned to it, the serial dictatorship mechanism is neither strategy proof nor Pareto efficient. We therefore propose a strategy-proof and Pareto-efficient mechanism.

Path auctions with multiple edge ownership

In this paper, we study path auction games in which multiple edges may be owned by the same agent. The edge costs and the set of edges owned by the same agent are privately known to the owner of the edges. In this setting, we show that, assuming that edges not on the winning path always get 0 payment, there is no individually rational, strategyproof mechanism in which only edge costs are reported. If the agents are asked to report costs as well as identity information, we show that there is no Pareto efficient mechanism that is false-name proof. We then study a first-price path auction in this model. We show that, in the special case of parallel-path graphs, there is always a Pareto efficient pure strategy ϵ -Nash equilibrium in bids. However, this result does not extend to general graph; we construct a graph in which there is no Pareto efficient pure strategy ϵ -Nash equilibrium.