Elsevier

Mathematical Social Sciences

Volume 90, November 2017, Pages 56-62
Mathematical Social Sciences

On characterizations of the probabilistic serial mechanism involving incentive and invariance properties

https://doi.org/10.1016/j.mathsocsci.2016.11.005Get rights and content

Highlights

  • Probabilistic serial is ordinal efficient, envy-free and weakly strategyproof.

  • PS is not characterized by the three properties.

  • Weak strategy proofness is logically independent of any invariance property.

Abstract

This paper studies the problem of assigning n indivisible objects to n agents when each agent consumes one object and monetary transfers are not allowed. Bogomolnaia and Moulin (2001) proved that for n=3, the probabilistic serial mechanism is characterized by the three axioms of ordinal efficiency, envy-freeness, and weak strategy-proofness. We show that this characterization does not extend to problems of arbitrary size; in particular, it does not hold for any n5. A number of general characterizations of the probabilistic serial mechanism have been obtained in the recent literature by replacing weak strategy-proofness with various invariance axioms while retaining ordinal efficiency and envy-freeness. We show that weak strategy-proofness is in fact logically independent of all invariance axioms used in these characterizations.

Introduction

Many real-life problems such as school choice, organ transplantation, and on-campus housing involve the assignment of discrete indivisible objects without the use of monetary transfers. We consider the simplest discrete resource allocation problem in which n objects are assigned to n agents who have strict preferences over objects. A mechanism is a rule that specifies a stochastic assignment of objects to agents based on their reported preferences. The widely-used mechanism for this type of problems in practice is the random serial dictatorship (RSD) mechanism: randomly order the agents and let them sequentially choose their favorite objects. RSD is well-known for its strategy-proofness and ex-post efficiency. In their seminal paper, Bogomolnaia and Moulin (2001) (BM hereafter) showed that RSD is neither ordinally efficient nor envy-free, but is weakly envy-free.

BM introduced the probabilistic serial (PS) mechanism as a major competitor to RSD. The outcome of PS is computed via the simultaneous eating algorithm (SEA): Imagine that each object is a continuum of probability shares. Let agents simultaneously “eat away” from their favorite objects at the same speed; once the favorite object of an agent is gone, she turns to her next favorite object, and so on. We interpret the share of an object eaten away by an agent throughout the process as the probability PS assigns her that object.

PS is ordinally efficient and envy-free. This surprising observation in turn led to much attention being devoted to PS and its various extensions1 and characterizations. Unlike RSD which is strategy-proof, PS is weakly strategy-proof. BM provided a first characterization of PS through ordinal efficiency, envy-freeness, and weak strategy-proofness with the added condition that there are three agents. A generalization of the original BM characterization to an arbitrary number of agents/objects has thus far been elusive. We specifically ask whether the BM characterization result holds for problems of arbitrary size and give a negative answer to this question. In particular, we construct a highly non-trivial mechanism for the case of five agents, different from PS, which satisfies ordinal efficiency, envy-freeness, and weak strategy-proofness (Lemma 1). Building on this construction, we show that, when there are at least five agents, it is possible to obtain a mechanism different from PS, which satisfies the three properties (Proposition 1).

Recently several papers provide various PS characterizations for more general settings with arbitrary number of agents and possibly for multiple copies of objects. A common theme in these characterizations is the use of ordinal efficiency and envy-freeness along with an invariance/monotonicity type property that requires the robustness of the random assignment to certain perturbations of agents’ preferences.2 In light of Proposition 1, it may be tempting to think that the invariance properties used in these characterizations are stronger than weak strategy-proofness. We show that no such conclusion is true. Indeed, we show that there is no logical connection between weak strategy-proofness and any of these invariance properties (Proposition 2).

Section  2 describes the formal model and Section  3 provides the main result. Section  4 concludes.

Section snippets

Model

A discrete resource allocation problem (cf.  Hylland and Zeckhauser, 1979, Shapley and Scarf, 1974), or simply a problem, is a list (N,A,) where N={1,,n} is a finite set of agents; A is a finite set of objects with |A|=|N|=n; and =(i)iN is a preference profile where i is the strict preference relation of agent i on A. Let P be the set of preferences for any agent. Let i denote the weak preference relation induced by i. We assume that preferences are linear orders on A, i.e., for all a,b

The main results

BM provide a complete characterization of PS by ordinal efficiency, envy-freeness, and weak strategy-proof ness when there are three agents and three objects. Our main result shows that this characterization no longer holds with five or more agents:

Proposition 1

When there are five or more agents, there exists a mechanism, different from PS, satisfying ordinal efficiency, envy-freeness, and weak strategy-proofness.

We prove this proposition through a counterexample, i.e., by providing a mechanism, different

Concluding remarks

We have shown that the characterization of PS by ordinal efficiency, envy-freeness, and weak strategy-proofness does not hold for the case of five or more agents, even though such a characterization is true for the case of three agents as BM showed. We have been unsuccessful in our attempts to address this problem in the case of four agents. We leave it as an open question for future investigation.

References (15)

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    Since their seminal contribution, the PS mechanism has been adapted for ordinal preferences allowing indifferences (Katta and Sethuraman, 2006), for multi-unit demand (Kojima, 2009), and for property rights necessitating individual rationality (Yılmaz, 2009, 2010) among many others. Bogomolnaia and Heo (2012) and Hashimoto et al. (2014) offer characterizations of the PS mechanism; see Kesten et al. (2017) and Chang and Chun (2017) for some recent related results. Kidney transplantation is generally the only long-term treatment for end-stage renal disease.

We would like to thank three anonymous referees and participants at Duke “Roth–Sotomayor: 20 Years After” Conference, CORE, Kyoto, Maastricht, Osaka, and Tsukuba for comments. Kurino acknowledges the financial support from JSPS KAKENHI Grant number 15K13002. Ünver acknowledges the research support of Microsoft Research Lab, New England.

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