On characterizations of the probabilistic serial mechanism involving incentive and invariance properties☆
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 objects are assigned to 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 where is a finite set of agents; is a finite set of objects with ; and is a preference profile where is the strict preference relation of agent on . Let be the set of preferences for any agent. Let denote the weak preference relation induced by . We assume that preferences are linear orders on , i.e., for all
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.
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Cited by (3)
Multi resource allocation with partial preferences
2023, Artificial IntelligenceShort trading cycles: Paired kidney exchange with strict ordinal preferences
2020, Mathematical Social SciencesCitation Excerpt :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.
Axiomatic characterizations of the constrained probabilistic serial mechanism
2023, Theory and Decision
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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.