Probabilistic assignment of objects: Characterizing the serial rule

Abstract

We study the problem of assigning a set of objects to a set of agents, when each agent receives one object and has strict preferences over the objects. In the absence of monetary transfers, we focus on the probabilistic rules, which take the ordinal preferences as input. We characterize the serial rule, proposed by Bogomolnaia and Moulin (2001) [2]: it is the only rule satisfying sd efficiency, sd no-envy, and bounded invariance. A special representation of feasible assignment matrices by means of consumption processes is the key to the simple and intuitive proof of our main result.

Introduction

We study the problem of assigning a set of “objects” (i.e., indivisible goods) to a set of agents, when each agent is to receive only one object and has strict preferences over the objects. In the absence of monetary transfers, randomization is the method of choice to guarantee fairness. An extensive recent literature, starting from Bogomolnaia and Moulin [2], is devoted to the study of probabilistic assignment rules in this setting. Earlier work (see Hylland and Zeckhauser [8], Zhou [13]) assumed that agents announce their cardinal preferences. The recent literature considers ordinal rules, which only take agentsʼ rankings over the objects as input.

The classical random priority rule orders agents using a uniform lottery, and lets them pick their most preferred objects in that order. An alternative is the “serial rule (S)”, introduced by Bogomolnaia and Moulin [2]. The rule is described by means of a “simultaneous eating” algorithm. Agents acquire probabilities of objects continuously over the unit interval of time $[0,1]$, simultaneously and at the same unit rate. Given a preference profile R, each agent starts with his most preferred object. When the object he consumes is exhausted, he switches to his next most preferred object among the ones that are still available. At each $\tau \in [0,1]$, $S(R)[\tau ]$ is a partial serial assignment that agents have acquired by τ. The final assignment $S(R)=S(R)[1]$ is given by vectors of probabilities agents have consumed.

The restriction to ordinal input calls for first order stochastic dominance as a way to compare assignments. Given agent iʼs ordinal preferences, one assignment (vector of probabilities receiving different objects) weakly dominates another for agent i, if and only if, for each object a, the total probability of his receiving objects that he prefers to a is at least as large under the first assignment as under the second.¹ In this spirit, Bogomolnaia and Moulin [2] define two natural requirements on a probabilistic assignment. “Sd efficiency”² requires that no other assignment weakly dominates the given one for all agents. “Sd no-envy” requires for each agentʼs assignment to dominate, in his opinion, the assignment of any other agent.

The serial rule fares better than the random priority rule in that it satisfies sd efficiency and sd no-envy, neither one of which is satisfied by the random priority rule, though those properties do not pin it down (see the remark in Section 3).³

Much work has been done over the last decade on probabilistic rules that only take ordinal preferences as input. The serial rule occupies a central place in this literature. However, until recently, its axiomatic characterization had been elusive. This paper proposes such a characterization, by means of sd efficiency, sd no-envy, and an axiom, which we call “bounded invariance”. This last requires that, whenever the preferences of one agent change so that the ranking over his upper counter set of an object, say a, remains the same, the probability share of a assigned to each agent remains the same. We show that this characterization holds even under weaker requirements. Moreover, we introduce a new, simple, and intuitive proof technique, which allows us to easily obtain an array of related characterizations.

The key to our result is a special representation of assignment matrices by means of “consumption processes”. This representation is introduced by Heo [6] as “preference-decreasing consumption schedules”: each assignment matrix P is represented as the output of a process along which agents continuously acquire probabilities of objects over the unit interval of time, $[0,1]$. It mimics the simultaneous eating algorithm: each agent consumes probabilities of objects at unit speed in decreasing order of his preferences. However, each agent may switch from one object to another even when the former object is not yet exhausted. Agent i ends consuming an object, say a, exactly when she acquires the probability $p_{ia}$, given by $P_i$. We represent this process by ${(P[\tau ])}_{\tau \in [0,1]}$. Given a rule satisfying the properties that we impose, we compare the simultaneous eating algorithm with the consumption processes representing assignment matrices selected by the rule. By imposing our properties, we obtain that the assignment matrix selected by the rule should coincide with the assignment matrix selected by the serial rule.

Three recent papers are closely related to the current manuscript. Heo [6] provides a characterization of the serial rule by means of sd efficiency, “sd equal-division lower bound”, “limited invariance”, and “consistency”, an axiom pertaining to variable populations.⁴ Independently, Kesten et al. [9] were the first to characterize the rule by means of sd efficiency, sd no-envy, and one additional invariance axiom, which they call “upper invariance”.⁵ Independently and concurrently, Hashimoto and Hirata [4] formulated another invariance axiom for a different domain on which the “null object”⁶ exists. Applying our result directly to the variants of the model in which several copies of objects may exist, or agents could receive the null object, we obtain these characterizations as corollaries, given that our invariance axiom implies theirs. The studies by Kesten et al. [9] and Hashimoto and Hirata [4] contain two additional characterizations that do not invoke invariance axioms and are distinct from ours (Theorem 1 in Kesten et al. [9] and Theorem 3 in Hashimoto and Hirata [4]). After we became aware of these results, we found that our proof technique also allows us to obtain straightforward proofs of these results.⁷

The paper is organized as follows. We describe the model, notation, and axioms in Section 2. We present our main result in Section 3. We show how all the related results, mentioned above, can be unified using our proof technique in Section 4 (we relegate a short argument, needed for one case, to Appendix A).