Random assignment: Redefining the serial rule

doi:10.1016/j.jet.2015.04.008

Journal of Economic Theory

Volume 158, Part A, July 2015, Pages 308-318

https://doi.org/10.1016/j.jet.2015.04.008 Get rights and content

Abstract

We provide a new, welfarist, interpretation of the well-known Serial rule in the random assignment problem, strikingly different from previous attempts to define or axiomatically characterize this rule.

For each agent i we define $t_{i} (k)$ to be the total share of objects from her first k indifference classes this agent i gets. Serial assignment is shown to be the unique one which leximin maximizes the vector of all such shares $(t_{i} (k))$ .

This result is very general; it applies to non-strict preferences, and/or non-integer quantities of objects, as well.

In addition, we characterize Serial rule as the unique one sd-efficient, sd-envy-free, and strategy-proof on the lexicographic preferences extension to lotteries.

Section snippets

Model and results

Let $N = {1, \dots, n}$ be the set of agents, and $A = {a_{1}, \dots, a_{m}}$ be the set of objects.⁵ Let $R$ be the set of all orderings over A. Each $i \in N$ has preferences $R_{i} \in R$ over A, and a preference profile is $R = {(R_{i})}_{i \in N}$ . Given a particular R, we denote by $U_{i} (a) = U (R_{i}, a) = {b \in A : b R_{i} a$ or $b = a}$ the upper contour set over a under preferences $R_{i}$

References (11)

A. Bogomolnaia et al.
Probabilistic assignment of objects: characterizing the serial rule
J. Econ. Theory
(2012)
A. Bogomolnaia et al.
New solution to the random assignment problem
J. Econ. Theory
(2001)
A.-K. Katta et al.
A solution to the random assignment problem on the full preference domain
J. Econ. Theory
(2006)
A. Bogomolnaia et al.
Random matching under dichotomous preferences
Econometrica
(2004)
Hashimoto, T., Hirata, D., 2011. Characterizations of the probabilistic serial mechanism....

There are more references available in the full text version of this article.

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We study the random assignment of indivisible objects among a set of agents with strict preferences. We show that there exists no mechanism which is unanimous, strategy-proof and envy-free. Weakening the first requirement to q-unanimity – i.e., when every agent ranks a different object at the top, then each agent shall receive his most-preferred object with probability of at least q – we show that a mechanism satisfying strategy-proofness, envy-freeness and ex-post weak non-wastefulness can be q-unanimous only for $q \leq \frac{2}{n}$ (where n is the number of agents). To demonstrate that this bound is tight, we introduce a new mechanism, Random-Dictatorship-cum-Equal-Division (RDcED), and show that it achieves this maximal bound when all objects are acceptable. In addition, for three agents, RDcED is characterized by the first three properties and ex-post weak efficiency. If objects may be unacceptable, strategy-proofness and envy-freeness are jointly incompatible even with ex-post weak non-wastefulness.
The vigilant eating rule: A general approach for probabilistic economic design with constraints
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We consider probabilistic allocation of objects under ordinal preferences and constraints on allocations. We devise an allocation mechanism, called the vigilant eating rule (VER), that applies to nearly arbitrary constraints. It is constrained ordinally efficient, can be computed efficiently for a large class of constraints, and treats agents equally if they have the same preferences and are subject to the same constraints. When the set of feasible allocations is convex, it is characterized by ordinal egalitarianism. As a case study, we assume objects have priorities for agents and apply VER to sets of probabilistic allocations that are constrained by stability. While VER always returns a stable and constrained efficient allocation, it fails to be strategyproof, unconstrained efficient, and envy-free. We show, however, that each of these three properties is incompatible with stability and constrained efficiency.
Constrained random matching
2022, Journal of Economic Theory
Citation Excerpt :
GCPS is also fair (Proposition 4). It is anonymous (intuitively, agents' identity is irrelevant to the mechanism), satisfies equal treatment of equals (if two agents face the same constraints and have the same preferences, GCPS assigns them identical probability-share allocations), and it selects the most ordinally egalitarian outcome given the constraints, where ordinal egalitarianism is a Rawls-style equity concept introduced by Bogomolnaia (2015, 2018). The GCPS mechanism is general: it can be used in any constrained matching environment.
This paper generalizes the Probabilistic Serial (PS) mechanism of Bogomolnaia and Moulin (2001) to matching markets with arbitrary constraints. The constraints are modeled as a set of permissible ex post allocations. A result of independent interest gives a simple geometric characterization of all lotteries over the set of permissible allocations: they are the ones that satisfy a collection of simple and tractable linear inequalities determined by the constraints. The inequalities correspond to the hyperplanes defining a convex polytope that is intuitively constructed from the given set. When a general version of the PS algorithm is executed under these inequalities, the outcome is an efficient and fair lottery over the set of permissible allocations. The method is general, can be applied to both one-sided and two-sided matching markets, and allows for multi-unit demand.
Random assignments on preference domains with a tier structure
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We address a standard random assignment problem (Bogomolnaia and Moulin, 2001). A weakly connected domain admitting an sd-strategy-proof, sd-efficient and equal-treatment-of-equals rule is characterized to be a restricted tier domain. Conversely, on such a domain, the probabilistic serial rule is uniquely characterized by either sd-strategy-proofness, sd-efficiency and equal treatment of equals, or sd-efficiency and sd-envy-freeness. Moreover, we provide an algorithm to construct unions of multiple restricted tier domains, each of which admits an sd-strategy-proof, sd-efficient and sd-envy-free rule.
Efficient rules for probabilistic assignment
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We propose an algorithm to construct sd-efficient probabilistic allocations of objects. Our algorithm proceeds in discrete rounds and distributes probability shares of objects sequentially yielding two natural degrees of freedom: selection rules governing the choice of objects in each round and distribution rules governing the dispensation of probability shares. The transparent procedure tightly connects properties of the rule to these choices making it ideally suited for practical applications. First proving that the algorithm identifies all sd-efficient allocations, we apply the technique to obtain a surprising and “decentralized” representation of the serial rule. Further applications construct new sd-efficient rules.
Probabilistic assignment of indivisible objects when agents have the same preferences except the ordinal ranking of one object
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Bogomolnaia and Moulin (2001) show that there is no rule satisfying equal treatment of equals, stochastic dominance efficiency, and stochastic dominance strategyproofness for a probabilistic assignment problem of indivisible objects. Kasajima (2013) shows that the incompatibility result still holds when agents are restricted to have single-peaked preferences. In this paper, we further restrict the domain and investigate the existence of rules satisfying the three properties. As it turns out, the three properties are still incompatible even if all agents have the same preferences except the ordinal ranking of one object.

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^☆: The author is grateful to Jay Sethuraman, William Thomson, and anonymous referees for helpful suggestions and comments.

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NotesRandom assignment: Redefining the serial rule☆

Abstract

Section snippets

Model and results

J. Econ. Theory

J. Econ. Theory

J. Econ. Theory

Random matching under dichotomous preferences

Econometrica

Notes
Random assignment: Redefining the serial rule☆