Size versus fairness in the assignment problem

doi:10.1016/j.geb.2014.11.006

Games and Economic Behavior

Volume 90, March 2015, Pages 119-127

https://doi.org/10.1016/j.geb.2014.11.006 Get rights and content

Abstract

When not all objects are acceptable to all agents, maximizing the number of objects actually assigned is an important design concern. We compute the guaranteed size ratio of the Probabilistic Serial mechanism, i.e., the worst ratio of the actual expected size to the maximal feasible size. It converges decreasingly to $1 - \frac{1}{e} ≃ 63.2 %$ as the maximal size increases. It is the best ratio of any Envy-Free assignment mechanism.

Section snippets

The problem and the punchline

Lotteries are commonly used to allocate indivisible resources (objects), especially so when monetary transfers are ruled out. Examples include the assignment of jobs to time-slots, of workers to tasks or offices, the allocation of seats in overdemanded public schools (Abdulkadiroğlu and Sonmez, 2003, Kojima and Unver, 2014), of students to dormitory rooms or courses, etc. An excellent survey is the work of Sonmez and Unver (2011). Using cash transfers and prices in such problems skews the

Related literature

1) The first random assignment mechanism in Hylland and Zeckhauser (1979) is a competitive equilibrium with fiat money to buy lotteries, and relies on cardinal (von Neuman Morgenstern) individual utilities over objects. Such individual reports are too complex in practice, so attention turned to the more realistic ordinal mechanisms where a report is simply a ranking of the acceptable objects. The most natural ordinal mechanism is the time honored Random Priority (RP), a.k.a. serial

Random assignment with outside options

Fix N the set of agents and A of objects, with respective cardinalities n and q. A preference $R_{i}$ of agent $i \in N$ is a possibly empty ordered subset of A, written $R_{i} = (a_{1}, a_{2}, \dots, a_{k})$ where $a_{1}$ is the best object for i and $a_{k}$ her least preferred acceptable object. We write $R_{i} = \emptyset$ if no object is acceptable to i, and $a \in R_{i}$ means that a is an acceptable object for i. The set of individual preferences is $R (A)$ .

A profile of preferences $R \in R (A)$ defines a compatibility bipartite graph $E \subseteq N \times A$ : $i a \in E (R) \Leftrightarrow a \in R_{i}$ ,

Efficiency and guaranteed size

Given a problem Δ and two deterministic assignments $P, P^{'} \in P^{d} (E (Δ))$ , we say that $P^{'}$ Pareto dominates P if $P \neq P^{'}$ and for all $a, b$ ${p_{i a}^{'} = 1 and p_{i b} = 1} \Rightarrow a R_{i} b$ ${p_{i a}^{'} = 0 for all a} \Rightarrow {p_{i a} = 0 for all a}$ An efficient (Pareto optimal) deterministic assignment is one that is not Pareto dominated.

In any problem $Δ \in A^{m}$ there is at least one efficient deterministic assignment of size m (i.e., the maximum possible size). This follows because if an assignment $P \in P^{d} (E)$ is Pareto dominated by $P^{'}$ , then $s (P) \leq s (P^{'})$ . On the other hand

Three axioms and two mechanisms

Given a problem Δ, agent i compares two feasible assignments $P, P^{'} \in P (E (Δ))$ by means of her own allocations $p (i) = {(p_{i a})}_{a \in A}$ and $p^{'} (i)$ , the i-th rows of P and $P^{'}$ respectively. We define a critical incomplete preference relation for agent i with preferences $R_{i} = (a_{1}, \dots, a_{k})$ , $1 \leq k \leq q$ . We say that $p (i)$ is sd-preferred to $p^{'} (i)$ (where sd stands for stochastic dominance) if $\sum_{1}^{t} p_{i a_{t}} \geq \sum_{1}^{t} p_{i a_{t}}^{'} for all t, 1 \leq t \leq k$ and we write $p (i) \overset{s d_{i}}{⪰} p^{'} (i)$ (this relation is empty if $R_{i} = \emptyset$ ). Note that sd-indifference is just equality. We

The result

For any two integers $k, m$ such that $1 \leq k < m$ we define $S (m, k) = \frac{1}{k + 1} + \frac{1}{k + 2} + \dots + \frac{1}{m}$ Noticing that $S (m, k)$ decreases in k, we define for any $m \geq 2$ the integer $k_{m}$ by the inequalities $S (m, k_{m}) \leq 1 < S (m, k_{m} - 1)$ Finally we set $r_{m} = 1 - \frac{k_{m}}{m} S (m, k_{m})$ .

For large $k, m$ we have $S (m, k) ≃ \ln (\frac{m}{k})$ hence for large m: $\ln (\frac{m}{k_{m}}) ≃ 1 ⟺ k_{m} ≃ \frac{1}{e} m$ , and finally $r_{m} ≃ 1 - \frac{1}{e}$ . We can say more:

Lemma 1

The sequence $r_{m}$ is decreasing and converges to $1 - \frac{1}{e} = 0.632$ at the speed $O (\frac{1}{n})$ . For instance $r_{2} = 0.750$ , $r_{3} = 0.722$ , $r_{4} = 0.708$ , $r_{5} = 0.687$ , $r_{10} = 0.662$ , $r_{20} = 0.648$ . (proof in

Concluding comments

1. There are inefficient Envy-Free mechanisms with a worse performance than PS, that are still better than throwing away all objects all the time. For instance we can draw objects uniformly and offer them sequentially, uniformly among all the still unmatched agents, throwing the winner and the object away if she does not accept it. This is clearly an envy-free mechanism because once an object is drawn, it is lost to agents other than the winner, therefore the distribution of objects a given

References (26)

P. Biró et al.
Size versus stability in the marriage problem
Theor. Comput. Sci.
(2010)
A. Bogomolnaia et al.
Probabilistic assignment of objects: characterizing the serial rule
J. Econ. Theory
(2012)
A. Bogomolnaia et al.
A new solution to the random assignment problem
J. Econ. Theory
(2001)
B. Kalyanasundaram et al.
An optimal deterministic algorithm for online b-matching
Theor. Comp. Sci.
(2000)
A.K. Katta et al.
A solution to the random assignment problem on the full preference domain
J. Econ. Theory
(2006)
T. Sonmez et al.
Matching, allocation, and exchange of discrete resources
A. Abdulkadiroğlu et al.
Random serial dictatorship and the core from random endowments in house allocation problems
Econometrica
(May 1998)
A. Abdulkadiroğlu et al.
School choice: a mechanism design approach
Amer. Econ. Rev.
(June 2003)
A. Bhalgat et al.
Social welfare in one-sided matching markets without money
B. Birnbaum et al.
On-line bipartite matching made simple
ACM SIGACT News
(March 2008)

Bogomolnaia, A. Random assignment: redefining the serial rule. Glasgow University....

A. Bogomolnaia et al.

A simple random assignment problem with a unique solution

Econ. Theory

(2002)

E. Budish et al.

Designing random allocation mechanisms: theory and applications

Amer. Econ. Rev.

(2013)

Cited by (25)

Bounded incentives in manipulating the probabilistic serial rule
2024, Journal of Computer and System Sciences
The Probabilistic Serial mechanism is valued for its fairness and efficiency in addressing the random assignment problem. However, it lacks truthfulness, meaning it works well only when agents' stated preferences match their true ones. Significant utility gains from strategic actions may lead self-interested agents to manipulate the mechanism, undermining its practical adoption. To gauge the potential for manipulation, we explore an extreme scenario where a manipulator has complete knowledge of other agents' reports and unlimited computational resources to find their best strategy. We establish tight incentive ratio bounds of the mechanism. Furthermore, we complement these worst-case guarantees by conducting experiments to assess an agent's average utility gain through manipulation. The findings reveal that the incentive for manipulation is very small. These results offer insights into the mechanism's resilience against strategic manipulation, moving beyond the recognition of its lack of incentive compatibility.
A planner-optimal matching mechanism and its incentive compatibility in a restricted domain
2023, Games and Economic Behavior
In many random assignment problems, the central planner pursues their own policy objective, such as matching size and minimum quota fulfillment. Several practically important policy objectives do not align with agents' preferences and are known to be incompatible with strategy-proofness. This paper demonstrates that such policy objectives can be attained using mechanisms that satisfy Bayesian incentive compatibility within a restricted domain of von Neumann Morgenstern utilities. We establish that a mechanism satisfies Bayesian incentive compatibility in an inverse-bounded-indifference domain if and only if the mechanism satisfies the three axioms of swap monotonicity, lower invariance, and interior upper variance. We apply this axiomatic characterization to analyze the incentive property of the constrained random serial dictatorship mechanism (CRSD). CRSD is designed to generate an individually rational assignment that optimizes the central planner's policy objective function. Since CRSD satisfies these axioms, it is Bayesian incentive compatible within an IBI domain.
Strategy-proof and envy-free random assignment
2023, Journal of Economic Theory
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.
A pessimist's approach to one-sided matching
2023, European Journal of Operational Research
Inspired by real-world applications such as the assignment of pupils to schools or the allocation of social housing, the one-sided matching problem studies how a set of agents can be assigned to a set of objects when the agents have preferences over the objects, but not vice versa. For fairness reasons, most mechanisms use randomness, and therefore result in a probabilistic assignment. We study the problem of decomposing these probabilistic assignments into a weighted sum of ex-post (Pareto-)efficient matchings, while maximizing the worst-case number of assigned agents. This decomposition preserves all the assignments’ desirable properties, most notably strategy-proofness. Next to discussing the complexity of the problem, we obtain tight lower and upper bounds on the optimal worst-case number of assigned agents. Moreover, we propose two alternative column generation frameworks for the introduced problem, which prove to be capable of finding decompositions with the theoretically best possible worst-case number of assigned agents, both for randomly generated data, and for real-world school choice data from the Belgian cities Antwerp and Ghent. Lastly, the proposed column generation frameworks are inherently flexible, and can therefore also be applied to settings where other ex-post criteria are desirable, or to find decompositions that satisfy other worst-case measures.
Tight social welfare approximation of probabilistic serial
2022, Theoretical Computer Science
We study the problem of approximate social welfare maximization (without money) in assignment problems for n agents who have cardinal preferences over a finite set of items. Probabilistic Serial (PS) mechanism is known to be ordinally efficient and envy-free. We prove that its social welfare approximation ratio is $Θ (n^{- \frac{1}{2}})$ while no envy-free mechanism can achieve an approximation ratio better than $O (n^{- \frac{1}{2}})$ . In addition, our proof also implies a $Θ (n^{- \frac{1}{2}})$ rank approximation of PS, answering an open question posted in [8].
Minimal envy and popular matchings
2022, European Journal of Operational Research
Citation Excerpt :
Frequent non-existence of popular matchings is a major problem of this otherwise desirable and natural concept. Provided existence, popularity stands in same row with Pareto efficiency and its other refinements such as the maximal-size matchings (Afacan, Bó, & Turhan, 2018; Afacan & Dur, 2018; Bogomolnaia & Moulin, 2004; 2015), the rank-efficient matchings (Abdulkadiroğlu & Sönmez, 2003; Featherstone, 2015) and welfare-maximizing matchings (Biró & Gudmundsson, 2020). We propose a constructive extension of popular matchings that always exists.
We study ex-post fairness and efficiency in the object allocation problem. A matching is individually fair if it minimizes the number of envying agents, we call it minimal envy matching, and conditional on being minimal envy also minimizes the number of envying agents in a reduced problem, we call it minimal envy-2 matching. A matching is socially fair if supported by the majority of agents against any other matching – popular matching. We observe that when a popular matching exists it is equivalent to a minimal envy-2 matching.
We show the equivalence between global and local popularity: a matching is popular if and only if in no group of size up to 3 agents a majority can benefit by exchanging objects, keeping the remaining matching fixed. We algorithmically show that an arbitrary matching is path-connected with a popular matching by locally-popular exchanges in small groups. Thus a corresponding market converges to a popular matching.
Popular matchings often do not exist. We define most popular matching as a matching that is popular among the largest (by cardinality) subset of agents. We show that a matching is minimal envy-2 if and only if it is minimal envy and most popular, and propose a polynomial-time algorithm to find a Pareto efficient minimal envy-2 matching.

View all citing articles on Scopus

^☆: We are especially grateful to our colleagues David Manlove and Baharak Rastegari for introducing us to their research project EPSRC Grant# EP/K010042/1, and sharing the results already obtained with their colleagues at the University of Liverpool. Special thanks also to Jay Sethuraman for guiding us through the literature on online matching, and to Bettina Klaus, Tadashi Hashimoto, and seminar participants in Warwick, Manchester and Toulouse, for stimulating conversations. The comments of three anonymous referees have been very helpful.

View full text

Size versus fairness in the assignment problem☆

Abstract

Section snippets

The problem and the punchline

Related literature

Random assignment with outside options

Efficiency and guaranteed size

Three axioms and two mechanisms

The result

Concluding comments

Theor. Comput. Sci.

J. Econ. Theory

J. Econ. Theory

Theor. Comp. Sci.

J. Econ. Theory

Random serial dictatorship and the core from random endowments in house allocation problems

Econometrica

School choice: a mechanism design approach

Amer. Econ. Rev.

Social welfare in one-sided matching markets without money

On-line bipartite matching made simple

ACM SIGACT News

A simple random assignment problem with a unique solution

Econ. Theory

Designing random allocation mechanisms: theory and applications

Amer. Econ. Rev.