Risk aversion in one-sided matching
aa r X i v : . [ c s . D S ] J a n Risk aversion in one-sided matching
How to find desirable decompositions of probabilistic assignmentsTom Demeulemeester a , Dries Goossens b , Ben Hermans a , and RoelLeus a a Research Center for Operations Research & Statistics, KU Leuven, Belgium b Department of Business Informatics and Operations Management, GhentUniversity, Belgium
January 5, 2021
Abstract
Inspired by real-world applications such as the assignment of pupils toschools or the allocation of social housing, the one-sided matching prob-lem studies how a set of agents can be assigned to a set of objects whenthe agents have preferences over the objects, but not vice versa. For fair-ness reasons, most mechanisms use randomness, and therefore result ina probabilistic assignment. We study the problem of decomposing theseprobabilistic assignments into a weighted sum of ex-post (Pareto-)efficientmatchings, while maximizing the worst-case number of assigned agents.This decomposition preserves all the assignments’ desirable properties,most notably strategy-proofness. For a specific class of probabilistic as-signments, including the assignment by the Probabilistic Serial mecha-nism, we propose a polynomial-time algorithm for this problem that ob-tains a decomposition in which all matchings assign at least the expectednumber of assigned agents by the probabilistic assignment, rounded down,thus achieving the theoretically best possible guarantee. For general prob-abilistic assignments, the problem becomes
N P -hard. For the RandomSerial Dictatorship (RSD) mechanism, we show that the worst-case num-ber of assigned agents by RSD is at least half of the optimal, and thatthis bound is asymptotically tight. Lastly, we propose a column gener-ation framework for the introduced problem, which we evaluate both onrandomly generated data, and on real-world school choice data from theBelgian cities Antwerp and Ghent. Introduction
We study assignment problems in which a set of agents have preferences over aset of indivisible objects , but not vice versa, while the agents can be assigned toat most one object. This problem models a wide range of applications, rangingfrom the assignment of pupils to schools to the allocation of social housing (see[9] and [15] for an overview of applications). In this type of problem, which isgenerally referred to as bipartite matching with one-sided preferences or (Capac-itated) House Allocation, fairness plays a crucial role, and monetary transfersare generally not allowed. Most mechanisms used in practice adopt randomnessto ensure that all agents are treated equally. As a result, we obtain a proba-bilistic assignment that determines for each agent-object pair the probability ofbeing assigned to each other.In order to implement this probabilistic assignment, however, we need todecompose it into a set of corresponding matchings , which state for each agent-object pair whether they are assigned to each other or not. We will focus ontwo aspects of finding such a decomposition that have a great practical appealto decision makers, but that have not yet been studied extensively.First of all, we make the realistic assumption that decision makers are risk-averse, and want to avoid ending up with an “unlucky” random draw that re-sults, for example, in a matching that assigns exceptionally few agents to anobject. We therefore analyze the problem of maximizing the worst-case num-ber of assigned agents by all matchings in the decomposition of a probabilisticassignment. A second natural requirement is that all matchings in the decompo-sition of a probabilistic assignment be desirable on their own. A crucial propertyin the context of one-sided matching is ex-post (Pareto-)efficiency : a matchingis ex-post efficient if there is no other matching that is at least as preferredby all agents and strictly more preferred by at least one agent. If agents wantto exchange their assigned objects, for example, this matching is not ex-postefficient.Importantly, we do not want to harm the desirable properties, such as strategy-proofness , that are related to the probabilistic assignment that is decom-posed. We therefore focus on decomposing the original probabilistic assignmentexactly, rather than to approximate it (contrary to, e.g., [3], [11]).Given a probabilistic assignment that satisfies certain fairness criteria, thewell-known
Birkhoff-von Neumann theorem [8, 20] guarantees that a decom-position over matchings exists. This decomposition, however, is generally notunique, and not all decompositions are equally desirable. The problem of se-lecting the most desirable among all feasible decompositions has received littleattention in the literature. To the best of our knowledge, a similar problemwas only studied by Budish et al. [12], who proposed a method to minimize thevariance in the utility that is experienced by the agents between the differentmatchings in the decomposition. Nevertheless, their method is not capable ofimposing certain desirable criteria, such as ex-post efficiency, upon the match-ings in the decomposition.Our contributions are both theoretical and practical. From a theoretical per-2pective, we introduce two new problems: (1) the problem of finding a decom-position of a probabilistic assignment X that maximizes the worst-case numberof assigned agents, which we refer to as MD ( X ) , and (2) the same problem withthe additional requirement that all matchings in the decomposition be ex-postefficient, which we refer to as MD-SD ( X ) . We propose an algorithm to solveMD ( X ) in polynomial time, and prove that the optimal value of MD ( X ) willalways be equal to the expected number of assigned agents by X , rounded down.When solving MD-SD ( X ) , however, we find that this algorithm can only be usedfor a specific class of probabilistic assignments, which includes the well-studied Probabilistic Serial (PS) [10] mechanism, but not the often used
Random Se-rial Dictatorship (RSD) [1]. For general probabilistic assignments, MD-SD ( X ) becomes N P -hard. Specifically for the decomposition of the probabilistic as-signment by the RSD mechanism, we show that instances exist for which RSDfinds the optimal worst-case number of assigned agents by MD-SD ( X RSD ) , whilein other instances the worst-case number of assigned agents by RSD is only halfof the optimal. Lastly, we propose a column generation framework to find anoptimal decomposition for MD-SD ( X ) .From a practical perspective, we evaluate the proposed methods both ongenerated data sets as well as on two real-world school choice instances fromthe Belgian cities Antwerp and Ghent, and we show that the gain in the worst-case number of assigned agents, in comparison to RSD, supports the adoptionof our methods in practical applications.The remainder of this paper is structured as follows. Section 2 formally de-fines one-sided matching. Next, Section 3 introduces the problems MD ( X ) andMD-SD ( X ) , whereas the complexity and the optimal values of both problemsare discussed in Section 4. Specifically for MD-SD ( X ) , Section 5 introducesa column generation framework, while Section 6 analyzes the performance ofthe proposed methods on existing and randomly generated one-sided matchinginstances. Lastly, Section 7 concludes. A one-sided matching problem is defined by a four-tuple ( N, O, >, q ) , where N is a finite set of agents, and O is a finite set of objects to which the agents in N want to be assigned. The preference profile > = ( > , . . . , > | N | ) contains foreach agent i ∈ N a strict ordering > i over the objects in O ∪ ∅ , where ∅ refersto the outside option of not being assigned to any object. We write j > i k ifagent i ∈ N prefers object j to object k , and ∅ > i l if she prefers the outsideoption over object l , with j, k, l ∈ O . Moreover, the capacities q = ( q , . . . , q | O | ) determine for each object j ∈ O the maximum number of agents q j ∈ N thatcan be assigned to it. Denote the set of all one-sided matching instances by I .A matching M = [ m ij ] is a binary matrix, indexed by all agents and objects,in which m ij = 1 if M assigns agent i ∈ N to object j ∈ O , and m ij = 0 otherwise. An alternative way to indicate that agent i ∈ N is assigned to object j ∈ O is to state that M ( i ) = j . Similarly, a probabilistic assignment X = [ x ij ] ,3ith x ij ∈ [0 , , can be interpreted as the probabilities with which the agentsin N are assigned to the objects in O . A probabilistic assignment is feasible if the sum of the allocation probabilities of each agent is at most one, and thesum of the allocation probabilities of each object does not exceed the object’scapacity, i.e.,(i) P j ∈ O x ij ≤ , ∀ i ∈ N ,(ii) P i ∈ N x ij ≤ q j , ∀ j ∈ O .Denote the set of all feasible matchings by M , and the set of all feasibleprobabilistic assignments by ∆ M , with M ⊆ ∆ M . Moreover, denote the ex-pected number of assigned agents in a probabilistic assignment X by µ ( X ) = P ( i,j ) ∈ N × O x ij . Note that the number of agents that is assigned to an objectin matching M ∈ M equals µ ( M ) . In most practical applications, one does not make a decision by using the proba-bilistic assignment directly. Instead, the assignment is typically implemented bya lottery over matchings, in which the weights are chosen such that the expectedresult is equivalent to the initial probabilistic assignment. A natural questionthat arises is over which sets of feasible matchings a probabilistic assignmentcan be implemented in this way. Budish et al. [12] formalized this using thefollowing definition:
Definition 1.
A feasible probabilistic assignment X ∈ ∆ M is implementableover M ′ , where M ′ = { M t } Tt =1 ⊆ M is a subset of T feasible matchings, ifthere exist non-negative numbers λ = { λ t } Tt =1 , with P Tt =1 λ t = 1 , such that X = T X t =1 λ t M t . Given an arbitrary feasible probabilistic assignment X ∈ ∆ M , the well-known Birkhoff-von Neumann theorem ensures that there will always exist asubset of feasible matchings over which X can be implemented. Birkhoff [8]and von Neumann [20] first proved this result for instances where the numberof unit-capacity objects is equal to the number of agents. Budish et al. [12]later generalized their result to settings with an arbitrary number of agents andobjects, and arbitrary capacities: Theorem 1 (Birkhoff-von Neumann theorem [8, 12, 20]) . All feasible proba-bilistic assignments X ∈ ∆ M are implementable over M . This means that each feasible probabilistic assignment X can be written asa convex combination of a set of feasible matchings { M t } Tt =1 ⊆ M . In orderto implement X , it suffices to randomize over the matchings in this convexcombination in such a way that matching M t is selected as the final matching4ith a probability equal to its weight λ t in the convex combination. Considerthe following definition, which formalizes the difference between implementingand decomposing a probabilistic assignment: Definition 2.
Given a probabilistic assignment X ∈ ∆ M that is implementableover M ′ = { M t } Tt =1 ⊆ M with weights λ = { λ t } Tt =1 ,(i) an implementation of X is an algorithm that randomly selects a singlematching M t ∈ M ′ according to the probability distribution defined bythe weights λ ;(ii) a decomposition of X is the tuple ( M ′ , λ ) .In other words, an implementation of a probabilistic assignment is an algo-rithm that implicitly describes an underlying decomposition by randomly select-ing each of the matchings with a probability that is equal to the correspondingweight in the decomposition. In most practical applications, therefore, an im-plementation of a probabilistic assignment is needed to make a decision, andnot its entire decomposition. To illustrate the introduced terminology, considerthe following example. Example 1.
Let N = { , , , } be a set of agents and O = { a, b, c } be a setof objects. The capacities of the objects are equal to q = (2 , , . Considerthe following preference lists > i of the agents i ∈ N over the objects j ∈ O , and consider the feasible probabilistic assignment X (in which the agentscorrespond to the rows and the objects to the columns):agents , a > b > c > ∅ agents , a > ∅ > b > c X = / /
12 1 / / /
12 1 / / / Agents 1 and 2 prefer object a to object b , and object b to object c , whileagents 3 and 4 only prefer object a to the outside option of not being assignedto any object. Moreover, agent 1 has a probability of of being assigned toobject b in X . We can rewrite X in the following way: X = 512 + 512 + 112 + 112 . Denote the four matchings in the decomposition of X by M , M , M and M .Thus, probabilistic assignment X is implementable over { M t } t =1 . Observethat µ ( X ) = 3 and that every matching M t also assigns µ ( M t ) = 3 agents toan object, for t = 1 , . . . , . In one-sided matching, objects are indifferent about which agents will be as-signed to them. Therefore, there may exist ties between agents that have to be5roken in order to obtain a matching. Define a tie-breaking rule σ = [ σ j ] for j ∈ O , where σ j = ( σ j (1) , . . . , σ j ( | N | )) is a strict ordering over the agents in N .When object j ’s capacity is insufficient, tie-breaking rule σ j is used to decidewhich agent should be assigned to object j , among all agents who prefer thatobject equally. Denote the set of all tie-breaking rules by Σ .A deterministic mechanism π : I × Σ
7→ M is a function that returns afeasible matching M π ( I,σ ) ∈ M for each one-sided matching instance I ∈ I ,and for each tie-breaking rule σ ∈ Σ . Similarly, a probabilistic mechanism ψ : I 7→ ∆ M returns a feasible probabilistic assignment X ψ ( I ) ∈ ∆ M for eachone-sided matching instance I ∈ I . Note that a probabilistic mechanism onlyreturns a probabilistic assignment, and not necessarily its implementation ordecomposition. Unless stated differently, the term “mechanism” will refer toprobabilistic mechanisms in the remainder of this paper. Moreover, we willsimply refer to X ψ ( I ) by X ψ when the instance I is clear from the context.When designing a mechanism, there are several desirable properties onewants to satisfy. An optimality concept that has received broad attention inthe context of one-sided matching (e.g., [1], [2]) is ex-post (Pareto-)efficiency . Amatching M ∈ M is ex-post efficient if there is no other matching M ′ ∈ M thatis at least as preferred as M by all agents and strictly more preferred than M by at least one agent. The Serial Dictatorship (SD) mechanism is a surprisinglysimple deterministic mechanism to obtain an ex-post efficient matching. Denotethe set of all tie-breaking rules with identical orderings for all objects by Σ ′ ⊆ Σ .Given a one-sided matching instance I ∈ I and a strict ordering σ ∈ Σ ′ of theagents, the SD mechanism will first assign the first-ranked agent σ (1) to hermost preferred object, then the second-ranked agent σ (2) to her most preferredobject among the objects with remaining capacity, etc. The resulting matchingwill be SD ( I, σ ) = M SD ( I,σ ) . The SD mechanism possesses several desirableproperties beside ex-post efficiency [18]: it is neutral (invariant to relabelingthe objects), nonbossy (no agent can change another agent’s allocation withoutchanging her own), strategy-proof (submitting true preferences is optimal for allagents), and easy to compute. Moreover, given a one-sided matching instance I ∈ I , the SD mechanism can generate any ex-post efficient matching in I [1, 2].We will therefore denote the set of all ex-post efficient matchings by M SD ⊆ M .Despite its desirable properties, the SD mechanism is rarely used in real-world applications, because it is not anonymous : two agents with the samepreference list might be treated differently, simply because one of them appearsbefore the other one in the ordering of the agents that serves as an input for SD.In order to overcome this issue, Abdulkadiroğlu and Sönmez [1] introduced the Random Serial Dictatorship (RSD) mechanism, which will randomly select anordering to which the SD mechanism is applied. Applying the RSD mechanismto a one-sided matching instance I = ( N, O, >, q ) ∈ I will result in a proba-bilistic assignment X RSD ( I ) , which is the equally weighted sum of the matchingsobtained by the SD mechanism for all different orderings σ ∈ Σ ′ : X RSD ( I ) = RSD ( I ) = 1 | N | ! X σ ∈ Σ ′ SD ( I, σ ) . (1)6n the remainder of the paper, we will simply refer to X RSD ( I ) as X RSD if the in-stance is clear from the context. Note that the definition of the RSD mechanismis directly linked to an implementation of X RSD in the sense of Definition 2. Inthe remainder of this paper, we will use the term
RSD mechanism to refer to thefunction that maps an instance I to the probabilistic assignment X RSD ( I ) , andthe term RSD algorithm to refer to the implementation of X RSD ( I ) which resultsin the matching SD ( I, σ ) by randomly selecting a tie-breaking rule σ ∈ Σ ′ .Although the RSD algorithm is widely used for real-world one-sided match-ing problems, Bogomolnaia and Moulin [10] showed that it fails to satisfy thefollowing notion of efficiency that is stronger than ex-post efficiency. Definition 3.
A probabilistic assignment X ∈ ∆ M is ordinally efficient if andonly if there is no probabilistic assignment X ′ ∈ ∆ M that is at least as preferredas X by all agents and strictly more preferred than X by at least one agent.Contrary to the RSD mechanism, the Probabilistic Serial (PS) mechanism,that was developed by Bogomolnaia and Moulin [10], will always result in anordinally efficient assignment. To obtain this probabilistic assignment, assumethat time t runs continuously from 0 to 1. At each point in time, each agent“eats” with a uniform eating speed of one from her most preferred object that hasremaining capacity. At time t = 1 , we obtain the ordinally efficient probabilisticassignment X PS . Note that an implementation of X PS is not specified by itsconstruction, in contrast to X RSD .Next to resulting in an ordinally efficient probabilistic assignment, the PSmechanism satisfies several other desirable properties [10]. Similarly to the RSDmechanism, PS will be anonymous. Moreover, PS will be envy-free (no agent willprefer the assignment probabilities of another agent, for any compatible utilityfunction), while RSD is not. In contrast to the RSD mechanism, however, PSwill not be strategy-proof. In fact, Bogomolnaia and Moulin [10] showed thatno mechanism can satisfy ordinal efficiency, strategy-proofness and anonymityat the same time. More generally, the extent to which probabilistic assignmentscan satisfy certain combinations of properties has been widely studied in theliterature, leading to an interesting series of impossibility results [4, 10, 16, 17,21].One crucial observation in the context of this paper is that many relevantproperties such as ordinal efficiency, anonymity, and strategy-proofness are in-herent to the probabilistic assignment X ∈ ∆ M . Hence, all such propertiesare retained regardless of the decomposition of X . Other properties, such asex-post efficiency, are defined on the level of the matchings in the decompositionof X . To ensure that a decomposition of X satisfies a property of the lattercategory, we thus have to explicitly enforce this property on all matchings inthe decomposition. 7 Maximin decomposition
Although the Birkhoff-von Neumann theorem (Theorem 1) is an important re-sult, it does not guarantee desirable outcomes in practical applications. Fromthe perspective of an individual agent, a decomposition might be undesirablebecause one of the matchings in the decomposition is undesirable in itself (Ex-ample 2). Decision makers might also prefer one decomposition to anotherbecause they are risk-averse (Example 3).
Example 2.
Reconsider the decomposition of probabilistic assignment X inExample 1. The third matching in the decomposition, M , is clearly sub-optimalas it is not ex-post efficient. We know that agent 2 prefers object b to object c .Nevertheless, she is assigned to object c while object b has unused capacity. Asimilar argument holds for agent 1 in M . Example 3.
Consider an instance with four agents and two objects. The ca-pacities of the objects are equal to q = (2 , , and the agents’ preferences are asindicated below. X is a feasible probabilistic assignment for this instance.agents , a > b > ∅ agents , a > ∅ > b X = / / / / / / It can be easily verified that the following two decompositions of X are bothfeasible: X = 12 + 12 = 12 + 12 . Although both decompositions result in the same probabilistic assignment X ,a risk-averse decision maker will probably prefer the second decomposition overthe first one: the second decomposition of X is guaranteed to always assignthree agents to an object, whereas there is a risk that only two agents will beassigned to an object by the first decomposition of X .Inspired by Examples 2 and 3, we identify two criteria that impact thedesirability of a decomposition. First of all, the agents generally have a setof properties that, depending on the context, they want to see satisfied by allmatchings in the decomposition (Example 2). These properties could include,for example, ex-post efficiency, or group-specific quota. Secondly, we argue thatrisk-averse decision makers want to minimize the risk of ending up with anundesirable matching when implementing a probabilistic assignment, so as toavoid, for example, a low average utility of the agents, a low average preferencefor the received object, or a low total number of assigned agents (Example 3).In this paper, we focus on maximizing the worst-case number of assigned agents.Our column generation framework in Section 5, however, can be easily adaptedto find a decomposition that optimizes other worst-case criteria.8ore specifically, we study the two following, closely related problems. First,consider the problem of maximizing the worst-case number of assigned agentsin the decomposition of a probabilistic assignment X ∈ ∆ M , while allowing allfeasible matchings in M to be used in this decomposition. We will refer to thisproblem as Maximin Decomposition of X , or simply MD ( X ) . Denoting allmatchings in M that assign at least k ∈ Z agents to an object by M k ⊆ M , weformally define MD ( X ) as follows: Definition 4.
Maximin Decomposition of X (MD ( X ) ). Given a one-sidedmatching instance I = ( N, O, >, q ) ∈ I , and a probabilistic assignment X ∈ ∆ M , find a decomposition of X over M k that maximizes k .Second, we define a problem similar to MD ( X ) , but with the additionalrequirement that all matchings in the decomposition of X be ex-post efficient.Denote the set of all ex-post efficient matchings that assign at least k ∈ Z agentsto an object by M SD k ⊆ M SD . Definition 5.
Maximin Decomposition of X over M SD (MD-SD ( X ) ).Given a one-sided matching instance I = ( N, O, >, q ) ∈ I , and a probabilisticassignment X ∈ ∆ M , find a decomposition of X over M SD k that maximizes k .In the remainder of this paper, we will refer to the largest value of k forwhich X is implementable over M k , resp. M SD k , as the optimal value of MD ( X ) ,resp. MD-SD ( X ) , and we will denote the optimal value of MD-SD ( X ) by z ( X ) .Denote the floor-operator by ⌊·⌋ , and the ceiling-operator by ⌈·⌉ . We canidentify the following intuitive upper bound on the optimal values of MD ( X ) and MD-SD ( X ) , which is based on the observation that it is not possible todecompose a probabilistic assignment X by only using matchings that assignstrictly more agents to an object than the expected number of assigned agentsin X . Proposition 1.
The optimal values of MD ( X ) and MD-SD ( X ) are upper boundedby ⌊ µ ( X ) ⌋ , for all X ∈ ∆ M . In this section, we examine the complexity and the optimal values of MD ( X ) and MD-SD ( X ) . We show in Section 4.1 that the optimal value of MD ( X ) is ⌊ µ ( X ) ⌋ , for all X ∈ ∆ M , and we propose a polynomial-time algorithm to find acorresponding decomposition. In Section 4.2, we establish that for MD-SD ( X ) the same result only holds for a specific class of probabilistic matchings, whereasMD-SD ( X ) is N P -hard for general probabilistic assignments. Specifically forMD-SD ( X RSD ) , in Section 4.3 we provide a lower and upper bound on theoptimal gain in worst-case number of assigned agents compared to the RSDalgorithm. 9 .1 Complexity of MD ( X ) The following theorem states that obtaining a decomposition of X that attainsthe upper bound ⌊ µ ( X ) ⌋ of Proposition 1 can be achieved in polynomial time.The existence of such a decomposition follows from a result by Budish et al.[12, Theorem 1] for a more general constraint structure known as a bihierarchy .Budish et al. further describe an implementation in the sense of Definition 2that runs in polynomial time. The main novelty of our result is that it extendsthe method of Budish et al. such that it yields a complete decomposition inpolynomial time. Theorem 2.
For every probabilistic assignment X ∈ ∆ M , we can find a de-composition ( M ′ , λ ) of X in polynomial time in which each matching M ∈ M ′ assigns either ⌊ µ ( X ) ⌋ or ⌈ µ ( X ) ⌉ agents.Proof. Consider a probabilistic assignment X ∈ ∆ M and assume that µ ( X ) ∈ N . We will describe an iterative approach to obtain a decomposition of X in which all matchings assign exactly µ ( X ) agents. The case where µ ( X ) isfractional can be dealt with by adding an additional row and column to X ,corresponding to a dummy agent a and a unit-capacity object o , where theunique non-zero element ( a, o ) has an assignment probability equal to ⌈ µ ( X ) ⌉ − µ ( X ) . A decomposition that always assigns ⌈ µ ( X ) ⌉ agents in this modifiedinstance, then guarantees to assign either ⌊ µ ( X ) ⌋ or ⌈ µ ( X ) ⌉ non-dummy agentsin the original instance.Following Budish et al. [12], we can rewrite the conditions for X to be a fea-sible probabilistic assignment using a constraint structure H ⊂ N × O , where H contains a constraint set S ⊆ N × O with associated quota q S for each agent,for each object, and for each agent-object pair. In particular, define H = [ ( i,j ) ∈ N × O { ( i, j ) } ∪ [ i ∈ N ( { i } × O ) ∪ [ j ∈ O ( N × { j } ) , and let q S = q j if S = N × { j } for some j ∈ O , and q S = 1 for all other S ∈ H .A probabilistic assignment X is then feasible if and only if ≤ P ( i,j ) ∈ S x ij ≤ q S for every S ∈ H . Observe that |H| = | N | · | O | + | N | + | O | = O ( | N | · | O | ) .Each iteration t of our iterative approach starts from a given probabilisticassignment X t , where X t = X if t = 1 , and X t is defined in the previousiteration otherwise. If X t is a matching, i.e., if X t ∈ M , then the procedureterminates and we define M t = X t . Otherwise, we apply to X t a modifiedversion of the algorithm by Budish et al. [12], which we describe in Appendix A,in order to obtain a matching M t ∈ M with the following two properties:(i) µ ( M t ) = µ ( X t ) ;(ii) if P ( i,j ) ∈ S x tij is integer, then P ( i,j ) ∈ S m tij = P ( i,j ) ∈ S x tij , for each S ∈ H .Next, let λ t be the largest value for λ ∈ R such that X t + λ ( X t − M t ) ∈ ∆ M ,where λ t can be determined by checking which of the constraint sets S ∈ H X t +1 = X t + λ t ( X t − M t ) ,and proceeds to the next iteration.Now assume that our procedure terminates in T iterations. It follows byexpanding the above recursion that the matchings { M t } Tt =1 and weights ˆ λ = { ˆ λ t } Tt =1 , with ˆ λ t = λ t λ t if t = 1 , λ t λ t Q t − u =1 11+ λ u if t ∈ { , . . . , T − } , Q t − u =1 11+ λ u if t = T ,yield a decomposition of X in which all matchings assign µ ( X ) agents. Thislatter guarantee follows from Property (i) of M t and by definition of X t +1 foreach t = 1 , . . . , T − .We claim that our procedure terminates in T ≤ |H| iterations. For eachprobabilistic assignment X ′ ∈ ∆ M , denote by τ ( X ′ ) = |{ S ∈ H : P ( i,j ) ∈ S x ′ ij ∈ N }| the number of constraint sets S ∈ H that are integer in X ′ . Observe that τ ( X ′ ) ≤|H| , where equality is reached if and only if X ′ ∈ M . To prove our claim, itthus suffices to show that, for every iteration t with X t / ∈ M , it holds that τ ( X t +1 ) > τ ( X t ) .Consider an arbitrary iteration t in which X t / ∈ M , and denote R t = X t − M t . Observe that R t = 0 since X t = M t . By Property (ii) of M t , it holds forevery S ∈ H for which P ( i,j ) ∈ S x tij is integer that P ( i,j ) ∈ S r tij = 0 and thus, bydefinition of X t +1 , that P ( i,j ) ∈ S x t +1 ij = P ( i,j ) ∈ S x tij . Property (ii) of M t alsoimplies that P ( i,j ) ∈ S x tij is fractional for every S ∈ H with P ( i,j ) ∈ S r tij = 0 ,and thus there exists a λ > for which X t + λR t ∈ ∆ M . The maximalchoice λ t for λ is then such that there is a S ′ ∈ H with P ( i,j ) ∈ S x tij fractionaland P ( i,j ) ∈ S x t +1 ij integer (because a new constraint becomes binding). Thisyields τ ( X t +1 ) > τ ( X t ) , which proves our claim.Since the modified algorithm of Budish et al. [12], described in Appendix A,runs in time O ( |H| min {| N | , | O |} ) and since finding λ t in a given iteration t < T can be done in time O ( |H| ) , we can obtain our desired decomposition of X intime O ( |H| min {| N | , | O |} ) .Theorem 2 implies that the optimal value of MD ( X ) is equal to ⌊ µ ( X ) ⌋ ,for any X ∈ ∆ M , and that the corresponding decomposition can be found inpolynomial time. In other words, if the decision maker does not require specificproperties to be satisfied by all matchings in the decomposition of a probabilisticassignment X , it is always possible to decompose X by only using matchingsthat assign at least ⌊ µ ( X ) ⌋ agents to an object. X ) In contrast to MD ( X ) , the complexity of MD-SD ( X ) depends on the propertiesof the probabilistic assignment X . Consider the following property.11 efinition 6. (Aziz et al. [6]) A probabilistic assignment X ∈ ∆ M is robustex-post efficient if and only if every possible decomposition of X only containsex-post efficient matchings.Denote the set of robust ex-post efficient probabilistic assignments by ∆ M R ⊆ ∆ M . Define a probabilistic assignment X ∈ ∆ M to be ex-post efficient if adecomposition of X exists that only contains ex-post efficient matchings. Thefollowing proposition shows the relationship between the three efficiency con-cepts for probabilistic assignments that have been mentioned in this paper. Proposition 2. (Aziz et al. [6]) Ordinal efficiency of a probabilistic assignmentimplies robust ex-post efficiency, which in turn implies ex-post efficiency.
Proposition 2 has the following implications for the PS mechanism.
Corollary 1.
The PS mechanism will always return a probabilistic assignmentthat is robust ex-post efficient.
Moreover, with respect to the RSD mechanism, Aziz et al. [6] obtained thefollowing result (see Example 2, where X = X RSD , and M , M / ∈ M SD ). Proposition 3. (Aziz et al. [6]) The RSD mechanism may return a probabilisticassignment that is not robust ex-post efficient.
Using these results, we can now identify an important class of probabilisticassignments for which we obtain the same results as in Theorem 2, i.e., for whichthe optimal value of MD-SD ( X ) is guaranteed to be equal to its upper bound ⌊ µ ( X ) ⌋ , and for which we can find a decomposition in polynomial time. Theorem 3.
For every robust ex-post efficient probabilistic assignment X ∈ ∆ M R , we can find a decomposition ( M ′ , λ ) of X in polynomial time in whicheach matching M ∈ M ′ assigns either ⌊ µ ( X ) ⌋ of ⌈ µ ( X ) ⌉ agents.Proof. Consider an arbitrary robust ex-post efficient probabilistic assignment X ∈ ∆ M R . By using the polynomial-time algorithm from the proof of Theo-rem 2, we can find a decomposition of X in which all matchings assign either ⌊ µ ( X ) ⌋ or ⌈ µ ( X ) ⌉ agents to an object. By definition of robust ex-post efficiency,all matchings in this decomposition will be ex-post efficient.Combining this result with Corollary 1, we can conclude that we can decom-pose the probabilistic assignment X PS in polynomial time over a set of ex-postefficient matchings that assign at least ⌊ µ ( X PS ) ⌋ agents to an object. FromProposition 3, however, we know that we cannot guarantee the ex-post effi-ciency of the matchings in the decomposition if we apply the same approachto X RSD .Although Theorem 3 showed that MD-SD ( X ) is polynomially solvable when X is robust ex-post efficient, the following result holds for general probabilistic as-signments. Theorem 4.
MD-SD ( X ) is N P -hard in general. roof. The result immediately follows from the fact that it is
N P -complete todetermine whether an arbitrary probabilistic assignment can be implementedover ex-post efficient matchings [6].Note that the proof of Theorem 4 does not apply for X RSD , because X RSD isguaranteed to have at least one decomposition over ex-post efficient matchings,given by Equation (1). Hence, the complexity status of MD-SD ( X RSD ) is open. X RSD ) By Theorem 3, we know that the optimal value z ( X ) of MD-SD ( X ) equals itsupper bound ⌊ µ ( X ) ⌋ when X is robust ex-post efficient. In this section, we ana-lyze the possible range of values of z ( X ) when X / ∈ ∆ M R . More specifically, wefocus on the RSD mechanism, given its practical and theoretical importance: weestablish that ⌊ µ ( X RSD ) ⌋ is an asymptotically tight lower bound on z ( X RSD ) ,whereas twice the worst-case number of assigned agents by any ex-post efficientmatching is an asymptotically tight upper bound on z ( X RSD ) .Denoting the cardinality of a minimum-cardinality ex-post efficient matchingin a one-sided matching instance I ∈ I by p − ( I ) , we can identify the followingbounds on z ( X RSD ( I ) ) . Proposition 4. ⌊ µ ( X RSD ( I ) ) ⌋ < z ( X RSD ( I ) ) < p − ( I ) , for all I ∈ I .Proof. By definition of MD-SD ( X RSD ( I ) ) , and given the upper bound on z ( X RSD ( I ) ) from Proposition 1, we know that p − ( I ) ≤ z ( X RSD ( I ) ) ≤ ⌊ µ ( X RSD ( I ) ) ⌋ . Hence,what remains to be shown is that ⌊ µ ( X RSD ( I ) ⌋ < p − ( I ) .Denote by p + ( I ) the cardinality of a maximum-cardinality ex-post efficientmatching in a one-sided matching instance I ∈ I , and note that p − ( I ) ≤ p + ( I ) .We will consider two cases. First, if p − ( I ) = p + ( I ) , clearly ⌊ µ ( X RSD ( I ) ) ⌋
There exists a family of instances I k ∈ I , with k ≥ an integer,for which lim k →∞ z ( X RSD ( I k ) ) ⌊ µ ( X RSD ( I k ) ) ⌋ = 12 . Proof.
Consider a family of one-sided matching instances I k = ( N, O, >, q ) ,with k ≥ an integer. Let N = { , , . . . , k } be a set of k agents, and let O = { o , o , . . . , o k +1 } be a set of k + 1 objects. Let the capacities of the objects13n O be equal to q = ( k, , , . . . , . Moreover, let the preference list > i of eachagent i ∈ N be equal to > i = ( o > o > . . . > o k +1 > ∅ if i ≤ k,o > ∅ > o > . . . > o k +1 if i > k. We now show that µ ( X RSD ( I k ) ) = 2 k − and z ( X RSD ( I k ) ) = k for every k ≥ . First, consider an arbitrary strict ordering σ ∈ Σ ′ of the agents that is used bythe SD mechanism to construct an ex-post efficient matching SD ( I k , σ ) ∈ M SD .The preference lists are such that each agent i ≤ k always receives an objectand such that an agent i > k only receives an object if it is among the first k agents in the ordering σ . Hence, if we denote by τ i the probability that agent i receives an object by applying the SD mechanism, then we obtain that τ i = 1 if i ≤ k , and, since there are k agents, that τ i = k/k = 1 /k if i > k . It followsthat µ ( X RSD ( I k ) ) = k X i =1 τ i = k X i =1 τ i + k X i = k +1 τ i = k + k − kk = 2 k − . Second, to prove that z ( X RSD ( I k ) ) = k , recall that the set of all ex-postefficient matchings that assign at least r ∈ Z agents to an object is denoted by M SD r ⊆ M SD . In instance I k , denote the unique ex-post efficient matching thatassigns k agents to an object by M ∈ M SD , and note that all other matchingsin M SD assign strictly more than k agents to an object. Moreover, note thatmatching SD ( I k , σ ) will only be equal to M if all agents i ≤ k are ranked beforeall k − k other agents i > k in ordering σ . Denote the probability that the RSDalgorithm will obtain matching M in instance I k by θ k = (cid:0) k k (cid:1) − > . Denotingthe weight of matching M t ∈ M SD by λ t ≥ , every feasible decomposition of X RSD ( I k ) should satisfy X t : M t ∈M SD k +1 λ t = k X i =1 x RSD ( I k ) i = 1 − θ k < , (2)because we know that every matching in M SD k +1 assigns exactly one agent i ≤ k to object o . As M SD = M SD k +1 ∪ M and M / ∈ M SD k +1 , Equation (2) impliesthat λ > . This, in turn, implies that z ( X RSD ( I k ) ) = k , because every feasibledecomposition of X RSD ( I k ) over ex-post efficient matchings has a strictly positiveweight for the matching M that assigns exactly k agents to an object.Combining equalities µ ( X RSD ( I k ) ) = 2 k − and z ( X RSD ( I k ) ) = k , we obtainthat lim k →∞ z ( X RSD ( I k ) ) ⌊ µ ( X RSD ( I k ) ) ⌋ = lim k →∞ k k − , which proves the theorem. 14econdly, we show that there exist instances where z ( X RSD ) realizes themaximum possible improvement in the worst-case number of assigned agentswith respect to the RSD algorithm, as the ratio of z ( X RSD ) over p − ( I ) convergesto 2. Theorem 6.
There exists a family of instances I ℓ ∈ I , with ℓ ≥ an integer,for which lim ℓ →∞ z ( X RSD ( I ℓ ) ) p − ( I ℓ ) = 2 . Proof.
Consider a family of one-sided matching instances I ℓ = ( N, O, >, q ) ,with ℓ ≥ an integer. Let N = { , , . . . , ℓ } be a set of ℓ agents, and let O = { o , o } be a set of two objects. Let the capacities of the objects in O beequal to q = ( ℓ, ℓ ) . Moreover, let the preference list > i of each agent i ∈ N beequal to > i = ( o > o > ∅ if i ≤ ℓ,o > ∅ > o if i > ℓ. First, the minimum-cardinality ex-post efficient matching in this instancewill be equal to SD ( I ℓ , σ ) for every strict ordering σ ∈ Σ ′ that ranks all agents i ≤ ℓ before agents i > ℓ . Hence, p − ( I ℓ ) = ℓ .Second, we show that we can decompose X RSD ( I ℓ ) over ex-post efficientmatchings that all assign ℓ − agents to an object, i.e., z ( X RSD ( I ℓ ) ) = 2 ℓ − .To construct a decomposition of X RSD ( I ℓ ) over M SD ℓ − , consider the followingset of ℓ matchings { M t } ℓt =1 , where M t ( i ) = o if i = t,o if i ≤ ℓ and i = t,o if i ∈ ∪ ℓγ =2 { ( ℓ − t + γ } , ∅ otherwise . Clearly, all matchings in { M t } ℓt =1 are ex-post efficient and assign exactly ℓ − agents to an object. Note that each agent is assigned to o by exactly onematching in { M t } ℓt =1 , and that each agent i ≤ ℓ will always be assigned to anobject in these matchings. Moreover, note that x RSD ( I ℓ ) ij = / ℓ if j = 1 , − / ℓ if j = 2 and i ≤ ℓ, if j = 2 and i > ℓ. Therefore, a feasible decomposition of X RSD ( I ℓ ) over M SD ℓ − can be constructedby giving matching M t a weight of λ t = / ℓ , for each t = 1 , . . . , ℓ .Combining both results, we obtain that lim ℓ →∞ z ( X RSD ( I ℓ ) ) p − ( I ℓ ) = lim ℓ →∞ ℓ − ℓ = 2 , which proves the theorem. 15 Column generation
In order to be able to solve MD-SD ( X ) to optimality for reasonably large in-stances when X is not robust ex-post efficient, we introduce a column generation framework. Recall that the total number of tie-breaking rules | Σ ′ | that can byused as an input for the SD mechanism to obtain an ex-post efficient matchingis equal to | N | ! , which implies that the number of ex-post efficient matchings |M SD | can be exponential in the number of agents. A linear programming modelthat determines the weights of each matching in the decomposition would there-fore contain an exponential number of decision variables. A common approachto tackle this issue is column generation (e.g., [7]). The idea is to first solve themodel with a subset of the variables in the restricted master problem . Subse-quently, we check whether this leads to an optimal solution over all variables.If this is not the case, a new variable will be added to the model, which is thensolved again until the restricted master problem returns an optimal solutionover all variables. New variables are generated by a separate subproblem calledthe pricing problem , which identifies a variable with a negative reduced cost orshows that no such variable exists. We solve MD-SD ( X ) by means of a binary search for z ( X ) , in which eachstep consists of solving a linear program to check whether X = [ x ij ] , with ( i, j ) ∈ N × O , is implementable over M SD k . The restricted master problem fora given value of k in this binary search is the following linear program [ RMP ] .Denote ˜ M SD k ⊆ M SD k to be the subset of the variables that are considered inthe restricted master problem. Moreover, define decision variables λ t to be theweight of matching M t ∈ ˜ M SD k in the decomposition of X , and let s be anauxiliary variable. [ RMP ] min s (3a)s.t. X t : M t ∈ ˜ M SD k λ t m tij − x ij ≥ ∀ ( i, j ) ∈ N × O, (3b) X t : M t ∈ ˜ M SD k λ t m tij − x ij ≤ s ∀ ( i, j ) ∈ N × O, (3c) X t : M t ∈ ˜ M SD k λ t = 1 , (3d) λ t ≥ ∀ t : M t ∈ ˜ M SD k . (3e)Constraints (3b) impose that the probability of agent i being assigned toobject j is at least as high in the solution as in the original probabilistic assign-ment X . Constraints (3c) ensure, together with the objective function, that s will be equal to the maximum difference between the constructed probabilisticassignment and X , over all agent-object pairs. As a result, a feasible decom-position of X over ˜ M SD k has been found if the objective value of [ RMP ] equals16ero. Furthermore, constraints (3d) and (3e) state that the weights in the de-composition are non-negative and sum up to one. To ensure the feasibility of [ RMP ] for any subset ˜ M SD k , we can include an artificial super-column M inwhich m ij = 1 for all agent-object pairs ( i, j ) ∈ N × O . Moreover, to reduce thenumber of constraints, we can remove redundant constraints (3b) when x ij = 0 ,and constraints (3c) when x ij = 1 .Denote the dual variables related to the constraints (3b), (3c), and (3d) by u ij ≥ , v ij ≤ , and w , respectively, for each ( i, j ) ∈ N × O . Then a solutionof [ RMP ] with dual values u ∗ = [ u ∗ ij ] , v ∗ = [ v ∗ ij ] , and w ∗ is optimal over allvariables in M SD k if no matching M ∈ M SD k has a negative reduced cost, i.e., iffor all matchings M ∈ M SD k X ( i,j ) ∈ N × O − m tij ( u ∗ ij + v ∗ ij ) − w ∗ ≥ . (4) In order to determine the existence of a matching that violates (4), we formulatea pricing problem to find a feasible ex-post efficient matching M ∈ M SD k thatminimizes the left-hand side of inequality (4) for given dual values u ∗ , v ∗ , and w ∗ . This pricing problem contains two sets of constraints: constraints (5b)-(5e)ensure that the constructed matching is feasible and assigns at least k agentsto an object, while constraints (5f)-(5j) guarantee that the matching is ex-postefficient.First, define binary decision variables m ij , where m ij = 1 if agent i ∈ N isassigned to object j ∈ O , and m ij = 0 otherwise. Then, the following integerlinear program [ PP ] will find a minimum-weight matching that assigns at least k agents to an object. [ PP ] min X i ∈ N X j ∈ O − m ij (cid:0) u ∗ ij + v ∗ ij (cid:17) − w ∗ (5a)s.t. X j ∈ O m ij ≤ ∀ i ∈ N, (5b) X i ∈ N m ij ≤ q j ∀ j ∈ O, (5c) X i ∈ N X j ∈ O m ij ≥ k, (5d) m ij ∈ { , } ∀ ( i, j ) ∈ N × O. (5e)The objective function of [ PP ] minimizes the left-hand side of inequality (4),and will find a matching with a negative reduced cost if and only if the objectivevalue is strictly negative. In order to reduce the number of variables in [ PP ] we can fix m ij to one (zero) if agent i is always (never) assigned to object j bythe probabilistic assignment X that we want to decompose. Note that [ PP ] is, except for the inequality in constraint (5d), identical to the formulation by17ell’Amico and Martello [14] for the k - cardinality Assignment Problem ( k -AP), which finds a minimum-weight matching that assigns exactly k agentsto an object. They showed that k -AP is polynomially solvable because of thetotal unimodularity of the constraint matrix. The same holds for [ PP ] if we donot require the final matching to be ex-post efficient.Second, we incorporate the three necessary and sufficient conditions for amatching to be ex-post efficient that were introduced by Abraham et al. [2].Firstly, we call a matching M ∈ M maximal if there is no agent-object pair ( i, j ) ∈ N × O for which i is not assigned to any object in M , j has unassignedcapacity in M , and i prefers j to the outside option of not being assigned. Sec-ondly, a matching M is trade-in-free if there is no agent-object pair ( i, j ) ∈ N × O such that i is assigned in M , j has unassigned capacity in M , and i prefers j to M ( i ) . Lastly, M is cyclic coalition-free if no sequence C = h i , i , . . . , i r − i exists (with r ≥ ) for which i l is assigned in M for all l = 0 , , . . . , r − , and i l prefers M ( i l +1 ) to M ( i l ) , for ≤ l ≤ r − (all subscripts are taken modulo r ).The following result holds: Proposition 5. (Abraham et al. [2]) A matching M ∈ M is ex-post efficientif and only if M is maximal, trade-in-free, and cyclic coalition-free. Define an auxiliary binary variable r j for each object j ∈ O . We append thefollowing constraints to [ PP ] . [ PP PE ] X i ∈ N m ij q j ≥ r j ∀ j ∈ O, (5f) − X l ∈ O m il ≤ r j ∀ ( i, j ) ∈ N × O : j > i ∅ , (5g) m il ≤ r j ∀ i ∈ N, j, l ∈ O : j > i l > i ∅ , (5h) X ( i,j ) ∈ C m ij ≤ | C | − for all cycles C in G ( M ) , (5i) r j ∈ { , } ∀ j ∈ O. (5j)The first set of constraints (5f) ensure that r j equals zero when object j hasunused capacity. Secondly, constraints (5g) make sure that the final matchingis maximal: if agent i is not assigned to any object, then none of the objectsthat she prefers to the outside option can have unused capacity. Similarly,constraints (5h) guarantee that the final matching is trade-in-free. Lastly, con-straints (5i) ensure that the final matching is cyclic coalition-free. These con-straints make use of the envy graph of a matching. Definition 7.
The envy graph G ( M ) of a matching M ∈ M in a one-sidedmatching instance I = ( N, O, >, q ) ∈ I is a directed graph that contains avertex for each agent i ∈ N , and an edge from agent i to agent t ∈ N ifagent i would rather be assigned to the object to which agent t is assigned, i.e., M ( t ) > i M ( i ) . 18learly, M is cyclic coalition-free if and only if its envy graph G ( M ) isacyclic. Constraints (5i) achieve this by forbidding each potential cycle C inthe envy graph G ( M ) . Due to the exponential number of cycle-eliminationconstraints (5i), we adopt a cutting plane approach: each time the pricingproblem finds an optimal integer solution M , we construct the envy graph G ( M ) and check whether it contains a cycle C ′ . If this is the case, we add a cycle-elimination constraint for C ′ to [ PP PE ] .Whereas we argued that the basic pricing problem [ PP ] can be solved inpolynomial time, the addition of constraints (5f)-(5j) to [ PP ] causes the prob-lem to become computationally hard. Denote by [ PP ] the combination of for-mulations [ PP ] and [ PP PE ] . Consider the following problem definition. Definition 8. ex-post Efficient at Least k -cardinality Assignment (EL k -AP). Given a one-sided matching instance I = ( N, O, >, q ) ∈ I , an integer k ≤ | N | , and weights w ij ∈ R for all agent-object pairs ( i, j ) ∈ N × O , find aminimum-weight ex-post efficient matching X ∈ M SD k .The following result establishes the complexity of EL k -AP. Proposition 6. EL k -AP is N P -hard.Proof.
Saban and Sethuraman [18] showed that determining whether or notthere exists an ex-post efficient matching that assigns a given agent i ∈ N to agiven object j ∈ O is N P -complete. Denote this problem by
SD Feasibility .We will show that there exist parameter values for which
SD Feasibility canbe solved by the [ PP ] formulation for EL k -AP. Consider an arbitrary agent-object pair ( i ′ , j ′ ) ∈ N × O . Note that for k = 0 , and for dual values ˜ u ∗ , ˜ v ∗ ,and ˜ w ∗ , with ˜ w ∗ = 0 , and ˜ u ∗ ij + ˜ v ∗ ij = ( if ( i, j ) = ( i ′ , j ′ ) , otherwise,the objective function of [ PP ] becomes min − m i ′ j ′ . In other words, an ex-postefficient matching that assigns agent i ′ to object j ′ will exist if and only if theoptimal objective value of [ PP ] equals minus one, for k = 0 , and for dual values ˜ u ∗ , ˜ v ∗ , and ˜ w ∗ . In this section, we discuss the computational aspects of decomposing X RSD ,which we test both on existing and on newly generated instances. The choice toapply our methods on the RSD mechanism is motivated by its widespread usein practical applications. In particular, these experiments allow us to gain thetwo following main insights. Firstly, despite the worst-case result of Theorem 5,the optimal value of MD-SD ( X RSD ) is equal to its upper bound ⌊ µ ( X RSD ) ⌋ forall generated instances that were solved to optimality. Secondly, we observethat even when the column generation framework does not obtain a guaranteed19ptimal solution within the imposed runtime limit, its results can still be highlyvaluable in real-world applications, and we provide two real-world school choicecases from the Belgian cities of Antwerp and Ghent as examples. All experiments were implemented with C++, compiled with Microsoft VisualC++ 2019, and run on an Intel Core i7-7700 processor running at 3.60 GHz,with 16GB of RAM memory on a Windows 10 64-bit OS. All linear and inte-ger programs are solved using IBM ILOG CPLEX 12.9, implemented in C++with Concert Technology, with default parameter settings, and with a precisionof − to avoid numerical issues.To the best of our knowledge, no set of benchmark instances exists for one-sided matching. We have therefore developed a parameterized data generationtool that is based on the properties of two real-world data sets from the schoolchoice problem in the Belgian cities Antwerp and Ghent, and we generated in-stances using the default parameter values (Table 2). Appendix B contains adetailed description of the data generation, and its repositories and the evaluatedinstances are available online ( https://github.com/DemeulemeesterT/GOSMI.git ).Moreover, in order to decompose X RSD , this matrix first needs to be com-puted. Although the RSD algorithm can be easily executed by randomly se-lecting an ordering σ ∈ Σ ′ , Aziz et al. [5] showed that computing X RSD is P -complete, and thus intractable. Moreover, Saban and Sethuraman [18]found that X RSD cannot even be approximated efficiently. We therefore esti-mate X RSD in this section by computing Equation (1) over a random sample ˆΣ ′ ⊆ Σ ′ of the orderings, and we opted for sample size | ˆΣ ′ | = 10 , . Because ˆΣ ′ is a random sample, both the resulting estimate of X RSD and the solution ofthe column generation framework are unbiased.Similarly, given the
N P -hardness of finding a minimum-cardinality ex-postefficient matching [2], we estimate p − ( I ) , i.e., the cardinality of the minimum-cardinality ex-post efficient matching in a one-sided matching instance I ∈ I ,by taking the minimum-cardinality matching among the matchings that aregenerated by the SD mechanism for the tie-breaking rules in ˆΣ ′ .In all experiments, we include an initial subset of matchings that assign atleast k agents to an object as variables in the initial restricted master prob-lem [ RMP ] , when solving the column generation framework for k ∈ Z . Here, atrade-off emerges: whereas a larger sample size is likely to decrease the numberof times the pricing problem will be called, it also tends to increase the solu-tion time of the restricted master problem. To balance these two effects, weadopt a sample size of , random orderings, among which we retain thematchings that assign at least the desired number of agents to an object. Whensolving [ RMP ] for ⌊ µ ( X RSD ⌋ , for example, generally slightly more than half ofthe sampled matchings are retained.The cutting plane algorithm for constraints (5i) has been implemented us-ing lazy constraints. Each time the pricing problem [ PP ] finds an integer so-lution M ∈ M k , the corresponding envy graph G ( M ) is constructed, and is20hen decomposed into strongly connected components (SCC) using Tarjan’s al-gorithm [19]. Next, we identify cycles in each SCC until no more cycles exist,or until all agents are in exactly one detected cycle, and we add the correspond-ing constraints to [ PP ] . During this cycle identification process, we aim to findcycles of a short length by closing the cycle whenever possible. Table 1 summarizes the main findings of our computational experiments for gen-erated data. First of all, we can conclude that the column generation frameworkmanages to solve relatively large instances to optimality in acceptable runtimes.We observe that both an increase in the number of agents, and an increase inthe number of objects per agent cause the runtimes to rise.Table 1 also shows that, for all instances that were solved to optimality, thedifference between z ( X RSD ) and the worst-case number of assigned agents bythe RSD algorithm is substantial, with an average increase of around 3% of thetotal number of agents. In fact, this is the maximal possible increase, as weobserve that for all optimally solved instances, z ( X RSD ) is equal to its upperbound ⌊ µ ( X RSD ) ⌋ . This can be explained by the results of Che and Kojima [13],who showed that X PS and X RSD become equivalent when the market becomeslarge, and we know from Theorem 3 that z ( X PS ) = ⌊ µ ( X PS ) ⌋ .For instances that could not be solved to optimality within the runtime limitof one hour, the objective value s ∗ of the restricted master problem [ RMP ] isgenerally very small. The maximum value of s ∗ over all unsolved instances is . · − , for example, whereas the average value of s ∗ equals . · − overall unsolved instances excluding the ones with 1,500 agents and 150 or 1,500objects. Recall that s ∗ represents the maximum difference for any agent-objectpair between the generated probabilistic assignment by [ RMP ] and X RSD . Thismeans that even when s ∗ is strictly greater than zero, we still obtain a desirabledecomposition that approximates X RSD , and that only contains ex-post efficientmatchings. All matchings in this decomposition furthermore assign at least k agents to an object when solving [ RMP ] for X RSD over M SD k . Althoughthe generated probabilistic assignment will be a small overestimation of X RSD because of constraints (3b), we argue that the resulting decomposition is stillvaluable for real-world applications.We have also evaluated the algorithm we introduced in the proof of Theo-rem 2 to decompose X RSD such that all matchings in the decomposition assignat least z ( X RSD ) agents to an object. For the instances of Table 1, the al-gorithm only resulted in an ex-post efficient decomposition for some instanceswith a small number of objects, namely for 20 among the 75 instances where ( | N | , | O | ) ∈ { (10 , , (50 , , (100 , } . The algorithm from Theorem 2 is there-fore not suited to find an optimal decomposition for MD-SD ( X ) when X is notrobust ex-post efficient. 21 able 1: Average performance (CPU in s) of the column generation framework for MD-SD ( X RSD ) when optimality was obtained, number of instances ( X ) out of 25 solved to optimal-ity within one hour, the number of solved instances ( < UB) for which z ( X RSD ) < ⌊ µ ( X RSD ) ⌋ ,and the average relative improvement ( ∆ , % of | N | ) of z ( X RSD ) compared to the worst-casenumber of assigned agents by RSD | N | / | O | = 1 | N | / | O | = 10 | N | / | O | = 50 | N | CPU X < UB ∆ (%) CPU X < UB ∆ (%) CPU X < UB ∆ (%)10 0 .
08 25 0 2 . .
65 25 0 5 .
68 0 .
22 25 0 3 . .
02 25 0 5 .
40 0 .
59 25 0 4 .
08 0 .
35 25 0 3 . .
35 25 0 2 .
76 44 .
11 25 0 2 .
26 6 .
80 25 0 1 . ,
000 1 , .
44 6 0 2 .
00 1 , .
21 12 0 1 .
58 255 .
97 22 0 1 . , − − − − − − , .
08 4 0 1 . Next to evaluating our methods on generated data, we also consider two real-world school choice data sets from the cities of Antwerp and Ghent. The datafrom Antwerp corresponds to the primary school enrollment in the scholas-tic year of 2014-2015, and contains 4,236 students and 186 schools. The dataof Ghent corresponds to the secondary school enrollment in 2018-2019, andcontains 3,081 students and 64 schools. For confidentiality reasons, the datasets cannot be made publicly available. In this section, we sampled the worst-case number of assigned agents p − ( I ) by the RSD algorithm over 1,000,000tie-breaking rules.In both data sets, the possible increase that we can realize in the worst-casenumber of assigned agents is substantial, compared to the RSD algorithm. ForAntwerp, this increase is at least equal to 46 students, which is an increase by1.08% of the total number of students, while the increase in Ghent equals atleast 39 students, or 1.27%. Note that the actual increase can even be larger,because we only sampled p − ( I ) .When applying the column generation framework of Section 5 to both in-stances, we notice that a larger initial number of included matchings in [ RMP ] leads to better overall solutions, which is probably due to the generally longcomputation time needed to solve the pricing problem. For Ghent, for exam-ple, the best found objective value of [ RMP ] is equal to 0.00245 after fifteenhours of computing when starting from an initial sample size of 10,000 match-ings, among which the matchings that assigned more than ⌊ µ ( X RSD ) ⌋ studentswere retained. When starting from 50,000 matchings, however, an optimal de-composition is found in the first iteration after almost two hours. Similarly,for Antwerp the objective value decreases from 0.01029 when sampling 10,000initial matchings to . · − for an initial sample of 50,000 matchings.22 Conclusion
In this paper, we have studied the problem of decomposing a probabilistic as-signment for one-sided matching over ex-post efficient matchings while maxi-mizing the worst-case number of assigned agents. With respect to the two moststudied mechanisms for one-sided matching, PS and RSD, we have obtained thefollowing insights.We have established that it is always possible to decompose the probabilis-tic assignment by the PS mechanism in polynomial time over ex-post efficientmatchings that all assign the expected number of assigned agents by PS, eitherrounded up or down. The same result does not hold for the RSD mechanism,however. On the one hand, it is possible for the RSD algorithm to assign onlyhalf of the optimal worst-case number of agents to an object. On the other hand,instances exist for which we cannot realize any improvement in the worst-casenumber of assigned agents when decomposing the probabilistic assignment byRSD over ex-post efficient matchings.In our computational experiments, we have found that for all generatedinstances that were solved to optimality, the optimal worst-case number of as-signed agents when decomposing the probabilistic assignment by RSD over ex-post efficient matchings is equal to the expected number of assigned agents byRSD, rounded down. This promising result encourages the adoption of oursolution methods in practical applications. By applying the column genera-tion framework that we have introduced to real-world school choice data sets,we have illustrated that even if optimality cannot be obtained in the desiredruntime, the found solution can still be highly valuable in practice.
Acknowledgements
Tom Demeulemeester is funded by PhD fellowship 11J8721N ofResearch Foundation - Flanders. Ben Hermans is funded by post-doc fellowship 12ZZI21Nof Research Foundation - Flanders. We are grateful to Steven Penneman from the city ofAntwerp, and Pieter De Wilde from the city of Ghent for their collaboration and for makingtheir data sets available to us.
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This appendix describes a modified version of the algorithm of Budish et al. [12]that, given a probabilistic assignment X ∈ ∆ M with µ ( X ) ∈ N , constructs amatching M ∈ M that satisfies Properties (i)-(ii) as stated in the proof ofTheorem 2. Our approach simplifies the one of Budish et al. [12] as it considersonly the specific constraint structure H with quotas ( q S ) S ∈H correspondingto ∆ M , as introduced in the proof of Theorem 2, rather than the more generalbihierarchical structure considered by Budish et al. [12]. This restriction enablesus to reduce the O ( |H| ) running time of the algorithm by Budish et al. [12] to O ( |H| min {| N | , | O |} ) .Given an assignment X ∈ ∆ M , consider a bipartite graph B ( X ) = ( V, E ( X )) with vertices V = N ∪ O ∪ { s, t } and a set of edges E ( X ) consisting of(i) an edge { i, j } for every ( i, j ) ∈ N × O for which x ij is fractional,(ii) an edge { s, i } for every i ∈ N for which P j ∈ O x ij is fractional,(iii) and an edge { j, t } for every j ∈ O for which P i ∈ N x ij is fractional.Every iteration u of the algorithm starts from a given probabilistic assign-ment X u and a graph B ( X u ) , where X u = X if u = 1 , and X u is defined inthe previous iteration otherwise. If X u is a matching, i.e., if X u ∈ M , then theprocedure outputs M = X u and terminates. Otherwise, we determine a cycle C = h i , j , i , j , . . . , i r , j r i in B ( X u ) with { i k , j k } , { j k , i k +1 } ∈ E ( X u ) for ev-ery k = 1 , . . . , r and i r +1 = i , where the indices are such that i k ∈ N ∪ { t } and j k ∈ O ∪ { s } for every k = 1 , . . . , r . Next, we let α be the largest realnumber such that the probabilistic assignment X u +1 with x u +1 ij = x uij + α if ( i, j ) = ( i k , j k ) for some k = 1 , . . . , r , x uij − α if ( i, j ) = ( i k +1 , j k ) for some k = 1 , . . . , r , x uij otherwise,is feasible, update the graph B ( X u +1 ) , and proceed to the next iteration.To see that the algorithm yields a matching with the desired properties intime O ( |H| min {| N | , | O |} ) , observe first that in every iteration u the constructedassignment X u ∈ ∆ M satisfies Properties (i)-(ii) as stated in the proof of The-orem 2. Moreover, it follows from the construction of B ( X u ) that the degree ofeach vertex in B ( X u ) is either zero or at least two and, since B ( X u ) is bipartite,25hat each cycle contains at most {| N ∪ { t }| , | O ∪ { s }|} edges. Hence, ifthere is at least one ( i, j ) ∈ N × O for which x uij is fractional, then we can obtaina cycle of the desired form, compute α , and determine X u +1 and B ( X u +1 ) intime O (min {| N | , | O |} ) . Finally, since α is chosen such that τ ( X u +1 ) > τ ( X u ) ,where τ is defined as in the proof of Theorem 2, the algorithm terminates afterat most |H| iterations. B Data generation