Constrained Serial Rule on the Full Preference Domain
aa r X i v : . [ ec on . T H ] N ov Constrained Serial Rule on the Full PreferenceDomain
Priyanka Shende ∗ Date: November 3, 2020
Abstract
We study the problem of assigning objects to agents in the presence of arbitrary linear con-straints when agents are allowed to be indifferent between objects. Our main contribution is thegeneralization of the (Extended) Probabilistic Serial mechanism via a new mechanism calledthe Constrained Serial Rule. This mechanism is computationally efficient and maintains desir-able efficiency and fairness properties namely constrained ordinal efficiency and envy-freenessamong agents of the same type. Our mechanism is based on a linear programming approachthat accounts for all constraints and provides a re-interpretation of the “bottleneck” set of agentsthat form a crucial part of the Extended Probabilistic Serial mechanism.
Keywords: random assignment; probabilistic serial mechanism; constrained ordinal effi-ciency; indifferences ∗ University of California, Berkeley; Email: [email protected] Introduction
Allocating a number of indivisible objects among a set of agents in a fair and efficient manner is one of the most fundamental problems in applied and theoretical economics. Classi-cal models for object allocation without monetary transfers such as the house allocation model(Hylland and Zeckhauser (1979)) are now well understood and mechanisms that guarantee anymaximal subset of attainable properties along the dimensions of fairness , efficiency , and incentivecompatibility are well known. However, many applications in practice impose additional constraintson the set of allowed allocations and designing allocation mechanisms in such constrained settingsremains an ongoing challenge. In this work, we present a novel Constrained Serial Rule mechanismthat always obtains efficient and fair outcomes for object allocation under a large class of generalconstraints.In the traditional object allocation model, a finite set of indivisible objects must be allocatedto a set of agents based on the agents’ preferences over those objects. In this paper, we restrict ourattention to ordinal mechanisms where the agents report only a preference ranking over objects .This setting models a large number of real-world applications such as placement of studentsto public schools (Abdulkadiroğlu and Sönmez, 2003), course allocation (Budish, 2011), organdonation (Roth et al., 2005), and on-campus housing allocation (Chen and Sönmez, 2002) amongothers. In many of these applications, it is highly desirable that the mechanism is fair , i.e. no agentis discriminated against and also efficient , i.e. there is no other outcome that is preferred by allagents. Unfortunately, since objects are indivisible, it can be easily observed that no mechanismcan be perceived as being fair ex-post. Randomization is, therefore, often used as a tool to restorefairness from an ex-ante perspective.In the context of house allocation, Bogomolnaia and Moulin (2001) introduced the notion of ordinal efficiency . A random assignment is said to be ordinally efficient if there exists no otherassignment that stochastically dominates it. Indeed, they showed that the well-studied RandomPriority mechanism that orders agents in a uniformly random order and then allocates each agent hermost preferred object from the set of remaining objects is not ordinally efficient and only satisfies aweaker notion of ex-post efficiency. The notion of envy-freeness that requires each agent to prefer herown allocation to anyone else’s allocation is often considered as the gold standard of fairness in manydifferent settings such as resource allocation (Foley, 1967), cake-cutting (Robertson and Webb,1998), and rent division (Edward Su, 1999). In the context of random assignment mechanisms, This is in contrast with cardinal mechanisms where the agents report cardinal utilities for each object.
Probabilistic Serial mechanism that alwaysproduces ordinally efficient and envy-free outcomes. The probabilistic serial mechanism can bedescribed as follows. At time zero, each agent begins “eating” her most preferred object. An objectbecomes unavailable once the total time spend by agents eating it equals one. Once an objectbecomes unavailable, all agents that were eating it, switch to eating their most preferred objectamong the ones still available. Finally, the probability that an agent receives some object is thetime spent by that agent to eat the object.Unfortunately, the probabilistic serial mechanism assumes that agents have strict prefer-ences over objects, which is a fairly restrictive assumption in practice. Indeed, as discussedby Katta and Sethuraman (2006) and Erdil and Ergin (2017), ties in preferences are widespreadin many practical applications. For example, agents may treat some objects as identical. Evenwhen objects are all distinct, evaluating and ranking all objects may be computationally pro-hibitive, and agents may only reveal coarse rankings with indifferences. In their influential paper,Katta and Sethuraman (2006) present the
Extended Probabilistic Serial mechanism that generalizesthe probabilistic serial mechanism to the full preference domain and retains the desirable propertiesof ordinal efficiency and envy-freeness.Widespread applicability of these mechanisms is hindered by the fact that these mechanismsassume that every random assignment is feasible. In a large number of practical applications, legaland policy requirements necessitate studying mechanisms where the set of feasible assignmentsis constrained in some way. For example, in the course allocation problem, there are oftenrequirements on the minimum (and maximum) number of students assigned to a course. Similarly,in school choice applications, it is required to find assignments that maintain a minimum levelof diversity (Ehlers et al. (2014)). In resident matching, it is often necessary for the allocation ofdoctors to hospitals to satisfy geographic constraints (Kamada and Kojima (2015)). In the refugeeresettlement problem, objects represent settlement facilities and a feasible assignment must besuch that the total demands of all agents assigned to a facility must be met by the total supplyof resources at that facility (Delacrétaz et al. (2016)). Similarly, in kidney matching applications(Roth et al. (2005)), blood-type compatibility imposes constraints on feasible matchings. In thispaper, we study the object allocation problem with arbitrary linear constraints on the set of feasibleprobabilistic assignments. Following the work of Balbuzanov (2019), this formalization supportsan arbitrary set of constraints on ex-post allocations.3 .1 Our Contributions
Our primary contribution is to generalize the probabilistic serial mechanism to the full prefer-ence domain and support arbitrary linear constraints on the set of feasible random assignments viaa new mechanism called the
Constrained Serial Rule . Our mechanism is computationally efficientand only requires a running time that is polynomial in the number of constraints, agents, andobjects. For the classical unconstrained house allocation setup on the full preference domain, ourmechanism coincides with the extended probabilistic serial mechanism of Katta and Sethuraman(2006). Various generalizations of the probabilistic serial mechanism have been proposed formulti-unit demand (Kojima (2009)), specific type of constraints such as bi-hierarchical constraints(Budish et al., 2013), type-dependent distributional constraints (Ashlagi et al., 2020), combinato-rial demand (Nguyen et al., 2016), property rights with individual rationality (Yılmaz, 2010), andeven arbitrary constraints on the ex-post allocations (Balbuzanov, 2019). Our constrained serialrule unifies this literature and provides a common generalization of all these mechanisms and alsoprovides an extension to the full preference domain.We show that the constrained serial rule maintains the desirable efficiency and fairnessproperties of the probabilistic serial mechanism even in our general constrained setting. In particular,our mechanism is constrained ordinally efficient. While it is easy to observe that arbitrary constraintsrule out the existence of envy-free mechanisms, we show that the constrained serial rule maintains acompelling notion of fairness. Intuitively, we say agents 𝑖 and 𝑗 are of the same type if the constraintstructure does not distinguish between the two agents. We show that the constrained serial rulemechanism guarantees envy-freeness among any pair of agents of the same type. However, ourmechanism is not strategyproof or even weak-strategyproof. This is unsurprising since even inthe unconstrained setting, weak-strategyproofness is incompatible with ordinal efficiency and envy-freeness on the full preference domain (Katta and Sethuraman, 2006). There is a growing body of literature on assignment and matching mechanisms subject to con-straints. Several studies have considered floor and ceiling constraints in the context of controlledschool choice, college admissions, and affirmative action (Ashlagi et al., 2020; Biró et al., 2010;Echenique and Yenmez, 2015; Ehlers et al., 2014; Fleiner and Kamiyama, 2016; Fragiadakis and Troyan,2017; Goto et al., 2015; Hafalir et al., 2013; Hamada et al., 2016; Kojima, 2012; Kominers and Sönmez,2013; Westkamp, 2013). Echenique et al. (2019) consider arbitrary ex-post constraints as in4albuzanov (2019) and provide a pseudo-market equilibrium solution that is constrained ex-anteefficient and fair.Akbarpour and Nikzad (2014); Budish et al. (2013); Pycia and Ünver (2015) have studied theimplementability of random matching mechanisms. Budish et al. (2013) identify bi-hierarchicalconstraint structures as a necessary and sufficient condition for implementing a random assign-ment using lottery of feasible assignments. They also provide a generalization of the (extended)probabilistic serial mechanism in the case when there are no floor constraints. Indeed, our mech-anism is able to accommodate bi-hierarchical constraint inequalities in the presence of non-zerofloor constraints. Akbarpour and Nikzad (2014) consider more general constraints beyond bi-hierarchical structures and show how feasible random assignments can be implemented approxi-mately. Pycia and Ünver (2015) provide sufficient conditions on the properties of random mecha-nisms that continue to be satisfied on the deterministic mechanisms when random mechanisms aredecomposed as a lottery over these deterministic mechanisms. While the focus of our paper is noton implementability, we provide a small discussion of this in Section 3.4.
We consider a finite set 𝑁 of agents and a finite set 𝑂 of objects. Let 𝑛 = | 𝑁 | be the number ofdistinct agents and 𝜌 = | 𝑂 | be the number of distinct objects. Every agent has a unit demand andeach object 𝑜 ∈ 𝑂 is supplied in 𝑞 𝑜 ∈ N copies. When objects are scarce, we can include the nullobject, ∅ , in the set 𝑂 , which is supplied in a quantity sufficient to meet the demand of all agents.That is, 𝑞 ∅ ≥ | 𝑁 | . We can, therefore, without loss of generality, assume that P 𝑜 ∈ 𝑂 𝑞 𝑜 ≥ 𝑛 . Eachagent 𝑖 ∈ 𝑁 has a preference relation (cid:23) 𝑖 on the set of objects in 𝑂 . The preference (cid:23) 𝑖 is assumedto be complete and transitive. In particular, we allow agents to be indifferent between any pair ofobjects in 𝑂 . Let 𝐸 ((cid:23)) be the number of indifference classes within the preference (cid:23) . For any ℓ ≤ 𝐸 ((cid:23) 𝑖 ) , let 𝑇 𝑖 ( ℓ ) be the set of objects in the first ℓ indifference classes of the preference (cid:23) 𝑖 . Aset of individual preferences of all agents constitutes a preference profile (cid:23) = ((cid:23) 𝑖 ) 𝑖 ∈ 𝑁 . Let R denotethe set of all complete and transitive relations on 𝑂 and R 𝑛 be the set of all possible preferenceprofiles.A random assignment of objects to agents is given by a vector x = ( 𝑥 𝑖,𝑜 ) 𝑖 ∈ 𝑁,𝑜 ∈ 𝑂 ∈ [ , ] 𝑛𝜌 Our results also generalize to the case when all agents demand 𝑑 ≥ X 𝑜 ∈ 𝑂 𝑥 𝑖,𝑜 = ∀ 𝑖 ∈ 𝑁 X 𝑖 ∈ 𝑁 𝑥 𝑖,𝑜 ≤ 𝑞 𝑜 ∀ 𝑜 ∈ 𝑂 In assignment x , every agent 𝑖 ’s allocation is given by the sub-vector x 𝑖 = ( 𝑥 𝑖,𝑜 ) 𝑜 ∈ 𝑂 , where thequantity 𝑥 𝑖,𝑜 is interpreted to be the probability with which object 𝑜 is assigned to agent 𝑖 . Let 𝑥 𝑖 ( 𝑆 ) = P 𝑜 ∈ 𝑆 𝑥 𝑖,𝑜 be agent 𝑖 ’s cumulative allocation for the set of objects in set 𝑆 . An assignmentis deterministic whenever 𝑥 𝑖,𝑜 ∈ { , } , i.e, every agent is assigned a single object with probability1. Let D denote the set of all deterministic assignments and Δ D denote the set of all randomassignments. A random assignment mechanism is a mapping, 𝜑 : R 𝑛 → Δ D , that associates eachpreference profile (cid:23) ∈ R 𝑛 with some random assignment x ∈ Δ D .We extend agents’ preferences from the set of objects to the set of random allocations using thestochastic dominance relation. Given two random assignments x and y , allocation x 𝑖 stochasticallydominates allocation y 𝑖 with respect to (cid:23) 𝑖 , denoted by x 𝑖 𝑠𝑑 ((cid:23) 𝑖 ) y 𝑖 , if and only if P 𝑜 ′ (cid:23) 𝑖 𝑜 𝑥 𝑖,𝑜 ′ ≥ P 𝑜 ′ (cid:23) 𝑖 𝑜 𝑦 𝑖,𝑜 ′ for all 𝑜 ∈ 𝑂 . If the inequality is strict for some 𝑜 ∈ 𝑂 , then x 𝑖 strictly stochasticallydominates y 𝑖 , in which case we denote it by x 𝑖 𝑠𝑑 (≻ 𝑖 ) y 𝑖 .We now introduce a general class of constraints into our model. At any given preferenceprofile, we assume that the set of feasible random assignments can be described as a convexpolytope. Formally, at preference profile (cid:23) , the set of feasible random assignments Δ C ( (cid:23) ) isparameterized by a matrix 𝐴 = [ 𝑎 𝑐𝑖,𝑜 ] ≤ 𝑐 ≤ 𝑚, { 𝑖,𝑜 }∈ 𝑁 × 𝑂 ∈ R 𝑚 × 𝑛𝜌 and a vector 𝒃 = [ 𝑏 𝑐 ] ≤ 𝑐 ≤ 𝑚 ∈ R 𝑚 ,where 𝑐 is a generic constraint and 𝑚 is the number of constraints, and is defined as: Δ C ( (cid:23) ) = { x ∈ Δ D | 𝐴 x ≤ 𝒃 } We assume that at each preference profile (cid:23) , the set of feasible random assignments is non-empty. That is, Δ C ( (cid:23) ) ≠ ∅ . Such a formulation of the constraints enables us to apply our modelto many different specific applications that we describe in Section 4. Let Δ C = { Δ 𝐶 ( (cid:23) )} (cid:23) ∈R 𝑛 bethe collection of constraint polytopes for all preference profiles. Given a collection of constraints Δ C , a mechanism is feasible if at every preference profile (cid:23) , 𝜑 ( (cid:23) ) ∈ Δ C ( (cid:23) ) .We next define the normative properties of efficiency and fairness in the presence of constraints. Definition 2.1 (Constrained Ordinal Efficiency) . A random assignment x is constrained ordinallyefficient at a preference profile (cid:23) and constraint set Δ C ( (cid:23) ) if there does not exist another randomassignment x ′ ∈ Δ C ( (cid:23) ) such that x ′ 𝑖 𝑠𝑑 ((cid:23) 𝑖 ) x 𝑖 for all 𝑖 ∈ 𝑁 , with x ′ 𝑖 𝑠𝑑 (≻ 𝑖 ) x 𝑖 for at least one 𝑖 ∈ 𝑁 . Amechanism 𝜑 is constrained ordinally efficient if for every preference profile (cid:23) , 𝜑 ( (cid:23) ) is constrained6rdinally efficient.The classic notion of fairness requires that no agent should envy the allocation received bysome other agent. When faced with arbitrary feasibility constraints, it is easy to see that one cannotguarantee the existence of envy-free assignments. Therefore, we restrict ourselves to fairnesscomparisons between agents that belong to the same type. For a given constraint matrix 𝐴 , we sayagents 𝑖 and 𝑗 belong to the same type if for every object 𝑜 , the variables 𝑥 𝑖,𝑜 and 𝑥 𝑗,𝑜 have the samecoefficients in every constraint in 𝐴 . Definition 2.2 (Agent Type) . Let Δ C ( (cid:23) ) = { x ∈ Δ D | 𝐴 x ≤ 𝒃 } where 𝐴 = [ 𝑎 𝑐𝑖,𝑜 ] ≤ 𝑐 ≤ 𝑚, { 𝑖,𝑜 }∈ 𝑁 × 𝑂 denote the constraint set a preference profile (cid:23) . Two agents 𝑖 and 𝑗 are said to be of the same type at this profile, if for every constraint 1 ≤ 𝑐 ≤ 𝑚 , and for every object 𝑜 ∈ 𝑂 , we have 𝑎 𝑐𝑖,𝑜 = 𝑎 𝑐𝑗,𝑜 . Definition 2.3 (Envy-freeness among agents of the same type) . A random assignment x is envy-freeamong agents of the same type if for every pair of agents 𝑖, 𝑗 ∈ 𝑁 of the same type, x 𝑖 𝑠𝑑 ((cid:23) 𝑖 ) x 𝑗 .A mechanism 𝜑 is envy-free for agents of the same type if for every preference profile (cid:23) , 𝜑 ( (cid:23) ) isenvy-free among agents of the same type. We first give a brief intuitive description of the classical probabilistic serial mechanism (Bogomolnaia and Moulin,2001) in the simple house allocation model . The mechanism is best described as a continuoustime procedure for 𝑡 ∈ [ , ] : at each infinitesimal time interval [ 𝑡, 𝑡 + 𝑑𝑡 ) , each agent 𝑖 consumes 𝑑𝑡 amount of her most preferred object among the set of objects currently available. When thisprocedure terminates, the probability that an agent is assigned an object is given by the fraction ofthe object consumed by the agent.While attempting to extend the probabilistic serial mechanism to our general model on thefull preference domain and with arbitrary constraints on the eventual random assignment, onefaces two key challenges. First, when agents have strict preferences, each agent at any point intime has a unique most preferred object and hence the mechanism can simply allocate that objectto the agent. On the other hand, when agents are indifferent between two or more objects, themechanism can no longer uniquely identify an object to assign. Katta and Sethuraman (2006)deal with this difficulty by constructing a flow graph where every agent points to her set of most Agents have strict preferences over objects and there are no additional constraints on the assignment. bottleneck agents andobjects. Intuitively, the set of bottleneck agents are those that compete the most among themselvesand the bottleneck objects are those desired by bottleneck agents. Once the bottleneck agents havebeen identified, the extended probabilistic serial mechanism allocates all bottleneck objects amongthese agents uniformly. As in the classic probabilistic serial rule, the mechanism then proceedsby each bottleneck agent simply pointing to her next most preferred object. A key observation isthat the (extended) probabilistic serial mechanism attempts to assign each agent her most preferredobject for as long as possible. In fact, as observed by Bogomolnaia (2015), one can provide awelfarist interpretation of the probabilistic serial rule as follows: for any agent 𝑖 , let 𝑥 𝑖 ( ℓ ) be thetotal probability share of objects that agent 𝑖 receives for her top ℓ indifference classes; then theprobabilistic serial mechanism leximin maximizes the vector of all such shares ( 𝑥 𝑖 ( ℓ )) 𝑖 ∈ 𝑁,𝑘 ≤ 𝐸 ((cid:23) 𝑖 ) .We crucially use this observation in our constrained serial rule mechanism.The second challenge arises due to the presence of arbitrary constraints on the space of feasiblerandom assignments. We observe that at any step of the mechanism, the (extended) probabilisticserial mechanism always assigns each agent her most preferred object (at that time). In the presenceof constraints, however, it is essential to not allow agents to obtain their most preferred object ifsuch an allocation leads to infeasibility. More precisely, the mechanism needs to look ahead intime as it builds up a partial allocation to ensure that there is at least one way to extend the partialallocation to a feasible assignment. Our constrained serial rule mechanism uses a linear programthat explicitly accounts for all constraints and at every step maintains a feasible solution.We now describe the linear program that is used crucially by the mechanism. Let 𝑆 ⊆ 𝑁 denoteany subset of agents and let ℓ 𝑖 ∈ { , , . . . , 𝐸 ((cid:23) 𝑖 )} for each agent 𝑖 ∈ 𝑁 denote an indifferencethreshold. Let 𝐹 be a set of triples that denotes prior promised assignments. Formally, a triple ( 𝑖, ℓ, 𝛾 ) ∈ 𝐹 indicates that agent 𝑖 must receive a total probability share of at least 𝛾 from her top ℓ indifference classes. The linear program 𝐿𝑃 ( 𝑆, 𝐹, ( ℓ 𝑖 ) 𝑖 ∈ 𝑁 ) specified in Figure 1 finds a randomassignment x ∈ R 𝑛𝜌 + that satisfies all constraints specified by 𝐹 in addition to the imposed feasibilityconstraints and maximizes the total probability share that each agent 𝑖 ∈ 𝑆 receives from her top ℓ 𝑖 indifference classes. The variables ℎ 𝑖 for each 𝑖 ∈ 𝑁 represent the total probability share received byagent 𝑖 for objects in her top ℓ 𝑖 indifference classes. Constraints (1) and (2) enforce the requirementthat the linear program maximizes min 𝑖 ∈ 𝑆 ℎ 𝑖 . Constraints (3) and (4) enforce that the obtainedrandom assignment is feasible, and finally constraint (5) requires the assignment to be consistentwith the requirements specified by the triples in 𝐹 .8 𝑃 ( 𝑆, 𝐹, ( ℓ 𝑖 ) 𝑖 ∈ 𝑆 ) = maximize x , h ,𝜆 𝜆 s . t . ℎ 𝑖 ≥ 𝜆 ∀ 𝑖 ∈ 𝑆 (1) X 𝑜 ∈ 𝑇 𝑖 ( ℓ 𝑖 ) 𝑥 𝑖,𝑜 ≥ ℎ 𝑖 ∀ 𝑖 ∈ 𝑁 (2) 𝐴 x ≤ 𝒃 (3) x ∈ Δ D (4) X 𝑜 ∈ 𝑇 𝑖 ( ℓ ) 𝑥 𝑖,𝑜 ≥ 𝛾 ∀( 𝑖, ℓ, 𝛾 ) ∈ 𝐹 (5) h , 𝜆 ≥ Figure 1:
Linear Program used by the Constrained Serial Rule
We are now ready to describe the constrained serial rule mechanism formally. The mechanismproceeds in multiple rounds. We initialize 𝐹 = ∅ and for each agent 𝑖 ∈ 𝑁 , we initialize ℓ 𝑖 = ℓ 𝑡𝑖 denotes the threshold indifference class for agent 𝑖 in round 𝑡 . In other words, inround 𝑡 , we consider the total probability share of objects in 𝑇 𝑖 ( ℓ 𝑡𝑖 ) assigned to agent 𝑖 . Let ℎ 𝑡𝑖 denotethe total probability share of objects in the top ℓ 𝑡𝑖 indifference classes assigned to agent 𝑖 . We usethe linear program described in Figure 1 to find a feasible random assignment such that min 𝑖 ℎ 𝑡𝑖 ismaximized. The mechanism then identifies a set 𝐵 𝑡 of bottleneck agents. Intuitively, these are theset of agents who are responsible for the 𝐿𝑃 objective in this round to be only 𝜆 𝑡 . Since we aredealing with arbitrary linear constraints, our definition of bottleneck agents needs to be more subtlethan that of Katta and Sethuraman (2006). We define 𝐵 𝑡 to be a minimal set of agents such thatsolving the linear program while only attempting to maximize the utility of agents in that set alsoyields the same objective value of 𝜆 𝑡 . Our definition of bottleneck agents is central to the validity ofthe mechanism as well as its efficiency and fairness properties. Finally, once the bottleneck set ofagents has been identified, we update 𝐹 𝑡 to guarantee that in future rounds each agent 𝑖 ∈ 𝐵 𝑡 obtainsat least the promised 𝜆 𝑡 probability share from her top ℓ 𝑡𝑖 indifference classes and then incrementthe threshold ℓ 𝑡𝑖 for all such agents. The mechanism then proceeds to the next round and the processcontinues until every agent receives a total probability share of 1. Algorithm 1 provides a completeformal description of the algorithm.We provide a simple example run of Algorithm 1 on a constrained allocation problem to9 nitialize: 𝑡 ← ℓ 𝑡𝑖 ← , ∀ 𝑖 ∈ 𝑁 ; 𝐹 𝑡 ← ∅ ; for 𝑡 = , , . . . do ( x 𝑡 , h 𝑡 , 𝜆 𝑡 ) ← 𝐿𝑃 ( 𝑁, 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) ; if 𝜆 𝑡 = thenx ← x 𝑡 ;terminate; else Find a minimal set 𝐵 𝑡 such that 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) has objective value 𝜆 𝑡 ;Update 𝐹 𝑡 + = 𝐹 𝑡 ∪ {( 𝑖, ℓ 𝑡𝑖 , 𝜆 𝑡 ) | 𝑖 ∈ 𝐵 𝑡 } ;Update ℓ 𝑡 + 𝑖 = ℓ 𝑡𝑖 + ∀ 𝑖 ∈ 𝐵 𝑡 ℓ 𝑡𝑖 otherwise endend Algorithm 1: The Constrained Serial Ruleillustrate our constrained serial rule.
Example 3.1.
Consider an allocation problem with three agents 𝑁 = { , , } and three objects 𝑂 = { 𝑎, 𝑏, 𝑐 } . Our goal is to obtain an assignment that satisfies the usual bistochastic constraints,i.e. P 𝑜 ∈ 𝑂 𝑥 𝑖,𝑎 = ∀ 𝑖 ∈ 𝑁 and P 𝑖 ∈ 𝑁 𝑥 𝑖,𝑎 = ∀ 𝑜 ∈ 𝑂 . In addition, there are two additionalconstraints as follows: 𝑥 ,𝑎 + 𝑥 ,𝑎 ≤ . 𝑥 ,𝑐 + 𝑥 ,𝑐 ≥ .
5. The agents’ preferences are given asfollows. ≻ : 𝑎 𝑏 𝑐 ≻ : { 𝑎, 𝑏 } 𝑐 ≻ : 𝑐 𝑏 𝑎 We illustrate how our mechanism works through this example. In the first round, we initialize ℓ = ℓ = ℓ = 𝜆 = min { 𝑥 ,𝑎 , 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑐 } . With the given constraints, one potential optimum solution assigns 𝑥 ,𝑎 = 𝑥 ,𝑏 = 𝑥 ,𝑐 = . 𝜆 = .
5. We then proceed to find the set of bottleneck agents.In this example, either of the singleton sets with agents 1 or 3 could be the bottleneck set. This is10ecause the constraint 𝑥 ,𝑎 + 𝑥 ,𝑎 ≤ . 𝑎 . Similarly, the constraint 𝑥 ,𝑐 + 𝑥 ,𝑐 ≥ . 𝑐 . Suppose we select agent 1 to be the bottleneck agent. Then, weincrement ℓ = ℓ = ℓ =
1. We also set 𝐹 = ( , , . ) to signify that agent 1must continue to obtain 0 . 𝜆 = min { 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑐 } . As described earlier, agent 3 is unable to receivemore than 0 . 𝑐 in any feasible solution and hence we obtain 𝜆 = .
5. In thiscase, agent 3 is the unique bottleneck agent and as earlier we increment her indifference thresholdand add the triple ( , , . ) to 𝐹 . Similarly, in the third round, we solve the linear program tofind a feasible solution that maximizes 𝜆 = min { 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑐 + 𝑥 ,𝑏 } . However, theconstraints 𝑥 ,𝑎 + 𝑥 ,𝑎 ≤ . 𝑥 ,𝑎 + 𝑥 ,𝑎 + 𝑥 ,𝑎 = 𝑥 ,𝑎 ≥ . 𝑥 ,𝑐 + 𝑥 ,𝑏 ≤ .
5. So yet again, we have 𝜆 = . ( , , . ) to 𝐹 .In the fourth round, the linear program attempts to find a feasible solution that respects all theconstraints in 𝐹 and maximizes 𝜆 = min { 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑎 + 𝑥 ,𝑏 , 𝑥 ,𝑐 + 𝑥 ,𝑏 + 𝑥 ,𝑎 } . In this case,one potential optimum solution assigns 𝑥 ,𝑎 = . , 𝑥 ,𝑏 = . , 𝑥 ,𝑏 = . , 𝑥 ,𝑐 = . , 𝑥 ,𝑎 = . 𝜆 = .
75. Since the constraint 𝑥 ,𝑎 + 𝑥 ,𝑎 = . 𝑏 has beenfully allocated, both agents 1 and 2 are in the bottleneck set in this round. We increment theindifference threshold of both the agents to obtain ℓ = ℓ = ( , , . ) and ( , , . ) to 𝐹 .Finally, in the fifth round, we again solve the linear program to find a feasible solution thatrespects all the constraints in 𝐹 and maximizes 𝜆 = min 𝑖 ∈ 𝑁 { 𝑥 𝑖,𝑎 + 𝑥 𝑖,𝑏 + 𝑥 𝑖,𝑐 } . Since any feasiblesolution satisfies 𝑥 𝑖,𝑎 + 𝑥 𝑖,𝑏 + 𝑥 𝑖,𝑐 =
1, we obtain 𝜆 = 𝐹 . For this example, theunique such solution is given by the following random assignment. a b c1 0.5 0.25 0.252 0 0.75 0.253 0.5 0 0.5 .2 Properties We first show that the constrained serial rule presented in Algorithm 1 is well-defined andalways produces a feasible random assignment.
Proposition 3.1.
Algorithm 1 terminates and always produces a feasible random assignment.Proof . We first observe that in any step 𝑡 , the 𝐿𝑃 always has a feasible solution. This is because,for any 𝑡 >
1, the solution ( 𝑥 𝑡 − , ℎ 𝑡 − , 𝜆 𝑡 − ) from the previous iteration continues to be a feasiblesolution; whereas for 𝑡 =
1, the existence of a feasible solution is guaranteed since the assignmentconstraints are assumed to be satisfiable.In any step 𝑡 , it is easy to see that the bottleneck set of agents 𝐵 𝑡 is guaranteed to existby observing that the set of all agents 𝑁 satisfies the required constraint by definition. Further,we observe that any agent 𝑖 ∗ ∈ 𝑁 such that ℓ 𝑡𝑖 ∗ = 𝐸 ((cid:23) 𝑖 ∗ ) does not appear in the bottleneckset 𝐵 𝑡 . This is because, for any such agent 𝑖 ∗ ∈ 𝑁 , the linear programs 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) and 𝐿𝑃 ( 𝐵 𝑡 \ { 𝑖 ∗ } , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) are actually identical and by definition 𝐵 𝑡 is the minimal set thatsatisfies the condition. Thus in any step 𝑡 , we increment the indifference threshold of at leastone agent. Finally, by definition of feasible assignment, when ℓ 𝑡𝑖 = 𝐸 ((cid:23) 𝑖 ) , ∀ 𝑖 ∈ 𝑁 , we may set ℎ 𝑡𝑖 = P 𝑜 ∈ 𝑂 𝑥 𝑡𝑖,𝑜 = 𝑖 ∈ 𝑁 and hence obtain 𝜆 𝑡 = 𝑖 ∈ 𝑁 has 𝐸 ((cid:23) 𝑖 ) ≤ 𝜌 , the mechanism terminates in at most 𝑛𝜌 rounds.Finally, since the linear program in Figure 1 explicitly maintains the constraints 𝐴 x ≤ 𝒃 and 𝑥 ∈ Δ D , the outcome is guaranteed to be feasible. (cid:3) We then proceed to show two technical lemmas. The first lemma shows that even thoughwe allow arbitrary linear constraints on the random assignment, the projection of the feasibilitypolytope on the h and 𝜆 variables satisfies a desirable monotonicity property. Specifically, theconstraints on the h variables can be expressed as a set of linear upper-bound constraints with onlynon-negative coefficients. Lemma 3.2.
Fix any step 𝑡 of the algorithm. Let 𝑃 be the polytope defined by the constraints in 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) and let 𝑄 = {( h , 𝜆 ) | ∃( x , h , 𝜆 ) ∈ 𝑃 } be its projection. Then there exists anon-negative matrix ˜ 𝐴 and a non-negative vector ˜ 𝑏 such that 𝑄 = ℎ 𝑖 ≥ 𝜆, ∀ 𝑖 ∈ 𝐵 𝑡 ˜ 𝐴 h ≤ ˜ 𝒃 h , 𝜆 ≥ roof . By the Fourier-Motzkin theorem, we know that 𝑄 is also a polytope that can be obtainedfrom 𝑃 using Fourier-Motzkin elimination. Since the only constraints involving 𝜆 in 𝑃 do notcontain any x variables, those constraints remain unchanged in polytope 𝑄 . Let ˜ 𝐴 = [ ˜ 𝑎 𝑐𝑖 ] and˜ 𝑏 = [ ˜ 𝑏 𝑐 ] denote the minimal linear constraints on the variables h obtained after Fourier-Motzkinelimination. We now need to show that ˜ 𝐴 and ˜ 𝑏 are non-negative.We first observe that the polytope 𝑄 is downward closed on the h variables, i.e. for any h ′ ≤ h ,if ( h , 𝜆 ) ∈ 𝑄 then there exists a 𝜆 ′ ≤ 𝜆 such that ( h ′ , 𝜆 ′ ) ∈ 𝑄 . This is because, by definition, if ( h , 𝜆 ) ∈ 𝑄 then there exists an x such that ( x , h , 𝜆 ) ∈ 𝑃 . Further, observing the polytope 𝑃 (refer toFigure 1), it is clear that ( x , h ′ , 𝜆 ′ ) also belongs to 𝑃 where 𝜆 ′ = min 𝑖 ∈ 𝐵 𝑡 ℎ ′ 𝑖 , and thus ( h ′ , 𝜆 ′ ) ∈ 𝑄 .Now, suppose for contradiction that there exists a constraint 𝑐 such that the entry ˜ 𝑎 𝑐𝑗 < 𝑗 ∈ 𝑁 . Since this constraint is not redundant, there exists a vector ˜ h ∈ R 𝑛 + such that the 𝑐 thconstraint is the only binding constraint, i.e., P 𝑖 ˜ 𝑎 𝑐𝑖 ˜ ℎ 𝑖 = ˜ 𝑏 𝑐 and further ˜ ℎ 𝑖 > 𝑖 ∈ 𝑁 . Define h ′ ∈ R 𝑛 + as ℎ ′ 𝑖 = ˜ ℎ 𝑖 , ∀ 𝑖 ∈ 𝑁 \ { 𝑗 } and ℎ ′ 𝑗 =
0. By definition h ′ < ˜ h , and yet P 𝑖 ˜ 𝑎 𝑐𝑖 ℎ ′ 𝑖 > ˜ 𝑏 𝑐 and thus h ′ ∉ 𝑄 which is a contradiction.Finally if the matrix ˜ 𝐴 is non-negative and the vector ˜ 𝑏 has a negative entry, then the polytope 𝑄 must be empty. Since 𝑄 is guaranteed to be non-empty, we must have that ˜ 𝑏 is non-negative. (cid:3) The following lemma demonstrates the importance of our definition of bottleneck agents.Informally, it states that if 𝐵 𝑡 is the set of bottleneck agents at some step 𝑡 of the algorithm, then noagent in 𝐵 𝑡 can obtain a higher total allocation for her top ℓ 𝑡𝑖 indifference classes without hurtingsome other agent in 𝐵 𝑡 . This lemma is crucial to proving the constrained serial rule producesconstrained ordinally efficient outcomes, and also to demonstrate its fairness properties. Lemma 3.3.
Fix any step 𝑡 of the algorithm and let ( x 𝑡 , h 𝑡 , 𝜆 𝑡 ) denote an optimal solution to 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) . Then š y = ( 𝑦 𝑖,𝑜 ) 𝑖 ∈ 𝑁,𝑜 ∈ 𝑂 ∈ R 𝑛𝜌 + such that1. 𝐴 y ≤ 𝑏 , y ∈ Δ D 𝑦 𝑖 ( 𝑇 𝑖 ( ℓ )) ≥ 𝛾, ∀( 𝑖, ℓ, 𝛾 ) ∈ 𝐹 𝑡 𝑦 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) ≥ 𝜆 𝑡 , ∀ 𝑖 ∈ 𝐵 𝑡 with atleast one strict inequality.Proof . Let 𝑃 be the polytope defined by the constraints in 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) . Let 𝑄 = {( h , 𝜆 ) |∃( x , h , 𝜆 ) ∈ 𝑃 )} be the projection of 𝑃 on the h and 𝜆 variables. By Lemma 3.2, there exists anon-negative matrix ˜ 𝐴 = [ ˜ 𝑎 𝑐𝑖 ] and a non-negative vector ˜ 𝑏 such that 𝑄 = ℎ 𝑖 ≥ 𝜆, ∀ 𝑖 ∈ 𝐵 𝑡 ˜ 𝐴 h ≤ ˜ 𝒃 h , 𝜆 ≥ ( x 𝑡 , h 𝑡 , 𝜆 𝑡 ) is an optimal solution to 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) . Let ˜ h be defined as˜ ℎ 𝑖 = 𝜆 𝑡 for all 𝑖 ∈ 𝐵 𝑡 and ˜ ℎ 𝑖 = 𝑖 ∉ 𝐵 𝑡 . We now have ( x 𝑡 , ˜ h , 𝜆 𝑡 ) is also an optimal solutionto 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) . Thus, ( ˜ h , 𝜆 𝑡 ) does not lie in the interior of the polytope 𝑄 . Thus at leastone of the constraints ˜ 𝐴 h ≤ ˜ 𝑏 must be tight at this point, i.e., there exists a constraint 𝑐 such that P 𝑖 ∈ 𝐵 𝑡 ˜ 𝑎 𝑐𝑖 ˜ ℎ 𝑖 = ˜ 𝑏 𝑐 . We now consider two cases. Case 1 : ˜ 𝑎 𝑐𝑖 > 𝑖 ∈ 𝐵 𝑡 . Suppose for contradiction that there exists a y that satisfies thepremises of the lemma. For any agent 𝑖 ∈ 𝑁 , let ℎ ′ 𝑖 = 𝑦 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) , so we have ℎ ′ 𝑖 ≥ 𝜆 𝑡 for all 𝑖 ∈ 𝐵 𝑡 with at least one strict inequality. Thus since ( y , h ′ , 𝜆 𝑡 ) is a feasible solution to 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) ,we have ( y , h ′ , 𝜆 𝑡 ) ∈ 𝑃 . By definition of 𝑄 , we have ( h ′ , 𝜆 𝑡 ) ∈ 𝑄 . However, we have X 𝑖 ∈ 𝐵 𝑡 ˜ 𝑎 𝑐𝑖 ℎ ′ 𝑖 > X 𝑖 ∈ 𝐵 𝑡 ˜ 𝑎 𝑐𝑖 𝜆 𝑡 = X 𝑖 ∈ 𝐵 𝑡 𝑎 𝑐𝑖 ˜ ℎ 𝑖 = ˜ 𝑏 𝑐 which is a contradiction to the statement that ( h ′ , 𝜆 𝑡 ) ∈ 𝑄 . Case 2 : ˜ 𝑎 𝑐𝑗 = 𝑗 ∈ 𝐵 𝑡 . Consider the set 𝐵 ′ = 𝐵 𝑡 \ { 𝑗 } . We now have P 𝑖 ∈ 𝐵 ′ ˜ 𝑎 𝑐𝑖 ˜ ℎ 𝑖 = ˜ 𝑏 𝑐 .Let 𝑃 ′ be the polytope defined by the constraints in 𝐿𝑃 ( 𝐵 ′ , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) and 𝑄 ′ be its projection.Since 𝐵 𝑡 is the minimal bottleneck set, there exists a ( y ′ , ℎ ′ , 𝜆 ′ ) ∈ 𝑃 ′ such that 𝜆 ′ > 𝜆 𝑡 and ℎ ′ 𝑖 = 𝜆 ′ , ∀ 𝑖 ∈ 𝐵 ′ . Hence, we have X 𝑖 ∈ 𝐵 ′ ˜ 𝑎 𝑐𝑖 ℎ ′ 𝑖 > X 𝑖 ∈ 𝐵 ′ ˜ 𝑎 𝑐𝑖 𝜆 𝑡 = X 𝑖 ∈ 𝐵 ′ 𝑎 𝑐𝑖 ˜ ℎ 𝑖 = ˜ 𝑏 𝑐 and thus ( ℎ ′ , 𝜆 ′ ) ∉ 𝑄 ′ . But, this contradicts the fact that ( y ′ , ℎ ′ , 𝜆 ′ ) ∈ 𝑃 ′ . (cid:3) We are now ready to prove that the constrained serial rule algorithm finds a constrainedordinally efficient assignment.
Theorem 3.4.
For any preference profile (cid:23) and constraint set Δ C ( (cid:23) ) , the outcome of Algorithm 1is constrained ordinally efficient.Proof . Let x be the solution of the algorithm for any preference profile (cid:23) and constraint set Δ C ( (cid:23) ) .In order to prove that x is constrained ordinally efficient it suffices to show that if there exists x ′ ∈ Δ C ( (cid:23) ) such that x ′ 𝑖 𝑠𝑑 ((cid:23) 𝑖 ) x 𝑖 for all 𝑖 ∈ 𝑁 , then 𝑥 ′ 𝑖 ( 𝑇 𝑖 ( ℓ )) = 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ )) , for all 𝑖 ∈ 𝑁 , for all ℓ ∈ { , , ..., 𝐸 ((cid:23) 𝑖 )} . We prove this using contradiction.For any round 𝑡 , let ( 𝑥 𝑡 , ℎ 𝑡 , 𝜆 𝑡 ) denote the optimal solution to 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) . For anyagent 𝑖 in the bottleneck set 𝐵 𝑡 , the mechanism fixes the cumulative allocation received by agent 𝑖 forher top ℓ 𝑡𝑖 indifference classes. Hence we have 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) = 𝑥 𝑡𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) = 𝜆 𝑡 . Towards a contradiction,let 𝑡 be the first step in the algorithm such that 𝑥 ′ 𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) ≠ 𝑥 𝑡𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) for some agent 𝑗 ∈ 𝐵 𝑡 .Since x ′ 𝑖 𝑠𝑑 ((cid:23) 𝑖 ) x 𝑖 for all 𝑖 ∈ 𝑁 , it must be that 𝑥 ′ 𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) > 𝑥 𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) = 𝑥 𝑡𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) = 𝜆 𝑡 . Also,we have 𝑥 ′ 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) ≥ 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) = 𝜆 𝑡 for all agents 𝑖 ∈ 𝐵 𝑡 with 𝑖 ≠ 𝑗 . Further, since x ′ is a feasible14andom assignment, we have x ′ ∈ Δ D and 𝐴 x ′ ≤ 𝒃 . Lastly, since round 𝑡 is the first time x ′ differsfrom x , 𝑥 ′ 𝑖 ( 𝑇 𝑖 ( ℓ )) = 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ )) ≥ 𝛾 , ∀( 𝑖, ℓ, 𝛾 ) ∈ 𝐹 𝑡 . This is in direct contradiction to Lemma 3.3.Therefore, x is constrained ordinally efficient. (cid:3) In the presence of arbitrary constraints, the existence of envy maybe inevitable between anypair of agents if one agent is more constrained than the other. However, as the next theorem showswe can guarantee envy-freeness among any pair of agents of identical type.
Theorem 3.5.
At any preference profile (cid:23) , the constrained serial rule guarantees envy-freenessamong agents of the same type.Proof . Let x denote the outcome of the algorithm. Consider any pair of agents 𝑖 and 𝑗 that are ofthe same type. Suppose for contradiction that agent 𝑖 envies 𝑗 , i.e. there exists an indifference class ℓ < 𝐸 ((cid:23) 𝑖 ) such that 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ )) < 𝑥 𝑗 ( 𝑇 𝑖 ( ℓ )) . Let 𝑡 denote the step of the algorithm when ℓ 𝑡𝑖 = ℓ andagent 𝑖 is in the bottleneck set 𝐵 𝑡 . Let ( x 𝑡 , h 𝑡 , 𝜆 𝑡 ) denote the optimal solution to 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) .Since agent 𝑖 is in the bottleneck set 𝐵 𝑡 , the mechanism fixes the cumulative allocation received byagent 𝑖 for her top ℓ indifference classes. Hence we have 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ )) = 𝑥 𝑡𝑖 ( 𝑇 𝑖 ( ℓ )) = 𝜆 𝑡 .Let’s consider two cases: Case 1:
Suppose agent 𝑗 ∈ 𝐵 𝑡 . We have 𝑥 𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) = 𝑥 𝑡𝑗 ( 𝑇 𝑗 ( ℓ 𝑡𝑗 )) = 𝜆 𝑡 . However, as 𝑥 𝑗 ( 𝑇 𝑖 ( ℓ )) > 𝜆 𝑡 , there must exist some object 𝑜 ∈ 𝑇 𝑖 ( ℓ ) \ 𝑇 𝑗 ( ℓ 𝑡𝑗 ) such that 𝑥 𝑗,𝑜 > Case 2:
Suppose agent 𝑗 ∉ 𝐵 𝑡 . By construction, for any triple ( 𝑗 , ℓ ′ , 𝛾 ) ∈ 𝐹 𝑡 , we have ℓ ′ ≤ ℓ 𝑡 − 𝑗 and 𝑥 𝑗 ( 𝑇 𝑗 ( ℓ ′ )) = 𝛾 ≤ 𝜆 𝑡 . Thus, there must exist some object 𝑜 ∈ 𝑇 𝑖 ( ℓ ) \ 𝑇 𝑗 ( ℓ 𝑡 − 𝑗 ) suchthat 𝑥 𝑗,𝑜 > 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ )) = 𝜆 𝑡 <
1, there exists some object 𝑝 ∉ 𝑇 𝑖 ( ℓ ) where 𝑥 𝑖,𝑝 >
0. Let0 < 𝜀 < min { 𝑥 𝑗,𝑜 , 𝑥 𝑖,𝑝 } be some fixed constant. We can now define a new outcome y as follows. Let 𝑦 𝑖 ′ ,𝑜 ′ = 𝑥 𝑖 ′ ,𝑜 ′ for all objects 𝑜 ′ ∈ 𝑂 and agents 𝑖 ′ ∉ { 𝑖, 𝑗 } . For agents 𝑖 ′ ∈ { 𝑖, 𝑗 } , let 𝑦 𝑖 ′ ,𝑜 ′ = 𝑥 𝑖 ′ ,𝑜 ′ forall objects 𝑜 ′ ∉ { 𝑜, 𝑝 } . Let 𝑦 𝑖,𝑜 = 𝑥 𝑖,𝑜 + 𝜀 , 𝑦 𝑖,𝑝 = 𝑥 𝑖,𝑝 − 𝜀 , and 𝑦 𝑗,𝑜 = 𝑥 𝑗,𝑜 − 𝜀 , 𝑦 𝑗,𝑝 = 𝑥 𝑗,𝑝 + 𝜀 . Sinceagents 𝑖 and 𝑗 are of the same type and we have 𝑦 𝑖,𝑜 ′ + 𝑦 𝑗,𝑜 ′ = 𝑥 𝑖,𝑜 ′ + 𝑥 𝑗,𝑜 ′ for all objects 𝑜 ′ ∈ 𝑂 , wemust have 𝐴 y = 𝐴 x ≤ b . Further, by construction we have y ∈ Δ D , i.e. y is a feasible outcome. Inaddition, by our choice of objects 𝑜 and 𝑝 , we have 𝑦 𝑘 ( 𝑇 𝑘 ( ℓ )) ≥ 𝑥 𝑘 ( 𝑇 𝑘 ( ℓ )) ≥ 𝛾 for all ( 𝑘, ℓ, 𝛾 ) ∈ F 𝑡 and also 𝑦 𝑘 ( 𝑇 𝑘 ( ℓ 𝑡𝑘 )) ≥ 𝑥 𝑘 ( 𝑇 𝑘 ( ℓ 𝑡𝑘 )) for all 𝑘 ∈ 𝐵 𝑡 . However, since 𝑦 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) > 𝑥 𝑖 ( 𝑇 𝑖 ( ℓ 𝑡𝑖 )) ≥ 𝜆 𝑡 , thiscontradicts Lemma 3.3. (cid:3) .3 Computational Complexity As shown in Proposition 3.1, the mechanism terminates in at most 𝑛𝜌 rounds where 𝑛 and 𝜌 denote the number of distinct agents and objects respectively. In each round 𝑡 , the algorithm solvesone instance of the linear program to compute the value of 𝜆 𝑡 . Further, the algorithm needs to finda set of bottleneck agents 𝐵 𝑡 . We now show that 𝐵 𝑡 can be found in polynomial time by solving asequence of at most 𝑛 linear programs.We recall that 𝐵 𝑡 is defined as any minimal set of agents such that the objective value of 𝐿𝑃 ( 𝐵 𝑡 , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) equals 𝜆 𝑡 . Algorithm 2 provides a simple iterative procedure to find such aminimal set. We first initialize 𝐵 𝑡 to be the set of all agents. In each step, the algorithm considersremoving an agent 𝑖 from 𝐵 𝑡 . If removing such an agent allows the linear program to obtain ahigher objective value, then clearly agent 𝑖 must belong to the bottleneck set. On the other hand, ifthe linear program obtains an objective value of only 𝜆 𝑡 , then agent 𝑖 can be safely removed fromconsideration since by definition 𝐵 𝑡 \ { 𝑖 } is a smaller candidate set. Input: 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 , 𝜆 𝑡 as defined in Algorithm 1 Result: 𝐵 𝑡 : Set of bottleneck agents 𝐵 𝑡 ← 𝑁 ; for 𝑖 = , , . . . , 𝑛 do 𝜆 ← objective value of 𝐿𝑃 ( 𝐵 𝑡 \ { 𝑖 } , 𝐹 𝑡 , ( ℓ 𝑡𝑖 ) 𝑖 ∈ 𝑁 ) ; if 𝜆 == 𝜆 𝑡 then 𝐵 𝑡 ← 𝐵 𝑡 \ { 𝑖 } endend Return 𝐵 𝑡 Algorithm 2:
Procedure to find the bottleneck setAlgorithm 2 terminates in at most 𝑛 iterations and thus in total each round of the constrainedserial rule requires solving at most ( 𝑛 + ) linear programs. Algorithm 1 can thus be executed intime that is polynomial in the size of the constraints, number of agents and objects. Randomization in object allocation mechanisms is often used as a tool to incorporate fairnessfrom an ex-ante perspective. The outcome of the random assignment mechanism is treated as a16robability distribution over deterministic outcomes and an outcome drawn from this distributionis what gets implemented in practice. By the Birkhoff-von Neumann theorem, it is well knownthat every bistochastic random assignment can be implemented efficiently as a lottery over feasibledeterministic assignments, i.e., a deterministic assignment where every agent is assigned one objectand each object is assigned to one agent. However, in the presence of arbitrary constraints on therandom assignment, such a decomposition into a lottery over deterministic assignments that satisfythose constraints may not exist. The following example illustrates such a situation.
Example 3.2.
Consider a simple example with one agent, 𝑁 = { } , and three objects 𝑂 = { 𝑎, 𝑏, 𝑐 } .In the presence of constraints 𝑥 ,𝑎 + 𝑥 ,𝑏 ≤ / 𝑥 ,𝑏 + 𝑥 ,𝑐 ≤ /
3, and 𝑥 ,𝑎 + 𝑥 ,𝑐 ≤ /
3, a potentialfeasible solution to the random assignment problem is to set 𝑥 ,𝑎 = 𝑥 ,𝑏 = 𝑥 ,𝑐 = /
3. However, itcan be readily seen that there exists no deterministic assignment 𝑋 that satisfies all three constraintsand still obtains 𝑋 ,𝑎 + 𝑋 ,𝑏 + 𝑋 ,𝑐 = 𝑘 where 𝑘 denotes the size of the largest bundle. We discuss suchimplementation details where applicable in the specific applications in Section 4.On the other hand, our model of imposing constraints on the random assignment solutiongeneralizes the approach of imposing constraints on the ex-post deterministic outcomes. As shownby Balbuzanov (2019), any set of arbitary constraints on the ex-post outcomes can be representedby a set of linear inequalities on the random assignment. While such a reduction always exists, wenote that it may not be computationally efficient. In this section, we discuss how the constrained serial rule can be applied for several concreteapplications of constrained object allocation. 17 .1 Unconstrained Object Allocation
We can apply our model to the unconstrained object assignment problem by simply set-ting Δ C ( (cid:23) ) = Δ D for all (cid:23) ∈ R 𝑛 . In this case, the random assignment output by the con-strained serial rule coincides with that given by the extended probabilistic serial algorithm ofKatta and Sethuraman (2006).We first briefly discuss the extended probabilistic serial algorithm. At every round 𝑡 , theextended probabilistic serial algorithm constructs a flow network where agents point to their mostpreferred objects among the set of objects available at that round. Using the parametric max-flowalgorithm, the algorithm then identifies a bottleneck set of agents 𝑋 to be those that satisfy: 𝑋 = arg min 𝑌 ⊆ 𝑁 | Γ ( 𝑌 ) || 𝑌 | where Γ ( 𝑌 ) denotes the set of objects that are most preferred by atleast one agent in 𝑌 . We note thatthis set of agents 𝑋 is precisely the most constrained set in this round, i.e., an agent 𝑖 ∈ 𝑋 can getexactly | Γ ( 𝑋 )|| 𝑋 | (and not more) total probability share from her preferred objects. In the constrainedserial rule algorithm, since there are not other constraints restricting the random assignments, thissame set 𝑋 of agents will be chosen as the bottleneck set 𝐵 𝑡 . Budish et al. (2013) considers the object allocation problem with general quota constraintsand identified a large class of constraints called ‘bi-hierarchical constraints’ that are universallyimplementable, i.e., any random assignment satisfying these constraints can be implemented as alottery over deterministic assignments that satisfy the same constraints.A constraint in their setup is of the form 𝑞 𝑆 ≤ P ( 𝑖,𝑜 )∈ 𝑆 𝑥 𝑖,𝑜 ≤ 𝑞 𝑆 , where 𝑆 ⊆ 𝑁 × 𝑂 is a setof agent-object pairs, x is a deterministic assignment, and 𝑞 𝑆 , 𝑞 𝑆 are both integers. A constraintstructure H = ( 𝑆, 𝒒 𝑆 ) comprises of collection of such constraints. An additional requirement on H is that it must include all singleton sets. A constraint structure H is a hierarchy if for every 𝑆, 𝑆 ′ ∈ H , either 𝑆 ⊆ 𝑆 ′ or 𝑆 ′ ⊆ 𝑆 or 𝑆 ∩ 𝑆 ′ = ∅ . Finally, H is a bi-hierarchy if there existhierarchies H , H such that H = H ∪ H and H ∩ H = ∅ . Budish et al. (2013) proposedthe Generalized Probabilistic Serial mechanism for object allocation with bi-hierarchical quotaconstraints. However, their mechanism only works for upper-bound quotas and assumes that alllower bounds 𝑞 𝑆 = H in our constrainedserial rule algorithm (even in the presence of indifferences in the preference relations) by defining:18 C ( (cid:23) ) = { x ∈ Δ D | 𝑞 𝑆 ≤ X ( 𝑖,𝑜 )∈ 𝑆 𝑥 𝑖,𝑜 ≤ 𝑞 𝑆 for all ( 𝑆, 𝒒 𝑆 ) ∈ H } for all (cid:23) ∈ R 𝑛 Thus our algorithm generalizes the approach of Budish et al. even for lower bound quota constraints.The key technical innovation that allows us to do so lies in the fact that our algorithm looks ahead intime to ensure that the partial solution obtained at any time leads to a feasible random assignment.
Ashlagi et al. (2020) study type dependent distributional constraints that do not conform to abi-hierarchical structure. In this setup, every agent 𝑖 ∈ 𝑁 is associated with a type 𝑡 𝑖 ∈ 𝑇 where 𝑇 denotes a finite set of types. Let 𝑅 ⊆ 𝑇 denote an arbitrary set of agent types and let 𝑜 ∈ 𝑂 denote an arbitrary object. A single constraint is of the form 𝑞 𝑅,𝑜 ≤ P 𝑖 ∈ 𝑁 | 𝑡 𝑖 ∈ 𝑅 𝑥 𝑖,𝑜 ≤ 𝑞 𝑅,𝑜 , i.e.,the mechanism imposes floor and ceiling quotas on the total allocation of all agents belonging to aspecific set of agent types at a given object.As earlier, it can be readily seen that such distributional constraints can be easily represented inour framework. Since the constraints do not conform to a bi-hierarchical structure, the outcome ofour mechanism cannot always be implemented as a lottery over feasible deterministic assignments.However, as shown by Ashlagi et al. (2020), any random assignment satisfying these constraintscan be decomposed into a distribution over almost feasible deterministic outcomes where everyfloor and ceiling constraint is violated by at most | 𝑇 | . Balbuzanov (2019) considers the problem of random object assignment when we are givenan explicit list of ex-post feasible allocations and the random assignment must be implemented asa lottery over these allocations. For a preference profile (cid:23) , let 𝐶 ( (cid:23) ) be the set of all permissibledeterministic assignments and Δ C ( (cid:23) ) be the convex hull of the set C . He shows that for every 𝐶 ( (cid:23) ) , there exists a minimal set of constraints parameterized by the matrix 𝐴 , with 𝑎 𝑐𝑖,𝑜 ≥ ( 𝑖, 𝑜 ) ∈ 𝑁 × 𝑂 and constraint 𝑐 , and the vector 𝒃 ≥ such that Δ C ( (cid:23) ) = { x ∈ Δ D | 𝐴 x ≤ 𝒃 } . Hegeneralizes the probabilistic serial mechanism to incorporate these inequalities for the case whenagents have strict preferences. The constrained serial rule algorithm generalizes his mechanism tothe full preference domain. 19 .5 Combinatorial Assignment Another class of problems where our mechanism can be applied to is the problem of allocatingbundles of indivisible objects to agents when preferences exhibit complementarities (Budish (2011);Budish and Cantillon (2012); Nguyen et al. (2016)). Formally, let 𝐺 be an underlying set of objects,where each object 𝑔 ∈ 𝐺 is supplied in 𝑞 𝑔 copies. A bundle of objects can be represented by avector in N | 𝐺 |+ , where the 𝑡 th co-ordinate of this vector corresponds to the number of copies of theobject 𝑡 and N + = N ∪ { } . Let 𝑂 = { 𝑜 ∈ N | 𝐺 |+ | P 𝑔 ∈ 𝐺 𝑜 𝑔 ≤ 𝑘 } now be the set of all bundlesof size at most 𝑘 . We assume that each bundle is available in a single copy. Each agent 𝑖 ∈ 𝑁 isinterested in consuming one bundle from the set 𝑂 and has a complete and transitive preference (cid:23) 𝑖 on the set 𝑂 . A common application that fits in this class of problems is the course allocationproblem. Every student is to be assigned a schedule of at most 𝑘 courses, where each course 𝑔 hasa finite number of seats 𝑞 𝑔 .The set of feasible random allocations can be described by the following set of constraintsthat enforce that the total amount allocated of any object 𝑔 ∈ 𝐺 is at most its supply 𝑞 𝑔 . At everypreference profile (cid:23) , we have: Δ C ( (cid:23) ) = Δ C = { 𝑥 ∈ Δ D | X 𝑖 ∈ 𝑁 X 𝑜 ∈ 𝑂 𝑜 𝑔 · 𝑥 𝑖,𝑜 ≤ 𝑞 𝑔 , ∀ 𝑔 ∈ 𝐺 } From Definition 2.2, it is easy to see that all agents under these feasibility constraints are ofthe same type. Therefore, the constrained serial rule guarantees constrained ordinally efficient andenvy-free outcomes. Further, as discussed by Nguyen et al. (2016), any outcome of the mechanismcan be implemented as a lottery over deterministic assignments that violate the supply constraintsby at most 𝑘 −
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