Sequential Mechanisms for Multi-type Resource Allocation
Sujoy Sikdar, Xiaoxi Guo, Haibin Wang, Lirong Xia, Yongzhi Cao
SSequential Mechanisms for Multi-type ResourceAllocation
Sujoy Sikdar , Xiaoxi Guo , Haibin Wang , Lirong Xia , and YongzhiCao Binghamton University, [email protected] Peking University, [email protected], [email protected] Rensselaer Polytechnic Institute, [email protected] Peking University, [email protected]
February 23, 2021
Abstract
Several resource allocation problems involve multiple types of re-sources, with a different agency being responsible for “locally” allocat-ing the resources of each type, while a central planner wishes to providea guarantee on the properties of the final allocation given agents’ pref-erences. We study the relationship between properties of the localmechanisms, each responsible for assigning all of the resources of adesignated type, and the properties of a sequential mechanism whichis composed of these local mechanisms, one for each type, appliedsequentially, under lexicographic preferences , a well studied model ofpreferences over multiple types of resources in artificial intelligenceand economics. We show that when preferences are O -legal, meaningthat agents share a common importance order on the types, sequentialmechanisms satisfy the desirable properties of anonymity, neutrality,non-bossiness, or Pareto-optimality if and only if every local mecha-nism also satisfies the same property, and they are applied sequentiallyaccording to the order O . Our main results are that under O -legal lex-icographic preferences, every mechanism satisfying strategyproofnessand a combination of these properties must be a sequential composi-tion of local mechanisms that are also strategyproof, and satisfy thesame combinations of properties. Consider the example of a hospital where patients must be allocated sur-geons and nurses with different specialties, medical equipment of differenttypes, and a room Huh et al. (2013). This example illustrates multi-type1 a r X i v : . [ c s . G T ] F e b esource allocation problems (MTRAs), first introduced by Moulin (1995),where there are p ≥ indivisible items which are not interchange-able, and a group of agents having heterogeneous preferences over receivingcombinations of an item of each type. The goal is to design a mechanismwhich allocates each agent with a bundle consisting of an item of each type.Often, a different agency is responsible for the allocation of each typeof item in a distributed manner, using possibly different local mechanisms,while a central planner wishes that the mechanism composed of these lo-cal mechanisms satisfies certain desirable properties. For example, differentdepartments may be responsible for the allocation of each type of medicalresources, while the hospital wishes to deliver a high standard of patient careand satisfaction given the patients’ preferences and medical conditions; inan enterprise, clients have heterogeneous preferences over cloud computingresources like computation and storage Ghodsi et al. (2011, 2012); Bhat-tacharya et al. (2013), possibly provided by different vendors; in a university,students must be assigned to different types of courses handled by differentdepartments; in a seminar class, the research papers and time slots Mackinand Xia (2016) may be assigned separately by the instructor and a teach-ing assistant respectively, and in rationing Elster (1992), different agenciesmay be responsible for allocating different types of rations such as food andshelter.Unfortunately, as Svensson (1999) shows, even when there is a single typeof items and each agent is to be assigned a single item, serial dictatorshipsare the only strategyproof mechanisms which are non-bossy , meaning thatno agent can falsely report her preferences to change the outcome withoutalso affecting her own allocation, and neutral , meaning that the outcomeis independent of the names of the items. In a serial dictatorship, agentsare assigned their favorite remaining items one after another according to afixed priority ordering of the agents. P´apai (2001) shows a similar result forthe multiple assignment problem , where agents may be assigned more thanone item, that the only mechanisms which are strategyproof, non-bossy, and Pareto-optimal are sequential dictatorships , where agents pick a favorite re-maining item one at a time according to a hierarchical picking sequence,where the next agent to pick an item depends only on the allocations madein previous steps. Pareto-optimality is the property that there is no otherallocation which benefits an agent without making at least one agent worseoff. More recently, Hosseini and Larson (2019) show that even under lex-icographic preferences, the only mechanisms for the multiple assignmentproblem that are strategyproof, non-bossy, neutral and Pareto-optimal areserial dictatorships with a quota for each agent.Mackin and Xia (2016) study MTRAs in a slightly different setting toours: a monolithic central planner controls the allocation of all types ofitems. They characterize serial dictatorships under the unrestricted domainof strict preferences over bundles with strategyproofness, non-bossiness, and2ype-wise neutrality, a weaker notion of neutrality where the outcome is in-dependent of permutations on the names of items within each type. Perhapsin light of this and other negative results described above, there has beenlittle further work on strategyproof mechanisms for MTRAs. This is theproblem we address in this paper.We study the design of strategyproof sequential mechanisms for MTRAswith p ≥ p local mechanisms, one for eachtype, applied sequentially one after the other, to allocate all of the itemsof the type, under the assumption that agents’ preferences are lexicographic and O -legal.For MTRAs, lexicographic preferences are a natural, and well-studiedassumption for reasoning about ordering alternatives based on multiplecriteria in social science Gigerenzer and Goldstein (1996). In artificialintelligence, lexicographic preferences have been studied extensively, forMTRAs Sikdar et al. (2017, 2019); Sun et al. (2015); Wang et al. (2020); Guoet al. (2020), multiple assignment problems Hosseini and Larson (2019); Fu-jita et al. (2015), voting over multiple issues Lang and Xia (2009); Xia et al.(2011), and committee selection Sekar et al. (2017), since lexicographic pref-erences allow reasoning about and representing preferences in a structuredand compact manner. In MTRAs, lexicographic preferences are defined byan importance order over the types of items, and local preferences over itemsof each type. The preference relation over any pair of bundles is decided infavor of the bundle that has the more preferred item of the most importanttype at which the pair of bundles differ, and this decision depends only onthe items of more important types.In several problems, it is natural to assume that every agent shares acommon importance order. For example, when rationing Elster (1992), itmay be natural to assume that every agent thinks food is more importantthan shelter, and in a hospital Huh et al. (2013), all patients may considertheir allocation of surgeons and nurses to be more important than the med-ical equipment and room. O -legal lexicographic preference profiles, whereevery agent has a common importance order O over the types, have beenstudied recently by Lang and Xia (2009); Xia et al. (2011) for the multi-issuevoting problem. When agents’ preferences are O -legal and lexicographic, itis natural to ask the following questions about sequential mechanisms thatdecide the allocation of each type sequentially using a possibly different localmechanism according to O , which we address in this paper: (1) if every localmechanism satisfies property X , does the sequential mechanism composed ofthese local mechanisms also satisfy X ? , and (2) what properties must everylocal mechanism satisfy so that their sequential composition satisfies property X ? .1 Contributions For O -legal preferences, a property X ∈ { anonymity, type-wise neutrality,non-bossiness, monotonicity, Pareto-optimality } , and any sequential mecha-nism f O = ( f , . . . , f p ) which applies each local mechanism f i one at a timeaccording to the importance order O , we show in Theorem 1 and Theorem 2that f O satisfies X if and only if every local mechanism it is composed ofsatisfies X .However, sequential compositions of locally strategyproof mechanismsare not guaranteed to be strategyproof, which raises the question: underwhat conditions are sequential mechanisms strategyproof? We begin byshowing in Proposition 1, that when agents preferences are lexicographic,but agents have different importance orders, sequential mechanisms com-posed of locally strategyproof mechanisms are, unfortunately, not guaran-teed to be strategyproof. In contrast, we show in Proposition 2 that se-quential composition of strategyproof mechanisms are indeed strategyproofwhen either: (1) agents’ preferences are separable and lexicographic, evenwhen different agents may have different importance orders, or (2) agents’preferences are lexicographic and O -legal and all of the local mechanismsare also non-bossy.Our main results characterize the class of mechanisms that satisfy strat-egyproofness, along with different combinations of non-bossiness, neutrality,and Pareto-optimality under O -legal preferences as O -legal sequential mech-anisms. We show: • In Theorem 3, that under O -legal lexicographic preferences, the class ofmechanisms satisfying strategyproofness and non-bossiness is exactly theclass of mechanisms that can be decomposed into multiple locally strate-gyproof and non-bossy mechanisms, one for each combination of type andallocated items of more important types. This class of mechanisms is exactlythe class of O -legal conditional rule nets (CR-nets) Lang and Xia (2009); • In Theorem 4, that a mechanism is strategyproof, non-bossy, and type-wise neutral if and only if it is an O -legal sequential composition of serialdictatorships; • In Theorem 5, that a mechanism is strategyproof, non-bossy, and Pareto-optimal if and only if it is an O -legal CR-net composed of serial dictatorships.Finally, we show that despite the negative result in Proposition 1 thatwhen agents’ preferences do not share a common importance order on thetypes, sequential compositions of locally strategyproof mechanisms may notsatisfy strategyproofness, we show in Theorem 6, that computing beneficialmanipulations w.r.t. a sequential mechanism is NP-complete.4 Related Work and Discussion
The MTRA problem was introduced by Moulin (1995). More recently, itwas studied by Mackin and Xia (2016), who characterize the class of strate-gyproof and non-bossy mechanisms under the unrestricted domain of strictpreferences over bundles as the class of serial dictatorships. However, as theynote, it may be unreasonable to expect agents to express preferences as com-plete rankings over all possible bundles, besides the obvious communicationand complexity issues arising from agents’ preferences being represented bycomplete rankings.The literature on characterizations of strategyproof mechanisms Svens-son (1999); P´apai (2001); Hosseini and Larson (2019) for resource allocationproblems belong to the line of research initiated by the famous Gibbard-Satterthwaite Theorem Gibbard (1973); Satterthwaite (1975) which showedthat dictatorships are the only strategyproof voting rules which satisfy non-imposition, which means that every alternative is selected under some pref-erence profile. Several following works have focused on circumventing thesenegative results by identifying reasonable and natural restrictions on the do-main of preferences. For voting, Moulin (1980) provide non-dictatorial rulessatisfying strategyproofness and non-imposition under single-peaked Black(1948) preferences. Our work follows in this vein and is closely related tothe works by Le Breton and Sen (1999), who assume that agents’ prefer-ences are separable, and more recently, Lang and Xia (2009) who considerthe multi-issue voting problem under the restriction of O -legal lexicographicpreferences, allowing for conditional preferences given by CP-nets similar toour work. Xia and Conitzer (2010) consider a weaker and more expressive do-main of lexicographic preferences allowing for conditional preferences. Here,agents have a common importance order on the issues, and the agents pref-erences over any issue is conditioned only on the outcome of more importantissues. They characterize the class of voting rules satisfying strategyproof-ness and non-imposition as being exactly the class of all CR-nets. CR-netsdefine a hierarchy of voting rules, where the voting rule for the most impor-tant issue is fixed, and the voting rule for every subsequent issue dependsonly on the outcome of the previous issues. Similar results were shownearlier by Barbera et al. (1993, 1997, 1991).In a similar vein, Sikdar et al. (2017, 2019) consider the multi-type vari-ant of the classic housing market Shapley and Scarf (1974), first proposedby Moulin (1995), and Fujita et al. (2015) consider the variant where agentscan receive multiple items. These works circumvent previous negative re-sults on the existence of strategyproof and core-selecting mechanisms underthe assumption of lexicographic extensions of CP-nets, and lexicographicpreferences over bundles consisting of multiple items of a single type respec-tively. Wang et al. (2020); Guo et al. (2020) study MTRAs with divisible andindivisible items, and provide mechanisms that are fair and efficient under5he notion of stochastic dominance by extending the famous probabilisticserial Bogomolnaia and Moulin (2001) and random priority Abdulkadiro˘gluand S¨onmez (1998) mechanisms, and show that while their mechanisms donot satisfy strategyproof in general, under the domain restriction of lexico-graphic preferences, strategyproofness is restored, and stronger notions ofefficiency can be satisfied. A multi-type resource allocation problem (MTRA) Mackin and Xia (2016),is given by a tuple (
N, M, P ). Here, (1) N = { , . . . , n } is a set of agents,(2) M = D ∪ · · · ∪ D p is a set of items of p types, where for each i ≤ p , D i is a set of n items of type i , and (3) P = ( (cid:31) j ) j ≤ n is a preference profile ,where for each j ≤ n , (cid:31) j represents the preferences of agent j over the setof all possible bundles D = D × · · · × D p . For any type i ≤ p , we use k i torefer to the k -th item of type i , and we define T = { D , . . . , D p } . We alsouse D i refers to { D i +1 , . . . , D p } ,and D ≤ i , D ≥ i are in the same manner. For any profile P , and agent j ≤ n ,we define P − j = ( (cid:31) k ) k ≤ n,k (cid:54) = j , and P = ( P − j , (cid:31) j ). Bundles.
Each bundle x ∈ D is a p -vector, where for each type i ≤ p , [ x ] i denotes the item of type i . We use a ∈ x to indicate that bundle x containsitem a . For any S ⊆ T , we define D S = × D ∈ S D , and − S = T \ S . Forany S ⊆ T , any bundle x ∈ D S , for any D ∈ − S , and item a ∈ D , ( a, x )denotes the bundle consisting of a and the items in x , and similarly, for any U ⊆ − S , and any bundle y ∈ D U , we use ( x , y ) to represent the bundleconsisting of the items in x and y . For any S ⊆ T , we use x ⊥ S to denotethe items in x restricted to the types in S . Allocations. An allocation A : N → D is a one-to-one mapping fromagents to bundles such that no item is assigned to more than one agent. A denotes the set of all possible allocations. Given an allocation A ∈ A , A ( j ) denotes the bundle allocated to agent j . For any S ⊆ T , we use A ⊥ S : N → D S to denote the allocation of items restricted to the types in S . CP-nets and O -legal Lexicographic Preferences. An acyclic CP-net Boutilier et al. (2004) N over D is defined by (i) a directed graph G = ( T, E ) called the dependency graph , and (ii) for each type i ≤ p , a conditional preference table CP T ( D i ) that contains a linear order (cid:31) x over D i for each x ∈ D P a ( D i ) , where P a ( D i ) is the set of types correspondingto the parents of D i in G . A CP-net N represents a partial order over D which is the transitive closure of the preference relations represented by all6igure 1: An O -legal lexicographic preference with an underlying CP-net,where O = [ D (cid:66) D ].of the CP T entries which are { ( a i , u , z ) (cid:31) ( b i , u , z ) : i ≤ p, a i , b i ∈ D i , u ∈D P a ( D i ) , z ∈ D − P a ( D i ) \{ D i } } .Let O = [ D (cid:66) · · · (cid:66) D p ] be a linear order over the types. A CP-net is O -legal if there is no edge ( D i , D l ) with i > l in its dependency graph. A lexicographic extension of an O -legal CP-net N is a linear order (cid:31) over D ,such that for any i ≤ p , any x ∈ D D i ,if a i (cid:31) x b i in N , then, ( x , a i , y ) (cid:31) ( x , b i , z ). The linear order O over typesis called an importance order , and D is the most important type, D is thesecond most important type, etc. We use O to denote the set of all possibleimportance orders over types.Given an important order O , we use L O to denote the set of all possiblelinear orders that can be induced by lexicographic extensions of O -legalCP-nets as defined above. A preference relation (cid:31)∈ L O is said to be an O -legal lexicographic preference relation, and a profile P ∈ L nO is an O -legal lexicographic profile. An O -legal preference relation is separable , if thedependency graph of the underlying CP-net has no edges. We will assumethat all preferences are O -legal lexicographic preferences throughout thispaper unless specified otherwise. Example 1.
Here we show how to compare bundles under an O -legal lexico-graphic preference with CP-net. In Figure 1(a) is a dependency graph whichshows that D depends on D . Figure 1(b) is the CP T for both types, whichimplies (1 , ) (cid:31) (1 , ) , (2 , ) (cid:31) (2 , ) . Figure 1(c) gives the impor-tance order O = [ D (cid:66) D ] . With O we can compare some bundles directly.For example, (1 , ) (cid:31) (2 , ) , (1 , ) (cid:31) (2 , ) because the most impor-tant type with different allocations is D and (cid:31) ∅ . Finally, Figure 1(d)shows the relations among all the bundles. We note that any lexicographic extension of an O -legal CP-net accordingto the order O does not violate any of the relations induced by the originalCP-net, and always induces a linear order over all possible bundles unlike7P-nets which may induce partial orders.For any O -legal lexicographic preference relation (cid:31) over D , and givenany x ∈ D D
An allocation mechanism f satisfies: • anonymity , if for any permutation Π on the names of agents, and anyprofile P , f (Π( P )) = Π( f ( P )); • type-wise neutrality , if for any permutation Π = (Π , . . . , Π p ), where forany i ≤ p , Π only permutes the names of the items of type i according to apermutation Π i , and any profile P , f (Π( P )) = Π( f ( P )); • Pareto-optimality , if for every allocation A such that there exists an agent j such that A ( j ) (cid:31) j f ( P )( j ), there is another agent k such that f ( P )( k ) (cid:31) j A ( k ). • non-bossiness , if no agent can misreport her preferences and change theallocation of other agents without also changing her own allocation, i.e.there does not exist any pair ( P, (cid:31) (cid:48) j ) where P is a profile and (cid:31) (cid:48) j is themisreported preferences of agent j such that f ( P )( j ) = f ( P − j , (cid:31) (cid:48) j )( j ) andfor some agent k (cid:54) = j , f ( P )( k ) (cid:54) = f ( P − j , (cid:31) (cid:48) j )( k ). • non-bossiness of more important types , if no agent j can misreport herlocal preferences for less important types and change the allocation of moreimportant types to other agents without also changing her own allocation ofmore important types. i.e. for every profile P , every agent j ≤ n , every type D i , i ≤ p , and every misreport of agent j ’s preferences (cid:31) (cid:48) j where for every h < i , every u ∈ P ar ( D h ), (cid:31) (cid:48) j ⊥ D h ,u = (cid:31) j ⊥ D h ,u , it holds that if for someagent k (cid:54) = j , f ( P − j , (cid:31) (cid:48) j )( k ) ⊥ D ≤ i (cid:54) = f ( P )( k ) ⊥ D ≤ i , then f ( P − j , (cid:31) (cid:48) j )( j ) ⊥ D ≤ i (cid:54) = f ( P )( j ) ⊥ D ≤ i . 8 monotonicity , for any agent j , any profile P , let (cid:31) (cid:48) j be a misreportpreference such that if Y ⊆ D is the set of all bundles whose ranks areraised and it holds that for every x , z ∈ Y , x (cid:31) j z = ⇒ x (cid:31) (cid:48) j z , then, f ( P − j , (cid:31) (cid:48) j )( j ) ∈ { f ( P )( j ) } ∪ Y . • strategyproofness , if no agent has a beneficial manipulation i.e. there isno pair ( P, (cid:31) (cid:48) j ) where P is a profile and (cid:31) (cid:48) j is a manipulation of agent j ’spreferences such that f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) j f ( P )( j ). Theorem 1.
For any importance order O ∈ O , any X ∈ { anonymity,type-wise neutrality, non-bossiness, monotonicity, Pareto-optimality } , and f O = ( f , . . . , f p ) be any O -legal sequential mechanism. Then, for O -legalpreferences, if for every i ≤ p , the local mechanism f i satisfies X , then f O satisfies X .Proof. (Sketch) Throughout, we will assume that O = [ D (cid:66) · · · (cid:66) D p ], andthat P is an arbitrary O -legal preference profile over p types. For any i ≤ p ,we define g i to be the sequential mechanism ( f , . . . , f i ). The proofs ofanonymity and type-wise neutrality are relegated to the appendix. non-bossiness. Let us assume for the sake of contradiction that the claimis false, i.e. there exists a profile P , an agent j and a misreport (cid:31) (cid:48) j suchthat for P (cid:48) = ( (cid:31) − j , (cid:31) (cid:48) j ), f O ( P )( j ) = f O ( P (cid:48) )( j ), and f O ( P ) (cid:54) = f O ( P (cid:48) ).Then, there is a type i ≤ p such that, f O ( P ) ⊥ D
Suppose the claim is true for p ≤ k types. Let P bean O -legal lexicographic profile over k + 1 types, and f O = ( f i ) i ≤ k +1 is asequential composition of Pareto-optimal local mechanisms. Suppose for thesake of contradiction that there exists an allocation B such that some agentsstrictly better off compared to f O ( P ), and no agent is worse off. Then, byour assumption of lexicographic preferences, for every agent k who is notstrictly better off, B ( k ) = f O ( P )( k ), and for every agent j who is strictlybetter off, one of two cases must hold. (1) B ( j ) ⊥ D (cid:31) j f O ( P )( j ) ⊥ D , or (2) B ( j ) ⊥ D = f O ( P )( j ) ⊥ D . (1): If there exists an agent such that B ( j ) ⊥ D (cid:31) j f O ( P )( j ) ⊥ D , this is a contradiction to our assumption that f is Pareto-optimal. (2): Suppose B ( j ) ⊥ D = f O ( P )( j ) ⊥ D for all agents who arestrictly better off. Let g = ( f , . . . , f k +1 ). W.l.o.g. let agent 1 strictly prefer B (1) to f O ( P )(1). Then, g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )(1) (cid:31) B (1) ⊥ D ≤ k +1 \ D ,and for every other agent l (cid:54) = 1, either g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )( l ) (cid:31) l B ( l ) ⊥ D ≤ k +1 \ D , or g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )( l ) = B ( l ) ⊥ D ≤ k +1 \ D , which is acontradiction to our induction assumption. Theorem 2.
For any importance order O ∈ O , X ∈ { anonymity, type-wise neutrality, non-bossiness, monotonicity, Pareto-optimality } , and f O =( f , . . . , f p ) be any O -legal sequential mechanism. For O -legal preferences,if f O satisfies X , then for every i ≤ p , f i satisfies X .Proof. (Sketch) We only provide the proof of non-bossiness here. The restof the proofs are in the appendix. non-bossiness. Assume for the sake of contradiction that k ≤ p is themost important type such that f k does not satisfy non-bossiness. Then,there exists a preference profile Q = ( (cid:31) k ) j ≤ n over D k , and a bossy agent l and a misreport Q (cid:48) = ( (cid:31) k − l , ¯ (cid:31) kl ), such that f k ( Q (cid:48) )( l ) = f k ( Q )( l ), but f k ( Q (cid:48) ) (cid:54) = f k ( Q ). Now, consider the O -legal separable lexicographic profile P , where for any type i ≤ p , the preferences over type D i is denoted P ⊥ D i and P ⊥ D k = Q , and the profile P (cid:48) obtained from P by replacing (cid:31) l with (cid:31) (cid:48) l ,which in turn is obtained from (cid:31) l by replacing (cid:31) l ⊥ D k with ¯ (cid:31) kl . It is easy tosee that f O ( P (cid:48) ) ⊥ D
For any importance order O ∈ O , when the preferencesare not O -legal, and agents are either optimistic or pessimistic, a sequen-tial mechanism f O composed of strategyproof mechanisms is not necessarilystrategyproof.Proof. When preferences are lexicographic, and not O -legal, a sequentialmechanism composed of locally strategyproof mechanisms is not necessarilystrategyproof, when agents are either optimistic or pessimistic, as we showwith counterexamples. Consider the profile with two agents and two types H and C . Agent 1’s importance order is H (cid:66) C , preferences over H is 1 H (cid:31) H and over C is conditioned on the assignment of house 1 H : 1 C (cid:31) C , H :2 C (cid:31) C . Agent 2 has importance order C (cid:66) H and separable preferenceswith order on cars being 2 C (cid:31) C , and order on houses 1 H (cid:31) H . Considerthe sequential mechanism composed of serial dictatorships where H (cid:66) C andfor houses the picking order over agents is (2 , , H C and 1 H C respectively to agents 1 and 2. When agent 2 misreportsher preferences over houses as 2 H (cid:31) H , and agent 1 is truthful and eitheroptimistic or pessimistic, the allocation is 1 H C and 2 H C to agents 1 and2 respectively, a beneficial misreport for agent 2.In contrast, sequential mechanisms composed of locally strategyproofmechanisms are guaranteed to be strategyproof under two natural restric-tions on the domain of lexicographic preferences: (1) when agents’ pref-erences are lexicographic and separable, but not necessarily O -legal w.r.t.a common importance order O , and (2) when agents’ have O -legal lexico-graphic preferences, and the local mechanisms also satisfy non-bossiness.11 roposition 2. For any importance order O ∈ O , a sequential mechanismcomposed of strategyproof local mechanisms is strategyproof,(1) when agents are either optimistic or pessimistic, and their preferencesare separable and lexicographic, or(2) when agents’ preferences are lexicographic and O -legal and the localmechanisms also satisfy non-bossiness.Proof. (1): Let P be a profile of separable lexicographic preferences. Sup-pose for the sake of contradiction that an agent j has a beneficial mis-report (cid:31) (cid:48) j , and let P (cid:48) = ( P − j , (cid:31) (cid:48) j ). Let k be the type of highest impor-tance to j for which [ f O ( P (cid:48) )( j )] k (cid:54) = [ f O ( P )( j )] k . Then, by our assump-tion that preferences are lexicographic, k being the most important typefor j where her allocated item differs, and that P (cid:48) is a beneficial manip-ulation, it must hold that [ f O ( P (cid:48) )( j )] k (cid:31) [ f O ( P )( j )] k . Since, preferencesare separable, [ f ( P (cid:48) )] k = f k ( P (cid:48)⊥{ D k } ). Since every other agent is truthful, P (cid:48)⊥{ D k } = ( P − j ⊥{ D k } , (cid:31) (cid:48) j ⊥ D k , ) , and (cid:31) (cid:48) j ⊥ D k (cid:54) = (cid:31) j ⊥ D k is a beneficial manip-ulation, which implies that f k is not strategyproof, a contradiction to ourassumption.(2) Now, we consider the case where the profile of truthful preferences P is an arbitrary O -legal and lexicographic profile of preferences that may notbe separable, and the local mechanisms are non-bossy and strategyproof.Suppose for the sake of contradiction that an agent j has a beneficial mis-report (cid:31) (cid:48) j , and let P (cid:48) = ( P − j , (cid:31) (cid:48) j ). W.l.o.g. let O = [1 (cid:66) · · · (cid:66) p ].Let k be the most important type for which agent j receives a differentitem. We begin by showing that by our assumption that the local mech-anisms are non-bossy, and our assumption of O -legal lexicographic prefer-ences, it holds that for every i < k according to O , f i ( P (cid:48) ) ⊥ D i = f i ( P ) ⊥ D i .For the sake of contradiction, let h < k be the first type for whichsome agent l receives a different item, i.e. [ f ( P (cid:48) )( l )] h (cid:54) = [ f ( P )( l )] h , and f ( P (cid:48) ) ⊥ D A (directed) conditional rule net (CR-net) M over D is defined by(i) a directed graph G = ( { D , ..., D p } , E ) , called the dependency graph ,and(ii) for each D i , there is a conditional rule table CRT i that contains a mech-anism denoted M ⊥ D i ,A for D i for each allocation A of all items of types thatare parents of D i in G , denoted Pa ( D i ) .Let O = [ D (cid:66) · · · (cid:66) D p ] , then a CR-net is O -legal if there is no edge ( D i , D l ) in its dependency with i > l . Example 2. We note that the local mechanisms in a CR-net may be anymechanism that can allocate n items to n agents given strict preferences.In Figure 2, we show a CR-net f where all the local mechanisms are serialdictatorships. The directed graph is shown in Figure 2(a), which implies D depends on D . Figure 2(b) shows the CRT of f . In the CRT, f : ( b, a ) means that in the serial dictatorship f , agent b picks her most preferreditem first followed by agent a , and it is similar for f , f (cid:48) . The conditionsin the CR-net, which are partial allocations are represented by mappings,for example, ( a → ) means agent a gets . Figure 2 (c) and (d) are the O -legal preferences of agents a and b , respectively, where O = [ D (cid:66) D ] .According to f , first we apply f on D , and we have a → , b → . Then, y CRT of f we use f for D , and we have a → , b → . Therefore f outputs an allocation where a → (1 , ) , b → (2 , ) . Lemma 1. When agents’ preferences are restricted to the O -legal lexico-graphic preference domain, for any strategyproof mechanism f , any profile P , and any pair ( P − j , (cid:31) (cid:48) j ) obtained by agent j misreporting her preferencesby raising the rank of f ( P )( j ) such that for any bundle b , f ( P )( j ) (cid:31) j b = ⇒ f ( P )( j ) (cid:31) (cid:48) j b , it holds that f ( P − j , (cid:31) (cid:48) j )( j ) = f ( P )( j ) .Proof. Suppose for the sake of contradiction that f is a strategyproof mech-anism that does not satisfy monotonicity. Let P = ( (cid:31) j ) j ≤ n be a profile, j be an agent who misreports her preferences as (cid:31) (cid:48) j obtained from (cid:31) j by raising the rank of f ( P )( j ), specifically, for any bundle b , f ( P )( j ) (cid:31) j b = ⇒ f ( P )( j ) (cid:31) (cid:48) j b . Then, either: (1) f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) (cid:48) j f ( P )( j ), or (2) f ( P )( j ) (cid:31) (cid:48) j f ( P − j , (cid:31) (cid:48) j )( j ).(1) Suppose f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) (cid:48) j f ( P )( j ). First, we claim that if f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) (cid:48) j f ( P )( j ), then f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) j f ( P )( j ). Suppose for the sake ofcontradiction that this were not true, then f ( P )( j ) (cid:31) j f ( P − j , (cid:31) (cid:48) j )( j ) and f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) (cid:48) j f ( P )( j ). This is a contradiction to our assumption on (cid:31) (cid:48) j .This implies that f ( P − j , (cid:31) (cid:48) j )( j ) (cid:31) j f ( P )( j ) and (cid:31) (cid:48) j is a beneficial misreportfor agent j , a contradiction to our assumption that f is strategyproof.(2) If f ( P )( j ) (cid:31) (cid:48) j f ( P − j , (cid:31) (cid:48) j )( j ), then (cid:31) j is a beneficial misreport foragent j w.r.t. P (cid:48) , a contradiction to our assumption that f is strategyproof. Theorem 3. For any importance order O , a mechanism satisfies strate-gyproofness and non-bossiness of more important types under the O -legallexicographic preference domain if and only if it is an O -legal locally strate-gyproof and non-bossy CR-net.Proof. The if part is obvious (and is proved in Proposition 2). We prove theonly if part by induction. Claim 1. If an allocation mechanism satisfies non-bossiness of more im-portant types and strategyproofness, then it can be decomposed into a locallystrategyproof and non-bossy CR-net. Proof by induction on the number of types. The claim is trivially true forthe base case with p = 1 type. Suppose the claim holds true for p = k typesi.e. when there are at most k types, if an allocation mechanism is non-bossyin more important types and strategyproof, then it can be decomposed intolocally strategyproof and non-bossy mechanisms.When p = k + 1, we prove that any non-bossy and strategyproof allo-cation mechanism f for a basic type-wise allocation problem can be decom-posed into two parts by Step 1: 14. Applying a local allocation mechanism f to D to compute allocation[ A ] .2. Applying an allocation mechanism f ⊥ D − , [ A ] to types D − . • Step 1. For any strategyproof allocation mechanism satisfying non-bossiness of more important types, allocations for type 1 depend only onpreferences restricted to D . Claim 2. For any pair of profiles P = ( (cid:31) j ) j ≤ n , Q = ( (cid:31) (cid:48) j ) j ≤ n , and P ⊥ D = Q ⊥ D , we must have that f ( P ) ⊥ D = f ( Q ) ⊥ D .Proof. Suppose for sake of contradiction that f ( P ) ⊥ D (cid:54) = f ( Q ) ⊥ D . For any0 ≤ j ≤ n , define P j = ( (cid:31) (cid:48) , . . . , (cid:31) (cid:48) j , (cid:31) j +1 , . . . , (cid:31) n ) and suppose f ( P j ) ⊥ D (cid:54) = f ( P j +1 ) ⊥ D for some j ≤ n − 1. Let [ A ] = f ( P j )( j + 1) ⊥ D and [ B ] = f ( P j +1 )( j + 1) ⊥ D . Now, suppose thatCase 1: [ A ] = [ B ] , but for some other agent ˆ j , f ( P j )(ˆ j ) ⊥ D (cid:54) = f ( P j +1 )(ˆ j ) ⊥ D . This is a direct violation of non-bossiness of more importanttypes because P j ⊥ D = P j +1 ⊥ D by construction.Case 2: [ A ] (cid:54) = [ B ] . If [ B ] (cid:31) j +1 ⊥ D [ A ] , then ( P j , (cid:31) (cid:48) j +1 ) is a bene-ficial manipulation due to agents’ lexicographic preferences. Otherwise, if[ A ] (cid:31) j +1 ⊥ D [ B ] , then ( P j +1 , (cid:31) j +1 ) is a beneficial manipulation due to ourassumption that (cid:31) j +1 ⊥ D = (cid:31) (cid:48) j +1 ⊥ D and agents’ lexicographic preferences.This contradicts the strategyproofness of f . • Step 2. Show that f satisfies strategyproofness and non-bossiness.First, we show that f must satisfy strategyproofness by contradiction.Suppose for the sake of contradiction that f is strategyproof but f is notstrategyproof. Let P = ( (cid:31) j ) j ≤ n be a profile of agents’ preferences over D .Then, there exists an agent j ∗ with a beneficial manipulation (cid:31) (cid:48) j ∗ . Now,consider a profile Q = ( ¯ (cid:31) j ) j ≤ n where for every agent j, ¯ (cid:31) j ⊥ D = (cid:31) j and themechanism f whose local mechanism for D is f . We know from Step 1that f ( Q ) ⊥ D = f ( Q ⊥ D ) = f ( P ). However, in that case, because agents’preferences are lexicographic with D being the most important type, agent j ∗ has a successful manipulation ¯ (cid:31) (cid:48) j ∗ where ¯ (cid:31) (cid:48) j ∗ ⊥ D = (cid:31) (cid:48) j ∗ since the resultingallocation of f ( ¯ (cid:31) − j ∗ , ¯ (cid:31) (cid:48) j ∗ ) is a strictly preferred item of type D . This is acontradiction to our assumption on the strategyproofness of f .Then, we also show that f satisfies non-bossiness. Suppose for the sakeof contradiction that f is not non-bossy. Let P = ( (cid:31) j ) j ≤ n be a profile ofagents’ preferences over D . Then, there exists an agent j ∗ with a bossypreference (cid:31) (cid:48) j ∗ such that for P (cid:48) = ( (cid:31) − j ∗ , (cid:31) (cid:48) j ∗ ), f ( P )( j ∗ ) = f ( P (cid:48) )( j ∗ ) while f ( P )( j ) (cid:54) = f ( P (cid:48) )( j ) for some j . Now, consider a profile Q = ( ¯ (cid:31) j ) j ≤ n wherefor every agent j, ¯ (cid:31) j ⊥ D = (cid:31) j and the mechanism f whose local mechanismfor D is f . We know from Step 1 that f ( Q ) ⊥ D = f ( Q ⊥ D ) = f ( P ).However, in that case, because agents’ preferences are lexicographic with15 being the most important type, agent j ∗ has a bossy preference ¯ (cid:31) (cid:48) j ∗ where ¯ (cid:31) (cid:48) j ∗ ⊥ D = (cid:31) (cid:48) j ∗ such that f ( Q )( j ∗ ) ⊥ D = f ( ¯ (cid:31) − j ∗ , ¯ (cid:31) (cid:48) j ∗ )( j ∗ ) ⊥ D while f ( Q )( j ) ⊥ D (cid:54) = f ( ¯ (cid:31) − j ∗ , ¯ (cid:31) (cid:48) j ∗ )( j ) ⊥ D for some j . This is a contradiction to ourassumption that f satisfies non-bossiness of more important types. • Step 3. The allocations for the remaining types only depend on theallocations for D . Claim 3. Consider any pair of profiles P , P such that [ A ] = f ( P ⊥ D ) = f ( P ⊥ D ) , and P ⊥ D − , [ A ] = P ⊥ D − , [ A ] , then f ( P ) = f ( P ) .Proof. We prove the claim by constructing a profile P such that f ( P ) = f ( P ) = f ( P ).Let P = ( (cid:31) j ) j ≤ n , P = ( ¯ (cid:31) j ) j ≤ n and P = ( ˆ (cid:31) j ) j ≤ n . Let ˆ (cid:31) j be obtainedfrom (cid:31) j by changing the preferences over D by raising [ A ] ( j ) to the topposition. Agents’ preference over D − are ˆ (cid:31) j ⊥ D − , [ A ] = (cid:31) j ⊥ D − , [ A ] (=¯ (cid:31) j ⊥ D − , [ A ] ). It is easy to check that for every bundle b , f ( P )( j ) (cid:31) j b = ⇒ f ( P )( j ) ˆ (cid:31) j b . By applying Lemma 1 sequentially to every agent, f ( P ) = f ( P ). Similarly, f ( P ) = f ( P ). It follows that for any allocation[ A ] of items of type D , there exists a mechanism f ⊥ D − , [ A ] such that forany profile P , we can write f ( P ) as ( f ( P ⊥ D ) , f ⊥ D − , [ A ] ( P ⊥ D − , [ A ] )). • Step 4. Show that f ⊥ D − , [ A ] satisfies strategyproofness and non-bossinessof important types for any allocation [ A ] of D .Suppose for the sake of contradiction that f ⊥ D − , [ A ] is not strategyprooffor some profile P ⊥ D − , [ A ] . Then, for P = ( (cid:31) j ) j ≤ n there is an agent j ∗ witha beneficial manipulation w.r.t. P and [ A ] , (cid:31) (cid:48) j ∗ ⊥ D − , [ A ] (cid:54) = (cid:31) j ∗ ⊥ D − , [ A ] and (cid:31) (cid:48) j ∗ ⊥ D = (cid:31) j ∗ ⊥ D . Let Q = ( (cid:31) − j ∗ , (cid:31) (cid:48) j ∗ ). Then, f ( Q )( j ) =([ A ] , f ⊥ D − , [ A ] ( Q ⊥ D − , [ A ] ))( j ) (cid:31) j ([ A ] , f ⊥ D − , [ A ] ( P ⊥ D − , [ A ] ))( j ) = f ( P )( j ). This is a contradiction to the strategyproofness of f .Suppose for sake of contradiction that f ⊥ D − , [ A ] does not satisfy non-bossiness of important types. Then, there is a profile P = ( (cid:31) j ) j ≤ n , and anagent j ∗ with a bossy manipulation of her preferences (cid:31) j ∗ ⊥ D − , [ A ] . Then,it is easy to verify that f also does not satisfy non-bossiness of importanttypes.In Step 1, we showed that the allocation for D only depends on therestriction of agents’ preferences to D i.e. over P ⊥ D . In Step 3 we showedthat f ( P ) can be decomposed as ( f ( P ⊥ D ) , f ⊥ D − , [ A ] ( P ⊥ D − , [ A ] )) where[ A ] = f ( P ⊥ D ). In Steps 2 we showed that f must be strategyproof andnon-bossy. In Step 4, we showed that for any output [ A ] of f , the mecha-nism f ⊥ D − , [ A ] satisfies both strategyproofness and non-bossiness of impor-tant types i.e. that we can apply the induction assumption that f ⊥ D − , [ A ] is a locally strategyproof and non-bossy CR-net of allocation mechanisms.Together with the statement of Step 2, this completes the inductive argu-ment. 16n Theorem 4, we characterize the class of strategyproof, non-bossy ofmore important types, and type-wise neutral mechanisms under O -legal lexi-cographic preferences, as the class of O -legal sequential compositions of serialdictatorships. The proof relies on Theorem 3 and Claim 4, where we showthat any CR-net mechanism that satisfies type-wise neutrality is an O -legalsequential composition of neutral mechanisms, one for each type. Claim 4. For any importance order O , an O -legal CR-net with type-wiseneutrality is an O -legal sequential composition of neutral mechanisms.Proof. We prove the claim by induction. Suppose f is such a CR-net. Fromthe decomposition in the proof of Claim 1, we observe that the mechanismused for type i depends on f ( P ) ⊥ D ≤ i . From this observation, and the im-portance order O , we can deduce that the mechanism for type 1 dependson no other type, and therefore there is only one mechanism for type 1, say, f . First we show that f is neutral. Otherwise, there exists a permuta-tion Π over D , f (Π ( P ⊥ D )) (cid:54) = Π ( f ( P ⊥ D )). Let I = ( I i ) i ≤ p where I i is the identity permutation for type i . Then for Π = (Π , I − ), we have f (Π( P )) ⊥ D = f (Π ( P ⊥ D )) (cid:54) = Π ( f ( P ⊥ D )) = Π( f ( P )) ⊥ D , a contradic-tion.Now, suppose that for a given i , there is only one mechanism f i (cid:48) for each type i (cid:48) ≤ i , and each f i (cid:48) is neutral. Let Π = (Π ≤ i , I >i ) andwe have f (Π( P )) ⊥ D ≤ i = Π( f ( P )) ⊥ D ≤ i . Let A = f ( P ) ⊥ D ≤ i and B = f (Π( P )) ⊥ D ≤ i = Π ≤ i ( A ). Because P is chosen arbitrarily, A and B arealso arbitrary outputs of mechanism f over D ≤ i . Let f i +1 = f ⊥ D i +1 ,A ,and f (cid:48) i +1 = f ⊥ D i +1 ,B . Similarly both f i +1 and f (cid:48) i +1 are arbitrary mechanismsin CRT . Because f is neutral, we have f (Π( P )) ⊥ D i +1 = Π( f ( P )) ⊥ D i +1 ,i.e. f i +1 ( P ⊥ D i +1 ,A ) = f (cid:48) i +1 ( M ( P ) ⊥ D i +1 ,B ). By assumption we know thatΠ i +1 = I i +1 , so P ⊥ D i +1 ,A = Π( P ) ⊥ D i +1 ,B . That means f i +1 and f (cid:48) i +1 canreplace each other in CRT of f for type i + 1. Therefore in fact there is onlyone mechanism f i +1 for type i + 1 in CRT .Moreover f i +1 must be neutral. Otherwise, there must be some permu-tation Π i +1 over D i +1 , f i +1 (Π i +1 ( P ⊥ D i +1 ,A )) (cid:54) = Π i +1 ( f i +1 ( P ⊥ D i +1 ,A )). Thenfor Π = (Π ≤ i +1 , I >i +1 ), we have f (Π( P )) ⊥ D i +1 = f i +1 (Π( P ) ⊥ D i +1 ,B ) = f i +1 (Π i +1 ( P ⊥ D i +1 ,A )) (cid:54) = Π i +1 ( f i +1 ( P ⊥ D i +1 ,A )) = Π( f ( P )) ⊥ D i +1 , a contra-diction. Theorem 4. For any importance order O , under the O -legal lexicographicpreference domain, an allocation mechanism satisfies strategyproofness, non-bossiness of more important types, and type-wise neutrality if and only if itis an O -legal sequential composition of serial dictatorships.Proof. Let O = [ D (cid:66) D (cid:66) · · · (cid:66) D p ]. When p = 1, we know that serialdictatorship is characterized by strategyproofness, non-bossiness, and neu-17rality Mackin and Xia (2016). Let P = ( (cid:31) j ) j ≤ n be an arbitrary O -legallexicographic preference profile. ⇒ : Let f O = ( f , . . . , f p ). It follows from Theorem 3 that if each f i satis-fies strategyproofness and non-bossiness, then f O satisfies strategyproofnessand non-bossiness of more important types, because f O can be regarded asa CR-net with no dependency among types. If each f i satisfies neutrality,then by Theorem 1 we have that f satisfies type-wise neutrality. There-fore, since each f i is a serial dictatorship, which implies that it satisfiesstrategyproofness, non-bossiness, and neutrality, we have that f O satisfiesstrategyproofness, non-bossiness of more important types, and type-wiseneutrality. ⇐ : We now prove the converse. Let f be a strategyproof and non-bossymechanism under O -legal lexicographic preferences. Then by Theorem 3,we have that f is an O -legal strategyproof and non-bossy CR-net. The restof the proof depends on the following claim:Claim 4 implies that there is only one mechanism f i for each type i in CRT , and f i is neutral. Therefore with Theorem 3 and Claim 4, if f satisfies strategyproofness, non-bossiness of more important types, andtype-wise neutrality, we have that f is an O -legal sequential compositionof local mechanisms that are strategyproof, non-bossy, and neutral, whichimplies that they are serial dictatorships Mackin and Xia (2016). Theorem 5. For any arbitrary importance order O , under the O -legallexicographic preference domain, an allocation mechanism satisfies strate-gyproofness, non-bossiness of more important types, and Pareto-optimalityif and only if it is an O -legal CR-net composed of serial dictatorships.Proof. (Sketch) For a single type, we know that serial dictatorship is char-acterized by strategyproofness, Pareto-optimality, and non-bossiness P´apai(2001). The proof is similar to Theorem 4, and uses a similar argument toTheorems 1 and 2, to show that an O -legal CR-net is Pareto-optimal if andonly if every local mechanism is Pareto-optimal. The details are providedin the apoendix.Finally, we revisit the question of strategyproofness when preferencesare not O -legal w.r.t. a common importance order. We show in Theorem 6that even when agents’ preferences are restricted to lexicographic prefer-ences, there is a computational barrier against manipulation; determiningwhether there exists a beneficial manipulation w.r.t. a sequential mechanismis NP-complete for MTRAs, even when agents’ preferences are lexicographic.Details and the full proof are relegated to the appendix. Definition 2. Given an MTRA ( N, M, P ) , where P is a profile of lexico-graphic preferences, and a sequential mechanism f O . in BeneficialMa- ipulation , we are asked whether there exists an agent j and an O -legallexicographic preference relation (cid:31) (cid:48) j such that f O (( P − j , (cid:31) (cid:48) j ))( j ) (cid:31) j f O ( P )( j ) . Theorem 6. BeneficialManipulation is NP-complete when preferencesare not O -legal. We studied the design of strategyproof sequential mechanisms for MTRAsunder O -legal lexicographic preferences, and showed the relationship be-tween properties of sequential mechanisms and the local mechanisms thatthey are composed of. In doing so, we obtained strong characterizationresults showing that any mechanism satisfying strategyproofness, and com-binations of appropriate notions of non-bossiness, neutrality, and Pareto-optimality for MTRAs must be a sequential composition of local mecha-nisms. This decomposability of strategyproof mechanisms for MTRAs pro-vides a fresh hope for the design of decentralized mechanisms for MTRAsand multiple assignment problems. Going forward, there are several inter-esting open questions such as whether it is possible to design decentralizedmechanisms for MTRAs that are fair, efficient, and strategyproof under dif-ferent preference domains. Acknowledgments LX acknowledges support from NSF References Atila Abdulkadiro˘glu and Tayfun S¨onmez. 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Throughout, we will assume that O = [ D (cid:66) · · · (cid:66) D p ], and that P isan arbitrary O -legal preference profile over p types. For any i ≤ p , we define g i to be the sequential mechanism ( f , . . . , f i ). anonymity . It is easy to see that the claim is true when p = 1.Now, suppose that claim is true for all p ≤ k . Let P be an ar-bitrary profile over k + 1 types. Let g = ( f , . . . , f k +1 ). Now,Π( f O ( P )) = (Π( f ( P ⊥ D )) , Π( g ( P ⊥ D ≤ k +1 \ D , Π( f ( P ⊥ D )) ))), and f O (Π( P )) =( f (Π( P ⊥ D )) , g (Π( P ⊥ D ≤ k +1 \ D ,f (Π( P ⊥ D )) ))). Since f is anonymous,Π( f ( P ⊥ D )) = f (Π( P ⊥ D )). Therefore, P ⊥ D ≤ k +1 \ D , Π( f ( P ⊥ D )) = P ⊥ D ≤ k +1 \ D ,f (Π( P ⊥ D )) . Then, by the induction assumption, g satisfies anonymity, and we have Π( g ( P ⊥ D ≤ k +1 \ D , Π( f ( P ⊥ D )) )) = g (Π( P ⊥ D ≤ k +1 \ D ,f (Π( P ⊥ D )) )). It follows that Π( f O ( P )) = f O (Π( P )). type-wise neutrality. We show only the induction step. Supposethat the claim is always true when p ≤ k . Let P be an arbi-trary profile over k + 1 types. Let g = ( f , . . . , f k +1 ), and Π − =(Π , . . . , Π k +1 ). Let A = Π ( f O ( P ⊥ D )) and B = f (Π ( P ⊥ D )).Now, Π( f O ( P )) = ( A , Π − ( g ( P ⊥ D ≤ k +1 \ D ,A ))), and f O (Π( P )) =( B , g (Π − ( P ⊥ D ≤ k +1 \ D ,B ))). Since f is neutral, A = B . Then, P ⊥ D ≤ k +1 \ D ,A = P ⊥ D ≤ k +1 \ D ,B . Then, by the induction assump-tion, g satisfies type-wise neutrality, and Π − ( g ( P ⊥ D ≤ k +1 \ D ,A )) = g (Π − ( P ⊥ D ≤ k +1 \ D ,B )). It follows that Π( f O ( P )) = f O (Π( P )). non-bossiness. Let us assume for the sake of contradiction that the claimis false, i.e. there exists a profile P , an agent j and a misreport (cid:31) (cid:48) j suchthat for P (cid:48) = ( (cid:31) − j , (cid:31) (cid:48) j ), f O ( P )( j ) = f O ( P (cid:48) )( j ), and f O ( P ) (cid:54) = f O ( P (cid:48) ).Then, there is a type i ≤ p such that, f O ( P ) ⊥ D
Suppose the claim is true for p ≤ k types. Let P bean O -legal lexicographic profile over k + 1 types, and f O = ( f i ) i ≤ k +1 is asequential composition of Pareto-optimal local mechanisms. Suppose for thesake of contradiction that there exists an allocation B such that some agentsstrictly better off compared to f O ( P ), and no agent is worse off. Then, byour assumption of lexicographic preferences, for every agent k who is notstrictly better off, B ( k ) = f O ( P )( k ), and for every agent j who is strictlybetter off, one of two cases must hold. (1) B ( j ) ⊥ D (cid:31) j f O ( P )( j ) ⊥ D , or (2) B ( j ) ⊥ D = f O ( P )( j ) ⊥ D . (1): If there exists an agent such that B ( j ) ⊥ D (cid:31) j f O ( P )( j ) ⊥ D , this is a contradiction to our assumption that f is Pareto-optimal. (2): Suppose B ( j ) ⊥ D = f O ( P )( j ) ⊥ D for all agents who arestrictly better off. Let g = ( f , . . . , f k +1 ). W.l.o.g. let agent 1 strictly prefer B (1) to f O ( P )(1). Then, g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )(1) (cid:31) B (1) ⊥ D ≤ k +1 \ D ,and for every other agent l (cid:54) = 1, either g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )( l ) (cid:31) l B ( l ) ⊥ D ≤ k +1 \ D , or g ( P ⊥ D ≤ k +1 \ D ,f O ( P ) ⊥ D )( l ) = B ( l ) ⊥ D ≤ k +1 \ D , which is acontradiction to our induction assumption. Theorem 2. For any importance order O ∈ O , X ∈ { anonymity, type-wise neutrality, non-bossiness, monotonicity, Pareto-optimality } , and f O =( f , . . . , f p ) be any O -legal sequential mechanism. For O -legal preferences,if f O satisfies X , then for every i ≤ p , f i satisfies X .Proof. anonymity. Suppose that for some k ≤ p , f k does not sat-isfy anonymity. Then, there exists a profile P k on D k such that forsome permutation Π on agents f k (Π( P k ) (cid:54) = Π( f k ( P k )). Now, considerthe O -legal separable lexicographic profile P , where for any type i ≤ p ,the preferences over type D i is denoted P ⊥ D i and P ⊥ D k = P k . Itis easy to see that, f O (Π( P )) = ( f i (Π( P ⊥ D i ))) i ≤ p , and Π( f O ( P )) =Π( f ( P ⊥ D ) , . . . , f p ( P ⊥ D p )) = (Π( f ( P ⊥ D )) , . . . , Π( f p ( P ⊥ D p )))). Byanonymity of f , f O (Π( P )) = Π( f O ( P )), which implies that f k (Π( P ⊥ D k )) =Π( f k ( P ⊥ D k )), which is a contradiction. type-wise neutrality. Suppose that some k ≤ p , f k does not satisfy neu-trality. Then, there exists a profile P k on D k such that for some permutation24 k on D k f k (Π k ( P k ) (cid:54) = Π k ( f k ( P k )). Now, consider the O -legal separable lex-icographic profile P , where for any type i ≤ p , the preferences over type D i is denoted P ⊥ D i and P ⊥ D k = P k , and let Π = (Π , . . . , Π k , . . . , Π p ) be apermutation over D by applying P i i on D i for each type i ≤ p . f O (Π( P )) =( f i (Π i ( P ⊥ D i ))) i ≤ p , and Π( f O ( P )) = (Π ( f ( P ⊥ D )) , . . . , Π p ( f p ( P ⊥ D p )))).By type-wise neutrality of f O , f O (Π( P )) = Π( f O ( P )). This implies that f k (Π k ( P ⊥ D k )) = Π k ( f k ( P ⊥ D k )), where P ⊥ D k ) = P k , which is a contradic-tion. non-bossiness. Assume for the sake of contradiction that k ≤ p is the mostimportant type such that f k does not satisfy non-bossiness. Then, thereexists a preference profile Q = ( (cid:31) k ) j ≤ n over D k , and a bossy agent l anda misreport Q (cid:48) = ( (cid:31) k − l , ¯ (cid:31) kl ), such that f k ( Q (cid:48) )( l ) = f k ( Q )( l ), but f k ( Q (cid:48) ) (cid:54) = f k ( Q ). Now, consider the O -legal separable lexicographic profile P , wherefor any type i ≤ p , the preferences over type D i is denoted P ⊥ D i and P ⊥ D k = Q , and the profile P (cid:48) obtained from P by replacing (cid:31) l with (cid:31) (cid:48) l , which inturn is obtained from (cid:31) l by replacing (cid:31) l ⊥ D k with ¯ (cid:31) kl . It is easy to seethat f O ( P (cid:48) ) ⊥ D Suppose that some k ≤ p , f k does not satisfy Pareto-optimality. Then, there exists a profile P k such that f k ( P k ) is Pareto-dominated by an allocation B of D k . Now, consider the O -legal separa-ble lexicographic profile P , where for any type i ≤ p , the preferences overtype D i is denoted P ⊥ D i and P ⊥ D k = P k . Then, f O ( P ) = ( f i ( P ⊥ D i , ) ) i ≤ p is Pareto-dominated by the allocation B of all types, where for all types i (cid:54) = k , B ⊥ D i = f i ( P ⊥ D i ), and B ⊥ D k = A , which is a contradiction to theassumption that f O is Pareto-optimal.25 .3 Proof of Theorem 5 Theorem 5. For any arbitrary importance order O , under the O -legallexicographic preference domain, an allocation mechanism satisfies strate-gyproofness, non-bossiness of more important types, and Pareto-optimalityif and only if it is an O -legal CR-net composed of serial dictatorships.Proof. Let O = [ D (cid:66) D (cid:66) · · · (cid:66) D p ]. Under single type, we know thatserial dictatorship is characterized by strategyproofness, Pareto-Optimality,and non-bossiness P´apai (2001). Let P = ( (cid:31) j ) j ≤ n be an arbitrary O -legallexicographic preference profile. ⇒ : Let f be an O -legal CR-net. From Theorem 3 we know that if eachlocal mechanism of f satisfies strategyproofness and non-bossiness, then f satisfies strategyproofness and non-bossiness of more important types.We now prove that if each local mechanism is Pareto-Optimal, then f isPareto-optimal, similarly to Theorem 1. Suppose for the sake of contradic-tion that f is not Pareto-optimal, i.e. for some P , the allocation B = ( B i ) i ≤ p Pareto-dominates f ( P ) = A = ( A i ) i ≤ p . Let i be the most important typethat A and B and different allocuation, and we have A
BeneficialManipulation is NP-complete when preferencesare not O -legal.Proof. We show a reduction from 3-SAT. In an instance I of 3-SAT involv-ing s Boolean variables { x , . . . , x s } , and a formula F involving t clauses { c , . . . , c t } in 3-CNF, we are asked if F is satisfiable. Given such an arbi-trary instance I of 3-SAT, we construct an instance J of BeneficialMa-nipulation in polynomial time. We will show that I is a Yes instance of26-SAT if and only if J is a Yes instance of BeneficialManipulation . Foreach j ≤ t , we label the three literals in clause j as l jj , l jj , and l jj where j < j < j . We construct instance J of BeneficialManipulation tohave: Types: s + 1 types. Agents: • For every variable i ≤ s , and every clause j ≤ t , two agents 0 ji , ji , and adummy agent d ji . • For every clause j , an agent c j . • A special agent 0. Items: For every agent a and every type k ≤ s + 1, an item named [ a ] k . Preferences: