aa r X i v : . [ ec on . T H ] S e p Strategy-proof allocation with outside option * Jun Zhang
Institute for Social and Economic Research, Nanjing Audit UniversityEmail: [email protected]
September 14, 2020
Abstract
We consider strategy-proof mechanisms to solve allocation problems where agentscan choose outside options if they wish. Mechanisms could return an allocation or arandomization over allocations. We prove two useful theorems, relying on an invari-ance property of the allocations found by strategy-proof mechanisms when agentsvary the ranking of outside options in their preferences. The first theorem provesthat, among individually rational and strategy-proof mechanisms, pinning down ev-ery agent’s probability of choosing outside option in every preference profile is equiv-alent to pinning down a mechanism. The second theorem provides a su ffi cient condi-tion for proving equivalence between two strategy-proof mechanisms when the num-ber of possible allocations is finite. The two theorems provide a unified foundationfor existing results in several distinct models and imply new results in some models. Keywords : strategy-proofness; individual rationality; outside option; stochastic domi-nance; size dominance; contraction-invariance
JEL Classification : C71, C78, D78 * I am grateful to seminar and conference participants in Waseda University, Shanghai University ofFinance and Economics, 2018 Nanjing International Conference on Game Theory, and 2019 Conference onEconomic Design in Budapest. This paper is supported by National Natural Science Foundation of China(Grant No. 71903093). An earlier version was circulated under the title “Truncation-invariant allocation.”
Introduction
Allocation problems where agents have outside options are prevalent in real life andeconomic models. In marriage markets (Gale and Shapley, 1962) men and women havethe outside option of being alone. In school choice (Abdulkadiro ˘glu and S¨onmez, 2003)children have the outside option of attending private schools. In auction markets (Vick-rey, 1961) bidders have the outside of not bidding. To solve such problems, participationconstraints require mechanism designers to choose individually rational (IR) mechanisms,which ensure that participants are no worse o ff than choosing outside options. Incentiveconcerns motivate mechanism designers to choose strategy-proof mechanisms, which giveagents the right incentive to reveal true preferences. This paper demonstrates the restric-tion of IR and strategy-proofness in the presence of outside options. We prove two simplebut useful theorems for strategy-proof mechanisms, relying on an invariance property ofthe allocations found by such mechanisms when agents vary the ranking of outside op-tions in their preferences. Applying them to several distinct models, we provide a unifiedfoundation for existing results and obtain new results in some models.We present our theorems in an abstract allocation model that is proposed by Alva andManjunath (2019). In the model finite agents have ordinal preferences, which need notbe strict, over a set of allocations. The model is abstract because it does not specify whatallocations mean and how they are generated. It only requires that each allocation have awell-defined set of participants; non-participants consume outside options. This modelcan accommodate many environments. We consider direct revelation mechanisms thatelicit agents’ preferences and return an allocation or a randomization over allocations.Those returning an allocation are called deterministic mechanisms , while those returninga randomization over allocations are called random mechanisms . Randomization couldbe used for di ff erent purposes in di ff erent environments. For example, in market designrandomization is often used to restore fairness. Our theorems are applicable to determin-istic mechanisms as well as to random mechanisms. For ease of exposition, we presentour theorems for random mechanisms and regard deterministic mechanisms as specialcases. With agents’ ordinal preferences, we use (first-order) stochastic dominance to makewelfare comparison between random allocations/mechanisms. A mechanism is strategy-proof if, fixing the others’ preferences, when an agent misreports preferences, the foundrandom allocation is weakly stochastically dominated (with respect to the agent’s truepreferences) by the found random allocation when the agent reports true preferences.When an agent varies the ranking of outside option in his preferences, a type of strate-1ies attracting our attention is what we call contraction : an agent moves his outside optionupwards and places it immediately below an allocation that was far better than his out-side option, but the upper contour set of the allocation is invariant (i.e., those weaklybetter than the allocation remains weakly better than the allocation in his new prefer-ences). A special case of contraction is known as truncation in the literature: an agentimproves the ranking of his outside option without changing preferences over the otherallocations. In a strategy-proof mechanism, if an agent contracts preferences, the totalprobability mass of the allocations in the invariant upper contour set mentioned abovemust be invariant . We call this property contraction-invariance . When we only considertruncation strategies, we call the property truncation-invariance .Our Theorem 1 proves that, if agents can arbitrarily contract preferences and neverregard allocations they participate in as indi ff erent with outside options, for any two IRand strategy-proof mechanisms, comparing agents’ probabilities of being participants inthe two mechanisms is equivalent to making welfare comparison between the two mech-anisms. If every agent has a higher probability of participating in one mechanism thanof participating in the other, then the former mechanism stochastically dominates thelatter. This result looks surprising at first glance, because intuitively comparing wel-fare requires more information than comparing participation probabilities. But in an IRand strategy-proof mechanism, relying on contraction-invariance, we can recover everyagent’s welfare in a preference profile from his participation probability in all possiblepreference profiles where he contracts preferences. This enables us to obtain Theorem 1.When the set of allocations in our model is finite, Theorem 2 presents a su ffi cient con-dition for proving equivalence between strategy-proof mechanisms. It will be useful forproving uniqueness of strategy-proof mechanisms among a class of desirable mechanismsin applications. Specifically, we prove that when agents can arbitrarily contract prefer-ences, if the random allocations found by two strategy-proof mechanisms for any pref-erence profile are not di ff erent in the probabilities of allocations far better than agents’outside options only if their found random allocations are welfare-equivalent, then thetwo mechanisms must find welfare-equivalent random allocations for every preferenceprofile. This theorem relies on the finite allocations assumption, because the assumptionensures the existence of preference profiles where there do not exist allocations far bet-ter than agents’ outside options in their preferences. If two strategy-proof mechanismsare welfare-equivalent in every such preference profile, then contraction-invariance will We say an allocation is far better than an agent’s outside option if it is above his outside option but isnot immediately above his outside option in his preferences. ff er-ences between the foundations of AM’s theorem and ours. In our model, agents’ prefer-ences over allocations are complete but their extended preferences over random alloca-tions by stochastic dominance are incomplete. Somehow strategy-proofness is a strongerrestriction on deterministic mechanisms than on random mechanisms. We have to relyon contraction-invariance to demonstrate the structure of random allocations found bystrategy-proof mechanisms. This requires us to consider variations of agents’ preferenceswith invariant upper contour sets. There is no counterpart of Theorem 2 in AM’s pa-per. We use Theorem 2 to unify the results in several models and sometimes obtain newresults. Finally, clarifying the role of contraction-invariance reveals that our theoremshold for non-strategy-proof mechanisms that satisfy contraction-invariance. In particu-lar, our theorems hold for truncation-invariant mechanisms. In applications, some desir-able mechanisms are naturally truncation-invariant although they are not strategy-proof.Our model can accommodate various environments. In this paper we mainly presentapplications to market design problems. In market design it is common to use directmechanisms to elicit agents’ ordinal preferences. The usefulness of Theorem 1 for de-terministic mechanisms has been well demonstrated by AM. To show its usefulness forrandom mechanisms, we apply Theorem 1 to the random assignment model studied byHylland and Zeckhauser (1979), Bogomolnaia and Moulin (2001) and many others. Inthat model, finite indivisible objects are randomly assigned to finite agents. Theorem 1implies that making welfare comparison between two IR and strategy-proof mechanismsis equivalent to comparing the probability mass of objects assigned to every agent in thetwo mechanisms. In particular, strategy-proof improvement can be done if and only ifby assigning more objects to every agent. Theorem 1 also suggests a property weakerthan non-wastefulness that is su ffi cient for a mechanism to be on the e ffi ciency frontierof IR and strategy-proof mechanisms. These results explain Erdil’s (2014) findings that anecessary condition for strategy-proof improvement is to assign more objects in total to Hylland and Zeckhauser (1979) propose the pseudo-market mechanism to elicit agents’ cardinal utili-ties. So their setup is beyond our model, and their mechanism is not considered in this paper. size improvement . It requires thatthe new mechanism should increase the number of assignments for at least one prefer-ence profile, and when the new mechanism does not increase the number of assignments,every agent who obtains an assignment in the existing mechanism should still obtain anassignment in the new one. So the new mechanism respects agents’ welfare in the existingmechanism in a very weak sense. Applying Theorem 2, we prove that size improvementis impossible if the two mechanisms are truncation-invariant and the existing mechanismis not too ine ffi cient. By presenting the result for truncation-invariant mechanisms, weintend to apply it to mechanisms in the literature that may not be strategy-proof.In the remaining applications, we show that Theorem 2 provides a unified foundationfor the results of Hirata and Kasuya (2017) in the many-to-one matching with contractsmodel (Hatfield and Milgrom, 2005), a characterization result of Combe et al. (2020) intheir teacher reassignment model, and the main results of S¨onmez (1999) and Ehlers(2018) in a generalized object assignment model. In these applications mechanisms aredeterministic. Under weak assumptions that do not ensure the existence of stable match-ings, Hirata and Kasuya (2017) prove that there is at most one stable and strategy-proofmechanism, and if the doctor-optimal stable mechanism exists, it is the only candidate.Combe et al. (2020) propose a class of mechanisms to solve their teacher reassignmentmodel. In the one-to-one environment, they prove that a mechanism they recommend isthe only strategy-proof mechanism in that class. In his abstract allocation model, S¨onmez(1999) proves that there exists an IR, Pareto e ffi cient and strategy-proof mechanism onlyif the core is essentially single-valued and the mechanism selects a core allocation whenthe core is nonempty. This finding unifies the results in several distinct models, includinghousing market, marriage and roommate problem, coalition formation problem, and so-cial network. Ehlers (2018) extends S¨onmez’s result by replacing the core with a supersetcalled the individually-rational-core. Theorem 2 explains all of these results.After discussing related literature in the rest of this section, we present our model in It is known that substitutability and other assumptions are necessary to ensure the existence of stablematchings and the existence and strategy-proofness of the doctor-optimal stable mechanism.
Related literature.
Outside options enrich agents’ manipulation strategies, and there-fore strengthen the restriction of strategy-proofness. Our theorems demonstrate implica-tions of this strengthened restriction. Several results in the market design literature havesimilarly relied on the role of outside options. In the school choice with coarse prioritiesmodel, Abdulkadiro ˘glu et al. (2009) prove that when students have outside options, nostrategy-proof deterministic mechanism Pareto improves the deferred acceptance algo-rithm (DA) equipped with any tie-breaking rule. As Alva and Manjunath (2019) haveexplained, this result is implied by Theorem 1. Kesten and Kurino (2019) prove thateven if exogenous outside options are eliminated, as long as students can arbitrarily rankschools in preferences, some schools will endogenously play the role of outside options. In the random assignment model, Martini (2016) proves that every IR and strategy-proofmechanism that treats equals equally is wasteful. This result answers an open questionof Erdil (2014). In Section 4.1, we prove a result that also shows the tension betweene ffi ciency, fairness, and strategy-proofness, and explains another result of Erdil (2014). Itsimilarly relies on the joint restriction of these axioms in the presence of outside options.When agents can arbitrarily truncate preferences, strategy-proofness in our theoremscan be replaced by truncation-invariance. In the literature similar invariance notionshave been proposed in di ff erent models. In the object assignment model, Hashimoto etal. (2014) define a notion called weak truncation robustness . In footnote 7 we explain thatour truncation-invariance is equivalent to their notion in the object assignment model.Ehlers and Klaus (2014, 2016) define a di ff erent invariance notion for deterministic mech-anisms. It requires that if an agent truncates preferences at a point that is worse than hisassignment, then the allocation for the whole market remain unchanged. This require-ment is much stronger than truncation-invariance. In the matching with contracts model, When priorities are strict and students may not have outside options, Kesten (2010) has proved thatno strategy-proof and Pareto e ffi cient deterministic mechanism Pareto improves DA. truncation-consistency for deterministic mech-anisms. It requires that if an agent truncates preferences at a point that is worse than hisassignment, then his assignment remain unchanged. Truncation-consistency essentiallycoincides with truncation-invariance for deterministic mechanisms.In the object assignment model, we propose a criterion to compare the numbers ofassignments in a benchmark mechanism and in a new mechanism that is supposed toreplace the benchmark. Our criterion requires that the new mechanism respect agents’welfare in the benchmark in a very weak sense. Afacan et al. (2018) and Afacan and Dur(2018) propose a di ff erent criterion to compare the number of assignments in two deter-ministic mechanisms. In their criterion the two mechanisms are symmetric. Afacan etal.’s (2018) aim to find mechanisms to maximize the number of assignments in the schoolchoice model. They show that maximality is incompatible with fairness and strategy-proofness. Afacan and Dur (2018) prove two impossibility results in the same flavor asours. Compared with their results, ours uses weaker axioms, allows for weak preferences,and holds for both deterministic mechanisms and random mechanisms. We use the abstract allocation model of Alva and Manjunath (2019). The model con-sists of a finite set of agents I and a nonempty set of allocations A . For ease of exposition,we assume that A is finite. We discuss the case of an infinite A in Section 5. We denotean agent by i or j , and an allocation by a or b . Each allocation a is associated with a setof participants , which is a subset of agents and denoted by I ( a ). For each agent i , define A i ≔ { a ∈ A : i ∈ I ( a ) } to be the set of allocations where i is a participant. If i does notparticipate in an allocation a , i consumes his outside option denoted by o ∗ i . So i consumeshis outside option in every a ∈ A\A i .Each agent i has a complete, reflexive, and transitive preference relation % i on A i ∪{ o ∗ i } .This setup means that i is indi ff erent between any two allocations where he consumes hisoutside option. We allow i to have non-trivial indi ff erent preferences. That is, i couldbe indi ff erent between two allocations where he is a participant, and could be indi ff erentbetween an allocation where he is a participant and an allocation where he consumeshis outside option. Let ≻ i denote the asymmetric component of % i and ∼ i denote the Note that allocations and outside options in our model are abstract. If an agent cares about the alloca-tion assigned to the other agents, he should be regarded as a participant in the allocation. % i . Let R i denote i ’s preference domain. Let % I ≔ ( % i ) i ∈ I denotea preference profile, and R ≔ × i ∈ I R i denote the domain of preference profiles.For each agent i , we say an allocation a is acceptable to i if a % i o ∗ i , and is strictly accept-able to i if a ≻ i o ∗ i . An allocation a is individually rational (IR) if a % i o ∗ i for all i . We assumethat there exists an allocation where every i consumes o ∗ i . It ensures that there exists anIR allocation for every preference profile.For any two distinct allocations a and b , we say b weakly Pareto dominates a if b % i a forall i ; if there further exists j such that b ≻ j a , we say b strictly Pareto dominates a ; if a ∼ i b for all i , we say a and b are welfare-equivalent . An allocation is Pareto e ffi cient if it is notstrictly Pareto dominated by any other allocation.The following notations will simplify our exposition. For every a ∈ A and every i ∈ I ,we define A ( % i , a ) ≔ { b ∈ A : b % i a } , A ( ≻ i , a ) ≔ { b ∈ A : b ≻ i a } , and A ( ∼ i , a ) ≔ { b ∈ A : b ∼ i a } .We call A ( % i , a ) the upper contour set of a in % i , A ( ≻ i , a ) the strict upper contour set of a in % i , and A ( ∼ i , a ) the indi ff erence class of a in % i . We similarly define A i ( % i , a ), A i ( ≻ i , a ),and A i ( ∼ i , a ) for every a ∈ A and every i ∈ I . So A i ( ≻ i , o ∗ i ) is the set of strictly acceptableallocations for i , and A i ( ∼ i , o ∗ i ) is the set of allocations in A i that i regards as indi ff erentwith o ∗ i . For every i and every % i ∈ R i , we define A i ( Ï i , o ∗ i ) ≔ { a ∈ A i : there exists b ∈ A i such that a ≻ i b % i o ∗ i } . If A i ( ∼ i , o ∗ i ) , ∅ , then A i ( Ï i , o ∗ i ) = A i ( ≻ i , o ∗ i ). If A i ( ∼ i , o ∗ i ) = ∅ , then A i ( ≻ i , o ∗ i ) \A i ( Ï i , o ∗ i ) isthe worst indi ff erence class among strictly acceptable allocations. So in words, A i ( Ï i , o ∗ i )is the part of A i ( % i , o ∗ i ) that is not closest to o ∗ i in % i . We regard the allocations in A as deterministic . A random allocation , denoted by p , isa probability distribution on A . For every a , let p a denote the probability of a in p . Forevery A ⊂ A , we define p [ A ] ≔ P a ∈ A p a . Let ∆ ( A ) denote the set of all random allocations.Because every deterministic allocation can be regarded as a degenerate random alloca-tion, we make the convention that A ⊂ ∆ ( A ). With this convention, in the rest of thepaper we will simply call the elements of ∆ ( A ) allocations , and when necessary we will A deterministic allocation a can be described as probability distribution p such that p a = 1 and p b = 0for every b , a . So for every A ⊂ A , p [ A ] = 1 if and only if a ∈ A . A deterministic to avoid confusion. An allocation p is individually ratio-nal (IR) if, for every a with p a > a is IR as a deterministic allocation. In other words, p is IR if and only if for every i , p [ A ( % i , o ∗ i )] = 1.We use (first-order) stochastic dominance to extend agents’ preferences from deter-ministic allocations to random allocations. Definition 1 (Stochastic dominance) . For any two allocations p and p ′ ,1. p ′ weakly stochastically dominates p for agent i , denoted by p ′ % sdi p , if, for all a ∈ A , p ′ [ A ( % i , a )] = X b % i a p ′ a ≥ X b % i a p a = p [ A ( % i , a )] . If the inequality is strict for some a , then p ′ strictly stochastically dominates p for i .If all inequalities are actually equalities, then p ′ and p are welfare-equivalent for i .We denote them respectively by p ′ ≻ sdi p and p ′ ∼ sdi p .2. p ′ weakly stochastically dominates p if p ′ % sdi p for all i .If there exists j such that p ′ ≻ sdj p , then p ′ strictly stochastically dominates p .If p ′ ∼ sdi p for all i , then p ′ and p are welfare-equivalent . For deterministic allocations, stochastic dominance reduces to Pareto dominance.For every allocation p and every i , p [ A i ] is i ’s probability of being a participant in p .We call it i ’s participation size in p . For example, in the object assignment model in Section4.1, a random allocation p specifies each agent’s probability of obtaining each object, and p [ A i ] is i ’s probability of obtaining an object. We define a criterion to compare agents’participation sizes in two di ff erent allocations. Definition 2 (Size dominance) . For any two allocations p and p ′ , p ′ weakly size dominates p if p ′ [ A i ] ≥ p [ A i ] for all i . If the inequality is strict for some j , then p ′ strictly size dominates p . If p ′ [ A i ] = p [ A i ] for all i , then p ′ and p are size-equivalent . Agents’ participation sizes in an allocation are independent of their preferences, butwelfare comparison betweeen two allocations depends on agents’ preferences. So stochas-tic dominance and size dominance are independent concepts. But when we impose the IRconstraint, an agent’s participation size contains information about his welfare. Specif-ically, for every IR allocation p and every i , p [ A i ] = p [ A i ( ≻ i , o ∗ i )] + p [ A i ( ∼ i , o ∗ i )]. So p [ A i ]8s an upper bound on i ’s probability of being better o ff than choosing his outside option.If A i ( ∼ i , o ∗ i ) = ∅ , then p [ A i ] is exactly i ’s probability of being better o ff than choosing hisoutside option. In that case, weak stochastic dominance implies weak size dominance forIR allocations (yet strict stochastic dominance does not imply strict size dominance). Thismotivates the following assumption. Assumption 1 (No indi ff erence with outside option (NI)) . For every i and every % i ∈ R i , A i ( ∼ i , o ∗ i ) = ∅ . NI holds when agents’ preferences are strict. It also holds where agents may regardallocations they participate in as indi ff erent, but regard them as fundamentally di ff erentfrom outside options. For example, kidney patients may regard compatible donors asindi ff erent, but they do not regard compatible donors as indi ff erent with no transplant.Our theorems rely on agents’ freedom of arbitrarily ranking outside options in prefer-ences. This is formalized as a richness assumption on agents’ preference domains. Recallthat for every % i , A i ( Ï i , o ∗ i ) is the part of A i ( % i , o ∗ i ) that is not closest to o ∗ i in % i . Assumption 2 (Richness) . For every i , every % i ∈ R i , and every a ∈ A i ( Ï i , o ∗ i ) , there exists % ′ i ∈ R i such that, for all b ∈ A i , b % i a = ⇒ b % ′ i a , and a ≻ i b = ⇒ a ≻ ′ i o ∗ i ≻ ′ i b . Put di ff erently, Richness requires that for every % i ∈ R i and every a ∈ A i ( Ï i , o ∗ i ), thereexist % ′ i ∈ R i such that A i ( % i , a ) = A i ( % ′ i , a ) = A i ( ≻ ′ i , o ∗ i ) and A i ( ∼ ′ i , o ∗ i ) = ∅ . In words, o ∗ i isranked immediately below a in % ′ i , the upper contour set of a is invariant from % i to % ′ i ,and there is no indi ff erence between every a ∈ A i and o ∗ i in % ′ i (so NI is assumed for % ′ i ).Because the allocations in A i ( % i , o ∗ i ) \A i ( % i , a ) become not acceptable in % ′ i , we call everysuch % ′ i a contraction of % i at a . If some % ′ i ranks the allocations in A i in the same way as % i does, we call it the truncation of % i at a and denote it by % ai . If R i contains the truncationof every % i ∈ R i at every a ∈ A i ( Ï i , o ∗ i ), we say R i accommodates truncation.Finally, we define a notion that coarsens welfare-equivalence. Recall that two alloca-tions p and p ′ are welfare-equivalent if for every i and every a ∈ A , p ′ [ A ( % i , a )] = p [ A ( % i , a )]. The notion only requires p ′ [ A ( % i , a )] = p [ A ( % i , a )] for every i and every a ∈ A i ( Ï i , o ∗ i ).Because A i ( Ï i , o ∗ i ) is the upper part of A in % i , we call this notion upper-equivalence . Definition 3 (Upper-equivalence) . Two allocations p and p ′ are upper-equivalent if for ev-ery i and every a ∈ A i ( Ï i , o ∗ i ) , p ′ [ A i ( % i , a )] = p [ A i ( % i , a )] . .2 Mechanism A mechanism is a function ψ : R → ∆ ( A ) that finds an allocation for every preferenceprofile. If ψ always finds a deterministic allocation, then it is a deterministic mechanism . Amechanism ψ is said to satisfy a property we have defined for allocations if, for all % I ∈ R , ψ ( % I ) satisfies the property.The criteria used to compare allocations are extended to mechanisms. For any twomechanisms ψ ′ and ψ , we say ψ ′ weakly stochastically dominates ψ if, for all % I ∈ R , ψ ′ ( % I ) weakly stochastically dominates ψ ( % I ). If there further exists some % I where ψ ′ ( % I ) strictly stochastically dominates ψ ( % I ), we say ψ ′ strictly stochastically dominates ψ .We say ψ ′ and ψ are welfare-equivalent if, for all % I ∈ R , ψ ′ ( % I ) and ψ ( % I ) are welfare-equivalent. Size dominance and size-equivalence between mechanisms are similar.By using direct mechanisms, we often want to incentivize agents to report true pref-erences. This property is called strategy-proofness . For every % I ∈ R and every i , let % − i denote the preference profile of the agents other than i . Definition 4.
A mechanism ψ is strategy-proof if, for all % I ∈ R , all i ∈ I and all % ′ i ∈ R i \{ % i } , ψ ( % I ) % sdi ψ ( % ′ i , % − i ) . Let ψ be a strategy-proof mechanism. In any preference profile % I , suppose some i contracts his preferences % i at some a ∈ A i ( Ï i , o ∗ i ) and denote the contraction by % ′ i . Thenstrategy-proofness requires that ψ ( % I ) % sdi ψ ( % ′ i , % − i ) and ψ ( % ′ i , % − i ) % ′ sdi ψ ( % I ). Because A i ( % i , a ) = A i ( % ′ i , a ), the two conditions require that ψ ( % I )[ A i ( % i , a )] = ψ ( % ′ i , % − i )[ A i ( % i , a )]. That is, the probability mass of the upper contour set of a in % i is invariant when i contracts his preferences at a . We call this property contraction-invariance . Definition 5.
A mechanism ψ is contraction-invariant if, for all % I ∈ R , all i ∈ I , and all a ∈ A i ( Ï i , o ∗ i ) , if % ′ i is a contraction of % i at a , then ψ ( % I )[ A i ( % i , a )] = ψ ( % ′ i , % − i )[ A i ( % i , a )] . If a contraction-invariant mechanism ψ is IR, then for all % I , all i , and all contraction % ′ i of % i at all a ∈ A i ( Ï i , o ∗ i ), because A i ( % i , a ) = A i ( % ′ i , a ) = A i ( % ′ i , o ∗ i ) (since A i ( ∼ ′ i , o ∗ i ) = ∅ ), ψ ( % ′ i , % − i )[ A i ] = ψ ( % ′ i , % − i )[ A i ( % ′ i , o ∗ i )] = ψ ( % ′ i , % − i )[ A i ( % i , a )] = ψ ( % I )[ A i ( % i , a )] . In words, when i contracts his preferences at a , ψ ( % I )[ A i ( % i , a )] is equal to i ’s participationsize in the contracted preference profile. Our theorems will rely on this property.When every R i accommodates truncation, a weaker property than contraction-invarianceis called truncation-invariance . It only requires that the invariance property hold when10gents truncates preferences. Definition 6.
Let every R i accommodate truncation. A mechanism ψ is truncation-invariant if, for all % I ∈ R , all i ∈ I , and all a ∈ A i ( Ï i , o ∗ i ) , ψ ( % I )[ A i ( % i , a )] = ψ ( % ai , % − i )[ A i ( % i , a )] . In a deterministic mechanism ψ , ψ ( % I )[ A i ( % i , a )] = 1 if ψ ( % I ) ∈ A i ( % i , a ), and otherwise ψ ( % I )[ A i ( % i , a )] = 0. So ψ is truncation-invariant if for all % I , all i , and all a ∈ A i ( Ï i , o ∗ i ), ψ ( % ai , % − i ) % ai a if ψ ( % I ) % i a , and otherwise a ≻ ai ψ ( % ai , % − i ). Our first theorem presents the relation between stochastic dominance and size domi-nance for IR and strategy-proof mechanisms.
Theorem 1.
Let ψ and ψ ′ be two IR and strategy-proof mechanisms.1. Under NI, ψ ′ weakly stochastically dominates ψ = ⇒ ψ ′ weakly size dominates ψ .2. Under Richness, ψ ′ weakly size dominates ψ = ⇒ ψ ′ weakly stochastically dominates ψ . Proof. (1) Fix a preference profile % I . Suppose p ′ and p are two IR allocations and p ′ weakly stochastically dominates p . Then for every i , p [ A i ] = p [ A i ( ≻ i , o ∗ i )] + p [ A i ( ∼ i , o ∗ i )]and p ′ [ A i ] = p ′ [ A i ( ≻ i , o ∗ i )] + p ′ [ A i ( ∼ i , o ∗ i )]. NI requires A i ( ∼ i , o ∗ i ) = ∅ and weak stochas-tic dominance requires p ′ [ A i ( ≻ i , o ∗ i )] ≥ p [ A i ( ≻ i , o ∗ i )]. So p ′ [ A i ] ≥ p [ A i ], meaning that p ′ weakly size dominates p .(2) Suppose ψ ′ weakly size dominates ψ , but ψ ′ does not weakly stochastically dom-inate ψ . It means that there exists % I and i such that ψ ′ ( % I ) (cid:31) sdi ψ ( % I ). Then there mustexist a ∈ A such that ψ ( % I )[ A ( % i , a )] > ψ ′ ( % I )[ A ( % i , a )]. Because ψ and ψ ′ are IR, it must bethat a ≻ i o ∗ i and thus A ( % i , a ) = A i ( % i , a ). Because ψ ′ weakly size dominates ψ , it must bethat a ∈ A i ( Ï i , o ∗ i ); otherwise ψ ′ ( % I )[ A i ( % i , a )] = ψ ′ ( % I )[ A i ] ≥ ψ ( % I )[ A i ] = ψ ( % I )[ A i ( % i , a )].Under Richness, there exists a contraction % ′ i of % i at a . Because ψ and ψ ′ are strategy-proof, they are contraction-invariant. So ψ ( % ′ i , % − i )[ A i ] = ψ ( % I )[ A ( % i , a )] and ψ ′ ( % ′ i , % − i )[ A i ] = ψ ′ ( % I )[ A ( % i , a )]. This means that ψ ( % ′ i , % − i )[ A i ] > ψ ′ ( % ′ i , % − i )[ A i ]. But it contradictsthe assumption that ψ ′ weakly size dominates ψ . Truncation-invariance is equivalent to a seemingly stronger requirement that, for all % I ∈ R , all i ∈ I ,and all a ∈ A i ( Ï i , o ∗ i ), ψ ( % I )[ A i ( ∼ i , b )] = ψ ( % ai , % − i )[ A i ( ∼ i , b )] for all b ∈ A i ( % i , a ). To see this, note that forevery b ∈ A i ( ≻ i , a ), % bi is the truncation of both % i and % ai at b . By truncation-invariance, ψ ( % I )[ A i ( % i , b )] = ψ ( % bi , % − i )[ A i ( % i , b )] = ψ ( % ai , % − i )[ A i ( % i , b )]. In the object assignment model of Section 4.1, when agents havestrict preferences, Hashimoto et al. (2014) call such a stronger requirement weak truncation robustness .
11s we have explained in Section 2.1, when we impose the IR constraint and assumeNI, an agent’s participation size is his probability of being better o ff than choosing hisoutside option. So weak stochastic dominance implies weak size dominance for IR alloca-tions. The first part of Theorem 1 is unrelated to strategy-proofness of mechanisms. Theinsight behind the second part of Theorem 1 is that, for any IR and strategy-proof mech-anism ψ , when agents can arbitrarily contract preferences, relying on the contraction-invariance property we can recover every i ’s welfare in every % I from his participationsize in all possible contracted preference profiles. Specifically, to pin down i ’s welfare in ψ ( % I ), we need to know ψ ( % I )[ A ( % i , a )] for every a ∈ A ( % i , o ∗ i ) (because ψ is IR). First, forevery a ∈ A ( ∼ i , o ∗ i ), IR implies that ψ ( % I )[ A ( % i , a )] = 1. Second, for every a ∈ A i ( Ï i , o ∗ i ),by contraction-invariance, ψ ( % I )[ A ( % i , a )] is equal to i ’s participation size in ψ ( % ′ i , % − i )where % ′ i is any contraction of % i at a . Last, if A i ( ≻ i , o ∗ i ) , A i ( Ï i , o ∗ i ), which means that A i ( ∼ i , o ∗ i ) = ∅ , for every a ∈ A i ( ≻ i , o ∗ i ) \A i ( Ï i , o ∗ i ), ψ ( % I )[ A ( % i , a )] is equal to i ’s participationsize in ψ ( % I ). Therefore, for IR and strategy-proof mechanisms, weak size dominanceimplies weak stochastic dominance.An immediate corollary of Theorem 1 is that, when NI and Richness both hold, stochas-tic dominance is equivalent to size dominance for IR and strategy-proof mechanisms. Corollary 1.
Let ψ and ψ ′ be two IR and strategy-proof mechanisms. Under NI and Richness:1. ψ ′ strictly stochastically dominates ψ ⇔ ψ ′ strictly size dominates ψ .2. ψ ′ is welfare-equivalent to ψ ⇔ ψ ′ is size-equivalent to ψ . Our second theorem presents an implication of strategy-proofness given a finite setof allocations in our model. We prove that, under Richness, for any two strategy-proofmechanisms, if their found allocations for a preference profile are upper-equivalent onlyif the found allocations are actually welfare-equivalent, then the two mechanisms mustfind welfare-equivalent allocations for every preference profile.
Theorem 2.
Let ψ and ψ ′ be two strategy-proof mechanisms. Under Richness, if for every % I ∈ R , ψ ( % I ) and ψ ′ ( % I ) are upper-equivalent = ⇒ ψ ( % I ) and ψ ′ ( % I ) are welfare-equivalent,then ψ and ψ ′ are welfare-equivalent. Proof.
Suppose towards a contradiction that ψ and ψ ′ are not welfare-equivalent. De-fine R ∗ ≔ { % I ∈ R : ψ ( % I ) and ψ ′ ( % I ) are not welfare-equivalent } . Then for every % I ∈ R ∗ , ψ ( % I ) and ψ ′ ( % I ) are not upper-equivalent. Arbitrarily choose % I ∈ R ∗ and fix it. Be-cause ψ ( % I ) and ψ ′ ( % I ) are not upper-equivalent, there exist i and a ∈ A i ( Ï i , o ∗ i ) such that12 ( % I )[ A i ( % i , a )] , ψ ′ ( % I )[ A i ( % i , a )]. Under Richness, let % ′ i be any contraction of % i at a .Because ψ and ψ ′ are strategy-proof, ψ ( % ′ i , % − i )[ A i ( % ′ i , a )] = ψ ( % I )[ A i ( % i , a )] and ψ ′ ( % ′ i , % − i )[ A i ( % ′ i , a )] = ψ ′ ( % I )[ A i ( % i , a )]. So ψ ( % ′ i , % − i )[ A i ( % ′ i , a )] , ψ ′ ( % ′ i , % − i )[ A i ( % ′ i , a )], meaning that ψ ( % ′ i , % − i ) and ψ ′ ( % ′ i , % − i ) are not welfare-equivalent. Then ψ ( % ′ i , % − i ) and ψ ′ ( % ′ i , % − i ) arenot upper-equivalent. Using the above argument again, we can contract some agent’spreferences in ( % ′ i , % − i ) to obtain another preference profile where the allocations foundby ψ and ψ ′ are not welfare-equivalent. Iteratively using this argument means repeat-edly contracting agents’ preferences. But because I and A are finite, we cannot contractagents’ preferences for infinitely many times. This is a contradiction.An equivalent formulation of Theorem 2 is that, for any two strategy-proof mech-anisms ψ and ψ ′ , if for every % I , ψ ( % I ) and ψ ′ ( % I ) are not welfare-equivalent only ifthey are not upper-equivalent, then ψ and ψ ′ must be welfare-equivalent. In Section 4we present several applications of this formulation. To prove ψ ( % I ) and ψ ′ ( % I ) are notupper-equivalent, we need to find i and a ∈ A i ( Ï i , o ∗ i ) such that ψ ( % I )[ A i ( % i , a )] , ψ ′ ( % I )[ A i ( % i , a )]. If ψ and ψ ′ are deterministic mechanisms, we need to find i and a ∈ A i ( Ï i , o ∗ i )such that ψ ( % I ) % i a ≻ i ψ ′ ( % I ) or ψ ′ ( % I ) % i a ≻ i ψ ( % I ).We point out two di ff erences between Theorem 1 and Theorem 2. First, Theorem2 relies on the assumption of a finite A , whereas Theorem 1 actually does not rely on it.Theorem 1 holds when A is infinite, as long as relevant concepts are properly rephrased todeal with the measurability issue. But Theorem 2 relies on the fact that, with a finite A ,starting with any preference profile % I , by iteratively letting agents contract preferences,we can find a sequence of preference profiles ( % I , % I , . . . , % nI ) such that % I = % I and in % nI , forevery i , A i ( Ï ni , o ∗ i ) = ∅ . Obviously, the allocations found by any two strategy-proof mech-anisms ψ and ψ ′ for every such % nI are upper-equivalent. The essence of Theorem 2 isthat, if upper-equivalence in any preference profile implies welfare-equivalence, then, bycontraction-invariance, the welfare-equivalence between ψ ( % nI ) and ψ ′ ( % nI ) implies upper-equivalence and therefore welfare-equivalence between ψ ( % n − I ) and ψ ′ ( % n − I ). By induc-tively applying this argument, we prove that for every % I , ψ ( % I ) and ψ ′ ( % I ) are welfare-equivalent. Second, Theorem 2 holds for strategy-proof mechanisms that need not to beIR, while Theorem 1 requires IR. This is because Theorem 1 relates agents’ participa-tion sizes to their welfare, but Theorem 2 does not use agents’ participation sizes. As we When A is infinite, we assume that for every % i ∈ R i and every a ∈ A , all of A ( % i , a ), A ( ≻ i , a ) and A ( ∼ i , a ) are measurable. A random allocation is a probability measure. A probability measure p ′ weaklystochastically dominates anothe probability measure p for agent i if, for all a ∈ A , p ′ [ A ( % i , a )] ≥ p [ A ( % i , a )].Then the proof of Theorem 1 remains as before as long as Richness and NI are assumed. ff erent from AM’s richness assumption. AM’s richnessassumes that, for every % i ∈ R i and every a, b ∈ A i with b ≻ i a % i o ∗ i , there exists % ′ i ∈ R i such that b ≻ ′ i o ∗ i ≻ ′ i a , and for all c ∈ A i , c % ′ i o ∗ i ⇒ c ≻ i a . In some sense % ′ i shrinks the set ofacceptable allocations in % i . The choice of % ′ i depends on a and b , the upper contour set of b needs not to be invariant from % i to % ′ i , and when it is invariant, it needs not to be equalto the strict upper contour set of o ∗ i in % ′ i . So AM’s richness is more relaxed than ours.AM’s richness is su ffi cient to obtain Theorem 1 for deterministic mechanisms, becauseagents’ participation sizes in deterministic allocations are degenerate and their prefer-ences over deterministic allocations are complete. When we use stochastic dominance tocompare welfare in random allocations and to define strategy-proofness, we have to relyon contraction-invariance to demonstrate the structure of allocations found by strategy-proof mechanisms. This requires some invariant upper contour set in agents’ preferenceswhen we connect di ff erent preference profiles. This requirement is incorporated into ourRichness assumption. There is no counterpart of Theorem 2 in AM’s paper.In Section 3.1 we present useful corollaries of our theorems. In Section 3.2 we discusstruncation-invariant mechanisms. ffi ciency frontier of IR and strategy-proof mechanisms Within any class of mechanisms, a member is on the e ffi ciency frontier if it is notstrictly stochastically dominated by any other member. In the object assignment model(see Section 4.1), Bogomolnaia and Moulin (2001) call a mechanism ordinally e ffi cient ifit is not strictly stochastically dominated by any other mechanism. Clearly, within anyclass of mechanisms, its ordinally e ffi cient members must be on the e ffi ciency frontier.But Erdil (2014) proves that, within the class of IR and strategy-proof mechanisms, non-wastefulness, which is weaker than ordinal e ffi ciency, is su ffi cient for a member to be onthe e ffi ciency frontier. We can similarly define ordinal e ffi ciency in our model. But belowwe present some properties weaker than ordinal e ffi ciency that are su ffi cient for a mech-anism to be on the e ffi ciency frontier of IR and strategy-proof mechanisms in our model.In Section 4.1, we will show that these properties are weaker than non-wastefulness inthe object assignment model. 14 efinition 7 (Bidominance) . For any two allocations p and p ′ , p ′ bidominates p if p ′ strictlystochastically dominates p and also strictly size dominates p ; p ′ strongly bidominates p if p ′ bidominates p and for every i , p ′ ≻ sdi p ⇒ p ′ [ A i ] > p [ A i ] . An allocation p is unbidominated if it is not bidominated by any other allocation, and is not-strongly-bidominated if it is notstrongly bidominated by any other allocation. If p ′ bidominates p , p ′ cannot strictly stochastically dominate p through reassigningprobabilities of deterministic allocations without increasing agents’ participation sizes.If p ′ strongly bidominates p , for every better o ff agent in p ′ , his participation size isincreased in p ′ . We say a mechanism is unbidominated/not-strongly-bidominated if italways finds allocations of such properties. Example 1 illustrates these concepts. Example 1 (Illustration of bidominance) . Suppose I = { i, j } and A = { a, b, c, o ∗ } . The par-ticipants in each deterministic allocation are as follows: I ( a ) = I ( b ) = { i, j } , I ( c ) = { i } , and I ( o ∗ ) = ∅ . So i participates in a, b, c , and j participates in a, b . Agents have the followingpreferences: b ≻ i c ≻ i a ≻ i o ∗ and b ≻ j a ≻ j c ∼ j o ∗ . Consider the following five allocations: p p p p p / a, / o ∗ / b, / o ∗ / b, / c b a All of p , p , p , p strictly stochastically dominate p . p does not bidominate p becauseno agent’s participation size is increased in p . p bidominates p because i ’s participationsize is increased in p . But p does not strongly bidominate p because j is better o ff in p yethis participation size is not increased in p . p and p strongly bidominate p because everyagent’s participation size is increased.Because every agent most prefers b , p is the only ordinally e ffi cient allocation. p is strictlystochastically dominated by p . But because every agent’s participation size in p is one, p is unbidominated. p is bidominated by p , because both agents become strictly better o ff and j ’s participation size is increased in p . But p is not-strongly-bidominated, because the onlyway to strictly stochastically dominate p is to move probabilities from c to b and this makes i strictly better o ff but does not increase i ’s participation size. We first present a corollary of Theorem 1. If an allocation p is unbidominated, thenany allocation p ′ that strictly stochastically dominates p has to be size-equivalent to p . Soif two strategy-proof mechanisms ψ ′ and ψ ′′ strictly stochastically dominate some IR andunbidominated mechanism ψ , then ψ ′ and ψ ′′ are size-equivalent to ψ and are thereforesize-equivalent to each other. By Theorem 1, under Richness, ψ ′ and ψ ′′ are welfare-equivalent. It means that in terms of welfare at most one strategy-proof mechanism can15trictly stochastically dominate an IR and unbidominated mechanism. Furthermore, if ψ is strategy-proof, then ψ ′ and ψ ′′ have to be welfare-equivalent to ψ , which contradictsthe assumption that they strictly stochastically dominate ψ . So under Richness, unbidom-inated mechanisms are on the e ffi ciency frontier of IR and strategy-proof mechanisms. Corollary 2.
Let ψ be an IR and unbidominated mechanism. Under Richness:1. If ψ ′ and ψ ′′ are two strategy-proof mechanisms that strictly stochastically dominate ψ ,then ψ ′ and ψ ′′ are welfare-equivalent.2. If ψ is strategy-proof, then ψ is not strictly stochastically dominated by any other strategy-proof mechanism. We then present a corollary of Theorem 2. If an IR and not-strongly-bidominated al-location p is strictly stochastically dominated by an allocation p ′ , then there exists i suchthat p ′ ≻ sdi p and p ′ [ A i ] ≤ p [ A i ]. Suppose NI holds. Then p ′ ≻ sdi p implies p ′ [ A i ] ≥ p [ A i ].So we must have p ′ [ A i ] = p [ A i ]. Then p ′ ≻ sdi p requires that there exist a ∈ A i ( Ï i , o ∗ i )such that p ′ [ A i ( % i , a )] > p [ A i ( % i , a )]. That is, p and p ′ are not upper-equivalent. This ob-servation means that, under NI, if an IR, not-strongly-bidominated and strategy-proofmechanism ψ is strictly stochastically dominated by another strategy-proof mechanism ψ ′ , then for every % I where ψ ′ ( % I ) and ψ ( % I ) are not welfare-equivalent, they are neitherupper-equivalent. If Richness also holds, by Theorem 2, ψ and ψ ′ are welfare-equivalent,which is a contradiction. So under NI and Richness, not-strongly-bidominated mecha-nisms are on the e ffi ciency frontier of IR and strategy-proof mechanisms. Corollary 3.
Under NI and Richness, an IR, not-strongly-bidominated and strategy-proofmechanism is not strictly stochastically dominated by any other strategy-proof mechanism.
Because unbidominated mechanisms are not-strongly-bidominated, but the converseis not true, Corollary 3 is stronger than the second statement of Corollary 2 when NIand Richness both hold. However, Corollary 3 relies on a finite A , but Corollary 2 doesnot. When we are interested in a subclass of IR and strategy-proof mechanisms, we candefine unbidominated/not-strongly-bidominated mechanisms within the subclass. Thenthe second statement of Corollary 2 and Corollary 3 hold within the subclass. We willpresent an application in Section 4.Alva and Manjunath (2019) call a ∈ A Pareto-constrained participation-maximal if theredoes not exist b ∈ A that weakly Pareto dominates a and strictly participation expands a . AM say b strictly participation expands a if I ( a ) ( I ( b ). It is the degeneration of strict size dominance. a is unbidominated if there does not exist b that strictly Pareto dominates a and strictly participation expands a . Under NI, this di ff erence disappears because if b weakly Pareto dominates a and strictly participation expands a , then b must strictlyPareto dominate a . This is why AM assume both NI and richness in their results forPareto-constrained participation-maximality, while our Corollary 2 only assumes Rich-ness. There is no counterpart of strong bidominance and Corollary 3 in AM’s paper. Though our theorems are presented for strategy-proof mechanisms, their proofs relyonly on contraction-invariance, which is weaker than strategy-proofness. When pref-erence domains accommodate truncation, our theorems hold for truncation-invariantmechanisms that are not strategy-proof. Truncation-invariance describes a structure ofthe allocations found by a mechanism when agents truncate preferences. It can be im-plied by properties unrelated to incentive. For example, in the matching with contractsmodel (Hatfield and Milgrom, 2005), if substitutability and law of aggregate demand arenot assumed for hospitals’ choice functions, the doctor-optimal stable mechanism maynot exist, and if it exists, it may not be strategy-proof. But in Section 4.3 we prove that ifit exists, doctor-optimal stability implies that it is truncation-invariant. This enables usto explain the results of Hirata and Kasuya (2017) using our terminology and theorems.Some other mechanisms in matching theory are not strategy-proof, but their algorithmicprocedures naturally imply truncation-invariance. For example, when agents have strictpreferences, in the
Boston mechanism (BM) in the school choice model (Abdulkadiro ˘gluand S¨onmez, 2003) and the probabilistic serial mechanism (PS) in the object assignmentmodel (Bogomolnaia and Moulin, 2001), agents report favorite schools/objects step bystep, and the allocation found at each step is permanent and depends only on agents’preferences revealed by the step. This means that BM and PS are truncation-invariant.Of course, when strategy-proofness is given up, it raises the concern that agents maymisreport preferences. If two IR and truncation-invariant mechanisms are observed to besize-equivalent, we are not sure whether they are size-equivalent with respect to agents’true preferences. Then we are hesitant to conclude that the two mechanisms are welfare-equivalent. This is why we focus on strategy-proof mechanisms in our theorems. But in In every round of BM, students apply for favorite schools that have not rejected them, and schools withempty seats admit new applicants according to priority rankings. In PS, agents consume probability sharesof objects with equal rates. ffi ciency, fairness,and strategy-proofness (Zhou, 1990; Bogomolnaia and Moulin, 2001; Martini, 2016; Nes-terov, 2017). When e ffi ciency and fairness are primary goals, we have to give up strategy-proofness. We should also be aware that recent studies suggest that agents may not reporttrue preferences in some strategy-proof mechanisms, and that some strategy-proof mech-anisms are more likely to incentivize truth-telling than the others (Li, 2017; Ashlagi andGonczarowski, 2018; Rees-Jones and Skowronek, 2018; Hassidim et al., 2017). Whetheragents will report true preferences in a mechanism in a specific environment is an empir-ical question. Our theorems are useful to reveal the structure of non-strategy-proof buttruncation-invarianct mechanisms when incentive is not a concern. We present four applications to show the usefulness of our theorems. Alva and Man-junath (2019) have presented several applications to show its usefulness of Theorem 1for deterministic mechanisms. In the first application we show its usefulness for ran-dom mechanisms. We apply Theorem 1 to the object assignment model and obtain newfindings. These findings strengthen and explain Erdil (2014). We use the remaining ap-plications to show the usefulness of Theorem 2, because it is a new result for deterministicmechanisms as well as for random mechanisms. In the second application we propose asize improvement relation between mechanisms in the object assignment model to ad-dress the question that when it is possible to replace an existing mechanism with a newone to increase the number of assignments. This question is relevant in real-life marketswhere increasing the number of assignments is a desirable goal. We apply Theorem 2 toobtain an impossibility result. In third application we consider two di ff erent two-sidedmatching models. We show that Theorem 2 can explain all results of Hirata and Kasuya(2017) in the many-to-one matching with contracts model, and a characterization resultof Combe et al. (2020) in their teacher reassignment model. In the last application weprovide a new proof of the main theorems of S¨onmez (1999) and Ehlers (2018). We showthat the logic in Theorem 2 lies behind their theorems.18 .1 Stochastic dominance in the object assignment problem In the object assignment model, a finite set of indivisible objects O are assigned to afinite set of agents I . Each object o has q o ∈ N copies, and each agent i demands a copy ofan object and has a strict preference relation % i over ˜ O ≔ O ∪ {∅} , where ∅ denotes everyagent’s outside option of obtaining nothing. The preference domain of each agent is theset of all strict preferences P . So NI and Richness are satisfied.An allocation is represented by a nonnegative matrix p = ( p i,o ) i ∈ I,o ∈ ˜ O where P i ∈ I p i,o ≤ q o for all o ∈ O and P o ∈ ˜ O p i,o = 1 for all i ∈ I . Every p i,o denotes the probability that i obtains a copy of o , and p i = ( p i,o ) o ∈ ˜ O is the lottery assigned to i . An allocation p is deterministic if every p i,o ∈ { , } . The participants in every deterministic allocation arethose who obtain objects. For every i and every o , we define k p i k ≔ P o ∈ O p i,o , k p o k ≔ P i ∈ I p i,o , and k p k ≔ P i ∈ I k p i k . So k p i k is i ’s participation size in p , and k p k is the totalamount of objects assigned to agents in p . We call k p k the size of p . Stochastic dominanceand size dominance between allocations/mechanisms are defined as in our model.An allocation p is IR if for all i ∈ I and all o ∈ O , p i,o > ⇒ o ≻ i ∅ . p is non-wasteful if itis IR and there do not exist i ∈ I and o ∈ O such that k p o k < q o and o ≻ i o ′ for some o ′ ∈ ˜ O with p i,o ′ >
0. A non-wasteful allocation must be unbidominated, but an unbidominatedallocation may be wasteful. After defining the above notions, we state the corollary of Theorem 1. Because prefer-ences in this model are strict, welfare-equivalence means coincidence.
Corollary 4.
In the object assignment model with strict preferences, for any two IR and strategy-proof mechanisms ψ and ψ ′ ,1. ψ ′ strictly size dominates ψ ⇐⇒ ψ ′ strictly stochastically dominates ψ ;2. ψ ′ is size-equivalent to ψ ⇐⇒ ψ ′ = ψ ;3. if ψ is not-strongly-bidominated, ψ is not strictly stochastically dominated by ψ ′ . By the Birkho ff -von Neumann theorem and its generalization (Birkho ff , 1946; Von Neumann, 1953;Kojima and Manea, 2010), every allocation is a probability distribution over deterministic allocations. Suppose a non-wasteful allocation p is bidominated by another allocation p ′ . Then it must be that k p ′ k > k p k . So there exist o ∈ O and i ∈ I such that k p o k < k p ′ o k ≤ q o and p i,o < p ′ i,o . Because p is non-wasteful,it must be that k p i k = 1. But because i is weakly better o ff in p ′ , there must exist o ′ ∈ O such that o ≻ i o ′ and p i,o ′ >
0. However, this means that p is wasteful.On the other hand, suppose there are two agents i, j and two objects o, o ′ , each having two copies. Bothagents prefer o to o ′ and prefer o ′ to ∅ . The allocation where both agents obtain (1 / o, / o ′ ) is unbidomi-nated because every agent’s participation size is one. But it is wasteful because o is wasted. ψ ′ is of greater size than another mechanism ψ if,for every % I , k ψ ′ ( % I ) k ≥ k ψ ( % I ) k , and for some % ′ I , k ψ ′ ( % ′ I ) k > k ψ ( % ′ I ) k . Erdil proves that,within the class of IR and strategy-proof mechanisms, a mechanism strictly stochasti-cally dominates another mechanism only if the former is of greater size than the latter,and non-wasteful mechanisms are on the e ffi ciency frontier. Clearly, size dominance im-plies greater size. So Corollary 4 explains and strengthens Erdil’s results. We know thatCorollary 4 actually holds for IR and truncation-invariant mechanisms. So it implies thata non-wasteful and strategy-proof random mechanism is not strictly stochastically dom-inated by PS, and a non-wasteful and strategy-proof deterministic mechanism (e.g., DA)is not strictly Pareto dominated by BM. Being not-strongly-bidominated is su ffi cient for a mechanism to be on the e ffi ciencyfrontier of IR and strategy-proof mechanisms, but it is not necessary. This is shown byProposition 1 below. Among IR and strategy-proof mechanisms, the random prioritymechanism (RP) is desirable because it is not only strategy-proof but also satisfies ex-postPareto e ffi ciency (ExPE) and equal treatment of equals (ETE). But Erdil finds that althoughRP is a randomization over non-wasteful deterministic mechanisms, RP is wasteful asa random mechanism. When there is one copy of each object, Erdil constructs a strategy-proof mechanism to strictly stochastically dominate RP. However, Erdil shows that waste-fulness of RP cannot be eliminated through strategy-proof improvement: any strategy-proof mechanism that strictly stochastically dominates RP is still wasteful. Proposition 1shows that this is not due to RP as a benchmark mechanism. We prove that any IR andstrategy-proof mechanism ψ that satisfies ExPE and ETE is strongly bidominated andany strategy-proof mechanism ψ ′ that strictly stochastically dominates ψ is still stronglybidominated. This result is not implied by our theorems, so we prove it in Appendix A. Proposition 1.
In the object assignment model with strict preferences, suppose | I | ≥ , | O | ≥ ,and q o = 1 for all o ∈ O . Let ψ be an IR and strategy-proof mechanism that satisfies ETE andExPE, and ψ ′ be a strategy-proof mechanism that strictly stochastically dominates ψ . Thenboth ψ and ψ ′ are strongly bidominated. Proof.
In Appendix A, we introduce another invariance property we call size-invariance . DA and BM can be regarded as mechanisms to solve the object assignment model where priorities usedby objects to rank agents are regarded as a part of the definition of these mechanisms. An allocation satisfies ExPE if it can be written as a convex combination of Pareto e ffi cient deterministicallocations, and it satisfies ETE if agents of equal preferences obtain equal lotteries. RP is the randomization over the serial dictatorship mechanisms by drawing the ordering of agentsuniformly at random.
20t requires that if an agent changes his preferences without changing the set of accept-able objects, his participation size should be invariant. Truncation-invariance and size-invariance capture di ff erent restrictions of strategy-proofness. The proof of Proposition1 relies only on the two invariance properties.Martini (2016) has proved that an IR and strategy-proof mechanism that satisfies ETEmust be wasteful. In Proposition 1, ExPE of ψ is necessary for proving that ψ ′ is stronglybidominated. To see it, consider the mechanism that always assigns ∅ to agents. Thismechanism is IR and strategy-proof and satisfies ETE. But it violates ExPE. It is strictlyPareto dominated by the serial dictatorship mechanism (with any ordering of agents),which is strategy-proof and non-wasteful. In this subsection we propose a size improvement relation between mechanisms in theobject assignment model. We ask the following question. Suppose a mechanism has beenused in a market, but the market designer wants to replace it with a new one to increasethe total number of objects assigned to agents (i.e., the size of the allocation). We wantto know when the replacement is (im)possible if the two mechanisms satisfy some desir-able properties. Kidney exchange and refugee resettlement are exemplary markets thatmotivate our question. Kidney exchange becomes widespread in the last decade to over-come the shortage of deceased-donor kidneys and the medical incompatibility betweenliving donors and their intended recipients. Resettlement is the transfer of refugees froman asylum country to another country that agrees to admit them and ultimately grantthem permanent settlement. At least two features in these markets motivate our ques-tion. First, not all resources in these markets are acceptable to agents. A donated kidneymay be medically incompatible with a patient, and a refugee may be unable to live in acountry/city for various reasons. Second, it is far more important for an agent in thesemarkets to get an acceptable resource than to get a better resource when an acceptableresource has been guaranteed. Being unmatched is very undesirable. This explains whythe market designer wants to increase the number of acceptable assignments in thesemarkets. There are other markets where our question makes sense. Optimization and market design techniques have been used in kidney exchange and refugee resettle-ment to maximize the number of acceptable assignments. See Roth et al. (2005); S¨onmez et al. (forthcom-ing); Kratz and Andersson (2019); Andersson and Ehlers (2019).
21n our question, the old mechanism will restrict the replacement because agents’ ini-tial welfare needs to be respected. In practice, when a market designer wants to imple-ment a new mechanism that harms some people, those people will complain/oppose thereplacement, which becomes an obstacle to the replacement. In our question, if the newmechanism Pareto improves the old, all agents will be glad to support the replacement.But if some agents are harmed in the new mechanism, they may oppose the replacement.We assume that agents are altruistic and agree that increasing the number of acceptableassignments is desirable for the society as a whole. But because being unmatched is veryundesirable, an agent will oppose the replacement if and only if he is matched in theold mechanism but become unmatched in the new one. We assume that in this case theonly excuse the market designer can use to justify the replacement is that the number ofacceptable assignments is strictly increased in the new mechanism. In other words, if thenew mechanism does not strictly increase the number of acceptable assignments, everymatched agent in the old mechanism must also be matched in the new mechanism, whichmeans that the allocations found by the two mechanisms are size-equivalent. When ran-dom mechanisms are used, agents compare their participation sizes before and after thereplacement.Formally, we study the object assignment model in Section 4.1, but now we allow forweak preferences. In kidney exchange, a patient prefers compatible donors to incompat-ible donors, and, among compatible donors, prefers live donors to cadavers because theformer have a higher chance of success. But a patient is indi ff erent between donors of thesame type. Similarly, a refugee is indi ff erent between resettlement locations of similarcharacteristics. We assume that agents are never indi ff erent between a real object and thestatus of being unmatched. Let B denote the set of all complete, reflexive, and transitivepreference relation % on ˜ O ≔ O ∪ {∅} such that for every o ∈ O , either o ≻ ∅ or ∅ ≻ o . B isthe preference domain of all agents. So NI and Richness are satisfied. Allowing for weakpreferences enables us to accommodate more environments, but as we will explain, ourresult holds when agents’ preferences are strict.Now we define the size improvement relation between mechanisms. Definition 8.
A new mechanism ψ ′ size improves an old mechanism ψ if1. ∀ % I , k ψ ′ ( % I ) k ≥ k ψ ( % I ) k , and ∃ % ′ I , k ψ ′ ( % ′ I ) k > k ψ ( % ′ I ) k ;2. ∀ % I , k ψ ′ ( % I ) k = k ψ ( % I ) k = ⇒ ψ ′ ( % I ) is size-equivalent to ψ ( % I ) . ψ and ψ ′ , if ψ ′ strict size dominates ψ , then ψ ′ size improves ψ . But the converse is not true. So the following result is not implied by Theorem 1. Proposition 2.
In the object assignment model with weak preferences, an IR, unbidominated,and truncation-invariant mechanism is not size improved by any other IR and truncation-invariant mechanism.
By presenting Proposition 2 for truncation-invariant mechanisms, we intend to applyit to non-strategy-proof mechanisms. In the proof of the proposition, a crucial step is toprove the following lemma. This lemma enables us to apply Theorem 2.
Lemma 1.
Fixing % I , for any two IR allocations p and p ′ , if p is unbidominated and k p ′ k > k p k ,then p and p ′ are not upper-equivalent. Proof.
The proof is in Appendix B.
Proof of Proposition 2.
Suppose ψ and ψ ′ are two IR and truncation-invariant mecha-nisms, ψ is unbidominated, and ψ ′ size improves ψ . For every % I , if ψ ( % I ) and ψ ′ ( % I ) arenot size-equivalent, then it must be that k ψ ′ ( % I ) k > k ψ ( % I ) k . By Lemma 1, ψ ′ ( % I ) and ψ ( % I )are not upper-equivalent. Because NI holds, if ψ ( % I ) and ψ ′ ( % I ) are not size-equivalent,then they are not welfare-equivalent. So ψ ( % I ) and ψ ′ ( % I ) are not welfare-equivalent onlyif they are not upper-equivalent. By Theorem 2, ψ ′ and ψ are welfare-equivalent. So theyare size-equivalent, which is a contradiction.If agents have strict preferences, Lemma 1 holds and its proof can be simplified. SoProposition 2 holds when agents have strict preferences, with P being the domain.The implication of Proposition 2 can be understood from two perspectives. Supposethe old mechanism is not too ine ffi cient (being unbidominated), and the old and the newmechanisms satisfy properties that imply truncation-invariance. If it is important to re-spect agents’ initial welfare, then the old mechanism cannot be replaced by the new one.While if the market designer has enough power to implement the replacement regardlessof agents’ initial welfare, then it is inevitable that sometimes the replacement harms someagents in favor of some others, but does not help the society as a whole.We apply Proposition 2 to well-known deterministic mechanisms in the literaturewhere agents’ preferences are strict: Deferred Acceptance (DA; Gale and Shapley, 1962),Boston Mechanism (BM; Abdulkadiro ˘glu et al., 2005), Top Trading Cycle (TTC; Shapleyand Scarf, 1974), Serial Dictatorship (SD; Satterthwaite and Sonnenschein, 1981), Hier-archical Exchange (HE; P ´apai, 2000), and Trading Cycles (TC; Pycia and ¨Unver, 2017).23ome of these mechanisms rely on priorities used by objects to rank agents. Here we re-gard priorities as a part of the definitions of these mechanisms. All of these mechanismsexcept BM are strategy-proof. As we have explained in Section 3.2, the algorithmicprocedure of BM implies truncation-invariance. All of these mechanisms except DA isPareto e ffi cient. DA is unbidominated because it is non-wasteful. So we obtain the fol-lowing corollary of Proposition 2. Corollary 5.
In the object assignment model with strict preferences, none of DA, BM, TTC,SD, HE, and TC is size improved by any other truncation-invariant mechanism. In particular,they do not size improve each other.
Proposition 2 is also applicable to random mechanisms. For example, when agentshave strict preferences, because PS is IR, non-wasteful and truncation-invariant, PS isnot size improved by any other truncation-invariant mechanism. In Appendix C, weconstruct a mechanism that strictly size dominates PS. The mechanism is not truncation-invariant, but it is ordinally e ffi cient and envy-free, which are the flagship properties ofPS. This answers an open question raised by Huang and Tian (2017) on the existence ofan ordinally e ffi cient and envy-free mechanism that improves the size of PS. It also showsthat truncation-invariance of the improving mechanism is necessary for Proposition 2 tohold. In Appendix C, we construct another IR and unbidominated mechanism that isnot truncation-invariant but is strictly size dominated by PS. It means that truncation-invariance of the old mechanism is also necessary for Proposition 2 to hold. Last, becauseErdil (2014) has constructed a strategy-proof mechanism to bidominate RP, being un-bidominated of the old mechanism is also necessary for Proposition 2 to hold. We use Theorem 2 and its corollaries to explain existing results in two di ff erent match-ing models. In many-to-one matching with contracts, the doctor-optimal stable mecha-nism (DOSM) finds a stable matching that Pareto dominates every other stable matching.It is known that substitutability, law of aggregate demand (LAD), and irrelevance of re-jected contracts (IRC) for hospitals’ choice functions are su ffi cient and in some sense alsonecessary for the existence of stable matchings and the existence and strategy-proofnessof DOSM (Hatfield and Milgrom, 2005; Ayg ¨un and S¨onmez, 2013). Actually, under those Pycia and ¨Unver (2011) extend TC and HE to the environment where each object may have severalcopies, and prove that the extended mechanisms are Pareto e ffi cient and strategy-proof. non-wastefulness (di ff erent than the notion in the object assignment model), and provethat any non-wasteful and strategy-proof mechanism is not strictly Pareto dominated byany other individually rational (for both doctors and hospitals) and strategy-proof mech-anism. Under IRC, stable mechanisms are non-wasteful. Proposition 3 (Hirata and Kasuya (2017)) . In many-to-one matching with contracts, underIRC, (1) there is at most one stable and strategy-proof mechanism, and if the doctor-optimalstable mechanism exists, it is the only candidate; (2) a non-wasteful and strategy-proof mecha-nism is not Pareto dominated by any other individually rational and strategy-proof mechanism.
Because we will explain the above result using our terminology in detail, let us brieflydefine HK’s model. A finite set of doctors D sign contracts with a finite set of hospitals H . Each contract x involves a doctor d ( x ) and a hospital h ( x ). Let X be the finite setof contracts. A subset of contracts X ′ is a matching if it includes at most one contractfor each doctor; denote by x ( d, X ′ ) the contract involving d and by x ( h, X ′ ) the set ofcontracts involving h . Each d ∈ D has a strict preference relation % d over { x ∈ X : d ( x ) = d } ∪ {∅} , where ∅ is the outside option of not signing any contract. All strict preferencesare possible. Each h ∈ H has a choice function C h such that for all X ′ ⊂ X , C h ( X ′ ) = C h ( x ( h, X ′ )) ⊂ x ( h, X ′ ). C h satisfies IRC if for all X ′ ⊂ X and x ∈ X , x < C h ( X ′ ∪ { x } ) implies C h ( X ′ ∪ { x } ) = C h ( X ′ ). A matching X ′ is individually rational if x ( d, X ′ ) % d ∅ for all d and C h ( X ′ ) = x ( h, X ′ ) for all h . X ′ is blocked by ( h, X ′′ ) with h ∈ H and X ′′ ⊂ X if X ′′ ∩ X ′ = ∅ , x ( h, X ′′ ) ⊂ C h ( X ′ ∪ X ′′ ) and x ( d, X ′′ ) ≻ d x ( d, X ′ ) for all d ∈ { d ( x ) } x ∈ X ′′ . X ′ is stable if it isindividually rational and unblocked. X ′ is non-wasteful if it is individually rational andthere does not exist another individually rational matching X ′′ such that X ′ ( X ′′ .To prove Proposition 3, HK prove the following lemma. In our terminology, the lemmaproves that if two stable mechanisms find di ff erent matchings for a preference profile ofdoctors, the matchings are not upper-equivalent. By Theorem 2, it implies that any two Alva (2018) extends IRC to choice correspondences and shows its equivalence to weak axiom of re-vealed preference.
Lemma 2 (Hirata and Kasuya (2017)) . In many-to-one matching with contracts, under IRC,if there exist two stable mechanisms ψ and ψ ′ , then for every preference profile % D where ψ ( % D ) , ψ ′ ( % D ) , there exists d ∈ D such that ∅ , x ( d, X ′ ) , x ( d, X ′′ ) , ∅ . Proof.
Consider any % D where ψ ( % D ) , ψ ′ ( % D ). Let ψ ( % D ) = X ′ and ψ ′ ( % D ) = X ′′ . Wewant to prove that there exists d ∈ D such that ∅ , x ( d, X ′ ) , x ( d, X ′′ ) , ∅ . Because X ′ , X ′′ ,either X ′ \ X ′′ , ∅ or X ′′ \ X ′ , ∅ . Suppose without loss of generality that X ′ \ X ′′ , ∅ . Ifthere exist x ∈ X ′ \ X ′′ and x ′ ∈ X ′′ such that d ( x ) = d ( x ′ ), we are done because x , x ′ .Otherwise, for every d ∈ { d ( x ) } x ∈ X ′ \ X ′′ , it must be that x ( d, X ′ ) ≻ d x ( d, X ′′ ) = ∅ . For every h ∈ H such that x ( h, X ′ \ X ′′ ) , ∅ , we prove that C h ( X ′ ∪ X ′′ ) = x ( h, X ′′ ) and x ( h, X ′′ ) \ x ( h, X ′ ) , ∅ . Suppose C h ( X ′ ∪ X ′′ ) , x ( h, X ′′ ) for some h . By IRC, C h ( X ′ ∪ X ′′ ) ∩ ( X ′ \ X ′′ ) , ∅ . It meansthat ( h, C h ( X ′ ∪ X ′′ ) ∩ ( X ′ \ X ′′ )) can block X ′′ , which is a contradiction. Since C h ( X ′ ) = x ( h, X ′ ) , x ( h, X ′′ ), IRC requires that x ( h, X ′′ ) \ x ( h, X ′ ) , ∅ . But this means that X ′′ \ X ′ , ∅ and x ( h, X ′′ \ X ′ ) , ∅ . By a similar argument as above, we can prove that C h ( X ′ ∪ X ′′ ) = x ( h, X ′ ), which is a contradiction since x ( h, X ′′ ) , x ( h, X ′ ).Below we prove that when DOSM exists, though it may not be strategy-proof, doctor-optimal stability implies that it is truncation-invariant. Because Theorem 2 actually holdsfor truncation-invariant mechanisms, if a stable and strategy-proof mechanism exists, itmust coincide with DOSM. Lemma 3.
In many-to-one matching with contracts, under IRC, if the doctor-optimal stablemechanism exists, it is truncation-invariant.
Proof.
Let ψ denote the doctor-optimal stable mechanism. For all % D , all d ∈ D and all x ∈ X with d ( x ) = d such that x ≻ d ∅ :1. If x ≻ d ψ d ( % D ), we prove that ψ d ( % xd , % − d ) = ∅ , where % xd is the truncation of % d at x . Suppose ψ d ( % xd , % − d ) , ∅ . Since ψ is individually rational, ψ d ( % xd , % − d ) % d x ≻ d ψ d ( % D ).Since ψ ( % xd , % − d ) is stable under ( % xd , % − d ) and ψ d ( % xd , % − d ) % d x , ψ ( % xd , % − d ) must be stableunder % D . But this contradicts the doctor-optimal stability of ψ ( % D ) under % D because ψ ( % D ) does not Pareto dominate ψ ( % xd , % − d ).2. If ψ d ( % D ) % d x , we prove that ψ d ( % xd , % − d ) % xd x . Since ψ ( % D ) is stable under % D and ψ d ( % D ) % d x , it must be stable under ( % xd , % − d ). By the doctor-optimal stability of ψ ( % xd , % − d ) under ( % xd , % − d ), ψ d ( % xd , % − d ) % xd ψ d ( % D ). So ψ d ( % xd , % − d ) % xd x .26inally, we prove that non-wasteful matchings defined by HK are exactly not-strongly-bidominated matchings within the set of individually rational matchings. Lemma 4.
In many-to-one matching with contracts, a matching X ′ is non-wasteful if and onlyif it is not-strongly-bidominated among individually rational matchings. Proof.
For any two individually rational matchings X ′ and X ′′ , we prove that X ′ is stronglybidominated by X ′′ if and only if X ′ ( X ′′ . X ′ is strongly bidominated by X ′′ if X ′′ strictlyPareto dominates X ′ and for all d ∈ D , x ( d, X ′′ ) ≻ d x ( d, X ′ ) implies x ( d, X ′ ) = ∅ . So for all d ∈ D with x ( d, X ′ ) , ∅ , x ( d, X ′′ ) = x ( d, X ′ ). It means that X ′ ( X ′′ .Conversely, if X ′ ( X ′′ , for all d ∈ D with x ( d, X ′ ) , ∅ , x ( d, X ′′ ) = x ( d, X ′ ) since eachdoctor signs at most one contract. For all d ∈ D with x ( d, X ′ ) = ∅ , x ( d, X ′′ ), individualrationality implies x ( d, X ′′ ) ≻ d x ( d, X ′ ). So X ′′ strongly bidominates X ′ . Proof of Proposition 3.
Suppose there exist two stable and strategy-proof mechanisms.By Lemma 2, if they find matchings that are not welfare-equivalent, the matchings arealso not upper-equivalent. By Theorem 2, the two mechanisms coincide. By Lemma3, when DOSM exists, it is truncation-invariant. Because Theorem 2 actually holds fortruncation-invariant mechanisms, when stable and strategy-proof mechanisms exist, theycoincide with DOSM. With Lemma 4, we can apply Corollary 3 to conclude that any non-wasteful and strategy-proof mechanism is not strictly Pareto dominated by any otherindividually rational and strategy-proof mechanism.The second matching model we consider is the teacher reassignment model proposedby Combe et al. (2020). In the model every teacher is initially matched to a school, andteachers and schools have strict preferences over each other. The question is how torematch teachers and schools to improve their welfare. Combe et al. propose a classof mechanisms called
Block Exchange (BE) to find the set of desirable matchings. Theyfurther recommend a subclass of BE called
Teacher Optimal BE (TO-BE). Every TO-BEis strategy-proof for teachers. In the one-to-one environment, TO-BE is single-valued,and Combe et al. prove that TO-BE is the only strategy-proof mechanism among BE.
Proposition 4 (Combe et al. (2020)) . In the one-to-one environment of the teacher reassign-ment model, TO-BE is the unique strategy-proof mechanism in the class of BE.
We will not go into details about the model and mechanisms. Briefly, the one-to-oneenvironment includes equal numbers of teachers T and schools S . Every i ∈ S ∪ T has Because the preferences of schools are governed by policies, schools are not strategic agents. % i over agents on the other side. A matching is a one-to-one mapping µ : T → S . Let µ denote the initial matching. This environment can be accommodatedby our model where every teacher’s outside option is his initial assignment. BE is a gen-eralization of TTC by taking preferences of both sides into account. Combe et al. proveseveral lemmas to obtain Proposition 4. In our terminology, one of their lemmas, calledLemma 5 below, proves that if any BE and TO-BE find di ff erent matchings for a prefer-ence profile of teachers, the matchings are not upper-equivalent. Then Theorem 2 directlyimplies Proposition 4. For brevity, we do not present the proof of Lemma 5. Combe et al.need other lemmas and use a logic di ff erent from Theorem 2 to obtain Proposition 4. Lemma 5 (Combe et al. (2020)) . In the one-to-one environment of the teacher reassignmentmodel, let ψ be any selection of BE. For any preference profile of teachers % T , if x = TO-BE ( % T ) , ψ ( % T ) = y , then there exists t ∈ T such that x ( t ) ≻ t y ( t ) ≻ t µ ( t ) . In a generalized object assignment model, S¨onmez (1999) proves that there exists anIR, Pareto e ffi cient and strategy-proof mechanism only if the core is single-valued and themechanism selects a core allocation when the core is nonempty. Ehlers (2018) extendsS¨onmez’s theorem by replacing the core with a superset called the individually-rational-core (IR-core). The IR-core is more likely to be nonempty than the core. In this subsectionwe show that the logic in Theorem 2 lies behind the theorems of S¨onmez and Ehlers. Let C ir ( % I ) denote the IR-core when the preference profile of agents in S¨onmez’s model is % I . Proposition 5 (S¨onmez (1999); Ehlers (2018)) . In Sonmez’s model, if there exists an IR,Pareto e ffi cient and strategy-proof mechanism ψ , then:1. for all % I and all a, b ∈ C ir ( % I ) , a ∼ i b for all i ∈ I ;2. for all % I with C ir ( % I ) , ∅ , ψ ( % I ) ∈ C ir ( % I ) . To prove Proposition 5, we briefly introduce S¨onmez’s model and the two core notions.In the model every agent is endowed with a finite set of indivisible objects. An allocationassigns all objects to agents and every object is assigned to one agent. Let a ( i ) denote theset of objects assigned to agent i in an allocation a . Let ω denote the initial allocation (i.e.,the endowment). S¨onmez makes two assumptions on agents’ preferences:• Assumption A: for all % i ∈ R i and all a ∈ A , a ∼ i ω ⇔ a ( i ) = ω ( i ).28 Assumption B: for all % i ∈ R i and all a ∈ A ( ≻ i , ω ), there exists % ′ i ∈ R i such that A ( % i , a ) = A ( ≻ ′ i , ω ) = A ( % ′ i , a ) and A ( ≻ i , a ) = A ( ≻ ′ i , a ).For all T ⊂ I , define ω ( T ) ≔ ∪ i ∈ T ω ( i ). We say an allocation a weakly dominates an al-location b via the coalition T ⊂ I under % I , denoted by a wdom T b , if a ( i ) ⊂ ω ( T ) for all i ∈ T , a % i b for all i ∈ T , and a ≻ j b for some j ∈ T . The core under % I consists of allundominated allocations, whereas the IR-core under % I is defined to be C ir ( % I ) ≔ { b ∈ A : ∄ T and a ∈ A with a wdom T b and a ( i ) = ω ( i ) for all i ∈ I \ T } . The IR-core may be empty,but when it is nonempty, every IR-core allocation is IR and Pareto e ffi cient.Clearly, S¨onmez’s model is a special case of ours by letting every agent’s outside optionbe his endowment. S¨onmez’s two assumptions on preferences are stronger than NI andRichness. Below we prove that the IR-core has the contraction-invariance property. Lemma 6.
In Sonmez’s model, for every % I ∈ R with C ir ( % I ) , ∅ , every a ∈ C ir ( % I ) , every i ,and every contraction % ′ i of % i at a that satisfies Sonmez’s Assumption B, a ∈ C ir ( % ′ i , % − i ) . Proof.
Suppose a < C ir ( % ′ i , % − i ). Then under ( % ′ i , % − i ), there exist T ⊂ I and c ∈ A such that c wdom T a and c ( k ) = ω ( k ) for all k ∈ I \ T . So c % j a for all j ∈ T \{ i } , c % ′ i a , and either c ≻ j a for some j ∈ T \{ i } , or c ≻ ′ i a . By Assumption B, c % ′ i a implies c % i a and c ≻ ′ i a implies c ≻ i a . So under % I , we must have c wdom T a and c ( k ) = ω ( k ) for all k ∈ I \ T . So a < C ir ( % I ),which is a contradiction.Now suppose there exists an IR, Pareto e ffi cient and strategy-proof mechanism ψ . Iffor all % I , C ir ( % I ) = ∅ , then the proposition holds trivially. Otherwise, we define R ∗ ≔ { % I ∈ R : C ir ( % I ) , ∅} , which is nonempty. Arbitrarily choose % ∗ I ∈ R ∗ and a ∈ C ir ( % ∗ I ), andfix them. Then we construct a mechanism ψ ′ according to the following steps:1. Let ψ ′ ( % ∗ I ) = a . So ψ ′ ( % ∗ I ) ∈ C ir ( % ∗ I ).2. For every i such that a ≻ ∗ i c ≻ ∗ i ω for some c ∈ A , let % ′ i be any contraction of % ∗ i at a satisfying Assumption B. Let ψ ′ ( % ′ i , % ∗− i ) = a . By Lemma 6, ψ ′ ( % ′ i , % ∗− i ) ∈ C ir ( % ′ i , % ∗− i ).3. For every % I considered in step 2 and every i such that a ≻ i c ≻ i ω for some c ∈ A ,let % ′ i be any contraction of % i at a satisfying Assumption B. Let ψ ′ ( % ′ i , % − i ) = a . ByLemma 6, ψ ′ ( % ′ i , % − i ) ∈ C ir ( % ′ i , % − i ).4. Repeat the above operation until we cannot find a new preference profile. Since inevery step we contract a preference profile in the previous step, we must stop infinite steps. Denote by R ′ the set of preference profiles (including % ∗ I ) considered inthese steps. We know that for all % I ∈ R ′ , ψ ′ ( % I ) = a ∈ C ir ( % I ).29. For all % I ∈ R\R ′ , let ψ ′ ( % I ) = ψ ( % I ).So in every preference profile, ψ ′ either selects an IR-core allocation or coincides with ψ . Thus, ψ ′ is IR and Pareto e ffi cient. The following lemma proves that ψ ′ and ψ find dif-ferent allocations for a preference profile only if the allocations are not upper-equivalent. Lemma 7.
For every % I ∈ R with ψ ′ ( % I ) , ψ ( % I ) , there exists i such that ψ ′ ( % I ) ≻ i ψ ( % I ) ≻ i ω . Proof.
For every % I ∈ R such that ψ ′ ( % I ) , ψ ( % I ), it must be that ψ ′ ( % I ) ∈ C ir ( % I ). Sinceboth ψ ′ ( % I ) and ψ ( % I ) are IR and Pareto e ffi cient, there must exist distinct i, j such that ψ ( % I ) ≻ i ψ ′ ( % I ) and ψ ′ ( % I ) ≻ j ψ ( % I ). Suppose for all k ∈ I ′ ≔ { i ∈ I : ψ ′ ( % I ) ≻ i ψ ( % I ) } , ψ ( % I ) ∼ k ω . By Assumption A, ψ k ( % I ) = ω ( k ) for all k ∈ I ′ . Then ψ ( % I ) can weakly block ψ ′ ( % I ) via I \ I ′ under % I , since all agents in I \ I ′ weakly prefer ψ ( % I ) to ψ ′ ( % I ), and i ∈ I \ I ′ strictly prefers ψ ( % I ) to ψ ′ ( % I ). But it contradicts the fact that ψ ′ ( % I ) ∈ C ir ( % I ). So theremust exist k ∈ I ′ such that ψ ′ ( % I ) ≻ k ψ ( % I ) ≻ k ω . Proof of Proposition 5.
For every % I ∈ R with ψ ′ ( % I ) , ψ ( % I ), ψ ′ ( % I ) = a ∈ C ir ( % I ). ByLemma 7, there exists i such that a ≻ i ψ ( % I ) ≻ i ω . Let % ′ i be any contraction of % i at a satisfying Assumption B. So ψ ′ ( % ′ i , % − i ) = a ≻ ′ i ω . Strategy-proofness of ψ requires that ψ ( % ′ i , % − i ) ∼ ′ i ω . So ψ ′ ( % ′ i , % − i ) , ψ ( % ′ i , % − i ). Then, as in the proof of Theorem 2, this meansthat we can iteratively contract the preference profile, which is a contradiction. So it mustbe that ψ = ψ ′ . Note that ψ ′ can be constructed for every % ∗ I ∈ R ∗ and every a ∈ C ir ( % ∗ I ). Soall allocations in C ir ( % ∗ I ) are welfare-equivalent to ψ ( % ∗ I ). It means that C ir ( % ∗ I ) is essen-tially single-valued and ψ selects an IR-core allocation when the IR-core is nonempty. In Section 2 we assume that there are finite deterministic allocations in our model.This assumption simplifies our exposition and is relied upon by Theorem 2. It is alsosatisfied by all applications presented in Section 4. Yet as we have explained in footnote8, Theorem 1 holds no matter A is finite or infinite, as long as NI and Richness are as-sumed and relevant definitions are adjusted to accommodate the measurability issue. Fordeterministic mechanisms, Theorem 1 reduces to Alva and Manjunath’s (2019) theoremwhere there is no assumption on the cardinality of A .But there are environments where A is infinite and NI and Richness are violated. SoTheorem 1 is not applicable to those environments. After using an example to explainthese cases, we change our exposition and present an alternative version of Theorem 1. It30onveys the same insight as Theorem 1 does, but it is applicable to more environments.However, we will explain why the earlier exposition in Section 3 is desirable.Recall that NI assumes that for every % i ∈ R i , A i ( ∼ i , o ∗ i ) = ∅ . Richness assumes that forevery % i ∈ R i and every a ∈ A i ( Ï i , o ∗ i ), there exists % ′ i ∈ R i such that A i ( % i , a ) = A i ( % ′ i , a ) = A i ( ≻ ′ i , o ∗ i ) and A i ( ∼ ′ i , o ∗ i ) = ∅ . So o ∗ i is placed immediately below a in % ′ i , and the uppercontour set of a is invariant from % i to % ′ i . In the following example, we present a trans-ferable utility environment where both NI and Richness are violated. Agents’ preferencesare continuous in transfers so that for every % i and every a ∈ A i ( % i , o ∗ i ), A i ( % i , a ) is a closedset whereas A i ( ≻ i , o ∗ i ) is an open set. Example 2 (NI and Richness are violated when A is infinite) . Consider the allocation ofa single object among finite agents of independent and private valuations. Let v i denote i ’svaluation, with the domain R + . An allocation a = ( f a , t a ) consists of an assignment function f a : I → { , } and a payment function t a : I → R + , where f ai = 1 if and only if i wins the object,and t ai is i ’s payment. At most one agent wins the object, and every payment is nonnegative.Because transfers are continuous, there are infinitely many allocations in this problem. Every i ’s utility in an allocation a is v i · f ai − t ai , and his preferences % i over allocations are induced byutilities. Every i ’s outside option o ∗ i is not bidding and paying zero. So for any a , a ≻ i o ∗ i if andonly if f ai = 1 and t ai < v i , and if a ≻ i o ∗ i , there exists b such that a ≻ i b ≻ i o ∗ i (by choosing f bi = 1 and t ai < t bi < v i ). Thus, for every % i , there does not exist a such that A i ( % i , a ) = A i ( ≻ i , o ∗ i ) . SoRichness is violated. NI is also violated because for any a such that ( f ai , t ai ) = (1 , v i ) , a ∼ i o ∗ i . The earlier definition of contraction-invariance requires invariant upper contour setsin variations of agents’ preferences. When NI and Richness are violated, we change tofocus on strict upper contour sets. Specifically, we no longer assume NI and replaceRichness with the following assumption.
Assumption 3 (Richness*) . For every % i ∈ R i and every a ∈ A i ( ≻ i , o ∗ i ) , there exists % ′ i ∈ R i such that A i ( ≻ i , a ) = A i ( ≻ ′ i , o ∗ i ) . Richness* requires that the strict upper contour set of a in % i become the set of strictlyacceptable allocations in % ′ i . In particular, o ∗ i is placed weakly above a in % ′ i . So now,we call every such % ′ i a contraction of % i above a . Richness* is satisfied in Example 2,because for every a ≻ i o ∗ i , % ′ i could be induced by the valuation v ′ i = t ai , so that a ∼ ′ i o ∗ i and A i ( ≻ i , a ) = A i ( ≻ ′ i , o ∗ i ). When A is finite, Richness implies Richness*. When A is finite, for any % i ∈ R i and any a ∈ A i ( ≻ i , o ∗ i ), as long as a is not one of the best allocations ψ contraction-invariant* if for every % I , every i ∈ I , andevery a ∈ A i ( ≻ i , o ∗ i ), if % ′ i is any contraction of % i above a , then ψ ( % I )[ A i ( ≻ i , a )] = ψ ( % ′ i , % − i )[ A i ( ≻ i , a )] . Because A i ( ≻ i , a ) = A i ( ≻ ′ i , o ∗ i ), we also have ψ ( % I )[ A i ( ≻ i , a )] = ψ ( % ′ i , % − i )[ A i ( ≻ ′ i , o ∗ i )]. As be-fore, strategy-proof mechanisms must be contraction-invariant*.For all % i ∈ R i and all a ∈ A , we assume that A ( % i , a ) and A ( ≻ i , a ) are measurable.A (random) allocation p is a probability measure. So p is IR if, for all i , p [ A ( % i , o ∗ i )] =1. Now we say an allocation p ′ weakly stochastically dominates another allocation p foragent i if, for all a ∈ A , p ′ [ A ( ≻ i , a )] ≥ p [ A ( ≻ i , a )]. Strict stochastic dominance and welfare-equivalence are defined similarly, and they are extended to mechanisms as before. When A is finite, these definitions are essentially the same as before.In an IR allocation p , for any i , p [ A i ] = p [ A i ( ≻ i , o ∗ i )] + p [ A i ( ∼ i , o ∗ i )]. Without assumingNI, it could happen that p [ A i ] , p [ A i ( ≻ i , o ∗ i )]. So welfare comparison no longer impliesparticipation size comparison. Now we change to focus on p [ A i ( ≻ i , o ∗ i )] and call it the welfare size of i in p . It is i ’s probability of being better o ff than consuming outside optionin p . We say that an allocation p ′ weakly welfare size dominates another allocation p if, forall i , p ′ [ A i ( ≻ i , o ∗ i )] ≥ p [ A i ( ≻ i , o ∗ i )]. By this definition, weak stochastic dominance impliesweak welfare size dominance. Then we prove an alternative version of Theorem 1. Theorem 1*.
Under Richness*, let ψ and ψ ′ be two IR and strategy-proof mechanisms. ψ ′ weakly stochastically dominates ψ ⇐⇒ ψ ′ weakly welfare size dominates ψ . Proof.
The ⇒ direction holds by the definitions of stochastic dominance and welfaresize dominance. So we only prove the ⇐ direction. Suppose ψ ′ weakly welfare sizedominates ψ but does not weakly stochastically dominate ψ . It means that there exist % I and i such that ψ ′ ( % I ) (cid:31) sdi ψ ( % I ). Because ψ and ψ ′ are IR, there exists a ∈ A ( % i , o ∗ i )such that ψ ( % I )[ A ( ≻ i , a )] > ψ ′ ( % I )[ A ( ≻ i , a )]. The fact that ψ ′ weakly welfare size domi-nates ψ requires a ≻ i ω . Under Richness*, let % ′ i be any contraction of % i above a . Bycontraction-invariance*, ψ ( % ′ i , % − i )[ A i ( ≻ ′ i , o ∗ i )] > ψ ′ ( % ′ i , % − i )[ A i ( ≻ ′ i , o ∗ i )]. But it contradictsthe assumption that ψ ′ weakly welfare size dominates ψ . in % i , there exists an allocation b such that A i ( % i , b ) = A i ( ≻ i , a ). Let % ′ i be any contraction of % i at b . Then A i ( ≻ i , a ) = A i ( % i , b ) = A i ( ≻ ′ i , o ∗ i ). So % ′ i is a contraction of % i above a . If a is one of the best allocations in % i ,in the proof of Theorem 1*, we will never consider the contraction of % i above a . So the di ff erence betweenRichness and Richness* for such a is inessential. % i ∈ R i and every a, b ∈ A i with b ≻ i a % i o ∗ i , there exists % ′ i ∈ R i such that b ≻ ′ i o ∗ i ≻ ′ i a , andfor all c ∈ A i , c % ′ i o ∗ i ⇒ c ≻ i a . In Example 2, for any b ≻ i a % i o ∗ i , % ′ i can be induced by anyvaluation v ′ i ∈ ( t bi , t ai ), so that A i ( % i , b ) = A i ( % ′ i , b ) ( A i ( ≻ ′ i , o ∗ i ) ( A i ( % ′ i , a ). Our theorems rely on the existence of outside options and agents’ freedom of arbitrar-ily ranking outside options in their prefererences. When such freedom is restricted, thesecond part of Theorem 1 does not hold because we can no longer recover an agent’s wel-fare in a preference profile from his participation size in other preference profiles. Theo-rem 2 neither holds because there may no longer exist a preference profile for which theallocations find by any two strategy-proof mechanisms are upper-equivalent, and eventhough such preference profiles exist, we cannot use contraction-invariance to connectthe allocations in di ff erent preference profiles. Our model is general enough to accom-modate various environments. We have presented several applications in market design.It will be interesting to find more applications in other fields of economics. A Proof of Proposition 1
In the object assignment model where agents have strict preferences, for every agent i , let O ( ≻ i , ∅ ) denote the set of i ’s acceptable objects when his preference relation is % i .Below we define another invariance property of mechanisms that is weaker than strategy-proofness. An IR mechanism is size-invariant if when an agent changes preferences with-33ut changing the set of acceptable objects, the total probability mass of acceptable objectsassigned to the agent is invariant. Definition 9.
An IR mechanism ψ is size-invariant if, for all ≻ I ∈ R , all i ∈ I , and all % ′ i ∈P \{ % i } with O ( ≻ ′ i , ∅ ) = O ( ≻ i , ∅ ) , k ψ i ( ≻ ′ i , ≻ − i ) k = k ψ i ( ≻ I ) k . We will prove the following result in this appendix. Proposition 1 follows.
Lemma 8.
In the object assignment model with strict preferences, suppose | I | ≥ , | O | ≥ ,and q o = 1 for all o ∈ O . Let ψ be an IR, truncation-invariant and size-invariant mechanismthat satisfies ETE and ExPE, and ψ ′ be a size-invariant mechanism that strictly stochasticallydominates ψ . Then both ψ and ψ ′ are strongly bidominated. For any two mechanism ψ and ψ ′ in Lemma 8, we proceed by a few steps to find theallocations found by ψ for a few preference profiles. For one of them, the allocation foundby ψ is strongly bidominated, and ψ ′ finds the same allocation for the preference profile.So both ψ and ψ ′ are strongly bidominated.Specifically, let { , , , } denote four agents and { a, b, c } denote three objects. We con-sider preference profiles in which the agents in { , , , } do not accept objects other than { a, b, c } , and the other agents, if any, most prefer ∅ . So we essentially consider problemsconsisting of four agents and three objects.We first prove a claim that will be repeatedly used. Claim 1.
If there exist I ′ ⊆ { , , , } and O ′ ⊆ { a, b, c } such that every i ∈ I ′ prefers everyobject in O ′ to every object not in O ′ and | I ′ | ≥ | O ′ | , then in every IR and ExPE allocation p , for all o ∈ O ′ , k p o k = 1. Proof.
In any IR deterministic allocation p , if k p o k = 0 for some o ∈ O ′ , then there mustexist i ∈ I ′ who obtains an object that is worse than o , which means that p is not Paretoe ffi cient. So in any IR and Pareto e ffi cient deterministic allocation, all objects in O ′ mustbe exhausted. Thus, in any IR and ExPE allocation p , all objects in O ′ are exhausted.By Claim 1, if an object in { a, b, c } is most preferred by some agent, the object must beexhausted in any IR and ExPE allocation. Step 1.
We consider the following four preference profiles A , A ′ , A ′′ , and A ′′′ . We willfind allocations by ψ for them. For convenience, we omit unacceptable objects for eachagent and use o x to mean that the relevant agent obtains a probability share x of object o .34 % % % % a / a / a / c a b A ′ % % % % a / a / a / c b / a c b A ′′ % % % % a / a / a / a / b / b / b / c / c / c / A ′′′ % % % % a / a / a / c / b / b / a c / c / b We first derive the allocations found by ψ for A and A ′ . By Claim 1, a is exhausted inall four preference profiles. In A , ExPE implies that 4 obtains c . ETE implies that 1, 2 and3 each obtain 1 / a . Similarly, in A ′ , since 4 prefers c to a, b and 2 prefers a, b to c , ExPEimplies that 4 obtains c . By comparing A with A ′ , truncation-invariance implies that 2obtains 1 / a in A ′ . Since a is exhausted, ETE implies that 1 and 3 each obtain 1 / a . Then,ExPE implies that 2 obtains 2 / b .To derive the allocation for A ′′ , we prove Claim 2. Claim 2. In A ′′ each agent obtains 1 / a . Proof.
Consider the four preference profiles A ′′ , A ′′ , A ′′ and A ′′ . Note that A ′′ = A ′′ .We only consider the assignment of a . By Claim 1, a is exhausted in all four preferenceprofiles. In A ′′ , by ETE, all agents obtain 1 / a . By truncation-invariance, in A ′′ , 1 obtains1 / a . Then by ETE, each of the other agents also obtains 1 / a . By comparing A ′′ and A ′′ ,truncation-invariance implies that 2 obtains 1 / a in A ′′ . By ETE, 1 also obtains 1 / a . ByETE, 3 and 4 each obtain 1 / a . By comparing A ′′ and A ′′ , truncation-invariance impliesthat 4 obtains 1 / a in A ′′ . By ETE, 1 and 2 each obtain 1 / a in A ′′ . So 3 obtains 1 / a . A ′′ % % % % a / a / a / a / b / b / b / c / A ′′ % % % % a / a / a / a / b b / c / cA ′′ % % % % a / a / a / a / b b c / c c b A ′′ % % % % a / a / a / a / b b bb c c A ′′ , by Claim 1, all objects must be exhausted. By ETE, 1, 2 and 4 each obtain 1 / b and 1 / c . Note that the total amount of objects assigned to 4 is 11 / A ′′′ with A ′ , truncation-invariance implies that 1 obtains 1 / a in A ′′′ .By ETE, 2 also obtains 1 / a . By ExPE, 4 never obtains a and b . So 3 obtains 1 / a . ByClaim 1, b is exhausted, and, by ETE, 1 and 2 each obtain 1 / b . By comparing A ′′′ and A ′′ , size-invariance implies that 4 obtains 11 / c in A ′′′ . So 1 and 2 each obtain 1 / c . Step 2.
We consider the following three preference profiles B , B ′ , and B ′′ . B % % % % a / a / a / c / b / b / c / c / B ′ % % % % c / c / a / c / b / b / a / a / B ′′ % % % % a / c / a / c / b / b / c a By comparing B with A ′′′ , truncation-invariance implies that 4 obtains 11 / c in B . Byusing similar arguments as in Claim 2, 1 , , / a in B . By Claim 1, b and c must be exhausted. So 1 and 2 each obtain 1 / b and 1 / c . By symmetric arguments, weobtain the allocation for B ′ .In B ′′ , since 1 prefers b to c and c to a , 2 prefers c to b and b to a , and b is acceptableonly to 1 and 2, ExPE implies that 1 never obtains c and 2 never obtains a . By usingsimilar arguments as in Claim 2, we can show that in B ′′ , 1 and 3 each obtain 1 / a , and2 and 4 each obtain 1 / c . By comparing B ′′ with B , size-invariance implies that the totalamount of objects that 2 obtains is 7 /
8. So 2 obtains 3 / b . Symmetrically, by comparing B ′′ with B ′ , size-invariance implies that 1 obtains 3 / b . So in the allocation for B ′′ , b is notexhausted and the total amount of objects obtained by each of 1 and 2 is less than one.This allocation is strongly bidominated by another allocation where 1 and 2 each obtain1 / b . This means that ψ is strongly bidominated.Suppose ψ ′ is a size-invariant mechanism that strictly stochastically dominates ψ .Since all objects are exhausted in ψ ( B ) and ψ ( B ′ ), it must be that ψ ( B ) = ψ ′ ( B ) and ψ ( B ′ ) = ψ ′ ( B ′ ). Since ψ ′ ( B ′′ ) weakly stochastically dominates ψ ( B ′′ ), in ψ ′ ( B ′′ ), it must be that 1and 3 each obtain 1 / a , and 2 and 4 each obtain 1 / c . So in ψ ′ ( B ′′ ), 1 never obtains c and2 never obtains a . Because ψ ( B ) = ψ ′ ( B ) and ψ ( B ′ ) = ψ ′ ( B ′ ), by size-invariance of ψ ′ , 1 and2 each obtain 3 / b in ψ ′ ( B ′′ ). So ψ ′ ( B ′′ ) = ψ ( B ′′ ). It means that ψ ′ is strongly bidominated.36 Proof of Lemma 1
We first define some notations. For each o ∈ ˜ O ≔ O ∪ {∅} and each % i ∈ B , we define O ( % i , o ) ≔ { o ′ ∈ O : o ′ % i o } ; O ( ≻ i , o ) and O ( ∼ i , o ) are similar. We also define O ( Ï i , ∅ ) ≔ { o ∈ O : o ≻ i o ′ ≻ i ∅ for some o ′ ∈ O } . Let O ( % i ) , O ( % i ) , . . . , O K i ( % i ) be the partition of O ( ≻ i , ∅ )induced by ∼ i . So each O k ( % i ) is an indi ff erence class for i , and we make the conventionthat the objects in O k ( % i ) are better than those in O k +1 ( % i ). We suppress the dependenceof K i on % i . For every allocation p , O ′ ⊂ O and i ∈ I , we define p i [ O ′ ] ≔ P o ∈ O ′ p i,o .We prove the lemma by contradiction. Fix a preference profile % I and two IR alloca-tions p and p ′ such that p is unbidominated and k p ′ k > k p k . Define A ≔ { o ∈ O : k p ′ o k > k p o k} . Since k p ′ k > k p k , A is nonempty. For each o ∈ A , there exists i ∈ I such that p ′ i,o > p i,o .Suppose p and p ′ are upper-equivalent. It means that for all i ∈ I and all o ∈ O ( Ï i , ∅ ), P o ′ % i o p i,o ′ = P o ′ % i o p ′ i,o ′ . In other words, for all i and all 1 ≤ k < K i , p i [ O k ( % i )] = p ′ i [ O k ( % i )].We will proceed by a few steps to find a contradiction. Step one:
We generate a directed network according to the following procedure.
Step 1 : Let each o ∈ A point to each i ∈ I such that p ′ i,o > p i,o . It means that we createan edge o → i . Denote by I the set of agents who are pointed by an object in A . We haveexplained that I is nonempty.Let each i ∈ I point to each o ′ ∈ O such that (1) p ′ i,o ′ < p i,o ′ and (2) there exists o ∈ A such that o → i and o % i o ′ . It means that we create an edge i → o ′ . Denote by O the setof objects pointed by agents in I . It may happen that O ∩ A , ∅ . Step 2 : Let each o ∈ O point to each i ∈ I , if o has not pointed to i , such that p ′ i,o > p i,o .Denote the set of such i by I . It may happen that I ∩ I , ∅ . Let each i ∈ I point to each o ′ ∈ O , if i has not pointed to o ′ , such that (1) p ′ i,o ′ < p i,o ′ and (2) there exists o ∈ A ∪ O such that o → i and o % i o ′ . Denote the set of such o ′ by O . . . . Step k ≥
3: Let each o ∈ O k − point to each i ∈ I , if o has not pointed to i , such that p ′ i,o > p i,o . Denote the set of such i by I k . Let each i ∈ I k point to each o ′ ∈ O , if i has notpointed to o ′ , such that (1) p ′ i,o ′ < p i,o ′ and (2) there exists o ∈ A ∪ [ ∪ k − ℓ =1 O ℓ ] such that o → i and o % i o ′ . Denote the set of such o ′ by O k . Stop : The procedure stops when there are no new edges created in some step.Since there are finite agents and finite objects, the procedure must stop in finite steps.In the generated directed network, denote the set of agents by I ′ and the set of objects by O ′ . So A ⊆ O ′ . From the procedure we know that for every i ∈ I ′ , there exists an object37 ∈ A and a directed path from o to i , written as o → i → o → i → o → · · · → i n → o n → i, such that, for all ℓ ∈ { , . . . , n } , p ′ i ℓ ,o ℓ − > p i ℓ ,o ℓ − , p ′ i ℓ ,o ℓ < p i ℓ ,o ℓ , and o ℓ − % i ℓ o ℓ ( o = o ), and p ′ i,o n > p i,o n . If n = 0, o directly points to i . Step two:
We prove that for all i ∈ I ′ , k p i k = 1. Suppose for some i ∈ I ′ , k p i k < o ∈ A to i . Given that p and p ′ are IR, for all ℓ ∈{ , . . . , n } , both o ℓ − and o ℓ are acceptable to i ℓ because p ′ i ℓ ,o ℓ − > p i ℓ ,o ℓ >
0, and o n is acceptable to i because p ′ i,o n >
0. Since k p o k < k p ′ o k , it must be that k p o k < q o . So for asu ffi ciently small ǫ >
0, in the above path if we increase every p i ℓ ,o ℓ − by ǫ , decrease every p i ℓ ,o ℓ by ǫ , increase p i,o n by ǫ , and do not change the other probabilities, then we obtain anew allocation that bidominates p , contradicting the assumption that p is unbidominated. Step three:
We prove that O \ O ′ , ∅ . The fact that k p i k = 1 for all i ∈ I ′ impliesthat P i ∈ I ′ k p i k ≥ P i ∈ I ′ k p ′ i k . Because k p k < k p ′ k , we must have I \ I ′ , ∅ and P i ∈ I \ I ′ k p i k < P i ∈ I \ I ′ k p ′ i k . For every i ∈ I \ I ′ and every o ∈ O ′ , because o does not point to i , p i,o ≥ p ′ i,o . If O ′ = O , we should have P i ∈ I \ I ′ k p i k ≥ P i ∈ I \ I ′ k p ′ i k , which is a contradiction. Step four:
We prove that there exists o ∈ O \ O ′ such that o ∈ A ⊆ O ′ , which will be acontradiction. For every i ∈ I ′ , let o i be one of the best objects in O ′ such that o i → i . Thatis, there does not exist o ′ ∈ O ′ such that o ′ ≻ i o i and o ′ → i . For every o ∈ O \ O ′ , if o is notacceptable to i , then p i,o = p ′ i,o = 0, whereas if o is acceptable to i , then either o i % i o or o ≻ i o i . If o i % i o , because i does not point to o , it must be that p i,o ≤ p ′ i,o . So,(1) X o ∈ O \ O ′ : o i % i o p i,o ≤ X o ∈ O \ O ′ : o i % i o p ′ i,o . If o ≻ i o i , suppose o ∈ O k ( % i ) for some k . It must be that k < K i . For all o ′ ∈ O ′ ∩ O k ( % i ),since o ′ does not point to i (because o ′ ≻ i o i ), we must have p i,o ′ ≥ p ′ i,o ′ . So X o ′ ∈ O k ( % i ) ∩ O ′ p i,o ′ ≥ X o ′ ∈ O k ( % i ) ∩ O ′ p ′ i,o ′ . Recall that we have assumed that p i [ O k ( % i )] = p ′ i [ O k ( % i )], which is equivalent to X o ′ ∈ O k ( % i ) ∩ O ′ p i,o ′ + X o ′ ∈ O k ( % i ) ∩ ( O \ O ′ ) p i,o ′ = X o ′ ∈ O k ( % i ) ∩ O ′ p ′ i,o ′ + X o ′ ∈ O k ( % i ) ∩ ( O \ O ′ ) p ′ i,o ′ . P o ′ ∈ O k ( % i ) ∩ O ′ p i,o ′ ≥ P o ′ ∈ O k ( % i ) ∩ O ′ p ′ i,o ′ , we obtain X o ′ ∈ O k ( % i ) ∩ ( O \ O ′ ) p i,o ′ ≤ X o ′ ∈ O k ( % i ) ∩ ( O \ O ′ ) p ′ i,o ′ . Summarizing over all o ′ ∈ O k ( % i ) ∩ ( O \ O ′ ) for all k such that o ′ ≻ i o i , we obtain(2) X o ′ ∈ O \ O ′ : o ′ ≻ i o i p i,o ′ ≤ X o ′ ∈ O \ O ′ : o ′ ≻ i o i p ′ i,o ′ . Summarizing (1) and (2), we obtain X o ′ ∈ O \ O ′ p i,o ′ ≤ X o ′ ∈ O \ O ′ p ′ i,o ′ . Summarizing over all i ∈ I ′ , we obtain(3) X i ∈ I ′ X o ′ ∈ O \ O ′ p i,o ′ ≤ X i ∈ I ′ X o ′ ∈ O \ O ′ p ′ i,o ′ . We have proved that P i ∈ I \ I ′ k p i k < P i ∈ I \ I ′ k p ′ i k , which is equivalent to X i ∈ I \ I ′ X o ∈ O p i,o < X i ∈ I \ I ′ X o ∈ O p ′ i,o , or, X i ∈ I \ I ′ X o ∈ O ′ p i,o + X i ∈ I \ I ′ X o ∈ O \ O ′ p i,o < X i ∈ I \ I ′ X o ∈ O ′ p ′ i,o + X i ∈ I \ I ′ X o ∈ O \ O ′ p ′ i,o . For every i ∈ I \ I ′ and every o ∈ O ′ , because o does not point to i , p i,o ≥ p ′ i,o . So X i ∈ I \ I ′ X o ∈ O ′ p i,o ≥ X i ∈ I \ I ′ X o ∈ O ′ p ′ i,o . So we must have(4) X i ∈ I \ I ′ X o ∈ O \ O ′ p i,o < X i ∈ I \ I ′ X o ∈ O \ O ′ p ′ i,o . X i ∈ I X o ∈ O \ O ′ p i,o < X i ∈ I X o ∈ O \ O ′ p ′ i,o , or, X o ∈ O \ O ′ k p o k < X o ∈ O \ O ′ k p ′ o k . So there exists o ∈ O \ O ′ such that k p o k < k p ′ o k . But this means that o ∈ A ⊆ O ′ , which isa contradiction. C Examples for Section 4.2
In Example 3, we construct a mechanism that strictly size dominates PS. The mecha-nism is not truncation-invariant, but it is ordinally e ffi cient and envy-free. Example 3 (PS is strictly size dominated by an ordinally e ffi cient and envy-free mecha-nism) . There are four agents , , , and four objects a, b, c, d . Consider a mechanism ψ thatis di ff erent from PS in the preference profile % I in the following table, and coincides with PS inthe other preference profiles. In the table we use o x to mean that the relevant agent obtains aprobability share x of object o . Unacceptable objects are omitted from agents’ preference lists. % % % % a / a / b / b / c / c / c / c / d / d / a a (a) PS ( % I ) % % % % a / a / b / b / c / c / c c d / d / a a (b) ψ ( % I ) It is easy to see that ψ ( % I ) is ordinally e ffi cient and envy-free. So ψ is an ordinally e ffi cientand envy-free mechanism. From PS ( % I ) to ψ ( % I ) , and obtain more amounts of objects, and and obtain the same amounts of objects as before. So ψ strictly size dominates PS. But ψ is not truncation-invariant. If truncates preferences at c in % I , he will obtain the lottery (1 / b, / c ) , instead of the lottery (1 / b ) that is required by truncation-invariance. In Example 4, we construct an IR and unbidominated mechanism that is not truncation-invariant but is strictly size dominated by PS.40 xample 4 (An IR and unbidominated mechanism is strictly size dominated by PS) . Thereare four agents , , , and four objects a, b, c, d . Consider a mechanism ψ that is di ff erentfrom PS in the preference profile % I in the following table, and coincides with PS in the otherpreference profiles. In the table we use o x to mean that the relevant agent obtains a probabilityshare x of object o . Unacceptable objects are omitted from agents’ preference lists. % % % % a / a / b / b / c / c / c / c / d / d / a a (a) PS ( % I ) % % % % a / a / b / b / c c c / c / d d a a (b) ψ ( % I ) Because ψ ( % I ) is non-wasteful, ψ is unbidominated. From ψ ( % I ) to PS ( % I ) , and obtainmore amounts of objects, and and obtain the same amounts of objects as before. So ψ isstrictly size dominated by PS. But ψ is not truncation-invariant. If truncates preferences at c in % I , he will obtain the lottery (1 / b, / c ) , instead of the lottery (1 / b, / c ) that is requiredby truncation-invariance. References
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