Ordinal Bayesian incentive compatibility in random assignment model
aa r X i v : . [ ec on . T H ] S e p Ordinal Bayesian incentive compatibilityin random assignment model ∗ Sulagna Dasgupta and Debasis Mishra † September 29, 2020
Abstract
We explore the consequences of weakening the notion of incentive compatibility fromstrategy-proofness to ordinal Bayesian incentive compatibility (OBIC) in the randomassignment model. If the common prior of the agents is a uniform prior , then a largeclass of random mechanisms are OBIC with respect to this prior – this includes theprobabilistic serial mechanism. We then introduce a robust version of OBIC: a mech-anism is locally robust OBIC if it is OBIC with respect all independent priors in someneighborhood of a given independent prior. We show that every locally robust OBICmechanism satisfying a mild property called elementary monotonicity is strategy-proof.This leads to a strengthening of the impossibility result in Bogomolnaia and Moulin(2001): if there are at least four agents, there is no locally robust OBIC and ordinallyefficient mechanism satisfying equal treatment of equals.
Keywords. ordinal Bayesian incentive compatibility, random assignment, probabilis-tic serial mechanism.
JEL Code.
D47, D82 ∗ We are grateful to Sven Seuken, Timo Mennle, Arunava Sen, Dipjyoti Majumdar, Souvik Roy, andWonki Cho for their comments. † Dasgupta: University of Chicago, ; Mishra: Indian Statistical Institute, Delhi, [email protected] Introduction
The paper explores the consequences of weakening incentive compatibility from strategy-proofness to ordinal Bayesian incentive compatibility in the random assignment model (one-sided matching model). Ordinal Bayesian incentive compatibility (OBIC) requires thatthe truth-telling expected share vector of an agent first-order stochastically dominates theexpected share vector from reporting any other preference. This weakening of strategy-proofness in mechanism design models without transfers was proposed by d’Aspremont and Peleg(1988). We study OBIC by considering mechanisms that allow for randomization in the as-signment model.In the random assignment model, the set of mechanisms satisfying ex-post efficiencyand strategy-proofness is quite rich. Despite satisfying such strong incentive properties,all of them either fail to satisfy equal treatment of equals , a weak notion of fairness, or ordinal efficiency , a stronger but natural notion of efficiency than ex-post efficiency. Indeed,Bogomolnaia and Moulin (2001) propose a new mechanism, called the probabilistic serialmechanism , which satisfies equal treatment of equals and ordinal efficiency. However, theyshow that it fails strategy-proofness, and no mechanism can satisfy all these three propertiessimultaneously if there are at least four agents. A primary motivation for weakening thenotion of incentive compatibility to OBIC is to investigate if we can escape this impossibilityresult.We show two types of results. First, if the (common) prior is a uniform probabilitydistribution over the set of possible preferences, then every neutral mechanism satisfying amild property called elementary monotonicity is OBIC. An example of such a mechanismis the probabilistic serial mechanism. This is a positive result and provides a strategicfoundation for the probabilistic serial mechanism. In particular, it shows that there existsordinally efficient mechanisms satisfying equal treatment of equals which are OBIC withrespect to the uniform prior.Second, we explore the implications of strengthening OBIC as follows. A mechanism is Pycia and ¨Unver (2017) characterize the set of deterministic, strategy-proof, Pareto efficient, and non-bossy mechanisms in this model. This includes generalizations of the top-trading-cycle mechanism. Neutrality is a standard axiom in social choice theory which requires that objects are treated symmet-rically. Elementary monotonicity is a mild monotonicity requirement of a mechanism. We define it formallyin Section 4. ocally robust OBIC (LROBIC) with respect to an independent prior if it is OBIC with respectto every independent prior in its “neighborhood”. The motivation for such requirementof robustness in the mechanism design literature is now well-known, and referred to asthe Wilson doctrine (Wilson, 1987). We show that every LROBIC mechanism satisfyingelementary monotonicity is strategy-proof. An immediate corollary of this result is thatthe probabilistic serial mechanism is not LROBIC (though it is OBIC with respect to theuniform prior). As a corollary, we can show that when there are at least four agents, thereis no LROBIC and ordinally efficient mechanism satisfying equal treatment of equals. Thisstrengthens the seminal impossibility result of Bogomolnaia and Moulin (2001) by replacingstrategy-proofness with LROBIC.Both our results point to very different implications of OBIC in the presence of elementarymonotonicity – if the prior is uniform, this notion of incentive compatibility is very permissive;but if we require OBIC with respect to a set of independent priors in any neighborhood of agiven prior, this notion of incentive compatibility is very restrictive. As we discuss in Section6, such implications have been known for deterministic voting models (Majumdar and Sen,2004). But ours is the first paper to point this out for the random assignment model. Assignments.
There are n agents and n objects. Let N := { , . . . , n } be the set of agentsand A be the set of objects. We define the notion of a feasible assignment first. Definition An n × n matrix L is an assignment if L ia ∈ [0 , ∀ i ∈ N, ∀ a ∈ A X a ∈ A L ia = 1 ∀ i ∈ N X i ∈ N L ia = 1 ∀ a ∈ A For any assignment L , we write L i as the share vector of agent i . Formally, a share vector isa probability distribution over the set of objects. For any i ∈ N and any a ∈ A , L ia denotes All our results extend even if the number of objects is not the same as the number of agents. We assumethis only to compare our results with the random assignment literature, where this assumption is common.Also, whenever we say an assignment, we mean a random assignment from now on. i of object a . The second constraint of the assignment definition requiresthat the total share of every agent is 1. The third constraint of the assignment requires thatevery object is completely assigned. Let L be the set of all assignments.An assignment L is deterministic if L ia ∈ { , } for all i ∈ N and for all a ∈ A . Let L d be the set of all deterministic assignments. By the Birkohff-von-Neumann theorem, forevery L ∈ L , there exists a set of deterministic assignments in L d whose convex combinationgenerates L . Preferences.
The preference (a strict ordering) of an agent i over A will be denotedby P i . The set of all strict preferences over A is denoted by P . A preference profile is P ≡ ( P , . . . , P n ), and we will denote by P − i the preference profile P excluding the preference P i of agent i . Prior.
We assume that the preference of each agent is independently and identicallydrawn using a common prior µ , which is a probability distribution over P n . We will denoteby µ ( P i ) the probability with which agent i has preference P i . With some abuse of notation,we will denote the probability with which agents in N \ { i } have preference profile P − i as µ ( P − i ). Note that by independence, µ ( P − i ) = × j = i µ ( P j ). Our solution concept is Bayes-Nash equilibrium but we restrict attention to ordinal mecha-nisms, i.e., mechanisms where we only elicit ranking over objects from each agent. Hence,whenever we say mechanism , we refer to such ordinal mechanisms. Formally, a mechanism is a map Q : P n → L . A mechanism Q assigns a share vector Q i ( P ) to agent i at everypreference profile P .Before discussing the notions of incentive compatibility, it is useful to think how agentscompare share vectors in our model. Fix agent i with a preference P i over the set of objects A . Denote the k -th ranked object in P i as P i ( k ). Consider two share vectors π, π ′ . For every a ∈ A , we will denote by π a and π ′ a the share assigned to object a in π and π ′ respectively. The restriction to not consider cardinal mechanisms is arguably arbitrary. But it is consistent with theliterature on random assignment models. We do not know how the set of incentive compatible mechanismsexpand if we consider cardinal mechanisms.
4e will say π first-order-stochastically-dominates (FOSD) π ′ according to P i if ℓ X k =1 π P i ( k ) ≥ ℓ X k =1 π ′ P i ( k ) ∀ ℓ ∈ { , . . . , n } . In this case, we will write π ≻ P i π ′ . Notice that ≻ P i is not a complete relation over theoutcomes. An equivalent (and well known) definition of ≻ P i relation is that for every von-Neumann-Morgenstern utility representation of P i , the expected utility from π is at least asmuch as π ′ .The most standard notion of incentive compatibility is strategy-proofness (dominantstrategy incentive compatibility), which uses the FOSD relation to compare share vectors. Definition A mechanism Q is strategy-proof if for every i ∈ N , every P − i ∈ P n − ,and every P i , P ′ i ∈ P , we have Q i ( P i , P − i ) ≻ P i Q i ( P ′ i , P − i ) . The interpretation of this definition is that fixing the preferences of other agents, the truth-telling share vector must FOSD other share vectors that can be obtained by deviation. Thisdefinition of strategy-proofness appeared in Gibbard (1977) for voting problems, and has beenthe standard notion in the literature on random voting and random assignment problems.The ordinal Bayesian incentive compatibility notion is an adaptation of this by changingthe solution concept to Bayes-Nash equilibrium. It was first introduced and studied ina voting committee model in d’Aspremont and Peleg (1988), and was later used in manyvoting models (Majumdar and Sen, 2004). To define it formally, we introduce the notionof an interim share vector. Fix an agent i with preference P i . Given a mechanism Q , the interim share of object a for agent i by reporting P ′ i is: q ia ( P ′ i ) = X P − i ∈P n − µ ( P − i ) Q ia ( P ′ i , P − i ) . The interim share vector of agent i by reporting P ′ i will be denoted as q i ( P ′ i ). Definition A mechanism Q is ordinally Bayesian incentive compatible (OBIC) (with respect to prior µ ) if for every i ∈ N and every P i , P ′ i ∈ P , we have q i ( P i ) ≻ P i q i ( P ′ i ) . P P P ′ P P a ; a ; c ; c ; a ; c ; c ; b ; a ; 0 a ; b ; a ; b ; c ; 0 b ; b ; c ; 0 b ; Table 1: Manipulation by agent 1.It is immediate that if a mechanism Q is strategy-proof it is OBIC with respect to any prior.Conversely, if a mechanism is OBIC with respect to all priors (including correlated priors),then it is strategy-proof. We investigate a simple example to understand the implications of strategy-proofness andOBIC for the probabilistic serial mechanism. Suppose n = 3 with three objects { a, b, c } .Consider the preference profiles ( P , P , P ) and ( P ′ , P , P ) shown in Table 3.1 – the ta-ble also shows the share vector of each agent in the probabilistic serial mechanism ofBogomolnaia and Moulin (2001). In the probabilistic serial mechanism, each agent starts“eating” her favorite object simultaneously till the object is finished. Then, she moves tothe best available object according to her preference and so on. Table 3.1 shows the out-put of the probabilistic serial mechanism for preference profiles ( P , P , P ) and ( P ′ , P , P ).Since Q a ( P , P , P ) + Q c ( P , P , P ) > Q a ( P ′ , P , P ) + Q c ( P ′ , P , P ), we conclude that Q ( P ′ , P , P ) ⊁ P ′ Q ( P , P , P ). Hence, agent 1 can manipulate from P ′ to P , when agents2 and 3 have preferences ( P , P ).When can such a manipulation be prevented by OBIC? Note that P is generated from P ′ by permuting a and c . Suppose we permute P and P also to get P ′ and P ′ respectively: c P ′ b P ′ a and a P ′ c P ′ b. Since the probabilistic serial mechanism is neutral (with respect to objects), the share vectorof agent 1 at ( P ′ , P , P ) is an ( a, c ) permutation of its share vector at ( P , P ′ , P ′ ). Further,when all the preferences are equally likely, the probability of ( P , P ) is equal to the proba-bility of ( P ′ , P ′ ). So, the total expected probability of a and c for agent 1 at P and P ′ isthe same (where expectation is taken over ( P , P ) and ( P ′ , P ′ )). As we show below, this6rgument generalizes and the expected share vector at P ′ first-order-stochastic-dominatesthe expected share vector at P when the true preference is P ′ and prior is uniform. In this section, we present our first result which shows that the set of OBIC mechanisms ismuch larger than the set of strategy-proof mechanisms if the prior is the uniform prior. Aprior µ is the uniform prior if µ ( P i ) = |P| = n ! for each P i ∈ P . Uniform prior puts equalprobability on each of the possible preferences. We call a mechanism U-OBIC if it is OBICwith respect to the uniform prior.We show that there is a large class of mechanisms which are U-OBIC - this will includesome well-known mechanisms which are known to be not strategy-proof. This class is char-acterized by two axioms, neutrality and elementary monotonicity, which we define next. Todefine neutrality, consider any permutation σ : A → A of the set of objects. For everypreference P i , let P σi be the preference generated when the permutation σ is applied to P i .Let P σ be the preference profile generated by permuting each preference in the preferenceprofile P by the permutation σ . Definition A mechanism Q is neutral if for every P and every permutation σ , Q ia ( P ) = Q iσ ( a ) ( P σ ) ∀ i ∈ N, ∀ a ∈ A. Neutrality requires that objects be treated symmetrically by the mechanism. Any mechanismwhich does not use the “names” of the objects is neutral – this includes the random prior-ity mechanisms, the simultaneous eating algorithm mechanisms (including the probabilisticserial mechanism) in Bogomolnaia and Moulin (2001).Our next axiom is elementary monotonicity, an axiom which requires a mild form ofmonotonicity. To define it, we need the notion of “adjacency” of preferences. We say prefer-ences P i and P ′ i are adjacent if there exists a k ∈ { , . . . , n − } such that P i ( k ) = P ′ i ( k + 1) , P i ( k + 1) = P ′ i ( k ) , and P i ( k ′ ) = P ′ i ( k ′ ) ∀ k ′ / ∈ { k, k + 1 } . In other words, P ′ i is obtained by swapping consecutively ranked objects in P i . Here, if P i ( k ) = a and P i ( k + 1) = b , we say that P ′ i is an ( a, b )-swap of P i .7 efinition A mechanism Q satisfies elementary monotonicity if for every i ∈ N ,every P − i ∈ P n − , and every P i , P ′ i ∈ P such that P ′ i is an ( a, b ) -swap of P i , we have Q ib ( P ′ i , P − i ) ≥ Q ib ( P i , P − i ) Q ia ( P ′ i , P − i ) ≤ Q ib ( P i , P − i ) . In other words, as agent i lifts alternative b in ranking by one position by swapping itwith a (and keeping the ranking of every other object the same), elementary monotonicityrequires that the share of object b should weakly increase for agent i , while share of object a should weakly decrease. It is not difficult to see that elementary monotonicity is a necessarycondition for strategy-proofness. As we show later, elementary monotonicity is satisfied by avariety of mechanisms - including those which are not strategy-proof. However, every neutralmechanism satisfying elementary monotonicity is U-OBIC. Theorem Every neutral mechanism satisfying elementary monotonicity is U-OBIC.
Proof : Fix a neutral mechanism Q satisfying elementary monotonicity. The proof goes invarious steps. Step 1.
Pick an agent i and two preferences P i and P ′ i . Pick any k ∈ { , . . . , n } and suppose P i ( k ) = a and P ′ i ( k ) = b . We show that q ia ( P i ) = q ib ( P ′ i ). This is a consequence of uniformprior and neutrality. To see this, let P ′ i = P σi for some permutation σ of objects in A . Then, b = σ ( a ) and hence, for every P − i , we have Q ia ( P i , P − i ) = Q iσ ( a ) ( P σi , P σ − i ) = Q ib ( P ′ i , P σ − i ) . Due to uniform prior and using the above expression, q ia ( P i ) = 1 n ! X P − i Q ia ( P i , P − i ) = 1 n ! X P − i Q ib ( P ′ i , P σ − i ) = 1 n ! X P − i Q ib ( P ′ i , P − i ) = q ib ( P ′ i ) , where the third equality follows from the fact that { P − i : P − i ∈ P n − } = { P σ − i : P − i ∈ P n − } .In view of step 1, with some abuse of notation, we write q ik to denote the interim shareof the object at rank k in the preference. We call q i the interim rank vector of agent i . Step 2.
Pick an agent i and a preference P i . We show that q ik ≥ q i ( k +1) for all k ∈{ , . . . , n − } (i.e., interim shares are non-decreasing with rank). Fix a k and let P i ( k ) = a P i ( k + 1) = b . Then, consider the preference P ′ i , which is an ( a, b )-swap of P i . For every P − i , elementary monotonicity implies Q ia ( P i , P − i ) ≥ Q ia ( P ′ i , P − i ). Due to uniform prior, q ia ( P i ) ≥ q ia ( P ′ i ). But by Step 1, q ik = q ia ( P i ) ≥ q ia ( P ′ i ) = q i ( k +1) . Step 3.
We show that Q is OBIC with respect to uniform priors. Suppose agent i haspreference P i . By Steps 1 and 2, he gets interim rank vector ( q i , . . . , q in ) by reporting P i with q ij ≥ q ij +1 for all j ∈ { , . . . , n − } . Suppose she reports P ′ i = P σi , where σ is somepermutation of set of objects. By Steps 1 and 2, the interim share vector is a permutation ofinterim rank vector q i . Using non-decreasingness of this interim share vector with respectto ranks, we get q i ( P i ) ≻ P i q i ( P ′ i ). Hence, Q is OBIC with respect to uniform prior. (cid:4) Theorem 1 significantly generalizes, an analogous result in Majumdar and Sen (2004),who consider the voting problem and only deterministic mechanisms. They arrive at thesame conclusion as Theorem 1 in their model. Theorem 1 shows that their result holds evenin the random assignment problem.
Bogomolnaia and Moulin (2001) define a family of mechanisms, which they call the simul-taneous eating algorithms (SEA). Though the SEAs are not strategy-proof, they satisfycompelling efficiency and fairness properties, which we discuss in Section 5. We informallyintroduce the SEAs – for a formal discussion, see Bogomolnaia and Moulin (2001).Each SEA is defined by a (possibly time-varying) eating speed function for each agent.At every preference profile, agents simultaneously start “eating” their favorite objects at arate equal to their eating speed. Once an object is completely eaten (i.e., the entire shareof 1 is consumed), the amount eaten by each agent is the share of that agent of that object.Once an object completely eaten, agents go to their next preferred object and so on.If the eating speed of each agent is the same, then the simultaneous eating algorithm is anonymous . Bogomolnaia and Moulin (2001) call the unique anonymous SEA, the proba-bilistic serial mechanism. For axiomatic characterization of the PS mechanism, see Bogomolnaia and Heo (2012) andHashimoto et al. (2014). orollary Every simultaneous eating algorithm mechanism is U-OBIC.
Proof : Clearly, the SEAs are neutral. The SEAs also satisfy elementary monotonicity (Cho,2018; Mennle and Seuken, 2018). Hence, by Theorem 1, we are done. (cid:4)
While the uniform prior is an important prior in decision theory, it is natural to ask ifTheorem 1 extends to other “generic” priors. Though we do not have a full answer to thisquestion, we have been able to answer this question in negative under a natural robustnessrequirement. Our robustness requirement is local . Take any independent prior µ , and let µ ′ be any independent prior in the ǫ -radius ball around µ (where ǫ > || µ ( P ) − µ ′ ( P ) || < ǫ for all P ∈ P . In this case, we write µ ′ ∈ B ǫ ( µ ). Our local robustness requirement is thefollowing. Definition A mechanism Q is locally robust OBIC (LROBIC) with respect to anindependent prior µ if there exists an ǫ > such that for every independent prior µ ′ ∈ B ǫ ( µ ) , Q is OBIC with respect to µ ′ . It is well known that Bayesian incentive compatibility with respect to all priors lead tostrategy-proofness (Ledyard, 1978). Here, we require OBIC with respect to all independentpriors in the ǫ -neighborhood of an independent prior. Bhargava et al. (2015) study a versionof LROBIC with respect to uniform prior but their robustness also allows the mechanismto be OBIC with respect to correlated priors. They show that a large class of voting rulessatisfy their notion of OBIC. We show that in the random assignment model, LROBIC withrespect to any independent prior has a very different implication. Theorem A mechanism is LROBIC with respect to an independent prior and satisfieselementary monotonicity if and only if it is strategy-proof.
Proof : Every strategy-proof mechanism is OBIC with respect to any prior. A strategy-proof mechanism satisfies elementary monotonicity (Mishra, 2016). So, we now focus on See Theorem 3 and the discussions following Theorem 3 in Cho (2018) Q be an LROBIC mechanism with respect to an inde-pendent prior µ . Suppose Q satisfies elementary monotonicity. We do the proof in two steps. Step 1.
In this step, we decompose OBIC into three conditions. This decomposition issimilar to the decomposition of strategy-proofness in Mennle and Seuken (2018) – there aresome minor differences in axioms and we look at interim share vectors whereas they look atex-post share vectors.Our decomposition of OBIC uses the following three axioms.
Definition A mechanism Q satisfies interim elementary monotonicity if for every i ∈ N and every P i , P ′ i such that P ′ i is an ( a, b ) -swap of P i , we have q ib ( P ′ i ) ≥ q ib ( P i ) q ia ( P ′ i ) ≤ q ia ( P i ) . Give a preference ordering P i of agent i and an object a ∈ A , define U ( a, P i ) := { x ∈ A : x P i a } and L ( a, P i ) := { x ∈ A : a P i x } . Definition A mechanism Q satisfies interim upper invariance if for every i ∈ N andevery P i , P ′ i such that P ′ i is an ( a, b ) -swap of P i , and for every x ∈ U ( a, P i ) , we have q ix ( P ′ i ) = q ix ( P i ) . Definition A mechanism Q satisfies interim lower invariance if for every i ∈ N andevery P i , P ′ i such that P ′ i is an ( a, b ) -swap of P i , and for every x ∈ L ( b, P i ) , we have q ix ( P ′ i ) = q ix ( P i ) . The following proposition characterizes OBIC using these axioms.
Proposition A mechanism Q is OBIC with respect to a common independent prior if andonly if it satisfies interim elementary monotonicity, interim upper invariance, and interimlower invariance. Since the proof of Proposition 1 is similar to the characteration of strategy-proofness inMennle and Seuken (2018), we postpone its proof to the Supplementary Appendix A.11 tep 2.
Using Proposition 1, we now complete the proof. Pick an agent i ∈ N and P i , P ′ i ∈ P such that P ′ i is an ( a, b )-swap of P i . By Proposition 1, Q satisfies interim upperinvariance and interim lower invariance. Hence, we know that for all c / ∈ { a, b } , we get X P − i µ ( P − i ) h Q ic ( P i , P − i ) − Q ic ( P ′ i , P − i ) i = 0 . (1)Since µ is a probability distribution over P , we can treat it as a vector in R n ! − . Using µ ( P − i ) ≡ × j = i µ ( P j ), we note that the LHS of the Equation 1 is a polynomial function of { µ ( P ) } P ∈P . The equation describes the zero set of this polynomial function, which hasmeasure zero (Caron and Traynor, 2005). Hence, given any independent prior µ ∗ and ǫ > all µ ∈ B ǫ ( µ ∗ ) (which has non-zero measure), then Q ic ( P i , P − i ) = Q ic ( P ′ i , P − i ) for all c / ∈ { a, b } .Hence, for any LROBIC mechanism Q , for every i ∈ N , for every P − i , if we consider twopreferences P i and P ′ i such that P ′ i is an ( a, b ) swap of P i , we see that for all c / ∈ { a, b } , wehave Q ic ( P i , P − i ) = Q ic ( P ′ i , P − i ). Hence, we have shown that Q satisfies ex-post versions ofour interim lower invariance and interim upper invariance. Mennle and Seuken (2018) referto these properties as upper invariance and lower invariance (see also Cho (2018)). Theyshow that upper invariance, lower invariance, and elementary monotonicity are equivalent tostrategy-proofness. Since Q satisfies elementary monotonicity, it is strategy-proof. (cid:4) We now explore the compatibility of LOBIC and ordinal efficiency.
Definition A mechansim Q is ordinally efficient if at every preference profile P thereexists no assignment L such that L i ≻ P i Q i ( P ) ∀ i ∈ N, with L i = Q i ( P ) for some i . Bogomolnaia and Moulin (2001) show that every ordinally efficient mechanism is ex-postefficient but the converse is not true if n ≥
4. In fact, for n ≥
4, strategy-proofness isincompatible with ordinally efficiency along with the following weak fairness criterion.
Definition A mechanism Q satisfies equal treatment of equals if at every preferenceprofile P and for every i, j ∈ N , we have h P i = P j i ⇒ h Q i ( P ) = Q j ( P ) i Corollary Suppose n ≥ . Then, there is no locally robust OBIC and ordinally efficientmechanism satisfying equal treatment of equals. Proof : By Theorem 2, a locally robust OBIC mechanism satisfies ex-post versions of up-per invariance and lower invariance. Mennle and Seuken (2017) show that the proof inBogomolnaia and Moulin (2001) can be adapted by replacing strategy-proofness with ex-post versions of upper invariance and lower invariance. Hence, these two properties areincompatible with ordinal efficiency and equal treatment of equals for n ≥
4, and we aredone. (cid:4)
There is fairly large literature on random assignment problems. We briefly summarize themand relate them to our results. We also discuss the literature on ordinal Bayesian incentivecompatibility.
Ordinal Bayesian incentive compatibility.
Strategy-proofness is a demanding re-quirement in the voting models. Specially, when the domain of preferences is unrestricted,strategy-proofness and unanimity lead to dictatorship (Gibbard, 1973; Satterthwaite, 1975;Gibbard, 1977). Majumdar and Sen (2004) weaken strategy-proofness to OBIC in the votingmodels but restrict attention to deterministic mechanisms. They show that every determin-istic neutral voting mechanism satisfying elementary monotonicity is OBIC with respect touniform priors. Our Theorem 1 shows that this result generalizes to the random assignmentmodel. Majumdar and Sen (2004) also show that with “generic” priors, every deterministicOBIC mechanism satisfying unanimity is a dictatorship in the unrestricted domain. Mishra(2016) generalizes this result to some restricted domains of voting (like the single peakeddomain). He shows that in the deterministic voting model, elementary monotonicity andOBIC with respect to “generic” prior is equivalent to strategy-proofness in a variety of re-stricted domains – see also Hong and Kim (2018) for a strengthening of this result. Though13hese results are similar to our Theorem 2, there are significant differences. First, we con-sider randomization while these results are only for deterministic mechanisms. Our notionof locally robust OBIC is stronger than OBIC with respect to generic priors used in thesepapers. Second, ours is a model of private good allocation (random assignment), while thesepapers deal with the voting model.
Random assignment model.
The deterministic assignment model stands on the pillarsof two mechanisms: (a) serial dictatorship and (b) top trading cycle. These mechanisms areex-post efficient and strategy-proof. In fact, these mechanisms satisfy a stronger incentiveproperty called group strategy-proofness , which is equivalent to strategy-proofness and non-bossiness . For characterizations of deterministic mechanisms satisfying Pareto efficiency andgroup strategy-proofness, see Pycia and ¨Unver (2017); P´apai (2000); Svensson (1999).A natural motivation for studying random mechanisms is fairness. This motivates thestudy of random serial dictatorship or, as they are commonly called, the random prior-ity mechanisms. The random priority mechanisms are fair but they fail a natural efficiencycriteria called ordinal efficiency. Bogomolnaia and Moulin (2001) introduce a family of mech-anisms called simultaneous eating algorithms which generate ordianally efficient random as-signments. The probabilistic serial mechanism belongs to this family and it is anonymous.However, it is not strategy-proof. In fact, Bogomolnaia and Moulin (2001) show that thereis no ordinally efficient and strategy-proof mechanism satisfying equal treatment of equalswhen there are at least four agents. There is a small literature that provides strategic foundations to the probabilistic se-rial (PS) mechanism. Bogomolnaia and Moulin (2001) show that the PS mechanism sat-isfies weak -strategy-proofness. Their notion of weak strategy-proofness requires that themanipulation share vector cannot first-order-stochastic-dominate the truth-telling share vec-tor. Bogomolnaia and Moulin (2002) study a problem where agents have an outside option .When agents have the same ordinal ranking over objects but the position of outside optionin the ranking of objects is the only private information, they show that the PS mech-anism is strategy-proof. In fact, they show that the PS mechanism is characterized bystrategy-proofness, ordinal efficiency, and equal treatment of equals. This can be viewed Katta and Sethuraman (2006) extend the simultaneous eating algorithm to allow for ties in preferences. With three agent, the random priority mechanism satisfies these properties.
14s a domain restriction to achieve strategy-proofness of the PS mechanism. Other contri-butions in this direction include Liu (2019); Liu and Zeng (2019), who identifies domainswhere the probabilistic serial mechanism is strategy-proof. Che and Kojima (2010) showthat the PS mechanism and the random priority mechanism (which is strategy-proof) areasymptotically equivalent. Similarly, Kojima and Manea (2010) show that when sufficientlymany copies of an object are present, then the PS mechanism is strategy-proof. Thus, inlarge economies, the PS mechanism is strategy-proof. Balbuzanov (2016) introduce a notionof strategy-proofness which is stronger than weak strategy-proofness and show that the PSmechanism satisfies it. His notion of strategy-proofness is based on the “convex” dominationof lotteries, and hence, called convex strategy-proofness . Mennle and Seuken (2018) definea notion called partial strategy-proofness , which is weaker than strategy-proofness and showthat the PS mechanism satisfies it. They show that strategy-proofness is equivalent to up-per invariant, lower invariant and elementary monotonicity (they call it swap monotonicity).Their notion of partial strategy-proofness is equivalent to upper invariance and elementarymonotonicity, and hence, it is weaker than strategy-proofness. The main difference betweenthese weakenings of strategy-proofness and ours is that OBIC is a prior-based notion of in-centive compatibility. It is the natural analogue of Bayesian incentive compatibility in anordinal environment. Ehlers and Mass´o (2007) study OBIC in a two-sided matching prob-lem. Their main focus is on stable mechanisms. They characterize the beliefs for which astable mechanism is OBIC.
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A Supplementary Appendix
Proof of Proposition 1
Proof : The proof uses locally
OBIC.
Definition A mechanism Q is locally ordinally Bayesian incentive compatible(locally OBIC) with respect to an independent common prior µ if for every i ∈ N andevery P i , P ′ i ∈ P such that P ′ i is an ( a, b ) -swap of P i , we have q i ( P i ) ≻ P i q i ( P ′ i ) . Locally OBIC only considers a subset of incentive constraints. Using Carroll (2012); Cho(2016), we conclude that locally OBIC implies OBIC in our model. Hence, it is enoughto show that interim elementary monotonicity, interim upper invariance, and interim lowerinvariance are equivalent to locally OBIC.Suppose Q is locally OBIC. Fix agent i ∈ N and consider P i and P ′ i such that P ′ i is an( a, b )-swap of P i . interim upper invariance. If U ( a, P i ) = ∅ , then there is nothing to show. Else, let U ( a, P i ) = { a , . . . , a k } with a P i a P i . . . P i a k . Note that since P ′ i is an ( a, b )-swap of P i , we have a P ′ i a P ′ i . . . P ′ i a k . Using OBIC, we get q ia ( P i ) ≥ q ia ( P ′ i ) and q ia ( P ′ i ) ≥ q ia ( P i ) . (2)Hence, q ia ( P i ) = q ia ( P ′ i ). Now, suppose q ia j ( P i ) = q ia j ( P ′ i ) for all j ∈ { , . . . , ℓ − } , where ℓ ≤ k . Then, using OBIC again, we get ℓ X h =1 q ia h ( P i ) ≥ ℓ X h =1 q ia h ( P ′ i ) and ℓ X h =1 q ia h ( P ′ i ) ≥ ℓ X h =1 q ia h ( P i ) . P ℓh =1 q ia h ( P i ) = P ℓh =1 q ia h ( P ′ i ). But q ia j ( P i ) = q ia j ( P ′ i ) for all j ∈ { , . . . , ℓ − } implies that q ia ℓ ( P i ) = q ia ℓ ( P ′ i ). By induction, we conclude that Q satisfies interim upperinvariance. interim elementary monotonicity. Interim elementary monotonicity follows by con-sidering OBIC from P ′ i to P i : X x ∈ U ( a,P ′ i ) q ix ( P ′ i ) + q ib ( P ′ i ) ≥ X x ∈ U ( a,P ′ i ) q ix ( P i ) + q ib ( P i ) . But interim upper invariance implies that q ib ( P ′ i ) ≥ q ib ( P i ) as desired. A similar proof shows q ia ( P ′ i ) ≤ q ia ( P i ). interim lower invariance. By OBIC, X x ∈ U ( a,P i ) q ix ( P i ) + q ia ( P i ) + q ib ( P i ) ≥ X x ∈ U ( a,P i ) q ix ( P ′ i ) + q ia ( P ′ i ) + q ib ( P ′ i )Using interim upper invariance of Q , we get q ia ( P i ) + q ib ( P i ) ≥ q ia ( P ′ i ) + q ib ( P ′ i ) . The OBIC constraint from P ′ i to P i is similar and gives the other inequality and we conclude q ia ( P i ) + q ib ( P i ) = q ia ( P ′ i ) + q ib ( P ′ i ) . With this, we can repeat the argument of interim upper invariance to get interim lowerinvariance.The converse statement that any Q satisfying interim elementary monotonicity, interimupper invariance, and interim lower invariance is locally OBIC is straightforward, and leftto the reader. (cid:4)(cid:4)