Efficiency and strategy-proofness in object assignment problems with multi-demand preferences

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1 Efficiency and strategy-proofness in object assignment problems with multi-demand preferences Tomoya Kazumura and Shigehiro Serizawa February 7, 2016 Abstract We consider the problem of allocating sets of objects to agents and collecting payments. Each agent has a preference relation over the set of pairs consisting of a set of objects and a payment. Preferences are not necessarily quasi-linear. Non-quasi-linear preferences describe environments where the wealth effect is non-negligible: the payment level changes agents willingness to pay for swapping sets. We investigate the existence of efficient and strategy-proof rules. A preference relation is unit-demand if, given a payment level, receiving several objects are no better than receiving one of the them; it is multi-demand if, given a payment level, when an agent receives an object, receiving some additional object(s) makes him better off. We show that if a domain contains all the unit-demand preferences and at least one multi-demand preference relation, and if there are more agents than objects, then no rule satisfies efficiency, strategy-proofness, individual rationality, andno subsidy for losers on the domain. Keywords. Strategy-proofness, efficiency, multi-demand preferences, unit-demand preferences, non-quasi-linear preferences, minimum price Walrasian rule JEL Classification Numbers. D44, D71, D61, D82 The preliminary version of this article was presented at the IDGP 2015 Workshop, the 2015 Conference on Economic Design, and the II MOMA Meeting. The authors thank participants at those conferences for their comments. They are also grateful to seminar participants at Waseda University and Shanghai University of Finance and Economics for their comments. The authors specially thank William Thomson, Yuichiro Kamada and Ryan Tierney for their detailed comments. This research was supported by the Joint Usage/Research Center at ISER, Osaka University. The authors acknowledge the financial support from the Japan Society for the Promotion of Science (Kazumura, 14J05972; Serizawa, 15H03328). Graduate School of Economics, Osaka University 1-7 Machikaneyama, Toyonaka, Osaka , Japan. pge003kt@student.econ.osaka-u.ac.jp) Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki, Osaka , Japan. serizawa@iser.osaka-u.ac.jp 1

2 1 Introduction We consider an object assignment problem with money. Each agent receives a (possibly empty) set of objects and, possibly, pays money for the set. He has a preference relation over the set of pairs consisting of a set of objects and a payment. An allocation specifies how the objects are allocated and how much each agent pays. An (allocation) rule is a mapping from a class of admissible preference profiles, which we call a domain, to the set of allocations. An allocation is efficient if, without reducing the total payment, no other allocation makes all agents at least as well off and at least one agent better off. A rule is efficient if it always selects an efficient allocation. A rule is strategy-proof if, for each agent, it is a weakly dominant strategy to report his true preferences. We investigate the existence of efficient and strategy-proof rules. Our model can be viewed as a multi-object auction model. Much of the literature on auction theory assumes preferences to be quasi-linear. This means that the valuations over sets of objects are not affected by payment level. On the quasi-linear domain, the so-called VCG rules (Vickrey, 1961; Clarke, 1971; Groves, 1973) are efficient and strategy-proof, and they are the only rules satisfying these properties (Holmstöm, 1979). As Marshall (1920) demonstrates, preferences are approximately quasi-linear if payments are sufficiently low. However, in important applications of auction theory such as spectrum license allocation, house allocation, etc., prices are often equal to or exceed agents annual revenues. Excessive payments for objects may impair an agent s ability to purchase complements for an effective use of the objects, and thus may influence the benefit the agent derives from the objects. Another reason why preferences may not be quasi-linear is that an agent may need a loan to be able to pay high prices, and typically financial costs are nonlinear in borrowing. 12 Another common assumption is the unit-demand property. 3 It says that given a payment level, whenever an agent receives a set consisting of several objects, he finds it at most as good as one of them. For unit-demand preferences, there exists the minimum Walrasian equilibrium price, which is lower than any other Walrasian equilibrium price.thus, on the unit-demand domain, minimum price Walrasian (MPW) rules are well-defined. A minimum price Walrasian (MPW ) rule always selects an allocation associated with the minimum price Walrasian equilibria for each preference profile. The MPW rules are strategy-proof on the unitdemand domain (Demange and Gale, 1985). It is straightforward to see that the MPW rules satisfy the following additional two properties: One is individual rationality, which says that each agent finds his assignment at least as desirable as getting no object and paying nothing. The other is no subsidy for losers, which says that the payment of an agent who receives no object is nonnegative. On the unit-demand domain, when there are more agents than objects, the MPW rules are the only rules satisfying efficiency, strategy-proofness, individual rationality, and no subsidy for losers (Morimoto and Serizawa, 2015). 4 Although the unit-demand assumption is suitable in some important cases such as house allocation, etc., in many other cases, some agents may well wish to receive more than one 1 See Saitoh and Serizawa (2008) for numerical examples. 2 Ausubel and Milgrom (2002) also discuss the importance of the analysis under non-quasi-linear preferences. Also see Sakai (2008) and Baisa (2013) for more examples of non-quasi-linear preferences. 3 For example, see Andersson and Svensson (2014), Andersson et al. (2015), and Tierney (2015). 4 Note that the unit-demand domain contains non-quasilinear preferences, and thus the Hölmstrom (1979) result does not apply 2

3 object, and indeed, many authors have analyzed such situations. 5 Now, a natural question arises. On a domain that is neither quasi-linear nor unit-demand, do efficient and strategy-proof rules exist? This is the question we address. To state our result, we need an additional property of preferences. A preference relation satisfies the multi-demand property if, given a payment level, when an agent receives an object, receiving some additional object(s) makes him better off. We show that when there are more agents than objects, on any domain that includes the unit-demand domain and contains at least one multi-demand preference relation, no rule satisfies efficiency, strategy-proofness, individual rationality, and no subsidy for losers. This result shows the difficulty of designing efficient and strategy-proof rules unless a domain is contained in either the quasi-linear or the unit-demand domain. This article is organized as follows. In Section 2, we introduce the model and basic definitions. In Section 3, we define the minimum price Walrasian rule. In Section 4, we state our result and show the sketch of the proof. Section 5 concludes. All the proofs appear in Appendix. 2 The model and definitions There are n 2 agents and m 2 objects. We denote the set of agents by N {1,...,n} and the set of objects by M {1,...,m}. LetM be the power set of M. With abuse of notation, for each a M, wemaywritea to mean {a}. Each agent receives a subset of M and pays some amount of money. Thus, the agents common consumption set is M R and a generic (consumption) bundle for agent i is a pair z i =(A i,t i ) M R. Let0 (, 0). Each agent i has a complete and transitive preference relation R i over M R. LetP i and I i be the strict and indifference relations associated with R i. A typical class class of preferences is denoted by R. WecallR n a domain. The following are basic properties of preferences. Money monotonicity: For each A i Mand each pair t i,t i R with t i <t i,(a i,t i ) P i (A i,t i). First object monotonicity: For each ({a},t i ) M R, ({a},t i ) P i (,t i ). 6 Free disposal: For each (A i,t i ) M R and each A i Mwith A i A i,(a i,t i ) R i (A i,t i ). Possibility of compensation: For each (A i,t i ) M Rand each A i M,thereare t i,t i R such that (A i,t i ) R i (A i,t i)and(a i,t i ) R i (A i,t i ). Continuity: For each z i M R, theupper contour set at z i, UC i (z i ) {z i M R : z i R i z i },andthelower contour set at z i, LC i (z i ) {z i M R : z i R i z i}, are both closed. Definition 1 A preference relation is classical if it satisfies the five properties just defined. 7 Let R C be the class of classical preferences. We call (R C ) n the classical domain. Lemma 1 holds for classical preference. The proof is relegated to Appendix. 5 For example, see Gul and Stacchetti (1999, 2000), Bikhchandani and Ostroy (2002), Papai (2003), Ausubel (2004, 2006), Mishra and Parkes (2007), de Vries et al (2007), and Sun and Yang (2006, 2009, 2014). 6 This property is called desirability of objects in Morimoto and Serizawa (2015). 7 Morimoto and Serizawa (2015) define classical preferences as preferences satisfying the same properties except for free disposal, since in their model no agent can receive more than one object. 3

4 Lemma 1 Let R i R C, z i M R, and A i M. There is t i R such that z i I i (A i,t i ). For each R i R C,eachz i M R, andeacha i M,letV i (A i ; z i ) R be such that (A i,v i (A i ; z i )) I i z i.callv i (A i ; z i )thevaluation of A i at z i for R i. Bycontinuity,foreach R i R C,each(A i,t i ) M R, andeacha i M, V i (A i ;(A i,t i )) is continuous with respect to t i. By money monotonicity, for each R i R C and each pair (A i,t i ), (A i,t i) M R, (A i,t i ) R i (A i,t i) if and only if V i (A i;(a i,t i )) t i. Definition 2 A preference relation R i is quasi-linear if for each pair (A i,t i ), (A i,t i) M R and each t i R, (A i,t i ) I i (A i,t i) implies (A i,t i + t i ) I i (A i,t i + t i ). Let R Q be the class of classical and quasi-linear preferences. We call (R Q ) n the quasilinear domain. Obviously, R Q R C. Remark 1 Let R i R Q. Then, (i) there is a valuation function v i : M R + such that v i ( ) = 0, and for each pair (A i,t i ), (A i,t i) M R, (A i,t i ) R i (A i,t i) if and only if v i (A i) t i v i (A i ) t i,and (ii) for each (A i,t i ) M R and each A i M, V i (A i;(a i,t i )) t i = v i (A i) v i (A i ). Now we define important classes of preferences. The following property formalizes the notion that given a payment level, an agent desires to consume at most one object. Definition 3 A preference relation R i satisfies the unit-demand property if for each (A i,t i ) M R with A i > 1, there is a A i such that (a, t i ) R i (A i,t i ). 8 The condition says that for each payment level, if an agent receives a set consisting of several objects, there is an object in the set that makes him at least as good as receiving the set. Note that it is possible that when an agent with a unit-demand preferences receives an object, he is made better off by receiving an additional object without changing the payment level. However, an agent with a unit-demand preferences never find a set consisting several objects better than each object in the set. The unit-demand property is equivalent to the following: for each A i Mwith A i > 1, and each t i R, V i (A i ;(,t i )) = max a Ai V i (a;(,t i )). 9 Let R U be the class of classical and unit-demand preferences. We call (R U ) n the unit-demand domain. Obviously, R U R C. Remark 2 Let R i R U. By free disposal, for each (A i,t i ) M R with A i > 1, there is a A i such that (a, t i ) I i (A i,t i ) Figure 1 illustrates a unit-demand preference relation. ***** FIGURE 1 (Unit-demand preference relation) ENTERS HERE ***** We also consider a property that formalizes the notion that given a payment level, an agent desires to consume several objects. 8 Given a set X, X denotes the cardinality of X. 9 Gul and Stacchetti (1999) define the unit-demand property for quasi-linear preferences. In their model, a preference relation R i R Q satisfies the unit-demand property if for each A i Mwith A i > 1, v i (A i )= max a Ai v i (a). 4

5 Definition 4 A preference relation R i satisfies the multi-demand property if for each ({a},t i ) M R, there is A i Msuch that a A i and (A i,t i ) P i ({a},t i ). The condition says that given a payment level, when an agent receives an object, receiving some additional object(s) makes him better off. Note that given a payment level, even if an agent with a multi-demand preferences receives a set consisting of several objects, he may find it indifferent to each object in the set. The multi-demand property is equivalent to the following: for each a M and each t i R, there is A i Msuch that a A i and V i (a;(,t i )) <V i (A i ;(,t i )). Let R M be the class of classical and multi-demand preferences. We call (R M ) n the multi-demand domain. Remark 3 Let R i R M. By free disposal, for each ({a},t i ) M R, (M,t i ) P i ({a},t i ). Figure 2 illustrates a multi-demand preference relation. ***** FIGURE 2 (Multi-demand preference relation) ENTERS HERE ***** The following are examples of preferences satisfying the multi-demand property. Example 1: k-object-demand preferences. Given k {1,...,m}, a preference relation R i R C satisfies the k-object-demand property if (i) for each (A i,t i ) M R with A i <k,and each a M \ A i,(a i {a},t i ) P i (A i,t i ), and (ii) for each (A i,t i ) M R with A i k, there is A i A i with A i = k such that (A i,t i ) I i (A i,t i ). 10 Clearly, for each k {2,...,m}, preferences satisfying the k-object-demand property satisfy the multi-demand property. Example 2: Substitutes and complements. Suppose that the set of objects are divided into two non-empty sets K and L, and agent i with a preference relation R i views objects a and b as substitutes if both a and b are in the same set, and as complements if a and b are in different sets. For example, objects in K can be pens and objects in L can be notebooks. Formally, R i satisfies the following property: For each A i Mwith A i > 1andeacht i R, ifa i K or A i L, then there is a A i such that (A i,t i ) I i (a, t i ), and otherwise, for each a A i, (A i,t i ) P i (a, t i ). Clearly, this preference relation R i satisfies the multi-demand property. Some preferences in R C violate both of the unit-demand property and the multi-demand property. Example 3: (Figure 3) A preference relation violating the unit-demand property and the multidemand property. Let R i R C be such that for each a M and each t i R, V i (a;(,t i )) = t i +5,andforeachA i Mwith A i > 1, and each t i R, { t i +5 ift i 5, V i (A i ;(,t i )) = 1 (t 2 i + 5) otherwise. Then, for each pair a, b M and each t i R with t i < 5, V i ({a, b};(,t i )) = 1 2 (t i +5)> t i +5=V i (a;(,t i )) = V i (b;(,t i )), and thus, we have ({a, b},t i +5)P i (a, t i +5)I i (b, t i +5). Thus, R i does not satisfy the unit-demand property. Moreover, for each a M, eacha i M 10 In Gul and Stacchetti (1999), this notion is called k satiation 5

6 with a A i,andeacht i R with t i 5, V i (A i ;(,t i )) = t i +5=V i (a;(,t i )), and thus, we have (A i,t i +5)I i (a, t i +5). Thus,R i does not satisfy the multi-demand property. ***** FIGURE 3 (R i in Example 3) ENTERS HERE ***** An object allocation is an n-tuple A (A 1,,A n ) M n such that A i A j = for each i, j N with i j. We denote the set of object allocations by A. A(feasible) allocation is an n-tuple z (z 1,...,z n ) ((A 1,t 1 ),...,(A n,t n )) (M R) n such that (A 1,...,A n ) A. We denote the set of feasible allocations by Z. Given z Z, we denote the object allocation and the agents payments at z by A (A 1,...,A n )andt (t 1...,t n ), respectively, and we also write z =(A, t). A preference profile is an n-tuple R (R 1, R n ) R n.givenr R n and i N, let R i (R j ) j i. An allocation rule, or simply a rule on R n is a function f : R n Z. Given a rule f and R R n, we denote the bundle assigned to agent i by f i (R) and we write f i (R) =(A i (R),t i (R)). Now, we introduce standard properties of rules. An allocation z ((A i,t i )) i N Z is (Pareto-)efficient for R R n if there is no feasible allocation z ((A i,t i)) i N Z such that (i) for each i N, z i R i z i, (ii) for some j N,z j P i z j, and (iii) i N t i i N t i. The first property states that for each preference profile, a rule chooses an efficient allocation. Efficiency: For each R R n, f(r) is efficient for R. Remark 4 By money monotonicity and Lemma 1, the efficiency of allocation z is equivalent to the property that there is no allocation z ((A i,t i)) i N Z such that (i )foreachi N, z i I i z i,and(ii ) i N t i > i N t i. The second property states that no agent benefits from misrepresenting his preferences. Strategy-proofness: For each R R n,eachi N, andeachr i R, f i (R) R i f i (R i,r i ). The third property states that an agent is never assigned a bundle that makes him worse off than he would be if he had received no object and paid nothing. Individual rationality: For each R R n and each i N, f i (R) R i 0. The fourth property states that the payment of each agent is always nonnegative. No subsidy: For each R R n and each i N, t i (R) 0. The final property is a weaker variant of the fourth: If an agent receives no object, his payment is nonnegative. No subsidy for losers: For each R R n and each i N, ifa i (R) =, t i (R) 0. 3 Minimum price Walrasian rules In this section we define the minimum price Walrasian rules and state several facts related to them. Let p (p 1,...,p M ) R m + be a price vector. The budget set at p is defined as B(p) {(A i,t i ) M R : t i = a A i p a }.GivenR i R,thedemand set at p for R i is defined as D(R i,p) {z i B(p) :foreachz i B(p), z i R i z i}. 6

7 Lemma 2 Let R i R U and p R m +. (i) Suppose p R m ++. Then, for each (A i,t i ) D(R i,p), A i 1. (ii) Let A i Mbe such that (A i, a A i p a ) R i (A i, a A p a ) for each A i Mwith i A i 1. Then, (A i, a A i p a ) D(R i,p). Definition 5 Let R R n. A pair ((A, t),p) Z R m + is a Walrasian equilibrium (WE) for R if W-i: for each i N, (A i,t i ) D(R i,p), and W-ii: for each a M, if a/ A i for each i N, then, p a =0. Condition W-i says that each agent receives a bundle that he demands. Condition W-ii says that an object s price is zero if it is not assigned to anyone. Given R R n,letw (R) and P (R) be the sets of Walrasian equilibria and prices for R, respectively. Lemma 3 Let R (R U ) n and p P (R). (i) If n>m, then p a > 0 for each a M. (ii) There is ((A, t),p) W (R) such that A i 1 for each i N. In our model, each agent can receive several objects. On the other hand, some authors study a model in which each agent can receive at most one object. We call this model the unit-demand model. In the model, preferences are defined over M {0} R, where 0 means not receiving any object in M and is called null object. Money monotonicity, first object monotonicity, possibility of compensation, and continuity are defined in the unit-demand model in the same manner as defined in our model. In the unit-demand model, classical preferences are defined as preferences satisfying the four properties. Note that for each preference relation R i R U, there is a corresponding classical preference relation R i in the unit-demand model: for each pair ({a},t i ), ({b},s i ) M R, ({a},t i ) R i ({b},s i ) if and only if (a, t i ) R i (b, s i ); and for each pair ({a},t i ), (,s i ) M R, ({a},t i ) R i (,s i ) (resp. (,s i ) R i ({a},t i )) if and only if (a, t i ) R i (0,s i ) (resp. (0,s i ) R i (a, t i )). Let R (R U ) n and R be the preference profile in the unit-demand model which corresponds to R. For each p P (R), by (ii) of Lemma 3, there is an allocation (({a i },t i )) i N such that ((({a i },t i )) i N,p) W (R), and thus, (((a i,t i )) i N,p) is a WE for R. On the other hand, for each p R m +,ifpis a WE price vector for R, then there is an allocation ((a i,t i )) i N such that (((a i,t i )) i N,p) is a WE for R, and thus by (ii) of Lemma 2, ((({a i },t i )) i N,p) W (R). Therefore, the set of WE price vectors for R coincides with the set of WE price vectors for R. In the unit-demand model, several results on Walrasian equilibrium are established. By the preceding argument, the same results continue to hold in our model for preferences satisfying the unit-demand property. Fact 1 (Alkan and Gale, 1990) For each R (R U ) n, a Walrasian equilibrium for R exists. 11 Precisely, Alkan and Gale (1990) show the non-emptiness of the core in a two-sided matching model. However, the two-sided model includes the unit-demand model, and in the unit-demand model, non-emptiness of the core is equivalent to the existence of a Walrasian equilibrium. 12 Fact 1 is also shown by other authors. See, for example, Quinzi (1984), Gale (

8 Fact 2 (Demange and Gale, 1985) For each R (R U ) n, there is a unique minimum Walrasian equilibrium price vector, i.e., a vector p P (R) such that for each p P (R), p p. 13 A minimum price Walrasian equilibrium (MPWE) is a Walrasian equilibrium whose price is minimum. Given R R n,letp min (R) betheminimum Walrasian equilibrium price for R, andz W min(r) betheset of Walrasian equilibrium allocations associated with p min (R). Although there might be several minimum price Walrasian equilibria, they are indifferent for each agent, i.e., for each R R n, each pair z,z Z W min(r), and each i N, z i I i z i. Definition 6 A rule f on R n is a minimum price Walrasian (MPW) rule if for each R R n, f(r) Z W min(r). It is easy to show that the MPW rules on (R U ) n satisfy efficiency, individual rationality, and no subsidy. Demange and Gale (1985) show that the MPW rules are strategy-proof on the classical domain in the unit-demand model. Our arguments above allow us to convert each MPWE allocation in the unit-demand model into an MPWE for a unit-demand profile in our model. Moreover, all the minimum price Walrasian equilibria are indifferent for each agent. Thus, the result by Demange and Gale (1985) implies that the MPW rules on (R U ) n also satisfy strategy-proofness in our model. Morimoto and Serizawa (2015) show that in the unit-demand model, when n>m, only the MPW rules satisfy efficiency, strategy-proofness, individual rationality, andno subsidy for losers on the classical domain. The following lemma shows that in our model, when n>m, efficient rules never assign more than one object to agents whose preferences satisfy the unitdemand property. Lemma 4 (Single object assignment) Let n>m. Let R R U and let f be an efficient rule on R n.letr R n and i N. IfR i R U, A i (R) 1. By Lemma 4, when n>m, for each rule on (R U ) n satisfying efficiency, it always assigns each agent at most one object. Thus, when n>m, for each rule on (R U ) n satisfying efficiency, strategy-proofness, individual rationality, andno subsidy for losers, there is a corresponding rule in the unit-demand model, and moreover, it is easy to see that the corresponding rule also satisfies the four properties. Thus, the result by Morimoto and Serizawa (2016) continues to hold in our model. Fact 3 (Demange and Gale, 1985 for (i); Morimoto and Serizawa, 2015 for (ii)) (i) The minimum price Walrasian rules on (R U ) n satisfy efficiency, strategy-proofness, individual rationality and no subsidy. (ii) Let n>m. Then, the minimum price Walrasian rules are the only rules on (R U ) n satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers. 4 Main result In this section, first we state the main theorem. Next, we explain how we prove the theorem. 13 For each p, p R m, p p if and only if for each i {1,...,m}, p i p i. 8

9 4.1 Impossibility result We consider domains that result from adding multi-demand preferences to the unit-demand domain and we investigate whether efficient and strategy-proof rules still exist on such domains. In marked contrast to Fact 3 in Section 3, the results are negative. Namely, if there are more agents than objects, and if the domain includes the unit-demand domain and contains even a single multi-demand preference relation, then no rule satisfies efficiency, strategy-proofness, individual rationality and no subsidy for losers. Theorem Let n>m. Let R 0 R M and R be such that R R U {R 0 }. Then, no rule on R n satisfies efficiency, strategy-proofness, individual rationality and no subsidy for losers. Corollary Let n>m.letr = R U R M. Then, no rule on R n satisfies efficiency, strategyproofness, individual rationality and no subsidy for losers. Remark 5 The Corollary is a standard form of impossibility results on strategy-proofness in that since Gibbard (1973) and Satterthwaite (1975), many impossibility theorems on strategyproofness in this form are established. The Theorem can be applied to more variety of environments than Corollary. The Corollary cannot be applied unless all preferences in R M in addition to R U are deemed plausible. On the other hand, the Theorem can be applied once just one preference relation R 0 is arbitrarily chosen from R M in addition to R U is deemed plausible. For example, consider an environment where n =40,m = 20 and only preferences satisfying k-object-demand property for k {1, 2, 3, 4, 5}. The Theorem can be applied to this environment, but the Corollary cannot be. Remark 6 In this paper, we assume that preferences are drawn from a common class of R. If the preferences of each agent are drawn from a class R i that depends on the identity of the agent, our theorem can be strengthened as follows: Suppose that, for each i N, R i R U, and there is j N and R j R M such that R j R j. Then, no rule on i N R i satisfies efficiency, strategy-proofness, individual rationality and no subsidy for losers. Remark 7 In this paper, we assume preferences satisfy free disposal. However, this can be weakened without changing the result. Formally, let R C be a class of preferences satisfying money monotonicity, first object monotonicity, possibility of compensation, and continuity. And let R U be the class of preferences in R C satisfying the unit-demand properties, and R M the class of preferences in R C satisfying the multi-demand property. Then, all the lemmas and facts hold even if R U is replaced by R U, because the proofs do not depend on free disposal. Moreover, by slightly modifying the proof of the Theorem, we have the following result: If a class of preferences R is such that (i) R R U or R R U, and (ii) there is R 0 Rsuch that R 0 R M, then there is no rule on R n satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers. In footnotes, we explain how we modify the proof of the Theorem. 9

10 4.2 Sketch of the proof Preliminary results We state six lemmas which we use in the sketch of the proof and in the formal proof. The proof of each lemma is relegated to Appendix, or is omitted if it is straightforward. Let R R C be such that R R U. Let f be a rule on R n satisfying efficiency, strategyproofness, individual rationality and no subsidy for losers. Lemma 5 states that if an agent receives no object, then his payment is zero. This is immediate from individual rationality and no subsidy for losers. Thus we omit the proof. Lemma 5 (Zero payment for losers) Let R R n and i N. IfA i (R) =, t i (R) =0. Lemma 6 states that for each agent, his payment is at most the valuation, at 0, of the set of objects that he receives. This is immediate from individual rationality. Thus, we omit the proof. Lemma 6 For each R R n and each i N, t i (R) V i (A i (R); 0). Lemma 7 states that each object is assigned to some agent. This follows from efficiency, n>m, and first object monotonicity. We omit the proof. Lemma 7 (Full object assignment) For each R R n and each a M, there is i N such that a A i (R). Lemma 8 is a necessary condition for efficiency. Lemma 8 (Necessary condition for efficiency) Let R R n and i, j N with i j. Let A i,a j Mbe such that A i A j = and A i A j A i (R) A j (R). Then, V i (A i ; f i (R)) + V j (A j ; f j (R)) t i (R)+t j (R). By Lemma 4, Lemma 5, Lemma 8 and n > m, we can show that f satisfies no subsidy. Lemma 9 (No subsidy) For each R R n and each i N, t i (R) 0. Lemma 10 below states that f coincides with an MPW rule on (R U ) n. This is immediate from Fact 3 (ii). Thus we omit the proof. Lemma 10 For each R (R U ) n, f(r) Z W min(r). Given i N and R i R n 1, we define the option set of agent i for R i by o i (R i ) {z i M R : R i Rs.t. f i (R i,r i )=z i }. Lemma 11 states that (i) the option set does not contain more than one bundle with the same set of objects, and (ii) each agent receives one of the most preferred bundles in his option set. This is straightforward from strategy-proofness. Thus, we omit the proof. Lemma 11 Let i N and R i R n. (i) For each pair (A i,t i ), (A i,t i) o i (R i ),ifa i = A i, then t i = t i. (ii) For each R i Rand each z i o i (R i ), f i (R) R i z i. 10

11 4.2.2 Two-agent and three-object example Since the proof of the Theorem is very complicated, we relegate it to the Appendix. Here we demonstrate the ideas and techniques of the proof by applying them to a particular example in a two-agent and three-object setting. Let M = {a, b} and N = {1, 2, 3}. In the formal proof, the preference relation R 0 is an arbitrary element of R M, but here we pick R 0 from R Q R M. For concreteness, let v 0 (a) = 20, v 0 (a) = 18 and v 0 ({a, b}) =40. However, the idea of our proof does not depend on R 0 R Q. We assume R 0 R Q only for simplicity of expression. Let R R C be such that R R U {R 0 }. We suppose that there is a rule f on R 3 satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers, and derive a contradiction. Step A: Constructing a preference profile. Let R 1 = R 0. We construct R 2 R U and R 3 R U depending on R 1 so that a contradiction is derived. We define R 2 satisfying V 2 (a; 0) >v 1 ({a, b}) and V 2 (b; 0) V 2 ( ;(b, 0)) < min{v 1 ({a, b}) v 1 (a),v 1 ({a, b}) v 1 (b)}. 14 For example, let R 2 R U be such that for each A 2 M \ { }, 41 if A 2 = {a}, 9 if A 2 = {a}, V 2 (A 2 ; 0) = 12 if A 2 = {b}, V 2 (A 2 ;(, 1)) = 10 if A 2 = {b}, 41 if A 2 = {a, b}, 10 if A 2 = {a, b}, 1 if A 2 = {a}, 2 if A 2 = {a}, V 2 (A 2, (, 2)) = 9 if A 2 = {b}, V 2 (A 2 ;(, 3)) = 0 if A 2 = {b}, 9 if A 2 = {a, b}, 0 if A 2 = {a, b}. We define R 3 satisfying R 3 R U R Q,and v 3 (a) =v 3 (b) < 3 5 min{v 1({a, b}) v 1 (a),v 1 ({a, b}) v 1 (b)}. 14 In this sketch, we assume that M = {a, b} and R 0 R Q for the simplicity of expression. For a general multi-demand preference relation R 0, the RHS of the first inequality is set as the maximal value of the valuations of M at various t 1 in [0,V 1 (M; 0)]: max {t 1 V 1 ( ;(M,t 1 ))}. t 1 [0,V 1(M;0)] For a general multi-demand preference relation R 0, the RHS of the second inequality is set as the minimum value of the marginal valuations of the second object: min min{v 1({a, b};(a, t 1 )) t 1,V 1 ({a, b};(b, t 1 )) t 1 }. t 1 [0,V 1({a};0)] Moreover, for a general set M of objects, the RHS of the second inequality is defined as the minimum value of the maximal marginal valuations of additional sets of objects. Formally, it is defined as t 1 in the Appendix. 11

12 For example, let v 3 (a) =v 3 (b) = 9. Note that V 2 (a; 0) >v 3 ({a, b}). Let R (R 1,R 2,R 3 ). Figure 4 illustrates R. ***** FIGURE 4 (R =(R 1,R 2,R 3 )) ENTERS HERE ***** Step B:A 2 (R). Suppose by contradiction that A 2 (R) =. By Lemma 5, f 2 (R) =0. By Lemma 7, there is i 2suchthata A i (R). Let A i = A i (R) \{a} and A 2 = {a}. NotethatA i A 2 = and A i A 2 A i (R) A 2 (R). Also note that V i (A i ; f i (R)) + V 2 (A 2 ; f 2 (R)) = t i (R) v i (A i (R)) + v i (A i (R) \{a})+v 2 (a; 0) (by f 2 (R) =0) >t i (R)+t 2 (R) (by t 2 (R) =0&V 2 (a; 0) >v i ({a, b})) Thus, by Lemma 8, efficiency is violated, a contradiction. Step C: A 1 (R) =a. Substep C-1: (a, 9) o 1 (R 1 ). Let R 1 R U be such that V 1(a; 0) > min{v 2 (a; 0),V 3 (a; 0)} and V 1(b; 0) < min{v 2 (b; 0),V 2 (b; 0)}. For example, let R 1 R U be such that V 1(a; 0) = 50, V 1(b; 0) =1andV 1({a, b}; 0) =50. Since (R 1,R 1 ) (R U ) 3, by Lemma 10, f(r 1,R 1 ) Z W min(r 1,R 1 ). Let z Z be such that Figure 5 illustrates (R 1,R 1 )andz. z 1 =(a, 9), z 2 =(b, 9) and z 3 = 0. ***** FIGURE 5 ((R 1,R 1 )andz in Step C) ENTERS HERE ***** Let p (9, 9). Then, D(R 1,p)={{a}}, D(R 2,p)={{b}}, andd(r 3,p)={, {a}, {b}}. Thus (z,p) W (R 1,R 1 ), implying p min (R 1,R 1 ) p. Ifp a min(r 1,R 1 ) < 9orp b min(r 1,R 1 ) < 9, then 0 / D(R 1,p min (R 1,R 1 )) and for each i {2, 3}, 0/ D(R i,p min (R 1,R 1 )), which implies p min (R 1,R 1 ) / P (R 1,R 1 ), a contradiction. Thus, p min (R 1,R 1 )=(9, 9). Moreover, z is the only WE allocation supported by p min (R 1,R 1 ). Thus, f 1 (R 1,R 1 )=z, and hence, f 1 (R 1,R 1 )=(a, 9) o 1 (R 1 ). Substep C-2: (b, 10) o 1 (R 1 ). Let R 1 R U be such that V 1 (b; 0) > min{v 2 (b; 0),V 3 (b; 0)} and V 1 (a; 0) < min{v 2 (a; 0),V 2 (a; 0)}. 12

13 For example, let R 1 R U be such that V 1 (a; 0) =1,V 1 (b; 0) = 50 and V 1 ({a, b}; 0) =50. Since (R 1,R 1 ) (R U ) 3, by Lemma 10, f(r 1,R 1 ) Z W min(r 1,R 1 ). Let z Z be such that Figure 6 illustrates (R 1,R 1 )andz. z 1 =(b, 10), z 2 =(a, 9), and z 3 = 0. ***** FIGURE 6 ((R 1,R 1 )andz in Step C) ENTERS HERE ***** Let p (9, 10). Then, D(R 1,p ) = {{b}}, D(R 2,p ) = {{a}, {b}}, and D(R 3,p ) = {, {a}}. Thus(z,p ) W (R 1,R 1 ), implying p min (R 1,R 1 ) p.ifp a min(r 1,R 1 ) < 9, then 0 / D(R 1,p min (R 1,R 1 )) and for each i {2, 3}, 0/ D(R i,p min (R 1,R 1 )), which implies p min (R 1,R 1 ) / P (R 1,R 1 ), a contradiction. Thus, p a min(r 1,R 1 )=9. Ifp b min(r 1,R 1 ) < 10, then we have D(R 1,p min (R 1,R 1 )) = {{b}} and D(R 2,p min (R 1,R 1 )) = {{b}}, which further implies p min (R 1,R 1 ) / P (R 1,R 1 ), a contradiction. Thus, p min (R 1,R 1 )=(9, 10). Moreover, z is the only WE allocation which is supported by p min (R 1,R 1 ). Thus, f(r 1,R 1 )=z,and hence, f 1 (R 1,R 1 )=(b, 10) o 1 (R 1 ). Substep C-3: A 1 (R) =a. Since A 2 (R) by Step B, A 1 (R) 1. If A 1 (R) =, then by Lemma 5, we have f 1 (R) = 0, and thus, f 1 (R 1,R 1 ) P 1 f 1 (R), which contradicts strategy-proofness. Thus, A 1 (R) =a or b, and therefore, by (i) of Lemma 11, f 1 (R) =(a, 9) or (b, 10). Since (a, 9) P 1 (b, 10), (ii) of Lemma 11 implies f 1 (R) =(a, 9). Step D: f(r) is not efficient for R. By A 2 (R) and A 1 (R) =a, A 2 (R) =b. By Lemma 6, t 2 (R) V 2 (b; 0) = 12. By Lemma 9, t 2 (R) 0. Let z ((A i,t i )) i N Z be such that Figure 7 illustrates z. z 1 =({a, b}, 29), z 2 =(, 3), and z 3 = f 3 (R). ***** FIGURE 7 (z in Step D) ENTERS HERE ***** Since, V 1 ({a, b}; f 1 (R)) = V 1 ({a, b};(a, 9)) = 29, it is easy to see that z 1 I 1 f 1 (R) and z 3 I 3 f 3 (R). Also by t 2 (R) 0andA 2 (R) =b, z 2 =(, 3) I 2 (b, 0) R 2 f 2 (R). Moreover, by t 1 (R) =9andt 2 (R) 12, t i =29 3+t 3 (R) =26+t 3 (R) > t i (R), i N i M implying that f(r) isnotefficient for R, a contradiction. 13

14 We emphasize the difference between a (direct) proof of the Corollary that one might write, and the proof of the Theorem that we have shown. To prove the Corollary directly, we can freely pick preference profiles in R U R M to derive a contradiction. On the other hand, in the proof of the Theorem, we may only choose preferences from R U {R 0 }. Moreover, the preference relation R 0, which could be anything in R M, forces us to construct profiles depending on R 0, further complicating the process. In the above sketch, R 0 R M is assumed to be quasi-linear, but the basic logic of the sketch works even in the case R 0 R M \R Q. In the formal proof in the Appendix, we have six steps. Steps A, B, C, and D correspond to Steps 1, 3, 4, and 6, respectively, in the formal proof. Steps 2 and 5 in the formal proof are necessary only for the more general case, so they do not appear in the above sketch. 5 Concluding remarks In this article, we have considered an object assignment problem with money where each agent can receive more than one object. We focused on domains that include the unit-demand domain and contain some multi-demand preferences. We studied allocation rules satisfying efficiency, strategy-proofness, individual rationality, andno subsidy for losers, and showed that if the domain includes the unit-demand domain and contains at least one multi-demand preference relation, and there are more agents than objects, then no rule satisfies the four properties. As discussed in Section 1, we have been motivated by the search for efficient and strategy-proof rules on a domain which is not quasi-linear or unit-demand. Our result establishes the difficulty of designing efficient and strategy-proof rules on such a domain. We state two remarks on our result. Maximal domain. Some literature on strategy-proofness investigates the existence of maximal domains on which there are rules satisfying desirable properties. 15 A domain R n is a maximal domain for a list of properties if there is a rule on R n satisfying the properties, and for each R R, no rule on (R ) n satisfies the properties. Our result is rather closer to maximal domain results than impossibility results of the form of the Corollary. However, our result does not imply that the unit-demand domain is a maximal domain for the four properties in the Theorem, since we add only multi-demand preferences to the unit-demand domain and derive the non-existence of rules satisfying the four properties. In fact, what domains including (R U ) n are maximal domains for the four properties is an open question. However, we are sure that (R U ) n is not a maximal domain for the four properties. For example, consider R i in Example 3 and let R R U {R i }. Since R i does not satisfy the unit-demand property, R R U. Note that for each R j R,eachA j Mwith A j > 1and each t j R, ift j 0, then there is a A j such that (a, t j ) I j (A j,t j ). Thus, since each agent never pay negative amount of money under the minimum price Walrasian rules, and since they satisfy the four properties on the unit-demand domain, they also satisfy the four properties on R n. Hence, (R U ) n is not a maximal domain for the four properties. Although we do not find maximal domains for the four properties, the multi-demand class includes the most natural preferences outside the unit-demand class. Thus, our result implies that on the most of natural domains including the unit-demand domain, if there are more 15 For example, see Ching and Serizawa (1998), Berga and Serizawa (2000), Massó and Neme (2001), Ehlers (2002), etc. 14

15 agents than objects, we have an impossibility of designing rules satisfying the four properties. Identical objects. Some literature on object assignment problems also study the case in which the objects are identical. 16 In this paper, we do not make this assumption. When objects are not identical, the domain includes a greater variety of preference profiles than when objects are identical. This variety plays an important role in our proof. Therefore, our theorem does not exclude the possibility that when objects are identical, multi-demand preferences can be added to the unit-demand domain without preventing the existence of rules satisfy the four properties. Appendix: Proofs A Proofs of Lemmas Proof of Lemma 1: Since R i is complete, transitive, and continuous, there is a continuous utility function u i : M R R representing R i. By possibility of compensation, there are t i R and t i R such that (A i,t i) R i z i R i (A i,t i ), that is, u i (A i,t i) u i (z i ) u i (A i,t i ). If we have either (A i,t i) I i z i or z i I i (A i,t i ), then we are done. Thus, suppose (A i,t i) P i z i and z i P i (A i,t i ), that is, u i (A i,t i) >u i (z i ) >u i (A i,t i ). Since u i is continuous, u i (A i, ) is continuous in R. Moreover, by money monotonicity, t i <t i. Since [t i,t i ] is a closed interval, intermediate value theorem implies that there is t i (t i,t i ) such that u i (A i,t i )=u i (z i ), that is (A i,t i ) I i z i. Proof of (i) of Lemma 2: Suppose there is (A i,t i ) D(R i,p) such that A i > 1. By R i R U, there is a A i such that (a, t i ) R i (A i,t i ). By (A i,t i ) B(p), t i = b A i p b.byp R m ++ and A i > 1, p a < b A i p b = t i. Thus, by money monotonicity, (a, p a ) P i (a, t i ) R i (A i,t i ), which contradicts (A i,t i ) D(R i,p). Proof of (ii) of Lemma 2: Let A i M.If A i 1, then by the def. of A i,(a i, a A i p a ) R i (A i, a A p a ). Suppose A i > 1. By R i R U, there is a A i such that (a, p a ) R i i (A i, b A p b ). By p a 0 and by money monotonicity, (a, p a ) R i (a, i b A p b ) R i (A i, i b A p b ). i Thus, by the def. of A i, (A i, p b ) R i (a, p a ) R i (A i, p b ). b A i b A i Thus, (A i, b A i p b ) D(R i,p). Proof of (i) of Lemma 3: By contradiction, suppose that n>mand p a =0forsomea M. Then, by money monotonicity, for each i N, (a, p a ) P i 0. Thus, for each i N, 0 / D(R i,p), 16 For example, see Saitoh and Serizawa (2008), Ashlagi and Serizawa (2012), Adachi (2014), etc. 15

16 which implies that for each ((A, t),p) W (R), A i. However, this contradicts n>m. Proof of (ii) of Lemma 3: By p P (R), there is (A, t) Z such that ((A, t),p) W (R). Let N = {i N : A i > 1} and i N. By R i R U, there is a A i such that (a, t i ) R i (A i,t i ). By (A i,t i ) B(p), t i = b A i p b.byp R m +, p a b A i p b = t i. Thus, by money monotonicity, (a, p a ) R i (a, t i ) R i (A i,t i ), which implies (a, p a ) D(R i,p). Hence, for each i N, there is a i A i such that (a i,p a i ) D(R i,p). Let z Z be such that for each i N, z i =(a i,p a i ), and for each i N \N, z i =(A i,t i ). Then, each agent receives at most one object at z, and clearly, (z,p) W (R). Proof of Lemma 4: Suppose by contradiction that R i R U and A i (R) > 1. Then, there is a A i (R) such that (a, t i (R)) I i f i (R). By A i (R) > 1, we can take b A i (R) \{a}. By n>m, there is j N \{i} such that A j (R) =. Let z ((A k,t k )) k N Z be such that z i =(a, t i (R)), z j =(b, t j (R)), and z k = f k (R) foreachk N \{i, j}. Clearly, k N t k = k N t k(r), and for each k \{i, j}, z k I k f k (R). Moreover, z i = (a, t i (R)) R i f i (R), and by first object monotonicity, z j =(b, t j (R)) P j f j (R). This contradicts efficiency. Proof of Lemma 8: Let R R C be such that R R U,andf a rule on R n satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers. Suppose by contradiction that V i (A i ; f i (R)) + V j (A j ; f j (R)) >t i (R) +t j (R). Let z Z be such that z i =(A i,v i (A i ; f i (R))), z j =(A j,v j (A j ; f j (R))), and for each k N \{i, j}, z k = f k(r). Then z k I k f k (R) foreachk N. Moreover, V i (A i ; f i (R)) + V j (A j ; f j (R)) + k i,j t k(r) > k N t k(r). By Remark 4, this contradicts efficiency. Proof of Lemma 9: (Figure 8.) Let R R C be such that R R U,andf a rule on R n satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers. Suppose by contradiction that t i (R) < 0forsomeR R n and i N. Let vi R ++ be such that vi < min a M min j N V j (a; 0). Let R i R U R Q be such that for each A i M\{ }, v i(a i )=vi. First, suppose A i (R i,r i )=. By Lemma 5, f i (R i,r i )=0. Thus, by v i(a i (R)) > 0 > t i (R) and money monotonicity, f i (R) =(A i (R),t i (R)) P i (A i (R),v i(a i (R))) I i 0 = f i (R i,r i ). This contradicts strategy-proofness. Hence, A i (R i,r i ) By R i R U, A i (R i,r i ), and Lemma 4, there is a M such that A i (R i,r i )=a. Since n>mand A i (R i,r i ), there is j N \{i} such that A j (R i,r i )=. By Lemma 5, 16

17 f j (R i,r i )=0. Then, V i ( ; f i (R i,r i )) + V j (a; f j (R i,r i )) = V i ( ; f i (R i,r i )) + V j (a; 0) (by f j (R i,r i )=0) >V i ( ; f i (R i,r i )) + v i(a) (by the def. of R i) = t i (R i,r i ) v i(a)+v i(a) (by (ii) of Remark 1) = t i (R i,r i ) = t i (R i,r i )+t j (R i,r i ). (by t j (R i,r i )=0) This contradicts Lemma 8. ***** FIGURE 8 (Illustration of proof of Lemma 9)) ENTERS HERE ***** B Proof of Theorem The proof of the Theorem has six steps. Step 1: Constructing a preference profile. Let R 1 R 0. Claim 1: There is t 1 R such that t 1 > 0 and t 1 = min a M min t 1 [0,V 1 ({a};0)] {V 1 (M;(a, t 1 )) t 1 }. 17 Proof: Note that for each a M, V 1 (M;(a, t 1 )) t 1 is continuous with respect to t For each a M, since [0,V 1 (a; 0)] is compact, there is ˆt a 1 [0,V 1 (a; 0)] such that V 1 (M;(a, ˆt a 1)) ˆt a 1 = min t1 [0,V 1 (a;0)]{v 1 (M;(a, t 1 )) t 1 }.Foreacha M, lett a 1 min t1 [0,V 1 (a;0)]{v 1 (M;(a, t 1 )) t 1 }. Since M is finite, min{t a 1 : a M} exists. Let t 1 min{t a 1 : a M}. Then, t 1 = min a M min t 1 [0,V 1 (a;0)] {V 1 (M;(a, t 1 )) t 1 }. Next, we show t 1 > 0. Let a M be such that t 1 = V 1 (M;(a, ˆt a 1)) ˆt a 1.ByRemark3, t 1 = V 1 (M;(a, ˆt a 1)) ˆt a 1 > As we discussed in Remark 7, the Theorem holds even without free disposability. To prove the Theorem without free disposability, t 1 needs to be defined as: t 1 =min min a M t 1 [0,V 1({a};0)] A 1 M max {V 1 (A 1 ;(a, t 1 )) t 1 }. a A 1 18 Note that Berge s maximum theorem implies that for each a M, max A1 M V 1 (A 1 ;(a, t 1 )) t 1 is continuous with respect to t 1 (See Berge,1963). Thus, the same argument holds for the case without free a A 1 disposability, 17

18 Claim 2: There is t 1 R such that t 1 t 1 and t 1 = max {t 1 V 1 ( ;(M,t 1 ))}. 19 t 1 [0,V 1 (M;0)] Proof: Note that t 1 V 1 ( ;(M,t 1 )) is continuous with respect to t 1. Since [0,V 1 (M; 0)] is compact, there exist ˆt 1 [0,V 1 (M; 0)] and t 1 R such that t 1 = ˆt 1 V 1 ( ;(M, ˆt 1 )) = max {t 1 V 1 ( ;(M,t 1 ))}. t 1 [0,V 1 (M;0)] Next, we show t 1 t 1.Leta M and t 1 [0,V 1 (a; 0)] be such that t 1 = V 1 (M;(a, t 1 )) t 1. Let ˆt 1 V 1 (M;(a, t 1 )). By t 1 [0,V 1 (a; 0)], ˆt 1 [0,V 1 (M; 0)]. Since V 1 ( ;(M, ˆt 1 )) = V 1 ( ;(a, t 1 )) and since first object monotonicity implies t 1 V 1 ( ;(a, t 1 )) 0, Thus, ˆt 1 V 1 ( ;(M, ˆt 1 )) = V 1 (M;(a, t 1 )) t 1 + t 1 V 1 ( ;(a, t 1 )) V 1 (M;(a, t 1 )) t 1. t 1 = max {t 1 V 1 ( ;(M,t 1))} t 1 [0,V 1(M;0)] ˆt 1 V 1 ( ;(M, ˆt 1 )) V 1 (A 1 ;(a, t 1 )) t 1 = t 1. Let d R ++ be such that d > t 1. By Claim 1, there is d R ++ such that 5(m 1)d <t 1 and for each a M, (a, 3d ) P 1 0. Note that by Claim 2, 5(m 1)d <t 1 t 1 <d. Let a M be such that for each a M \{a },(a, 3d ) R 1 (a, 3d ). Without loss of generality, assume a =1. For each i {2,...,m}, letr i R U satisfy the following conditions: d if a = i 1, V i (a; 0) = 4d if a = i, (ia) otherwise, d V i ( ;(i, 0)) = d, and for each a M \{i},v i (a;(i, 3d )) < 0. For each i N \{1,...,m}, letr i R U R Q be such that for each a M, v i (a) =3d. (ib) (ic) 19 To prove the Theorem without free disposability, t 1 needs to be defined as: t 1 = max A 1 M\{ } max {t 1 V 1 ( ;(A 1,t 1 ))}. t 1 [0,V 1(A ;0)] 18

19 Denote R (R 1,...,R n ). Figure 9 illustrates R i and R j,wherei {2,...,m} and j {m +1,...,n}. ***** FIGURE 9 (Illustration of R i (i {2,...,m}) andr j (j {m +1,...,m}).) ENTERS HERE ***** Step 2: For each i {2,...,m}, V i ( ; f i (R)) d. Let i {2,...,m}. If A i (R) =, then by Lemma 5, V i ( ; f i (R)) = t i (R) =0> d. Suppose A i (R). ByR i R U and Lemma 4, there is a M such that A i (R) =a. If a = i, then by Lemma 9 and money monotonicity, (i, 0) R j f j (R), and thus, by ib, V i ( ; f i (R)) V i ( ;(i, 0)) = d.ifa i, thenbyic and Lemma 9, V i (a;(i, 0)) <V i (a;(i, 3d )) < 0 t i (R), which implies (i, 0) P i f i (R), and thus, by ib V i ( ; f i (R)) >V i ( ;(i, 0)) = d. Step 3: For each i {2,...,m}, A i (R). Suppose by contradiction that there is an agent i {2,...,m} such that A i (R) =. By Lemma 5, t i (R) = 0. By Lemma 7, there is an agent j N \{i} such that i 1 A j (R). We show the following claim. Claim 1: t j (R) V j ( ; f j (R)) <d. Proof: We have three cases. Case 1: j =1. 20 By free disposal and individual rationality, (M,t 1 (R)) R 1 f 1 (R) R i 0, which implies t 1 (R) V 1 (M; 0). Thus, by Lemma 9, t 1 (R) [0,V 1 (M; 0)]. Therefore, by the def. of t 1, t 1 (R) V 1 ( ;(M,t 1 (R))) t 1. Moreover, (M,t 1 (R)) R 1 f 1 (R) implies V 1 ( ; f 1 (R)) V 1 ( ;(M,t 1 (R))). Thus, t 1 (R) V 1 ( ; f 1 (R)) t 1 (R) V 1 ( ;(M,t 1 (R))) t 1 <d. Case 2: j {2,...,m}. By R j R U and Lemma 4, A j (R) =i 1. By Lemma 6, t j (R) V j (i 1; 0). Since j 1 i 1, V j (i 1; 0) =4d or d.thus,v j (i 1; 0) 4d.Moreover, by Step 2, V j ( ; f j (R)) d. Therefore, by 5d < 5(m 1)d <d, t j (R) V j ( ; f j (R)) V j (i 1; 0)+d 5d <d. Case 3: j {m +1,...,n}. By the def. of R j and 3d < 5(m 1)d <d, t j (R) V j ( ; f j (R)) = v j (i 1) = 3d <d. 20 Without free disposability, the argument of Case 1 is modified as follows: By Lemma 6 and Lemma 9, t 1 (R) [0,V 1 (A 1 (R); 0)]. Thus, by the def. of t 1, t 1 (R) V 1 ( ; f 1 (R)) t 1 <d. 19

20 By f i (R) =0, V i (i 1; f i (R)) = V i (i 1; 0) =d. Thus, by Claim 1, V i (i 1; f i (R)) + V j ( ; f j (R)) >d + t j (R) d = t i (R)+t j (R). This contradicts Lemma 8. Step 4: A 1 (R) =1. First we show the following claim. Claim 1: For each a M, there is t 1 (a) R such that (a, t 1 (a)) o 1 (R 1 ) and { 3d if a =1, t(a) > 3d otherwise. Proof: Let a M and R 1 R U be such that Figure 10 illustrates R 1. V 1(a; 0) > max i(a; 0), i N (1a ) for each b M \{a},v 1(b; 0) < min i(b; 0), and i N (1b ) for each b M \{a},v 1(b;(a, 3d )) < 0. (1c ) ***** FIGURE 10 (Illustration of R 1) ENTERS HERE ***** By (R 1,R 1 ) (R U ) n and Lemma 10, f(r 1,R 1 ) Z W min(r 1,R 1 ). Let p p min (R 1,R 1 ). By (R 1,R 1 ) (R U ) n and Lemma 4, for each i N, A i (R 1,R 1 ) 1. In the following three paragraphs, we show (a, p a ) o 1 (R 1 ). First, suppose A 1 (R 1,R 1 )=. By Lemma 5, f 1 (R 1,R 1 )=0. By A 1 (R 1,R 1 )= and Lemma 7, there is an agent i N \{1} such that A i (R 1,R 1 )=a. Since f(r 1,R 1 ) Z W min(r 1,R 1 ), t i (R 1,R 1 )=p a. By Lemma 6, p a = t i (R) V i (a; 0). Since f(r 1,R 1 ) Z W min(r 1,R 1 ), f 1 (R 1,R 1 )=0 D(R 1,p), and thus, 0 R 1 (a, p a ). Therefore, V 1(a; 0) p a V i (a; 0). This contradicts 1a. Hence, A 1 (R 1,R 1 ). Next, suppose that for some b M\{a}, A 1 (R 1,R 1 )=b. Since f(r 1,R 1 ) Zmin(R W 1,R 1 ), t 1 (R 1,R 1 )=p b. By Lemma 6, p b V 1(b; 0). By A 1 (R 1,R 1 ) and n>m, there is an agent i N \{1} such that A i (R 1,R 1 )=. By Lemma 5, f i (R 1,R 1 )=0. Since f(r 1,R 1 ) Zmin(R W 1,R 1 ), f i (R 1,R 1 )=0 D(R i,p), and thus, 0 R i (b, p b ). Therefore, V i (b; 0) p b V 1(b; 0). This contradicts 1b.Thusforeachb M \{a}, A 1 (R 1,R 1 ) b. 20