Efficiency and Strategy-Proofness in Object Assignment Problems with Multi Demand Preferences

Consider the problem of allocating objects to agents and how much they should pay. Each agent has a preference relation over pairs of a set of objects and a payment. Preferences are not necessarily quasi-linear. Non-quasi-linear preferences describe environments where payments influence agents' abilities to utilize objects. This paper is to investigate the possibility of designing efficient and strategy-proof rules in such environments. A preference relation is single demand if an agent wishes to receive at most one object; it is multi demand if whenever an agent receives one object, an additional object makes him better off. We show that if a domain contains all the single demand preferences and at least one multi demand preference relation, and there are more agents than objects, then no rule satisfies efficiency, strategy-proofness, individual rationality, and no subsidy for losers on the domain.


Introduction
Consider an object assignment model with money. Each agent receives a set of objects, pays money, and has a preference relation over a set of objects and a payment. An allocation specifies how the objects are allocated to the agents and how much they pay. An (allocation) rule is a mapping from the class of admissible preference profiles, which we call "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. Strategy-proofness is a condition of incentive compatibility. It requires that each agent should have an incentive to report his true preferences. This paper is to investigate the possibility of designing efficient and strategyproofness rules.
Our model can be treated as one of the multi-object auction models. Much literature on auction theory makes an assumption on preferences, "quasi-linearity." It states that valuations over objects are not affected by payment level. On the quasi-linear domain, i.e., the class of quasi-linear preferences, rules so-called "VCG rules" (Vickrey, 1961;Clarke, 1971; Groves, 1973) satisfy efficiency and strategy-proofness, and they are only rules satisfying those properties (Holmstöm, 1979).
As Marshall (1920) demonstrates, preferences are approximately quasi-linear if payments for goods are sufficiently low. However, in important applications of auction theory such as spectrum license allocations, house allocations, etc., prices are often equal to or exceed agents' annual revenues. Excessive payments for the objects may damage agents' budgets to purchase complements for effective uses of the objects, and thus may influence the benefits from the objects. Or agents may need to obtain loans to pay high amounts, and typically financial costs are nonlinear in borrowings. This factor also makes agents' preferences non-quasi-linear. 1 In such important applications, quasi-linearity is not a suitable assumption. 2 Some authors studying object assignment problems do not assume quasi-linearity but make a different assumption on preferences, "single demand" property. 3 It states that an agent wishes to receive at most one object. On the single demand domain, i.e., the class of single demand preferences, it is known that the minimum price Walrasian rules are well-defined. 4 The minimum price Walrasian (MPW) rules are rules that assign an allocation associated with the minimum price Walrasian equilibria for each preference profile. Demange and Gale (1985) show that the MPW rules are strategy-proof on the single demand domain. It is straightforward that in addition to efficiency, the MPW rules satisfy two properties on this domain: individual rationality; no subsidy for losers. Individual rationality states that the bundle assigned to an agent is at least as good as getting no object and paying zero. No subsidy for losers states that the payment of an agent who receives no object is nonnegative. Morimoto and Serizawa (2015) show that on the single demand domain, the MPW rules are only rules satisfying efficiency, strategy-proofness, individual rationality, and no subsidy for losers. 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 Precisely, if agents have unit demand preferences, minimum price Walrasian equilibria exist. See Quinzi (1984), Gale (1984), and Alkan and Gale (1990).
Although the assumption of the single demand property is suitable for some important cases such as house allocation, etc., the number of such applications is limited. In many cases, there are agents who wish to receive more than one object, and indeed, many authors analyze such cases. 5 Now, one natural question arises. Is it possible to design efficient and strategy-proof rules on a domain which is not the quasi-linear domain or the single demand domain? This is the question we address in this paper. To state our result precisely, we define a property, which we call the "multi demand"property. A preference relation satisfies the multi demand property if when an agent receives an object, an additional object makes him better off. We start from the single demand domain, and expand the domain by adding multi demand preferences. We show that on any domain that includes the single 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 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 proof. Section 5 concludes.

The model and definitions
Consider an economy where there are n ≥ 2 agents and m ≥ 2 indivisible objects. We denote the set of agents by N ≡ {1, . . . , n} and the set of objects by M ≡ {1, . . . , m}. Let M be the power set of M . For each a ∈ M , with abuse of notation, we sometimes write a to denote {a}. Each agent receives a subset of M and pays some amount of money. Thus, the consumption set is M × R and a generic (consumption) bundle of agent i is denoted by Each agent i has a complete and transitive preference relation R i over M × R. Let P i and I i be the strict and indifferent relations associated with R i . A set of preferences is called a domain and a generic domain is denoted by R. 6 The following are basic properties of preferences.

Desirability of object:
Definition 1 A preference relation is classical if it satisfies money monotonicity, possibility of compensation, continuity, and desirability of object. 8 Let R C be the class of classical preferences, and we call it the classical domain. Throughout this paper, we assume that preferences are classical.
Note that by money monotonicity, possibility of compensation and continuity, for each We call this payment the compensated valuation of A i from z i for R i . Note that by money monotonicity, for each ( . Let R Q be the class of quasi-linear preferences, and we call it the quasi-linear domain.
. Now we define important classes of preferences. The following definition captures preferences of agents who desire to consume only one object.

Definition 3 A preference relation
Condition (i) says that an agent prefers any object to no object. Condition (ii) says that if an agent receives a set consisting of several objects, there is an object in the set that makes him indifferent to the set. Let R U be the class of single demand preferences and we call it the single demand domain. Obviously, R U ⊊ R C .
Note that the single demand property is equivalent to the following: (i ′ ) for each a ∈ M and each t i ∈ R, CV i (a; (∅, t i )) > t i , and (ii ′ ) for each A i ∈ M and each t i ∈ R, CV i (A i ; (∅, t i )) = max a∈A i CV i (a; (∅, t i )). 10 Figure 1 illustrates a single demand preference relation.
***** FIGURE 1 (Single demand preference relation) ENTERS HERE ***** We also consider preferences of agents who desire to consume more than one object. The following definition captures preferences of agents who desire to consume until a fixed number of objects. 8 Morimoto and Serizawa (2015) also define classical preferences as a preferences satisfying the same properties that we impose. However, because of condition (ii) of desirability of object, their definition is slightly different from ours. 9 Given a set X, |X| denotes the cardinality of X. 10 Gul and Stacchetti (1999) define the single demand property for quasi-linear preferences. However they do not require condition (i ′ ). In their model, a preference relation R i ∈ R Q satisfies the single demand property if for each . This condition corresponds to condition (ii ′ ).
. 11 Condition (i) says that an agent prefers an additional object until he gets k objects. Condition (ii) says that if an agent receives a set consisting of at least k objects, there is a subset consisting of at most k objects that makes him indifferent to the original set.
The k−objects demand property is equivalent to the following: Figure 2 illustrates a k−objects demand preference relation, when k = 2. ***** FIGURE 2 (2−objects demand preference relation) ENTERS HERE ***** The following class of preferences is larger than the classes of k-objects demand preferences. The definition requires only that when an agent receives an object, an additional one make him better off.
Let R M be the class of preferences satisfying the multi demand property and call it the multi demand domain. Note that for each k ∈ {2, . . . , m}, preferences satisfying the k−objects demand property satisfy the multi demand property. But there are multi demand preferences that do not satisfy the k−objects demand property for any k ∈ {2, . . . , m} (See Figure 3). Note also that no multi demand preference relation satisfies the single demand property, i.e., R M ∩ R U = ∅.
The multi demand property is equivalent to the following: for each a ∈ M , each Figure 3 illustrates a multi demand preference relation. Note that this preference relation does not satisfy the k−objects demand property for any k ∈ {2, . . . , m}.
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 ) and t ≡ (t 1 . . . , t n ), respectively.
A preference profile is an n-tuple R ≡ (R 1 , · · · R n ) ∈ R n . Given R ∈ R n and i ∈ N , let R −i ≡ (R j ) j̸ =i . 11 In Gul and Stacchetti (1999), this notion is called k−satiation. 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. The first property states that for each preference profile, a rule chooses an efficient allocation.
Remark 2 By money monotonicity and continuity, efficiency is equivalent to the condition that there is no feasible allocation The second property states that no agent benefits from misrepresenting his preferences.

Strategy-proofness: For each
. 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.
The fourth property states that the payment of each agent is always nonnegative.
The final property is a weaker variant of the fourth: if an agent receives no object, his payment is nonnegative.

Minimum price Walrasian rule
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 Condition (WE-i) says that each agent receives a bundle that he demands. Condition (WEii) says that an object's price is zero if it is not assigned to anyone. Given R ∈ R n , let W (R) and P (R) be the sets of Walrasian equilibria and prices for R, respectively.

Remark 4
Let R ∈ (R U ) n and p ∈ P (R). If n > m, then p a > 0 for each a ∈ M .
The facts below are results for models in which each agent can receive at most one object. On the other hand, each agent can receive several objects in our model. By condition (ii) of the single demand property, however, the same results hold for single demand preferences.

Fact 1
For each R ∈ (R U ) n , a Walrasian equilibrium for R exists.
Fact 1 is shown by several authors. 12 Fact 2 below states that for each preference profile, there is a unique minimum Walrasian equilibrium price vector.

Fact 2 (Demange and Gale, 1985)
For each R ∈ (R U ) n , there is a unique p ∈ P (R) such that for each p ′ ∈ P (R), p ≤ p ′ . 13 Let p min (R) denote this price vector for R. A minimum price Walrasian equilibrium (MPWE) is a Walrasian equilibrium associated with the minimum price. 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, p min (R)), (z ′ , p min (R)) ∈ W (R), and each i ∈ N , The fact below states that on the single demand domain, the minimum price Walrasian rules satisfy the properties stated in the fact, and that if there are more agents than objects, they are the unique rules satisfying them.

Main result
We extend domains from the single demand domain by adding multi demand preferences and investigate whether efficient and strategy-proof rules still exist on such domains. In marked contrast to Fact 3 in Section 3, the results on expanded domains are negative. Namely, if there are more agents than objects, and the domain includes the single demand domain and contains at least one multi demand preference relation, then there exits no rule satisfying 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. 12 See, for example, Quinzi (1984), Gale (1984), and Alkan and Gale (1990). 13 For each p, p ′ ∈ R m , p ≤ p ′ if and only if for each i ∈ {1, . . . , m}, p i ≤ p ′i .

Remark 5
In this paper, we assume that all the agents have the common domain R. If each agent i ∈ N has his own domain R i , Theorem can be strengthen as follows. Suppose R i ⊇ R U for each i ∈ N , and there is j ∈ N and R j ∈ R M such that R j ∈ R j . Then, there is no rule on ∏ i∈N R i satisfying efficiency, strategy-proofness, individual rationality and no subsidy for losers.
Proof: Suppose by contradiction that there is a rule f on R n satisfying the four properties.
Part I. We state six lemmas that are used in the proof. Lemma 1 below 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 1 (Zero payment for losers)
Lemma 2 below states that all objects are always assigned. This follows from efficiency, n > m, and desirability of objects. We omit the proof.

Lemma 2 (Full object assignment) For each R ∈ R n and each a ∈ M , there is
Lemma 3 below states that if an agent has a single demand preference relation, then he does not receive more than one object. This follows from efficiency, the single demand property, and n > m. We omit the proof.

Lemma 3 (Single object assignment) Let
Lemma 4 below is a necessary condition for efficiency.

Proof: Suppose by contradiction that
. By Remark 2, this is a contradiction to efficiency. 2 By Lemma 1, Lemma 4 and n > m, we can show that f satisfies no subsidy.

Lemma 5 (No subsidy ) For each
Proof: (Figure 4.) Suppose by contradiction that t i (R) < 0 for some R ∈ R n and i ∈ N . Let min j∈N CV j (a; 0). Note that by R ′ i ∈ R U and desirability of object, for each This is a contradiction to strategy-proofness. Hence,

By n > m and
This is a contradiction to Lemma 4. 2 ***** FIGURE 4 (Illustration of proof of Lemma 5)) ENTERS HERE ***** Lemma 6 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 6 For each
Part II. The proof of Theorem has five steps.
Step 1: Constructing a preference profile.
Suppose by contradiction that for some i ∈ {m + 2, . . . , n}, A i (R) ̸ = ∅. By R i ∈ R U and Lemma 3, there is a ∈ M such that A i (R) = a. Since there are only m objects, there is j ∈ {1, . . . , m + 1} such that A j (R) = ∅. By Lemma 1, t j (R) = 0. Let This is a contradiction to Lemma 4. 2 Given i ∈ N and R −i ∈ R n−1 , we define the option set of agent i for R −i by Step 3: Let a ∈ M . If a = 1, then (a, t a ) ∈ o 1 (R −1 ) for some t a ≤ p 1 . If a ̸ = 1, then (a, t a ) ∈ o 1 (R −1 ) for some t a ≥ p a .
This is a contradiction to Lemma 4. individual rationality, f i (R) R i 0, and thus, by (i − 1), t i (R) ≤ CV i (1; 0) < p 1 . Therefore, This is a contradiction to Lemma 4.
This is a contradiction to Lemma 4.
This is a contradiction to Lemma 4. 2

Concluding remarks
In this article, we considered an object assignment problem with money where each agent can receive more than one object. We focused on domains that include the single demand domain and contain some multi demand preferences. We studied allocation rules satisfying efficiency, strategy-proofness, individual rationality, and no subsidy for losers, and showed that if the domain includes the single demand domain and contains at least one multi demand preference relation, and there are more agents than objects, then no rule on the domain satisfies the four properties. As we discussed in Section 1, for the applicability to various important cases, we investigated the possibility of designing efficient and strategy-proof rules on a domain which is not the quasi-linear domain or the single demand domain. Our result suggests 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 addresses maximal domains on which there are rules satisfying desirable properties. 16 A domain R is a maximal domain for a list of property on rules 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 theorem almost implies that when the number of agents is greater than that of objects, the single demand domain is a maximal domain for efficiency, strategy-proofness, individual rationality, and no subsidy for losers. However, our theorem does not imply such a maximal domain result in that only multi demand preferences can be added to the single demand domain to derive the non-existence of rules satisfying the above properties. For example, consider a preference relation such that there is k ∈ {2, . . . , m} such that (i) for each (A i , t i ) ∈ M with |A i | ≤ k, there is a ∈ A i such that (A i , t i ) I i (a, t i ), and (ii) for each (A i , t i ) ∈ M with |A i | ≥ k, and each a ∈ M \ A i , (A i ∪ {a}) P i (A i , t i ). This preference relation does not satisfy the single demand property nor the k ′ −object demand property for any k ′ ∈ {2, . . . , m}. When such a preference relation is added to the single demand domain, our theorem does not exclude the possibility that some rules satisfy the four properties. Hence, it is an open question whether there exist rules satisfying the four properties when a non-multi demand preference is added to the single demand domain. Identical objects. Some literature on object assignment problems also study the case in which the objects are identical. 17 In this paper, we assume that the objects are not identical. And this assumption 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 single demand domain while keeping the existence of rules satisfy the four properties.