Manipulation via endowments in auctions with multiple goods

Mathematical Social Sciences 87 (2017) 75–84

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Manipulation via endowments in auctions with multiple goods✩ Nozomu Muto a , Yasuhiro Shirata b,∗ a

Department of Economics, Yokohama National University, 79-3 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan

b

Department of Economics, Otaru University of Commerce, 3-5-21 Midori, Otaru, Hokkaido, 047-8501, Japan

highlights • • • • •

We study manipulation via endowments in an auction setting with multiple goods. A mechanism immune to the manipulation via endowments is destruction-proof. In general, there exists no mechanism which is destruction-proof. We find a restricted domain where the VCG auction is destruction-proof. The restriction is met when each winner’s value is close to the next-highest value.

article

info

Article history: Received 16 May 2016 Received in revised form 19 December 2016 Accepted 3 March 2017 Available online 14 March 2017

abstract We study manipulation via endowments in a market in an auction setting with multiple goods. In the market, there are buyers whose valuations are their private information, and a seller whose set of endowments is her private information. A social planner, who wants to implement a socially desirable allocation, faces the seller’s manipulation via endowments, in addition to buyers’ manipulation of misreporting their valuations. We call a mechanism immune to the seller’s manipulation via endowments destruction-proof. In general, there exists no mechanism which is destruction-proof, together with strategy-proofness of the buyers, efficiency, and participation. Nevertheless, we find a restricted domain of the buyers’ valuation profiles satisfying a new condition called per-capita goods–buyer submodularity. We show that, in this domain, there exists a mechanism which is destruction-proof, together with the above properties. The restriction is likely to be met when each winner’s valuation is close to the nexthighest valuation. We also provide a relation to monopoly theory, and argue that per-capita goods–buyer submodularity is independent of the standard elasticity argument. © 2017 Elsevier B.V. All rights reserved.

1. Introduction

✩ We are grateful to Akira Okada for his guidance and encouragement. We thank two anonymous referees, whose comments and suggestions helped us significantly to improve the article. We also thank Shigehiro Serizawa, Taiki Todo, Takuro Yamashita, and participants in seminars at Hitotsubashi University, SWET at Hokkaido University, 11th Meeting of Society for Social Choice and Welfare, the 2012 Asian meeting of the Econometric Society, and the 19th Decentralization Conference in Japan for their helpful comments. Muto gratefully acknowledges support from the Spanish Ministry of Science and Innovation through grant ‘‘Consolidated Group-C’’ ECO2008-04756 and FEDER, and from JSPS KAKENHI 26780116. Shirata gratefully acknowledges support from the MEXT of Japan through the grant Global COE Hi-Stat and JSPS KAKENHI 24830004. The former title of the paper was ‘‘Goods Revenue Monotonicity in Combinatorial Auctions’’. ∗ Corresponding author. E-mail addresses: [email protected] (N. Muto), [email protected] (Y. Shirata).

http://dx.doi.org/10.1016/j.mathsocsci.2017.03.002 0165-4896/© 2017 Elsevier B.V. All rights reserved.

We consider an exchange market with multiple and heterogeneous goods in which agents’ endowments are their private information, and a mechanism designer, or a social planner, whose purpose is to achieve a socially desirable allocation. In addition to the standard manipulation of misreporting their valuations, the private information on the seller’s endowment may aggravate this mechanism design problem since the agents can manipulate the outcome of the mechanism by destroying a part of their endowments and reporting the sets of remaining goods. Such manipulation via endowments may cause misallocation of goods and undesirable social outcomes. We call a mechanism immune to it destruction-proof.1

1 This terminology follows Atlamaz and Klaus (2007).

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N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

For example, Food and Agriculture Organization of the United Nations (FAO) reports that a significant proportion of foods are lost or wasted at the initial agricultural production stage (FAO, 2011). FAO infers that the food losses and waste occurs because disposing of her crop is financially profitable if a farmer produces more than required.2 In addition to retailers’ preferences, we consider the number of crops is likely to be farmer’s private information. Since those are often distributed through an auction, we study an auction mechanism immune to manipulation via endowments, together with standard manipulation via preferences. Postlewaite (1979) first studied manipulation via endowments in a classical exchange market in which agents have continuous, strictly monotone, and strictly convex preferences, and found a mechanism that satisfies destruction-proofness together with efficiency and individual rationality. Such existence, however, fails in general. Atlamaz and Klaus (2007) show that no mechanism satisfies the above three conditions in an exchange market with indivisible goods and without monetary transfers when agents are assumed to have separable preferences. None of these papers studying the classical exchange market, however, considers mechanisms which are immune to misreporting the preferences in addition to manipulation via endowments, so that we cannot directly apply their results to an auction market, in which the designer has no information on buyers’ valuations. Therefore, unlike the above papers, we consider an auction market in which there are multiple buyers whose valuations of the goods are their private information and who have no endowments, and there is a seller whose set of endowments is her private information. Thus, in addition to the seller’s manipulation via endowments, the buyers can misreport their valuations, which is standard manipulation in auctions. The designer knows that no buyer has endowments, and can ignore the manipulation via endowments by the buyers. Therefore, the designer seeks a mechanism which is incentive compatible with respect to the buyers’ valuations, and immune to the seller’s manipulation via endowments. It is well-known that in auctions with multiple goods, large possibilities of combinations of objects aggravate a difficulty in designing a suitable mechanism. In the literature, the Vickrey–Clarke–Groves (VCG) mechanism is one of the most widely accepted as a candidate possessing desirable properties. In the VCG mechanism, no buyer obtains a negative payoff, no buyer has an incentive to misreport his true preference, and the outcome is always efficient in terms of the reported valuations. One can, however, observe that the VCG mechanism fails to satisfy destruction-proofness in the general domain of valuation profiles of the buyers. Indeed, we show in Proposition 1 that implementing destruction-proof outcomes together with strategyproofness, efficiency, and participation is impossible in any domain containing all single-unit demand valuations. Given this general impossibility, we consider under what condition the VCG mechanism is immune to manipulation via endowments.3 We introduce a restricted domain of valuation profiles satisfying a new condition, per-capita goods–buyer submodularity, which requires that the social welfare per capita be submodular with respect to goods and buyers. Our main result (Theorem 3) shows that the VCG mechanism is destruction-proof in this domain. Per-capita goods–buyer submodularity is likely to be satisfied when for each buyer who wins some of the goods, there exists

2 In Japan, disposing of crops is institutionalized by the Ministry of Agriculture, Forestry and Fisheries (MAFF) based on the Vegetable Production and Marketing Stabilization Act, in order to keep a price floor in a case of having excess crops more than expected. See Ito and Dyck (2002). 3 Another direction is to consider a second-best mechanism. We discuss this issue in multi-unit auctions in Muto et al. (2017).

another buyer who obtains nothing and has a valuation similar to that winner. This is intuitive; if the lowest-winning-bid is close to the highest-losing-bid, the competition between buyers is so strong that the seller earns a sufficiently large revenue. The seller has no incentive to throw away the goods in such cases.4 Although this similarity property does not always hold in practice, if the seller has a certain prior belief over the valuation profiles of the buyers, and destruction-proofness holds with high probability with respect to that belief, then it is difficult to imagine wasteful destruction by the seller. Such a situation arises in many practical auctions. For example, in many auctions such as treasury auctions, auction of drilling rights for oil, electricity auctions and others, it is documented that the valuations are positively correlated between buyers, and that the buyers’ valuations are likely to be similar to each other.5 Even when the valuations are independent, the buyers’ valuations are similar to each other if the distribution concentrates in a small region. We also demonstrate by examples that if per-capita goods–buyer submodularity holds, the valuations are relatively close to the additive. Thus, this domain is quite different from those in the existing literature on auctions, e.g. the substitutes domain and the complements domain. Our analysis of destruction-proofness in auction markets is related to standard monopoly theory. To discuss this relation, we investigate another domain in which all goods are homogeneous for all buyers, and thus the auction can be viewed as a multiunit auction. We show, thanks to the homogeneity, that the VCG mechanism in the multi-unit auction is destruction-proof if the marginal value elasticity of demand is higher than or equal to one. This result is parallel to the fact in monopoly theory that the monopolist’s revenue is monotonically increasing with respect to the quantity if the price elasticity of demand is higher than or equal to one. We argue, however, that per-capita goods–buyer submodularity is logically independent of the above elasticity in the market with homogeneous goods. Since cross elasticity matters, it is hard to generalize the elasticity argument to the auction market with heterogeneous goods. Hence, our per-capita goods–buyer submodularity is a new condition that guarantees no under-supply in the auction markets. 1.1. Related literature A number of existing papers have studied a version of manipulation via endowments in various markets. In an exchange market, as mentioned, Postlewaite (1979) and Atlamaz and Klaus (2007) study manipulation via endowments. Thomson (2008) finds that the Walrasian rule is manipulable via another type of manipulation via endowments, borrowing the endowments. Sertel (1994) studies an exchange market with a single public good and shows that the Lindahl rule is manipulable via endowments. Applying the notion of manipulation via endowments into a hospital-intern matching market, Sönmez (1997) studies hospital’s manipulation via capacities. He shows that any stable matching mechanism is manipulable via capacities. Konishi and Utku Ünver (2006) and Ehlers (2010) further study manipulability via

4 This argument is reminiscent of the result of Mezzetti and Tsetlin (2008), who show that in the uniform-price multi-unit auction, when the number of losers is large, the uniform-price auction in which the price equals the lowest-winning-bid approximates the standard uniform-price auction where the price is given by the highest-losing-bid. 5 Hortaçsu and McAdams (2010) empirically analyze the Turkish treasury auction assuming affiliated private values, and conclude that the expected revenue does not depend on the auction format. The estimated valuation is close to the bid in the data, whose variance was quite low (which was 0.035, while the market-clearing price was higher than 80). Li et al. (2000) and Campo et al. (2003) study the OCS auctions in US, and Hortaçsu and Puller (2008) study the electricity auctions in Texas.

N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

capacities of the optimal matching mechanisms. Sertel and ÖzkalSanver (2002); Fiestras-Janeiro et al. (2004), and Iriş and ÖzkalSanver (2011) study a two-sided matching market where each agent has endowments of goods, and show manipulability via endowments of the optimal matching. In the auction market with multiple goods, Ausubel and Milgrom (2002) first study revenue monotonicity, which we can interpret as a version of the seller’s manipulation via endowments. Suppose that the seller’s valuations are zero for any bundles of goods. Then, destruction-proofness is equivalent to a condition that the revenue of the seller is nondecreasing with respect to the set of goods for sale. This monotonicity is related to a well-known property of buyer revenue monotonicity (Milgrom, 2004, Chapter 2): A CA mechanism is buyer revenue monotone if the seller earns no more revenue by excluding some buyers. Ausubel and Milgrom (2002) show that the VCG mechanism is buyer revenue monotone at every valuation profile in the domain V if and only if V is contained in the substitute domain, and this is equivalent to the condition that the welfare function is submodular with respect to the set of buyers whenever there are four or more buyers. We will show in Section 3.1 that by Theorem 3, both destructionproofness and buyer revenue monotonicity are ensured in the intersection of the per-capita goods–buyer submodular domain and the buyer submodular domain with four or more buyers, and this intersection is nonempty. In contrast to buyer revenue monotonicity, destruction-proofness often fails for substitute valuations by Proposition 1. This implies that in the substitutes domain, destruction-proofness is stronger than buyer revenue monotonicity. There are other papers that discuss buyer revenue monotonicity: Rastegari et al. (2011) show that in the single-minded domain6 no mechanism satisfies buyer revenue monotonicity together with participation, consumer sovereignty, and a property that any good should be allocated to a buyer who positively values it. They provide inefficient mechanisms that satisfy buyer revenue monotonicity.7 Todo et al. (2009) characterize strategyproof and buyer revenue monotone auction mechanisms in a general domain of valuations. They also discuss relations between buyer revenue monotonicity and false-name-proofness. Lamy (2010) shows that there is no buyer-optimal core selecting auction which satisfies buyer revenue monotonicity if there are more than two goods for sale, while there exists one if there are only two goods. Beck and Ott (2009) introduce a condition stronger than both destruction-proofness and buyer revenue monotonicity; the revenue should not decrease if buyers report weakly higher valuations for all bundles. They show a necessary condition of this stronger monotonicity, and propose core-selecting mechanisms satisfying their monotonicity condition. Their monotonicity is much stronger than destruction-proofness. In fact, their necessary condition is so strong that it rules out most non-additive valuations. In contrast, our Theorem 3 provides a sufficient condition which is satisfied in a domain containing non-additive valuations with a nonempty interior. The rest of the paper is organized as follows. Section 2.1 defines a combinatorial auction mechanism, and introduces destructionproofness. In Section 2.2, we show an impossibility result. In

6 A valuation function v is single-minded if there is a particular bundle of goods i xi ⊆ G such that i wants only xi . That is, vi (yi ) = vi (xi ) if xi ⊆ yi ⊆ G, and vi (yi ) = 0 otherwise. Goods are not substitutes if the targeted bundle contains two or more goods. 7 Rastegari et al. (2011, Section 4.2) consider destruction-proofness in the singleminded domain. Their impossibility is, however, immediately followed from that with buyer revenue monotonicity since destroying a good g is equivalent to disqualifying every single-minded buyer with a target bundle including g.

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Section 3.1, we show a possibility in the per-capita goods–buyer submodular domain. Section 3.2 studies a multi-unit auction with homogeneous goods. In Section 3.3, we discuss another sort of manipulation in which an agent hides the endowment. Section 4 concludes. 2. Preliminaries 2.1. The model A mechanism designer (or a social planner) faces an auction design problem. There are indivisible goods which are potentially to be sold, and we denote the universal set of goods which are potentially to be sold by G. We assume that G contains at least two goods. We analyze problems for multiple sets of goods contained in G, as an auction mechanism generally works with distinct sets of goods. We denote the set of goods actually sold in the auction by G ⊆ G. We also denote the finite set of buyers who actually participate in the auction by N with |N | ≥ 2. A typical buyer is  denoted by i ∈ N, and let XG = (x1 , . . . , x|N | ) ⊆ G|N | | xi ∩ xj =

 ∅ for all i, j ∈ N (i ̸= j) be the set of feasible allocations when the set of goods to be sold is G ⊆ G.8 There is a single seller who owns endowments to be auctioned G while every buyer is endowed with no goods.9 Given the set of goods G, each buyer i ∈ N has a private valuation function vi : xi → vi (xi ) ∈ R over all bundles of goods xi ⊆ G, and the seller has a private valuation function v0 : x0 → v0 (x0 ) ∈ R over all bundles of goods x0 ⊆ G. We assume that the designer knows v0 , so the seller’s private information is only about her endowments G. We fix v0 throughout the paper.Let Vi be the set of valuation functions of buyer i ∈ N, and V ⊆ i∈N Vi be the set of valuation profiles of all buyers.1011 We always assume free-disposal; xi ⊆ x′i implies vi (xi ) ≤ vi (x′i ) for all i ∈ N ∪ {0} and all vi ∈ Vi . For normalization, we assume that all buyers and the seller value no goods at zero, i.e. vi (∅) = 0 for all i ∈ N ∪ {0} and all vi ∈ Vi . All buyers and the seller have quasi-linear payoff functions. If each buyer i obtains a bundle of goods xi ⊆ G in exchange of payment ti ∈ R, his payoff is vi (xi ) − ti , and the seller’s payoff is v0 (x0 ) + i∈N ti . Since the seller’s endowments are her private information, the designer asks the seller to report the set of endowments G. According to this report, an auction mechanism allocates endowments to buyers. By the revelation principle, we focus on the class of deterministic direct combinatorial auction mechanisms (CA mechanisms for short) M = (MG )G⊆G . A collection of auctions   M = x(G), t (G) G⊆G is a CA mechanism in which the seller reports

ˆ ⊆ G, and simultaneously, each buyer i ∈ N a set of goods G

8 If G is infinite, it is natural to assume measurability of feasible allocations with respect to a suitable sigma-algebra on G. We omit this mathematical argument because it does not enrich implications of our results. In Section 3.2, we will consider a multi-unit auction of perfectly divisible goods. In this case, we consider the Borel sigma-algebra on a closed interval. See footnote 17 for details. 9 For simplicity, we focus on an auction setting with a single seller. Our results can be generalized to markets with multiple sellers if their valuations to any bundles of goods are zero. 10 We do not write dependency of V on G explicitly. We note that V can be i

i

regarded as a set of valuation functions on G restricted to G. Furthermore, we allow the domain of the valuation profiles not to have a product structure. This may happen if valuations are correlated across buyers, although we do not formulate the prior probability distribution explicitly. 11 We often denote a valuation profile of buyers by v = (v , . . . , v ) ∈ 1



i∈N

Vi , and for a buyer i

(v1 , . . . , vi−1 , vi+1 , . . . , v|N | ).

|N |

∈ N, the profile of the other buyers by v−i =

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N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

ˆ 12 bids a valuation function vˆ i ∈ Vi under the set of goods G. The allocation of goods and monetary payments are determined ˆ and the valuation profile according to the reported set of goods G  vˆ = (ˆv1 , . . . vˆ |N | ). Given reported Gˆ ⊆ G, for each vˆ ∈ i∈N Vi , ˆ )(ˆv ) = x(G

ˆ )(ˆv ), . . . , x|N | (Gˆ )(ˆv ) x1 (G





∈ XG is the allocation   ˆ ˆ ˆ )(ˆv ) ∈ R|N | is the function, and t (G)(ˆv ) = t1 (G)(ˆv ), . . . , t|N | (G ˆ )(ˆv ) and t (Gˆ )(ˆv ) payment function. In what follows, we denote x(G ˆ ) and t (ˆv ; Gˆ ), respectively. We denote the set of goods that as x(ˆv ; G  ˆ ) = Gˆ \ i∈N xi (ˆv ; Gˆ ). the seller keeps by x0 (ˆv ; G  ˆ ⊆ G and the buyers report vˆ ∈ i∈N Vi , If the seller reports G then each buyer i ∈ N with true valuation vi obtains payoff vi (xi (ˆv ; Gˆ ))−ti (ˆ v ; Gˆ ) and the seller with valuation v0 obtains payoff  ˆ v0 (x0 (ˆv ; G)) + i∈N ti (ˆv ; Gˆ ).

We assume that the seller is unable to manipulate the ˆ including a good which does not mechanism by misreporting G ˆ \ G ̸= ∅. Since there exists belong to the true endowments G, i.e. G ˆ \ G ̸= ∅, misreporting Gˆ must be revealed a winner for good g ∈ G ex-post. The designer can easily prevent such a manipulation by incorporating a large penalty into the mechanism in advance. We next define properties of CA mechanisms. Definition 1. (i) CA mechanism M satisfies participation (or individual rationality) if a payment is zero forany buyer obtaining payoff zero. That is, for all (ˆvi , vˆ −i ) ∈ i∈N Vi , and all G ⊆ G, if vˆ i (xi (ˆvi , vˆ −i ; G)) = 0 then ti (ˆvi , vˆ −i ; G) = 0. (ii) CA mechanism M is strategy-proof (for buyers) if no buyer has an incentive to misreport his valuations in MG,N = (x, t ) for any  G ⊆ G. That is, for all i ∈ N, all vi ∈ Vi , all (ˆvi , vˆ −i ) ∈ i∈N Vi , and all G ⊆ G,

vi (xi (vi , vˆ −i ; G)) − ti (vi , vˆ −i ; G) ≥ vi (xi (ˆvi , vˆ −i ; G)) − ti (ˆvi , vˆ −i ; G). (iii) CA mechanism M is efficient if for all v ∈ G ⊆ G, x(v; G) ∈ argmax y∈XG

 i∈N



i∈N

Vi and all

   vi (yi ) + v0 G \ yi . i∈N

We assume that this is well-defined for all v .13 Now, we introduce a notion of strategic manipulation of the seller, which will play the central role in this paper. A CA mechanism M is not manipulable via endowments, or destructionˆ (G proof, if the seller has no downward-incentive to misreport G in M. Formally, we define this concept as follows: Definition 2. A CA mechanism M is destruction-proof if the seller has no incentive to destroy a part of endowments G, thereby ˆ ( G, in MG,N = (x, t ) for all G ⊆ G and all valuation reducing it to G

ˆ ⊆ G, and all v ∈ V , profiles v ∈ V . That is, for all G ⊆ G, all G       ˆ) + ˆ ). v0 x0 (v; G) + ti (v; G) ≥ v0 x0 (v; G ti (v; G i∈N

i∈N

This condition says that the seller’s payoff is non-decreasing with respect to the set of goods.

12 We assume that the seller and the buyers simultaneously make reports. If the seller reports the set of goods G and after that, the buyers make reports, then our arguments do not change because we will assume strategy-proofness for all buyers in each MG . 13 This maximum exists under a weak condition. A sufficient condition is that each

vi is upper semi-continuous and the domain is compact in a suitable topology (see Holmström, 1979, footnote 6).

To characterize desirable mechanisms, we introduce a social welfare function among the buyers and the seller. We denote the welfare for  coalition S ⊆ N with set of goods G ⊆ G at valuation  profile v ∈ i∈N Vi by w(v; G, S ) = maxx∈XG i∈S vi (xi ) + v0 G \





xi . A natural candidate satisfying the desirable properties defined in Definition 1 is the Vickrey–Clarke–Groves (VCG) mechanism. The VCG mechanism (MGVCG )G⊆G is an efficient CA mechanism in which the payment function is given by i∈S

tiVCG (ˆv ; G) = w(ˆv ; G, N \ {i}) −



vˆ j (xVCG (ˆv ; G)) j

j∈N \{i}

  − v0 xVCG v ; G) 0 (ˆ  for each i ∈ N, each vˆ ∈ i∈N Vi , and each G ⊆ G. The VCG

mechanism obviously satisfies participation, strategy-proofness, and efficiency for any environment. Bystrategy-proofness, the VCG payoff of the seller v0 (xVCG (v; G) for each 0 (v; G)) + i∈N ti reported G is computed with respect to the buyers’ true valuations. The following example demonstrates that the VCG mechanism is not destruction-proof in some environment.

Example 1. Let G = {a, b} be the set of two goods. The seller has no valuation over the goods, i.e. v0 (x0 ) = 0 for any bundle x0 . There are two buyers 1 and 2. For each buyer i = 1, 2, valuation vi is given as follows; v1 ({a}) = 7, v1 ({b}) = 3, v1 ({a, b}) = 8, v2 ({a}) = 3, v2 ({b}) = 7, and v2 ({a, b}) = 8. Suppose the seller reports {a, b}. Then, the outcome of the VCG mechanism is allocating a to 1 and b to 2, and payment 1 by both, i.e. xVCG (v; {a, b}) = ({a}, {b}) and t VCG (v; {a, b}) = (1, 1). Hence, the seller’s payoff is 1 + 1 = 2. On the other hand, if the seller destroys good a and reports only good b, the VCG outcome is allocating b to 2 with payment 3. The seller’s payoff is 3, which exceeds the payoff obtained from selling both goods a and b. Hence, the VCG mechanism is not destruction-proof and the misallocation occurs if V contains the above valuation profile. ♦ 2.2. An impossibility Destruction-proofness, as one may suppose, is a strong requirement in general. In this section, we prove an impossibility in any domain including all ‘‘single-unit demand’’ valuations. We say that a valuation function vi is single-unit demand if for all xi ⊆ G, vi (xi ) = supg ∈xi vi ({g }). Let VSUD be the set of valuation functions with a single-unit demand. This is an extreme case of substitutes since obtaining any combination of two bundles of goods causes no increase in valuations.



Proposition 1. Suppose that V = i∈N Vi , and Vi ⊇ VSUD for all i ∈ N. Then, any CA mechanism M that satisfies efficiency, strategyproofness, and participation is not destruction-proof. The proof is given in Appendix A. We then introduce a standard notion of substitutes. Suppose that G is finite and each good g ∈ G is sold separately at price pg . Then, the demand correspondence for each buyer i at price vector p = (pg )g ∈G is defined by



Di (p) = argmax vi (xi ) −



xi ⊆G

g ∈xi



pg .

Definition 3. Suppose that G is finite. Goods are substitutes for buyer i if for any p, p′ such that pg ≤ p′g for all g ∈ G and any xi ∈ Di (p), there exists an x′i ∈ Di (p′ ) such that {g | g ∈ xi , pg = p′g } ⊆ x′ .14

14 We can generalize the definition for the infinitely many goods straightforwardly.

N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

Note that since any buyer has a quasi-linear payoff function with no budget constraints, this condition is equivalent to the gross substitutes condition defined by Kelso and Crawford (1982). We denote the set of all substitute valuations by VSub . Since VSub ⊇ VSUD , we obtain the following corollary of Proposition 1:



Corollary 2. Suppose that G is finite. If V = i∈N Vi , and Vi ⊇ VSub for all i ∈ N, then any CA mechanism M that satisfies efficiency, strategy-proofness, and participation is not destruction-proof.

Proof. Fix G. In the VCG mechanism, the payment of each buyer i ∈ N is tiVCG (v; G) = w(v−i ; G, N \ {i})

− [w(v; G, N ) − vi (xVCG (v; G))]. i

(2)

By (2), the payoff of the seller is

v0 (xVCG 0 (v; G)) +



tiVCG (v; G)

i∈N

=

3. Main results

79



w(v−i ; G, N \ {i}) − |N |w(v; G, N )

i∈N

3.1. A possibility result: per-capita goods–buyer submodularity

+



vi (xVCG (v; G)) + v0 (xVCG i 0 (v; G))

i∈N

We showed an impossibility in Proposition 1 that if Vi ⊇ VSUD for all i, then no mechanism satisfies destruction-proofness together with efficiency, strategy-proofness, and participation. This section examines the existence of such mechanisms on a restricted domain of valuation profiles. We consider a restricted domain satisfying the following property: Let w(v; ˜ G, S ) = |1S | w(v; G, S ) be the welfare per capita for coalition S with ∅ ̸= S ⊆ N. Definition 4. A profile of valuation functions v is per-capita goods–buyer submodular if for all G′ ⊆ G ⊆ G and all i ∈ N,

w(v; ˜ G, N ) − w(v ˜ −i ; G, N \ {i}) ≤ w(v; ˜ G′ , N ) − w(v ˜ −i ; G′ , N \ {i}).

(1)



A domain of valuation profiles V = i∈N Vi is per-capita goods–buyer submodular if every valuation profile in V is percapita goods–buyer submodular.15 Remark that both the left-hand-side and the right-hand-side of inequality (1) can be negative. For example, consider an extreme case in which buyer i’s valuation is so low that no object is assigned to i at the efficient allocation. Then, it is immediate that the right-hand-side of (1) which equals − |N |(|N1 |−1) w(v; G′ , N ) is negative. It is also true that the left-hand-side of (1) which equals − |N |(|N1 |−1) w(v; G, N ) is negative. There exists a per-capita goods–buyer submodular domain with a nonempty interior; for example, suppose that v =(v1 , . . . , vn ) is such that vi is additive (i.e. satisfies vi (G) = g ∈G vi ({g })) for all i ∈ N, and vi = vj for all i, j ∈ N. Then, the total welfare w(v; G, N ) = vi (G) is independent of N, and thus w(v; ˜ G, N ) − w(v ˜ −i ; G, N \ {i}) = − |N |(|N1 |−1) w(v; G, N ) is nonincreasing in G. This monotonicity in G directly implies percapita goods–buyer submodularity. If vi is strictly increasing in G, the above monotonicity is also strict. In such a case, per-capita goods–buyer submodularity holds with strict inequalities for this v , and also for any valuation profiles close to v . In inequality (1), only the buyers whose allocation changes by dropping one of the buyers are relevant. As long as some buyer j’s valuation is so low that j wins nothing both in the auction with i ∈ N and in that without i, the change of j’s valuation does not affect the inequality. Similarity across buyers’ valuations matters except for such buyers with low valuations. We will discuss more details on this issue later. Theorem 3. Suppose that V is per-capita goods–buyer submodular. Then, the VCG mechanism is destruction-proof.

15 This terminology follows ‘‘bidder (or buyer) submodularity’’ (Ausubel and Milgrom, 2002), submodularity of the welfare function with respect to the set of buyers.

=



w(v−i ; G, N \ {i}) − (|N | − 1)w(v; G, N )

i∈N

=



(|N | − 1)w(v ˜ −i ; G, N \ {i}) − (|N | − 1)|N |w(v; ˜ G, N )

i∈N

= (|N | − 1)



 w(v ˜ −i ; G, N \ {i}) − w(v; ˜ G, N ) .

(3)

i∈N

The right-hand-side in (3) is nondecreasing in G since, by percapita goods–buyer submodularity, w(v; ˜ G, N \ {i}) − w(v; ˜ G, N ) is nondecreasing in G for each i ∈ N. It is obvious that this argument holds for all G ⊆ G. Hence, the seller cannot earn a larger payoff by destroying a part of the endowments.  We now discuss properties of per-capita goods–buyer submodularity. Roughly speaking, per-capita goods–buyer submodularity is likely to be satisfied when there is at least one losing buyer who has a similar valuation for each buyer winning some good. To see it, we introduce a replica of the set of buyers into the auction. For a positive integer K , a set of buyers N, and v ∈ V , let N (K ) = {(i, k) | i ∈ N , k = 1, . . . , K } be a set of K |N | buyers, and v K be the K -replica valuation profile with v(i,k) = vi for each (i, k) ∈ N (K ). Proposition 4. Suppose that G is finite.16 There exists K¯ such that for each valuation profile v ∈ V and each K ≥ K¯ , the K -replica valuation profile v K satisfies per-capita goods–buyer submodularity. Proof. Let K¯ = |G| + 1. Since for any K ≥ K¯ and any i ∈ N, there exists k = 1, . . . , K such that buyer (i, k) obtains no goods in an efficient allocation in the K -replica environment, dropping any buyer (i, k) does not influence the resulting efficient allocation. Therefore, for any K ≥ K¯ , any (i, k) ∈ N (K ), and any G′ ⊆ G, we have w(v; G′ , N (K )) = w(v−(i,k) ; G′ , N (K ) \ {(i, k)}). Since w(v; G′ , N (K )) ≥ 0 is nondecreasing in G′ ⊆ G, we obtain that w(v; ˜ G′ , N (K )) − w(v ˜ −(i,k) ; G′ , N (K ) \ {(i, k)}) = 1 ′ − (K |N |−1)K |N | w(v; G , N (K )) must be nonincreasing in G′ . Hence, per-capita goods–buyer submodularity is satisfied.  Proposition 4 implies that if the set of goods is finite, for any valuation function vi , a profile (vi , vi , . . . , vi ) satisfies per-capita goods–buyer submodularity whenever competition among buyers for each bundle is strong enough. As the number of replica K increases, the number of buyers whose demands are the same also increases. If there are many such buyers, price competition among them is extremely strong. Then, since each good is sold at a

16 For infinite goods, there is the following counter-example. Suppose that G is countably infinite and take any positive integer K . Then, profile (v1 , . . . , vK |N | ) with vi (x) = 1 for any x ̸= ∅ and any i satisfies w(v; ˜ G) − w(v; ˜ G, N \ {i}) = 0, but if we take G′ with |G′ | < |N |, w(v; ˜ G′ , N ) − w(v; ˜ G′ , N \ {i}) < 0.

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sufficiently high price, the seller has no incentive to destroy any of the goods in the VCG auction. This argument can be applied even if the replicated buyers do not have exactly the same valuation. When there are K buyers whose valuations are sufficiently similar to each other, an analogous argument shows that there exists K¯ such that the valuation profile with K ≥ K¯ always satisfies per-capita goods–buyer submodularity. Since some of K buyers obtain nothing in the auction, per-capita goods–buyer submodularity is likely to be satisfied when there is at least one losing buyer who has a similar valuation for each buyer winning some good. Conversely, even if there are many buyers, we can show that if the valuation of a buyer winning some goods is very far from the valuations of the others’, per-capita goods–buyer submodularity fails. Proposition 5. Let V be any domain of valuations for a set of buyers N. For any v ∈ V and any i ∈ N, there exists a valuation function v˜ i ∈ G

R2+ such that (˜vi , v−i ) is not per-capita goods–buyer submodular. Proof. Let us fix v ∈ V and i ∈ N. For a good a ∈ G, take a valuation v˜ i such that v˜ i (x) ≥ |N|N|−| 1 w(v−i ; G, N \{i})+1 if a ∈ x, and v˜ i (x) = 0

otherwise. Since v˜ i ({a}) > w(v−i ; G, N \ {i}), buyer i must obtain a in any efficient allocation under (˜vi , v−i ). Then,

w(˜ ˜ vi , v−i ; G, N ) − w(˜ ˜ vi , v−i ; G \ {a}, N ) − w(v ˜ −i ; G, N \ {i}) + w(v ˜ −i ; G \ {a}, N \ {i}) v˜ i ({a}) + w(v−i ; G \ {a}, N \ {i}) w(v−i ; G \ {a}, N \ {i}) − = |N | |N | w(v−i ; G, N \ {i}) w(v−i ; G \ {a}, N \ {i}) − + |N | − 1 |N | − 1 v˜ i ({a}) w(v−i ; G, N \ {i}) ≥ − |N | |N | − 1  1  |N | = w(v−i ; G, N \ {i}) v˜ i ({a}) − |N | |N | − 1 ≥

1

|N |

> 0.

Hence, per-capita goods–buyer submodularity fails.



Per-capita goods–buyer submodularity is a condition of a valuation profile, not of an individual valuation function. These two propositions suggest that it is not fit to discuss per-capita goods–buyer submodularity with respect to a valuation of an individual buyer, since similarity between some buyers is closely related to per-capita goods–buyer submodularity. We conclude this section by presenting a simple example of a per-capita goods–buyer submodular domain which has a nonempty intersection with the domain of substitutes. Example 2. Consider an environment with two buyers and two goods. Let N = {1, 2} and G = {a, b}. We assume substitutes, namely, vi ({a}) + vi ({b}) ≥ vi ({a, b}) for all vi ∈ Vi and i ∈ N. The valuation vi is additive if the equality holds. By the  definition, the free-disposal holds if and only if vi ({a, b}) ≥ max vi ({a}), vi ({b}) for all vi ∈ Vi and i ∈ N. The valuation vi is single-unit demand if the equality holds. The seller has no valuation over the goods, i.e. v0 (x0 ) = 0 for any bundle x0 . Then, in this environment, we can show that per-capita goods–buyer submodularity holds true if vi ({a}) ≤ 1, vi ({b}) ≤ 1, vi ({a, b}) ≥ 32 , vi ({a, b}) ≤ vi ({a}) + vi ({b}) for i = 1, 2. The domain satisfying this sufficient condition seems to be not very small. We remark that by max{vi ({a}), vi ({b})} ≤ 1 < 32 ≤ vi ({a, b}), the single-unit demand domain is not included as shown in Proposition 1.

To derive the above sufficient condition, we first consider the case in which allocation (x1 , x2 ) = ({a}, {b}) is efficient, v1 ({a}) ≥ v2 ({a}), and v1 ({b}) ≤ v2 ({b}). By (1), per-capita goods–buyer submodularity holds true if and only if αi ≤ vi ({a, b}) for i = 1, 2, where

1

 1 v1 ({a}) + v1 ({b}), v1 ({a}) + v2 ({b}) , 2 2  1 1 α2 = max v1 ({a}) + v2 ({b}), v2 ({a}) + v2 ({b}) .

α1 = max

2

2

Under the above sufficient condition, since 12 vi (x) + vj (x′ ) ≤ 32 for all i, j = 1, 2 and all x, x′ = {a}, {b}, these two inequalities hold true. The derivation in each other remaining case is given in Appendix B, in which we also provide a necessary and sufficient condition that per-capita goods–buyer submodularity holds true. To sum up, the valuations are not close to the single-unit demand domain and are close to the additive, if per-capita goods–buyer submodularity is satisfied. ♦ 3.2. Multi-unit auctions—a relation to monopoly theory To connect destruction-proofness with standard monopoly theory, this section studies a multi-unit auction where all goods are homogeneous. Of course, per-capita goods–buyer submodularity works also in this environment. We furthermore introduce another condition that ensures destruction-proofness. This condition is related to a well-known argument in monopoly theory that the monopolist’s revenue is increasing in the quantity if the price elasticity of demand is larger than one. We argue that this condition is independent of per-capita goods–buyer submodularity because the elasticity argument crucially relies on the homogeneity assumption. Therefore, per-capita goods–buyer submodularity is a new requirement of the domain of valuation functions. Let Q be the potential amount of the homogeneous goods, and Q be the total quantity of the homogeneous goods to be sold (Q ≤ Q). To convey an intuition we assume divisible goods, and discuss whether the seller’s revenue is nondecreasing with respect to the quantity Q of the goods. Every buyer i’s valuation function depends only on the quantity of goods allocated to i, denoted by qi . Let vi (qi ) be the valuation when i obtains qi ∈ [0, Q].17 We assume that for all i ∈ N and all vi ∈ Vi , valuation vi is twice continuously differentiable, vi′ ≥ 0, vi′′ < 0, and limqi →0 vi′ (qi ) = ∞. For simplicity, we assume that the seller’s valuation is zero, i.e. v0 (q0 ) = 0 for any q0 ≤ Q . Let q∗ (v; Q , S ) be an efficient allocation in a coalition S with ∅ ̸= S ⊆ N for a valuation profile v ∈ V such that qVCG (v; Q ) = i ∗ q ∈ V and all Q ≤ Q. By efficiency, i (v; Q , N ) for all v  ∗ i∈S qi (v; Q , S ) = Q , and there is a common marginal value p(v; Q , S ) such that p(v; Q , S ) = vi′ (q∗i (v; Q , S )) for all i ∈ S. p/Q

Denote by e(v; Q , S ) = − ∂ p/∂ Q ≥ 0, the marginal value elasticity of aggregated demand of coalition S with ∅ ̸= S ⊆ N. Then, we obtain the following result:

Proposition 6. In the multi-unit auction, if e(v−i ; Q , N \ {i}) ≥ 1 for all i ∈ N , v ∈ V , and all Q ∈ [0, Q], then the VCG mechanism satisfies destruction-proofness. Proof. Since w(v; G, S ) = of buyer i is tiVCG (v; Q ) =





i∈S

vi (q∗i (v; Q , S )), the VCG payment

 vj (q∗j (v−i ; Q , N \ {i})) − vj (q∗j (vi ; Q , N )) . (4)

j̸=i

17 One can accommodate this notion to the original model by letting G = [0, Q] endowed with the Borel sigma-algebra, and the domain Vi satisfy v˜ i (xi ) = v˜ i (x′i ) for all v˜ i ∈ Vi and all Borel-measurable xi , x′i ⊆ G ⊆ G with an equal Lebesgue measure qi .

N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

By (4), for all i ∈ N, we have

∂ VCG ∂  ti (v; Q ) = vj (q∗j (v−i ; Q , N \ {i})) ∂Q ∂ Q j̸=i ∂  − vj (q∗j (v; Q , N )) ∂ Q j̸=i  ∂ ∗ p(v−i ; Q , N \ {i}) = q (v−i ; Q , N \ {i}) ∂Q j j̸=i  ∂ ∗ qj (v; Q , N ) p(v; Q , N ) − ∂ Q j̸=i  ∂ ∂  Q − p(v; Q , N ) Q − q∗i (v; Q , N ) ∂Q ∂Q  ∂ q∗ (v; Q , N )  . (5) = p(v−i ; Q , N \ {i}) − p(v; Q , N ) 1 − i ∂Q

= p(v−i ; Q , N \ {i})

Summing Eqs. (5) up with respect to i ∈ N yields

 ∂  VCG ti (v; Q ) = p(v−i ; Q , N \ {i}) − p(v; Q , N ) ∂ Q i∈N i∈N  ∂ q∗i (v; Q , N )  × 1− ∂Q i∈N  p(v−i ; Q , N \ {i}) − (|N | − 1)p(v; Q , N ). =

(6)

Since e(v−i ; Q , N \ {i}) ≥ 1, the function pQ satisfies that for all j and all Q ,

∂ p(v−i ; Q , N \ {i})Q ∂Q = p(v−i ; Q , N \ {i}) + (∂ p/∂ Q ) · Q   1 = p(v−i ; Q , N \ {i}) 1 − ≥ 0. e(v−i ; Q , N \ {i})

(7)

Fix i. Since q∗j (v; Q , N ) = q∗j (v−i ; Q − q∗i (v; Q , N ), N \ {i}) for all j ̸= i, we have

= vj′ (q∗j (Q − q∗i (v−i ; Q , N ), N \ {i})) = p(v−i ; Q − qi (v; Q , N ), N \ {i}).

i∈N

By (6) and (10), ∂∂Q proofness. 



VCG i∈N ti

Remark 2. One might wonder if it is possible to generalize the elasticity argument to the general environment as in Sections 2 and 3. There is, however, no natural generalization of Proposition 6 in the environment with heterogeneous goods because cross price elasticity of demand matters when discussing the revenue monotonicity in quantity. ♦

Definition 5. A strategy-proof CA mechanism M is hiding-proof if ˆ ⊆ G, and all G ⊆ G, for all v ∈ V , all G

Summing Eqs. (9) up with respect to i ∈ N yields p(v−i ; Q , N \ {i}).

for all Q and all i ∈ N. Obviously this inequality (11) is logically independent of (9). Therefore, per-capita goods–buyer submodularity is not directly related to the standard argument of elasticity. We note that the VCG price, which may not be uniform among buyers, is different from the uniform monopoly price. Intuitively, this difference causes the independence between the elasticity condition supposed in Proposition 6 and per-capita goods–buyer submodularity. ♦

(10)

= p(v−i ; Q − q∗i (v; Q , N ), N \ {i})(Q − q∗i (v; Q , N ))



(11)

(9)

(8)

p(v; Q , N )(Q − q∗i (v; Q , N ))

(|N | − 1)p(v; Q , N ) ≤

|N | − 1 Q ≤ p(v; Q , N \ {i})Q |N |

In addition to the manipulation by destroying, Postlewaite (1979) introduces another type of manipulation via endowments— hiding-proofness. In words, hiding-proofness requires that the seller cannot have an incentive to hide a part of her endowments. The difference between destroying and hiding endowments lies in the seller’s ex-post allocation. If the seller hides a part of endowments, she obtains those goods in ex-post, in addition to her allocation that a mechanism assigns. Formally, this is defined as follows:

By (7), we have ∂(pQ )/∂ Q ≥ 0. Then, by (8),

≤ p(v−i ; Q , N \ {i})Q .

Remark 1. Per-capita goods–buyer submodularity and price elasticity of demand are logically independent. In this environment with homogeneous goods, a computation similar to that in the proof of Proposition 6 shows that per-capita goods–buyer submodularity holds if and only if

3.3. Other types of manipulation via endowments

p(v; Q , N ) = vj′ (q∗j (v; Q , N )) ∗

This corollary claims that if the valuation function of each buyer is a concave function which is close to linear, then destructionproofness is satisfied. In contrast to the per-capita goods–buyer submodular valuations, the price elasticity of aggregated demand can be decomposed to each individual buyer’s elasticity. To provide an economic intuition of Proposition 6, suppose that there are many buyers, and that the price remains almost the same if a single buyer i drops out of the auction. Then, we can interpret p(v; Q , N ) as the inverse aggregated demand function that the monopolist (seller) faces, and p(v; Q , N )Q as the monopolist’s revenue function. Monopoly theory shows that if the aggregated demand is elastic such that the price elasticity of aggregated demand e(v; Q , N ) ≥ 1, then the revenue is monotonically non∂(p(v;Q ,N )Q ) ≥ 0. Hence, the monopolist has no decreasing in Q , i.e. ∂Q incentive to under-supply, and destruction-proofness is satisfied.

p(v; Q , N )

i∈N

81

(v; Q ) ≥ 0. This implies destruction-

An immediate sufficient condition for the above is that each buyer’s elasticity is larger than or equal to one. Formally, let v ′ (qi )/qi

ei (vi ; qi ) = − iv ′′ (q ) be the marginal value elasticity of demand i i of buyer i. If ei (vi ; qi ) ≥ 1 for all i ∈ N and all qi ∈ [0, Q], then e(v; Q , S ) ≥ 1 for any coalition S. Then, we obtain the following: Corollary 7. In the multi-unit auction, if ei (vi ; qi ) ≥ 1 for all i ∈ N, all vi ∈ Vi , and all qi ∈ [0, Q], then the VCG mechanism satisfies destruction-proofness.

v0 (x0 (v; G)) +



ti (v; G)

i∈N

   ˆ ). ≥ v0 x0 (v; Gˆ ) ∪ (G \ Gˆ ) + ti (v; G i∈N

By definition, it is immediately obtained that if the seller has no reservation values, then hiding-proofness is equivalent to destruction-proofness. In general, since the seller’s valuation v0   ˆ ) ∪ G \ Gˆ ≥ v0 (x0 (v; Gˆ )). By satisfies free-disposal, v0 x0 (v; G Definition 2, this implies that if a mechanism is hiding-proof, then it is destruction-proof. Thus, our impossibility result of Proposition 1 holds by replacing destruction-proofness by hidingproofness.

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N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84



Corollary 8. Suppose that V = i∈N Vi , and Vi ⊇ VSUD for all i ∈ N. Then, any CA mechanism M that satisfies efficiency, strategyproofness, and participation is not hiding-proof. Hiding-proofness is stronger than destruction-proofness: In the environment in Example 2, suppose that the seller’s valuation function is replaced by v0 ({a}) = v0 ({b}) = v0 ({a, b}) = v¯ 0 where v¯ 0 satisfies max{v1 ({b}), v2 ({a})} < v¯ 0 < min{v1 ({a}), v2 ({b})}. Then, a straightforward computation shows that destructionproofness holds but hiding-proofness fails. Although hiding is logically stronger, we consider destroying a more serious manipulation via endowments for the following reasons: First, as argued by Postlewaite (1979, pp. 258–259), destroying is more harmful to social welfare than hiding in some cases. If the seller hides goods, then each good is allocated to some buyer or the seller. Thus, even when a socially efficient allocation is not achieved, there is no waste of goods. However, if the seller destroys goods, those destroyed goods are social waste. Thus, we focus on destruction-proofness to prevent at least such social wastefulness. Second, destroying is a safer manipulation for the seller. If the auction designer can monitor the seller and find hidden goods with some positive probability after the auction, the designer easily prevents seller’s manipulation via hiding endowments by imposing a large penalty if the manipulation is revealed. However, if the seller destroys some part of goods before the auction, it is easier to erase all evidence of manipulation and to elude the monitoring. Third, destroying may work as a commitment device for the seller. When the designer is not able to monitor the seller, it might be profitable for the seller to re-auction hidden goods after the auction ends. However, once buyers take into account that possibility, a whole auction game becomes a multi-stage sequential auction. Consider an auction of substitute goods. Then, in the first auction, buyers would bid less aggressively because they infer that they will have a second chance of winning the goods. However, if the seller destroys some of her goods, it is impossible to re-auction the destroyed goods, and the buyers bid aggressively rather than in the case of hiding. 4. Conclusion We have considered a problem of the seller’s manipulation via endowments in an auction market in which the buyers’ valuations to the bundles of the goods are private information. Destructionproofness requires that the seller earn no higher payoff by destroying a certain part of endowments. We first demonstrated that in the domain containing valuations with a single-unit demand, there exists no mechanism satisfying strategy-proofness, efficiency, participation, and destruction-proofness. This suggests that combinatorial auction design can be seriously affected by seller’s manipulation via endowments, even if goods are supposed to be substitutes for all buyers. Nevertheless, we found a possibility in a restricted domain of valuations satisfying per-capita goods–buyer submodularity. This condition means that the welfare per capita is submodular with respect to the set of buyers and goods, and is likely to hold when each winner’s valuation is similar to some loser’s valuation. We also investigated a relation to monopoly theory by discussing the multi-unit auction with homogeneous goods. In the multi-unit auction, the VCG mechanism is destruction-proof if the marginal value elasticity of the aggregated demand is larger than or equal to one. Such elasticity is meaningful thanks to the homogeneity of the goods. We showed, however, that this elasticity is logically independent of per-capita goods–buyer submodularity. This suggests that in the context of combinatorial auctions, per-capita

goods–buyer submodularity is a new condition under which the monopolist may not under-supply. Given our results, further investigations will be necessary to implement desirable allocations in a larger domain; e.g. a domain containing all substitute valuations. Since a combinatorial auction which is not destruction-proof would misallocate endowments of the seller, a designer should construct a destruction-proof auction mechanism to achieve the objective, weakening some other desirable properties. We propose two directions that would be interesting. One is to design a second-best destruction-proof auction in terms of efficiency, while maintaining strategy-proofness and participation. The other is to design a destruction-proof auction mechanism satisfying efficiency, participation, and Bayesian Nash incentive compatibility instead of strategy-proofness. These issues are left for future research.18 Appendix A. Proof of Proposition 1 Appendix A proves Proposition 1. When G is finite or countably infinite, the revenue equivalence shown by Chung and Olszewski (2007) enables us to skip Step 1 of the proof below. In a general environment with a possibly uncountable cardinality of the set of goods, we adopt the graph theoretic method developed by Heydenreich et al. (2009). Fix a buyer i ∈ N and a profile v−i , and let f (vi ) = xVCG (vi , v−i ; G) for all vi . We define Gf = (X , l) as the weighted i complete directed allocation graph, where X is theset of nodes  with X = f (Vi ), and l(x, y) = infvi ∈f −1 (y) vi (y)−vi (x) is the length function for x, y ∈ X . A path from node x to node y is defined as P = (x = a0 , a1 , . . . , ak = y) such that aj ∈ X for j = 0, . . . , k. Let P (x, y) be the set of all paths from x to y. Define the distance of  (x, y) as d(x, y) = infP ∈P (x,y) kj=−01 l(aj , aj+1 ). Proof of Proposition 1. Step 1: Prove the revenue equivalence. Suppose that Vi = VSUD for all i ∈ N. Take any buyer i ∈ N and N −1 any profile v−i ∈ VSUD , and consider the allocation graph Gf . First, we prove the revenue equivalence of f . Heydenreich et al. (2009, Theorem 1) show that a necessary and sufficient condition is d(x, y) + d(y, x) = 0 for all x, y ⊆ G. Since strategy-proofness generally ensures d(x, y) + d(y, x) ≥ 0 (Heydenreich et al., 2009, Observation 2), it suffices to show that d(x, y) + d(y, x) ≤ ε for all ε > 0. Take any vi0 ∈ f −1 (x) and vi3 ∈ f −1 (y). For any δ ∈ (0, ε/4], let x¯ (δ) = {g ∈ x | vi0 ({g }) ≥ vi0 (x) − δ}, and y¯ (δ) = {g ∈ y | vi3 ({g }) ≥ vi3 (y) − δ}. Note that these sets are nonempty since we consider single-unit demand valuations. First, suppose that x¯ (δ) ∩ y¯ (δ) ̸= ∅. Then, we have vi0 (y) ≥ vi0 (x) − δ and vi3 (x) ≥ vi3 (y) − δ , which imply d(x, y), d(y, x) ≤ δ . Therefore, d(x, y) + d(y, x) ≤ 2δ ≤ ε . Next, suppose that x¯ (δ) ∩ y¯ (δ) = ∅. Let us fix g ∈ x¯ (δ) and g ′ ∈ y¯ (δ) (g ̸= g ′ ). For α, β ≥ 0, we denote a single-unit α,β α,β demand valuation function by v¯ i ∈ Vi satisfying v¯ i ({g }) = α , α,β

v¯ i

α,β

({g ′ }) = β , and v¯ i

({g ′′ }) = 0 for all g ′′ ̸= g , g ′ . We consider two cases: α,β Case 1: Suppose that there do not exist α, β such that f (¯vi ) ⊇ ′ {g , g }. Since the efficient allocation assigns a good to buyer i who values the good very highly, there exists a large value C such that α,β α,β f (¯vi ) ∋ g for all α, β ≥ C with α ≥ β + C , and f (¯vi ) ∋ g ′ for all α, β ≥ C with β ≥ α + C . Therefore, there exist α, ˜ β˜ ≥ C and α+δ ˜ ′ ,β˜

˜ ′′ α, ˜ β+δ

δ ′ , δ ′′ ∈ (0, δ] such that for vi1 := v¯ i and vi2 := v¯ i , the 1 2 ′ allocations f (vi ) ∋ g and f (vi ) ∋ g . Let x˜ := f (vi1 ) and y˜ := f (vi2 ). Since the assumption implies x˜ ̸∋ g ′ and y˜ ̸∋ g, each length is

18 Muto et al. (2017) consider a second-best destruction-proof mechanism in an auction with homogeneous goods and single-unit demand valuations.

N. Muto, Y. Shirata / Mathematical Social Sciences 87 (2017) 75–84

bounded as follows:

We derive a necessary and sufficient condition of per-capita goods–buyer submodularity of valuation profile (v1 , v2 ). There are four cases in terms of efficiency when N = {1, 2} and G = {a, b}; (1) v1 ({a}) + v2 ({b}) = w(v; G, N ), (2) v1 ({b}) + v2 ({a}) = w(v; G, N ), (3) v1 ({a, b}) = w(v; G, N ), and (4) v2 ({a, b}) = w(v; G, N ). Case (1): v1 ({a})+v2 ({b}) ≥ max{v1 ({b})+v2 ({a}), v1 ({a, b}), v2 ({a, b})}. In this case, allocation (x1 , x2 ) = ({a}, {b}) is efficient. By (1), per-capita goods–buyer submodularity holds true if and only if

l(x, x˜ ) ≤ v (˜x) − v (x) = 0 1 i

1 i

l(˜x, y˜ ) ≤ vi2 (˜y) − vi2 (˜x) = β˜ − α˜ + δ ′′ ≤ β˜ − α˜ + δ l(˜y, y) ≤ vi3 (y) − vi3 (˜y) ≤ δ l(y, y˜ ) ≤ vi2 (˜y) − vi2 (y) = 0 l(˜y, x˜ ) ≤ vi1 (˜x) − vi1 (˜y) = α˜ − β˜ + δ ′ ≤ α˜ − β˜ + δ l(˜x, x) ≤ vi0 (x) − vi0 (˜x) ≤ δ.

max {v1 ({a}), v2 ({a})}

Hence, d(x, y) + d(y, x) ≤ l(x, x˜ ) + l(˜x, y˜ ) + l(˜y, y)



83

 v1 ({a}) + v2 ({b}) + 2v2 ({a}) − 2v2 ({a, b}), ≥ max , 3v1 ({a}) + v2 ({b}) − 2v1 ({a, b}) 



  + l(y, y˜ ) + l(˜y, x˜ ) + l(˜x, x)

(12)

max {v1 ({b}), v2 ({b})}

≤ 4δ ≤ ε. α,β

Case 2: Suppose that there exist α, β such that f (¯vi ) ⊇ {g , g ′ }. By efficiency, this means that no other buyers −i positively α,β value {g , g ′ } since v¯ i is a single-unit demand valuation. Then, for

vi1 := v¯ iδ,0 and vi2 := vi0,δ , we have x˜ := f (vi1 ) ∋ g and y˜ := f (vi2 ) ∋ g ′ . Therefore, applying the same computation as in Case 1 for α˜ = β˜ = 0, δ ′ = δ ′′ = δ , we have d(x, y) + d(y, x) ≤ 4δ ≤ ε .

This completes the proof of revenue equivalence. Step 2: Prove impossibility. Recall that Vi = VSUD for all i ∈ N. Then, Vi is connected, i.e., for any vi , v˜ i ∈ Vi there is a path in Vi connecting vi and v˜ i . Suppose that a mechanism MG = (x(ˆv ; G), t (ˆv ; G)) for G satisfies efficiency and strategy-proofness in the domain V = V1 × · · · × V . By the revenue equivalence, the revenue ti (v; G) equals n  VCG ti (v; G) + c, where c is a constant. Let v i ∈ Vi be the zero valuation function with v i (xi ) = 0 for all bundles xi ⊆ G. Since v i (xi (v i , v−i ; G)) = 0, participation implies ti (vi , v−i ; G) = tiVCG (vi , v−i ; G) for any vi ∈ Vi and any v−i ∈ V−i . Therefore, the constant c is zero if MG satisfies efficiency, strategyproofness, and participation. Consider the following valuation functions with a single-unit demand: vi (xi ) = v := 1 + v0 (G) for any bundle xi ̸= ∅ and any buyer 1 ≤ i ≤ min{|G|, |N |}, and vi (xi ) = 0 for any bundle xi ⊆ G and any buyer i > min{|G|, |N |}. Then, w(G, N ) = v min{|G|, |N |}, and w(G, N \ {i}) = w(G, N ) − v for all 1 ≤ i ≤ min{|G|, |N |} and w(G, N \ {i}) = w(G, N ) otherwise. Note that the VCG mechanism allocates no good to the seller. Thus, buyer i’s VCG  each VCG payment is tiVCG (v; G) = w(G, N \ {i}) − v ( x (G, N )) − j i j̸=i

v0 (xVCG all i ∈ N. Hence, the seller’s payoff is 0 (G, N )) = 0 for VCG (v; G )) + t (v; G) = 0. v0 (xVCG 0 i i

Suppose that the seller destroys some goods, and that a set of goods G′ ( G with |G′ | = min{|G|, |N |} − 1 (≥ 1) remains to be sold. Since w(G′ , N ) − w(G′ , N \ {i}) = 0 for all i ∈ N, the payment tiVCG (v; G′ ) = v for each buyer 1 ≤ i ≤ |G′ |, and 0 otherwise.  VCG (v; G) = Hence, the seller’s payoff is v0 (xVCG 0 (v; G)) + i ti v¯ |G′ | > 0. Therefore, the payoff of the seller increases when a part of the endowments is destroyed. This implies that the VCG mechanism is not destruction-proof. By revenue equivalence, M is not destruction-proof in the domain V if M satisfies efficiency, strategy-proofness, and participation. Hence, M cannot be destruction-proof in any larger domains if M satisfies efficiency, strategy-proofness, and participation.  Appendix B. Complete characterization of Example 2 In Example 2, there are two buyers N = {1, 2} and two goods G = {a, b}. We assume substitutes, i.e. vi ({a})+vi ({b}) ≥ vi ({a, b}) and free-disposal, i.e. vi ({a, b}) ≥ max{vi ({a}), vi ({b})} for all vi ∈ Vi and i = 1, 2. The seller has no valuation over the goods, i.e. v0 (x0 ) = 0 for any bundle x0 .

≥ max

  v1 ({a}) + v2 ({b}) + 2v1 ({b}) − 2v1 ({a, b}), . 3v2 ({b}) + v1 ({a}) − 2v2 ({a, b})

(13)

Then, there are four subcases; (1.a) v1 ({a}) ≥ v2 ({a}) and v2 ({b}) ≥ v1 ({b}), (1.b) v1 ({a}) ≥ v2 ({a}) and v2 ({b}) < v1 ({b}), (1.c) v1 ({a}) < v2 ({a}) and v2 ({b}) ≥ v1 ({b}), and (1.d) v1 ({a}) < v2 ({a}) and v2 ({b}) < v1 ({b}). In case (1.d), we obtain v1 ({a}) + v2 ({b}) < v2 ({a}) + v1 ({b}). Since this contradicts the assumption v1 ({a}) + v2 ({b}) ≥ v2 ({a}) + v1 ({b}), it suffice to consider the first three cases (1.a), (1.b), and (1.c). Case (1.a): v1 ({a}) ≥ v2 ({a}) and v1 ({b}) ≤ v2 ({b}). By (12) and (13), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if αi ≤ vi ({a, b}) for i = 1, 2, where

 1 v1 ({a}) + v1 ({b}), v1 ({a}) + v2 ({b}) , 2 2 1  1 α2 = max v1 ({a}) + v2 ({b}), v2 ({a}) + v2 ({b}) .

α1 = max

1

2

2

Case (1.b): v1 ({a}) ≥ v2 ({a}) and v1 ({b}) > v2 ({b}). By (12) and (13), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if βi ≤ vi ({a, b}) for i = 1, 2, where

 β1 = max v1 ({a}) +  β2 = max v2 ({a}) +

1

 1 1 1 v2 ({b}), v1 ({a}) + v1 ({b}) + v2 ({b}) , 2 2 2 2  1 1 1 3 v2 ({b}), v1 ({a}) − v1 ({b}) + v2 ({b}) . 2

2

2

2

Case (1.c): v1 ({a}) < v2 ({a}) and v1 ({b}) ≤ v2 ({b}). By (12) and (13), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if γi ≤ vi ({a, b}) for i = 1, 2, where

 3 1 1 v1 ({a}) + v1 ({b}), v1 ({a}) − v2 ({a}) + v2 ({b}) , 2 2 2 2 1  1 1 1 γ2 = max v1 ({a}) + v2 ({b}), v1 ({a}) + v2 ({a}) + v2 ({b}), . γ1 = max

1

2

2

2

2

Case (2): v1 ({b})+v2 ({a}) ≥ max{v1 ({a})+v2 ({b}), v1 ({a, b}), v2 ({a, b})}. In this case, allocation (x1 , x2 ) = ({b}, {a}) is efficient. By (1), per-capita goods–buyer submodularity holds true if and only if max {v1 ({a}), v2 ({a})}

  v2 ({a}) + v1 ({b}) + 2v1 ({a}) − 2v1 ({a, b}), ≥ max , 3v2 ({a}) + v1 ({b}) − 2v2 ({a, b})

(14)

max {v1 ({b}), v2 ({b})}

≥ max

  v2 ({a}) + v1 ({b}) + 2v2 ({b}) − 2v2 ({a, b}), . 3v1 ({b}) + v2 ({a}) − 2v1 ({a, b})

(15)

Then, there are four subcases; (2.a) v1 ({a}) > v2 ({a}) and v2 ({b}) > v1 ({b}), (2.b) v1 ({a}) > v2 ({a}) and v2 ({b}) ≤ v1 ({b}),

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(2.c) v1 ({a}) ≤ v2 ({a}) and v2 ({b}) > v1 ({b}), and (2.d) v1 ({a}) ≤ v2 ({a}) and v2 ({b}) ≤ v1 ({b}). In case (2.a), we obtain v1 ({a}) + v2 ({b}) > v2 ({a}) + v1 ({b}). Since this contradicts the assumption v1 ({a}) + v2 ({b}) ≤ v2 ({a}) + v1 ({b}), it suffice to consider the last three cases (2.b), (2.c), and (2.d). Case (2.b): v1 ({a}) > v2 ({a}) and v2 ({b}) ≤ v1 ({b}). By (14) and (15), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if βi′ ≤ vi ({a, b}) for i = 1, 2, where 1

1

1

1

 β1′ = max v2 ({a}) + v1 ({b}), v1 ({a}) + v2 ({a}) + v1 ({b}) , 2 2 2 2 1  1 3 1 ′ β2 = max v2 ({a}) + v2 ({b}), − v1 ({a}) + v2 ({a}) + v1 ({b}) . 2

2

2

2

By vi ({a, b}) ≤ vi ({a}) + vi ({b}) for i = 1, 2, such a domain is nonempty whenever 2v1 ({b}) > v2 ({a}), v1 ({a}) + v1 ({b}) > v2 ({a}), and v1 ({a}) + 2v2 ({b}) > v2 ({a}) + v1 ({b}). Case (2.c): v1 ({a}) ≤ v2 ({a}) and v2 ({b}) > v1 ({b}). By (14) and (15), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if γi′ ≤ vi ({a, b}) for i = 1, 2, where

 γ1′ = max v1 ({a}) +  γ2′ = max v2 ({a}) +

1

 3 1 1 v1 ({b}), v1 ({b}) + v2 ({a}) − v2 ({b}) , 2 2 2 2  1 1 1 1 v1 ({b}), v2 ({a}) + v2 ({b}) + v1 ({b}) . 2

2

2

2

Case (2.d): v1 ({a}) ≤ v2 ({a}) and v2 ({b}) ≤ v1 ({b}). By (14) and (15), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if δi′ ≤ vi ({a, b}) for i = 1, 2, where

 δ1′ = max v1 ({a}) +  δ2′ = max v2 ({a}) +

 1 v1 ({b}), v2 ({a}) + v1 ({b}) , 2 2  1 1 v1 ({b}), v2 ({a}) + v2 ({b}) . 1

2

2

Case (3): v1 ({a, b}) ≥ max{v1 ({a}) + v2 ({b}), v1 ({b}) + v2 ({a}), v2 ({a, b})}. In this case, allocation (x1 , x2 ) = ({a, b}, ∅) is efficient. By substitutability, we obtain v1 ({a}) ≥ v2 ({a}) and v1 ({b}) ≥ v2 ({b}). Then, by (1), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if

v1 ({a, b}) ≤ 2v2 ({a, b})   + min v1 ({a}) − 2v2 ({a}), v1 ({b}) − 2v2 ({b}) . Case (4): v2 ({a, b}) ≥ max{v1 ({a}) + v2 ({b}), v1 ({b}) + v2 ({a}), v1 ({a, b})}. In this case, allocation (x1 , x2 ) = (∅, {a, b}) is efficient. By substitutability, we obtain v2 ({a}) ≥ v1 ({a}) and v2 ({b}) ≥ v1 ({b}). Then, by (1), a straight-forward computation proves that per-capita goods–buyer submodularity holds true if and only if

v2 ({a, b}) ≤ 2v1 ({a, b})   + min v2 ({a}) − 2v1 ({a}), v2 ({b}) − 2v1 ({b}) .

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