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An Impossibility Result for Ex-Post Implementable Multi-Item Auctions with Private Values Ron Lavi Industrial Engineering and Management The Technion – Israel Institute of Technology [email protected] Ahuva Mu’alem School of Computer Science and Engineering The Hebrew University, Jerusalem, Israel [email protected] Noam Nisan School of Computer Science and Engineering The Hebrew University, Jerusalem, Israel [email protected] October 1, 2007

Abstract We analyze ex-post implementable social choice functions for private-value and quasi-linear settings over restricted domains of preferences, the leading example being multi-item auctions (with either heterogeneous or homogeneous goods). Our work generalizes the characterization of Roberts (1979) who characterized ex-post implementability over unrestricted domains. We show that ex-post implementability for multi-item auctions (and related restricted domains) implies weighted welfare maximization, if the given function also satisfies four additional social choice requirements. The most significant requirement is similar to Arrow’s IIA condition, adjusted to the quasi-linear case, and we study its connection to various existing monotonicity properties.

JEL Classification Numbers: C70, D44 Keywords: Ex-post implementation, multi-item auctions, Vickrey-Clarke-Groves Mechanisms, Roberts’ theorem

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1

Introduction

This paper studies the class of ex-post implementable multi-item auctions, for quasi-linear and private values. Perhaps the most well-known mechanism in this context is the celebrated VickreyClarke-Groves (VCG) mechanism (due to Vickrey (1961); Clarke (1971); Groves (1973)), that implements in dominant-strategies the efficient, welfare-maximizing social choice function. We ask if there exist any other social choice function that is ex-post implementable? While efficiency is considered one of the central economic goals, there are many other desirable goals that a mechanism designer might wish to implement, e.g. maximizing the revenue, maintaining some sort of fairness in the allocation, or maintaining budget-balancedness, to name just a few other possible canonical goals. In this context, a broader question is the exact characterization of the possibilities – impossibilities border of ex-post implementation for quasi-linear and private values. The transition from the possibility of ex-post implementing many social goals, to the impossibility of doing so, turns out to be strongly related to the dimensionality of the domain of preferences being considered. On the one hand, Myerson’s simple monotonicity condition (Myerson, 1981), that completely characterizes ex-post implementability in single-dimensional domains, gives rise to many implementable social choice functions. For example, this monotonicity enables revenue-maximization in single-item auctions (Myerson, 1981), asymptotically efficient and dominant-strategy double auctions (McAfee, 1992), dominant-strategy and budget-balanced cost sharing mechanisms (Groves and Ledyard, 1977), and competitive auctions for unlimited supply of items (Goldberg, Hartline, Karlin, Saks and Wright, 2006), among many other positive results that we do not list. Therefore, in singledimensional domains, there is a possibility to ex-post implement a variety of social goals, other than welfare-maximization. On the other hand, if we allow full dimensionality, so that the domain becomes unrestricted, then the impossibility emerges: Roberts (1979) shows that every ex-post implementable social choice function for an unrestricted domain, with a range that contains at least three alternatives, must be a weighted welfare maximizer. Thus, all the above-mentioned social goals cannot be ex-post implemented in the unrestricted domain, hence the impossibility. However, most interesting domains lie somewhere between these two extremes of unrestricted and single dimensional domains. This intermediate range of multi-dimensional restricted domains includes most multi-item auction types, including general heterogeneous-items (“combinatorial”) auctions and identical-items (“multi-unit”) auctions. Almost nothing is known about the abovestated questions, for the intermediate range of restricted domains in general, and for the important range of multi-dimensional auctions, in particular. Our main result is an impossibility theorem, in the spirit of Roberts’ theorem: Every ex-post implementable multi-item auction, with a dense range, that satisfies unanimity, decisiveness, and weak-IIA, must be a weighted welfare maximizer. Thus, the impossibility to ex-post implement 2

various social goals other than welfare maximization is extended to the restricted domain of multiitem auctions, under the addition of the four requirements (and the next paragraph gives more details on these). The theorem holds for the case of heterogeneous-items, as well as for the case of identical-items. It also holds for more restricted cases, for example when players are restricted to be “double-minded” (i.e. interested in only two possible bundles). We do need to assume, however, the existence of complementarities, and so our result does not hold for auctions that assume substitutabilities, e.g. unit-dmand auctions. We now briefly explain the additional requirements. The range of alternatives is “dense” if every two players may sometimes receive a non-empty bundle simultaneously. This excludes the option to bundle all items, and always allocating them together to one of the players, by this bringing back single-dimensionality. Unanimity states that if every player, i, is “single-minded” and desires only one specific subset of items Si , and furthermore there exists an alternative that simultaneously allocates Si to i (for all i), then this alternative will indeed be chosen. By being single-minded in a way that enables all players to simultaneously win, the players express unanimity about the most preferred allocation, and therefore this condition is a straight-forward translation of the classic unanimity condition from the non-quasi-linear social choice literature. Decisiveness states that, for every type declaration of the other players, there always exists a type declaration for player i that will award her all items, and another type declaration in which she will get nothing. While this condition does not naturally translate to the general social choice framework, in the context of auctions it seems very natural: the vast majority of the theoretical auction formats, as well as the common “real life” auctions, all seem to satisfy decisiveness. The most meaningful and interesting condition seems to be weak-IIA. To explain it, let f : V → A be a social choice function that determines an alternative a ∈ A according to players’ type declaration v ∈ V , where V = V1 × · · · × Vn is the domain of player types. Weak-IIA is a translation of Arrow’s well-known property (Arrow, 1951), from the non-quasi-linear setting to the quasi-linear world. It states that, if there exist two types v, v 0 such that f (v) = a and f (v 0 ) = b, then there must exist a player i such that vi (a) 6= vi0 (a) or vi (b) 6= vi0 (b), i.e. that at least one player changes her value for either a or b in the transition from v to v 0 . Arrow’s condition requires that the relative position of a vs. b will change for at least one player, while here, since we have cardinal preferences, we can require even less, as the values can change while maintaining the relative position. (Arrow’s definition assumes a social welfare function, however, and we expand on this issue in the next paragraph). At a first glance, weak-IIA seems simply like requiring a consistent tie-breaking, but in the paper body we show by an example that some violations of weak-IIA are more subtle than that. While we do not argue that this is a trivial condition to include, it seems that there is a consensus in the economics literature that the study of its consequences is valid, interesting, and non-trivial. The analysis is composed of two main parts. First, we show that any ex-post implementable

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function that satisfies the four requirements can be described using a collection of “score” functions {fa (·)}a∈A such that, for any v ∈ V , f (v) is an alternative a with a maximal score fa (v(a)). Interestingly, this reveals a social welfare function

1

“underlying” the given social choice function,

where the social welfare function orders the alternatives according to their scores, and always agrees with the social choice function on the “best” alternative. This underlying social welfare function satisfies the property that, if the function changes the relative position of a vs. b in the transition from v to v 0 , then it must be the case that at least one player changes her value for either a or b. This exactly captures the essence of IIA, and translates it to the case of cardinal preferences, further explaining our choice of terminology. This part of the analysis was inspired by the work of Archer and Tardos (2002), who studied frugality issues in the single-dimensional “path auctions” domain. The second part of the proof uses the incentive constraints and the four requirements to show that these score functions must be linear and identical, implying that f is a weighted welfare maximizer. At this point, it seems important to draw a comparison with the non-quasi-linear case. The situation in this setting is quite similar to what we have described above, and, in particular, the possibilities – impossibilities border again depends on the dimensionality of the domain. If preferences are severely restricted so as to make the domain single-dimensional, then several expost implementable functions are known to exist (for example, the majority rule over a single-peaked domain; see the survey of Le Breton and Weymark (2004) for more examples). On the other hand, if preferences are completely unrestricted, the classic Gibbard-Satterthwaite impossibility (Gibbard, 1973; Satterthwaite, 1975) states that dictatorship is the only implementable social choice function. The proof makes extensive use of Arrow’s theorem, and in fact follows a very similar pattern to the first part of our proof: given an implementable social choice function for the non-quasi-linear case, one can construct a social welfare function that satisfies Arrow’s conditions. Then, Arrow’s theorem is used to argue that the original function is a dictatorship. Thus, one can see the conceptual similarities between our proof and this classic proof, similarities that may shed some additional light on the usefulness of weak-IIA to our proof. Notably, there additionally exist many studies on various multi-dimensional restricted domains in the non-quasi-linear case, in sharp contrast to the situation in the quasi-linear and private values case, as was described above. Perhaps the most relevant example to this current paper is the work of Barbera and Jackson (1995), who analyze the allocation of goods in exchange economies, and show that only degenerate strategy-proof mechanisms exist. Since this domain is restricted, they need few additional qualifiers for their theorems to hold, exactly like we do here. In particular, there is a conceptual similarity between the “no-bossiness” and the “tie-freeness” conditions that they use, and our weak-IIA. We further touch on this subject in section 3. Thus, confronting the vast knowledge that the theory offers for the non-quasi-linear case with the large gaps that exist 1

A social welfare function derives a social ordering of the alternatives from the set of individual orderings of them, while a social choice function outputs a single (best) alternative.

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in the important setting of quasi-linear and private values, we argue again that the motivation for our study strengthens. We also mention three other strands of works that are relevant to this paper. First, we mention the recent developments in the more general setting of quasi-linear and interdependent values. There, too, single-dimensional domains admit implementability results (see e.g. Maskin (1992)). On the other hand, Jehiel et al. (2006) have recently shown a strong impossibility result for “generic” valuations. These include unrestricted domains, if one wishes to use the above terminology, as well as many restricted (though rich enough) domains. Nevertheless, the case of multi-item auctions has not been completely determined yet, as Bikhchandani (2006) notes. Second, we very briefly point out that the solution concept of ex-post implementation has received much attention in the recent literature, under the framework of “robust mechanism design” (Bergemann and Morris, 2005). Wilson’s critique (Wilson, 1987) on the alternative notion of Bayesian implementability states that the usual common-prior assumption is a too strong assumption to make, and in that sense detail-free mechanisms are better. In a recent series of papers, Bergemann and Morris address this critique, and examine the implications of a transition from Bayesian implementability to ex-post implementability, in a general mechanism design setting. For private values, ex-post implementability is equivalent to dominant-strategy implementability (as Bergemann and Morris (2005) note), making this solution concept even more convincing. Third, we should address the difference between characterizing ex-post implementability in terms of monotonicity conditions, and showing impossibility theorems. McAfee and McMillan (1988) consider convex domains with differentiable social choice functions, and give a characterization of incentive-compatibility that translates to the condition that the social choice function is a subgradient of a convex function. Rochet (1987) generalizes this, and gives a cycle-monotonicity condition that completely characterizes ex-post implementability on every domain (assuming quasi-linear and private values). Bikhchandani et al. (2006) simplify this condition to a weak monotonicity condition, that characterizes ex-post implementability for auction domains, and Saks and Yu (2005) have later shown that this weak monotonicity characterizes ex-post implementability on every convex domain. However, these various monotonicity conditions are rather subtle, and therefore it is not clear what social choice functions will satisfy them. Our impossibility can be interpreted as showing that the only monotone functions (in the sense of the above monotonicity conditions) that additionally satisfy our requirements are weighted welfare maximizers. In section 5 we address this point in more depth, and show that coupling ex-post implementability with weak-IIA becomes equivalent to a “strong monotonicity” condition, that is a slight (but significant) strengthening of the condition of Bikhchandani et al. (2006). On the other hand, the discussion at the beginning of the section shows that multi-dimensional restricted domains that do admit monotone functions are quite rare. One such domain was very recently considered by Lavi and Swamy (2007). The remainder of the paper is organized as follows. In Section 2 we give full details on our

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setting, and define the essentials. Section 3 precisely defines the four social choice requirements, and provides a discussion on their strengths and weaknesses. Section 4 outlines our impossibility result and its analysis, with the full proof given in Appendix A. Section 5 studies the connection of the various monotonicity conditions to the weak-IIA assumption, and a concluding discussion is given in Section 6.

2

The Setting

An auctioneer wishes to allocate a set Ω of m items to n bidders, where each item may be allocated to at most one bidder. Each bidder, i, has a valuation function vi : 2Ω → v10 (c1 ) > fa (v(a)), then the transition from v to v 0 contradicts the downward-stability property. Hence f (v) = a implies that v1 (c1 ) ≤ fa (v(a)). Generalizing to any two alternatives a, b is done as follows. Assume by contradiction that there exists some v ∈ V with f (v) = a, but fa (v(a)) < α < fb (v(b)). By the above observation we have that v1 (c1 ) ≤ α. Let v 0 be identical to v, except that v10 (c1 ) is raised to be α. A simple technical claim, proved in the appendix, shows that f (v 0 ) is either a or c1 . However we cannot have f (v 0 ) = a, since by the above this would imply α = v10 (c1 ) ≤ fa (v 0 (a) = v(a)), which we know is false. And we cannot have f (v 0 ) = c1 , since v 0 (b) = v(b), and having v10 (c1 ) = α < fb (v 0 (b)) contradicts the definition of fb (v 0 (b)). This shows that f (v) is an alternative a with maximal score fa (v(a)). Using similar ideas, one can advance even further, and get fa (v(a)) = v1 (a) + fa (0, v−i (a)) (there is an asymmetry between player 1 and the others, as she has a special role in the definition of fa (·)). To summarize, the first part of the proof brings us to the following conclusion: Lemma 1 If f is ex-post implementable, and satisfies requirements 1 – 4, then, for any v ∈ V , 1. f (v) ∈ argmaxa∈A fa (v(a)). 2. For any a ∈ A, fa (v(a)) = v1 (a) + fa (0, v−i (a)). 11

ℜ fci (α + δ’) fa( v’(a) + δ·ei )

fa(v’(a)) fci (α)

fa ( · )

f ci ( · )

Figure 1: The positioning of fa (v 0 (a)), fci (v 0 (ci )), fa (v 00 (a)), and fci (v 00 (ci )). The positioning of fa (v 0 (a)) vs. fci (v 0 (ci )) is made possible by fine-tuning v1 (a), that affect fa (v 0 (a)), but not fci (v 0 (ci )). The positioning of fa (v 00 (a)) vs. fci (v 00 (ci )) is then implied by inequality 1. Using this, the second part of the proof will show that fa (0, v−i (a)) is linear. This is shown using the following characterization of a linear function: Claim 1 Let h : 1, if v1 (c1 ) ≥ 1, choose the allocation aj . However, if v1 (c1 ) < 1, choose the allocation b ∈ A such that b1 = ∅, and b maximizes Pn i=2 wi vi (b) (for some non-identical wi ’s, that may depend on the valuation declaration of player 1). Allocate bj to player j.

First, let us verify that f chooses a feasible allocation: if v1 (c1 ) < 1 then it must be the case that player 1 gets the empty set in the allocation a, and so we are free to choose any allocation b in the second phase, and allocate the items to the other player without considering player 1. Second, we verify that this auction is dominant-strategy implementable: For player 1, we can P set the payment to be ni=2 vi (a) + γa (according to Groves’ scheme), as, from the point of view of P player 1, we always choose an alternative a that maximizes ni=1 vi (a) + γa . For any other player

j, we set the payments again according to Groves’ scheme, but depending on v1 (c1 ). Since the choice between the two different formats depends only on player 1’s declaration, this does not hurt dominant-strategies. To see that f violates weak-IIA, consider the following two types of player 1: at first, v1 (c1 ) = 1 + , but the others declare high enough values, so 1 gets nothing. Now, if 1 lowers all his values

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by , the allocation changes since now f maximizes

Pn

i=2 wi vi (a),

but player 1 still gets the empty

bundle, and therefore her value for the new allocation remains unchanged. It is not hard to verify that f is still player decisive, and that A is dense. Thus, one can indeed violate weak-IIA and unanimity, and get an implementable non-welfare-maximizer, but it does not seem that this technique is helpful in any particular way. These examples demonstrate that the four requirements are necessary for the theorem to hold. Of-course, one should also consider the content of the requirements. While requirements 2 – 4 are rather straight-forward, and one can easily grasp their consequences and disadvantages, the weak-IIA is more subtle. The similarity to Arrow’s IIA and to Maskin’s monotonicity, as well as the above example, all hint that the requirement is not simply a “tie-breaking” rule, as one may be tempted to believe. In the next section we give a more thorough technical discussion on this requirement and its implications.

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Some Related Monotonicity Properties

Monotonicity requirements are well known to be closely related to implementability properties, and Bikhchandani et al. (2006) describe the exact monotonicity condition that characterizes expost implementability in private-value multi-item auction domains: Definition 2 (Weak monotonicity) A social choice function f : V → A satisfies Weak Monotonicity if, for any v ∈ V , player i, and vi0 ∈ Vi : f (v) = a and f (vi0 , v−i ) = b implies that vi0 (b) − vi (b) ≥ vi0 (a) − vi (a). In other words, this monotonicity property requires that, if player i shifts the social outcome of f from a to b by changing his type from vi to vi0 , then it must be that i’s value for b has increased at least as i’s value for a. Bikhchandani et al. (2006) show that a multi-item auction rule is ex-post implementable if and only if it satisfies weak monotonicity, i.e. that it completely characterizes ex-post implementability for multi-item auctions. We would therefore like to examine the connection of weak monotonicity to weak-IIA, and to demonstrate that one can slightly strengthen the condition so that it will naturally imply weak-IIA: Definition 3 (Strong monotonicity) A social choice function f satisfies Strong Monotonicity if for any v ∈ V , player i, and vi0 ∈ Vi : f (v) = a and f (vi0 , v−i ) = b 6= a imply that vi0 (b) − vi (b) > vi0 (a) − vi (a). In both weak and strong monotonicity, we have the situation that i’s valuation changed from vi to vi0 and this caused the outcome of f to change from a to b. Strong monotonicity asserts that this 15

implies that i’s valuation of b had to increase strictly more than did the valuation of a, while weak monotonicity requires that it did not increase less. This initially may seem like a slight change, but it is in fact crucial. The following definition characterizes the difference: Definition 4 (Quasilinear-IIA) f satisfies quasilinear-IIA if for any v, v 0 ∈ V , if f (v) = a and f (v 0 ) = b 6= a then there exists a player i such that vi0 (a) − vi0 (b) 6= vi (a) − vi (b). While the condition of weak-IIA from section 3 examines absolute values, quasilinear-IIA extends weak-IIA to consider value differences as well. This turns out to completely characterize the difference between the two monotonicity conditions: Proposition 2 A multi-item auction satisfies strong monotonicity if and only if it satisfies weak monotonicity and quasilinear-IIA. Hence, if one replaces weak monotonicity by strong monotonicity in the statement of the main theorem, then weak-IIA becomes redundant. In light of this, two questions should be asked. The first is whether one can actually construct implementable multi-item auctions that violate strong monotonicity. The answer is of-course yes, as we have shown that there exists an implementable auction that violates weak-IIA, and therefore also strong monotonicity. Then, a second question is whether there exist other problem domains for which weak monotonicity and strong monotonicity are equivalent. Quite surprisingly, the answer to this second question is also positive. In a companion paper we show that the two monotonicity conditions are equivalent on every open domain. Thus, for example, in the unrestricted domain (which is trivially an open set), weak-IIA can be assumed without loss of generality for every ex-post implementable function. We also show that in 2-player auctions where all items must always be allocated weak-IIA can be assumed without loss of generality, and this leads us to some computational impossibilities about 2-player auctions, that rely on the characterization that we have provided here.

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Discussion

The proof of the main theorem does not utilize the entire combinatorial structure of the multiitem auction setting, and so the characterization holds for other related restricted domains as well. The full technical details are given in the appendix, but, in general, any domain that is latticeshaped (and in particular has a maximal and a minimal element) admits the result, using a natural modification of requirements 1 – 4. For example, the result holds for multi-item auctions with “known double minded players” – each player desires exactly one of two bundles (out of which we need one bundle to be the entire set of items), and these bundles are public information. Thus, even if the secret information of the players contains two numbers, the characterization holds. 16

In this respect, it is interesting to contrast our result with the result of Lehmann, O’Callaghan and Shoham (2002), that gives a dominant-strategy auction for single-minded players. This auction is designed to be “fast”, i.e. computationally efficient, even if there are many items and many players, and operates as follows: each player reports a bundle Si and a value vi = vi (Si ); each bundle receives a rank vi (Si )/|Si |α (for some 0 < α ≤ 1) that takes into account both the value and the size of the bundle, and the “greedy” allocation procedure goes over all bundles, from the highest ranked and downwards, and allocates each bundle to its bidder if and only if it does not intersect previously assigned bundles. Interestingly, this allocation rule satisfies requirements 1 – 4. There are many instances for which the resulting allocation of this procedure does not maximize the welfare, but nevertheless Lehmann et al. (2002) describe a dominant-strategy mechanism that implements it. Our characterization indicates that this mechanism inherently relies on the fact that players are single-minded. The greedy allocation procedure by itself can be easily extended to the case where each player desires two (or more) bundles5 , but, unfortunately, this is no longer ex-post implementable! Lehmann et al. (2002) prove this directly, and our result indicates that this is not an accident. Any ex-post implementable allocation rule for double-minded players must violate at least one of the requirements 1 – 4, while this greedy procedure satisfies all requirements, even if players are “k-minded”. However, our proof does rely on the above mentioned lattice structure, and in particular on the existence of an alternative that its value can be freely increased by the player. Because of this, the proof does not carry through to multi-item auctions where some additional restrictions on the valuations are present. For example, one may consider some sort of a “no-complementarities” condition that rules out the existence of complementarities in the valuation, such as sub-modular valuations (see Lehmann, Lehmann and Nisan (2006) for a discussion on this definition, and on other possible no-complementarities conditions). Unfortunately, our proof does not shed any light on these restricted valuation classes. Another weakness of our result is the assumption of a finite alternative space, A. This rules out randomized social choice functions. Indeed, recent papers in the computer science literature suggest to use randomization in order to design computationally-efficient and incentive-compatible combinatorial auctions that are not welfare-maximizers (see e.g. Lavi and Swamy (2005) and Dobzinski et al. (2006)). More generally, however, the exact power of randomized mechanisms (while still keeping the mechanism “detail-free”) is yet to be characterized. For example, it will be interesting to know if various randomization methods can help in designing implementable non-welfare-maximizers for unrestricted domains, or perhaps Roberts’ impossibility carries through (in a certain formal way) to the randomized case. The entire paper is concerned with the difficulties of implementability in multi-item auctions. Another bothering aspect of these auction formats is their communicational and computational 5

The player reports two bundles and two values, a score is assigned to each bundle separately, and the greedy allocation procedure is performed as before.

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burden. This aspect is studied e.g. in the papers by Nisan and Segal (2006), Holzman and Monderer (2004), and Holzman, Kfir-Dahav, Monderer and Tennenholtz (2004).

A

Proof of Main Theorem

In this section we give the proof to theorem 1. The proof is described in several parts. First, we list the exact structural properties of the social choice setting that the proof uses, to make explicit the fact that the entire structure of multi-item auctions is not needed. This enables us to include more restricted domains, like multi-unit auctions, and auctions with double-minded players, in the theorem. See the discussion above (section 6) for more details.

A.1

Impossibility requirements

Consider the social choice function f : V → A, where V = V1 × · · · × Vn , and A is finite. Let 0i = {a ∈ A | ∀vi ∈ Vi , vi (a) = 0 }. In an auction domain, 0i is exactly all the alternatives a for which ai = ∅. The proof relies exactly on the following five assumptions about the setting, and these five assumptions are a consequence of our four social choice requiremnts. 1. The function f satisfies Weak Monotonicity (W-MON) and weak-IIA (W-IIA), and the range is dense, i.e. for every player i > 1 there exists an alternative a ∈ A such that a1 ∈ / 01 and ai ∈ / 0i . 2. The domain V is closed under the min and max operators, i.e. if v, v 0 ∈ V then min(v, v 0 ) ∈ V and max(v, v 0 ) ∈ V (where the min and max are taken coordinate-wise). In addition, all valuation are non-negative, i.e. for any v ∈ V , any player i, and any alternative a, vi (a) ≥ 0. Notice that this is implied by the combinatorial structure of multi-item auctions. 3. (shift) Let (vi ↑ δ) denote the valuation vi0 such that vi0 (a) = vi (a) + δ for every alternative a∈ / 0i , and vi0 (a) = 0 for any a ∈ 0i . The shift property assumes that, for any vi ∈ Vi and δ > 0, (vi ↑ δ) ∈ Vi . Again, this is immediately implied by the combinatorial structure of multi-item auctions. 4. (top alternative) For every player, i, there exists an alternative ci ∈ A that satisfies the following several properties. For a multi-item auction, this will be the alternative in which the player receives all items, and for the more abstract definition we need some notations: i

for any v ∈ V , we denote by v|c +=δ the valuation vector v 0 = (vi0 , v−i ), where vi0 satisfies vi0 (a) = vi (a) for any alternative a 6= ci , and vi0 (ci ) = vi (ci ) + δ. Similarly, v|c valuation vi0 (ci )

v0

=

(vi0 , v−i ),

where

vi0

satisfies

= α. Now, a top alternative satisfies:

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vi0 (a)

i →α

= vi (a) for any alternative a 6=

is the

ci ,

and

• For any v−i ∈ V−i there exists vi ∈ Vi such that f (vi , v−i ) = ci . This property is implied from the decisiveness requirement. • For any i and vi ∈ Vi , vi (ci ) ≥ maxa∈A vi (a). For a multi-item auction this is of-course satisfied by the alternative that allocates all items to i, and this is easily seen also for the next three properties. i

• For any v ∈ V , and every player i, v|c +=δ ∈ V for every δ > 0. • If vi (ci ) − δ > maxa∈A,a6=ci vi (a), δ > 0, then v|c

i +=(−δ)

∈V.

• For any two players i 6= j, and any vi ∈ Vi , vi (cj ) = 0. 5. (joint-decisiveness) The following property is a formal translation of what we really need out of the unanimity property. For every alternative a ∈ A, every n positive real numbers α1 , ..., αn , and every  > 0, there exists v ∈ V (that depends on a, the α’s, and ), such that: • vi (a) = αi for every player i such that a ∈ / 0i , vi (ci ) < vi (a) + , and f (v) = a. For a multi-item auction that satisfies unanimity, this property is satisfied by simply fixing every player i with ai 6= ∅ to be single-minded for ai , with vi (ai ) = αi . • For any player that receives a non-empty bundle in a, we can raise her value for a by δ, and the other values by at most that, and the result will remain a. Formally, fix some player i such that a ∈ / 0i . Then, for every δ > 0 there exists vi0 ∈ Vi such that vi0 ≤ vi ↑ δ, vi0 (a) = vi (a) + δ, and f (vi0 , v−i ) = a. As before, this property is satisfied for a multi-item auction by simply fixing every player i with ai 6= ∅ to be single-minded for ai , with vi (ai ) = αi + δ. Thus, we get Claim 2 Any multi-item auction that satisfies the social choice requiremnts 1 – 4 also satisfies the above five properties.

A.2

Proof

Claim 3 For any v ∈ V , a ∈ A, and  > 0, 1. If f (v) = ci then f (v|c

i +=

) = ci .

i

2. If f (v) = a then f (v|c += ) ∈ {a, ci }. proof: If f (v) = ci and player i increases his value for ci by , while keeping the other values unchanged, then by W-MON the chosen alternative must remain ci . Now suppose f (v) = a, and i

i

let w = v|c += . Suppose by contradiction that f (v|c += ) = b ∈ / {a, ci }. Let w0 = min(v, w). Since

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w0 (a) = v(a) and w0 ≤ v then by W-IIA and W-MON, through Prop. 1 about downward-stability, f (w0 ) = a. On the other hand, since w0 (b) = w(b) and w0 ≤ w then f (w0 ) = b, a contradiction. Given some valuation v ∈ V , we start by assigning a “score” to each alternative (as over-viewed in section 4). Definition 5

fa (v(a)) =

 1 1 1   inf {w1 (c ) | w ∈ V s.t. f (w) = c and w(a) = v(a)} a 6= c  

v1 (c1 )

a = c1 .

Note that fa (·) is well defined for any v ∈ V and a ∈ A, as by the properties of the top alternative the infimum is a real number. Claim 4 If f (v) = b then fb (v(b)) ≥ v1 (c1 ). proof: Suppose by contradiction that fb (v(b)) < v1 (c1 ). By Def. 5 there exists w ∈ V such that f (w) = c1 , w(b) = v(b), and w1 (c1 ) < v1 (c1 ). Let w0 = min(w, v). By W-IIA, since w(b) = v(b) = w0 (b), it follows that f (w0 ) = b. But by W-IIA again, since w0 (c1 ) = w(c1 ), it follows that f (w0 ) = c1 , a contradiction. Claim 5 f (v) ∈ argmaxa∈A { fa (v(a)) } for every v ∈ V . proof: Suppose by contradiction that there exists v ∈ V such that f (v) = b and fb (v(b)) < fa (v(a)) for some a ∈ A. By the previous claim it follows that a 6= c1 . Fix some α such that fb (v(b)) < α < fa (v(a)), and let w = v|c

1 →α

. By claim 3, f (w) ∈ {b, c1 }. If f (w) = c1 then we get a contradiction

to the definition of fa (v(a)), as w(a) = v(a), and w1 (c1 ) < fa (w(a)). If f (w) = b 6= c1 then note that w(b) = v(b), therefore fb (w(b)) < α = w1 (c1 ), and we get a contradiction to claim 4. Claim 6 fa (·) is monotone non-decreasing, for any a ∈ A. proof: Suppose v(a) ≤ v 0 (a) (coordinate-wise), but by contradiction fa (v(a)) > fa (v 0 (a)). By Def. 5 there exist w, w0 ∈ V such that f (w) = f (w0 ) = c1 , w(a) = v(a), w0 (a) = v 0 (a), and w0 (c1 ) < fa (v(a)). Define w00 = min(w, w0 ). By W-MON and W-IIA, f (w00 ) = c1 . Note that w100 (c1 ) = w0 (c1 ) < fa (v(a)). But since w00 (a) = w(a) = v(a), we get a contradiction to the definition of fa (v(a)). Claim 7 limα→∞ fci (α) = ∞. proof: By definition fc1 (v(c1 )) = v1 (c1 ). Using the fact that V (c1 ) is not bounded we get that fc1 is unbounded, as well. For i > 1, player i can ensure the choice of ci , for every value v1 (c1 ) of player 1. By claim 5, this implies that fci (v(ci )) ≥ fc1 (v(c1 )), which implies the claim for i. 20

In what follows, we use the term fa (0, v−1 (a)). Though this is natural in our multi-item auction domain, the general domain may not include any v1 with v1 (a) = 0. In this case we take fa (0, v−1 (a)) = inf >0 fa (, v−1 (a)), and the proof goes through as is. Note that by the joint decisiveness assumption, fa (, v−1 (a)) is well defined for any  > 0. Claim 8 For any a 6= c1 , fa (v(a)) ≤ v1 (a) + fa (0, v−1 (a)). proof: If a1 ∈ 01 then v1 (a) = 0 and the claim is trivially true. Otherwise, fix some  ≥ 0 and some w ∈ V such that f (w) = c1 , w(a) = (, v−1 (a)), and w1 (c1 ) ≤ fa (, v−1 (a)) + 0 for some arbitrary small 0 > 0. Let w10 = w1 ↑ (v1 (a) − ), and w100 = w10 |c

1 +=+0

. Let w00 = (w100 , w−i ). By W-MON,

since f (w) = c1 , the transition from w to w00 implies that f (w00 ) = c1 . Since w00 (a) = v(a) we get that fa (v(a)) ≤ w100 (c1 ) = w1 (c1 ) + v1 (a) + 0 ≤ fa (, v−1 (a)) + v1 (a) + 20 . This is true for any 0 > 0 and therefore fa (v(a)) ≤ fa (, v−1 (a)) + v1 (a). In turn, this inequality is true for any  ≥ 0 (and in particular for  = 0 if fa (0, v−1 (a)) is well-defined), and therefore fa (v(a)) ≤ fa (0, v−1 (a)) + v1 (a), as claimed. Claim 9 For any a 6= c1 , fa (v(a)) ≥ v1 (a) + fa (0, v−1 (a)). proof: Suppose by contradiction that fa (v(a)) < v1 (a) + fa (0, v−1 (a)), and fix some α such that fa (v(a)) < α < v1 (a) + fa (0, v−1 (a)). Let w be some valuation such that w1 (c1 ) = α, w(a) = v(a), and f (w) = c1 . Fix some  > 0. By the joint-decisiveness assumption, there exists some valuation z ∈ V such 0 = min(w−1 , z−1 ). that z1 (a) = , z1 (c1 ) ≤ 1.5, z−1 (a) = w−1 (a) = v−1 (a), and f (z) = a. Let w−1 1 0 1 1 0 Since w−1 (c ) = z−1 (c ) = ~0, it follows that f (w1 , w−1 ) = c . Since w−1 = w−1 (a) = z−1 (a) = v(a), 0 ) = a. it follows that f (z1 , w−1

Now consider z10 = z1 |c

1 →α−v (a)+2· 1

(note that z10 (c1 ) > z1 (c1 )). By claim 3, the transition z1

0 ) ∈ {a, c1 }. On the other hand, the transition from w to z 0 implies, to z10 implies that f (z10 , w−1 1 1

by W-MON, that f (z1 , w−1 ) 6= a, as z10 (c1 ) − w1 (c1 ) = −v1 (a) + 2 ·  > −v1 (a) +  ≥ z10 (a) − w1 (a). 0 ) = c1 , and so f (, v (a)) ≤ α − v (a) + 2 · . Therefore we get f (z10 , w−1 a −1 1

To conclude, we get fa (0, v−1 (a)) ≤ inf >0 fa (, v−1 (a)) ≤ α − v1 (a). But this contradicts the choice of α, and the claim follows. Corollary 1 fa (v(a)) = v1 (a) + fa (0, v−1 (a)). We now continue to show that fa (0, v−1 (a)) is an affine function, which will conclude the proof. To simplify notation we denote ga (v−1 (a) ≡ fa (0, v−1 (a)). The hard part is to show that ga (·) is affine in the “interior” of the domain (this term is an abuse of notation, and is not the usual topological interior):

21

Definition 6 (The Interior) The interior of V , denoted by V ◦ , is defined by: V ◦ = {v ∈ V | vi (bi ) > 0 for every player i and every bi such that bi ∈ / 0i }. In words, v is in the interior if every player has a strictly positive value for any alternative in which she gets a non-empty bundle. We will also need, for a given alternative b ∈ A, the set V ◦ (b) = {v(b)| exists v 0 ∈ V ◦ such that v 0 (b) = v(b)}. Fix any alternative a such that a1 ∈ / 01 , and take i such that ai ∈ / 0i . Note that there exists at least one such alternative, for every player i > 1, due to the dense range assumption. Take any i

i

v, v 0 ∈ V such that v(a), v 0 (a) ∈ V ◦ (a), and fix some α ∈ < such that f (v|c →α ) = f (v 0 |c →α ) = ci . By the properties of a top alternative, such α always exists. Since gci (·) is monotone non-decreasing and unbounded, we can assume without loss of generality that: 1. α > max{v1 (a), v10 (a)}. 2. α is a continuity point of gci (·). 0 (a)). 3. gci ((α)(ci )) > ga (v−1 (a)) and gci ((α)(ci )) > ga (v−1

(this follows since we can always increase α, while keeping the outcome to be ci , until we reach such a point). The key observation to the entire proof is the following claim, that essentially says that the difference ga ((vi + δ, v−1,−i )(a)) − ga ((vi , v−1,−i )(a)) is independent of vi . Claim 10 For any δ > 0, ga ((vi + δ, v−1,−i )(a)) − ga ((vi , v−1,−i )(a)) = ga ((vi0 + δ, v−1,−i )(a)) − ga ((vi0 , v−1,−i )(a)) = gci ((α + δ)(ci )) − gci ((α)(ci )). proof: For convenience we drop the term v−1,−i (a) for the entire proof. Suppose by contradiction that ga ((vi (a) + δ)(a)) − ga (vi (a)) > gci ((α + δ)(ci )) − gci ((α)(ci )). By continuity at α, there are δ0 > δ and  > 0 such that ga ((vi + δ)(a)) − ga (vi (a)) >  + gci ((α + δ0 )(ci )) − gci ((α)(ci )). Define v1 (a) = gci ((α)(ci )) − ga (vi (a)) − /2. By the joint-decisiveness assumption, take some i

valuation w ∈ V such that w(a) = (v1 (a), v−1 (a)) and f (w) = a. Let u = w|c →α . Thus, f (u) ∈ {a, ci } by claim 3. We first show that a cannot be chosen. By claim 5, it is enough to show that fa (u(a)) < fci (u(ci )). Indeed, we have fa (u(a)) − fci (u(ci )) = v1 (a) + ga (vi (a)) − gci ((α)(ci )) = −/2 < 0. We thus get that f (u) = ci .

22

Now assume that i raises his value for a by δ and his value for ci by δ0 and denote this valuation as u0 . Formally, u0i = (ui ↑ δ)|c

i +=(δ 0 −δ)

, and u0 = (u0i , u−i ). By W-MON the chosen alternative

remains ci . However, we shall show that fa (u0 (a)) > fci (u0 (ci )), and thus a contradiction. Indeed: fa (u0 (a))−fci (u0 (ci )) = v1 (a)+ga ((vi +δ)(a))−gci ((α+δ0 )(ci )) > v1 (a)+ga (vi (a))−gci ((α)(ci ))+ > 0. For the other direction, we use a similar idea. Suppose by contradiction that ga ((vi + δ)(a)) − ga (vi (a)) < gci ((α + δ)(ci )) − gci ((α)(ci )). By continuity at α, there are δ0 > δ and  > 0 such that ga ((vi + δ0 )(a)) − ga (vi (a)) +  < gci ((α + δ)(ci )) − gci ((α)(ci )). Furthermore, by the fact that α > vi (a), we can choose δ0 such that α + δ > vi (a) + δ0 + ∗ , for a small enough ∗ > 0. Define v1 (a) = gci ((α)(ci )) − ga (vi (a)) + /2, take some valuation w ∈ V such that w(a) = i

(v1 (a), v−1 (a)), f (w) = a, and maxb∈A wi (b) < vi (a)+ ∗ . Let u = w|c →α . As above, f (u) ∈ {a, ci }, and, by the choice of v1 (a), we have fa (u(a)) > fci (u(ci )). Therefore f (u) = a. Now assume that i raises his value for a by δ0 and his value for ci by δ. Formally, by the joint-decisiveness property, there exists u0i ∈ V such that u0i ≤ ui ↑ δ0 , u0i (a) = ui (a) + δ0 , and i

f (u0i , u−i ) = a. Now let u00i = u0i |c →ui (c

i )+δ

(note that by the choice of δ0 , it is valid to increase ci

less than all other coordinates). By W-MON plus W-IIA, f (u00i , u−i ) = a as well. However, similar to above, fa (u0 (a)) < fci (u0 (ci )), and thus a contradiction. Therefore we get the claim for v(a). Exactly the same argument follows for vi0 (a). The above claim will immediately imply that ga (·) is affine, by using the following fact from real analysis (we provide a proof at the end of the section): Technical Claim:

Fix some monotone function g : 0. It remains to show that γa = 0. It must be the case that γa ≥ 0, otherwise we get a contradiction to the joint-decisiveness property, when fixing v(a) ∈ V ◦ (a) such that maxi vi (a) <  – for some small enough  we will get fa (v(a)) < fc1 (v(c1 )), contradicting f (v) = a. Similarly if γa > 0, then when fixing v(c1 ) ∈ V ◦ (c1 ) such that v1 (c1 ) < γa we will get fa (v(a)) > fc1 (v(c1 )), contradicting f (v) = c1 . It remains to handle the alternatives a ∈ 01 . We first handle the top alternatives ci for i > 1. Claim 12 For every player i and every v(ci ) ∈ V ◦ (ci ), gci (v(ci )) = wi · vi (ci ). proof: Note that vj (ci ) = 0 whenever j = 6 i, and therefore gci (v(ci )) depends only on vi (ci ). First take some α such that gci (α) > 0, and assume that α is a continuity point for gci (·). Fix some alternative a ∈ / 01 ∪ 0i , and some v ∈ V such that v(a) ∈ V ◦ (a), f (v) = a, and fci (α) > fa (v(a)). Such a v exists, since we can take v(a) to be as close to zero as we want (i.e. maxi vi (a) <  for any  > 0 that we choose), and therefore by using the fact that fa (v(a)) is affine and approaches zero as maxi vi (a) approaches zero, we can indeed satisfy the last property. f (v) = a implies that i

fa (v(a)) ≥ fb (v(b)), and since fci (α) > fa (v(a)) it follows that f (v|c →α ) = ci . Now notice that all the requirements of claim 10 regarding α and v(a) are satisfied, and therefore, for any δ > 0, wi · δ = ga ((v(a) + δ · ei ) − ga (v(a)) = gci (α + δ) − gci (α). For α which is not a continuity point of gci (·), choose some α0 < α that is a continuity point, and gci (α0 ) > 0. By the above argument, we know that for any δ0 > 0, gci (α0 + δ0 ) − gci (α0 ) = wi · δ0 . Now for any δ > 0, let δ0 = α − α0 + δ, and δ00 = α − α0 . We get: gci (α + δ) − gci (α) = gci (α0 + δ0 ) − gci (α0 ) − [gci (α0 + δ00 ) − gci (α0 )] = wi · δ0 − wi · δ00 = wi · δ. Thus we get gci (α) = wi · α (by using the technical claim), for any α such that gci (α) ≥ 0. By arguments similar to those in claim 11, we get that for any valid value for vi (ci ), gci (vi (ci )) ≥ 0 (i.e. if gci (vi (ci )) < 0 for some v ∈ V then decisiveness is violated when i values only ci , by vi (ci ), and the other players have zero valuations), and the claim follows. Claim 13 For any a ∈ 01 and any v(a) ∈ V ◦ (a), ga (v(a)) =

Pn

i=2 wi

· vi (a).

proof: We use a similar argument to that of claim 10, replacing the separation fa (v(a)) = v1 (a) + ga (v−1 (a)) by the fact that we now know that gci (·) is linear. Fix any v(a) ∈ V ◦ (a) and any player 24

i such that a ∈ / 0i . Suppose by contradiction that ga ((v−1 (a) + δ · ei ) − ga (v−1 (a)) > wi · δ. The linearity of gci (·) implies that, for any α > 0, there exists δ0 > δ and  > 0 such that, ga (v−1 (a)+ δ ·ei )− ga (v−1 (a)) >  + gci (α + δ0 ) − gci (α). Fix α such that gci (α) > ga (v−1 (a)) and gci (α + δ0 ) < ga (v−1 (a) + δ · ei ). Fix w ∈ V according to the joint-decisiveness property such that w(a) = v(a) and f (w) = a. Let u = w|c

i →α

. Thus f (u) ∈ {a, ci }, and a cannot be chosen since fa (u(a)) = ga (v−1 (a)) < gci (α).

Thus f (u) = ci . Now assume that i raises his value for a by δ and his value for ci by δ0 and denote this valuation as u0 . By W-MON the chosen alternative remains ci . However, now gci (α+δ0 ) < ga (v−1 (a)+δ·ei ) = fa (u0 (a)), a contradiction. If ga (v−1 (a) + δ · ei ) − ga (v−1 (a)) < wi · δ then we use a similar argument. Fix a valuation u such that u(a) = v(a) and f (u) = a. Thus gci (ui (ci )) ≤ ga (v−1 (a)) (recall that v1 (a) = 0). We first contradict the case of an equality. In this case, gci (ui (ci ) + δ) = gci (ui (ci )) + wi · δ, and therefore ga (v−1 (a) + δ · ei ) < gci (ui (ci ) + δ). By joint decisiveness, however, there exists a valuation u0 such that f (u0 ) = a, u0 (a) = u(a) + δ · ei , and u0i (ci ) = ui (ci ) + δ, a contradiction to claim 5. Now, assume that gci (ui (ci )) < ga (v−1 (a)). In this case choose δ0 < δ and α > ui (ci ) ≥ ui (a) such that: (1) α + δ0 > ui (a) + δ, (2) gci (α) < ga (v−1 (a)), and (3)gci (α + δ0 ) > ga (v−1 (a) + δ · ei ). i

Let u0 = u|c →α . As above, f (u0 ) = a. Now assume that i raises his value for a by δ and his value for ci by δ0 . Formally, by the joint-decisiveness property, there exists u00i ∈ V such that u00i ≤ ui ↑ δ, i

0

u00i (a) = ui (a) + δ, and f (u00i , u−i ) = a. Now let wi = u00i |c →α+δ , i.e. wi (ci ) < u00i (ci ) (note that by the choice of α and δ0 , it is valid to increase ci less than all other coordinates). By W-MON plus W-IIA, f (wi , u−i ) = a as well. However, fa (wi (a), u−i (a)) < fci (wi (ci )), and thus a contradiction. Thus ga (v−1 (a) + δ · ei ) − ga (v−1 (a)) = wi · δ for any v(a) ∈ V ◦ (a), and we conclude that ga (·) is affine. It remains to show that γa = 0. If γa < 0 we get a contradiction to the joint-decisiveness property, when fixing v(a) ∈ V ◦ (a) such that maxi vi (a) <  – for some small enough  we will get fa (v(a)) < 0 ≤ fci (v(ci )), contradicting f (v) = a. Similarly if γa > 0, then when fixing v(c1 ) ∈ V ◦ (c1 ) such that v1 (c1 ) < γa we will get fa (v(a)) > fc1 (v(c1 )), contradicting f (v) = c1 . Corollary 2 For every social choice function f : V → A that satisfies the requirements of section A.1, there exist positive weights w1 , ..., wn , such that, for every v ∈ V ◦ , f (v) ∈ argmaxa∈A

n X

wi · vi (a).

i=1

It thus only remains to handle the case of a valuation v ∈ / V ◦ . For this we will need the second part of the decisiveness requirement, that was not used until now, and that states the following. For any player i, and any v−i ∈ V−i , there exists vi ∈ Vi such that f (vi , v−i ) ∈ 0i . In essence, this says that any player can guarantee a “no participation” outcome from her point of view. For

25

simplicity, we assume that ~0 ∈ Vi , and therefore we get that f (~0, v−i ) ∈ 0i for all i and v−i ∈ V−i . This implies: Claim 14 For any v ∈ V , if f (v) = a then v(a) ∈ V ◦ (a). proof: Suppose by contradiction that f (v) = a and there exists some i such that a ∈ / 0i , but ~ vi (a) = 0. Thus, by W-MON and W-IIA, f (0, v−i ) = a ∈ / 0i , contradicting decisiveness. Claim 15 For every v ∈ V , f (v) ∈ argmaxa∈A

n X

wi · vi (a)

i=1

proof: Up to now we showed that if v(a) ∈ V ◦ (a), v(b) ∈ V ◦ (b) and

P

i wi vi (a)


0 then v(ci ) ∈ V ◦ (ci ). Now, fix some α1 such P P 1 that i wi vi (a) < w1 α1 < i wi vi (b), and let v10 = v1 |c →α1 . Since α1 > v1 (c1 ) (by the above i

observation), f (v10 , v−1 ) ∈ {c1 , a}. But since (v10 (a), v−1 ) ∈ V ◦ (a) and v10 (c1 ) ∈ V ◦ (c1 ) we get that

f (v10 , v−1 ) 6= a, and therefore f (v10 , v−1 ) = c1 . Increase v10 value for all non-empty bundles by  and v10 (c1 ) by 2. Call this v100 . By W-MON f (v100 , v−1 ) = c1 , and v100 ∈ V1◦ . Now fix some α2 such that P w1 α1 < w2 α2 < i wi vi (b), and repeat the above so that we have v200 ∈ V2◦ , f (v100 , v200 , v−1,2 ) = c2 , P 00 and w2 α2 < i wi vi (b). After repeating this for all players i = 1, ..., n, we get v such that P v 00 (b) ∈ V ◦ (b), wn vn00 (cn ) < i wi vi00 (b), and f (v 00 ) = cn , contradicting corollary 2 This concludes the proof of the theorem.

Proof of Technical Claim: We split the claim to two: Claim 16 Suppose m : R+ → R is monotonically non-decreasing and there exists h : R+ → R+ such that m(x+δ)−m(x) = h(δ) for any x, δ ∈ R+ . Then there exist ω ∈ R+ such that h(δ) = ω ·δ. proof: Let ω = h(1) (note that ω ≥ 0 since m is non-decreasing). First we claim that for any two Pq−1 m((i + 1)/q) − m(i/q) = integers p, q, h(p/q) = ω · (p/q). Note that h(1) = m(1) − m(0) = i=0 P q · h(1/q). Thus h(1/q) = (1/q) · h(1). Similarly, h(p/q) = m(p/q) − m(0) = p−1 i=0 m((i + 1)/q) −

m(i/q) = p · h(1/q) = (p/q) · h(1) = (p/q) · ω. Now we claim that for any real δ, h(δ) = δ · ω.

Notice that since m is monotonically non-decreasing then h must be monotonically non-decreasing as well. Suppose by contradiction that h(δ) > δ · ω. Choose some rational r > δ close enough to δ such that h(δ) > r · ω. Since h is monotone and r > δ then h(r) ≥ h(δ), but since r is rational, h(r) = r · ω < h(δ), a contradiction. A similar argument holds if h(δ) < δ · ω. 26

Claim 17 Suppose that X ⊆ Rn has the property that x ∈ X and y ≥ x implies y ∈ X. Let m : X → R be monotonically non-decreasing, and suppose there exist ω1 , . . . , ωn such that m(x + δ · ei ) − m(x) = ωi · δ for any i, x ∈ X, and δ > 0. Then there exist γ ∈ R such that m(x) = Pn i=1 ωi · xi + γ. proof: First we claim that for any x, y ∈ X such that yi ≥ xi for all i, it is the case that P m(y) = m(x) + ni=1 ωi · (yi − xi ). Notice that (y1 , x2 , . . . , xn ) ∈ X, and m(y1 , x2 , . . . , xn ) = P m(x) + h1 (y1 − x1 ). Repeating this step n times we get m(y) = m(x) + ni=1 ωi · (yi − xi ). P Now fix some x∗ ∈ X. We claim that for any x ∈ X, m(x) = m(x∗ ) + ni=1 ωi · (xi − x∗i ). To P see this, choose some y such that yi ≥ xi , x∗i for all i. Thus m(y) = m(x) + ni=1 ωi · (yi − xi ) and P also m(y) = m(x∗ ) + ni=1 ωi · (yi − x∗i ), therefore the claim follows.

B

Other Proofs

Proposition 1 Every ex-post implementable social choice function (for the private values setting) that additionally satisfies weak-IIA also satisfies the following downward-stability property: For every v, v 0 ∈ V such that v 0 ≤ v (coordinate-wise), if f (v) = a, and v(a) = v 0 (a) (coordinate-wise), then f (v 0 ) = a. proof: We first show that f (v10 , v−1 ) = a. Suppose by contradiction that f (v10 , v−1 ) = b 6= a. Recall that weak monotonicity (definition 2) is necessary for ex-post implementability on every domain. Therefore it follows that v10 (b)−v1 (b) ≥ v10 (a)−v1 (a) = 0. Since v10 ≤ v1 , it follows that v10 (b) = v1 (b), and we get a contradiction to weak-IIA. Thus f (v10 , v−1 ) = a. Similarly, f (v10 , v20 , v−1,−2 ) = a, and so on and so forth. Thus f (v 0 ) = a as claimed.

Proposition 2 A multi-item auction satisfies Strong Monotonicity (S-MON) if and only if it satisfies Weak Monotonicity and Quasilinear-IIA. proof: We prove the proposition using several claims: Claim 18 If f satisfies W-MON and Quasilinear-IIA then f satisfies S-MON. proof: Fix any v ∈ V , player i, and ui ∈ Vi . Suppose f (v) = a and f (ui , v−i ) = b. We need to show that ui (b) − vi (b) > ui (a) − vi (a). By W-MON it follows that ui (b) − vi (b) ≥ ui (a) − vi (a). Suppose by contradiction that ui (b) − vi (b) = ui (a) − vi (a). But then, denote u = (ui , v−i ), and we have f (v) = a, f (u) = b, and for any player j, vj (a) − vj (b) = uj (a) − uj (b), thus contradicting Quasilinear-IIA. For the other direction, we first construct a non-normalized auction domain V˜i , i.e. a domain for

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which the valuations need not necessarily have vi (∅) = 0, by shifting all valuations up-wards: V˜i = {˜ vi ∈ ui (a) − vi (a). Therefore u ˜i (b) − v˜i (b) > u ˜i (a) − v˜i (a), and thus f˜ satisfies S-MON. For the second part, simply notice that, since V ⊆ V˜ , contradicting Quasilinear-IIA for f implies contradicting Quasilinear-IIA for f˜. Claim 20 (Dependence on Differences (DOD)) If f˜ satisfies S-MON then, f˜(˜ v ) = f˜(˜ vi + δ · ˜ (1, ..., 1), v˜−i ), for any v˜ ∈ V , δ ∈ u ˜i (a) − v˜i (a). proof: By claim 20 we can assume w.l.o.g. that u ˜i (a) = v˜i (a) for every i: otherwise let v˜0 i = v˜i + [˜ ui (a)−˜ vi (a)]·(1, ..., 1). Therefore f˜(v˜0 ) = a, and finding i such that u ˜i (b)− v˜0 i (b) > u ˜i (a)− v˜0 i (a) = 0 implies that u ˜i (b) − v˜i (b) > u ˜i (a) − v˜i (a). Now, we “move” from v˜ to u ˜ by L “elementary steps” v˜ = v 1 , v 2 , . . . , v L = u ˜, such that: (1) for any index j there exists a player i and d ∈ A such that vij+1 (a) = vij (a) for any a 6= d, and vij+1 (d) = vij (d) + u ˜i (d) − v˜i (d); (2) every pair (i, d) appears only once in the sequence; and (3) there

exists an index l∗ such that for any l ≤ l∗ , u ˜i (d) − v˜i (d) < 0, and for any l > l∗ , u ˜i (d) − v˜i (d) > 0 ˜ (note that we can indeed construct such elementary steps since V is an auction domain). By S∗ MON, f˜(v l ) = a, and for any l > l∗ , if f˜(v l ) = c then f˜(v l+1 ) ∈ {c, d} (where d is the alternative that changes from v l to v l+1 ). Therefore, if f˜(v L ) = b it follows that there exists i such that ui (b) − vi (b) > 0, as claimed. 28

The above sequence of claims proves the proposition: By the assumption of the proposition, f satisfies S-MON, and therefore so does f˜ (by claim 19). By claim 21, this means that f˜ satisfies Quasilinear-IIA (note that the condition in the claim implies Quasilinear-IIA). Finally, again using claim 19, this implies that f satisfies Quasilinear-IIA, as we need. It is interesting to note that S-MON does not always imply Quasilinear-IIA, only for specific domains (like our auction domains). For example, suppose there are four alternatives (A = {a, b, c, d}) and two players, each one with two possible types vi , ui such that: u1 (c) − v1 (c) > u1 (a) − v1 (a) = u1 (b)−v1 (b) > u1 (d)−v1 (d), and u2 (d)−v2 (d) > u2 (a)−v2 (a) = u2 (b)−v2 (b) > u2 (c)−v2 (c). Define f as follows: f (v1 , v2 ) = a, f (u1 , u2 ) = b, f (u1 , v2 ) = c, and f (v1 , u2 ) = d. It is not hard to verify that S-MON holds (there are four inequalities to check, all of them follow from the way the types are defined). Quasilinear-IIA does not hold since f (v) = a, f (u) = b, but u(a) − u(b) = v(a) − v(b).

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