1 Introduction
Imagine that you are a social planner wanting to auctionoff the seats of a local stadium at an extremely wealthy neighborhood (i.e., people have no budget constraints for the seats) for a big concert. As a social planner, your goal is to allocate the seats in a way that maximizes (or, at least, approximates as closely as possible) the happiness of the people interested in these seats. However, different people have different seat preferences; some people are happy with two consecutive seats anywhere in the stadium, some might be happy with only one seat in front of the stage, and some might want a whole row. Phrased in mechanism design language, this is a Combinatorial Auction, where you seek to optimize the Social Welfare by a truthful mechanism. Combinatorial Auctions, like the one above, appear in many contexts (e.g., spectrum auctions, network routing auctions [19], airport timeslot auctions [24], etc.) and have been extensively studied by both Economists and Computer Scientists (see e.g.,[6] for a survey).
As if this problem was not hard enough to solve, imagine that you find out two unfortunate events; the stadium is in fact at a workingmiddle class neighborhood (i.e., people do have budget constraints) and your boss is concerned about the effect of these budget constraints on the potential revenue. Now, the objective function should balance between the willingness and the ability of the people to pay for their seats. Motivated by usual discrepancies between the auction participants’ ability and willingness to pay, Dobzinski and Leme [10] introduced the notion of Liquid Welfare, which is the minimum of an agent’s budget and valuation for a bundle of goods. As such, maximizing the Liquid Welfare achieves a reasonable compromise between social efficiency and potential for revenue extraction (which is constrained by the budgets).
Problem Definition. More formally, a Combinatorial Auction (CA) consists of a set of items to be allocated to bidders. Each bidder has a valuation function . Valuation functions, , are assumed to be nondecreasing, i.e., , for all , and normalized . For the objective of Social Welfare (SW), the goal is to compute a partitioning of the set of items, , that maximizes . For the objective of Liquid Welfare (LW), we assume that each bidder also has a budget and the liquid welfare that can be extracted from agent for each set of items is ^{1}^{1}1Slightly abusing the terminology, we refer to as agent ’s liquid valuation.. Under this objective, the goal is to compute a partitioning of that maximizes .
We focus on CAs with submodular or XOS bidders. A set function is submodular if for every , and subadditive if . A set function is XOS (a.k.a. fractionally subadditive, see [17]) if there exist additive functions such that for every , . The class of submodular functions is a proper subset of the class of XOS functions, which is a proper subset of the class of subadditive functions.
Since bidder valuations have exponential size, a polynomial (in and ) algorithm must have oracle access to them. A value query specifies a set and receives the value . A demand query, denoted by , specifies a valuation function , a set of available items and a price for each available item , and receives the set (or bundle) maximizing , i.e., the set of available items that maximizes bidder’s utility at these prices. For brevity, we often write to denote the price of a bundle . Demand queries are strictly more powerful than value queries. Value queries can be simulated by polynomially many demand queries, and in terms of communication cost, demand queries are exponentially stronger than value queries [3].
1.1 Previous Work on Social Welfare
Truthful maximization of SW in CAs with submodular or XOS bidders has been a central problem in Algorithmic Mechanism Design, with many strong and deep results. Due to space restrictions, we only discuss results most relevant to our work. While discussing previous work below, we assume XOS bidders and polynomialtime randomized truthful mechanisms that approximate the SW, by accessing valuations through demand queries, unless mentioned otherwise.
In the worstcase setting, where we do not make any further assumptions on bidder valuations, Dobzinski et al. [11] presented the first truthful mechanism with a nontrivial approximation guarantee of . Dobzinski [7] improved the approximation ratio to for the more general class of subadditive valuations. Subsequently, Krysta and Vöcking [20] provided an elegant randomized online mechanism that achieves an approximation ratio of for XOS valuations. Dobzinski [9] broke the logarithmic barrier for XOS valuations, by providing an approximation guarantee of . We highlight that accessing valuations through demand queries is essential for these strong positive results. Dobzinski [8] proved that any truthful mechanism for submodular CAs with approximation ratio better than must use exponentially many value queries.
In the Bayesian setting, bidder valuations are drawn as independent samples from a known distribution. Feldman et al. [18] showed how to obtain item prices that provide a constant approximation ratio for XOS valuations. These results were significantly extended and strengthened in the recent work of Düetting et al. [14].
1.2 Intuition, Main Ideas, and Contribution
Our aim is to extend these results to the objective of LW. To this end, we exploit the fact that most of the mechanisms above (and the mechanisms of Krysta and Vöcking [20] and Feldman et al. [18], in particular) follow a simple pattern: first, by exploring either part of the instance in [20] or the knowledge about the valuation distribution in [18], the mechanism computes appropriate (a.k.a. supporting^{2}^{2}2
A price vector
supports allocation , if .) prices for all items. Then, these prices are “posted” to the bidders, who arrive onebyone and select their utilitymaximizing bundle, through a demand query, from the set of available items (see Algorithm 1).The technical intuition behind the high level approach above is nicely explained in [9, Section 1.2]. Let be an optimal solution for the SW (in fact, any constant factor approximation suffices). The supporting price of item in is , where is the additive valuation determining the value (recall that valuation functions are XOS). Intuitively, is how much item contributes to the social welfare of . Then, a price of for each item is appropriate in the sense that a constant approximation to can be obtained by letting the bidders arrive onebyone, in an arbitrary order, and allocating to each bidder her utility maximizing bundle, chosen from the set of available items by a demand query (see [9, Lemma 4.2]).
Hence, approximating the SW by demand queries boils down to computing such prices . In the Bayesian setting, prices can be obtained by drawing samples from the valuation disribution and computing the expected contribution of each item to a constant factor approximation of the optimal allocation (see Section 3 and Lemma 3.4 in [18]
). Similarly, the idea of estimating the contribution of the items would work under some market uniformity assumption, as the one introduced in Definition
5.1. In the worstcase setting, if we assume integral and polynomiallybounded valuations (i.e., that , for some constant ), a uniform price for all items selected at random from results in an logarithmic approximation ratio. Krysta and Vöcking [20] show how to estimate supporting prices online, by combining binary search and randomized rounding. Importantly, as long as each bidder does not affect the prices offered to her, this general approach results in (randomized universally) truthful mechanisms.Towards extending the above approach and results to the LW, our first observation (Lemma 3.1) is that if a valuation function is submodular (resp. XOS), then the corresponding liquid valuation function is also submodular (resp. XOS). Then, one can directly use the mechanisms of e.g., [20, 9, 18] with valuation functions and demand queries of the form: (i.e., wrt. the liquid valuation of the bidders) and obtain the same approximation guarantees but now for the LW. However, the resulting mechanisms are no longer truthful; bidders still seek to maximize their utility (i.e., value minus price) from the bundle that they get, subject to their budget constraint, rather than their liquid utility (i.e., liquid value minus price). Specifically, given a set of items available at prices , , a budgetconstrained bidder wants to receive the bundle , and might not be happy with the bundle computed by the demand query for the liquid valuation^{3}^{3}3For a concrete example, consider a bidder with budget and two items and available at prices and . Assume that the bidder’s valuation function is , (and therefore, her liquid valuation is ). The bidder wants to get item at price , which gives her utility . However, the demand query for her liquid valuation function allocates item , which gives her a utility of . Clearly, in this example, the bidder would have incentive to misreport her preferences to the demand query oracle..
To restore truthfulness, we replace demand queries with budgetconstrained demand queries. A budgetconstrained demand query, denoted by , specifies a valuation function , a set of available items , a price for each and a budget , and receives the set maximizing , subject to , i.e., the set of available items that maximizes bidder’s utility subject to her budget constraint.
To establish the approximation ratio, we first observe that the fact that liquid valuations are XOS suffices for estimating supporting prices, as in previous work on the SW. Additionally, we show that the bundles allocated by approximately satisfy the efficiency guarantees on the liquid welfare and the liquid utility of the allocated bundles (see Lemma 3.3). Specifically, we observe that the approximation guarantees of mechanisms for the SW mostly follow from the fact that a demand query guarantees that for the allocated bundle and for any , , and . In Lemma 3.3, we show that a budgetconstrained demand query, , guarantees that for the allocated bundle and any , , and . Using this property, we can prove the equivalent of [9, Lemma 4.2] (for completeness, we provide the proof in the Appendix, Lemma B.2) and also the approximation guarantees of the mechanisms in Krysta and Vöcking [20], Feldman et al. [18] but for the LW.
Contribution.
Formalizing the intuition above, we obtain a randomized universally truthful mechanism that approximates the LW within a factor of (Section 4), and a postedprice mechanism that approximates the LW within a constant factor when bidder valuations are drawn as independent samples from a known distribution (Section 6). Both mechanisms assume XOS bidder valuations; the former is based on the mechanism of Krysta and Vöcking [20] and the latter on the mechanism of Feldman et al. [18]. Motivated by large market assumptions often used in Algorithmic Mechanism Design (see e.g., [4, 16, 23] and the references therein), we introduce a competitive market assumption in Section 5. The main idea is that when there is an abundance of bidders, even if we remove a random half of them, the optimal LW does not decrease by much. Then, computing supporting prices for all items based on a randomly chosen half of the bidders, and offering these prices through budgetconstrained demand queries to the other half, yields a universally truthful mechanism that approximates LW within a constant factor (Theorem 5.5). Conceptually, in this work, we present a general approach through which known truthful approximations to the SW, that access valuations through demand queries, can be adapted so that they retain truthfulness and achieve similar approximation guarantees for the LW. The important properties required are that liquid valuation functions belong to the same class as valuation functions (proven for submodular, XOS and subadditive valuations), and that the efficiency guarantees of budgetconstrained demand queries on liquid welfare and liquid utility are similar to the corresponding efficiency guarantees of standard demand queries for liquid valuations (proven for all classes of valuations functions). Indeed, applying this approach to the mechanism of Dobzinski [9], we obtain a universally truthful mechanism that approximates the LW for CAs with XOS bidders within a factor of (the details are omitted due to space constraints). Similarly, we can take advantage of the improved results of Düetting et al. [14] in the Bayesian setting.
1.3 Previous Work on Liquid Welfare
Liquid Welfare was introduced as an efficiency measure for auctions when bidders are budget constrained in [10], since it was known that getting any nontrivial approximation for the SW in these cases is impossible. Moreover, Dobzinski and Leme [10] proved a (resp. )approximation to the optimal LW for the case of a single divisible item and submodular (resp. subadditive) bidders. Dobzinski and Leme [10] and Lu and Xiao [22] proved that the optimal LW can be approximated truthfully within constant factor for a single divisible good, additive bidder valuations and public budgets. Closer to our setting, Lu and Xiao [23] provided a truthful mechanism that achieves a constant factor approximation to the LW for multiitem auctions with divisible auctions, under a large market assumption. Under similar large market assumptions, Eden et al. [16] obtained mechanisms that approximate the optimal revenue within a constant factor for multiunit online auctions with divisible and indivisible items, and a mechanism that achieves a constant approximation to the optimal LW for general valuations over divisible items. However, prior to our work, there was no work on approximating the LW in CAs (in fact, that was one of the open problems in [10]).
Our work is remotely related to the literature of budget feasible mechanism design, a topic introduced by Singer [25] and studied in e.g., [13, 5, 2, 1, 26]. Budget feasible mechanism design focuses on payment optimization in reverse auctions, a setting almost orthogonal to the setting we consider in this work.
2 Notation and Preliminaries
The problem and most of the terminology and the notation are introduced in Section 1. In this section, we introduce some additional notation required for the technical part.
We use
to denote the expectation of a random variable
andto denote the probability of an event
. Let OPT (resp. ) denote the optimal SW (resp. LW)^{4}^{4}4The instance is clear from the context.. For some , which may depend on and , we say that a mechanism is approximate for the optimal SW (resp. LW) if it produces an allocation with (resp. ).Let a social choice function , which maps the set of liquid valuation functions of the bidders, , to an allocation, , and a payment scheme for this allocation. A deterministic mechanism is defined by the pair . Our mechanisms in this work are going to be randomized
, i.e., they are probability distributions over
deterministic mechanisms. The incentives desiderata for randomized mechanisms are usually either universal truthfulness (when for all the deterministic mechanisms, the bidders’ dominant strategy is truthfulness) or truthfulness in expectation [12, 15] (when bidders’ expected utilities are maximized under truthful reporting of their private information). In this work, we are focusing on the former, stronger notion; the one of universal truthfulness, under the bidders’ budget constraints.Definition 2.1 (Universal Truthfulness under Budget Constraints).
Let be a randomized mechanism over a set of deterministic mechanisms . Mechanism is universally truthful if for all and for any and any , such that and , it holds that: v_i(f^κ(v_i,B_i, v_i,B_i))  q^κ(v_i,B_i, v_i, B_i) ≥v_i(f^κ(v_i’, B_i’, v_i,B_i))  q^κ(v_i’, B_i’, v_i, B_i)
3 Approach
First, we show that if the if the bidder valuations are submodular (resp. XOS, subadditive), then their liquid valuations are submodular (resp. XOS, subadditive) as well. The proof of the following Lemma can be found in Appendix A.
Lemma 3.1.
Let be a nonnegative monotone submodular (resp. XOS, subadditive) function. Then, for any , is also monotone submodular (resp. XOS, subadditive).
In Algorithm 1, we present a universally truthful (since the prices offered to each bidder do not depend on her declaration and demand queries maximize bidders’ utility) mechanism, which is a simplified version of the mechanism in [20] for approximating SW in CAs. Since for the LW, bidders have budgets, we replace the demand queries in line 4 with budget constrained demand queries . As explained in Section 1.2, Algorithm 1 with s remains universally truthful for budgetconstrained bidders.
Lemma 3.2 (Truthfulness of BCDQs).
For budgetconstrained bidders, Algorithm 1 with s in line 4, is universally truthful.
The lemma follows directly from Definition 2.1. Nevertheless, universal truthfulness is not our sole desideratum; in each of the three settings analyzed in the following sections, we show why mechanisms similar in spirit to Algorithm 1 with s, yield good approximation guarantees for the LW. Before the settingspecific analyses, we relate the efficiency of to the efficiency of standard s for liquid valutions.
Lemma 3.3.
Let be the set allocated by the BCDQ for a bidder with valuation and budget . Then, for every other , the following hold: .
Proof.
We will prove each claim of the lemma separately. For claim 1, notice that if , then the Right Hand Side (RHS) of the inequality will be negative and thus, the inequality trivially holds. So, we will focus on the case where . We distinguish the following cases:

( and .) Hence, . Bundle was considered at the time of the query and yet, the query returned set . Thus: .

( and ) Then, the inequality trivially holds since: and prices are nonnegative.

( and ) The inequality holds since: .

( and ) Hence, . Bundle was considered at the time of the query and yet, the query returned set . Thus, .
This concludes our proof for claim 1.
4 WorstCase Setting
For the worstcase instances, adapting appropriately our Core Mechanism, we present Algorithm 2 (based again, on the mechanism of [20]). The only difference is that budgetconstrained bidders in Algorithm 2 are restricted to using s, instead of s, thus making the mechanism universally truthful (see Section 3). Resembling the analysis of [20], we show that for , Algorithm 2 achieves an approximation ratio of for the LW. First, we note that parameter^{5}^{5}5 can be computed with standard techniques, as explained in [20]. is selected so that there exists only one bidder whose liquid valuation for (weakly) exceeds it.
Theorem 4.1.
Algorithm 2 is universally truthful and for , achieves an approximation ratio of for the LW.
We present a series of Lemmas that will lead us naturally to the proof of the Theorem. Let and the provisional and the final allocation of Algorithm 2 respectively. First, we provide two useful bounds on . We find it important to also discuss an overselling variant of Algorithm 2. In the Overselling variant, allow us to assume that for Step 5 of Algorithm 2, (i.e., is allocated to bidder with certainty) and (thus the name of the variant). The Overselling variant allocates at most copies of each item and collects a liquid welfare within a constant factor of the optimal LW. To see that, observe that for , after allocating copies of some item , ’s price becomes . Then, there is only one agent with liquid valuation larger than who can get a copy of .
Lemma 4.2.
Let denote the final price of each item . Then, for any sets of items available when the bidders arrive, Algorithm 2 with satisfies .
Proof.
Since bidders are individually rational and do not violate their budget constraints, for every bidder it holds that and . The rest of the proof is identical to the proof of [20, Lemma 2] for . Specifically, let be the number of copies of item allocated just before bidder arrives, and let be the total number of copies of item allocated by Algorithm 2 with . Then,
where we have changed the order of summation and we have used the fact that . ∎
Lemma 4.3.
For sets , the Overselling variant of Algorithm 2 with satisfies .
Proof.
Let be the optimal allocation. From Lemma 3.3, we get that for each bidder , , where we use that the final price of each item is the largest one. Summing over all bidders, we have that , where the last inequality uses the fact that the optimal solution is feasible and thus, each item is allocated at most once in . ∎
Lemma 4.4.
The Overselling variant of Algorithm 2 with allocates at most copies of each item and computes an allocation with liquid welfare .
Of course, the allocation in Lemma 4.4 is highly infeasible, since it allocates a logarithmic number of copies of each item. The remedy is to use an allocation probability . For such values of , we can plugin the proof of [20, Lemma 6] (which just uses that the valuation functions are fractionally subadditive) and show that for each agent and for all , . We are now ready to conclude the proof of Theorem 4.1.
Lemma 4.5.
For Algorithm 2 with , it holds that and .
Proof.
Let be the optimal allocation. For each bidder , Lemma 3.3 implies that the response of satisfies , for any resulted from the outcome of the random coin flips. Therefore, . By the choice of , for any bidder , . Then, working with the expectations as in the proofs of Lemma 4.2, Lemma 4.3 and Lemma 4.4, we can show that . Finally, one can use linearity of expectation and show that . The details are omitted, due to lack of space, and can be found in [20, Lemma 4]. ∎
5 Competitive Market
Borgs et al. [4] were the first ones to define a budget dominance parameter that corresponded to the ratio of the maximum budget of all the bidders to the value of the optimum SW in the context of multiunit auctions with budgetconstrained bidders. More recently, Eden et al. [16] and Lu and Xiao [23] used similar notions of budget dominance^{6}^{6}6Namely, that , where is a large constant. (termed large market assumptions) as a means to achieve constant factor approximation to the LW in multiunit auctions and auctions with divisible items respectively. However, for the case of divisible items, it is clear that the definition of a large market used in the previous cases, becomes almost void (see Appendix C for a discussion). Below, we first introduce our definition of Competitive Markets for indivisible goods and then, show how one can obtain a constant factor approximation of the optimal LW, when bidders have XOS liquid valuations.
Definition 5.1 (  Competitive Market).
Let and a constant . A market is called  Competitive Market, if for any randomly removed set of bidders, , with cardinality , then for the remaining set of bidders, , it holds that:
(1) 
where by we denote the optimal LW achieved by bidders in set .
Proposition 5.2.
In an  Competitive Market, let be randomly chosen s.t. and let . Then:
Proof.
Let the event that and the event that . Then, we have:
where the inequality follows from the Union Bound. ∎
We are now ready to state our Competitive Market mechanism that will be used for approximating the optimal LW. We note here that the greedy algorithm is due to Lehmann et al. [21].
As usual, we denote the final allocation from mechanism presented in Algorithm 3. Valuations of bidders are XOS (and so are the liquid valuations (Lemma 3.1)); let be the maximizing clause of in the liquid valuation of bidder . Since ’s are additive, for each bidder and let . Notice that . We denote by , where is the contribution of item in . We divide the set of all items into two sets; the set of competitive items, denoted by and the set of noncompetitive items, denoted by . The following lemma upper bounds the contribution of noncompetitive items in the optimal solution.
Lemma 5.3.
Let for constant . Then, and .
Proof.
From Definition 5.1, it holds with constant probability (w.c.p) that: . Let be the set of the bidders that are allocated the noncompetitive items from the greedy algorithm when running on set . Then, in the augmented set , there exists an allocation ^{7}^{7}7Allocation is realized by allocating all items in to bidders in that also had them in the allocation and all items in to the bidders in that had them in the allocation of the greedy . The claim is completed by submodularity. with liquid valuation,
(2) 
and therefore we have w.c.p:
Rearranging the latter and using the fact that:
As a result, for the items in it holds w.c.p that: . ∎
In the next Lemma, we prove a lower bound on the contribution of competitive items to the solution obtained by the greedy algorithm, with respect to .
Lemma 5.4.
.
Proof.
Theorem 5.5.
The CM Algorithm is universally truthful and achieves, on expectation, a constant approximation to the optimal LW, i.e., .
Proof.
Since the bidders that control the prices being posted belong to set and they never get any item, it is their (weakly) dominant strategy to report their valuations and their budgets truthfully. Furthermore, the bidders that are buying under the said posted prices belong to set and they make BCDQs, which we shown to be truthful. Finally, the bidders are uniformly at random split at sets and .
For each item we have . Therefore, there exists an allocation for bidders in and items in that is supported by prices , where . Thus, from a modification of [9, Lemma 4.2] (formally presented in Lemma B.2), setting , for each , and running a fixed price auction in with prices , we get that: . Using the latter, along with the prices of the items, we have:
where the last inequality is due to Lemma 5.4. Thus, we conclude that:
∎
6 Bayesian Setting
The Bayesian Setting offers a great middle ground between the unstructured worstcase instances and the very structured Competitive Markets. In this setting, let be a profile of bidder valuations and a profile of bidder budgets. Assume that the bidders’ valuations are drawn independently from distributions and the budgets from distributions . For simplicity, let us assume that their liquid valuations are drawn independently from distributions . We will denote by the product distribution where liquid valuations profiles, , are independently drawn from.
We are going to show that the results presented in Feldman et al. [18] can be extended for budgetconstrained bidders. Specifically, we are going to show that, if liquid valuations are fractionally subadditive, then we can create appropriate prices such that, when presented to the bidders in a postedprice mechanism and bidders are making BCDQs, then we can obtain universally truthful constantfactor approximation mechanisms for the LW in Bayesian CAs. Our Lemma 6.2 establishes the existence of such appropriately scaled prices. The key component activating our results is that instead of reasoning about the utility achieved from the bundle purchased by bidder (as received by the BCDQ), we instead have to use Lemma 3.3. We state the Theorem and the Lemma below but we refer the reader to Appendix D for the full proofs. We also note that using our techniques one could even achieve the better approximation guarantees presented by Düetting et al. [14]. Their analysis is significantly more complex, however, and we omit it in the interest of space. The proofs of this Section can be found in Appendix D.
Theorem 6.1.
Let distribution over XOS liquid valuation profiles be given via a sample access to . Suppose that for every , we have: blackbox access to a LW maximization algorithm, ALG^{8}^{8}8ALG can be any algorithm that provides a approximation to the optimal LW, since we do not care about incentives (access to ALG will only happen for ghost samples). For example, it could be the greedy algorithm by Lehmann et al. [21]. for CAs. an XOS value query oracle (for liquid valuations sampled from )^{9}^{9}9An XOS value oracle takes as input a set and returns the corresponding additive representative function for the set , i.e., an additive function , such that (i) for any and (ii) .. Then, for any , we can compute item prices in time such that, for any bidder arrival order, the expected liquid welfare of the posted price mechanism is at least , where by we denote the solution produced by algorithm ALG.
Lemma 6.2.
Given a distribution over XOS liquid valuations, let be the price vector s.t. . Let be any price vector such that for all . Then, for any arrival order, , bidders buying bundles by making BCDQs under prices results in expected liquid welfare at least .
7 Conclusion and Open Questions
In this work, we showed how some of the best known truthful mechanisms that approximate the SW, can be adapted in order to yield the same order approximations for the LW, when bidders are budgetconstrained in the worstcase and Bayesian instances. Additionally, we introduced a notion of market competitiveness, for markets with indivisible goods and provided a constant factor approximation to the LW in this case. The most meaningful question that arises from our work (apart, of course, from the ever existent one of lowering the approximation guarantee in worstcase instances) is related to the competitive markets. We conjecture that the condition that we provide can be made even weaker, and leave it to future research.
We hope that the results and the techniques presented in this work, will serve future researchers in obtaining improved same order approximations for both the SW and the LW.
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Appendix A Supplementary Material for Section 3.
Proof of Lemma 3.1.
Clearly, capping valuation with budget does not affect monotonicity. We provide the proof for each case (i.e., submodular, XOS, subbaditive) separately.

(submodular) Let be a monotone submodular set function. Then, by the definition of submodularity, for sets and we have:
(5) Further, since is monotone: , which implies that . We distinguish the following cases:

If . Then, for the liquid valuations we have: , where the first inequality is due to monotonicity.

If . Then, .

If . This breaks down to the following two cases; on the one hand, if then, . On the other hand, if , then . Finally, we remark that due to monotonicity, these cases are the only possible ones.


(XOS) Let be an XOS set function; there exist additive functions s.t. . In order for to XOS, we need to prove that there exist additive functions s.t. . For each function we are going to define functions, one for each permutation of the items. Suppose a specific ordering of the items and let be the position of item in ordering . We define as: , if or , if . First, we are going to prove that for each .
By the definition of , it is clear to see that . Therefore, summing upon all items in (since we have additive functions), we get that:
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