Combinatorial Auctions with Endowment Effect
aa r X i v : . [ c s . G T ] M a y Combinatorial Auctions with Endowment Effect
Moshe Babaioff Shahar Dobzinski Sigal OrenMay 29, 2018
Abstract
We study combinatorial auctions with bidders that exhibit endowment effect. In most of theprevious work on cognitive biases in algorithmic game theory (e.g., [Kleinberg and Oren, EC’14]and its follow-ups) the focus was on analyzing the implications and mitigating their negativeconsequences. In contrast, in this paper we show how in some cases cognitive biases can beharnessed to obtain better outcomes.Specifically, we study Walrasian equilibria in combinatorial markets. It is well known thatWalrasian equilibria exist only in limited settings, e.g., when all valuations are gross substitutes,but fails to exist in more general settings, e.g., when the valuations are submodular. We considercombinatorial settings in which bidders exhibit the endowment effect , that is, their value for itemsincreases with ownership.Our main result shows that when the valuations are submodular, even a mild degree of endow-ment effect is sufficient to guarantee the existence of Walrasian equilibria. In fact, we show thatin contrast to Walrasian equilibria with standard utility maximizing bidders – in which the equi-librium allocation must be efficient – when bidders exhibit endowment effect any local optimumcan be an equilibrium allocation.Our techniques reveal interesting connections between the LP relaxation of combinatorialauctions and local maxima. We also provide lower bounds on the intensity of the endowmenteffect that the bidders must have in order to guarantee the existence of a Walrasian equilibriumin various settings.
Research in algorithmic mechanism design typically assumes that bidders are utility maximizers,i.e., they maximize their value for the chosen alternative minus their payment. However, empiricalevidence from behavioral economics suggests that this assumption is often inaccurate. In practice,individuals tend to exhibit different cognitive biases that are not captured by the classic modelof utility maximization. For example, the price tag might affect their value for a bottle of wine(irrational value assessment); or they may attribute higher values to items once they own them(endowment effect).A recent line of work in algorithmic game theory (Kleinberg and Oren [22] and their follow-ups[33, 1, 14, 23, 24, 3, 2]) mathematically models and analyzes the behavior of agents that exhibitcognitive biases in planning settings. In contrast, in this paper we initiate the study of cognitivebiases in the central setting of algorithmic mechanism design: combinatorial auctions. Furthermore,this line of work mostly focused on analyzing the implications of cognitive biases and mitigating their negative consequences. In this paper we take a different approach and show that in some settingscognitive biases can be harnessed to obtain better outcomes. In the context of algorithmic mechanism design, the only paper that incorporates behavioral assumptions that weare aware of is [8]. The paper applies prospect theory to a crowdsourcing setting.
Endowment Effect.
Do owning items makes them more valuable to us? The Nobel Laureateeconomist Richard Thaler coined the term endowment effect [34] to explain situations in which themere possession of an item makes it more valuable. An experiment by Knetsch [25] provides a starkillustration: two groups of students received goods in return for filling out a questionnaire. Onegroup received mugs and the other chocolate bars. Next, the students were given the option to swapitems. Since the items were allocated to the students randomly, we expect that about half of themwould have received the less desirable item and would like to switch. In contrary to this logic, 90%of the students in each group decided to keep their endowed item.A classic experiment by Kahneman, Knetsch and Thaler [19] attempts to quantify the endowmenteffect. In this experiment half of the students in a Law and Economics class at Cornell Universityreceived a $6 mug. After examining the mug, the students who received the mug were asked at whichprice are they willing to sell the mug (willingness to accept). The students who did not receive themug were asked for the price they are willing to pay for the mug (willingness to pay). As in theKnetch’s experiment, one could have expected that about 50% of the mug owners could be matchedwith a buyer that values the mug more than they do. However, in the experiment only 20% of themug owners sold their mug. Moreover, there was a significant gap between the median price sellerswere willing to sell at ($5 . . losing the mug). K˝oszegi and Rabin [27] incorporate reference points and expectations into prospect theory.Their framework suggests that the endowment effect will only appear in settings in which individualsdo not expect to trade. Masatlioglu and Ok [29] propose a model in which the individuals make arational choice on a constrained set of alternatives defined according to the status quo.Several papers study endowment effect in auction settings (e.g., [26, 17]). However, apart from[5] that discusses the endowment effect with multiple identical goods, all considered the single itemsetting, leaving open the question of incorporating the endowment effect into the more general com-binatorial auctions setting. See [28, 9] for some experimental work supporting this prediction. he Model. We consider a combinatorial auction with n bidders and a set M of m items. Eachbidder i has a valuation function v i : 2 M → R + that specifies his value for every bundle. Thevaluation function is non-decreasing and normalized ( v i ( ∅ ) = 0).Our modeling of the endowment effect is driven by the essence of this effect: an individualattributes an item a higher value after owning it; his value for items that he does not own remainsthe same. Formally, bidder i who previously valued item a at v i ( a ) will now value it at α i · v i ( a )for some α i ≥ Similarly, when turning to the more general setting of combinatorial auctions wedefine the value of a bundle S i that bidder i owns as α i · v i ( S i ). Note that it is insufficient to definethe new value of bidder i only for the bundle S i , we also need to define how the value of any otherbundle T changes when bidder i owns S i . We follow a similar line of reasoning: the value of theitems in S i ∩ T is multiplied by α i , while the marginal contribution of the remaining items T − S i remains the same. Formally, we define the endowed valuation of player i with α i who is endowedwith bundle S i to be: ∀ T ⊆ M, v S i ,α i i ( T ) = α i · v i ( S i ∩ T ) + v i ( T − S i | S i ∩ T )where v ( X | Y ) = v ( X ∪ Y ) − v ( Y ) denotes the marginal contribution of the bundle X given Y .Recall that a Walrasian equilibrium consists of an allocation ( S , . . . , S n ) and prices ( p , . . . , p m )such that:1. For each player i , v i ( S i ) − P j ∈ S i p j ≥ v i ( T ) − P j ∈ T p j , for every bundle T ⊆ M .2. If j / ∈ ∪ i S i then p j = 0.Several works attempted to relax the definition of a Walrasian Equilibrium in order to ensureexistence [12, 13] . Our approach is different; Given that the endowment effect changes the valuationsanyways, we are simply interested in a Walrasian equilibrium with respect to the endowed valuations:an allocation S and non-negative item prices ( p , . . . , p m ) form an ( α , . . . , α n ) -endowed equilibrium inan instance ( v , . . . , v n ) if they constitute a Walrasian equilibrium in the instance ( v S ,α , . . . , v S n ,α n n ).When α = α = . . . = α n we use the compact notation α -endowed equilibrium . Let S be theallocation and ( p , . . . , p m ) be the prices of some ( α , . . . , α n )-endowed equilibrium. Observe that S and ( p , . . . , p m ) also form an ( α, α, . . . , α )-endowed equilibrium, for α = max i α i . The reason is thatwhen S i maximizes the profit of v S i ,α i i , it also maximizes the profit of v S i ,α ′ i , for every α ′ > α i . Thus,in the rest of the paper we let α = max i α i and focus on α -endowed equilibria.Note that for α = 1 we recover the classic notion of Walrasian Equilibrium while, roughly speak-ing, as α goes to infinity the players insist more and more on keeping the bundle they were allocated. Throughout the paper we say that α supports an allocation S if there exist item prices that togetherwith S form an α -endowed equilibrium. The case 0 ≤ α i < One can also give a “classic” interpretation that does not involve cognitive biases to this transformation: consideran environment with transaction costs, e.g., a company that acquires a new location might be subject to propertytaxes and/or have to invest in new infrastructure. Similarly, a shop that moves to a new location might have to starta costly advertising campaign to inform its costumers about the move. In all these cases after owning their resourcesthe preference of the agents to the new status quo allocation increases. Feldman et al. [12] use bundle prices instead of individual item prices and do not require market clearance; Fuet al. [13] define a conditional equilibrium where each player does not wish to add any items to the bundle he wasallocated. Also related are works on simultaneous first [16] and second price [7] auctions. When α is very large this is similar to conditional equilibrium [13] as the bidders do not want to discard any itemwith marginal value greater than 0. esults. Analyzing Walrasian equilibria in the context of endowed valuations brings with it anatural question: when does an endowed equilibrium exist? The simple answer is always ; we showthat in any instance there exists an α > α -endowed equilibriumexists. However, this answer completely misses the point as the value of α for which an endowedequilibrium exists might be huge. Recall that the value of α corresponds to the intensity of theendowment effect, thus we expect α -endowed equilibria that arise in practice to have small value of α (in the experiment of [19] mentioned above, for example, it seems like α was about 2). This leads usto the definition of the endowment gap – the minimal value of α for which a Walrasian Equilibriumis guaranteed to exist. Thus, the main question that we ask in this paper is what is the endowmentgap for different valuation classes? Our main result shows that for submodular valuations there is always an allocation that is sup-ported by α = 2. That is, in combinatorial auctions with submodular valuations the endowment gapis at most . The proof of the theorem is constructive: an allocation is a local maximum if the welfare ofthe allocation cannot be improved by moving a single item from one player to another. We showthat in any instance of combinatorial auctions with submodular valuations any local maximum issupported by α = 2. In contrast, the First Welfare Theorem asserts that the allocation in a Walrasianequilibrium is a global optimum. Hence, a local maximum that is not a global maximum can be partof a 2-endowed equilibrium, but cannot be a part of a Walrasian equilibrium.Our work reveals interesting connections between the integrality gap of the linear program relax-ation for combinatorial auctions and the endowment gap. Nisan and Segal [30] show that a Walrasianequilibrium exists if and only if the integrality gap of a natural relaxation of the LP for combinato-rial auctions (the “configuration LP”) is 1, in other words, if and only if there is an optimal integralsolution to the LP. This in turn implies that an (integral) allocation S is supported by α if and onlyif it is an optimal solution of the (fractional) LP with respect to the perturbed valuations.An equivalent geometric interpretation is the following: consider the polytope defined by a giveninstance of a combinatorial auction. With submodular valuations, there might not be any optimalintegral solution on a vertex of the polytope. Fix some allocation and consider the change to theobjective function of the LP as α grows (changing the corresponding endowed valuations.) As α grows, the direction changes, rotating towards the direction of the endowed allocation.The allocation can be supported by the minimal value of α (if exists) for which the (integral)endowment allocation becomes an optimal solution to the LP. For submodular valuations, our resultshows that this happens quickly: when the allocation is a local maximum then a value of α of only2 suffices.It is not hard to see that the endowment gap is at least the integrality gap. However, note thatthe endowment gap is typically strictly larger. More generally, by analyzing the LP we give a precisedefinition for the minimal value of α required to support a given allocation. Roughly speaking, thisminimal value of α that supports an allocation S must be bigger than some combination of the valueof any fractional solution plus the “intersection” of this fractional solution with S . See a formaltreatment in Section 3.1.In fact, the LP point of view provides some interesting implications of our main result beyond therealm of endowed valuations: it is implicit in previous work that a local maximum in combinatorialauctions with submodular bidders provides a 2 approximation to the welfare maximizing solution.One implication of our main result is that a local maximum provides a 2 approximation even withrespect to the optimal fractional social welfare. More generally, the equilibrium allocation in an α -endowed equilibrium provides an α approximation to the optimal fractional social welfare.We also show that our analysis of submodular valuations is tight in the following sense: thereis an instance in which any local maximum requires α ≥ α ). We also show that there is an instancewith just 2 bidders in which every allocation requires α ≥ . . α > α ′ > α . Open Questions.
This work models the endowment effect in combinatorial auctions and analyzesthe endowment gap. Our main finding is that every local maximum can be supported by α = 2when the valuations are submodular and that the endowment gap for submodular valuations is atleast 1 .
5. An obvious open question is to close this gap. A related open question is to analyze theendowment gap of subclasses of submodular valuations. For example, for budget additive valuationswe are able to show that the endowment gap is also at least 1 .
5, but we are unable to prove that theendowment gap is strictly smaller than 2 even for this restricted class.The focus of this paper is on characterizing the existence of α -endowed equilibria. An interestingfollow-up question is to understand the “computational endowment gap”: the minimal value of α forwhich an α -endowed equilibrium can be efficiently computed. One would hope that for submodularvaluations a local maximum can be efficiently computed. Unfortunately, we show that there are bothcommunication and computation hurdles in finding a local maximum: finding a local maximum incombinatorial auctions with submodular bidders requires an exponential number of value queries.Moreover, we present a family of succinctly represented submodular valuations for which finding alocal optimum is PLS hard. On the somewhat more positive side, we do know how to find withonly polynomially many value queries an allocation (not necessarily a local maximum) that can besupported by some α >
1. However, this value of α is typically huge. Are there other allocationsthat can both be efficiently computed and supported by a small value of α ? Remarkably, we areunable to provide an answer for this question for any reasonable value of α , not even for a restrictedclass like budget additive valuations. In fact, we do not know if finding a local maximum whenthe valuations are budget additive is computationally hard. This question might be of independentinterest, regardless of the specific application to the endowment gap.Another interesting question is to devise natural auction methods that end up with an endowedequilibrium. If the valuations are gross-substitutes, then there are natural ascending auctions thatend up with a Walrasian equilibrium (e.g., [15]). Are there natural ascending auctions that end upwith an endowed equilibrium when the valuations are submodular? One question that might arisewhile developing such ascending auctions is to understand the extent to which bidders exhibit anendowment effect with respect to items that are temporarily assigned to them during the auctionand take this temporary endowment effect – if exists – into account.Finally, a natural measure of how far the market is from equilibrium suggests itself. Recallthat a valuation v is c -approximated by a valuation v ′ if for every bundle S , v ′ ( S ) c ≤ v ( S ) ≤ v ′ ( S ).Given valuations v , . . . , v n , define the “distance to equilibrium” as the minimal c for which thereexist v ′ , . . . , v ′ n such that each v ′ i c -approximates v i and the instance v ′ , . . . , v ′ n admits a Walrasianequilibrium. Since the endowment gap for submodular valuations is at most 2, this means that thedistance to equilibrium of any instance with submodular valuations is at most 2. It will be interestingto see if this result can be improved, e.g., maybe by using valuations that are not endowed valuations.Similarly, what is the distance to equilibrium of instances with subadditive or XOS valuations?5 The Model
There are n players and a set M of m goods, each agent i has a combinatorial valuation function v i : 2 M → R + . We assume that for each player i , v i is monotone ( S ⊆ T implies v i ( S ) ≤ v i ( T )) andnormalized ( v i ( ∅ ) = 0). We use the notation v i ( T | S ) to denote v i ( S ∪ T ) − v i ( S ), the marginal valueof T given S .Each player i has a parameter α i that measures the intensity of his endowment effect. Specifically,if player i is endowed with a bundle S i , then his valuation function is v S i ,α i i ( T ) = α i · v i ( S i ∩ T ) + v i ( T − S i | S i ∩ T )= v i ( T ) + ( α i − · v i ( S i ∩ T )We will use both expressions interchangeably.An allocation S = ( S , . . . , S n ) assigns to each agent i a set S i such that for every i = j , S i ∩ S j = ∅ . An allocation ( S , . . . , S n ) and (non-negative) item prices ( p , . . . , p m ) constitute an( α , . . . , α n ) -endowed equilibrium if:1. For each player i , v S i ,α i i ( S i ) − P j ∈ S i p j ≥ v S i ,α i i ( T ) − P j ∈ T p j , for every bundle T ⊆ M .2. If j / ∈ ∪ i S i then p j = 0.When α = α = . . . = α n we shorten the name to α -endowed equilibrium . Let S be the allocation and( p , . . . , p m ) be the prices of some ( α , . . . , α n )-endowed equilibrium. Observe that S and ( p , . . . , p m )also form an ( α, α, . . . , α )-endowed equilibrium, for α = max i α i . The reason for this is that when S i maximizes the profit of v S i ,α i i , it also maximizes the profit of v S i ,α ′ i , for every α ′ > α i . Thus, fromthis point onwards we let α = max i α i and focus on α -endowed equilibrium.An allocation ( S , . . . , S n ) is supported by α if there exist prices ( p , . . . , p m ) such that the pricesand the allocation form an α -endowed equilibrium. In particular, in every instance in which a Wal-rasian equilibrium exists (e.g., every instance in which the valuation functions are gross substitutes),we obviously have an endowed equilibrium supported by α = 1. In instances where a Walrasianequilibrium does not necessarily exist, we will be looking for the minimal value of α for which an α -endowed equilibrium exists.The conceptually and technically interesting regime is when α >
1, that is, a player assigns highervalue for items in their endowment, but see Appendix B for the regime 0 ≤ α < • Additive valuations:
A valuation v is additive if for every S , v ( S ) = P j ∈ S v ( { j } ). • Budget additive valuations:
A valuation v is budget additive if there exists b such that for every S , v ( S ) = min { b, P j ∈ S v ( { j } ) } . • Submodular valuations: a valuation v is submodular if for every S, T , v ( S ) + v ( T ) ≥ v ( S ∪ T ) + v ( S ∩ T ). • XOS valuations: a valuation is XOS if there exists additive valuations { a , . . . , a l } such thatfor every bundle S , v ( S ) = max ≤ k ≤ l a k ( S ). • Subadditive valuations: a valuation v is subadditive if for every S, T , v ( S ) + v ( T ) ≥ v ( S ∪ T ).6 The Endowment Gap
Consider some instance of a combinatorial auction with n players with valuations v , . . . , v n and a set M of m items. For a given instance, the endowment gap is, roughly speaking, the minimal value of α for which an α -endowed equilibrium exists in that instance. We are interested in proving bounds onthe value of α for which an α -endowed equilibrium exists for all instances of classes of combinatorialvaluations (e.g. submodular, XOS, subadditive). Definition 3.1
The endowment gap of an instance ( v , . . . , v n ) with respect to an allocation A =( A , . . . , A n ) , denoted G A ( v , . . . , v n ) , is the infimum of the values of α that support A . We naturally extend the definition of an endowment gap to an instance and to a class of valuations:
Definition 3.2 • The endowment gap of an instance ( v , . . . , v n ) is the minimum value, over every allocation S ,of the endowment gap with respect to the allocation S : min S G S ( v , . . . , v n ) . • The endowment gap of a class of valuations V is the supremum over the endowment gaps overall valuations profiles in which each valuation belongs to the class V : sup ( v ,...,v n ) ∈V n min S G S ( v , . . . , v n )Next, we provide a simple characterization that shows that any allocation A that its social welfarecannot be improved by reallocating items that do not contribute to the social welfare of A , can besupported by some α > Definition 3.3
Let S = ( S , . . . , S n ) be some allocation. For every item define q j to be the marginalcontribution of item j to the allocation S as follows: if there exists i such that j ∈ S i , let q j = v i ( j | S i − { j } ) . For every item j that is not allocated, let q j = 0 . Let Z = { j | q j = 0 } . S is maximal if for every player i , v i ( S i ∪ Z ) = v i ( S i ) . Note that in particular, in a maximal allocation there is no bidder with zero marginal value for anitem, for which some other bidder has positive marginal value given his set.
Proposition 3.4
1. Every maximal allocation S = ( S , . . . , S n ) can be supported by some α > . Furthermore, fora given allocation, we can find some α that supports it and the prices with poly ( n, m ) valuequeries.2. If an allocation S = ( S , . . . , S n ) is not maximal then there is no α ≥ that supports it. Proof:
Let Q + = M − Z (the set of items j with positive marginal contribution. Let OP T be some upper bound on the value of the optimal solution (e.g., n · max i v i ( M ) or simply OP T ifcomputational considerations are irrelevant). We use the following prices to support S : p j = 2 · OP T for j ∈ Q + and p j = 0 for j ∈ Z . We will show that α = m · OP T min j ∈ Q + q j and these prices form an endowedequilibrium.We first show that a player will never drop items that are in his set but not in Z . I.e., if wedenote by T some bundle that maximizes the profit of player i then S i − Z ⊆ T . Specifically, wewill show that v i ( S i ∩ T ) = v i ( S i ). Observe that v i ( S i ∩ T ) = v i ( S i ) indeed implies that S i − T ⊆ Z ,7ince otherwise there exists j ∈ Q + such that j ∈ S i − T . We then get a contradiction since v i ( S i ) > v i ( S i − { j } ) ≥ v i ( S i ∩ T ), where the first inequality is because j ∈ Q + and the second oneis due to the monotonicity of v i and the fact that j / ∈ T .We now show that the profit of player i from a bundle T such that v i ( S i ∩ T ) < v i ( S i ) is strictlysmaller than the profit of the bundle T ∪ S i . v S i ,αi ( T ∪ S i ) − X j ′ ∈ T ∪ S i p j ′ = α · v i ( S i ) + v i ( T − S i | S i ) − X j ′ ∈ S i p j ′ − X j ′ ∈ T − S i p j ′ = ( α − · v i ( S i ) + v i ( T ) − X j ′ ∈ S i p j ′ − X j ′ ∈ T − S i p j ′ = ( α − · v i ( S i ∩ T ) + ( α − · v i ( S i − T | S i ∩ T ) + v i ( T ) − X j ′ ∈ T p j ′ − X j ′ ∈ S i − T p j ′ Observe that v i ( S i − T | S i ∩ T ) = v i ( S i ) − v i ( S i ∩ T ) ≥ q j ≥ min j ∈ Q + q j (which holds by our discussionabove since v i ( S i ) − v i ( S i ∩ T ) ≥ q j as j / ∈ T and since j ∈ Q + ) and ( α − · min j ∈ Q + q j =( m · OP T min j ∈ Q + q j − · min j ∈ Q + q j = 20 m · OP T − min j ∈ Q + q j ≥ m · OP T . Thus, we have that: v S i ,αi ( T ∪ S i ) − X j ′ ∈ T ∪ S i p j ′ ≥ ( α − · v i ( S i ∩ T ) + 19 m · OP T + v i ( T ) − X j ′ ∈ T p j ′ − m · OP T> ( α − · v i ( S i ∩ T ) + v i ( T ) − X j ′ ∈ T p j ′ = v S i ,αi ( T ) − X j ′ ∈ T p j ′ We next show that the demand of any player is always a subset of S i ∪ Z . That is, if T is somebundle that maximizes the profit of player i then T ⊆ S i ∪ Z . Let R = T − ( S i ∪ Z ) and supposetowards contradiction that R = ∅ . We claim that the profit of the bundle T − R for player i is strictlyhigher than that of T : v S i ,αi ( T ) − X j ∈ T p j = v S i ,αi (( T − R ) ∪ R ) − X j ∈ T p j = v S i ,αi ( T − R ) + v i ( R | ( T − R )) − X j ∈ T − R p j − X j ∈ R p j ≤ v S i ,αi ( T − R ) + OP T − X j ∈ T − R p j − OP T< v S i ,αi ( T − R ) − X j ∈ T − R p j Together with our first observation, we have that a bundle T that maximizes the profit of player i must satisfy T = S i ∪ Z ′ , for some Z ′ ⊆ Z . Recall that v i ( S i ∪ Z ′ ) ≤ v i ( S i ∪ Z ) = v i ( S i ) and thatthe price of every j ∈ Z is p j = 0 and we get that the profit from T is exactly the profit of S i . Thus,for every player i , S i is a profit maximizing bundle, as needed. Finally, notice that to compute theprices and some α > n · max i v i ( M ) takes n value queries) and the marginal contribution of every item j (2 queries foreach item).For the second part of the proof, consider an allocation S that is not maximal. We will seethat for any α there are no prices that α -support this allocation. The proof is based on the simple8bservation that in any endowed equilibrium the price of every item j ∈ Z must be 0: this must bethe case by definition for every item j that is not allocated. If item j ∈ S i and the price of j ∈ Z is positive, then the profit of player i from the bundle S i − { j } is greater than his profit from thebundle S i .Since S is not maximal, there is some player i such that v i ( S i ∪ Z ) > v i ( S i ). Using the simpleobservation, the price of every item j ∈ Z is 0, thus the profit of player i from the bundle v i ( S i ∪ Z )is strictly larger than the profit from the bundle v i ( S i ). Therefore, S cannot be α -supported, for any α . In Appendix A we use this characterization to show that not only in every instance there is anallocation that can be supported, but there is even some welfare maximizing allocation that can be α -supported by some α >
1. The caveat is that the α that we guarantee might be huge. In Claim 3.8we give the exact value of the minimal α that supports an allocation S . However, even this precisecharacterization might result in large values of α . This is no coincidence: Proposition 5.1 shows thatfor every fixed α there is an instance for which the endowment gap is strictly bigger than α . Thus inmost of this paper we restrict our attention to specific classes of valuations, aiming to find boundson the endowment gap that hold for all instances in the class and, ideally, find prominent classes ofvaluations for which the gap is small. In this section we explore the connections between the endowment gap and the following linearprogram relaxation for combinatorial auctions:
Maximize: P ni =1 P S ⊆ M x i,S · v i ( S ) Subject to: • For each item j : P ni =1 P S ⊆ M | j ∈ S x i,S ≤ • for each bidder i : P S ⊆ M x i,S ≤ • for each i , S : x i,S ≥ Walrasian equilibrium : Theorem 3.5 ([30])
For every instance ( v , . . . , v n ) , there exists a Walrasian Equilibrium in theinstance ( v , . . . , v n ) if and only if the integrality gap of the above linear program is . Moreover,an integral allocation is the allocation of some Walrasian Equilibrium if and only if it is an optimalsolution to the LP. When considering endowed valuations v A ,α , . . . , v A n ,αn the theorem immediately implies an analogousresult for α -endowed equilibrium: Corollary 3.6
In an instance ( v , . . . , v n ) an allocation A = ( A , . . . , A n ) is α -supported if and onlyif A is an optimal solution to the LP of the instance ( v A ,α , . . . , v A n ,αn ) (implying in particular thatthe integrality gap of the latter instance is .) Furthermore, by the first welfare theorem we get that A is welfare maximizing with respect to v A ,α , . . . , v A n ,αn . We can also relate the welfare of supported allocations to that of fractional allocations. InSubsection 4.1 we use the next corollary to improve the bounds on the welfare guaranteed by localmaxima in combinatorial auctions with submodular valuations.
Corollary 3.7
In an instance ( v , . . . , v n ) , if an allocation A = ( A , . . . , A n ) is α -supported then itprovides an α -approximation to the maximum fractional welfare of the instance ( v , . . . , v n ) . roof: Let { x i,S } be some fractional solution of the LP of the instance ( v A ,α , . . . , v A n ,αn ). As theallocation A = ( A , . . . , A n ) is α -supported, by Corollary 3.6 and the definition of endowed valuationswe have that: α n X i =1 v i ( A i ) ≥ n X i =1 X S ⊆ M x i,S · v A i ,αi ( S ) ≥ n X i =1 X S ⊆ M x i,S · v i ( S )In particular, this holds for the welfare maximizing fractional solution of the instance ( v , . . . , v n ),implying that A provides an α -approximation to the value of that fractional allocation, as needed.As we will see next, the endowment gap has some interesting and useful connections to theintegrality gap. For our first application, recall that Proposition 3.4 shows that an allocation can besupported by some α if and only if it is maximal. We now use the connection to the LP to determinethe minimal value of α that can support a maximal allocation. Claim 3.8
Let A = ( A , ..., A n ) be some allocation. Given a fractional solution { x i,S } to the LP,define ψ A, { x i,S } = n X i =1 X S ⊆ M x i,S · v i ( S ∩ A i ) Suppose that A is supported by α . Then,1. For every fractional solution { x i,S } , α · P ni =1 v i ( A i ) ≥ P ni =1 P S ⊆ M x i,S · v i ( S )+ ( α − ψ A, { x i,S } .2. The endowment gap with respect to A equals to sup (cid:26) P ni =1 P S ⊆ M x i,S · v i ( S ) − ψ A, { x i,S } P ni =1 v i ( A i ) − ψ A, { x i,S } (cid:12)(cid:12)(cid:12)(cid:12) { x i,S } s.t. n X i =1 v i ( A i ) − ψ A, { x i,S } > (cid:27) Proof:
By Corollary 3.6, A can be α -supported if and only if for every fractional solution { x i,S } : n X i =1 v A i ,αi ( A i ) ≥ n X i =1 X S ⊆ M x i,S · v A i ,αi ( S )Additionally, since for every bundle S , v A i ,αi ( S ) = v i ( S ) + ( α − · v i ( S ∩ A i ), for every fractionalsolution { x i,S } it holds that: n X i =1 X S ⊆ M x i,S · v A i ,αi ( S ) = n X i =1 X S ⊆ M x i,S · v i ( S ) + ( α − n X i =1 X S ⊆ M x i,S · v i ( S ∩ A i ) | {z } ψ A, { xi,S } Note that the combination of the above two facts already establishes that if A is α -supportedthen claim (1) holds.We now continue to prove the second part. Rearranging, we have that A is α -supported if andonly if for every fractional solution: α · n X i =1 v i ( A i ) ≥ α · ψ A, { x i,S } + n X i =1 X S ⊆ M x i,S · v i ( S ) − ψ A, { x i,S } (1)10onsider the expression λ A, { x i,S } = α · P ni =1 v i ( A i ) − α · ψ A, { x i,S } . Observe that λ A, { x i,S } ≥
0, simplybecause ψ A, { x i,S } is composed of a sum of linear combinations of subsets of A i , for each player i (andthe valuations are monotone).If λ A, { x i,S } > α such that inequality (1) holds.Suppose that λ A, { x i,S } = 0. Observe that P ni =1 P S ⊆ M x i,S · v i ( S ) ≥ ψ A, { x i,S } , simply because eachterm v i ( S ) in the LHS is replaced by v i ( S ∩ A i ) in the RHS and the valuations are monotone.If P ni =1 P S ⊆ M x i,S · v i ( S ) > ψ A, { x i,S } and λ A, { x i,S } = 0, which holds for any A that is not maximal,then no value of α makes inequality (1) hold and thus this allocation cannot be supported by any α .However, if P ni =1 P S ⊆ M x i,S · v i ( S ) = ψ A, { x i,S } then any value of α makes the inequality hold.We have thus identified that for an allocation not to be supported by any α it must be that thereis some fractional solution { x i,S } for which λ A, { x i,S } = 0 and P ni =1 P S ⊆ M x i,S · v i ( S ) > ψ A, { x i,S } . Ifthis is not the case then by rearranging inequality (1) we can determine the minimal value of α thatsupports A :sup (cid:26) P ni =1 P S ⊆ M x i,S · v i ( S ) − ψ A, { x i,S } P ni =1 v i ( A i ) − ψ A, { x i,S } (cid:12)(cid:12)(cid:12)(cid:12) { x i,S } s.t. n X i =1 v i ( A i ) − ψ A, { x i,S } > (cid:27) Note that the supremum is bounded, since we assume that A can be supported by some α .Claim 3.8 implies that the endowment gap is at least the integrality gap: take A to be anyallocation and { x i,S } to be a fractional welfare maximizing solution: α ≥ P ni =1 P S ⊆ M x i,S · v i ( S ) − ψ A, { x i,S } P ni =1 v i ( A i ) − ψ A, { x i,S } ≥ P ni =1 P S ⊆ M x i,S · v i ( S ) P ni =1 v i ( A i )and the right hand side is obviously at least the integrality gap.However, as we will see in the paper, the integrality gap is usually strictly larger than theendowment gap. We now show that this is generically true for every instance with subadditivevaluations with an integrality gap bigger than 1. Claim 3.9
Consider an instance with two subadditive valuations. Suppose that the integrality gapof this instance is y > . Then, for every small enough δ > there is an instance with integralitygap x = y · (1+ δ )(1+ δy ) in which the endowment gap is strictly bigger than x . Proof:
Let ( v ′ , v ′ ) be two subadditive valuations. Denote by OP T ′ the welfare of an optimalintegral solution and by OP T ′∗ the welfare of an optimal fractional solution { x i,S } with respect to( v ′ , v ′ ) (so OP T ′∗ OP T ′ = y ). For each bidder i , consider the valuation v i ( S ) = v ′ i ( S ) + | S | · ǫ , where ǫ = δ · OP T ′∗ m . Note that v i is still a subadditive function. For the instance ( v , v ), let OP T be thewelfare of the optimal integral solution and
OP T ∗ be the welfare of the optimal fractional solution.Observe that the welfare of any allocation S with respect to ( v , v ) is larger than the welfare of thatallocation with respect to ( v ′ , v ′ ) by exactly ǫ times the number of allocated items in S . Thus, anoptimal allocation (fractional or integral) in the instance ( v ′ , v ′ ) in which all items are allocated is alsooptimal for ( v , v ) and the difference in the welfare is exactly m · ǫ . Therefore, OP T = OP T ′ + m · ǫ and OP T ∗ = OP T ′∗ + m · ǫ . Let x denote the integrality gap of the instance ( v , v ), we have that x = OP T ∗ OP T = OP T ′∗ + m · ǫOP T ′ + m · ǫ = (1 + δ ) OP T ′∗ OP T ′ + δOP T ′∗ = (1 + δ ) OP T ′∗ (1 + δy ) OP T ′ = y (1 + δ )(1 + δy )Let ( A , A ) be some allocation that can be supported by α in the instance ( v , v ). We claimthat A ∪ A = M . Else, there is some item j that is not allocated and thus its price is 0. Observethat given any bundle, item j has a positive marginal value of at least ǫ = δ · OP T ′∗ m > A ∪ { j } has a strictly larger profit than his equilibrium allocation A , acontradiction.We will show that in the instance ( v , v ), for any integral solution A = ( A , A ) such that A ∪ A = M , ψ A, { x i,S } ≥ ( x − · OP T . We can then apply Claim 3.8 which says that theendowment gap is at least:
OP T ∗ − ψ A, { x i,S } OP T − ψ A, { x i,S } ≥ x · OP T − ( x − · OP TOP T − ( x − · OP T = 12 − x This completes the proof since − x > x = y · (1+ δ )(1+ δy ) , where we use the fact that for any instance withtwo subaddititve players the integrality gap is strictly smaller than 2 (see Appendix C).We next show that in the instance ( v , v ), for any integral solution A = ( A , A ) such that A ∪ A = M , ψ A, { x i,S } ≥ ( x − · OP T . Observe that by subadditivity v i ( S ∩ A i ) ≥ v i ( S ) − v i ( S − A i ),thus: ψ A, { x i,S } = X i =1 X S ⊆ M x i,S · v i ( S ∩ A i ) ≥ X i =1 X S ⊆ M x i,S v i ( S ) | {z } OP T ∗ = x · OP T − X i =1 X S ⊆ M x i,S · v i ( S − A i )To complete the proof, we show that P i =1 P S ⊆ M x i,S · v i ( S − A i ) ≤ OP T . Observe that since A ∪ A = M , in P i =1 P S ⊆ M x i,S · v i ( S − A i ) we only assign player 1 subsets of A and player 2subsets of A . Taking into account that for each player i , P S x i,S ≤
1, we get that P i =1 P S ⊆ M x i,S · v i ( S − A i ) ≤ v ( A ) + v ( A ) ≤ OP T .The claim provides a generic way of proving lower bounds on the endowment gap of subclassesof subadditive valuations: start with an instance with the maximal integrality gap in the subclass.The claim guarantees that there is an instance with an endowment gap that is strictly higher thanthe maximal integrality gap. A more careful look at the proof shows that the new instance belongsto the subclass as long as the subclass is closed under sum, like submodular and XOS valuations (aclass of valuations V is closed under sum if for each v, u ∈ V we also have that v + u ∈ V ). We notethat although the claim guarantees a generic method of proving lower bounds on the endowmentgap, in the specific settings we study in this paper we are able to beat these bounds by introducingspecific instances with stronger guarantees. In this section we prove our main positive result: the endowment gap for submodular valuations isat most 2. We prove this by showing that any allocation that is a ”local optimum” of the socialwelfare function can be supported for α = 2 with prices that are equal to the marginal value of theitems for the player that receives each item. We start by defining the notion of local optimum. Definition 4.1
An allocation ( O , . . . , O n ) is a local optimum if ∪ ni =1 O i = M , and for every pairof players i and i ′ and item j ∈ O i we have that v i ( O i ) + v i ′ ( O i ′ ) ≥ v i ( O i − { j } ) + v i ′ ( O i ′ ∪ { j } ) . In other words, in a local optimum every item is allocated to some player, and reallocating any singleitem does not improve the welfare. Note that any welfare maximizing allocation is in particular alocal optimum. We are now ready to state our main positive result.12 heorem 4.2
Let v , . . . , v n be submodular valuations. Let O = ( O , . . . , O n ) be a local optimum.Then O is supported by any α ≥ . As an immediate corollary, the endowment gap of every instancewith submodular valuations is at most . Proof:
We explicitly construct prices that show that O is supported by 2. For each item j ∈ O i we define its price to be p j = v i ( j | O i − j ). Using the following two claims we show that for α ≥ p , . . . , p m ) and the allocation ( O , . . . , O n ) form an α -endowed equilibrium. Later we willobserve that our proofs hold even for lower prices.We start by showing that with these prices and α ≥
2, no player can gain by discarding itemsfrom his endowment.
Claim 4.3
Consider player i that is allocated bundle S i . Suppose that the price of each item j ∈ S i is p j = v i ( j | S i − { j } ) . Then, if α ≥ the profit of player i from every bundle S ′ is at most the profitof S i ∪ S ′ . I.e., v S i ,αi ( S ′ ) − P j ∈ S ′ p j ≤ v S i ,αi ( S i ∪ S ′ ) − P j ∈ S i ∪ S ′ p j . Proof:
We compare the profit of player i from bundle S ′ : v S i ,αi ( S ′ ) − X j ∈ S ′ p j = v i ( S ′ ) + ( α − · v i ( S i ∩ S ′ ) − X j ∈ S ′ p j to his profit from the bundle S ′ ∪ S i : v S i ,αi ( S i ∪ S ′ ) − X j ∈ S ′ ∪ S i p j = v i ( S ′ ∪ S i ) + ( α − · v i ( S i ) − X j ∈ S ′ ∪ S i p j Using the fact that v i ( S i ) = v i ( S i − S ′ | S i ∩ S ′ ) + v i ( S ′ ∪ S i ) and rearranging the last expression, weget that the profit of bundle S ′ ∪ S i equals: v i ( S ′ ∪ S i ) + ( α − · v i ( S i ∩ S ′ ) | {z } ≥ v Si,αi ( S ′ ) by monotonicity − X j ∈ S ′ p j + ( α − · v i ( S i − S ′ | S i ∩ S ′ ) − X j ∈ S i − S ′ p j . Thus, in order to show that v S i ,αi ( S i ∪ S ′ ) − P j ∈ S ′ ∪ S i p j ≥ v S i ,αi ( S ′ ) − P j ∈ S ′ p j , it suffices to showthat ( α − · v i ( S i − S ′ | S i ∩ S ′ ) − P j ∈ S i − S ′ p j ≥
0. Since α ≥
2, it holds that α − ≥ v i ( S i − S ′ | S i ∩ S ′ ) ≥ P j ∈ S i − S ′ p j for every submodularvaluation. Towards this end, denote the items in S i − S ′ by 1 , . . . , | S i − S ′ | . With this notationwe have that v i ( S i − S ′ | S i ∩ S ′ ) = P | S i − S ′ | j =1 v i ( j | ( S i ∩ S ′ ) ∪ { , . . . j − } ). Finally, observe thatby submodularity we have that for every 1 ≤ j ≤ | S i − S ′ | it holds that p j = v i ( j | S i − j ) ≤ v i ( j | S i ∪ S ′ − { j, j + 1 , . . . , | S i − S ′ |} ) = v i ( j | ( S i ∩ S ′ ) ∪ { , . . . j − } ).We next show that with these prices, an agent can never gain by adding items to his endowment. Claim 4.4
Let O = ( O , . . . , O n ) be a local optimum and suppose that the price of each item j ∈ O i is p j = v i ( j | O i − j ) . Then, for any α ≥ and bundle T the profit from O i is at least the profit from O i ∪ T . Proof:
Assume without loss of generality that O i ∩ T = ∅ . Observe that since the allocation O is a local optimum we have that for any j ∈ T , v i ( j | O i ) ≤ p j = v k ( j | O k − j ) for player k such that13 ∈ O k . Denote the items in T by 1 , . . . , | T | . Since the valuations are submodular we have that: α · v i ( O i ) + v i ( T − O i | O i ) − X j ∈ O i ∪ T p j = α · v i ( O i ) − X j ∈ O i p j + | T | X j =1 v i ( j | O i ∪ { , . . . , j − } ) − X j ∈ T p j ≤ α · v i ( O i ) − X j ∈ O i p j + | T | X j =1 v i ( j | O i ) − X j ∈ T p j ≤ α · v i ( O i ) − X j ∈ O i p j where in the second-to-last inequality we use the submodularity of v i to claim that v i ( j | O i ) ≥ v i ( j | O i ∪ { , . . . , j − } ).We apply claims 4.3 and 4.4 to conclude the proof of the theorem. Let O = ( O , . . . , O n ) be alocal optimum, recall that in a local optimum we have that ∪ ni =1 O i = M , and suppose that the priceof each item j ∈ O i is p j = v i ( j | O i − j ). To complete the proof we show that any player i demandsthe set O i at the these prices. By Claim 4.4, for α ≥ T : v O i ,αi ( O i ) − X j ∈ O i p j ≥ v O i ,αi ( O i ∪ T ) − X j ∈ O i ∪ T p j and by Claim 4.3 for α ≥ T : v O i ,αi ( O i ∪ T ) − X j ∈ O i ∪ T p j ≥ v O i ,αi ( T ) − X j ∈ T p j Combining the two inequalities we get that for any α ≥ T : v O i ,αi ( O i ) − X j ∈ O i p j ≥ v O i ,αi ( T ) − X j ∈ T p j as needed.We note that the proof still holds even if we reduce the prices such that the price of item j ∈ O i is max i ′ = i v i ′ ( j | O i ′ ) ≤ p j ≤ v i ( j | O i − { j } ). That is, the price p j can be reduced to the second highestmarginal value for the item. The reason is simple: the profit of player i from the bundle O i hasincreased at least as any other bundle in this reduction, so Claim 4.3 still holds. Claim 4.4 alsoholds, as for the proof to hold we only need the price of each item to be the second highest marginalvalue. One interesting corollary of the algorithm is not directly related to the endowment gap. In previouswork it was implicit that the welfare of any local optimum is at least half of the value of the welfaremaximizing integral solution. By a direct application of Corollary 3.7, we are able to strengthen thisresult and show that the welfare of any local optimum is at least half of the welfare of the welfaremaximizing fractional solution:
Proposition 4.5
Let ( O , . . . , O n ) be a local optimum. Let { x i,S } be some fractional solution forthe LP presented in Section 3.1. Then: · n X i =1 v i ( O i ) ≥ n X i =1 X S ⊆ M x i,S · v i ( S )14 .2 Tightness of Analysis The following proposition shows that considering local optima (or even global ones) will not help usprove a better bound than 2 on the endowment gap for submodular valuations:
Proposition 4.6
For any δ > , there is an instance with submodular valuations in which everylocal optimum maximizes the welfare and these local optima cannot be supported by α < − δ . Inthis instance there is another allocation that can be supported by α = 1 . δ . Proof:
Consider an instance with four submodular players ( a , a , b , b ) and 2 k + 1 items ( k ≥ X, Y, { c } , where X = { x , ..., x k } and Y = { y , ..., y k } . The valuations of the players are as follows: • a has a unit demand valuation with value k for each of the items in X (and 0 for the rest). • a has a unit demand valuation with value k for each of the items in Y (and 0 for the rest). • The valuations of b , b are defined as follows: given a set T , let v T be the budget additivevaluation with budget 1 that gives value k for every item in T and value of 1 for item { c } .Then the valuation of b is b ( S ) = v X ( S ) + | S | · ǫ and b ( S ) = v Y ( S ) + | S | · ǫ , where ǫ > b and b are submodular as they are the sum of two submodularvaluations.We next show that up to symmetry, there is only one locally optimal allocation. In every localmaximum item c must be allocated to either b or b as its marginal value for both is positive givenany bundle, and a , a have a value of 0 for item c . Without loss of generality assume it is allocatedto b . Now, it is not possible that in a local optimum b is allocated X ∪ { c } as his marginal valuefor each item in X is only ǫ , while a values each item in X at 1 /k . Thus, for ǫ < /k player a must receive an item from X , without loss of generality item x . Now, given any subset of X − { x } , b has positive value for any additional item in X − { x } , and is the only player with such positivevalue, so he must get X ∪ { c } − { x } . Finally, all items in Y must be allocated to b as the marginalvalue of any item in Y (given any subset of Y ) is larger for b than for any other player, in particular a . We conclude that in a local maximum, without loss of generality, a is allocated { x } , b isallocated X ∪ { c } − { x } and b is allocated Y .Fix some value of α that supports this allocation. First, we observe that for any j ∈ X − { x } , b ( j | X ∪{ c }−{ x , j } ) = ǫ , thus p j ≤ α · ǫ (otherwise the profit of b from the bundle X ∪{ c }−{ x }−{ j } is bigger than the profit of his equilibrium allocation). Similarly, p c ≤ α · ( k + ǫ ).Then, it must be that:1. b prefers his allocation over item c : (1 + k · ǫ ) · α − P y ∈ Y p y ≥ ǫ − p c .2. a that has zero profit in equilibrium has a non-positive profit from items in Y : for every y ∈ Y ,0 ≥ k − p y .Summing up all these inequalities with p c ≤ α · ( k + ǫ ), we get that: α · (1 + k · ǫ ) ≥ ǫ − α · ( k + ǫ ).That is, α ≥ ǫ k · ǫ +( k + ǫ ) , which approaches 2 for k that approaches ∞ and a choice of ǫ = k .As for the second part of the proposition, we will now see that the allocation in which a isallocated { x } , b is allocated X − { x } ∪ { c } , b is allocated Y − { y } and a is allocated { y } is supported by 1 . ǫ . To see this, we simply note that with this value of α the following pricesconstitute an equilibrium with respect to this allocation: p c = k + ǫ , p x = ǫ (for each x ∈ X ), p y = k + ǫ (for each y ∈ Y − { y } ), and p y = k + ǫ .15 .3 The Complexity of Finding a Local Optimum Our proof that the endowment gap for submodular valuations is at most 2 starts with a local maxi-mum (local optimum) ( O , . . . , O n ). Can such a local optimum be efficiently found? We next showthat there are both communication and computation hurdles in finding a local optimum: We firstshow that finding a local optimum in combinatorial auctions with submodular valuations requiresexponential number of value queries. In addition, we show that there exists some family of succinctlyrepresented submodular valuations for which finding a local optimum is PLS hard. Both resultshold even if there are only two players.In light of the hardness of finding exact local optimum, one might be tempted to use an approx-imate local optimum instead, as it is computationally feasible to find, and hope it can be supportedby a small α . However, it is easy to provide examples in which a (1 − ǫ )-approximate local optimumcannot be supported by any α . Consider a setting with two additive bidders and two items. The firstadditive bidder has a value of 1 − ǫ for item a , and ǫ for item b . The second bidder has a value of 0for both items. Observe that allocating a to the first bidder and b to the second one is an (1 − ǫ )-localoptimum. However, this allocation cannot be supported by any α : for the bundle { b } to be in thedemand of the second player, the price of item b must be 0. However, the profit of the bundle { ab } for player 1 is then strictly bigger than that of { a } . We thus leave as an open question the problem ofefficiently computing an α -endowed equilibrium for combinatorial auctions with submodular biddersfor a small value of α (Proposition A.1 shows that it is possible to efficiently compute an allocationthat is supported by some huge α ≥ Proposition 4.7
Finding a local optimum in a combinatorial auction with two submodular valua-tions requires exponentially many value queries, even with randomized algorithms.
Proof:
Let the number of items be m = 2 k + 1 for some integer k ≥
1. In the proof the valuation ofeach player i will belong to the following family of valuations parametrized by c iS satisfying 0 ≤ c iS < for every set S with | S | = k + 1: v i ( S ) = | S | , | S | ≤ kk + + c iS , | S | = k + 1 k + 1 , | S | ≥ k + 2 . Notice that v i is a monotone submodular valuation.Given two submodular valuations v and v from the family above, what can we say about theirlocal optimum? Consider an allocation ( S, M − S ). If the allocation is a local optimum, then clearly | S | = k or | M − S | = k as otherwise reallocating a single item from the larger bundle decreases thevalue of the large bundle by at most − c S and increases the value of the small bundle by 1, so theoverall welfare has strictly increased. Thus, only allocations ( S, M − S ) where one of the bundleshas size k can potentially be a local optimum.Recall that a local maximum of a labeled graph is a vertex whose label is at least as large asthe labels of its neighbors. For our reduction we use odd graphs. An odd graph is constructed by We will not give a precise definition here (see [18] for a formal definition), but the PLS class intuitively capturesproblems which admit a local search algorithm. An example for a PLS complete problem is finding a pure Nashequilibrium in congestions games [10]. The textbook definition of an odd graph starts with a set M of size 2 k + 1 and associates each of the vertices witha set S ⊆ M of size k . Two vertices v S and v S ′ are connected if and only if S ∩ S ′ = ∅ . The definition we give above iseasier to work with. Note that that this definition and the one we use above are equivalent as can be seen by changingthe label from S in the textbook definition to M − S . M of size 2 k + 1 and associating each of the vertices with a set S ⊆ M of size k + 1. Two vertices v S and v S ′ are connected if and only if | S ∩ S ′ | = 1.We will show that finding a local optimum when the valuations belong to the family definedabove is equivalent to finding a local maximum of an odd graph. This is enough to complete theproof as the results of Santha and Szegedy [32] imply that the number of queries needed to find alocal optimum in an odd graph is exp ( m ). This result is also holds for randomized algorithms (andin fact also for quantum algorithms).Given a labeled odd graph with M of size 2 k + 1, we reduce it to combinatorial auctions withsubmodular valuations as follows. Assume without loss of generality that the labels on the odd graphare less than (this can be achieved, e.g., by dividing all numbers by a large enough number). Wedefine identical valuations for the two players to be the above parametrized valuations where theparameter c S = c S = c S is equal to the label of the unique vertex in the odd graph that is associatedwith S , for each bundle S with | S | = k + 1. We next show that local maxima in this combinatorialauction correspond to local maxima in the odd graph.Consider an allocation ( S, M − S ) with | S | = k + 1. The welfare of this allocation is v ( S ) + v ( M − S ) = k + + c iS + k = 2 k + + c S . Similarly, if | M − S | = k +1 then the welfare is 2 k + + c M − S .Recall that an allocation is a local maximum if and only if moving item j to the other player doesnot improve the welfare. Thus, 2 k + + c S ≥ k + + c ( M − S ) ∪{ j } for every j ∈ S . In other words, c S ≥ c ( M − S ) ∪{ j } for every j ∈ S . Recall that in the odd graph the vertex that corresponds to S isconnected exactly to all vertices which correspond to ( M − S ) ∪{ j } for every j ∈ S . This immediatelyimplies that an allocation ( S, M − S ) ( | S | = k + 1) in the combinatorial auction is a local maximumif and only if the vertex that corresponds to S is a local maximum in the odd graph. The proof forallocations ( S, M − S ) where | M − S | = k + 1 is symmetric.The proposition proves that finding a local maximum requires exponentially many value queries.The other common model for accessing the valuations is the more general communication model.That is, how many bits do the players need to exchange in order to compute a local optimum? Oneobstacle in proving hardness in the communication model is that the proof of Proposition 4.7 isbased on the result of [32] which proves the hardness of finding a local maximum on the odd graphusing value queries. An analogous result for the communication model was not known when the firstversion of the paper was written.However, inspired by our work, the paper [4] studies a communication variant of local search onthe odd graph. Using this result and a very similar reduction to that of Proposition 4.7, we show: Proposition 4.8
The communication complexity of finding a local optimum in a combinatorial auc-tion with two submodular valuations is exp ( m ) . This results holds even for randomized protocols. We give a sketch of the proof in Appendix D.
Proposition 4.9
There exists a family of succinctly described submodular functions for which com-puting a value query can be done in polynomial time but finding a local optimum in a combinatorialauction with two valuations that belong to this family is PLS-hard.
Proof:
We reduce from the PLS complete problem of finding a locally optimal cut in a graph G .In this problem, we are given an edge-weighted graph G = ( V, E ), and we are looking for a partitionof the vertices (
S, V − S ) such that the (weighted) cut cannot be improved by moving any singlevertex from one side of the cut to the other.We now reduce this problem to a combinatorial auction with two identical submodular valuations.The items in the combinatorial auction are the vertices of the graph. For each set of vertices S , we17et v ( S ) to be the sum of the weights of all edges that touch at least one vertex in S . It is not hardto see that this valuation is monotone and submodular.Consider an allocation ( S, M − S ). Observe that the welfare of ( S, M − S ) is exactly the sum ofthe weights of all edges in the graph plus the sum of edges in the cut: the weight of an edge e = ( v, u )is counted once if and only if both are in S or if both are in M − S . It is counted twice if and only if v ∈ S or u ∈ M − S or u ∈ S or v ∈ M − S . This implies that ( S, M − S ) is a local optimum in thecombinatorial auction if and only if ( S, V − S ) is a locally optimal cut in the graph, as needed. We now prove some lower bounds on the endowment gap. In Section 4 we have shown that theendowment gap for submodular valuations is at most 2. What about larger classes of valuations,such as XOS or subadditive valuations? In sharp contrast to the case of submodular valuations, weshow that with the larger classes, the endowment gap cannot be bounded uniformly for the entireclass, even if there are only two players. Note that this does not contradict our claim (PropositionA.1) that for any instance, there exists an allocation that is supported by some α , as we now onlyshow that for every fixed α there is some instance such that no allocation is supported by that valueof α . Proposition 5.1
Fix α > . There exists an instance that consists of only three identical items andtwo players with identical XOS valuations such that no allocation is supported by α . Proof:
Consider an instance with two bidders and three identical items x , x , x . The valuationsof the bidders are identical (since the items are identical we slightly abuse notation and use v ( x ) tospecify the value of every bundle S such that | S | = x ): v (1) = 1 , v (2) = 1 + α , v (3) = 1 + α . Weprove that this is indeed an XOS valuation by providing an explicit construction of the clauses of v :a clause ( x j = 1) for every item x j , a clause ( x j = + α , x j ′ = + α ), for every pair of x j ′ and x j , j ′ = j , and a clause ( x = , x = , x = α ).Note that in every equilibrium allocation all items must be allocated: if item x j is unallocated,then its price is 0. However, since the valuations of the players are strictly increasing, the profit ofhis equilibrium allocation with x j is strictly bigger than his profit from his equilibrium allocation.Thus, from now on we only consider allocations that allocate all items. There are two possibleallocations that allocate all items (up to symmetry). We will show that both allocations are notsupported by α . In the first allocation, one player, say player 1, gets all three items. The marginalvalue of an item for player 1 is α − α < α . Thus the price of every item in equilibrium is atmost α · α = (if there is an item with a higher price, the profit for player 1 of the bundle thatcontains the other two items is higher than the profit of the grand bundle). In this case the profitof player 2 from taking one item is positive: 1 − = . A contradiction to the assumption that theempty bundle maximizes the profit of player 2.The other allocation is when one player, say player 1, is allocated two items and player 2 receivesonly one item. This allocation is not supported by α as well: since v (1) = 1 and v (2) = 1 + α , themarginal value of player 1 for any item is α . Thus, taking the endowment effect into account, theprice of each item that is allocated to player 1 cannot exceed α . But now the profit of the grandbundle for player 2 is strictly higher than its current single item allocation: the marginal value ofplayer 2 for the two items that were allocated to player 1 is α , while the sum of prices of these itemsis at most α . We have reached a contradiction once again.For submodular valuations we can prove that the endowment gap is at least . This leaves uswith a small gap to the upper bound of 2. 18 roposition 5.2 There exists an instance of two players with submodular valuations where the en-dowment gap is at least . Proof:
Feige and Vondrak [11] show that the integrality gap of two players with submodularvaluations is at least . We now show that this example shows that the endowment gap is at least .As mentioned, we have two players, call them Alice and Bob. There are 4 items, the value ofeach singleton is 1 the value of each bundle of three or more items is 2. The values of pairs of itemsare: Alice Bob { ab } / { cd } / { ac } / { bd } / { ad } / / { bc } / / α < . We will use the fact that the value of the optimal fractional solution is4 (Alice receives of each of the sets { ab } , { cd } , Bob receives of each of the sets { ac } , { bd } : x A, { ab } = x A, { cd } = x B, { ac } = x B, { bd } = ).1. ( { abcd } , ∅ ): The value of this allocation is 2, and since there is a fractional solution with value4, by Claim 3.8 this allocation requires α ≥ { abc } , { d } ): Suppose that there is an equilibrium with some α > p a , p b , p c , p d . v A ( c |{ ab } ) = 0, hence p c = 0 (as if p c > { ab } is strictlybigger than that of { abc } .) However, this means that v B ( c |{ d } ) − p c = , thus α · v B ( { d } ) − p d < α · v B ( { d } ) − p d + v B ( c |{ d } ) − p c I.e., the profit of Bob from the bundle { cd } is strictly bigger than that of { d } . A contradictionto the assumption that there is an equilibrium.3. ( { ab } , { cd } ): Denote this allocation by A and note that its welfare is 10 /
3. Observe that: ψ A, { x i,S } = n X i =1 X S ⊆ M x i,S · v i ( S ∩ A i ) = 12 v A ( { ab } ) + 12 v B ( { c } ) + 12 v B ( { d } ) = 2Thus, by Claim 3.8, to support A we must have α ≥ − − = .4. ( { ad } , { bc } ): Denote this allocation by A and note that its welfare is 10 /
3. Observe that ψ A, { x i,S } = n X i =1 X S ⊆ M x i,S · v i ( S ∩ A i ) = 12 v A ( { a } ) + 12 v A ( { d } ) + 12 v B ( { b } ) + 12 v B ( { c } ) = 2 . Similarly to before, by Claim 3.8, to support A we must have α ≥ − − = .Finally, we note that the optimal integral allocation ( { ab } , { cd } ) is indeed supported by α = 3 / p a = p b = 1, p c = p d = 2 / . Next, we consider a more restricted class – budget additive valuations –and show that even in this much simpler class the endowment gap is essentially the same. However,we do need more players to show this. 19 roposition 5.3 For every ǫ > , there exists an instance of four players with budget additivevaluations in which the endowment gap is at least ǫ/ . Proof:
We consider a modification of the integrality gap example of Chakrabarty and Goel [6]. Wehave 4 players ( a , a , b , b ) and 5 items ( x , x , y , y , c ). a , a have budget 1, b , b have budget2 + ǫ , for some arbitrarily small ǫ > • Players a and b have value 1 for items x , x . • Players a and b have value 1 for items y , y . • Players b , b value 2 for item c . • The rest of the values are 0.To analyze the endowment gap we provide several lemmas that characterize the allocations thatcan be supported in an endowed equilibrium. We then show that each of these allocations can besupported only by α ≥ ǫ/ . Lemma 5.4
Without loss of generality, in an endowed equilibrium b is assigned item c . Proof:
We claim that in equilibrium item c must be allocated to one of the players b , b . If c isnot allocated, then its price must be 0. The same holds if c is allocated to player a or player a ,the value of both players for c is 0. Now observe that each player b i has three items with positivevalue, and that b ( c |{ x , x } ) = ǫ >
0. Thus, when p c = 0, the profit of player b from a bundle thatcontains c is always strictly bigger than his profit from a bundle without it. The argument for player b is identical, and thus without loss of generality in equilibrium b is assigned item c . Lemma 5.5
Without loss of generality, in an endowed equilibrium player b is assigned at least oneof y , y . Proof:
Observe that if player b was not assigned any of the items y , y , then his profit is 0,since he has positive value only for items c, y , y and did not get any of them. The only other playerwith positive value for y and y is a . However, if a receives the bundle { y , y } the marginalcontribution of any of these items is 0, thus the price of both of these two items is 0. In that case,the profit of player b from the bundle { y } is strictly positive and is bigger than his 0 profit inequilibrium, which is a contradiction. Lemma 5.6
Without loss of generality, in an endowed equilibrium a is assigned item x . Proof:
We can claim that player a is allocated at least one of x , x : the only other player withpositive value for these items is b . Now, b is already assigned item c , so the marginal contributionof the item is at most ǫ and thus the price is at most α · ǫ . For ǫ < α , α · ǫ < { x } has a positive profit for player a . This profit is bigger than his 0 profit in equilibrium. Lemma 5.7
Without loss of generality, in equilibrium player b is assigned x and p x < α · ǫ . roof: Observe that x can only contribute positively to players a , b and that a ( x |{ x } ) = 0.Thus, if x is allocated to a then its price must be 0, but then b ( x |{ c } ) >
0, which implies that b profit increases when he adds x to his bundle. Similarly to before, this means that this is not anequilibrium allocation. Also, similarly to the previous lemma, p x < α · ǫ .This leaves us with the following two allocations that can be supported (up to symmetry):1. b gets { c, x } , b gets { y , y } , a gets x : in this case we have:(a) b prefers his allocation over item c : 2 · α − p y − p y ≥ − p c .(b) a prefers his allocation over item x (recall that p x ≤ α · ǫ ): α − p x ≥ − ǫ · α .(c) b prefers his allocation over { x , x } : 2 · α − p c ≥ α − p x (by rearranging α − p c ≥ − p x ).(d) a that has zero profit in equilibrium has a non-positive profit from y and y : 0 ≥ − p y ,0 ≥ − p y .Summing these inequalities we get that α · (4 + ǫ ) ≥
6. I.e., to support this allocation we need α ≥ ǫ/ . As ǫ approaches 0 this ratio approaches .2. b gets { c, x } , b gets y , a gets x , a gets y :(a) b prefers his allocation over item c : α − p y ≥ − p c .(b) b prefers his allocation over { y , y } : α − p y ≥ α + 1 − p y − p y . Hence p y ≥ a prefers his allocation over item x (recall that p x ≤ α · ǫ ): α − p x ≥ − ǫ · α .(d) b prefers his allocation over { x , x } : 2 · α − p c ≥ α − p x .(e) a prefers his allocation over item y : α − p y ≥ − p y .Summing these inequalities we get that α · (4 + ǫ ) ≥
6. I.e., to support this allocation we need α ≥ ǫ/ . As ǫ approaches 0 this ratio approaches . Acknowledgments
The second author was partially supported by BSF grant no. 2016192.
References [1] Susanne Albers and Dennis Kraft. Motivating time-inconsistent agents: A computational ap-proach. In
Proc. 12th Workshop on Internet and Network Economics , 2016.[2] Susanne Albers and Dennis Kraft. On the value of penalties in time-inconsistent planning. In , pages 10:1–10:12, 2017.[3] Susanne Albers and Dennis Kraft. The price of uncertainty in present-biased planning. In
Proc.13th Workshop on Internet and Network Economics , 2017.[4] Yakov Babichenko, Shahar Dobzinski, and Noam Nisan. The communication complexity of localsearch. arXiv preprint arXiv:1804.02676 , 2018.215] Katherine Burson, David Faro, and Yuval Rottenstreich. Multiple-unit holdings yield attenuatedendowment effects.
Management Science , 59(3):545–555, 2013.[6] Deeparnab Chakrabarty and Gagan Goel. On the approximability of budgeted allocations andimproved lower bounds for submodular welfare maximization and gap.
SIAM Journal on Com-puting , 39(6):2189–2211, 2010.[7] George Christodoulou, Annam´aria Kov´acs, and Michael Schapira. Bayesian combinatorial auc-tions. In
International Colloquium on Automata, Languages, and Programming , pages 820–832.Springer, 2008.[8] David Easley and Arpita Ghosh. Behavioral mechanism design: Optimal crowdsourcing con-tracts and prospect theory. In
Proc. 16th ACM Conference on Electronic Commerce , 2015.[9] Dirk Engelmann and Guillaume Hollard. Reconsidering the effect of market experience on theendowment effect.
Econometrica , 78(6):2005–2019, 2010.[10] Alex Fabrikant, Christos Papadimitriou, and Kunal Talwar. The complexity of pure nash equi-libria. In
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing , pages604–612. ACM, 2004.[11] Uriel Feige and Jan Vondrak. Approximation algorithms for allocation problems: Improvingthe factor of 1-1/e. In
Foundations of Computer Science, 2006. FOCS’06. 47th Annual IEEESymposium on , pages 667–676. IEEE, 2006.[12] Michal Feldman, Nick Gravin, and Brendan Lucier. Combinatorial walrasian equilibrium.
SIAMJournal on Computing , 45(1):29–48, 2016.[13] Hu Fu, Robert Kleinberg, and Ron Lavi. Conditional equilibrium outcomes via ascending priceprocesses with applications to combinatorial auctions with item bidding. In
ACM Conferenceon Electronic Commerce , 2012.[14] Nick Gravin, Nicole Immorlica, Brendan Lucier, and Emmanouil Pountourakis. Procrastinationwith variable present bias. In
Proceedings of the 2016 ACM Conference on Economics andComputation , EC ’16, pages 361–361, New York, NY, USA, 2016. ACM.[15] Faruk Gul and Ennio Stacchetti. Walrasian equilibrium with gross substitutes.
Journal ofEconomic theory , 87(1):95–124, 1999.[16] Avinatan Hassidim, Haim Kaplan, Yishay Mansour, and Noam Nisan. Non-price equilibriain markets of discrete goods. In
Proceedings 12th ACM Conference on Electronic Commerce(EC-2011), San Jose, CA, USA, June 5-9, 2011 , pages 295–296, 2011.[17] James E Heyman, Yesim Orhun, and Dan Ariely. Auction fever: the effect of opponents andquasi-endowment on product valuations.
Journal of Interactive Marketing , 18(4):7–21, 2004.[18] David S Johnson, Christos H Papadimitriou, and Mihalis Yannakakis. How easy is local search?
Journal of computer and system sciences , 37(1):79–100, 1988.[19] Daniel Kahneman, Jack L Knetsch, and Richard H Thaler. Experimental tests of the endowmenteffect and the coase theorem.
Journal of political Economy , pages 1325–1348, 1990.[20] Daniel Kahneman, Jack L Knetsch, and Richard H Thaler. Anomalies: The endowment effect,loss aversion, and status quo bias.
The journal of economic perspectives , 5(1):193–206, 1991.2221] Daniel Kahneman and Amos Tversky. Prospect theory: An analysis of decision under risk.
Econometrica 47,2 , pages 263–291, 1979.[22] Jon Kleinberg and Sigal Oren. Time-inconsistent planning: A computational problem in behav-ioral economics. In
Proceedings of the Fifteenth ACM Conference on Economics and Computa-tion , EC ’14, pages 547–564, New York, NY, USA, 2014. ACM.[23] Jon Kleinberg, Sigal Oren, and Manish Raghavan. Planning problems for sophisticated agentswith present bias. In
Proceedings of the 2016 ACM Conference on Economics and Computation ,EC ’16, pages 343–360, New York, NY, USA, 2016. ACM.[24] Jon Kleinberg, Sigal Oren, and Manish Raghavan. Planning with multiple biases. In
Proceedingsof the 2017 ACM Conference on Economics and Computation , EC ’17, pages 567–584, New York,NY, USA, 2017. ACM.[25] Jack L Knetsch. The endowment effect and evidence of nonreversible indifference curves.
Theamerican Economic review , 79(5):1277–1284, 1989.[26] Jack L Knetsch, Fang-Fang Tang, and Richard H Thaler. The endowment effect and repeatedmarket trials: Is the vickrey auction demand revealing?
Experimental economics , 4(3):257–269,2001.[27] Botond K˝oszegi and Matthew Rabin. A model of reference-dependent preferences.
The QuarterlyJournal of Economics , pages 1133–1165, 2006.[28] John List. Does market experience eliminate market anomalies?
The Quarterly Journal ofEconomics , 118(1):41–71, 2003.[29] Yusufcan Masatlioglu and Efe A Ok. Rational choice with status quo bias.
Journal of EconomicTheory , 121(1):1–29, 2005.[30] Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and sup-porting prices.
Journal of Economic Theory , 129(1):192–224, 2006.[31] William Samuelson and Richard Zeckhauser. Status quo bias in decision making.
Journal ofrisk and uncertainty , 1(1):7–59, 1988.[32] Miklos Santha and Mario Szegedy. Quantum and classical query complexities of local search arepolynomially related. In
Proceedings of the thirty-sixth annual ACM symposium on Theory ofcomputing , pages 494–501. ACM, 2004.[33] Pingzhong Tang, Yifeng Teng, Zihe Wang, Shenke Xiao, and Yichong Xu. Computational issuesin time-inconsistent planning models. arXiv preprint arXiv:1411.7472 , 2014.[34] Richard Thaler. Toward a positive theory of consumer choice.
Journal of Economic Behavior& Organization , 1(1):39–60, 1980.
A Existence of Endowed Equilibrium
We now show that for every instance there is some allocation that can be α -supported with some α >
1. The caveat is that this α might be huge and instance dependent. We will bring two (similar)proofs for this: one that shows that there is a welfare maximizing allocation that can be supported,and another proof that shows how to find such allocation in a computationally efficient way.23he value of α used in Proposition A.1 is instance dependent and can be very large (depends onthe value of OP T , the maximal welfare for the instance) and thus does not provide a uniform upperbound that hold for all instances. We show (Proposition 5.1) that an upper bound that holds for allinstances does not exist.
Proposition A.1
1. In every instance there is some welfare maximizing allocation O = ( O , . . . , O n ) for which thereexists some α > that supports it.2. There exists an algorithm that uses poly ( m, n ) value queries that finds some allocation that issupported by some α > . Proof:
Both proofs rely on applying Proposition 3.4. For the first part, we show that there existsa welfare maximizing allocation that is maximal: start with some optimal allocation ( O , . . . , O n )and consider the following process: if there is some player i and item j with v i ( j | O i ) = 0, removeitem j from O i . Repeat this process until obtaining an allocation ( O ′ i , . . . , O ′ n ) where every allocateditem have a positive marginal value. Note that ( O ′ i , . . . , O ′ n ) has the same value as ( O i , . . . , O n ) andthus it is welfare maximizing.Let Z be the set of unallocated items. Observe that any other item has a posititve marginalcontribution to the allocation. Furthermore, notice that for every i we have that v i ( Z | O ′ i ) = 0, sinceotherwise the welfare of the allocation ( O ; , . . . , O ′ i − , O ′ i + Z, O ′ i +1 , . . . , O ′ n ) is strictly larger than OP T (this is a valid allocation, as items in Z are not allocated at all), which is a contradiction. Nowwe can apply Proposition 3.4 and get that ( O ′ i , . . . , O ′ n ) can be supported by some α > M , and remove from it some item with a zero marginal value for player 1: v ( j | M −{ j } ) = 0, if such exists. Then repeat this process for the next item that have a zero marginalvalue until obtaining some set S such that for every j ∈ S , v ( j | S − { j } ) > v ( S ) = v ( M ).Similarly, starting with the remaining items M − S , obtain a bundle S ⊆ M − S such that forevery j ∈ S , v ( j | S − { j } ) > v ( S ) = v ( M − S ). Repeat similarly with the remainingitems for the next players. Observe that ( S , . . . , S n ) is maximal. Hence by the lemma there is some α > poly ( n, m ).Not only are there allocations which cannot be supported by any α , even some welfare maximizing allocations cannot be supported by any α . Specifically, this is the case for some welfare maximizingallocations in which items of zero marginal value are allocated to agents. See Example A.2 below.Note that this is very different than in the case of Walrasian equilibrium (when α = 1) in whichallocating item of zero marginal value to the agents, and pricing them at zero, cannot destroy theequilibrium. Example A.2
Consider a setting with three items and two players, where the value of the playersfor a bundle is only a function of its size: for player , the value of any pair of items (and bymonotonicity also of the grand bundle) is , and for player the value of any pair of items is ǫ satisfying > ǫ > . An optimal allocation is to allocate all three items to player . In this case,the marginal value of each of the items is v (1 | . Thus the price of each of the items must be since if it is positive the profit of player for a bundle that contains two items is bigger than theprofit of his equilibrium allocation. However, for a price of for each item, any pair of items give apositive profit for player , which is bigger than his profit from the empty bundle.Note that in this example the welfare maximizing allocation that allocates items { a, b } to player and leaves item c unallocated is supported by, e.g., p a = 1 , p b = 1 , p c = 0 and α = 2 . α -Endowed Equilibrium for α < In this section we discuss the case of α <
1. We show that if an α -endowed equilibrium exists thenits allocation must maximize the welfare with respect to the original valuations. This implies that tobe able to present such an equilibrium, we must be able to compute a welfare maximizing allocation.Additionally, we show that even for unit-demand valuations, an α -endowed equilibrium might fail toexist for any 0 < α <
1, establishing that for gross-substitutes valuations the minimal α needed toalways support an α -endowed equilibrium is indeed 1. B.1 Any α -Endowed Equilibrium for < α < is Welfare Maximizing We first show that for 0 < α ≤ α -supported allocation is socially efficient. Theorem B.1
For any < α ≤ , if an α -endowed equilibrium exists then its allocation is welfaremaximizing with respect to the original valuations. Proof:
Consider an endowed equilibrium with allocation ( S , . . . , S n ) with prices p , . . . , p m . Let( O , . . . , O n ) be a welfare maximizing allocation. For each player i we have: α · v i ( S i ) − p ( S i ) ≥ α · v i ( S i ∩ O i ) + v i ( O i − S i | S i ∩ O i ) − X j ∈ O i p j = α · v i ( S i ∩ O i ) + v i ( O i ) − v i ( S i ∩ O i ) − X j ∈ O i p j = v i ( O i ) − (1 − α ) v i ( S i ∩ O i ) − X j ∈ O i p j ≥ v i ( O i ) − (1 − α ) v i ( O i ) − X j ∈ O i p j = α · v i ( O i ) − X j ∈ O i p j Where the last inequality is due to the fact that for all i it holds that v i ( O i ) ≥ v i ( S i ∩ O i ), and since1 − α ≥
0. Taking a sum over all the players and using P i P j ∈ S i p j = P i P j ∈ O i p j we get that: α X i v i ( S i ) ≥ α X i v i ( O i )We conclude that P i v i ( S i ) ≥ P i v i ( O i ), so ( S , . . . , S n ) is socially efficient. B.2 Any α -Endowed Equilibrium for α = 0 is Welfare Maximizing We next show that for α = 0 any α -supported allocation is socially efficient. Observation B.2
For α = 0 , the allocation of any α -endowed equilibrium is welfare maximizingwith respect to the original valuations. Proof:
Fix some α -endowed equilibrium S = ( S , . . . , S n ) for α = 0. Since α = 0, each player i has no endowed value for the items he gets, so he must be paying 0 for each of the items in S i (otherwise he prefers dropping all items with positive price). Thus the price of all items must be 0and the endowed profit is zero: v S i ,αi ( S i ) − Σ j ∈ S i p j = 0 · v i ( S i ) − i prefers S i to M it holdsthat v S i ,αi ( S i ) − Σ j ∈ S i p j = 0 ≥ · v i ( S i ) + v i ( M | S i ) − P j ∈ M p j = v i ( M | S i ) − v i ( M ) − v i ( S i ) ≥ v i ( M ) = v i ( S i ). 25e get that ( S , . . . , S n ) is an 0-endowed equilibrium if and only if v i ( M ) = v i ( S i ) for player i .We next show that this implies that S is welfare maximizing. Assume that it is not, and there is anallocation ( O , O , . . . , O n ) such that P i v i ( O i ) > P i v i ( S i ). For this to be possible, it must be thecase that for at least one player i it holds that v i ( O i ) > v i ( S i ). But this yields a contradiction as v i ( M ) ≥ v i ( O i ) > v i ( S i ) = v i ( M ). We conclude that if ( S , . . . , S n ) is the allocation of an α -endowedequilibrium for α = 0 then it is welfare maximizing with respect to the original valuations. B.3 Unit-demand Valuations are not α -supported for < α < Recall that an α -endowed equilibrium for α = 1 is simply a Walrasian equilibrium. When valuationsare gross-substitutes (e.g. unit demand) then it is well known that a Walrasian equilibrium exists.We next show that even for unit-demand valuations, an α -endowed equilibrium does not exist for α <
1, and thus for unit-demand valuations it is indeed required that α ≥ α -endowed equilibrium.When α < α -endowed equilibrium for α < n identical players and n identical items. Each player is unit demand and wants any singleitem for a value of 1. Observe that if there is an α -endowed equilibrium, then by individual rationalitythe price of each item is be at most α . Let player i be the player that is allocated the item j withthe highest price p j (if there are several such players, choose one arbitrarily). Observe that player i prefers any other item j ′ : α · v i ( { j } ) − p j < v ( { j ′ } ) − p j ′ . This is because α < p i ≥ p j .Thus, no α -endowed equilibrium exists in this instance. C The Integrality gap of -Player Instances with Subaddititve Val-uations We now show that the integrality gap of instances with two players, both with subadditive valuationsis strictly less than 2. We consider some fractional solution { x i,S } of the LP and show how to roundit to an integral solution that provides an approximation ratio better than 2. We will prove thethat there is an integral allocation ( S , S ) such that v ( S ) + v ( S ) ≥ ( + m )Σ i =1 Σ S x i,S v i ( S ), asneeded.Suppose, without loss of generality, that Σ S x ,S v ( S ) ≥ Σ S x ,S v ( S ). Sample a bundle S by thedistribution that assigned probability x ,S to each bundle S . Allocate S to bidder 1 and M − S tobidder 2. The expected value of this rounded solution is exactly Σ S x ,S · ( v ( S ) + v ( M − S )). It istherefore enough to prove that Σ S x ,S · v ( M − S ) ≥ Σ S x ,S v ( S ) m . This is so as this implies that thevalue of the solution is at least Σ S x ,S v ( S ) + Σ S x ,S v ( S ) m and Σ S x ,S v ( S ) ≥ Σ S x ,S v ( S ) so overallthe expected value is at least ( + m )Σ i =1 Σ S x i,S v i ( S ), as needed.Given player i and item j , let q ij = Σ S : j ∈ S x i,S . In particular, for any item j we have that q j + q j ≤
1. Observe that the probability that player 2 receives item j is 1 − q j ≥ q j . That is,let D be the distribution in which player 2 receives bundle S with probability x ,S and D ′ be thedistribution where player 2 receives bundle M − S with probability x ,S . The marginal distributionof player 2 receiving item j in D ′ is at least the marginal distribution of player 2 receiving item j in D . We have that: E S ∼D ′ [ v ( S )] ≥ E S ∼D ′ [Σ j ∈ S v ( { j } )] m = Σ j (1 − q j ) · v ( { j } ) m ≥ Σ j q j · v ( { j } ) m ≥ Σ S x ,S v ( S ) m = E S ∼D [ v ( S )] m v . D The Communication Complexity of Finding a Local Maximum
The paper [4] studies the following communication variant of local search: let G = ( V, E ) be a knowngraph. Alice holds a function f A : V → R and Bob holds a function f B : V → R . The goal is to finda local maximum of the graph: a vertex v such that f A ( v ) + f B ( v ) ≥ f A ( u ) + f B ( u ) for any neighbor u of v . In particular, it is shown in [4] that if G is the odd graph then finding a local maximumrequires | V | c bits of communication, for some constant c . This results holds even for randomizedprotocols.We utilize this communication hardness result to prove that the communication complexity offinding a local maximum in combinatorial auctions is exp ( m ). The proof is similar to the proof ofClaim 4.7. Again, let the number of items be m = 2 k + 1 for some integer k ≥
1. The valuation ofeach player i belongs to the following family: v i ( S ) = | S | , | S | ≤ k − k − + c iM − S , | S | = kk − + c iS , | S | = k + 1 k, | S | ≥ k + 2 . It is not hard to see that if for every S it holds that c iS ≤ then the valuations are monotone andsubmodular. Also, similarly to before, an allocation ( S, M − S ) can be a local maximum if either | S | = k or | M − S | = k .In our reduction from the communication hardness of local search on the k ’th odd graph, wehave m = 2 k + 1 items. We let the valuation of Alice belong to the family above with c AS = f A ( S ).Bob’s valuation is similarly defined with c BS = f B ( S ). The proof that every local maximum of theodd graph is a local maximum of the combinatorial auction is very similar to the corresponding partof the proof of Claim 4.7. We proceed similarly as in the proof of Claim 4.7, except that we use thefact that by construction of our valuations v ( S ) + v ( M − S ) = v ( M − S ) + v ( S ). That is, thevalue of every allocation ( S, M − S ) ( | S | = k ) is 2 k − + + c AS + c BS and the value of everyallocation ( M − S, S ) ( | S | = k ) is also 2 k − + + c AS + c BS . Moving a single item j increases thewelfare if and only if c AS + c BS ≥ c A ( M − S ) ∪{ j } + c B ( M − S ) ∪{ j } .This establishes that | V | c = exp ( mm