Combinatorial auctions appear in many different contexts (e.g., spectrum auctions , network routing auctions , airport time-slot auctions , etc.) and have been studied extensively (and virtually from every possible aspect) for a few decades (see e.g.,  and the references therein).
In a combinatorial auction, a set of items (or goods) is to be allocated to bidders. Each bidder has a valuation function that assigns a non-negative value to any and quantifies ’s preferences over item subsets. Valuation functions are assumed to be non-decreasing (free disposal), i.e., , for all , and normalized, i.e., . The goal is to compute a partitioning (a.k.a. allocation ) of that maximizes the social welfare .
Most of the previous work has focused on CAs with either submodular (and XOS) or complement-free valuations. A set function is submodular if for all , , and subadditive (a.k.a. complement-free ) if . A set function is XOS (a.k.a. fractionally subadditive, see ) if there are additive functions such that for every , . The class of submodular functions is a proper subset of the class of XOS functions, which in turn is a proper subset of subadditive functions.
Since bidder valuations have exponential size in , algorithmic efficiency requires that the bidders communicate their preferences through either value or demand queries. A value query specifies a bidder and a set (or bundle) and receives its value . A demand query specifies a bidder , a set of available items and a price for each available item , and receives a bundle that maximizes ’s utility from the set of available items at these prices. Demand queries are strictly more powerful than value queries, in the sense that value queries can be simulated by polynomially many demand queries, and in terms of communication cost, demand queries are exponentially stronger than value queries .
The approximability of social welfare maximization by polynomial-time algorithms and truthful mechanisms for CAs with submodular and subadditive bidders has been extensively studied by the communities of Approximation Algorithms and Algorithmic Mechanism Design in the last two decades and are practically well understood (see e.g., Section 1.3 for a selective list of references most relevant to our work).
1.1 Perturbation Stability in Combinatorial Auctions
Motivated by recent work on beyond worst-case analysis of algorithms [32, 33] and on endowed valuations for combinatorial markets [4, 16], in this work, we investigate whether strong performance guarantees for social welfare maximization (by polynomial-time algorithms and truthful mechanisms, or even at the equilibrium of simple auctions) can be achieved for a very restricted (though still natural) class of CAs with perturbation stable valuations, where the optimal solution is resilient to a small increase of any bidder’s valuation for any single item.
From a bird’s view, we follow the approach of beyond worst-case analysis (see e.g., [33, 32]), where we seek a theoretical understanding of the superior practical performance of certain algorithms by formally analyzing them on practically relevant instances. Hence, researchers restrict their attention to instances that satisfy certain application-area-specific assumptions, which are likely to be satisfied in practice. Such assumptions may be of stochastic (e.g., smoothed analysis of Simplex and local search [34, 35, 15]) or deterministic nature (e.g., perturbation stability in clustering [1, 3, 5, 6]).
The beyond worst-case approach is not anything new for (Algorithmic) Mechanism Design. Bayesian analysis, where the bidder valuations are drawn as independent samples from an arbitrary distribution known to the mechanism, is standard in revenue maximization  and has led to many strong and elegant results for social welfare maximization by truthful posted price mechanisms (see e.g., [19, 14]). However, in this work, we significantly deviate from Bayesian analysis, where the mechanism has a relatively accurate knowledge of the distribution of bidder valuations. Instead, we suggest a deterministic restriction on the class of instances (namely, perturbation stability) and investigate if there is a natural class of mechanisms (e.g., P2A) that are incentive-compatible and achieve optimality for CAs with stable submodular valuations.
Our focus on perturbation stable valuations was actually motivated by the recent work of Babaioff et al.  and Ezra et al.  on combinatorial markets where the valuations exhibit the endowment effect. The endowment effect was proposed by the Nobel Laureate Richard Thaler  to explain situations where owning a bundle of items causes its value to increase. Babaioff et al.  defined that if an allocation is -endowed, for some , in a CA with bidder valuations , then the valuation function of each bidder becomes
for all item sets . Roughly speaking, the value of (and its subsets) is inflated by a factor of due to the endowment effect. The main result of  is that for any combinatorial market with submodular valuations , any locally optimal allocation and any , the market with -endowed valuations for admits a Walrasian equilibrium (see Section 2 for the definition) where each bidder receives . In simple words, social welfare maximization in combinatorial markets with endowed valuations is polynomially solvable and the optimal allocation is supported by item prices. Subsequently, Ezra et al.  presented a general framework for endowed valuations and extended the above result to XOS valuations and general valuations, for a sufficiently large endowment (see also previous work on bundling equilibrium and conditional equilibrium [9, 20]).
Inflated valuations due to the endowment effect naturally occur in auctions that take place regularly over time. Imagine auctions for e.g., season tickets of an athletic club, spots in a parking lot, reserving timeslots for airport gates, vacation packages at resorts, etc., where regular participants tend to value more the bundles allocated to them in the past, due to the endowment effect (see also  for more examples). Given the strong positive results of [4, 16], a natural question is whether CAs with valuations inflated due to the endowment effect allow for stronger approximation guarantees in social welfare maximization and mechanism design.
Stable Combinatorial Auctions. To investigate the question above, we adopt a slightly stronger condition on valuation profiles, namely perturbation stability, which is inspired by (and bears a resemblance to) the definition of perturbation stable clustering instances (see e.g., [1, 3, 5, 6]).
For a constant , a -perturbation of a valuations profile on a bidder and an item is a new valuations profile , where for all bidders , and for all ,
A CA with valuations is -perturbation stable (or -stable) if the optimal allocation for is unique and remains unique for all -perturbations of .
Let Alice and Bob compete for 2 items, and , and have valuations and , and and . The (unique) optimal allocation is to give to Alice and to Bob, with social welfare . A perturbation with most potential to change the optimal solution is to inflate Alice’s value of by . Then, we get , and . The optimal solution remains unique for any . Hence the above CA is -stable, for any .∎
At the conceptual level, we feel that the condition of -stability is easier to grasp and to think about in the context of mechanism design for CAs (compared against considering valuation profiles resulting from the -endowment of an optimal solution to an initial valuations profile )111For a better understanding of the two conditions at a technical level, we note that a (technically very useful) necessary condition for a valuations profile to be -stable is that for the optimal allocation , any bidders and any item ,
We focus on deterministic algorithms and mechanisms. We first show that a simple greedy algorithm (Algorithm 1) that allocates each item to the bidder with maximum finds the optimal allocation for CAs with -stable subadditive valuations (Theorem 1). Moreover, similarly to , we show that for -stable submodular valuations, combining the optimal allocation with a second price approach, where each item gets a price of , results in a Walrasian equilibrium (Theorem 3).
On the negative side, we prove that our positive results above cannot be significantly strengthened. We first show that there is a simple -stable CA with submodular bidders where approximating the social welfare within any factor larger than requires at least value queries, for any integer (Theorem 2). Thus, a polynomial-time approximation scheme does not exist for -submodular valuations. Moreover, we show that for any , there is a -stable CA with a XOS bidder and a unit demand bidder that does not admit a Walrasian equilibrium (Lemma 2).
On the mechanism design part, in a nutshell, we show that (possibly appropriately modified) Parallel 2nd Price Auctions (P2A) behave very well for stable CAs. We should highlight that despite the fact that maximizing the social welfare for -stable subadditive CAs is easy, VCG is not an option for the design of computationally efficient incentive compatible mechanisms. The reason is that removing a single bidder from a -stable CA may result in an NP-hard (and hard to approximate) (sub)instance.
In Section 5, we show that a P2A followed by a demand query for each bidder is dominant strategy incentive compatible (DSIC) for all CAs with subadditive bidders and maximizes the social welfare if the valuations profile is submodular and -stable (Theorem 4). If demand queries are not available, the mechanism boils down to a simple P2A. We show that P2A is ex-post incentive compatible (EPIC) for -stable submodular valuations and that truthful bidding leads to the optimal allocation.
On the negative side and rather surprisingly, we show that demand queries are indeed necessary for computationally efficient mechanisms that are DSIC for all submodular valuations and maximize the social welfare if the instance is -stable (even if is arbitrarily large, Theorem 6). Our construction is an insightful adaptation of the elegant lower bound in [11, Theorem 1] to the case of stable submodular valuations. We show that any DSIC mechanism that computes the optimal allocation for stable CAs and does not use demand queries must use exponentially many value queries. The crux of the proof is that in certain instances, the bidders may find profitable to misreport and switch from a non-stable instance to a stable one.
In Section 6, we analyze the Price of Anarchy (PoA) of P2A and Parallel 1st Price Auctions (P1A). Our results demonstrate that the quality of equilibria of simple non-truthful auctions improves only for -stable valuations, with . We show that the PoA of P2A for CAs with -stable submodular valuations is (Theorem 7), while there are -stable CAs with PoA equal to (Lemma 3), which matches the PoA for CAs with general submodular valuations (see e.g., ). Moreover, we show that the PoA of both P2A and P1A for CAs with -stable XOS valuations is at least , for any (Theorem 8 and Theorem 9).
1.3 Previous Work
Social welfare maximization with submodular and subadditive valuations has been studied extensively. Submodular Welfare Maximization (SMOD-WM) is known to be -approximable with polynomially many value queries  and -approximable, for a fixed constant , with polynomially many demand queries . Moreover, a simple and natural greedy algorithm achieves an approximation ratio of using only value queries . The results about polynomial-time approximability with value queries are best possible, in the sense that approximating SMOD-WM within a factor of , for any constant , is NP-hard  and requires exponentially many value queries . Furthermore, there is a constant , such that approximating SMOD-WM within a factor of with demand queries is NP-hard . Subadditive Welfare Maximization (SADD-WM) is -approximable with polynomially many value queries (and this is best possible ) and -approximable with polynomially many demand queries .
Truthful maximization of social welfare in CAs with submodular (or XOS) bidders has been a central problem in Algorithmic Mechanism Design. In the worst-case setting, where we do not make any further assumptions on bidder valuations, Dobzinski et al.  presented the first truthful mechanism that uses polynomially many demand queries and achieves a non-trivial approximation guarantee of . Dobzinski  improved the approximation ratio to for the more general class of subadditive valuations. Subsequently, Krysta and Vöcking  provided an elegant randomized online mechanism with an approximation ratio of for XOS valuations. Dobzinski  broke the logarithmic barrier for XOS valuations, by showing an approximation guarantee of , which was recently improved to by Assadi and Singla . Accessing valuations through demand queries is essential for these strong positive results. Dobzinski  proved that any truthful mechanism for CAs with submodular bidders with approximation ratio better than must use exponentially many value queries. Truthful -approximate mechanisms that use polynomially many value queries are known even for the more general class of subadditive valuations (see e.g., ).
In the Bayesian setting, Feldman et al.  showed how to obtain item prices that provide a constant approximation ratio for XOS valuations. These results were significantly extended and strengthened by Düetting et al. .
Previous work has also shown strong Price of Anarchy (PoA) guarantees for CAs with submodular, XOS and subadditive bidders that can be achieved by simple (non-truthful) auctions, such as P2A and P1A (see e.g., [8, 36, 29, 31]).
Our notion of perturbation stability for CAs is inspired by conceptually similar notions of perturbation stability in clustering [3, 6]. Angelidakis et al.  presented a polynomial-time algorithm for -stable clustering instances with center-based objectives (e.g., -median, -means, -center), while Balcan et al.  proved that there is no polynomial-time algorithm for -stable instances of -center, unless NP = RP. To the best of our knowledge, this is the first time that the notion of perturbation stability has been applied to social welfare maximization and to algorithmic mechanism design for Combinatorial Auctions.
2 Notation and Preliminaries
The key notion of -perturbation stability (Definition 1) and a significant part of the terminology and the notation are introduced in Section 1. In this section, we introduce some additional terminology, notation and conventions used in the technical part.
We always let denote the optimal allocation for the instance at hand, and let be the bundle of bidder in . For convenience, we usually let an index also denote the singleton set (we write , , , etc., instead of , , ). We use both and for the set difference. We denote the marginal contribution of a bundle wrt. as .
In addition to submodular, XOS, and subadditive valuations, we consider the classed of additive and unit-demand valuations , where there exist , such that for any , and , respectively. A useful property of an XOS valuation is that for any , there is an additive valuation that supports , in the sense that and for any , .
We focus on deterministic algorithms and mechanisms and consider bidders with quasi-linear utilities, where the utility of bidder with valuation for a bundle at price is
. For a price vector, we often let denote the price of a bundle .
An allocation and a price vector form a Walrasian Equilibrium if all items with are allocated and each bidder gets a utility maximizing bundle (or, his demand) in , i.e., , .
A mechanism is dominant-strategy incentive compatible (DSIC) (or truthful) if for any valuations profile , answering (value or demand) queries truthfully is a dominant strategy and guarantees non-negative utility for all bidders. A mechanism is called ex-post incentive compatible (EPIC) if truthful bidding is an ex-post Nash equilibrium and guarantees non-negative utility for all bidders.
Let be a profile of distributions over valuation functions (i.e., over possible bids). In a mechanism with allocation rule and item pricing rule , forms a Mixed Nash Equilibrium (MNE), if no bidder has an incentive to unilaterally deviate from , i.e., for any bidder and any distribution over valuation functions,
If instead of distributions over valuation functions, we restrict each and to valuation functions (i.e., to pure strategies over possible bids), we get the definition of a Pure Nash Equilibrium (PNE).
In Section 6, we consider the Price of Anarchy (PoA) of Parallel 2nd Price Auctions (P2A) and Parallel 1st Price Auctions (P1A). In a Combinatorial Auction, the PoA of a mechanism is the ratio of (resp. expected) social welfare at the worst Pure (resp. Mixed) Nash Equilibrium to the social welfare of the optimal allocation. Formally, focusing on the more general case of Mixed Nash Equilibria:
Properties of Stable Valuations. The following shows a technically useful property of -stable CAs (see also Footnote 1).
Lemma 1 (Valuation Stability).
Let be -stable and subadditive valuations. Then, for all bidders and all items
Fix a bidder , an item and another bidder . Since the valuations are -stable, the optimal allocation remains unique and optimal, even if we inflate ’s value for item by . Such a -perturbation of results in and . Using the optimality of the allocation after the perturbation, we obtain that:
Using that is non-decreasing concludes the proof of the lemma. ∎
3 Social Welfare Maximization for Stable Valuations
We next consider the problem of social welfare maximization for -stable CAs. We first show that for -stable subadditive valuations, we can compute the optimal solution with value queries in polynomial time.
Let be a -stable subadditive valuations profile. Then Algorithm 1 outputs the optimal allocation using value queries.
The number of value queries follows directly from the description of the algorithm. As for optimality, we fix an item and let be the bidder that gets in the optimal solution. Because of Lemma 1 and the fact that is -stable, we know that , for any other bidder . Because Algorithm 1 allocates to the bidder with the highest singleton value, gets item in the allocation of Algorithm 1. ∎
On the negative side, we next show that a polynomial-time approximation scheme does not exist even for -stable submodular valuations.
For any , there exists a submodular -stable valuations profile such that for any integer , approximating the optimal allocation in within any factor larger than requires at least value queries.
Given a set we define the following class of valuations:
We first prove that the above valuation is submodular for any . To do that we examine the value of , given that :
This entails that for any sets , where , it holds that . This proves the submodularity, for any .
Now we note that given a valuation and value query access to it, finding the bundle requires at least value queries: Let be such that . Observe that the valuations and differ only in their value for and . Thus, a query for the value of a bundle only tells us whether the valuation is or not. In the worst case, we have to query the value of every bundle for which (except the “last” bundle) to determine .
Now we fix the number of items and the number of bidders such that and is an integer. We also fix a partition of the items , where and the valuation of each bidder to .
First we note that allocation is the optimal allocation, with welfare , because a higher welfare is impossible, as each bidder values for .
Now we will prove that the valuations are -stable. Notice that the allocation with the second highest welfare has welfare : A welfare of is not achievable as one bidder would need to contribute and all the others exactly . To do that we would have to give each bidder exactly items and exactly one bidder would need to not take his optimal set. This entails that a second bidder would also not get his optimal, thus making both bidders contribute to the total welfare.
Since the second best allocation has welfare and every singleton value is 1, the maximum value that the second highest allocation can achieve even with inflation for 1 item by is . Thus the valuations are -stable if . This entails that can take any value less 2.
Because of the structure of the valuations, finding the optimal allocation is hard. More specifically, in the worst case allocating any bidder his correct bundle requires value queries. This means that with less queries no bidder is going to get his optimal set. If no bidder gets his optimal set the highest achievable welfare is guaranteed when all the bidders get items. This leads to an approximation ratio
Thus we have proven that the inapproximability. Together with the submodularity and the stability of the valuations, this concludes the proof. ∎
4 Existence of Walrasian Equilibrium
Similarly to [4, Theorem 4.2], we next show that combinatorial markets with -stable submodular valuations admit a Walrasian Equilibrium.
Let be -stable submodular valuations. For every bidder every item , let be the price of that item for which: . Then, the prices form a Walrasian Equilibrium.
Fix a bidder and his optimal bundle . We note that the price of each item is well defined. Because of Lemma 1, 2-stability, and , .
We next show that for any , bidder is not interested in getting . Because of submodularity, ’s additional utility due to item is at most . Because , , making ’s utility from non-positive. Hence, ’s demand is a subset of .
To conclude the proof, we show that for any item , gets non-negative utility due to . Fix a bundle in the demand of bidder with . Note that we have already proven that . The utility gained by taking is , which is non-negative; submodularity makes . Hence, is the demand of bidder . ∎
We next show that Theorem 3 cannot be extended to stable XOS valuations. In the proof of Theorem 3, -stability and submodularity ensure that the prices cannot exceed the marginal increase of adding an item to some . For XOS valuations, however, the marginals may not be increasing with . As a result, the utility of a bidder may be maximized by a strict subset of his optimal bundle . This is the core idea for the proof of the following lemma.
For every , there exists a -stable valuations profile with a XOS bidder and a unit demand bidder which does not admit a Walrasian Equilibrium.
Fix a value of and suppose there are items, where . The XOS bidder’s valuation is simply a function of the carnality of his bundle:
where is a very large positive value. It is easy to verify that the above valuation is indeed monotone. To prove that it is also XOS we arbitrarily order the items and use the following additive functions
For , , values item for , and the rest of the items for 0, i.e. : .
The -th function, , values every item for , i.e. : .
We now notice that . We should add that this holds only if the value of is large enough.
The unit-demand bidder values each item for : . The optimal solution is to give the whole bundle to the XOS bidder. If we endow the valuation of the unit-demand bidder by for any item , it would make it . Even if the unit demand bidder had this valuation, then the optimal allocation still is to give the XOS bidder all the items, making the valuations -stable.
Now we are going to prove that this instance does not admit a Walrasian Equilibrium. To do that let us assume that there exist prices that form a Walrasian Equilibrium. Denote with the item whose price is minimum, i.e. . The XOS bidder’s utility for the cheapest item should be at most his utility for the grand bundle:
By reordering and using the fact that has the minimum price we get
Now we also use the fact the the utility of the unit-demand bidder for taking item should not be strictly positive, since we have a WE
5 Mechanism Design For Stable Combinatorial Auctions
In this section, we investigate truthful mechanism design for CAs with stable submodular valuations. We should emphasize that despite Theorem 1, VCG cannot be used as a computationally efficient DSIC mechanism for stable subadditive CAs, because the subinstances , whose optimal solutions determine the payments, may not be stable and may be NP-hard to solve optimally (e.g., adding a bidder with additive valuation that has a huge value for each singleton to any CA results in a stable instance).
We first present a truthful extension of Algorithm 1, which is implemented as a Parallel 2nd Price Auction (P2A) and also uses a demand query for each bidder.
First, we show that Mechanism 2 is DSIC for subadditive bidders. We focus on the bidding step, because assuming that each set is determined in a truthful way, it is always in each bidder’s best interest to respond to his demand query truthfully.
We observe that no bidder has incentive to bid lower than his singleton value for an item. Bidding lower could only lead to the bidder losing some items, thus restricting the set of items available to him through his demand query. Moreover, no bidder has incentive to bid higher than his singleton value for an item. This would only entail having access to an item that has price at least his actual singleton value. However, because of subadditivity, the bidder does not include such an item in his demand set.
Next, we show that a P2A (Mechanism 3), that uses only value queries, is ex-post incentive compatible when restricted to -stable submodular valuations profiles. The proof of the following is an immediate consequence of Theorem 3 and Theorem 1.
The number of value queries immediately comes from the mechanism’s description. The fact that it finds the optimal allocation comes directly from theorem 1; the same bundles are allocated, which was proven to be optimal even for subadditive bidders.
To prove that the mechanism is EPIC, we fix a bidder , his valuation and suppose that any other bidder bids his singleton value for each item . We need to prove that has nothing to gain if he bids untruthfully.
We will show this by contradiction. Suppose that makes a different bid and ends up with higher utility. In order to change his utility he must change either his allocated set or the payments. Since the payments are independent of his bids, he must to be allocated a set different than . First we prove that it must hold that .
Let’s assume the opposite, that there exists a for which and . Then ’s utility by dropping would decrease by , but it would increase by . Because belongs to some , by Lemma 1, and -stability . Also because of submodularity, . This shows that has strictly higher utility if he drops any items not in . Now we have to disprove that .
By contradiction, suppose that there exists a such that , but . Then ’s utility by acquiring would increase by and decrease by . From Lemma 1, we know that , which means that bidder gains strictly positive utility by obtaining any item in .
The theorem is now proven. We have shown that bidder strictly loses utility from items outside of and strictly gains utility from items in . ∎
Interestingly, Mechanism 3 is not DSIC even when restricted to submodular CAs. The reason is that even the true bidder valuations profile may be -stable, their bids might be not. Hence, it may happen that bidder bids higher than his real singleton value on some item , but is allocated to different bidder . This may increase to a level that is no longer profitable for bidder to get item (which is exactly the reason that we employ the demand queries in Mechanism 2).
The remark above naturally motivates the question about existence of a computationally efficient DISC mechanism that computes the optimal allocation for -stable submodular CAs using only value queries. Rather surprisingly, the following answers this question in the negative.
Let be any mechanism that is DSIC, uses only value queries and finds the optimal solution for -stable submodular valuations, for some . Then makes exponentially many value queries.
The proof is an interesting adaptation of the proof of [11, Theorem 3.1]. For the proof, we use instances with just bidders. Fixing one to be additive, the other may bid “stably” and get any bundle. However, due to the structure of his true valuation, finding his demand may be intractable, which makes misreporting a profitable strategy.
To reach a contradiction, we assume that is DSIC, makes polynomially many value queries and always finds the optimal solution for -stable submodular valuations, for some fixed . First we establish the following, which helps determining whether a set of additive valuations is stable.
Let be a profile with additive valuations. Then, for any , is -stable if for any item , the largest value for in differs from the second largest value for in by a factor larger than . Namely, if , then , for all bidders .
Proof of Proposition 1.
The proposition follows directly from the fact that endowing an additive bidder for an item keeps him additive and inflates his singleton value by a factor of . ∎
For the rest of the proof, we consider bidders and fix the valuation according to which bidder makes his bids: . is not necessarily the true valuation of bidder . We fix his true valuation to be also additive, with the value of each item large enough. This valuation, together with any other bounded and submodular valuation, results in a valuations profile that is submodular and stable (for a large enough stability factor).
We next prove that bidder can get any bundle.
For any bundle , bidder 2 will be allocated , if he bids according to
Proof of Proposition 2.
First we fix any bundle . By Proposition 1, valuations are -stable. Taking large enough makes the valuations -stable. Given that they are also additive, and thus submodular, we get that the mechanism (which, by hypothesis, computes the optimal allocation for -stable instances) allocates to bidder . ∎
Next, we show that the prices set by for bidder 2 are bounded and strictly increasing.
For any bundle and any , it holds that
where and are the prices of bundles and assigned from for bidder 2.
Proof of Proposition 3.
We examine what happens when bidder 2 bids according to (5). First we set , which means that bidder 2 will receive . Since is DSIC, bidder 2 should not prefer over , i.e., . This implies the rhs of (6), because and (since ).
We also note that setting in (6), we get that .
We now create an exponentially large structured submenu, as in [11, Definition 3.2]. This concludes the proof, since the existence of such a submenu entails that requires exponentially many value queries to find the demand of bidder 2, as shown in [11, Lemma 3.10]. For completeness, we recall that a collection of bundles comprises a structured submenu for bidder 2 if:
For all , bidder 2 can be allocated .
For each : and .
For all such that and : .
For all : .
Since bidder 2 can get any bundle, the first property is satisfied. Also by Proposition 3, the third property is satisfied, because for any : .
To create the structured submenu, we fix and consider all the different bundles of size . Our submenu is a subset of those bundles, which immediately satisfies the first part of the second property. Also since , the price of each bundle is at most , which implies the last property.
We need to show the last part of the second property. To this end, following the construction of [11, Section 3.1], we split the interval into bins. For each bundle of size , we put in the -th bin if . Since there are bins and bundles, one bin must have exponentially many bundles. Let be the set of bundles in such a bin. Notice that the bundles of the same bin have prices which differ less , thus satisfying the last part of the second property.
A natural question is whether one could also follow the first part of the proof of [11, Theorem 3.1], in order to get a much stronger inapproximability bound of , for any . Unfortunately the answer is negative, because the polar additive valuation profiles in [11, Section 3.1] are far from stable. This explains the necessity of our careful construction of stable valuation profiles, in the first part of the proof of Theorem 6.
6 Price of Anarchy in Stable Combinatorial Auctions
6.1 The Price of Anarchy in Parallel 2nd Price Auctions
For XOS valuations, the PoA of P2A is at least . We next show that even for -stable valuations, the PoA of P2A does not improve.
There exists a -stable profile with unit-demand valuations for which the PoA of P2A is .
The instance in Example 1 is -stable and has been used to show that the PoA of P2A is at most . More precisely, we observe that there is an equilibrium with social welfare : Alice bids for and for and Bob bids for and for . ∎
Interestingly, the previous result is tight. With -stable submodular valuations, every equilibrium is optimal. To prove this, we introduce a no-overbidding assumption, which is weaker than the usual Strong No Overbidding assumption (where each bidder’s value for any bundle is at most the sum of his bids for ). We call our assumption Singleton No Overbidding (SiNO), as it restricts the bids to be below each corresponding singleton value. We also note that the bidding profile in Lemma 3 SiNO.
Definition 2 (Singleton No Overbidding).
A bidding profile satisfies Singleton No Overbidding (SiNO) if for any bidder and item : .
Now we are ready to prove that with SiNO, PoA is always for CAs with -stable submodular valuations.
Let be a -stable submodular valuations profile, and let be a bidding profile that forms a Pure Nash Equilibrium for P2A and satisfies SiNO. Then the allocation at the equilibrium coincides with the optimal allocation.
We prove the theorem by contradiction. First, denote with the second highest singleton valuation for item , i.e. if , ( has the highest singleton valuation for , by Lemma 1). Fix a bidder who does not get allocated his optimal set, i.e. . Let be the items that is allocated at the equilibrium that are also in , i.e. .
We construct a deviating bid for : For items not in , bids . For items in , bids as before (which means that the prices for these items will be the same as before). Finally for items in , bids infinitesimally more than . Note that this bidding strategy conforms to SiNO. Also because of SiNO, it guarantees that receives the whole bundle . His new utility is
where the inequality holds because of SiNO: Every bid must be below the corresponding singleton value, which in turn is less than the maximum of the singleton values. Now we use that ’ utility in (7) cannot be greater than ’s utility at the equilibrium (for brevity, we let from that point on). Hence, we obtain that:
In (8), we can eliminate the term , because it is included in the term . Also, we can ignore the rest of the prices in the lhs, as they are positive. Hence, we obtain that:
Now using that , for items , and submodularity, namely that , and (9), we obtain the following:
Now because, by definition, both sets and consist of the items that were not optimally allocated (and only them), summing up (10) over all bidders , we get that:
For every , it holds that , which contradicts (12). The latter inequality is true because of submodularity. More specifically, we can index items in as and denote , with and . Hence, we obtain that: