Simple auctions are often preferred in practice over complex truthful auctions. There has been a vast literature in the last decade studying the performance of simple auctions, using the price of anarchy (PoA) measure. The PoA is the ratio between the worst performance of an auction in equilibrium (for different equilibrium notions, see below) and the optimal outcome, with respect to some objective function.
Starting with the seminal paper of Christodoulou et al. [4, 5], a lot of effort has been given to quantifying the performance of simultaneous item auctions, using the PoA measure with respect to the social welfare objective. In a simultaneous item auctions, bidders submit bids on individual items despite the fact that their valuation is non-linear and exhibit different levels of substitutes and complements. Clearly, these auctions are non-truthful; bidders don’t even have the language to express their true valuations. The main message from  and follow-up work [16; 1; 11; 17; 9; 19; 6; 18] is that the performance of simultaneous item auctions is nearly optimal as long as the valuations are subadditive (also known as complement free).
The PoA of simple auctions has been studied with respect to different equilibrium notions in both complete- and incomplete-information settings. The most natural equilibrium notion in complete-information setting is pure Nash equilibrium (PNE), but PoA results have been extended to mixed Nash equilibrium (MNE), correlated equilibrium (CE), coarse-correlated equilibrium (CCE) and learning-based equilibrium notions [2; 16; 1; 17; 9; 19; 6; 18]. For precise definitions of these equilibrium notions, see Section 2.1
. In the incomplete-information setting, the profile of bidder values is drawn from a known probability distribution. Every bidder knows her own value, but only the probability distribution of others’ values. TheBayesian PoA (BPoA) measures the performance of the worst Bayes Nash equilibrium, where every bidder maximizes her utility in expectation over the other bidders’ values.
In simultaneous item auctions with bidders and items, every bidder has a valuation function , where is the value bidder assigns to set . The valuation function is assumed to be monotone and normalized. Despite the combinatorial structure of the valuation, bidders submit bids on every item separately and simultaneously. Let
be the bid vector of bidder, where is the bid of bidder for item , and be the bid profile of all bidders. Every item is sold separately via an auction based on bids on item alone.
The two main auction formats studied in the literature are simultaneous first-price auctions (S1PA) and simultaneous second-price auctions (S2PA). As their names suggest, in S1PA, every item is sold in a 1st-price auction; i.e., highest bidder wins and pays her bid, whereas in S2PA every item is sold in a 2nd-price auction; i.e., highest bidder wins and pays the 2nd highest bid.
The PoA and BPoA of simultaneous item auctions depend on the structure of the valuation functions. A hierarchy of complement-free valuations is given in . Four important classes of valuations include unit-demand, submodular, xos, and subadditive valuations, with the following strict containment relation:
A unit-demand valuation is one where there exist values , and . A subadditive valuation is one where , also known as a complement-free valuation, as the value of the union of any two sets never exceeds the sum of their values. The other valuations are formally defined in Section 2.2. Clearly, the PoA can only degrade as one moves to a larger valuation class.
PoA and BPoA of simultaneous 2nd price item auctions (S2PA): previous results.
There are some pathological examples showing that the PoA of S2PA can be arbitrarily bad, even in the simplest scenario of single item auction . A common approach toward overcoming such pathological examples is the no overbidding (NOB) assumption, stating that the sum of player bids on the a set of items she wins under bid profile , , never exceeds its value . Consequently, all PoA and BPoA results for S2PA are derived under the NOB assumption.
The PoA of S2PA with NOB for subadditive valuations is . This is tight even with respect to unit-demand valuations. The BPoA of S2PA for xos valuations is . For subadditive valuations the BPoA is at least , and is strictly smaller than . Finally, going beyond subadditive valuation is hopeless: there are instances where the PoA can be as bad as [11; 9]. The above results are summarized in Table 1.
No Underbidding (NUB)
Consider the following example (taken from ), showing that the PoA for unit-demand valuations is .
2 bidders, and 2 items: . Bidder 1 is unit-demand with values . Bidder 2 is unit-demand with values . Consider the following bid profile (which is a PNE that adheres to NOB): , and . Under this bid profile, bidders 1 and 2 receive items and , respectively, for a social welfare of 2. The optimal welfare is 4.
In this equilibrium, bidder 1 prefers item , yet bids on item , and gets item instead. The same goes for bidder 2 with respect to item . In what sense is this an equilibrium? To answer this question, we should revisit the foundation of the notion of an equilibrium. A Nash equilibrium is a descriptive, static notion, where no player can make a profitable deviation given the strategies of others. While a NE is a static notion, it is based on the underlying assumption that players engage in some dynamics, where they keep best responding to the current situation until a stable outcome is reached.
Let us revisit the PNE bid profile in Example 1.1 in light of this interpretation. For two sets , the marginal value of given is defined as . Bidder 1’s marginal value for item , given her current allocation (item ) is 1. For bidder , bidding on item is weakly dominated by bidding the marginal value of item given the current outcome, which is . Indeed, if bidder 1 gets item , her marginal value is and she pays at most , so why not?
If a bidder bids on an item less than the item’s marginal value, we say that she underbids. Formally, a bidder is said to underbid on item in a bid profile if , where is the set of items wins under . For example, in Example 1.1, bidder 1 underbids on item (and similarly, bidder 2 underbids on item ). In Section 4 we show that underbidding in a 2nd price auction is weakly dominated in some precise sense.
No-underbidding is not only a mere theoretical exercise. In second price auctions a lot of empirical evidence suggest that bidders tend to overbid, but not underbid (e.g., Kagel and Levin , Harstad , Cooper and Fang  and Roider and Schmitz ). It seems that ”laboratory second-price auctions exhibit substantial and persistent overbidding, even with prior experience” . The no-underbidding assumption is also consistent with the assumption made by Nisan et al.  that bidders break tie in favor of the highest bid that does not exceed their value. We therefore believe that to get a more accurate measure of the performance of simultaneous 2nd price auctions, we should impose a no underbidding assumption. This begs the following natural question:
Main Question. What is the performance (measured by PoA/BPoA) of simultaneous 2nd price item auctions under a no underbidding assumption?
1.1 Our Contribution
We first introduce the notion of item no underbidding (iNUB), where no agent underbids on any item. One might think that by imposing both NOB and iNUB, the optimal welfare will be achieved. This is indeed the case for a single item auction (where the optimal welfare is achieved by imposing any one of these assumptions alone). However, even a simple scenario with 2 items and 2 unit-demand bidders can have a PNE with sub-optimal welfare. This is demonstrated in the following example.
2 bidders, and 2 items: . Bidder 1 is unit-demand with values . Bidder 2 is unit-demand with values . Consider the following PNE bid profile, which adheres to both NOB and iNUB: . Under this bid profile, bidders 1 and 2 receive items and , respectively, for a social welfare of 4. The optimal welfare is 6. Thus, the PoA is .
Our first result states that is the worst possible ratio for bid profiles satisfying both NOB and iNUB, even for submodular valuations and even in settings with incomplete information (with a product distribution over valuations). In fact, a weaker condition than iNUB suffices for this result, requiring no underbidding only on items , where is the set of items that bidder receives in an optimal allocation under valuation profile , and is the set of items she wins under bid profile .
Theorem [submodular valuations, NOB and iNUB]: For every market with submodular valuations,
The above results are tight, even with respect to PNE and even for unit-demand valuations.
Moreover, the last theorem extends to -submodular valuations, defined as for every . We show that the (B)PoA degrades gracefully with the parameter ; namely the PoA with respect to CCE and the BPoA are at least under iNUB and at least under NOB and iNUB.
Beyond (-)submodular valuations, however, these bounds break. In particular, in Example 6.1 we present an instance with xos bidders and items, where the PoA with iNUB is . We also prove that this is the worst possible PoA, showing that the (B)PoA with xos bidders and items is always at least (see Appendix E). Moreover, in Example E.3 we show an instance with items and xos bidders, where the PoA with NOB and iNUB is , which is no better than the guarantee obtained with NOB alone.
To the best of our knowledge, this is the first PoA separation between submodular and xos valuations in simultaneous item auctions. In fact, the PoA of simultaneous item auctions is often the same for the entire range between unit-demand and xos.
A few remarks are in order. First, the above separation suggests that xos is “far” from submodular. Indeed, in Appendix A we show that xos is not -submodular for any fixed , even in settings with identical items. Second, under iNUB alone (without NOB), the BPoA of -submodular valuations is at least ( for submodular valuations). For xos valuations, however, iNUB alone is not helpful; the PoA may be (see Appendix E).
To deal with valuations beyond submodular, we consider a different no underbidding assumption, which applies to sets of items. A bidder is said to not underbid on a set of items if . The new condition, set no underbidding (sNUB), imposes the set no underbidding condition on every bidder with respect to the set .
With the sNUB definition, the PoA extends to subadditive valuations in full information settings, and to xos valuations even in incomplete information settings (with product distributions).
Theorem [subadditive and xos valuations, NOB and sNUB]: For every market with subadditive valuations, the PoA with respect to CCE of S2PA under NOB and sNUB is at least (see Theorem 7.2). For every market with xos valuations, the BPoA (under product distribution) of S2PA under NOB and sNUB is at least (see Corollary 6.2, which is derived from Theorems 2.15, 4.6 and 3.6). Both results are tight.
For incomplete information we show that the BPoA of subadditive valuations is at least of the optimal social welfare and it can be obtained in a much stronger sense, namely for every bid profile with non-negative sum of utilities (even a non-equilibrium profile) satisfying sNUB. This also holds for markets with arbitrary monotone valuations.
Theorem [Arbitrary valuations, sNUB]:
For every market (arbitrary monotone valuations), the PoA with respect to CCE and the BPoA (for any joint distribution) of S2PA under sNUB is at least(see Corollary 4.7).
PoA results make sense only when the corresponding equilibrium exists. We show that every market with XOS valuations admits a PNE satisfying sNUB and NOB. For subadditive valuations, a PNE might not exist (even without any NOB or NUB conditions). However, under a finite discretized version of the auction, a mixed Bayes Nash equilibrium is guaranteed to exist, and we show that there is at least one bid profile that admits both sNUB and NOB with arbitrary monotone valuation functions.
Interestingly, our results shed new light on the comparison between simultaneous 1st and 2nd price auctions.
S1PA vs. S2PA
Table 2 specifies BPoA lower bounds for S1PA and S2PA under NOB, assuming independent valuation distributions. According to these results, one may conclude that S1PA perform better than their S2PA counterparts. Indeed, for S2PA, the PoA is even for unit-demand bidders and even with respect to PNE , whereas the BPoA of S1PA is at least , even for the much more general class of xos valuations and with respect to the much more general class of Bayesian NE . Similarly, the BPoA of S2PA may be smaller than for subadditive valuations , whereas it is always at least for S1PA.
Our new results shed more light on the relative performance of S2PA and S1PA. When considering both no overbidding and no underbidding, the situation flips, and S2PA are superior to S1PA. For xos valuations, the bound for S1PA persists, but for S2PA the bound improves from to . For subadditive valuations and independent valuation distributions, S2PA under sNUB performs as well as S1PA (achieving BPoA of ), however in S2PA the bound holds also for correlated valuation distributions. For valuations beyond subadditive, S2PA performs better ( for S2PA and less than for S1PA). Note that no underbidding is not a reasonable assumption in first price auctions, where bidders pay their bids, therefore no underbidding is only relevant in S2PA.
|S1PA||[19; 6]||[9; 6]|||
1.2 Our Techniques
The standard technique for establishing performance guarantees for equilibria of simple auctions (i.e., PoA results) is the smoothness framework (see the survey in ). Smoothness is a parameterized notion; an auction is said to be -smooth if for any valuation profile and any bid profile there exists a bid for each player , s.t. . It is quite straightforward to show that if an auction is -smooth, then its PoA with respect to PNE is at least .
The power of the smoothness framework is in its extendability. While a lower bound on the PoA with respect to PNE follows easily from the smoothness property, this lower bound extends beyond this equilibrium notion [16; 17; 18; 19]. The first extension theorem shows that smoothness leads to the same PoA bound even with respect to CCE (in full information settings). The second extension theorem shows that smoothness also leads to the same lower bound on the Bayesian PoA in games with incomplete information.
We introduce a new parameterized notion called revenue guaranteed. An auction is said to be -revenue guaranteed if for every valuation profile and bid profile the revenue of the auction is bounded below by .
We show that in every -revenue guaranteed auction, the social welfare in every bid profile with non-negative sum of utilities is at least a fraction of the optimal welfare. Similarly to the smoothness framework, we augment our results with two extension theorems, one for PoA with respect to CCE, and one for BPoA in settings with incomplete information. Moreover, this result holds also in cases where the joint distribution of bidder valuations is correlated (whereas previous BPoA results hold only under a product distribution over valuations).
Combining the two tools of smoothness and revenue guaranteed, we get an improved bound. In particular, we show that in every auction that is both -smooth and -revenue guaranteed, the PoA with respect to CCE is at least . The same holds for the BPoA under product valuation distributions.
With this tool in hand, we analyze simultaneous 2nd price auctions with different valuations functions and different no underbidding conditions, where the goal is to establish revenue-guaranteed parameters that would imply PoA and BPoA bounds.
We first consider submodular and -submodular valuations. We show that every S2PA with -submodular valuations satisfying iNUB is -revenue guaranteed. This directly gives a lower bound of on the BPoA of -submodular valuations (and for submodular valuations). We also show that S2PA with -submodular valuations satisfying NOB are -smooth. Combining -revenue guaranteed with -smoothness gives a bound of on the BPoA for every S2PA with -submodular valuations with NOB and iNUB. For submodular valuations this gives the tight bound.
For valuations beyond -submodular valuations, the iNUB condition is not helpful, so we turn to the stronger sNUB condition. We show that every S2PA with arbitrary monotone valuations satisfying sNUB is -revenue guaranteed for bid profiles with non-negative sum of utilities. This recovers the bound on PoA with respect to CCE and BPoA with correlated distributions for S2PA satisfying sNUB. For XOS valuations, we combine the last result with the known -smoothness to get the bound for S2PA satisfying NOB and sNUB (for PoA with respect to CCE and for BPoA with product distributions). For subadditive valuations, we apply the technique from  to yield a tight bound of on the PoA with respect to CCE.
In a combinatorial auction a set of non-identical items are sold to a group of players. Let be the set of possible allocations to player , the set of possible valuations of player , and the set of actions available to player . Similarly, we let be the allocation space of all players, be the valuation space, and be the action space. An allocation function maps an action profile to an allocation , where is the set of items allocated to player . A payment function maps an action profile to a non negative payment , where is the payment of player . We assume that the valuation function of a player , where , is monotone and normalized, i.e., and also . We let be the valuation profile. An outcome is a pair of allocation and payment and the revenue is the sum of all payments, i.e. . We assume a quasi-linear utility function, i.e. . We are interested in measuring the social welfare, which is the sum of bidder valuations, i.e., . Given a valuation profile , an optimal allocation is an allocation that maximizes the over all possible allocations. We denote by the social welfare value of an optimal allocation.
Simultaneous item bidding auction
In a simultaneous item bidding auction (simultaneous item auction, in short) each item is simultaneously sold in a separate auction. An action profile is a bid profile , where is an -vector s.t. is the bid of player to item . The allocation of each item is determined by the bids . We use to denote the items won by player and to denote the price paid by the winner of item . As allocation and payment are uniquely defined by the bid profile, we overload notation and write and .
In a simultaneous second price auction (S2PA), each item is allocated to the highest bidder, who pays the second highest bid, i.e., .
In a simultaneous first price auction (S1PA), each item is allocated to the highest bidder, who pays her bid for that item, i.e., .
Full information setting: solution concepts and PoA
In the full information setting, the valuation profile is known to all players. The standard equilibrium concepts in this setting are pure Nash equilibrium (PNE), mixed Nash equilibrium (MNE), correlated Nash equilibrium (CE) and coarse correlated Nash equilibrium (CCE), where . Following are the definitions of the equilibrium concepts. As standard, for a vector , we denote by the vector with the ith component removed. Also, we denote with the space of probability distributions over a finite set .
Definition 2.1 (Pure Nash Equilibriun (PNE)).
A bid profile is a PNE if for any and for any , .
Definition 2.2 (Mixed Nash Equilibriun (MNE)).
A bid profile of randomized bids is a MNE if for any and for any , .
Definition 2.3 (Correlated Nash Equilibriun (CE)).
A bid profile of randomized bids is a CE if for any and for any mapping , .
Definition 2.4 (Coarse Correlated Nash Equilibriun (CCE)).
A bid profile of randomized bids is a CCE if for any and for any , .
For a given instance of valuations , the price of anarchy (PoA) with respect to an equilibrium notion is defined as: . For example, the PoA with respect to PNE is . The PoA for the other equilibrium types are defined in a similar manner. For a family of valuations , .
The following lemma will be useful in subsequent sections of this paper.
Consider an S2PA and a valuation . Let be a welfare-maximizing allocation. Then, for every bid profile the following holds:
Let . Since payments are non-negative, , and each item is sold in a separate second price auction, we get:
Inequality (2.1) holds since is at most the second highest bid on item . Notice that the term in (2.1) considers for each player all the items she wins in bid profile , which are allocated to some other player in the optimal allocation. Instead, we can change the order of summation and consider for each player all the items which are allocated to her in the optimal allocation, but not in bid profile . This accounts for the last equality. ∎
Incomplete information setting: solution concepts and Bayesian PoA
In an incomplete information setting, player valuations are drawn from a commonly known, possibly correlated, joint distribution , and the valuation of each player is a private information which is known only to player . The strategy of player is a function . Let denote the strategy space of player and the strategy space of all players. We denote by the bid vector given a valuation profile .
In some cases, we assume that the joint distribution of the valuations is a product distribution, i.e., . In these cases, each valuation is independently drawn from the commonly known distribution .
The standard equilibrium concepts in the incomplete information setting are the Bayes Nash equilibrium (BNE) and the mixed Bayes Nash equilibrium (MBNE):
Definition 2.6 (Bayes Nash Equilibriun (BNE)).
A strategy profile is a BNE if for any , any and any ,
Definition 2.7 (Mixed Bayes Nash Equilibriun (MBNE)).
A randomized strategy profile is a MBNE if for any , any and any ,
The Bayes Nash price of anarchy is:
The mixed Bayes Nash price of anarchy is defined similarly w.r.t. MBNE.
2.2 Valuation Classes
In what follows we present the valuation functions considered in this paper. As standard, for a valuation , item and set , we denote the marginal value of item , given set , as ; i.e., . In a similar manner, the marginal value of a set , given a set , is . Following are the valuation classes we consider:
- unit-demand (UD):
A valuation function is UD if there exist values such that for every set , .
- submodular (SM):
A valuation function is SM if for every two sets and element , .
- xos (also known as fractionally subadditive):
A valuation function is XOS if there exists a set of additive valuations , such that for every set , .
- subadditive (SA):
A valuation function is SA if for any subsets , .
- monotone (MON):
A valuation function is MON if .
A strict containment hierarchy of the above valuation classes is known: . We now introduce a new class of valuation functions, parameterized by ‘how far’ they are from submodular valuations:
Definition 2.8 (submodular (Sm)).
A valuation function is SM, for , if for every two sets and element , .
For any SM function and any sets :
Let . As is SM, we have for every . Therefore, . The inequality follows from submodularity, and the last equality is due to telescoping sum. ∎
If a valuation function, , is SM, then there exists a set of additive valuations , such that for every set , and there exists at least one such that .
The proof is an extension of the proof in Lehmann et al.  that any submodular function is XOS. Define additive valuations , one for each permutation of the items in . Let , where is the set of items in permutation preceding item . For any permutation and set with item denoting the the th item of in the permutation ,
where the inequality follows from the definition of SM. For any permutation in which the items of are placed first, we have . ∎
By definition, any SM valuation is SM. On the other hand, there is no , such that XOS is SM. In Appendix A we give an example of identical items, XOS function, sets and such that for every .
2.3 Smooth Auctions
An auction is smooth for parameters with respect to a bid space , if for any valuation profile and any bid profile there exists a bid for each player , s.t.:
It is shown in [16; 18] that for every smooth auction, the social welfare of any pure NE is at least . Via extension theorems, this bound extends to CCE in full-information settings and to Bayes NE in settings with incomplete information. These theorems are stated below, and their proofs appear in Appendix B for completeness.
A standard assumption in essentially all previous work on the PoA of simultaneous second price item auction (e.g., , , , , ) is no overbidding, meaning that players do not overbid on items they win. Formally,
Definition 2.14 (No overbidding (NOB)).
Given a valuation profile , a bid profile is said to satisfy NOB if for every player the following holds,
We now extend the last result to -SM valuations.
S2PA with SM valuations is smooth, with respect to bid profiles satisfying NOB.
Let be an SM valuation profile and let be a PNE satisfying NOB. From Lemma 2.10, for every valuation there exists a set , such that for every set , and there exists such that . Let be a welfare maximizing allocation, and let be an additive valuation such that . Consider the following hypothetical deviation for player : if , and otherwise.
Now let us consider the utility of player when deviating. As for every item , each such item contributes non-negative utility to and we can ignore this contribution while lower bounding ’s utility under . Consider item . If , player wins item . Otherwise, does not win item , and the term is non-positive. Since for every set , and since , we get:
The third inequality follows from the choice of and by the payment structure of 2nd price. The forth inequality follows by the fact that all items are allocated in equilibrium. Finally, the last inequality follows from NOB.
3 Revenue Guaranteed Auctions
Following is the definition of revenue guaranteed auctions. We then discuss the implications of this property in both full information and incomplete information settings.
Definition 3.1 (Revenue guaranteed auction).
An auction is revenue guaranteed for some with respect to a bid space , if for any valuation profile and for any bid profile the revenue of the auction is at least .
3.1 Full Information: Revenue Guaranteed Auctions
The following theorem establishes welfare guarantees on every pure bid profile of a revenue guaranteed auction in which the sum of player utilities is non-negative.
If an auction is revenue guaranteed with respect to a bid space , then for any pure bid profile , in which the sum of player utilities is non-negative, the social welfare is at least of the optimal social welfare.
Using quasi-linear utilities and non-negative sum of player utilities, we get:
By the revenue guaranteed property,
Punting it all together, we get
Rearranging, we get: , as required. ∎
If an auction is revenue guaranteed with respect to a bid space , then for any bid profile , in which the sum of the expected utilities of the players is non-negative, the expected social welfare is at least of the optimal social welfare.
The proof is identical to the proof of Theorem 3.2, except adding expectation over to every term, using the fact the the auction is revenue guaranteed for every in the support of , and using linearity of expectation.
Clearly, in every equilibrium (including CCE) the expected utility of every player is non-negative. It therefore follows that the expected welfare in any CCE is at least of the optimal social welfare.
For an auction that is both smooth and revenue guaranteed, we give a better bound on the price of anarchy:
If an auction is smooth with respect to a bid space and revenue guaranteed with respect to a bid space , then the expected social welfare at any CCE of the auction is at least of the optimal social welfare.
3.2 Incomplete Information: Extension Theorem for Revenue Guaranteed Auctions
In a similar manner to the smoothness extension theorem, we can prove an extension theorem for the revenue guarantee property, which gives expected welfare guarantees for settings with incomplete information. However, this extension theorem is stronger, in the sense that it holds with respect to correlated distributions and not only product prior distributions.
If an auction is revenue guaranteed with respect to a bid space , then for every joint distribution , possibly correlated, and every strategy profile , in which the expected sum of player utilities is non-negative, the expected social welfare is at least of the expected optimal social welfare.
We give a proof for pure strategies. The proof for mixed strategies follows by adding in a straightforward way another expectation over the random actions chosen in the strategy profile . As the utility of each player is quasi-linear and the expected sum of player utilities is non-negative, we use linearity of expectation and get,