Bayesian Auctions with Efficient QueriesThis work has been partially supported by NSF CAREER Award (No. 1553385), National NSF of China (No. 61741209) and the Fundamental Research Funds for the Central Universities. Part of this work was done when the first three authors were visiting Shanghai University of Finance and Economics.

Bayesian Auctions with Efficient Queries††thanks: This work has been partially supported by NSF CAREER Award (No. 1553385), National NSF of China (No. 61741209) and the Fundamental Research Funds for the Central Universities. Part of this work was done when the first three authors were visiting Shanghai University of Finance and Economics.

Jing Chen           Bo Li           Yingkai Li           Pinyan Lu
Department of Computer Science, Stony Brook University
Stony Brook, NY 11794, USA
{jingchen, boli2, yingkli}@cs.stonybrook.edu
Institute for Theoretical Computer Science, Shanghai University of Finance and Economics
Shanghai 200433, China
lu.pinyan@mail.shufe.edu.cn
Abstract

Generating good revenue is one of the most important problems in Bayesian auction design, and many (approximately) optimal dominant-strategy incentive compatible (DSIC) Bayesian mechanisms have been constructed for various auction settings. However, most existing studies do not consider the complexity for the seller to carry out the mechanism. It is assumed that the seller knows “each single bit” of the distributions and is able to optimize perfectly based on the entire distributions. Unfortunately this is a strong assumption and may not hold in reality: for example, when the value distributions have exponentially large supports or do not have succinct representations.

In this work we consider, for the first time, the query complexity of Bayesian mechanisms. We only allow the seller to have limited oracle accesses to the players’ value distributions, via quantile queries and value queries. For a large class of auction settings, we prove logarithmic lower-bounds for the query complexity for any DSIC Bayesian mechanism to be of any constant approximation to the optimal revenue. For single-item auctions and multi-item auctions with unit-demand or additive valuation functions, we prove tight upper-bounds via efficient query schemes, without requiring the distributions to be regular or have monotone hazard rate. Thus, in those auction settings the seller needs to access much less than the full distributions in order to achieve approximately optimal revenue.

Keywords: mechanism design, the complexity of Bayesian mechanisms, query complexity, quantile queries, value queries

1 Introduction

An important problem in Bayesian mechanism design is to design auctions that (approximately) maximize the seller’s expected revenue. More precisely, in a Bayesian multi-item auction a seller has heterogenous items to sell to players. Each player has a private value for each item , ; and each is independently drawn from some prior distribution . When the prior distribution is of common knowledge to both the seller and the players, optimal Bayesian incentive-compatible (BIC) mechanisms have been discovered for various auction settings [31, 16, 7, 8], where all players reporting their true values forms a Bayesian Nash equilibrium. When there is no common prior but the seller knows , many (approximately) optimal dominant-strategy incentive-compatible (DSIC) Bayesian mechanisms have been designed [31, 32, 12, 29, 35, 9], where it is each player’s dominant strategy to report his true values.

However, the complexity for the seller to carry out such mechanisms is largely unconsidered in the literature. Most existing Bayesian mechanisms require that the seller has full access to the prior distribution and is able to carry out all required optimizations based on , so as to compute the allocation and the prices. Unfortunately the seller may not be so knowledgeable or powerful in real-world scenarios. If the supports of the distributions are exponentially large (in and ), or if the distributions are continuous and do not have succinct representations, it is hard for the seller to write out “each single bit” of the distributions or precisely carry out arbitrary optimization tasks based on them. Even with a single player and a single item, when the value distribution is irregular, computing the optimal price in time that is much smaller than the size of the support is not an easy task. Thus, a natural and important question to ask is how much the seller should know about the distributions in order to obtain approximately optimal revenue.

In this work we consider, for the first time, the query complexity of Bayesian mechanisms. In particular, the seller can only access the distributions by making oracle queries. Two natural types of queries are allowed, quantile queries and value queries. That is, the seller queries the oracle with specific quantiles (respectively, values), and the oracle returns the corresponding values (respectively, quantiles) in the underlying distributions. These two types of queries happen a lot in market study. Indeed, the seller may wish to know what is the price he should set so that half of the consumers would purchase his product; or if he sets the price to be 200 dollars, how many consumers would buy it. Another important scenario where such queries naturally come up is in databases. Indeed, although the seller may not know the distribution, some powerful institutes, say the Office for National Statistics, may have such information figured out and stored in its database. As in most database applications, it may be neither necessary nor feasible for the seller to download the whole distribution to his local machines. Rather, he would like to access the distribution via queries to the database. Other types of queries are of course possible, and will be considered in future works.

In this work we focus on non-adaptive queries. That is, the seller makes all oracle queries simultaneously, before the auction starts. This is also natural in both database and market study scenarios, and adaptive queries will be considered in future works. Due to lack of space, most proofs are provided in the appendix.

1.1 Main Results

We would like to understand both lower- and upper-bounds for the query complexity of approximately optimal Bayesian auctions. In this work, we mainly consider three widely studied settings: single-item auctions and multi-item auctions with unit-demand or additive valuation functions. Our main results are summarized in Table 1.

Note that we allow arbitrary unbounded distributions that satisfy small-tail assumptions, with formal definitions provided in Section 5.1. Similar assumptions are widely adopted in sampling mechanisms [33, 18], to deal with irregular distributions with unbounded supports. Since distributions with bounded supports automatically satisfy the small-tail assumptions, the lower-bounds listed for the former apply to the latter as well.

Also note that our lower- and upper-bounds on query complexity are tight for bounded distributions. As will become clear in Section 3 and Appendix A, our lower-bounds allow the seller to make both value and quantile queries, and apply to any multi-player multi-item auctions where each player’s valuation function is succinct sub-additive: formal definitions in Appendix A. The lower-bounds also allow randomized queries and randomized mechanisms.

For the upper-bounds, all our query schemes are deterministic and only make one type of queries: value queries for bounded distributions and quantile queries in the other cases; see Sections 4 and 5. We show that our schemes, despite of being very efficient, only loses a small fraction of revenue compared with when the seller has full access to the distributions.

1.2 Discussions

Sample Complexity.

A closely related area to our work is sampling mechanisms [15, 3, 28, 19, 30, 18, 21, 22, 6]. It assumes that the seller does not know but observes independent samples from before the auction begins. The sample complexity measures how many samples the seller needs so as to obtain a good approximation to the optimal Bayesian revenue. The best-known sample complexity results are summarized in Table 2.

Oracle queries can be seen as targeted samples, where the seller actively asks the information he needs rather than passively learns about it from random samples. As such, it is intuitive that queries are more efficient than samples, but it is a priori unclear how efficient queries can be. Our results answer this question quantitatively and show that query complexity can be exponentially smaller than sample complexity: the former is logarithmic in the “size” of the distributions, while the latter is polynomial.

Finally, the design of query mechanisms facilitates the design of sampling mechanisms. If the seller observes enough samples from , then he can mimic quantile queries and apply query mechanisms: see Appendix E for more details.

Parametric Auctions.

Parametric mechanisms [2, 1] assume the seller only knows some specific parameters about the distributions, such as the mean, the median (or a single quantile), and the variance. Note that using quantile or value queries, one can get the exact value of the median and the approximate value of the mean, and then apply parametric mechanisms. However, existing parametric mechanisms only consider single-parameter auctions, where the distributions are regular or have monotone hazard rate. Since our mechanisms make non-adaptive oracle queries, our results imply parametric mechanisms in multi-parameter settings with general distributions, where the “parameters” are the oracle’s answers to our query schemes. Our lower-bounds imply that knowing only the median is not enough to achieve the same approximation ratios as we do. Finally, it remains unknown whether constant approximations can be achieved for multi-parameter auctions or general distributions, knowing only the mean and the variance.

Distributions within Bounded Distance.

Recently, [6] studies auctions where the true prior distribution is unknown to the seller, but he is given a distribution that is close to the true prior, as measured by the Kolmogorov distance. On the one hand, the learnt distributions from our query schemes can be far from the true prior in terms of the Kolmogorov distance, thus their mechanisms do not apply. On the other hand, although a distribution close to the true prior may be learnt via sufficiently many oracle queries, our lower-bounds imply that the query complexity of this approach will not be better than ours.

Using Experts as Oracles.

If the players’ value distributions are known by some experts, then the seller can use the experts as oracles. Indeed, we are able to design proper scoring rules [5, 11] for the seller to elicit truthful answers from the experts for his queries. If the experts are actually players in the auctions, then they have their own stakes about the final allocation and prices, and it would be interesting to see how the seller can still use them as oracles and get truthful answers for his queries, while keeping them truthful about their own values. See [14] for more discussions on this front.

1.3 Other Related Works

The complexity of auctions is an important topic in the literature, and several complexity measures have been considered. Following the taxation principle [24, 23], [26] defines the menu complexity of truthful auctions. For a single additive buyer, [17] shows the optimal Bayesian auction for revenue can have an infinite menu size or a continuum of menu entries, and [4] shows a constant approximation under finite menu complexity. Recently, [20] considers the taxation, communication, query and menu complexities of truthful combinatorial auctions, and shows important connections among them. The queries considered there are totally different from ours: we are concerned with the complexity of accessing the players’ value distributions in Bayesian settings, while [20] is concerned with the complexity of accessing the players’ valuation functions in non-Bayesian settings.

1.4 Future Directions

Many interesting questions about the query complexity of Bayesian auctions are worth exploring. First, as mentioned, we focus on non-adaptive queries in this work. One can imagine more powerful mechanisms using adaptive queries, where the seller’s later queries depend on the oracle’s responses to former ones. It is intriguing to design approximately optimal Bayesian mechanisms with lower query complexity using adaptive queries, or prove that even with such queries, the query complexity cannot be much better than our lower-bounds. Another interesting direction is when the answers of the oracle contain noise. In this case, the distributions learnt by the seller may be within a small distance from the “true distributions” defined by oracle answers without noise. This is related to [6] and it would be interesting to design mechanisms to handle such noise.

2 Preliminaries

2.1 Bayesian Auctions

In a multi-item auction there are items, denoted by , and players, denoted by . Each player has a non-negative value for each item , , which is independently drawn from distribution . Player ’s true valuation is . To simplify the notations, we may write for and for . Letting and , we use to denote the corresponding Bayesian auction instance and the optimal BIC revenue of . When is clear from the context, we write for short.

We will consider several classes of widely studied auctions. A single-item auction has . When , a bidder being unit-demand means his value for a subset of items is , and a bidder being additive means his value for is . When all bidders are unit-demand (respectively, additive), we call such an auction a unit-demand auction (respectively, an additive auction) for short.

2.2 Query Complexity

In this work, we only allow the seller to access the prior distributions via two types of oracle queries: value queries and quantile queries. Given a distribution over reals, in a value query, the seller sends a value and the oracle returns the corresponding quantile . In a quantile query, the seller sends a quantile and the oracle returns the corresponding value such that . With non-adaptive queries, the seller first sends all his queries to the oracle, gets the answers back, and then runs the auction. The query complexity is the number of queries made by the seller.

Note that the answer to a value query is unique. The quantile queries are a bit tricky, as for discrete distributions there may be multiple values corresponding to the same quantile , or there may be none. When there are multiple values, to resolve the ambiguity, let the output of the oracle be the largest one: that is, . When there is no value corresponding to , the oracle returns the largest value whose corresponding quantile is larger than : that is, . So for any quantile query , in general. For any discrete distribution and quantile query , is always in the support of . When , may be .

3 Lower Bounds

In this section, we prove lower bounds for the query complexity of Bayesian mechanisms, and we focus on DSIC mechanisms. As a building block for our general lower bound, we first have the following for single-item single-player auctions, proved in Appendix A.1.

Lemma 1.

For any constant , there exists a constant such that, for any large enough , any DSIC Bayesian mechanism making less than (randomized) non-adaptive value and quantile queries to the oracle, there exists a single-player single-item Bayesian auction instance where the values are bounded in , such that .

We extend this lemma to arbitrary multi-player multi-item Bayesian auctions with succinct sub-additive valuations, as follows, with the corresponding definitions and the proof of the theorem deferred to Appendix A.2.

Theorem 1.

For any constant , there exists a constant such that, for any , any large enough , any succinct sub-additive valuation function profile , and any DSIC Bayesian mechanism making less than non-adaptive value and quantile queries to the oracle, there exists a multi-item Bayesian auction instance with valuation profile , where and the item values are bounded in , such that .

Succinct sub-additive valuations is a very broad class and contains single-item, unit-demand, and additive auctions as special cases. Thus Theorem 1 automatically applies to those cases. We also note that it is shown in [36] that the optimal BIC revenue exceeds the optimal DSIC revenue by a constant factor even for two i.i.d. additive bidders and two identical items. So even with infinite samples, there exist constants such that no -approximation to is possible. However, Theorem 1 is stronger: for every constant , one needs at least the given number of queries to get a -approximation.

4 The Query Complexity for Bounded Distributions

In this section, we consider settings where all distributions are bounded within , and we construct efficient query mechanisms whose query complexity matches our lower-bounds. We show that it is sufficient to use only value queries, and we define in Section 4.1 a universal query scheme , which will be used as a black-box in our mechanisms. The seller uses algorithm to learn a distribution that approximates the prior distribution and is stochastically dominated by . The seller then runs existing DSIC Bayesian mechanisms using , while the players’ values are drawn from . In this sense, all our mechanisms are simple, but they are not given a true Bayesian instance as input.

The multi-player single-item setting is already non-trivial, but still easy, since we have a good understanding of the optimal mechanism, which is Myerson’s auction [31]. In particular, in the analysis it suffices to apply the revenue monotonicity theorem of [18]. The situation for unit-demand auctions and additive auctions is much more subtle. The optimal auction could be very complicated and may involve lotteries and bundling, and revenue monotonicity may not hold [27]. Even (disregarding complexity issues and) assuming we can design an optimal Bayesian mechanism for , it is unclear how much revenue it guarantees when the players’ values come from the true distribution . To overcome this difficulty, we rely on recent developments on simple Bayesian mechanisms with approximately optimal revenue.

The mechanism for unit-demand auctions is sequential post-price [29] and the analysis is relatively easy. For additive auctions, the Bayesian mechanism either runs Myerson’s auction separately for each item or runs the VCG mechanism with a per-player entry fee [35, 9]. However, an easy and direct analysis would lose a factor of in the query complexity. To achieve a tight upper-bound, we need to really open the box and analyze the mechanism differently in several crucial places, exploring its behavior under oracle queries.

To sum up, given our query scheme, our mechanisms are black-box reductions to simple Bayesian mechanisms, thus are simple, natural, and easy to implement in practice, while the analysis is non-black-box, non-trivial and reveals interesting connections between Bayesian mechanisms and query schemes.

4.1 The Value-Query Algorithm

The query algorithm  is defined in Algorithm 1. Here is the distribution to be queried. The algorithm takes two parameters, the value bound and the precision factor , makes value queries to the oracle, and then returns a discrete distribution . It is easy to verify that is stochastically dominated by . Moreover, it is worth mentioning that the mean of approximates that of . Indeed, . Therefore, by directly applying the parametric mechanism in [1] with parameter (for single-parameter auctions where the distributions are regular or MHR), we will get at least a fraction of their revenue.

4.2 Single-Item Auctions and Unit-Demand Auctions

Denoting by Myerson’s mechanism for single-item auctions, Mechanism 2 defines our efficient value Myerson mechanism .

The query complexity of is , since each distribution  needs value queries in . When is sufficiently small, . Also, is DSIC since is so.

In this section and throughout the paper, we often analyze “mismatching” cases where a Bayesian mechanism uses distribution while the actual Bayesian instance is (i.e., the players’ true values are drawn from ). We use to denote the expected revenue in this case. By construction, .

Because the distribution constructed in is stochastically dominated by , letting be the Bayesian instance under , by revenue monotonicity [18] we have . By Lemma 5 of [18], . Thus we have the following simple result.

Theorem 2.

, for any single-item instance with values in , mechanism is DSIC, has query complexity , and .

The construction for unit-demand auctions is similar, except the seller uses as a blackbox the DSIC mechanism of [29], denoted by : see Mechanism 3.

The main difficulty for unit-demand auctions is that we no longer have revenue monotonicity as in single-item auctions. Our analysis then comes in a non-blackbox way and relies on the COPIES setting [12, 29], which provides an upper-bound for the optimal BIC revenue. By properly upper-bounding the optimal revenue in the COPIES setting under , we are able to upper-bound the optimal revenue in unit-demand auctions using the expected revenue of . More precisely, we have the following theorem, proved in Appendix B.1.

Theorem 3.

, for any unit-demand instance with values in , mechanism is DSIC, has query complexity , and .

Letting , we have the query complexity in Table 1.

For additive auctions, the DSIC Bayesian mechanism in [35, 9] chooses between two mechanisms, whichever generates higher expected revenue under the true prior . The first is the “individual Myerson” mechanism, denoted by , which sells each item separately using Myerson’s mechanism. The second is the VCG mechanism with optimal per-player entry fees, denoted by .

In our mechanism , the seller queries about using algorithm  with properly chosen parameters. Given the resulting distribution , the seller either runs or runs as a blackbox, resulting in query mechanisms and . We only define the latter in Mechanism 4, and the former simply replaces with . Note that and . However, the seller cannot compute these two revenue and choose the better one, because he does not know . Thus he randomly chooses between the two, according to probabilities defined in our analysis, to optimize the approximation ratio. We have the following theorem, proved in Appendix B.2.

Theorem 4.

, for any additive instance with values in , mechanism is DSIC, has query complexity , and .

Theorem 4 is harder to show. Indeed, one cannot use revenue monotonicity or the COPIES setting to easily upper-bound the optimal BIC revenue. Our analysis is based on the duality framework of [9] for Bayesian auctions, properly adapted for the query setting. Finally, letting , we have the query complexity in Table 1.

5 The Query Complexity for Unbounded Distributions

Next, we construct efficient query mechanisms for arbitrary distributions whose supports can be unbounded. For a mechanism to approximate the optimal Bayesian revenue using finite non-adaptive queries to such distributions, it is intuitive that some kind of small-tail assumption for the distributions is needed. Indeed, given any mechanism with query complexity , there always exists a distribution that has a sufficiently small probability mass around a sufficiently large value, such that the mechanism cannot find it using queries. If this probability mass is where all the revenue comes (e.g., all the remaining probability mass is around value 0), then the mechanism cannot be a good approximation to . Following the literature [33, 18], the small-tail assumptions are such that the expected revenue generated from the “tail” of the distributions is negligible compared to the optimal revenue; see Section 5.1. Distributions with bounded supports automatically satisfy these assumptions, so are regular distributions in single-item auctions.

Even with small-tail assumptions, it is hard to generate good revenue from unbounded distributions with finite value queries. Instead, we show it is sufficient to use only quantile queries. As before, the seller uses our quantile-query algorithm (defined in Section 5.2) to learn a distribution that approximates , and then reduces to simple mechanisms under . However, even for single-item auctions, it is not so simple to show why the combination of these two parts work. Indeed, under value queries it is easy to “under-price” the item so that the probability of sale is the same as in the optimal mechanism for . Under quantile queries, under-pricing may lose a large amount of revenue because, for given quantiles, there is no guarantee on where the corresponding values are. Instead, the main idea in using quantile queries is to “over-price” the item. This is risky in many auction design scenarios, because it may significantly reduce the probability of sale, and thus lose a lot of revenue. We prove a key technical lemma in Lemma 2 for single-item auctions, where we show that by discretizing the quantile space properly, we can over-price the item while almost preserving the probability of sale as in the optimal mechanism under . In Lemma 4 of Appendix C, we prove another technical lemma showing that proper over-pricing can also be done in additive auctions.

Note that we can get the median of a distribution simply by querying the quantile . Then, for single-parameter auctions with regular distributions, using the parametric mechanism in [1] we get the same revenue as theirs. However, our query mechanisms can handle multi-parameter auctions and irregular distributions.

5.1 Small-Tail Assumptions

A Bayesian auction instance satisfies the Small-Tail Assumption 1 if there exists a function111If computation complexity is a concern, then one can further require that the function is efficiently computable. such that, for any constant and any BIC mechanism , letting , we have

 Ev∼DI∃i,j,qij(vij)≤ϵ1Rev(M(v;I))≤δ1OPT(I). (1)

Here is the quantile of under distribution , is the revenue of under the Bayesian instance when the true valuation profile is , and is the indicator function. For discrete distributions, Equation 1 is imposed on the probability mass over the highest values.

Equation 1 immediately implies the following weaker Small-Tail Assumption 2: there exists a function such that, for any constant , letting , we have

 Ev∼DI∃i,j,qij(vij)≤ϵ1RevOPT(v;I)≤δ1OPT(I). (2)

Here is the revenue generated by the optimal BIC mechanism for when the true valuation profile is . Similar assumptions are widely adopted in sampling mechanisms to deal with irregular distributions with unbounded supports.

5.2 The Quantile-Query Algorithm

We define our quantile-query algorithm in Algorithm 5. As before, is the distribution to be queried. The algorithm takes two parameters, the tail length  and the precision factor , makes quantile queries to the oracle, and then returns a discrete distribution .

5.3 Single-Item Auctions

Mechanism 6 defines our efficient quantile Myerson mechanism .

Theorem 5.

, any single-item instance satisfying Small-Tail Assumption 2, is DSIC, has query complexity , and .

Before proving Theorem 5, we first claim the following key lemma, which we prove in Appendix C.1 via an imaginary Bayesian mechanism that “over-prices”. Recall is the instance under .

Proof of Theorem 5.

First, mechanism is DSIC because is DSIC. Second, the query complexity of is , because there are quantile queries for each player and there are players in total. By definition, . By construction, is stochastically dominated by . Thus by revenue monotonicity . Combining these two equations with Lemma 2, Theorem 5 holds. ∎

Mechanism and Theorem 5 immediately extend to single-parameter downward-closed settings. Finally, when the distributions are regular, we are able to prove an even better query complexity and a matching lower-bound; see Section 6.

5.4 Unit-Demand Auctions

The unit-demand mechanism is similar (see Mechanism 7), and we have the following.

Theorem 6.

, any unit-demand instance satisfying Small-Tail Assumption 2, is DSIC, has query complexity , and .

The proof of Theorem 6 is similar to that of Theorem 3, but Lemma 2 above is used instead of Lemma 5 of [18], and the round-down scheme is replaced by the randomized round-down scheme designed in the proof of Lemma 2. The details have been omitted.

For additive auctions, we cannot use Small-Tail Assumption 2, because it does not imply that the revenue loss on the tail by running is much less than the revenue of the optimal mechanism. To approximate , not only we need Small-Tail Assumption 1, but we also approximate by running the quantile-query algorithm with different parameters. The resulting mechanism is defined in Mechanism 8, and the mechanism simply replaces with . Again, in the final mechanism the seller randomly chooses between the two query mechanisms, according to probabilities defined in the analysis. We have the following theorem, proved in Appendix C.2.

Theorem 7.

, any additive instance satisfying Small-Tail Assumption 1, is DSIC, has query complexity , and .

The main advantage of using quantile queries is to handle unbounded distributions. In addition, we can use the resulting query mechanisms to construct sampling mechanisms; see Appendix E. As shown in Theorem 7, the query complexity of mechanism has an extra factor of  compared with that of (and the lower bound). It would be interesting to see whether our lower-bounds can be improved in this scenario.

5.6 Using Quantile Queries for Bounded Distributions

As a corollary, Theorems 5, 6 and 7 also provide another way to approximate the optimal BIC revenue using only quantile queries when the distributions are bounded. More precisely, we have the following, proved in Appendix C.3.

Corollary 1.

For any , , and prior distribution with each bounded within , there exist DSIC mechanisms that use quantile queries for single-item auctions and unit-demand auctions, and use quantile queries for additive auctions, whose approximation ratios to are respectively , and .

6 Single-Item Auctions with Regular Distributions

In this section, we show that when we only consider regular distributions for single-item auctions, the query complexity can be much lower. In fact, we no longer need the small-tail assumptions even when the supports are unbounded. Here our lower- and upper-bounds are tight upto a logarithmic factor, and require different techniques from previous sections.

For the lower-bound, recall that in Section 3 we allow the distributions to be irregular. To construct the desired distributions, we can first find the un-queried quantile interval and then move the probability mass from its end points to internal points. Because the distributions can be irregular, we have complete control on where to put the probability mass. However, if the distributions have to be regular then this cannot be done. Instead, we start from two different single-peak revenue curves and construct regular distributions from them. We still want to move probability mass from the end points of the un-queried quantile interval to internal points, but such moves must be continuous in order to preserve regularity.

For the upper-bound, we show that regular distributions satisfy the small-tail property with a properly defined tail function. Thus our techniques for distributions with small-tails directly apply here.

6.1 Lower Bound

With regular distributions, by [19] it is sufficient to use a single sample to achieve -approximation in revenue for single-player single-item auctions. Because every distribution is a uniform distribution in the quantile space, a sample for such auctions can be obtained by first choosing a quantile  uniformly at random from and then making a quantile query. Thus, a single query is also sufficient for -approximation in this case. As such, unlike Theorem 1 where we have proved lower bounds for the query complexity for arbitrary constant approximations, for regular distributions we consider lower bounds for -approximations, where is sufficiently small. More precisely, we have the following, proved in Appendix D.1.

Theorem 8.

For any constant , there exists a constant such that, for any , any DSIC Bayesian mechanism making less than non-adaptive value and quantile queries to the oracle, there exists a multi-player single-item Bayesian auction instance where and is regular, such that .

6.2 Upper Bound

Our mechanism (i.e., “Efficient quantile Myerson mechanism for Regular distributions”) first constructs the distribution that approximates using the quantile-query algorithm with parameters and ; and then runs Myerson’s mechanism on . Formally, we have the following theorem, proved in Appendix D.2.

Theorem 9.

, and for any single-item instance where is regular, mechanism is DSIC, has query complexity , and .

Following [15, 28, 18], the sample complexity for single-item auction with regular distributions is bounded between and . However, each sample is a valuation profile of the players, and thus contains values. When is small, the query complexity in this setting is . Thus the query complexity is still much lower than the sample complexity.

Appendix A Missing Materials for Section 3

a.1 Proof of Lemma 1

Lemma 1 (restated). For any constant , there exists a constant such that, for any large enough , any DSIC Bayesian mechanism making less than (randomized) non-adaptive value and quantile queries to the oracle, there exists a single-player single-item Bayesian auction instance where the values are bounded in , such that .

Proof.

Given , for any constant , let . When is large enough, we have

 k=⌊logcH4(4c+2)logc(8c)⌋≥1.

We divide the quantile interval and the value interval into sub-intervals, with their right-ends defined as follows: , for each , , and for each . It is easy to see

 \vspace−5ptq0=1(8c)k(4c+2)≥H−14 % and u0=H(4c)k(4c+2)≥H⋅q0≥H34.

From now on, we will ignore the intervals below and .

Let and . We have . Accordingly, for any DSIC Bayesian mechanism that makes less than non-adaptive value and quantile queries, there exist a value interval and a quantile interval such that, with probability at least , no value in is queried and no quantile in is queried either. Indeed, if this is not the case, then for any pair and , with probability greater than , either is queried or is queried. Since there are value intervals and quantile intervals, the expected total number of queries made by  is at least , a contradiction.

We now construct different single-player single-item Bayesian instances , where the distributions outside the value range and the quantile range are all the same. Given such ’s, with probability at least mechanism  cannot distinguish the ’s from each other. We then show that when this happens, mechanism cannot be a -approximation for all instances .

More precisely, the distribution for each is defined in Table 3 and illustrated in Figure 1. Here is a small constant whose value will be determined in the analysis.

It is easy to see that for each value query from , the returned quantile is the same for all ’s. Moreover, when a quantile query is from , the oracle’s answer is again the same for all ’s, as illustrated in Table 4. Accordingly, with probability at least , mechanism cannot distinguish ’s from each other, which means it cannot distinguish ’s from each other.

We now analyze the optimal BIC revenue for those instances. For any , Myerson’s mechanism is optimal: it sets a (randomized) threshold for the unique player, if the player bids at least the threshold then he gets the item and pays the threshold payment, otherwise the item is unsold. Letting , it is not hard to verify that for each .

Next, we analyze the revenue of . Since is DSIC, the allocation rule must be monotone in the player’s bid, and he will pay the threshold payment set by , denoted by . Here may also be randomized. Note that for all instances, setting is strictly worse than setting , and setting is strictly worse than setting . Also, for any instance and any , setting is strictly worse than setting . Thus, when mechanism cannot distinguish the ’s, it must use the same for all ’s, and the best it can do is to set with some probability for each . Because , there exists such that . Thus we have

 Rev(M(Iz∗)) ≤ 14c⋅(4c)z∗⋅us⋅(qt+1−δ)+(1−14c)(4c)z∗−1⋅us⋅(qt+1−δ) < 12c⋅(4c)z∗⋅us⋅(qt+1−δ)=12cOPT(Iz∗),

where the first inequality is because for any threshold other than , the resulting expected revenue is no larger than that with the threshold being