Improved Approximations for Free-Order Prophets and Second-Price Auctions

Improved Approximations for Free-Order Prophets and Second-Price Auctions

Hedyeh Beyhaghi Cornell University. hedyeh@cs.cornell.edu.    Negin Golrezaei Massachusetts Institute of Technology. golrezae@mit.edu    Renato Paes Leme Google Research. renatoppl@google.com    Martin Pál Google. mpal@google.com    Balasubramanian Sivan Google Research. balusivan@google.com
Abstract

We study the fundamental problem of selling a single indivisible item to one of buyers with independent and potentially nonidentical value distributions. We focus on two simple and widely used selling mechanisms: the second price auction with eager personalized reserve prices and the sequential posted price mechanism. Using a new approach, we improve the best-known performance guarantees for these mechanisms. We show that for every value of the number of buyers , the eager second price (ESP) auction and sequential posted price mechanisms respectively earn at least and fractions of the optimal revenue. We also provide improved performance guarantees for these mechanisms when the number of buyers is small, which is the more relevant regime for many applications of interest. This in particular implies an improved bound of for free-order prophet inequalities.

Motivated by our improved revenue bounds, we further study the problem of optimizing reserve prices in the ESP auctions when the sorted order of personalized reserve prices among bidders is exogenous. We show that this problem can be solved polynomially. In addition, by analyzing a real auction dataset from Google’s advertising exchange, we demonstrate the effectiveness of order-based pricing.

Introduction

One of the most fundamental problems in auction/economic theory is selling a single indivisible item to one of buyers with independent and possibly heterogeneous value distributions. Optimizing the performance of this auction has remarkable theoretical and practical impact. In online advertising markets alone, such problems need to be solved every fraction of a second to decide what ads to show to Internet users (WSJ 2017). An overwhelming majority of these decisions are made via running eager second price (ESP) auctions (Paes Leme et al. 2016, Dhangwatnotai et al. 2015, Chawla et al. 2014).111In “eager” second price auctions, first the buyers that have not cleared their reserve prices are filtered. Subsequently, the winner is determined among non-filtered buyers. In contrast, in “lazy” second price auctions, first the buyer with the highest submitted bid is chosen as the potential winner. Then, this buyer is announced as the winner if his bid exceeds his reserve price. Motivated by this, we seek to answer the following key question: What fraction of optimal revenue can the ESP auction obtain?

The first non-trivial performance guarantee for this auction was given by Chawla et al. (2010a), who prove a approximation to the optimal revenue. They do so by proving the same approximation for the sequential posted price mechanism (SPM) and making the important connection that the same approximation factor carries over to the ESP auction that uses the posted prices as personalized reserve prices. The sequential posted price problem is another fundamental selling mechanism studied in the mechanism design literature and is interesting in its own right. The mechanism computes prices, one per buyer (without soliciting any bids), and approaches each buyer in descending order of prices to make a take-it-or-leave-it offer while the supply lasts. As stated in Chawla et al. (2010a), there are several advantages to SPMs compared to a traditional auction, explaining their ubiquitous presence (Holahan 2008): (i) SPMs have trivial game dynamics: responding truthfully is a dominant strategy, i.e., a buyer purchases the item when his value exceeds the posted price. (ii) These mechanisms satisfy strong notions of collusion resistance, like group strategy-proofness: a buyer interested in helping another buyer has to decline an item with a price below his value, thereby hurting his own utility. Considering these properties, characterizing the revenue bounds of these robust mechanisms is independently worthy of study.

Since the result of Chawla et al. (2010a), the approximation (or, more generally, the approximation when the number of buyers is ) for the SPM was found to be obtainable with various techniques: pipage rounding (Calinescu et al. 2011), correlation gap (Agrawal et al. (2012) and Yan (2011)), prophet secretary (Esfandiari et al. 2017) and Bernoulli selection lemma (Correa et al. 2017a). The first improvement over this bound was due to the work of Azar et al. (2017), who show how to improve to for . For smaller values of , has remained the best known bound until our current paper.

We now summarize our main contributions.

Improved Revenue Bounds. We provide an improved universal bound over (i.e., valid for any number of buyers ) for both SPMs and ESP auctions. In particular, for SPMs and ESP auctions, we prove improved approximations of and to the optimal revenue, respectively. While the universal bounds of and are pessimistic (they are obtained when goes to infinity), they are useful in settings where either there is significant uncertainty in the number of buyers, or the number of buyers is rather large.

The approximation factors are noticeably larger for smaller values of . Table 1 presents our improved bounds for along with the value of for comparison. We stress that, thanks to our novel proof techniques, our bounds are valid even if the distributions of buyers’ value are irregular. The table shows that for SPMs and ESP auctions, our approach outperforms the bound of by up to and , respectively. We note that for the motivating application of display advertising markets, the number of buyers in each auction is typically small, and therefore, improvement for small is particularly relevant.

We want to highlight that our ESP auction results are strictly better than our SPM results, i.e., we do not just invoke the result of Chawla et al. (2010a) to reduce ESP auctions to SPM, but also use some properties specific to ESP auctions to get these better bounds. Prior to this work, the SPM approximation ratio always matched the ESP approximation ratio (including in the work of Azar et al. (2017)), as they all invoke Chawla et al. (2010a) for ESP auction to SPM reduction.

1 2 3 4 5 6 7 8 8 10
SPM
ESP
Table 1: Revenue bounds of our mechanisms for different number of buyers. All numbers are rounded up.

Novel Proof Techniques. To establish our revenue bound for both SPMs and ESP auctions, we consider the best of the two pricing rules: Uniform and Myersonian. In the uniform pricing rule, the mechanisms post the same price for every buyer. In the Myersonian pricing rule, the mechanisms aim to imitate the optimal mechanism proposed by Myerson (1981). We apply the simple concept of the taxation principle, which enables one to view any deterministic incentive-compatible mechanism as a posted price mechanism for each buyer, with the price being dependent on the bids of other buyers.222In our proof, we do not invoke virtual values at all. In particular, we use the fact that any deterministic incentive-compatible mechanism, including the optimal Myerson’s mechanism,333Myerson’s mechanism, even for irregular distributions, where it is usually thought to be randomized, can be implemented as a deterministic mechanism by breaking ties consistently (e.g. lexicographically) between buyers. See Chawla and Sivan (2014). can be seen as posting a price for each buyer, where the price is a function of other buyers’ bids.

Taking inspiration from this fact, the Myersonian pricing rule computes a price for each buyer by sampling the bids of other buyers from their respective distributions and then computes the posted price that Myerson’s mechanism would have come up with given these bids from other buyers. We perform fresh and independent sampling while computing the prices for different buyers. Thus, while the prices from Myerson’s mechanism are highly correlated, the prices that we compute are independent of each other, easing our analysis. Having considered the best of uniform and Myersonian pricing rules, we then show our improved revenue bound by constructing novel factor revealing linear programs (LPs).

Optimal Order-Based Eager Reserve Prices. While the discussion so far focused on obtaining improved revenue bounds via simple reserve pricing schemes, the question of computing the optimal vector of reserve prices for independent value distributions is still open.444For the correlated value distribution, Paes Leme et al. (2016) show that the problem of optimizing reserve prices in ESP auctions is NP-complete. In this paper, we make some progress towards answering this important question. We develop a dynamic programming method to show that if we are given a sorted ordering over the personalized reserve prices among the buyers, the problem of computing the optimal vector of personalized reserve prices obeying that order can be done in polynomial time. Besides being a relevant subproblem to the problem of computing optimal personalized reserve prices555For example, for small values of , enumerating over all the orderings of reserve prices to find the truly optimal vector of reserve prices is possible., order-based optimal reserves can be interesting in their own right. For example, in display advertising markets, web publishers may be willing to treat certain buyers more preferentially than other buyers due to long-term relationships. In ESP auctions, this would amount to giving smaller reserve prices for the more preferred buyers. Another motivation for this problem is to develop a new method to improve prior pricing heuristics such as Myerson’s monopoly reserve prices or those proposed by Ronen (2001), Paes Leme et al. (2016), and Roughgarden and Wang (2016). In particular, ESP auctions can obtain the order over reserve prices from one of these several heuristics and optimize its revenue subject to this order. This is the first work that compares the performance of various prominent reserve pricing heuristics in a very data rich real world environment, besides proposing a practical technique to improve them.

Empirical and Numerical Studies. We evaluate the performance of order-based ESP auctions on synthetic and fully-anonymized real auction datasets from Google’s advertising exchange. We present results for synthetic data so that our results are replicable, and the real auction dataset enables us to assess our order-based ESP auctions in a realistic environment that does not necessarily satisfy our independency assumption. Specifically, we show that although submitted bids in our dataset are not independent across buyers, the order-based ESP auctions perform really well.

We investigate the performance of the ESP auctions that obey the order of prices in the four main pricing heuristics from the literature: Myerson’s monopoly reserve prices and the respective pricing heuristics proposed by Ronen (2001), Paes Leme et al. (2016), and Roughgarden and Wang (2016). We note that to the best of our knowledge, the present paper is the first work that compares these heuristics in a realistic environment666We observe that the pricing heuristic of Paes Leme et al. (2016) outperforms the other aforementioned heuristics. and provides a practical technique to improve them. We further study the performance of the order-based ESP auctions that follow the order of the average and coefficient of variation (CV) of the submitted bids. These orderings are motivated by Golrezaei et al. (2017), who show that to extract more revenue from buyers, the mechanism should favor buyers with smaller CVs and average bids. We show that by obeying the order suggested by the prior pricing heuristics and optimizing reserve prices subject to the order, the revenue of ESP auctions increases by up to . In addition, we show that ordering based on the average and CV of submitted bids performs well.

The rest of the paper is organized as follows. In Section 1, we review the related literature. In Section 2, we present our model, and we formally define SPMs and ESP auctions. Sections 3 and 4 present our universal revenue bounds for SPMs and ESP auctions. In Section 5, we provide our revenue bounds that take into account the number of buyers . Sections 6 and 7 are respectively dedicated to the order-based reserve price optimization problem and our empirical studies. For the sake of brevity, we only include proofs of selected results in the main text; the detailed proofs of the rest of the statements are deferred to the appendix.

1 Related Work

Our work relates and contributes to the literature on optimal auction design. The seminal work of Myerson (1981) shows that when buyers’ value distributions are regular and homogeneous, the optimal mechanism can be implemented via a second price auction with reserve. However, the structure of the optimal mechanism can be complex when the value distributions are heterogeneous and irregular (Myerson 1981). Because of this, several papers have studied simpler auction formats, such as second price auctions with (personalized) reserve prices (Hartline and Roughgarden (2009), Paes Leme et al. (2016), Roughgarden and Wang (2016), and Allouah and Besbes (2018)), boosted second price auctions (Golrezaei et al. 2017), the BIN-TAC mechanism (Celis et al. 2014), and first price auctions (Bhalgat et al. 2012), to name a few.

Hartline and Roughgarden (2009) study the question of approximating the optimal revenue via a Vickrey auction with personalized reserve prices, and show that for regular distributions, the second price auction with so-called monopoly reserve prices yields a -approximation for regular distributions, and that for irregular distributions, no constant factor approximation is possible with the monopoly reserves. Paes Leme et al. (2016) consider second price auctions and study the question of computing the optimal personalized reserve prices in a correlated distribution setting, and they show that the problem is NP-complete.  Roughgarden and Wang (2016) show that this problem is APX-hard for correlated distributions and give a -approximation. Finally, Golrezaei et al. (2017)—using empirical and theoretical analyses—show that when buyers are heterogeneous, their proposed mechanism, called boosted second price auction, gets a high fraction (more than ) of the optimal revenue and outperforms the second price auction with reserve. In the current work, we provide an improved approximation factor for second price auctions and show that this auction format, despite its simple structure, performs well even when the distributions are heterogeneous and irregular.

Our work is also related to the literature on prophet inequalities (Krengel and Sucheston (1977, 1978) and Kennedy (1987)). Specifically, studying posted price mechanisms has been intimately connected to the work on prophet inequalities. In the classic prophet inequality setting, independent (but not necessarily identical) random variables arrive in an adversarial sequence, and after each random variable arrives, the gambler faces two choices: accept the random variable and stop, or reject and continue. The objective is to maximize the expected value of the random variable selected by the gambler. Performance is measured based on the ratio of the gambler’s choice to the expected value of the maximum of random variables (the objective that a prophet with complete foresight can obtain).

Adversarial Prophets.

Hill and Kertz (1981) show that when variables are independent but not identical and their orders are chosen adversarially, the gambler can obtain at least of the expected value obtained by a prophet; see also Samuel-Cahn (1984).777This constant cannot be improved, even if the prophet were allowed to use adaptive strategies and even if we make large market assumptions, like each distribution occurring at least times for any arbitrarily large  (Abolhassani et al. 2017). This -approximation was later used by Chawla et al. (2010a) to give a -approximation for the posted prices mechanism when the buyers arrive in an adversarial order. When the random variables are i.i.d. and their orders are adversarial, Hill and Kertz (1982) show that the gambler can obtain at least of the prophet’s value and also show examples that prove that one cannot obtain a factor beyond Kertz (1986) later conjecture that is the best possible approximation. The first formal proof that one can go beyond was given by Abolhassani et al. (2017), who give for all beyond a certain constant . Simultaneously and independently, Correa et al. (2017a) show a approximation for this problem, thereby completely closing the gap. We highlight that the 0.745 approximation factor proved by Correa et al. (2017a) is not applicable to our setting, as in our work, the values of buyers are not i.i.d.

Free-Order Prophets.

In another variation of prophet inequalities, the gambler can pick the random variables in her desired order. This variation is known as free-order prophets. We first note, as recently shown by Correa et al. (2017b), that any approximation for SPMs implies the same approximation for free-order prophets. The bound on the best possible approximation by Kertz (1986) also holds for this setting.  Chawla et al. (2010a) give a approximation for the SPM problem. Recall that under SPMs, the seller approaches the buyers in decreasing order of their prices. This bound of was recently surpassed by Azar et al. (2017) to , and their bound also extends to random order prophets, in which the random variables arrive in a uniformly random order. Our bound of for SPMs, which implies the same bound for free-order prophets by the result of Correa et al. (2017b), is the best known bound for free-order prophets so far. More precisely, just like for SPMs and ESP auctions, ours is the first paper to go beyond the bound for every in the free-order prophets setting.

Posted Prices, Prophet Inequalities and Generalizations.

The connection between prophet inequalities and mechanism design was initiated by Hajiaghayi et al. (2007), who interpret the prophet inequality algorithms as truthful mechanisms for online auctions. Chawla et al. (2008) give a -approximation to the single-agent -items unit-demand pricing problem, via upper bounding the revenue of the multi-parameter setting by that of the single-item -buyers single-parameter problem. Chawla et al. (2010a, b) further this connection, and develop constant fraction approximations for several multi-parameter unit-demand settings, by establishing constant factor approximations to the corresponding single-parameter posted price settings through connections to prophet inequalities. Yan (2011) makes a connection between the revenue of SPMs and correlation gap for submodular functions (Agrawal et al. 2012)Chakraborty et al. (2010) develop a PTAS for computing the optimal adaptive and non-adaptive SPMs in a -item single-parameter setting. Kleinberg and Weinberg (2012) generalize the prophet inequalities result to the matroidal environment, and use this to give improved approximations for the multi-parameter unit-demand mechanism design problem. For other generalizations of posted prices and prophet inequalities to combinatorial settings, see Dütting and Kleinberg (2015), Feldman et al. (2015), Rubinstein and Singla (2017), Ehsani et al. (2018), and Dütting et al. (2017).

2 Model

There is a single indivisible item to be sold to one of buyers indexed by , where . The value of buyer for the item, denoted by , is drawn independently from distribution with density function . For each , distribution is public information, while value is buyer ’s private information.

In the following, we formally define the sequential posted price mechanisms, denoted by , and we then proceed to define the eager second price auctions, denoted by . Here, is a vector of posted prices in and a vector of reserve prices in .

Before presenting these mechanisms, we note that in this work, we focus on the eager second price auctions. The lazy second price auctions are incomparable to the eager second price auctions for general correlated distributions but are within a factor of of each other (Paes Leme et al. 2016). Further, Paes Leme et al. (2016) show that the optimal revenue from the eager auction dominates the optimal revenue from the lazy auction when the value distributions are independent. Motivated by this, we study ESP auctions in the current work. We note that it is known from an example in Ronen (2001) that it is impossible to obtain a better than -approximation for the lazy auctions with respect to the optimal revenue.

Sequential Posted Price Mechanisms

  • Buyers are sorted in decreasing order of their posted prices , . Without loss of generality, we assume that .

  • The mechanism approaches buyers in decreasing order of their posted prices. Precisely, for , the mechanism offers price to buyer . If the buyer accepts the offer, i.e., , the item will be allocated to buyer at a price of , and the mechanism will stop. Otherwise, the mechanism proceeds to the next buyer with the highest posted price, i.e., buyer .

Eager Second Price Auctions

  • Each buyer submits his bid/value .888Since ESP auctions are truthful, buyers’ bids are equal to their values.

  • All the buyers with value are first eliminated. Let be the set of all the buyers who clear their reserve prices.

  • The item is then allocated to buyer , who has the highest value among all buyers in set , and he pays . For other buyers, their payment is zero.

The following lemma is an important observation about the revenue of and mechanisms made by Chawla et al. (2010a). Note that the revenue of a mechanism is the total (expected) payment that it charges the buyers, where the expectation is taken with respect to the randomness in buyers’ value.

Lemma 1 (ESP Dominates SPM).

For any vector of prices and any value distributions, the revenue of is at least the revenue of .

Proof.

Proof of Lemma 1: For any realization of buyers’ value, if has a winner, then must also have a winner. If the winner in the two auctions is the same, then the revenue of must be at least as large as that of . If the winner is different, then the winner of is paying at least the bid of the winner in , and the latter is at least the posted price in .

Throughout the proof, with a slight abuse of notation, we respectively denote the expected revenue of and by and , where the expectation is taken with respect to the randomness in the buyers’ value.

2.1 Optimal Revenue Benchmark

Our benchmark, which we refer to as , is an incentive-compatible (truthful) and individually rational revenue-optimal auction in the independent value setting. This mechanism was designed by Myerson (1981). For most of our results, the specific form of the optimal auction is irrelevant. We will use only the fact that it is a deterministic truthful auction, and hence, the taxation principle (Hammond 1979) gives a simple equivalent form of expressing such a mechanism. As mentioned earlier, even when the distributions are not regular, Myerson’s mechanism can be implemented as a deterministic mechanism; see Chawla and Sivan (2014).

The following lemma describes the taxation principle in any deterministic truthful mechanism. We do not prove it here: its proof can be derived from any standard auction theory text.

Lemma 2 (Taxation Principle).

Given a single-item deterministic truthful mechanism , there are threshold functions for each buyer , depending only on the bids of the other buyers , such that the allocation and payment of mechanism can be described as follows:

  • if , then the item is allocated to buyer , and he is charged the threshold .

  • if , then either the item is allocated to buyer , and he is charged his value , or the item is not allocated to him, and he is not charged.

  • if , then the item is not allocated to buyer , and he is not charged.

We note that the threshold functions ’s can be computed for any deterministic incentive-compatible mechanism. In such a mechanism, each buyer has a critical value —which only depends on other buyers’ values—such that he gets allocated and pays if his value . When , he does not get allocated and pays . When , the mechanism can either allocate the item to buyer and charge him , or not allocate the item to him and charge . The threshold function is defined as . We give two examples to illustrate how these thresholds are computed.

Example 3.

Second price auction with no reserve: The critical value for buyer is .

Example 4.

Optimal auction with uniform distributions: Suppose that there are two buyers with values and drawn independently from the uniform distributions in [0,1] and [0,2], respectively. In the optimal auction, the item is allocated to the buyer with the highest non-negative virtual value.999The virtual value of buyer with value is given by . The virtual values of buyers and are respectively and . Thus, buyer is allocated when , and buyer is allocated when . Therefore, the threshold functions are given by and .

Thresholds, , are constructed in such a way that at most one buyer is strictly above his threshold. However, it is possible that more than one buyer is at the threshold, in which case a tie-breaking rule needs to be determined to exactly describe the allocation rule of the mechanism.

There are two important cases in which the issues of tie breaking can be ignored. The first case is when the value distributions are independent and continuous, with no atoms: in this case, the probability that is zero. The second case is when the distributions have a finite discrete support: here, the thresholds for any deterministic auction can be constructed in such a way that there will be at most one buyer who meets/exceeds the threshold, i.e., , in which case that buyer gets allocated and pays his threshold. To avoid issues with ties, throughout the paper, we assume that distributions are discrete, and in such a case, thresholds are set in a manner such that there is at most one buyer who meets/exceeds the threshold in the auction. The continuous case can be handled by discretizing the distributions and taking the limit of the discretization.

2.2 Definitions and Notations

Thresholds:

In the rest of the paper, refers to the threshold of buyer corresponding to the optimal auction (see Lemma 2). Whenever it is clear from context, we abbreviate or the function by .

Re-sampled thresholds:

We will often refer to the thresholds computed from independently re-sampled values: namely, for each buyer , sample for all , and denote by the re-sampled threshold where . Observe that we do not reuse samples: for each buyer , we freshly re-sample the values of all other buyers. We abbreviate by whenever it is clear from the context. Note that although for each , the distribution of is the same as the distribution of , ’s are independent across ’s, while ’s are correlated.

Myersonian posted prices:

This refers to the tuple of posted prices, one per buyer, namely, the re-sampled threshold for buyer .

Uniform posted price:

This refers to the highest revenue yielding uniform posted price, namely, .

Revenue lower bound probabilities:

Let , , be the probability that buyer wins and pays at least in the optimal auction, where the expectation is taken w.r.t. and . Further, let be the probability that the winner pays at least . It follows immediately that is a weakly decreasing function whose integral defines the optimal auction’s revenue:

(1)

3 Universal Revenue Bound for Sequential Posted Price Mechanisms

In this section, we provide a universal approximation factor for the SPMs that hold for any value of . We note that our approximation factor is valid even if value distributions are not regular.

Theorem 5 (Revenue Bound of SPM).

There exists a vector of prices such that .

Relation to free-order prophets: We note that Theorem 5 also implies that an improved bound of for the free-order prophet problem Correa et al. (2017b). In this problem, there are independent random variables with known distributions . A decision maker knows the distributions but not the realizations and he chooses an order to inspect the variables. Upon inspecting a variable, he learns its realized value and needs to choose between stopping and obtaining its value as a reward or abandoning that variable forever and continuing to inspect other variables. An algorithm for this problem is a policy that determines an order to inspect the variables and for each variable inspected decides whether to stop and obtain that reward or continue inspecting. If is the index of the random variable chosen by the decision maker, the performance of the algorithm is . The goal is to compare the performance of the decision maker with a prophet that knows all the values in advance and therefore can obtain . Correa et al. (2017b) recently showed that this problem is equivalent to the sequential posted prices problem. In particular, each variable in the free-order prophet problem can be mapped to a buyer with value such that the virtual value associated with has distribution . Then, they showed that one can solve the sequential posted prices problem for ’s and then map the policy back to a free-order prophet policy. Using this mapping, any algorithm for sequential posted prices can be transformed (in a black-box manner) to an algorithm for the free-order prophets problem with the same approximation ratio.

The proof of Theorem 5 is based on a novel technique that uses the the best of two posted price mechanisms (and hence, the ultimate mechanism is also a posted price mechanism). The first one, which we call the Myersonian posted price mechanism, posts prices that mimic the threshold functions in the optimal auction, and the second one, called the uniform posted price mechanism, posts the same price for every buyer. We show that choosing the best of these two mechanisms gives us the desired bound.

Myersonian Posted Price Mechanism: Approach the buyers in decreasing order of their Myersonian posted prices, i.e., the re-sampled thresholds (defined in Section 2.2), and allocate to the first buyer whose value exceeds his threshold . Recall that ’s are independent random variables across ’s, and each is distributed identically to . Let denote the expected revenue of this mechanism, where the expectation is taken w.r.t. randomness in both the re-sampled posted prices and the buyers’ values.

Uniform Posted Price Mechanism: Approach buyers in an arbitrary order, and allocate to the first buyer whose value exceeds the price . Let be the expected revenue of this mechanism, where the expectation is taken with respect to the buyers’ values.

In Theorem 5, we show that is at least a fraction of the optimal revenue, denoted by . This implies that there exist thresholds such that .

We focus on the best of these two mechanisms because they complement each other, i.e., in the worst instances of Myersonian pricing, uniform pricing has a good performance.101010 To illustrate that complements , consider the setting where there are buyers whose values are independently drawn from the uniform distribution in for a tiny . Then, the optimal auction is simply the second price auction with a uniform reserve of , and the uniform pricing scheme that just posts a price of gets very close to this optimal auction. However, in the Myersonian posted price mechanism, each buyer is offered a random threshold that is the maximum of uniform variables. Thus, each buyer is above such a threshold with probability . Since they are all independent, with probability , no buyer is above the threshold. Thus, just makes a approximation for this particular choice of prices. We are now ready to prove Theorem 5 by constructing a factor revealing LP: the proof immediately follows from Claim 6.

Claim 6 (Detailed Statement of Theorem 5).

The maximum of Myersonian Posted Price Mechanism’s and Uniform Posted Price Mechanism’s revenue is at least , where LP-SPM is defined as:

(LP-SPM)

where .

Proof.

Proof of Claim 6: We first show that . Subsequently we prove that

Without loss of generality, assume that the revenue of the posted price mechanism that chooses the best of uniform and Myersonian prices is normalized to : i.e., (this can be done by scaling all values). Note that the objective function of LP-SPM is the optimal revenue (see Equation (1)). Thus, proving that the constraints of Problem LP-SPM follow from upper/lower bounds on along with will imply that . We now show that the constraints follow from upper and lower bounds on , .

Upper Bounds on (First Set of Constraints): Consider the posted price mechanism that posts a price of for every buyer. The revenue of this mechanism is equal to , which is at least . Therefore, for every ; that is,

(2)

where the second inequality follows from . Equation (2) leads to

(3)

In the inequality, we also used the fact that is a probability and is at most . Note that Equation (3) is the first set of constraints in LP-SPM.

Lower Bounds on (Second Constraint): Let be the probability that the Myersonian posted price mechanism sells with price at least , which is the probability that there is at least one buyer with , where is the posted-price for buyer in . Then, we have:

(4)

where the inequality follows from . Next, we present a lower bound on by connecting to probability that describes the optimal auction.

By construction of , the probability that , is the same probability as buyer wins in the optimal auction and is charged at least , which is . Since the buyers’ values are all independent, and the prices are all computed independently with fresh re-sampling for each buyer , we have:

(5)

where . The first inequality follows from and the last inequality follows from being decreasing in and that (see Equation (3)). Invoking Equations (4) and (5) leads to the second constraint in LP-SPM.

Next, we compute the objective value LP-SPM. It is not difficult to guess the optimal solution of Problem LP-SPM. Since is increasing in , an optimal solution must satisfy that whenever , it must be that for every . Hence, the optimal solution of Problem LP-SPM has the following form:

where is the unique threshold for which . This leads to

By solving the above equation numerically, we get , and the optimal solution of Problem LP-SPM is given by . To see why note that

Hence, the SPM that chooses the best of uniform and Myersonian pricing rule yields at least of , which is the bound in Theorem 5.

4 Universal Revenue Bound for Eager Second Price Auctions

The bound presented in Theorem 5 is also a valid bound for the ESP auctions; see Lemma 1. However, the ESP auctions can potentially earn higher revenue than SPM by leveraging the second highest bid. We now show how to exploit the second highest bid to obtain an improved bound for the ESP auction.

The following is the main result of this section.

Theorem 7 (Revenue Bound of ESP).

There exist reserve prices such that .

The proof, in spirit, is similar to that of Theorem 5. We consider an ESP auction that chooses the best of Myersonian and Uniform reserve prices. We construct a factor revealing LP to bound its performance.

Proof of Theorem 7

Similar to the proof of Theorem 5, we define the following two ESP auctions.

Myersonian ESP Auction: We run the auction with personalized reserve prices for each buyer, with buyer facing the re-sampled threshold as his reserve price (see the definition in Section 2.2). Let denote the expected revenue of this auction.

Uniform ESP Auction: We run the auction with a uniform reserve price of , where is the second highest bid (which is also equal to the second highest value in a truthful auction). We denote the revenue of this auction by .

In the following claim, we show that the best of two aforementioned ESP auctions has the approximation factor given in Theorem 7, concluding its proof.

Claim 8 (Detailed Statement of Theorem 7).

For every positive integer , the maximum of Myersonian ESP Auction’s and Uniform ESP Auction’s revenue is at least , where , for , and

(LP-ESP)

In particular, setting , the approximation factor is .

The factor revealing LP, given in Claim 8, does not have a closed form solution. In the following table, we present the value of for different values of . Since is a valid approximation factor for every , it follows that is a valid approximation factor. As it becomes clear in the proof, parameter determines the precision of our discretization. Larger values of imply more granular discretization.

50 100 200 400 800 1600 3200
0.6606 0.6613 0.6617 0.6618 0.6619 0.6620 0.6620
Table 2: for different values of .

Before presenting the proof of Claim 8, we briefly discuss  LP-ESP. The objective of the problem is the optimal revenue. Recall that by Equation (1), the optimal revenue is where is the probability that the optimal auction sells at a price at least . We define such that , . We then have

(6)

Equation (6) verifies that the objective function of Problem LP-ESP is . Next, we explain the constraints of the LP. The first set of constraints follows from lower and upper bounds on . The second constraint follows from bounding .

We now present the proof of Claim 8.

Proof.

Proof of Claim 8: The goal is to show that . Similar to the proof of Theorem 5, without loss of generality, let . By the definition of ’s, the objective function of Problem LP-ESP is equal to . This implies that to show Claim 8, it suffices to prove that the constraints of Problem LP-ESP follow from upper/lower bounds on and .

First Set of Constraints: To derive the first set of constraints, we use lower and upper bounds on . Let , (note that is different from the , defined earlier). In addition, with a slight abuse of notation, let be the revenue of the ESP auction that posts a uniform price of for all buyers. By definition of the uniform ESP auction, we have . We now bound for any .

We start by bounding . Define as the probability that ESP auction with uniform price sells with price at least . Then, . We next bound via bounding .

For , we bound via

(7)

This bound holds because (i) while the ESP auction with uniform price can sell the item with price at least if there exists at least one buyer with value , the optimal auction can sell at price at least only if there is at least one buyer with , and as a result, , and (ii) by definition of , we have ; to see this recall that . So when , by monotonicity of , it must be the case that . Further, if , we have . Thus, .

For we bound by noting that the ESP auction with uniform price can sell at price at least only if there are at least two buyers bidding above . Let be the cardinality of set and be the cardinality of set .

Then, we have:

(8)

Combining the bounds in (7) and (8), we get

(9)

We next bound . With a slight abuse of notation, let be the probability that the Myersonian ESP auction sells at price greater than or equal to . Then, by the definition of we have

(10)

Then, considering that and by using Equations (9) and (10), we get

(11)

where the first inequality follows from our assumption that . To simplify the r.h.s. of  (11), we make use of Lemma 9, stated at the end of this section, that says , and .

By Equation (11) and Lemma 9, we get

(12)

The above equation holds for any . Set . Then, we have

(13)

where the equality follows from the definition of ’s and , and the fact that at , . The second inequality follows from definition of ’s and ’s, and the fact that and are decreasing in (for proof, see Lemma 10, which is stated at the end of this section) and that itself is a decreasing function. Then, since , Equation (13) gives us the first set of constraints in Problem LP-ESP.

The Second Constraint: We now apply the same procedure to bound . By Equation (10), Lemma 9, and the fact that , we have

which results in the second constraint of Problem LP-ESP.

Lemma 9.

Let be the number of buyers with ; that is, . Then,

(14)
(15)

where and .

Lemma 10.

Functions and are decreasing in for every positive integer and every , where and . In addition, functions and are decreasing in and .111111Note that and .

Proofs of Lemmas 910 are given in the appendix.

5 Revenue Bounds for a Finite Number of Buyers

The bounds presented in Theorems 5 and 7 hold for any number of buyers . As stated earlier, in online advertising markets, due to targeting and heterogeneous preference of buyers (advertisers), the number of buyers is rather small. Inspired by this fact, in this section, we obtain improved bounds for the SPMs and ESP auctions when the number of buyers is small. We use the same ideas as in Theorems 5 and 7 to get these improvements, with the main new ingredient being the usage of -dependent bounds of Lemma 9, namely, and , rather than the -independent limiting versions of these quantities used in Theorems 7.

In Figure 1, we illustrate our improved bounds for the SPMs and ESP auctions. We also depict the best bound known for these mechanisms prior to this work, i.e., . As stated earlier, this bound is due to Chawla et al. (2010a). We observe that our bound for the SPMs and ESP auctions improves the prior bound by up to and 4%, respectively. Further, the revenue bounds increase as the number of buyers decreases. This justifies the widespread use of the ESP auctions in advertising exchanges where the number of buyers in each auction is rather small.

Figure 1: Comparing our bound with the best prior known bound for . The red and green dashed curves respectively present the bounds in Theorems 5 and Theorem 7. Recall that these bounds are valid for any .

In this section, to highlight the dependency of our bounds on the number of buyers , we denote the revenue of SPMs and ESP auctions with vector of prices by and , respectively. We further denote the optimal revenue by .

5.1 Posted Price Mechanism

The following theorem is the main result of this section.

Theorem 11 (SPM with a Finite Number of Buyers).

There exist reserve prices such that for every positive integer , where

(LP-SPM-n)

Here, , , and .

The proof of Theorem 11 is similar to that of Theorem 7, and is provided in the appendix.

Table 3 shows for and