FFinite-Sample Average Bid Auction ˚ Haitian Xie † August, 2020
Abstract
The paper studies the problem of auction design in a setting where the auctioneer accessesthe knowledge of the valuation distribution only through statistical samples. A new framework isestablished that combines the statistical decision theory with mechanism design. Two optimalitycriteria, maxmin, and equivariance, are studied along with their implications on the form ofauctions. The simplest form of the equivariant auction is the average bid auction, which setindividual reservation prices proportional to the average of other bids and historical samples.This form of auction can be motivated by the Gamma distribution, and it sheds new light onthe estimation of the optimal price, an irregular parameter. Theoretical results show that it isoften possible to use the regular parameter population mean to approximate the optimal price.An adaptive average bid estimator is developed under this idea, and it has the same asymptoticproperties as the empirical Myerson estimator. The new proposed estimator has a significantlybetter performance in terms of value at risk and expected shortfall when the sample size is small.
JEL Classification:
C44, C57, D44, D82
Keywords:
Statistical Auction Design, Regret Ratio, Equivariance, Average Bid Auction,Empirical Myerson Auction, Price-mean Ratio. ˚ I am indebted to Graham Elliott and Yixiao Sun for their constant couragements and insightful comments. I amdeeply grateful for the guidance from Ying Zhu, who leads me to the exploration of the statistical issue in auctions.I also thank Jin Xi, Wanchang Zhang, Yu-Chang Chen, and participants in the econometrics lunch seminar at UCSan Diego. † Department of Economics, University of California San Diego. Email: [email protected]. a r X i v : . [ ec on . E M ] A ug Introduction
This paper studies the problem of auction design under the assumption that the optimal reserva-tion price is unknown due to the lack of exact knowledge of the valuation distribution. Instead,independent samples recorded from past auctions are assumed to be available to the auctioneer.In the classical Bayesian mechanism design literature, the auctioneer knows the distributionfrom which the bidders’ valuations are drawn. In this case, the optimal auction is derived in theseminal work of Myerson (1981), which is in the form of a second-price auction with the optimalreservation price. The optimal reserve price depends on the valuation distribution through p p F q “ arg max p p p ´ F p p qq , (1)where F is the valuation distribution. The job of acquiring a reasonable estimate of the optimalprice is left to the statisticians and computer scientists. So the complete process of auction designis divided artificially into two steps: (i) deriving the optimal price as if the distribution were known,and then (ii) approximating this price with data.However, as shown in the paper, the price p p F q is not a pathwise differentiable functional of thedistribution F ; thus, it can only be estimated at rates slower than the n { -rate. This is becausethe optimal reserve price depends on the density, which can only be estimated at a slower rate if noparametric form is imposed. A natural question arises that, when considering the entire problem,whether it would be favorable to sacrifice the optimality in the first step (when the distribution isknown) in exchange for a faster convergence rate in the second step by targeting a suboptimal butregular parameter.The investigation begins by framing the problem inside the statistical decision framework, wherethe allocation rule and transfer payments of an auction are the statistical decision rules withthe bids and historical samples being the data inputs. In this framework of statistical auctiondesign under finite sample, methods that resemble statistical estimation are integrated into theclassical mechanism design theory that mainly deals with the incentive compatibility issues underasymmetric information. The maxmin and equivariance principles are studied. The frameworkencompasses two special cases. First, when there are no available historical samples, then theproblem becomes the classical mechanism design . Second, when there is only one bidder, then theproblem becomes monopolistic pricing with samples.As a result, a new form of auction, called the average bid auction, is proposed. It assigns theauction item to the bidder with the highest valuation. Each bidder faces an individual reserva-tion price that is proportional to the average of other bidders’ bids and historical samples. This isbecause, for each bidder, the other bids plays the same role as the sample observations in construct-ing the estimator of the optimal price. In contrast, his own bid has to be excluded for incentive The usage of “prior” and “Bayesian” in this literature is somewhat different from that in statistics. The buyers’valuations are modeled as random variables in this context, whose distribution is referred to as the prior, and Bayesianmeans that the designer knows this distribution. Even in this case, the paper still contributes to the literature by studying the equivariance principle. p p F q . A set of interesting findings shows that the population meanoften serves as a useful reference point for the optimal price. Based on this idea, the paper proposesan adaptive way to conduct the average bid procedure, which uses the a pilot estimator for thecoefficient in the average bid procedure. This adaptive estimator has the same asymptotic propertiesas the empirical Myerson estimator, and it has a significantly better performance in terms of valueat risk and expected shortfall when the sample size is small. In the economics literature, empirical estimates have been obtained for the independent privatevalue auctions. Paarsch (1997) uses a parametric approach. For nonparametric estimation, Atheyet al. (2002) considers solving the first-order condition of (1) after estimating the density of F .Another popular approach is to directly maxmimize the empirical version of (1). Studies related tothis approach include Segal (2003); Prasad (2008); Coey et al. (2020). This method can be referredto as the empirical Myerson auction because the price is set as if the empirical distribution is thetrue distribution F .Studies of similar topics can be found under the name of algorithmic mechanism design in thetheoretical computer science literature, including works by Cole and Roughgarden (2014); Dhang-watnotai et al. (2015); Huang et al. (2018); Guo et al. (2019). The primary solution understudyin this literature is the empirical Myerson auction and its variants. As summarized in Babaioffet al. (2018), the two directions studied so far in this literature are two opposite directions: that ofasymptotic results where only the rate of convergence is of concern, and that where only a singlesample is available. The adaptive average bid procedure proposed in the paper partially fills thisimportant gap between the large sample case and the one (or two) sample case.There is a massive amount of literature on the structural econometrics on auction, as recentlysurveyed by Perrigne and Vuong (2019). The main goal of this literature is to identify and estimatethe valuation distribution from the observed bids in first-price auctions. However, this paper as-sumes that the samples come from a truthful mechanism, where the bids are equal to the valuationsthemselves, so identification is no longer a problem. Instead, this paper pushes the task one stepfurther, asking the question on how to set the allocation rule with a finite sample when all relevantquantities are identified.The rest of the paper is organized as follows. Section 2 introduces the statistical auctiondesign framework, where statistical methods are integrated into the classical mechanism designtheory. Section 3 studies general results regarding the maxmin and equivariant principle. Section4 motivates the average bid auction under Gamma distribution. Section 5 proposes the adaptiveaverage bid procdure and studies its performance under general distributional assumptions boththeoretically and with simulations. Section 6 concludes. Appendix A contains tables for simulationresults. Proofs are collected in Appendix B. 3 Statistical Auction Design
There is a single item to sell. There are k bidders, whose valuations of the item are collected in V “ p V , ¨ ¨ ¨ , V k q . Each bidder knows his own valuation. The seller doesn’t observe the valuations,instead he observes a collection of n samples W “ p W , ¨ ¨ ¨ , W n q , possibly from past auctions runon similar items. The valuations V , ¨ ¨ ¨ , V k and samples W , ¨ ¨ ¨ , W n are i.i.d. draws from thesame distribution F . Throughout the paper, F is assumed to be absolute continuous, with support r v, ¯ v s or r v, where v ě Definition 1.
A statistical auction consists of functions q i and t i for i “ , ¨ ¨ ¨ , k , where q i : R k ` ˆ R n ` Ñ r , s , such that ř ni “ q i ď , and t i : R k ` ˆ R n ` Ñ R . The inputs of the statistical auction are bids b P R k ` and samples w P R n ` . Each q i p b , w q isthe probability of assigning the item to bidder i , and t i p b , w q is the payment from bidder i to theseller. The condition ř ki “ q i ď ´ ř ki “ q i is the probability that the item is not sold. Note that a classical auctionwithout the samples can be treated as a special case where n “
0. It is worthy to note that therevelation principle applies in this setting, so the restriction to the direct mechanism is withoutloss of generality.The expected revenue R p q, F q from implementing the rule q under the distribution F is definedas R p q, F q “ k ÿ i “ E r t i p V , W qs , (2)which is the sum of the expected payment from each bidder. The expectation is taken over boththe valuations V and samples W .Since the bidders are strategic players in the auction, there are some sensible restrictions to placeon the form of the statistical auction. Each bidder i reports bid b i based on his own valuations.The quasilinear form of utility is assumed, which for each bidder i is defined as v i q i p b , w q´ t i p b , w q .When bidder i is bidding his own valuation v i , he receives utility U i p v i , b ´ i , w q “ v i q i pp v i , b ´ i q , w q ´ t i pp v i , b ´ i q , w q . where b ´ i P R k ´ ` is the set of bids from bidders other than i , and p v i , b ´ i q is a complete set ofbids that combines v i and b ´ i . It is desired that the auction to be designed in a way that the The W ’s can also be considered as independent signals of the valuation distribution F . If they come from differentauctions, then F is the probability measure of the bigger population encompassing all these auctions. We do notneed to assume that the bidders observe W . F . In other words, bidding truthfully isa dominant strategy for each bidder, as defined in the following. Definition 2.
A statistical auction is dominant strategy incentive compatible (DSIC) if for anybidder i , it holds for all v i , b i , b ´ i , and w that U i p v i , b ´ i , w q ě v i q i pp b i , b ´ i q , w q ´ t i pp b i , b ´ i q , w q . The other important constraint is called individual rationality, which means that the bidders’utility cannot go below some lower bound so that they are willing to participate in the auction inthe first place.
Definition 3.
A statistical auction is individual rational (IR) if for any bidder i , it holds for all v i , b ´ i , and w that U i p v i , b ´ i , w q ě . Since the bids in DSIC auctions are the valuations, henceforth the notation v i will be usedto represent both the valuation and the bid for bidder i . A relationship between q and t can bederived for any auction that satisfies DSIC and IR, which is the well-known Revenue EquivalencePrinciple in mechanism design theory. The following lemma is the statistical version of this resultthat involves the sample W . Lemma 1. (Revenue Equivalence) A statistical auction p q, t q is DSIC if and only if, for each bidder i it holds that(i) q i pp v i , v ´ i q , w q is non-decreasing in v i .(ii) For every v i in the support of F , the transfer payment satisfies: t i pp v i , v ´ i q , w q “ v i q i p v , w q ´ ż v i q i pp u, v ´ i q , w q du ´ U i pp v, v ´ i q , w q . Moreover, if p q, t q maximizes the auctioneer’s expected revenue, then t i pp v i , v ´ i q , w q “ v i q i p v , w q ´ ż v i q i pp u, v ´ i q , w q du. (3)This lemma means that, without loss of optimality, the allocation rule q “ t q i u ki “ completelydetermines the transfer payment t “ t t i u ki “ , hence the statistical auction. Throughout the paper,statistical auctions are considered to be of the form in Equation (3), and the allocation rule is oftenused to denote the corresponding auction. Below are some examples of statistical auctions in viewof Lemma 1. Example 1.
Consider the case of one bidder and one sample, i.e. k “ n “ . Let the allocationrule be q p v , w q “ t v ą w u . Then the associated payment is t p v , w q “ v t v ą w u ´ ż v t u ą w u du “ $&% w , if v ą w , if v ă w . his is the single sample identity pricing rule, where the seller posts a price equal to the sampleobservation. Example 2.
When k “ and n “ , there are 2 bidders and 1 sample. Let the allocation rule be q i p v i , v i , w q “ t v i ą v i , v i ą w u , i ‰ i P t , u , which means the item is allocated to the bidder with bids higher than both the sample and the otherbid. The associated payment is t i p v i , v i , w q “ $’’’&’’’% v i , if v i ą v i ą w w , if v i ą w ą v i , if v i ă w . This is the second-price auction with a reservation price set equal to the sample observation.
The discussion up to this point does not involve the distribution F , because we are consideringdominant strategies. The next step is to characterize the optimal revenue when F is known to theseller, so as to provide a reference point for the revenue earned by any statistical auction. Twouseful definitions are introduced. Let f be the pdf of F . Definition 4.
For a absolute continuous distribution F , the virtual valuation is defined as φ F p v q “ v i ´ ´ F p v i q f p v i q . The distribution F is said to be regular if φ F is strictly increasing. Let p p F q be defined as theunique root of φ F p¨q “ . Combining Lemma 1 with Equation (2), the revenue has a more explicit representation as R p q, F q “ k ÿ i “ E r q i p V , W q φ F p V i qs . This is the statistical version of what is being referred to as the Myerson’s lemma in the literature,which is very useful in a lot of the derivations afterwards.When F is regular, Myerson (1981) showed that the optimal auction q F is the second priceauction with reserve price p p F q . Note that when F is known, the optimal auction no longerdepends on the sample W . The allocation rule for this auction is q Fi p v q “ t v i ą max i ‰ i v i , v i ą p p F qu . This is equivalent to the definition in Equation (1) Thoughout the paper, the case of multiple winners ( v i “ max i ‰ i v i ) is ignored, since the distributions underconsideration are absolute continuous. xample 3. Consider F to be the exponential distribution, i.e. f p v q “ e ´ v . The virtual valuationfunction φ F p v q “ v ´ is increasing. The optimal reserve price is p p F q “ . The Myerson optimalallocation rule is q Fi p v q “ t v i ą max i ‰ i v i , v i ą u . With a little abuse of notation, denote the optimal revenue under F as R p F q “ R p q F , F q . Theutility function of the auctioneer in the entire statistical auction design problem is modeled as theratio between the expected revenue and the optimal revenue. I call it the regret ratio, of statisticalauction q under a regular distribution F , r p q, F q “ R p q, F q R p F q . The name regret ratio follows from the fact that each auction is valued based not on the absoluterevenue gained but on the relative revenue compared to the best achievable revenue under the samedistribution. The regret ratio is a solid choice of the objective function from the decision-theoreticperspective, as explained in the next section, along with the maxmin criterion.
The maxmin principle is used to protect against the worst possible distribution. It ranks thedecision rules based on their worst-case performances. In statistics, it is often referred to as theminimax principle, for the utility function is expressed in terms of loss. Let Q denote the set ofall allocation rules that satisfies the first condition in Lemma 1. The formal definition of maxminauction is as follows. Definition 5.
Given a class of distributions F , a DSIC statistical auction q is maxmin if inf F P F r p q, F q “ sup q P Q inf F P F r p q , F q . (4)Since the objective is the regret ratio function, the principle can also be considered as maxmin-regret in this case. The maxmin-regret principle is commonly applied in the treatment assignmentliterature (Manski, 2004; Stoye, 2009, 2012; Tetenov, 2012). However, instead of using the ratioas the regret, those works use the difference between the best achievable and the achieved as theregret. To distinguish, the latter is referred to as regret difference. Arguably, the regret ratio is abetter quantity in measuring the relative gains. Because the range of the regret difference varieswith the magnitude of the best achievable revenue, while the regret ratio always lies within the unitinterval. This property renders the regret ratios comparable across different distributions. Despitethis imparity, the two types of regret play the common role in avoiding selecting the trivial rule that It is also sometimes called the competitive ratio. Besides these works on finite-sample maxmin regret analysis, Kitagawa and Tetenov (2018) also studies thewelfare regret from the perspective of non-asymptotic risk bound. A discussion on this issue can be found inSadler (2015). For a thorough comparison among maxmin, maxmin regret difference, and maxminregret ratio, from an axiomatic perspective, the readers are referred to Brafman and Tennenholtz(2000).The maxmin procedure can also be seen as a zero-sum game played by the auctioneer againstnature. The auctioneer picks a statistical auction q P Q , while nature picks a distribution F P F . The set Q is already convexified by the randomization as in Definition 1. Nature can alsorandomized. To avoid technical issues, assume F is finite-dimensional. The nature’s mixed strategyis a prior distribution π P ∆ p F q . Nature’s payoff is given by the negative of the regret ratio. Awell-known result is that if the game has a Nash equilibrium p q ˚ , π ˚ q , then q ˚ is a maxmin auction.The prior π ˚ is called least favorable prior. This fact alleviates the difficulty in proving maxminitysince now one only needs to show that q ˚ is the best response against π ˚ , and vice versa.The item in the auction often possesses attributes that covariate with its valuation. The at-tribute is modeled by a finite-valued covariate X supported on X . Denote F x P F as the conditionaldistribution of the valuation given X “ x . The sample data is the set tp W j , X j qu nj “ . The vectorof samples W can be partitioned according to the covariate, W “ t W x : x P X u , where each W x contains the sample valuations whose associated attribute is x . The elements in W x are iid drawsfrom F x . The covariate of the item in the current auction is also observed. When the attributeof the current item is observed to be x , the DSIC bids contained in V are considered as equal indistribution to elements in W x .An illustrative example is provided to enhance understanding. Consider the auction on anantique china teacup. Historical data contains transaction prices, from second-price auction con-ducted in the past, for both antique china teacups and plates. The attribute, in this case, is binary,representing teacup or plate. The bidders’ valuations for the teacup in this auction is assumedto share a common distribution with the historical prices of teacups in the data, while potentiallydistributed differently from the prices of plates. The next goal is to show that the maxmin auctiononly uses the data for teacups to estimate the optimal reserve price, and completely ignores thedata for plates.With more information at hand, the auctioneer can condition the allocation on the covariate.Denote this more general form of allocation rule by κ “ t κ x : x P X u , where each x signifies theattribute of the item in the current auction. Each conditional allocation κ x p V , W , X q depends onthe entire set of available information. The notation q p V , W q is retained for the previously definedunconditional auction. It is important to note that, even after fixing the value x , the auction κ x is still different from an unconditional one in the sense that the elements in W can be treateddifferently accordingly to the value of X . The expected revenue, for an item with attribute x , is R ´ κ x , ˜ F ¯ “ E « k ÿ i “ φ F x p V i q κ x p V , W , X q ff , See Manski (2004) and Savage (1954) for illustrative examples on this issue. F “ t F x : x P X u P F X . Dividing R ´ κ x , ˜ F ¯ by R ˚ p F x q gives the conditional version ofregret ratio r ´ κ x , ˜ F ¯ . The maxmin problem becomes finding a κ x for each x such thatinf ˜ F P F X r p κ x , ˜ F q “ sup κ x inf ˜ F P F X r p q, ˜ F q , (5)where κ x varies in the set of all conditional auctions, and nature is allowed to vary each F x in F without any restrictions. Note that the regret ratio also implicitly depends on the marginaldistribution of X , but nature is assumed not to manipulate that. In fact, as shown in the proof ofProposition 1 the marginal distribution of X does not affect the maxmin auction.For any unconditional rule q as described in Definition 1, one can define a set of associatedconditional auctions t κ x , x P X u , that uses no cross-covariate information, by letting κ x p V , W , X q “ q p V , W x q . (6)Essentially, the conditional auction κ x discards all samples with covariate value different than x ,and proceed with the allocation q . Proposition 1.
Assume that the unconditional rule q ˚ is the maxmin solution to (4), with the leastfavorable prior being π ˚ P ∆ F . Then the associated conditional auction κ ˚ x , as defined in Equation(6), is the maxmin solution to (5). This type of result first appears in Stoye (2009). The intuition is that nature can choose a priorthat renders the data W x uninformative about items with attribute x ‰ x . Then the auctioneer isbest responding by discarding irrelevant information. This result provides practical guidance fromthe maxmin principle that only the most relevant data should be used in designing an auction ona specific item. However, this means that, in each auction, the effective sample size is going tobe relatively small. To overcome this issue, a form of auction is introduced in the next sectionthat performed particularly well under small to moderate sample size. Before that, another usefulstatistical decision principle is studied. The equivariance principle, in statistical estimation problems, is a group-theoretic formalization ofcertain intuitively appealing decisions. This intuition, in the context of statistical auction design,is that the final allocation should not be altered if the monetary unit of the bids and samples arechanged. More specifically, if the auctioneer decides to allocate the item to a bidder when the inputdata are in dollars, then it is natural to assume that this decision remains the same when the inputdata are in Yen.Besides this intuitive argument, there are many practical reasons for employing the equivarianceprinciple. First, the equivariance simplies the maximization of the regret ratio. Proposition 2 showsthat the regret ratio of an equivariant auction is constant along each scale family. Second, as in9he estimation problem, the equivariance principle provides an effective way to restrict the set ofstatistical rules under consideration. As shown in Proposition 3, the optimal equivariant auctionderived under a specific distributional assumption does not ignore the data. For this purpose, theequivariance principle is similar to unbiasedness principle, except that there is no natural analog ofunbiasedness in auction design. Another reason is that the relationship between equivariance andmaxmin through the generalized Hunt-Stein theorem, which states that the maxmin equivariantrule is maxmin overall. This is addressed in the next subsection after the introduction of themaxmin principle.We first introduce the equivariance structure in the statistical auction design problem. Considerin turn the scale transformation on three types of objects: distribution of valuations, bids andsamples, and the payment transfer. For any regular F and θ ą
0, let F θ denote the CDF of θV ,i.e. F θ p v q “ F ` vθ ˘ . The corresponding density is f θ p v q “ θ f ` vθ ˘ . The following lemma shows thatthe optimal revenue, from the Myerson auction, is scale-equivariant. Lemma 2.
For any regular F and θ ą , we have p p F θ q “ θp p F q , and R p F θ q “ θR p F q . Next, define a special class of statistical allocation rules, that are invariant to the scale trans-formation on bids and samples.
Definition 6.
An allocation rule is (scale-)invariant if, for any θ ą , q p θ v , θ w q “ q p v , w q , v P R k ` , w P R n ` . (7) A statistical auction is (scale-)equivariant if its allocation is (scale-)invariant.
Lemma 3.
For a scale-invariant allocation rule, the associated payment transfer rule satisfies, forany θ ą and i “ , ¨ ¨ ¨ , n , t i p θ v , θ w q “ θt i p v , w q , v P R k ` , w P R n ` . Lemma 3 justifies the name “equivariant” by showing that for any scale-invariant allocation,the associated payment transfer is scale-equivariant. The behavior of the scale transformation issummarized by Lemma 2 and Lemma 3. There is a notable difference between these two lemmas.The revenue from the Myerson optimal auction is equivariant when the distribution scales by θ .The revenue from the equivariant statistical auction is equivariant when the input valuations andsamples scale by θ . In particular, fix a distribution F , the revenue from the Myerson optimalauction is not equivariant.In the literature, a discussion on the formalized equivariance principle can be found in Allouahand Besbes (2018) in the setting of auction design with two bidders (and no sample). Most commonauctions are in the equivariant form, including the first and second-price auctions. The equivariant The usage of “equivariant” and “invariant” in the text is the same as in “equivariant estimator” and “invarianttest” in traditional statistics literature. q ˆ v w n , ¨ ¨ ¨ , v k w n , w w n , ¨ ¨ ¨ , w n ´ w n , ˙ . The auctions in Example 1 and 2 are both equivariant. Because the allocation in Example 1 is q p v , w q “ q p v { w , q “ t v { w ą u , and the allocation in Example 2 is q p v , v , w q “ q p v { w , v { w , q “ t v { w ą , p v { w q{p v { w q ą u ,q p v , v , w q “ q p v { w , v { w , q “ t v { w ă , p v { w q{p v { w q ă u . To complete the equivariance structure, we show that the regret ratio of any equivariant auctionis constant with respect to the scale parameter of the valuation distribution. From Lemma 2 andLemma 3 state that when the distribution scales by θ , both the optimal revenue and the expectedrevenue from an equivariant statistical auction scale by θ . Thus, we have the following proposition. Proposition 2.
For any regular distribution F , and equivariant auction, it holds that r p q, F θ q “ r p q, F θ q , for any θ, θ ą . The statistical auction design problem is equivariant with respect to the group of scale trans-formations. In the group-theoretic language, the set t F θ : θ ą u is called an orbit, within whichthe regret ratio is constant. The most straightforward result from the equivariance principle is tosimplify the space of distributions by collapsing it into the sets of orbits.By the generalize the Hunt Stein theorem we can expect the maxmin equivariant auction to bethe overall maxmin auction. The mathematical statement issup q P Q e inf F P F r p q, F q “ sup q P Q inf F P F r p q, F q , where Q e Ă Q denotes the set of all equivariant auctions. This means in finding the maxmin sta-tistical auction, attentions can be restricted to the equivariant auctions. Bondar and Milnes (1981)shows that such result holds when the associated group transformation satisfies the amenabilitycondition. And they have also shown that the scale transformation group (possibly one of the sim-plest transformation group) is amenable. See Chapter 5 of Lehmann and Casella (2006), Bondarand Milnes (1981), and Wesler (1959) for more discussions of the generalized Hunt-Stein theorem.Next, we study the general structure of the optimal equivairant auction. The following propo-sition shows the common representation of the optimal equivariant auction against a given scalefamily of valuation distributions. 11 roposition 3 (General Representation for Equivariant Auctions) . For any regular distribution F , the optimal equivariant auction that maximizes the regret ratio for the scale family t F θ : θ ą u has the allocation rule of the form q F,i p V , W q “ " V i ą ρ F p V ´ i , W q , V i ą max i ‰ i V i * , i “ , ¨ ¨ ¨ , k, (8) where ρ F is symmetric and homogeneous of degree one. When min t k, n u Ñ 8 , ρ F p V ´ i , W q con-verges in probability to p p F q . There are several features to the form of auction in (8). Notice that the denominator of theregret ratio, the optimal revenue, is a constant in this case. So the auctioneer only needs to focuson optimizing the revenue. The resulting auction is standard in the sense that a bidder wins onlyif his bid is the highest, which is represented by the part “ V i ą max i ‰ i V i ”. There is an individualreservation price ρ F p V ´ i , W q for each bidder i . In order to be the winner, the bidder needs to havea valuation higher than this reservation price.In setting the price for each bidder i , the effective sample used is in fact p V ´ i , W q , where bidder i ’s own valuation V i is excluded for the incentive compatibility issue. So the effective sample sizeis k ` n ´
1. Using the law of iterated expectations, the revenue from a bidder i can be written as E r q i p V , W q φ F p V i qs“ E „ q i ˆ V W n , ¨ ¨ ¨ , V k W n , W W n , ¨ ¨ ¨ , W n ´ W n , ˙ ˜ φ F ˆ V W n , ¨ ¨ ¨ , V k W n , W W n , ¨ ¨ ¨ , W n ´ W n ˙ , (9)where ˜ φ F ˆ V W n , ¨ ¨ ¨ , V k W n , W W n , ¨ ¨ ¨ , W n ´ W n ˙ “ E « φ F p V i q ˇˇˇˇˇ V W n , ¨ ¨ ¨ , V k W n , W W n , ¨ ¨ ¨ , W n ´ W n ff (10)can be considered as the equivariant version of the virtual valuation. In Myerson (1981), the optimalreserve price is derived as the root of the virtual valuation function. Similarly, here the reserve price ρ F p V ´ i , W q is derived based on the root of the equivariant virtual valuation ˜ φ F . From the classicalestimation viewpoint, we can consider ρ F p V ´ i , W q as an equivariant estimator of the optimal price p p F q with the risk function being designed as the (negative) regret ratio.The estimator ρ F p V ´ i , W q consistently estimates the true optimal price when there is a largenumber of bidders or samples because then the auctioneer can estimate the valuation distributionaccurately. The function ρ F is symmetric so other bidders’ valuations V ´ i and the samples W playthe exact same role in setting the individual reservation price. The fact that the samples W areutilized already shows the power of the equivariance principle. Because the overall optimal auctionis not equivariant and completely ignores the samples W since it set the reservation price to be p p F q .When the sample size n is large, little is lost if we ignore the information in V ´ i and using W alone to estimate the reserve price. However, in the finite sample case, when k and n are of12he same magnitude, it is rather important to take advantage of the information coming from thecurrent auction. After discussion on the general representation of equivariant auctions, we study a specific formof the equivariant auction. The simpliest form of ρ F one can think of is perhaps the average¯ S ´ i “ k ` n ´ ´ř ki ‰ i V i ` ř nj “ W j ¯ over the effective sample p V ´ i , W q . Following this idea, theaverage bid (AB) auction q β is defined as: q β,i p V , W q “ " V i ą β ¯ S ´ i , V i ą max i ‰ i V i * , i “ , ¨ ¨ ¨ , k, (11)where β is a positive constant. The abbreviation AB- β is used when the coefficient β is underdiscussion. We can see that the AB- β auction sets the reserve price to be β times the sampleaverage.In the case of one bidder and one sample ( k “ n “ β auction sets the price tobe βW . It is shown in Huang et al. (2018) that, for distributions with increasing hazard rate p ´ F q{ f , the pricing rule 0 . W guarantees a regret ratio of 0 . .
5. Notice that, in this case, the empiricalMyerson pricing rule sets the price to the observed sample W , and it is also called the single sampleidentity pricing rule. When there are more than one bidder and sample observation, we proceedto show that the AB auction is optimal under the Gamma distribution. Then in the next section,we show that the AB auction is favorable in the finite-sample setting under general distributionalassumptions. Statisticians studying finite-sample estimation and hypothesis testing have long made progress byimposing parametric assumptions to make the problem tractable. Similarly, we make progress inthis section by restricting attention inside the Gamma family. The choice of Gamma distributionin the auction data analysis is not new (see e.g. Friedman, 1956; Hossein Bor, 1977; Skitmore, 2014;Takano et al., 2014). In a more recent paper, Ballesteros-P´erez and Skitmore (2017) shows Gammadistribution is among the distributions that provide the best fit to the empirical auction data.The Gamma family of distributions has densities of the form f p v, α, θ q “ θ α v α ´ e ´ v { θ Γ p α q , α ě , θ ą , where Γ is the Gamma function, and α, θ are respectively the shape and scale parameter. We firstfix α , and study the optimal equivariant auction for the scale family t f p¨ , α, θ q : θ ą u . Then wecan let θ “ θ due to Proposition 2. In the rest of this13ection, quantities that depends on the distribution F are indexed by α .Following Proposition 3 and the discussion after it, we know the form of the equivariant reserveprice ρ α depends on maxmizing the revenue from one bidder (without constraining the sum of q i ’sto be less than 1). Without loss of generality, we study the revenue from bidder 1 and normalizethe other valuations and samples by the valuation V : E r t p V , W qs “ ż q p v , w q φ α p v q k ź i “ f p v i , α, q n ź j “ f p w j , α, q d v d w “ ż q ˆ v v , v w ˙ φ α p v q k ź i “ f p v ¨ p v i { v q , α, q n ź j “ f p v ¨ p w j { v q , α, q d v d w “ ż q p σ q d σ ż φ α p v q k ` n ź i “ f p v σ i , α, q v n ` k ´ dv , (12)where the change of variable σ “ p , σ , ¨ ¨ ¨ , σ k ` n q “ p , v { v , ¨ ¨ ¨ , v k { v , w { v , ¨ ¨ ¨ , w n { v q is used.So q has been taken out from the first layer of integration. To study the first layer of integration,define a function ϕ m p σ , α q “ ż φ α p v q m ź i “ f p v σ i , α, q v m ´ dv “ Γ p ` mα q Γ p α q m ˜ m ź i “ σ i ¸ α ´ ˆ »–˜ ` m ÿ i “ σ i ¸ ´ ´ mα ´ F p ` p m ´ q α, ` mα, ` p m ´ q α, ´ p ř mi “ σ i qq ` p m ´ q α fifl . where F is the Gaussian hypergeometric function. Also define ϕ m as the factor of ϕ m that onlydepends on the sum ř mi “ σ i , ϕ m p s, α q “ Γ p ` p m ` q α q Γ p α q p m ` q „ p ` s q ´ ´p m ` q α ´ F p ` mα, ` p m ` q α, ` mα, ´ s q ` mα . We can see that the sign of ϕ m is completelt determined by the sign of ϕ m , that is,sgn p ϕ m p σ , α qq “ sgn ˜ ϕ m ˜ m ÿ i “ σ i , α ¸¸ . This equality can be verified using Mathematica. The hypergeometric function is defined as the analytic con-tinuation of the power series F p a, b, c, x q “ ÿ n “ p a q n p b q n p c q n x n n ! , | x | ă , where p a q n “ a p a ` q ¨ ¨ ¨ p a ` n ´ q t n ě u , and p b q n , p c q n defined similarly. q of (12) is only a function of ř k ` n ´ i “ σ i “ p ř ki “ v i ` ř nj “ w j q{ v . Combining this fact with the general representation of equivariant auction in Proposition 3, weknow the equivariant reserve ρ α must be proportional to the sample average ¯ S ´ i . Thus, we haveproved the following corollary. Corollary 1.
The AB- β auction is the optimal equivariant auction for the scale Gamma family t f p¨ , α, θ q : θ ą u , where the constant β depends on k, n, and α . Figure 1 graphs the functions ϕ m p s, α q for m “ , α “ . , . , .
5. Each curve starts asa decreasing function, crosses the horizontal axis once from above, and then starts increasing. Infact, the two axes are the two asymptotes of the curve. However, these properties are not requiredfor the proof of Corollary 1. The optimal equivariant auction is of the average bid form as long asthe sign of ϕ m only depends on the sum of ratios ř mi “ σ i .Figure 1: The graph of ϕ p s, α q and ϕ p s, α q for different values of α .Even though the knowledge of α is assumed, the resulting auction is still useful for practicalpurposes due to the equivariance restriction as mentioned before. The effective sample average¯ S ´ i is the method of moment estimator for α when θ “
1. Without the equivariance restriction,¯ S ´ i would be replaced by its true value α in the AB auction, then the resulting auction wouldnot be equivariant and is less practical since it requires the exact knowledge of α . Also, the useof estimator ¯ S ´ i in the optimal auction verifies the findings in Zaigraev and Podraza-Karakulska(2008) that, the moment based estimator performs better than the maximum likelihood estimator,in terms of both finite sample bias and variance for the estimation of α .If the AB- β auction is the unique optimal equivariant auction under α , then it is an admissibleequivariant auction among all regular distributions, which means its regret ratio cannot be entirelydominated by another equivariant auction. Such uniqueness is guaranteed if ϕ k ` n ą The condition v ą max i ‰ v i is not incorporated into the expression since ř i q i ď This property is shown in Figure 1 rather than being proved mathematically. β under different m and α .By the equivalence between ϕ k ` n ą v ą β ¯ S ´ i , we know p k ` n ´ q{ β is the root of ϕ k ` n p¨ , α q . Figure 2 shows the value of β computed by the numerical root of ϕ k ` n p¨ , α q . The β curve increases to the “true value” p p α q{ α very rapidly as k ` n increases. This observation, togetherwith the fact that ¯ S ´ i consistently estimates α , verifies the consistency statement in Proposition3. Figure 2 also shows another important fact that p p α q{ α is bounded in the tight r . , s for all α . This means that any choice of β in such region is a reasonable one for the Gamma family. The previous discussion is based on the case that the shape parameter α is known. Next, aninformal discussion is provided on how to find the maxmin auction under the Gamma distributionfamily with an unknown shape parameter α . As explained earlier, the maxmin solution can befound through a game-theoretic approach. The mixed-strategy Nash equilibrium corresponds tothe maxmin solution of the problem.The shape of the regret ratio curve can be derived from Figure 2. Consider setting β tobe somewhere in r . , . s , say, then as α increases from 1, the regret ratio first increases, thendecreases, and eventually increases again. So for a careful choice of β , it is possible that the regretratio have two minima, one at α “ α ˚ . When the regret ratio have two minima,any probability mixture between the two minima is a best response for nature. If for a certain16ixture q β is the auctioneer’s best response, then we have pinned down the maxmin auction.Figure 3: Flattened Two-Bidder Regret Ratio Curves.Figure 3 shows the simulated regret ratios curves of q β for the case of two bidders and multiplesamples with β selected so that the curves approximately has two minima. The smaller minimaare at 1, and the larger minima are around 10. This verifies the conjecture about the shape ofthe regret ratio curve. Nature’s best response is a prior over t , α ˚ u . Denote π as the probabilityof nature choosing 1, and 1 ´ π is the probability of nature choosing α ˚ . Given this prior, theauctioneer chooses the auction that maximizes the Bayesian regret ratio πr p q, q ` p ´ π q r p q, α ˚ q ,which is a linear combination of R p q, q and R p q, α ˚ q . So the optimal form of auction is still theAB auction. And with a careful choice of π , one can get the desired value of β as the best responseof the auctioneer.From a practical point of view, the goal of the maxmin procedure is to flatten the regret ratiocurve so the corresponding statistical auction has stable performance. This goal is clearly achievedas in Figure 3, where each regret ratio curve is approxmately flat. With only 5 samples, the AB- β auction (with β “ . The main feature of the AB auction is that it uses (a fraction of) the sample average to estimate theoptimal price p p F q . To make this feature more salient, we study the case of monopolistic pricing17here there is only one buyer. In this case, the AB- β auction amounts to setting the price to beˆ p β “ β ¯ W n “ βn n ÿ j “ W j . The alternative estimator for comparison purpose is the empirical Myerson (EM) estimator com-monly seen in the literature. The EM procedure estimates the reservation price by maximizing theexpected revenue over the empirical distribution of the sample W , that is,ˆ p EM “ arg max p p ˜ n n ÿ i “ t W i ě p u ¸ . (13)This can be considered as the empirical version of Equation (1). This estimator can be made morerobust against heavy tail distributions by trimming the large observations (see e.g. Dhangwatnotaiet al., 2015; Huang et al., 2018). The mechanism design theory suggests that p p F q is the optimal price. In other words, p p F q is thetarget parameter. The usual way to proceed would be to find an estimator that works based onasymptotic theory. For example, such estimator is often consistent and converges to the targetparameter at a certain rate. However, as shown by Prasad (2008), the convergence rate of theEM estimator ˆ p EM is n { , which is considerably slower than the usual n { rate. . Moreover, thefollowing result states that this slow convergence rate is not peculiar to the EM estimator. In fact, p p F q itself is not a regular parameter in the appropriate sense. Proposition 4.
The parameter p p F q as a functional of F is not pathwise differentiable in the senseof Van Der Vaart (1991), hence it is not ? n -estimable. Therefore, even though the EM estimator seems to be a natural choice from the asymptoticsviewpoint, it may not have desired performance when the sample size is finite due to its slowconvergence rate. Based on this observation, the estimation of p p F q by using ˆ p β can be consideredas a two-step approximation. In the first step, the irregular parameter p p F q is approximated bya regular parameter βµ p F q , where µ p F q denotes the population mean of the distribution F . Inthe second step, the parameter βµ p F q is approximated by the estimator ˆ p β . The sample averageis a regular estimator for the population mean. So if in the first step βµ p F q can approximate p p F q well, then the AB estimator ¯ p β can quickly converge to the neighborhood region of p p F q evenwith a small smaple size. So the two-step approximation can be seen as a regularization in thefinite-sample estimation of the irregular parameter p p F q . While the EM estimator outperforms theAB estimator eventually with a large sample size, the AB estimator may perform better when thesample size is small, which is the harder case for conducting statistical analysis. The limiting distribution of n { p ˆ p EM ´ p q is not the Normal distribution. The general theory of cube-rootasymptotics is developed in Kim and Pollard (1990) p p F q can be approximated by βµ p F q for general distributions.This amounts to whether the price-mean ratio, p p F q{ µ p F q , can be well-approximated by a constant β . We already know from Figure 2 that for the Gamma distributions, the price-mean ratio isbounded in r . , s , a rather tight interval. The next result shows the price-mean ratio can bebounded likewise under assumptions a little stronger than regularity. Proposition 5.
Let F be a regular distribution.(i) If the function f ´ F is concave, then p p F q{ µ p F q ď .(ii) If the function λv ´ ´ F p v q f p v q (14) is increasing w.r.t. v for some λ P r , q . Then the price-mean ratio can be bounded as p ´ λ q λ ´ F p p p F qq ď p p F q{ µ p F q ď p ´ λ q ´ λ , (15) where p ´ λ q λ is taken to be e ´ (the limit) when λ “ . This result demonstrates the theoretical possibility to bound the price-mean ratio. The condi-tion that (14) is increasing is called λ -regularity in Schweizer and Szech (2019). The lower boundin (15) contains the unknown p p F q , a crude bound can be obtained by replacing the denomina-tor 1 ´ F p p p F qq by 1. More discussions on the λ -regularity condition can be found in Cole andRoughgarden (2014); Cole and Rao (2017). In Kleinberg and Yuan (2013), the conditon that p p F q ě cµ p F q for some c ą c -boundedness. They show thatthis condition has attractive implications on the revenue to welfare ratio.Apart from the theoretical result, Table 1 shows the numerical ranges of the price-ratio computedfor common distributions. For these distributions, the price-mean ratio is bounded around one.For the log-normal distribution, the price-mean ratio can exceed one since the virtual valuation isnot always concave in this case. Note that the price-mean ratio is scale-invariant, thus the scaleparameter in parametric families can be set to 1 for the calculation. Consider an adaptive procedure for the AB estimator, where the coefficient β is estimated bya pilot estimator. Let ˜ β “ ˆ p EM { ¯ W n be the ratio between the EM estimator and the sampleaverage. Estimator ˜ β itself cannot be directly used as the pilot because that would result inthe EM estimator. Instead, a coasening operation is applied. Consider r β, ¯ β s as the intervalcontaining the true price-mean ratio, which can be derived based on Proposition 5 or Table 1. Let β “ b ă b ă ¨ ¨ ¨ ă b L n “ ¯ β be a set of L n partitoning points of r β, ¯ β s , which depends on the19istribution PDF Parameter p { µ Light tail Generalized Normal exp p´ v α q{ Γ p { α q α ě r . , s Gamma v α ´ e ´ v { Γ p α q α ě r . , s Weibull αv α ´ exp p´ v α q α ě r . , s Heavy tail Student t pp α ` q{ q? απ Γ p α { q ´ ` v α ¯ ´ α ` α ě r . , s Log-normal ? παv exp ´ ´ log p v q α ¯ α P p , . s r . , . s Pareto α p ` v q ´ α ´ α ą p ` v { α q ´ α ´ α ą n . The pilot estimator ˆ β is defined as the closest partitioning point to ˜ β , i.e.ˆ β “ b l , such that | b l ´ ˜ β | ď | b l ´ ˜ β | , for l “ , ¨ ¨ ¨ , L n . The pilot estimator ˆ β provides a simple way for choosing the coefficient for the AB estimator.The adapted AB estimator is defined as the plug-in estimator ˆ p ˆ β . If the partition gets dense in theinterval r β, ¯ β s in a suitable rate as n increases, then the pilot estimator ˆ β would become a n { -consistent estimator of the price-mean ratio. Consequently, the adapted estimator ˆ p ˆ β consistentlyestimates p p F q with the n { rate. So with this adaptive procedure, the AB estimator has the sameasymptotic properties as the EM estimator. This fact is summarized in the following proposition. Proposition 6.
Assume ˆ p EM is n { -consistent and n { p ˆ p EM ´ p p F qq converges in distribution toa absolute continous random variable. If the mesh of the partition satisfies n { max ď l ď L n | b l ´ b l ´ | Ñ , then both the pilot estimator ˆ β and the adapted AB estimator ˆ p ˆ β are n { -consistent for p p F q{ µ p F q and p p F q , respectively. Simulation studies are conducted for several regular distributions, including Gamma, Gener-alized Normal, Student- t , Lognormal, and Generalized Pareto. Here is a description of the simu-lation procedure. Under each distribution, the optimal revenue is computed first. Then the ABand EM estimators are simulated, each with 10 replications. The sample size n is chosen to be10 , , , , , r β, ¯ β s is chosen to be r , . s , and the mesh | b l ´ b l ´ | “ n ´ { {
4. The estimatesare then transformed into realized revenues using the true DGP, and regret ratios are computed. Ties only occur with zero probability, so any tie-breaking rule can be used. Primitive conditions for this assumption can be found in Prasad (2008). The limiting distribution is the uniquemaximizer of some Gaussian process (related to the Chernoff distribution), and is indeed absolute continuous. δ % is the lower δ quantile of the regret ratio. The expected shortfall ES δ % is the conditionalexpectation of the regret ratio given that it is lower than VaR δ % . Simulation results show that whenthe sample size is large ( n “ , The AB procedure gives rise to a novel methodology that “regularizes” the estimation of a irregularparameter. Such regularization depends on the existence of a regular parameter that universally ap-proximates the irregular parameter under reasonable assumptions, which is a rather hard problem.This method is new in the literature and further development of which is left for future works.The AB estimator is a more competitive alternative for EM when data is sparse. There arethree reasons for emphasizing small sample cases in pricing problems. First, as an application ofthe maxmin principle, a result was derived in the paper for cases with covariates, which showedthat only the sample data with the most relevant covariate value should be used for data analysis.The practical implication is that the effective samples are going to be very small for each specificauction. Second, for emerging markets, where the design of auction is most important, the amountof data is arguable at a small level. Lastly, if the sample size is significant, then the choice ofstatistical method is less critical for a good performance.
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Simulation Results
Median Mean SD VaR VaR ES Parameter n AB EM AB EM AB EM AB EM AB EM AB EM1 10 0.92 0.85 0.96 0.93 0.10 0.19 0.70 0.43 0.56 0.21 0.61 0.3020 0.93 0.89 0.96 0.95 0.08 0.15 0.77 0.54 0.67 0.31 0.71 0.4030 0.95 0.91 0.97 0.96 0.06 0.13 0.82 0.62 0.73 0.39 0.77 0.4840 0.95 0.93 0.97 0.97 0.05 0.11 0.84 0.69 0.77 0.47 0.79 0.5650 0.96 0.94 0.98 0.97 0.05 0.09 0.85 0.74 0.78 0.54 0.81 0.62100 0.97 0.96 0.98 0.98 0.04 0.06 0.90 0.85 0.85 0.72 0.87 0.77200 0.98 0.98 0.99 0.99 0.03 0.03 0.92 0.91 0.88 0.84 0.90 0.873 10 0.90 0.89 0.95 0.95 0.14 0.16 0.59 0.52 0.41 0.24 0.48 0.3620 0.93 0.93 0.97 0.97 0.10 0.10 0.72 0.72 0.54 0.50 0.61 0.5930 0.95 0.95 0.98 0.98 0.07 0.08 0.80 0.80 0.64 0.64 0.70 0.7040 0.96 0.96 0.98 0.98 0.06 0.06 0.83 0.83 0.70 0.70 0.75 0.7550 0.97 0.97 0.99 0.99 0.05 0.05 0.86 0.86 0.75 0.75 0.79 0.80100 0.98 0.98 0.99 0.99 0.03 0.03 0.92 0.92 0.85 0.85 0.88 0.88200 0.99 0.99 0.99 0.99 0.02 0.02 0.95 0.95 0.91 0.91 0.93 0.935 10 0.91 0.91 0.96 0.96 0.13 0.13 0.63 0.63 0.41 0.37 0.49 0.4820 0.94 0.95 0.98 0.98 0.08 0.08 0.78 0.78 0.60 0.60 0.67 0.6730 0.96 0.96 0.98 0.98 0.06 0.06 0.84 0.84 0.71 0.71 0.76 0.7640 0.97 0.97 0.99 0.99 0.05 0.05 0.87 0.87 0.76 0.77 0.80 0.8150 0.97 0.97 0.99 0.99 0.04 0.04 0.89 0.89 0.80 0.80 0.83 0.84100 0.98 0.98 0.99 0.99 0.02 0.02 0.93 0.94 0.88 0.89 0.90 0.90200 0.99 0.99 1.00 1.00 0.01 0.01 0.96 0.96 0.93 0.93 0.94 0.947 10 0.92 0.92 0.96 0.97 0.12 0.12 0.67 0.67 0.44 0.44 0.53 0.5320 0.95 0.95 0.98 0.98 0.07 0.07 0.80 0.81 0.65 0.65 0.71 0.7130 0.96 0.96 0.98 0.98 0.05 0.05 0.85 0.86 0.74 0.74 0.78 0.7940 0.97 0.97 0.99 0.99 0.04 0.04 0.88 0.89 0.79 0.79 0.83 0.8350 0.97 0.98 0.99 0.99 0.04 0.04 0.90 0.91 0.83 0.83 0.85 0.86100 0.98 0.99 0.99 0.99 0.02 0.02 0.94 0.94 0.90 0.90 0.91 0.92200 0.99 0.99 1.00 1.00 0.01 0.01 0.97 0.97 0.94 0.94 0.95 0.95Table 2: Simulation Results under Gamma Distribution.24edian Mean SD VaR VaR ES Parameter n AB EM AB EM AB EM AB EM AB EM AB EM2 10 0.92 0.88 0.96 0.93 0.10 0.13 0.70 0.60 0.55 0.43 0.61 0.5020 0.95 0.90 0.97 0.95 0.07 0.11 0.80 0.67 0.67 0.52 0.72 0.5830 0.96 0.91 0.98 0.95 0.06 0.10 0.83 0.70 0.73 0.57 0.77 0.6240 0.96 0.92 0.98 0.96 0.05 0.09 0.86 0.73 0.76 0.61 0.80 0.6550 0.97 0.93 0.98 0.96 0.05 0.09 0.87 0.74 0.79 0.64 0.82 0.68100 0.98 0.94 0.99 0.97 0.03 0.07 0.91 0.79 0.85 0.71 0.87 0.74200 0.98 0.96 0.99 0.98 0.02 0.05 0.93 0.84 0.89 0.77 0.91 0.805 10 0.92 0.85 0.96 0.92 0.10 0.17 0.70 0.47 0.55 0.27 0.60 0.3520 0.94 0.88 0.97 0.94 0.07 0.15 0.79 0.53 0.67 0.34 0.72 0.4230 0.95 0.90 0.97 0.96 0.06 0.13 0.83 0.59 0.74 0.41 0.78 0.4840 0.96 0.91 0.98 0.96 0.05 0.12 0.85 0.64 0.78 0.45 0.81 0.5350 0.96 0.92 0.98 0.97 0.04 0.11 0.88 0.67 0.81 0.49 0.84 0.56100 0.97 0.95 0.98 0.98 0.03 0.07 0.91 0.80 0.87 0.63 0.89 0.70200 0.98 0.97 0.99 0.99 0.02 0.04 0.93 0.89 0.91 0.79 0.92 0.837 10 0.92 0.85 0.96 0.92 0.10 0.18 0.70 0.45 0.55 0.24 0.61 0.3220 0.94 0.88 0.97 0.94 0.07 0.15 0.79 0.53 0.68 0.32 0.73 0.4130 0.95 0.90 0.97 0.96 0.06 0.14 0.84 0.59 0.75 0.39 0.78 0.4740 0.96 0.92 0.98 0.97 0.05 0.12 0.85 0.64 0.78 0.44 0.81 0.5250 0.96 0.93 0.98 0.97 0.04 0.11 0.88 0.69 0.81 0.49 0.84 0.57100 0.97 0.96 0.98 0.98 0.03 0.07 0.91 0.82 0.87 0.66 0.89 0.72200 0.98 0.97 0.99 0.99 0.02 0.04 0.93 0.90 0.91 0.81 0.92 0.8410 10 0.92 0.85 0.96 0.92 0.10 0.18 0.70 0.44 0.55 0.23 0.61 0.3120 0.94 0.88 0.97 0.95 0.07 0.16 0.79 0.52 0.69 0.31 0.73 0.4030 0.95 0.90 0.97 0.96 0.06 0.14 0.84 0.60 0.75 0.38 0.78 0.4740 0.96 0.92 0.98 0.97 0.05 0.12 0.86 0.66 0.79 0.44 0.81 0.5350 0.96 0.93 0.98 0.97 0.04 0.10 0.88 0.70 0.81 0.50 0.84 0.58100 0.97 0.96 0.98 0.98 0.03 0.06 0.91 0.83 0.87 0.68 0.89 0.74200 0.98 0.98 0.99 0.99 0.02 0.04 0.93 0.90 0.91 0.82 0.92 0.8515 10 0.92 0.85 0.96 0.92 0.10 0.19 0.70 0.43 0.55 0.22 0.61 0.3020 0.94 0.88 0.97 0.95 0.07 0.16 0.80 0.53 0.69 0.31 0.74 0.4030 0.95 0.91 0.97 0.96 0.06 0.13 0.84 0.60 0.76 0.39 0.79 0.4740 0.96 0.92 0.98 0.97 0.05 0.11 0.86 0.67 0.79 0.45 0.81 0.5450 0.96 0.93 0.98 0.97 0.04 0.10 0.88 0.72 0.82 0.52 0.84 0.60100 0.97 0.96 0.98 0.98 0.03 0.06 0.91 0.84 0.87 0.69 0.89 0.75200 0.98 0.98 0.99 0.99 0.02 0.04 0.93 0.91 0.91 0.83 0.92 0.86Table 3: Simulation Results under Generalized Normal Distribution.25edian Mean SD VaR VaR ES Parameter n AB EM AB EM AB EM AB EM AB EM AB EM1.5 10 0.91 0.88 0.95 0.93 0.11 0.14 0.69 0.59 0.54 0.44 0.59 0.5020 0.93 0.89 0.96 0.94 0.09 0.12 0.75 0.63 0.62 0.51 0.66 0.5530 0.94 0.90 0.96 0.95 0.08 0.11 0.78 0.66 0.67 0.55 0.70 0.5940 0.94 0.91 0.96 0.95 0.07 0.11 0.79 0.67 0.68 0.58 0.72 0.6150 0.94 0.91 0.97 0.96 0.07 0.10 0.81 0.69 0.71 0.60 0.74 0.63100 0.96 0.94 0.98 0.97 0.05 0.08 0.85 0.74 0.76 0.66 0.79 0.69200 0.97 0.96 0.99 0.99 0.04 0.06 0.88 0.81 0.81 0.71 0.83 0.752 10 0.90 0.85 0.95 0.92 0.12 0.18 0.64 0.46 0.46 0.30 0.53 0.3720 0.92 0.87 0.96 0.94 0.10 0.16 0.71 0.51 0.56 0.36 0.62 0.4230 0.93 0.89 0.96 0.95 0.08 0.14 0.76 0.56 0.62 0.41 0.67 0.4640 0.94 0.91 0.97 0.96 0.08 0.13 0.78 0.60 0.65 0.44 0.70 0.5050 0.94 0.92 0.97 0.97 0.07 0.12 0.80 0.63 0.69 0.48 0.74 0.54100 0.96 0.95 0.98 0.98 0.05 0.08 0.86 0.79 0.78 0.60 0.81 0.68200 0.98 0.97 0.99 0.99 0.04 0.04 0.90 0.89 0.84 0.78 0.86 0.825 10 0.90 0.85 0.95 0.93 0.12 0.19 0.65 0.40 0.47 0.19 0.54 0.2720 0.93 0.90 0.96 0.96 0.09 0.15 0.75 0.56 0.62 0.29 0.67 0.4030 0.94 0.92 0.97 0.97 0.07 0.12 0.80 0.68 0.69 0.41 0.73 0.5240 0.95 0.94 0.97 0.97 0.06 0.10 0.82 0.74 0.73 0.52 0.76 0.6150 0.96 0.95 0.98 0.98 0.05 0.08 0.85 0.79 0.77 0.61 0.80 0.68100 0.97 0.97 0.99 0.99 0.04 0.05 0.89 0.88 0.84 0.78 0.86 0.82200 0.98 0.98 0.99 0.99 0.03 0.03 0.93 0.93 0.88 0.87 0.90 0.8910 10 0.91 0.86 0.95 0.94 0.11 0.19 0.67 0.42 0.50 0.19 0.57 0.2820 0.93 0.91 0.97 0.96 0.08 0.14 0.77 0.61 0.65 0.34 0.69 0.4530 0.95 0.93 0.97 0.97 0.06 0.10 0.81 0.72 0.72 0.49 0.76 0.5840 0.95 0.94 0.98 0.98 0.06 0.08 0.83 0.78 0.75 0.59 0.78 0.6650 0.96 0.95 0.98 0.98 0.05 0.07 0.86 0.81 0.78 0.65 0.81 0.71100 0.97 0.97 0.99 0.99 0.03 0.04 0.90 0.89 0.85 0.80 0.87 0.83200 0.98 0.98 0.99 0.99 0.02 0.02 0.94 0.94 0.89 0.88 0.91 0.9015 10 0.91 0.86 0.95 0.94 0.11 0.18 0.67 0.44 0.51 0.19 0.58 0.2920 0.94 0.91 0.97 0.96 0.08 0.13 0.77 0.63 0.66 0.37 0.70 0.4830 0.95 0.93 0.97 0.97 0.06 0.10 0.82 0.73 0.72 0.52 0.76 0.6040 0.95 0.95 0.98 0.98 0.06 0.08 0.83 0.78 0.75 0.60 0.78 0.6750 0.96 0.95 0.98 0.98 0.05 0.07 0.86 0.82 0.79 0.67 0.81 0.72100 0.97 0.97 0.99 0.99 0.03 0.04 0.90 0.89 0.85 0.80 0.87 0.84200 0.98 0.98 0.99 0.99 0.02 0.02 0.94 0.94 0.89 0.89 0.91 0.90Table 4: Simulation Results under Student- t Distribution.26edian Mean SD VaR VaR ES Parameter n AB EM AB EM AB EM AB EM AB EM AB EM1.5 10 0.90 0.89 0.95 0.94 0.12 0.13 0.65 0.62 0.49 0.46 0.55 0.5220 0.94 0.92 0.97 0.96 0.08 0.09 0.77 0.72 0.63 0.59 0.69 0.6430 0.95 0.94 0.98 0.97 0.06 0.08 0.82 0.77 0.70 0.66 0.75 0.7140 0.96 0.94 0.98 0.97 0.06 0.07 0.84 0.80 0.74 0.70 0.78 0.7450 0.97 0.95 0.99 0.97 0.05 0.06 0.87 0.82 0.77 0.73 0.81 0.77100 0.98 0.96 0.99 0.98 0.03 0.04 0.91 0.87 0.84 0.81 0.87 0.84200 0.99 0.97 1.00 0.98 0.02 0.03 0.94 0.91 0.89 0.87 0.91 0.881.2 10 0.92 0.88 0.96 0.93 0.10 0.14 0.72 0.57 0.57 0.37 0.62 0.4520 0.94 0.89 0.97 0.94 0.07 0.13 0.79 0.63 0.67 0.46 0.72 0.5230 0.95 0.91 0.97 0.95 0.06 0.11 0.83 0.66 0.73 0.52 0.77 0.5740 0.96 0.91 0.97 0.96 0.05 0.10 0.85 0.69 0.77 0.56 0.80 0.6150 0.96 0.92 0.97 0.96 0.05 0.10 0.86 0.71 0.79 0.59 0.81 0.63100 0.97 0.94 0.98 0.97 0.03 0.08 0.90 0.77 0.85 0.67 0.87 0.71200 0.98 0.96 0.98 0.98 0.03 0.06 0.92 0.84 0.89 0.74 0.90 0.781.0 10 0.91 0.85 0.95 0.92 0.11 0.17 0.67 0.47 0.52 0.28 0.58 0.3520 0.92 0.88 0.95 0.94 0.09 0.16 0.73 0.52 0.61 0.34 0.66 0.4130 0.93 0.89 0.96 0.95 0.08 0.14 0.77 0.57 0.68 0.40 0.71 0.4740 0.94 0.91 0.96 0.96 0.07 0.13 0.79 0.61 0.71 0.45 0.74 0.5150 0.94 0.92 0.97 0.97 0.07 0.11 0.80 0.65 0.72 0.48 0.75 0.55100 0.96 0.95 0.98 0.98 0.05 0.08 0.85 0.79 0.80 0.61 0.82 0.69200 0.97 0.97 0.99 0.99 0.04 0.04 0.89 0.89 0.84 0.78 0.86 0.820.7 10 0.88 0.85 0.94 0.94 0.15 0.20 0.56 0.37 0.39 0.16 0.45 0.2420 0.91 0.90 0.96 0.96 0.12 0.15 0.64 0.58 0.48 0.28 0.54 0.4130 0.93 0.93 0.97 0.97 0.09 0.11 0.72 0.70 0.57 0.45 0.63 0.5540 0.95 0.94 0.98 0.98 0.08 0.09 0.77 0.77 0.62 0.56 0.68 0.6550 0.95 0.95 0.98 0.98 0.07 0.07 0.81 0.81 0.66 0.64 0.72 0.71100 0.97 0.97 0.99 0.99 0.04 0.04 0.89 0.90 0.80 0.80 0.84 0.84200 0.98 0.98 0.99 0.99 0.02 0.02 0.94 0.94 0.89 0.89 0.91 0.910.5 10 0.90 0.89 0.96 0.96 0.14 0.16 0.56 0.55 0.35 0.25 0.43 0.3720 0.94 0.94 0.97 0.97 0.09 0.10 0.74 0.74 0.53 0.53 0.62 0.6230 0.95 0.96 0.98 0.98 0.07 0.07 0.82 0.82 0.66 0.66 0.72 0.7240 0.96 0.96 0.98 0.98 0.06 0.05 0.85 0.86 0.73 0.74 0.78 0.7850 0.97 0.97 0.99 0.99 0.05 0.05 0.88 0.88 0.78 0.78 0.81 0.82100 0.98 0.98 0.99 0.99 0.03 0.03 0.93 0.93 0.87 0.87 0.89 0.90200 0.99 0.99 1.00 1.00 0.02 0.02 0.96 0.96 0.92 0.93 0.94 0.94Table 5: Simulation Results under Lognormal Distribution.27edian Mean SD VaR VaR ES Parameter n AB EM AB EM AB EM AB EM AB EM AB EM2 10 0.92 0.88 0.96 0.93 0.10 0.13 0.70 0.60 0.55 0.43 0.61 0.5020 0.95 0.90 0.97 0.95 0.07 0.11 0.80 0.67 0.67 0.52 0.72 0.5830 0.96 0.91 0.98 0.95 0.06 0.10 0.83 0.70 0.73 0.57 0.77 0.6240 0.96 0.92 0.98 0.96 0.05 0.09 0.86 0.73 0.76 0.61 0.80 0.6550 0.97 0.93 0.98 0.96 0.05 0.09 0.87 0.74 0.79 0.64 0.82 0.68100 0.98 0.94 0.99 0.97 0.03 0.07 0.91 0.79 0.85 0.71 0.87 0.74200 0.98 0.96 0.99 0.98 0.02 0.05 0.93 0.84 0.89 0.77 0.91 0.805 10 0.92 0.85 0.96 0.92 0.10 0.17 0.70 0.47 0.55 0.27 0.60 0.3520 0.94 0.88 0.97 0.94 0.07 0.15 0.79 0.53 0.67 0.34 0.72 0.4230 0.95 0.90 0.97 0.96 0.06 0.13 0.83 0.59 0.74 0.41 0.78 0.4840 0.96 0.91 0.98 0.96 0.05 0.12 0.85 0.64 0.78 0.45 0.81 0.5350 0.96 0.92 0.98 0.97 0.04 0.11 0.88 0.67 0.81 0.49 0.84 0.56100 0.97 0.95 0.98 0.98 0.03 0.07 0.91 0.80 0.87 0.63 0.89 0.70200 0.98 0.97 0.99 0.99 0.02 0.04 0.93 0.89 0.91 0.79 0.92 0.837 10 0.92 0.85 0.96 0.92 0.10 0.18 0.70 0.45 0.55 0.24 0.61 0.3220 0.94 0.88 0.97 0.94 0.07 0.15 0.79 0.53 0.68 0.32 0.73 0.4130 0.95 0.90 0.97 0.96 0.06 0.14 0.84 0.59 0.75 0.39 0.78 0.4740 0.96 0.92 0.98 0.97 0.05 0.12 0.85 0.64 0.78 0.44 0.81 0.5250 0.96 0.93 0.98 0.97 0.04 0.11 0.88 0.69 0.81 0.49 0.84 0.57100 0.97 0.96 0.98 0.98 0.03 0.07 0.91 0.82 0.87 0.66 0.89 0.72200 0.98 0.97 0.99 0.99 0.02 0.04 0.93 0.90 0.91 0.81 0.92 0.8410 10 0.92 0.85 0.96 0.92 0.10 0.18 0.70 0.44 0.55 0.23 0.61 0.3120 0.94 0.88 0.97 0.95 0.07 0.16 0.79 0.52 0.69 0.31 0.73 0.4030 0.95 0.90 0.97 0.96 0.06 0.14 0.84 0.60 0.75 0.38 0.78 0.4740 0.96 0.92 0.98 0.97 0.05 0.12 0.86 0.66 0.79 0.44 0.81 0.5350 0.96 0.93 0.98 0.97 0.04 0.10 0.88 0.70 0.81 0.50 0.84 0.58100 0.97 0.96 0.98 0.98 0.03 0.06 0.91 0.83 0.87 0.68 0.89 0.74200 0.98 0.98 0.99 0.99 0.02 0.04 0.93 0.90 0.91 0.82 0.92 0.8515 10 0.92 0.85 0.96 0.92 0.10 0.19 0.70 0.43 0.55 0.22 0.61 0.3020 0.94 0.88 0.97 0.95 0.07 0.16 0.80 0.53 0.69 0.31 0.74 0.4030 0.95 0.91 0.97 0.96 0.06 0.13 0.84 0.60 0.76 0.39 0.79 0.4740 0.96 0.92 0.98 0.97 0.05 0.11 0.86 0.67 0.79 0.45 0.81 0.5450 0.96 0.93 0.98 0.97 0.04 0.10 0.88 0.72 0.82 0.52 0.84 0.60100 0.97 0.96 0.98 0.98 0.03 0.06 0.91 0.84 0.87 0.69 0.89 0.75200 0.98 0.98 0.99 0.99 0.02 0.04 0.93 0.91 0.91 0.83 0.92 0.86Table 6: Simulation Results under Generalized Pareto Distribution.28
Proofs
Proof of Lemma 1.
This result follows from standard arguments for DSIC auction design. See, forexample, Chapter 4 in B¨orgers (2015).
Proof of Lemma 2.
By definition, φ F θ p θp p F qq “ θp ´ ´ F θ p θp p F qq f θ p θp p F qq “ θ ˆ p p F q ´ ´ F p p p F qq f p p p F qq ˙ “ . Thus p p F θ q “ θp p F q . The Myerson auction under F θ is q F θ i p v q “ t v i ą max i ‰ i v i , v i ą θp p F qu . The optimal revenue is R p F θ q “ k ÿ i “ ż R k ` t v i ą max i ‰ i v i , v i ą θp p F qu ˆ v i ´ ´ F θ p v i q f θ p v i q ˙ k ź i “ f θ p v i q d v “ k ÿ i “ ż R k ` " v i θ ą max i ‰ i v i θ , v i θ ą p p F q * θ ˆ v i θ ´ ´ F p v i θ q f p v i θ q ˙ k ź i “ f ´ v i θ ¯ d ˆ θ v ˙ “ θR p F q . Proof of Lemma 3.
Using the revenue equivalence result in Lemma 1, we have t i p θ v , θ w q “ θv i q i p θ v , θ w q ´ ż θv i q i pp u, θ v ´ i q , θ w q du “ θq i p θ v , θ w q ´ θ ż v i q i pp u, v ´ i q , w q du “ θt i p v , w q . Proof of Proposition 1.
The proof follows closely the proof of Proposition 3 in Stoye (2009). Definethe prior ˜ π ˚ P ∆ ` F X ˘ by ˜ π ˚ “ ś x P X π ˚ . By construction, the marginals of ˜ π ˚ are all identical to π ˚ , while the states t F x u are mutually independent. We want to show that p κ ˚ , ˜ π ˚ q forms a Nashequilibrium. For x ‰ x , the independence between F x and F x (under prior ˜ π ˚ ) implies that W x is uninformative about F x . Thus κ ˚ x is a best response against ˜ π ˚ . Next, given that the auctioneerpicks κ ˚ x , the conditional regret ratio is r ´ κ ˚ x , ˜ F ¯ . By the definition of κ ˚ x , this regret ratio equalsto the one from applying q ˚ to the tuple p V , W x q . Thus nature is best responding by using anyprior that has marginal distributions equal to π ˚ .29 roof of Propsition 3. Due to Proposition 2, the regret ratio is the same for all θ . So we can let θ “ F . Thus the denominator of the regret ratio can be omitted.The problem is to find the allocation q that maximizes the revenue:max q P Q R p q, F q “ k ÿ i “ E r q i p V , W q φ F p V i qs s.t. q i P r , s , ÿ i q i ď
1, and q i scale-invariant . We can restrict the q i ’s to be either 1 or 0 because a q i taking values in p , q can never be optimalunless there are equal bids, which happens with probability zero. The maxmization can be achievedin two steps. In the first step, we solve the maxmization of (9) for each q i without imposing therestriction that ř i q i ď
1. By monotonicity of q i in V i and scale-invariance, the solution is of theform t V i ě ρ F p V ´ i , W qu , where ρ F is homogeneous of degree 1. ρ F is symmetric and does notdepend on i due to the symmetry in the maximization problem. Then in the second step we canpick among the non-zero q i ’s a unique one to be 1. Since φ F is increasing, we should choose the i with the largest V i , which leads to the expression t V i ą max i ‰ i V i u . Notice that this expressionsatisfies the monotonicity and scale-invariance properties. Combining the two steps, we get theexpression (8).For the last statement, consider the average bid auction defined by Equation (11). When β “ p p F q{ µ p F q is the true price-mean ratio, β ¯ S ´ i is consistent for p p F q as min t k, n u Ñ 8 . In thiscase, the regret ratio of the AB- β auction converges to 1. Now since the auction with ρ F maximizesthe regret ratio among all equivariant auctions, its regret ratio must also converge to 1. This means ρ F must converges in probability to the true optimal price p p F q since ρ F p V ´ i , W q is independentof V i . Proof of Proposition 4.
Consider a one-dimensional regular parametric submodel with parameter β . The PDF and CDP are parametrized as f p¨ , η q and F p¨ , η q respectively. The true probabilitydistribution is indexed by η . The parameter p p F q is implicitly defined through p p η q f p p p η q , η q ´ p ´ F p p p η q , η qq “ . Denote the true optimal price by p “ p p η q . By the implicit function theorem, the derivative of p with respect to η is ddη p p η q| η “ η “ ´ F η p p , η q ` p f η p p , η q f p p , η q ` p f p p p , η q . (16)The goal is to turn the above expression into a linear function of the score function s p¨ , η q “ f η p¨ ,η q f p¨ ,η q .The denominator in Equation (16) is constant when the true distribution f p¨ , η q is fixed, thus can30e ignored. The first term in the numerator, F η , can be written as F η p p , η q “ BB η ż r v,p s p u q f p u, η q du ˇˇˇ η “ η “ ż r v,p s p u q f η p u, η q du “ E η “ r v,p s p V q s p V, η q ‰ , which is a continuous, linear functional of s p¨ , η q . The second term in the numerator is p f η p p , η q ,where p is a constant. The term f η p p , η q “ s p p , η q f p p , η q , where f p p , η q is again a constant. It boils down to the term s p p , η q , which can be thought ofas the Dirac delta δ p applied to the score s p¨ , η q , where δ p p s q evaluates a function s at p . Thefunctional δ p is indeed linear. However, it is not bounded in the Hilbert space L p F p¨ , η qq , thusnot continuous. Hence the RHS of Equation (16) is not a continuous, linear operator of the scorefunction. Thus the parameter p p F q is not pathwise differentiable. Proof of Proposition 5.
When the hazard rate is concave, so is the virtual valuation function φ .By Jensen’s inequality, we have φ p p q “ “ E r φ p V qs ď φ p µ q . Thus p ď µ . For the second result,the lower bound follows directly from Proposition 7 in Schweizer and Szech (2019). For the upperbound, we have µ p F q “ ż ´ F p v q dv ě p p F qp ´ F p p p F qqq ě p p F qp ´ λ q { λ , where the last inequality follows from Lemma 3 in Schweizer and Szech (2019). Proof of Proposition 6.
By the assumption on ˆ p EM , we know ˜ β is n { -consistent for p { µ and n { p ˜ β ´ p { µ q converges in distribution to a absolute continous random variable. For ˆ β , considerany (cid:15) ą
0, we have P ´ n { | ˆ β ´ p { µ | ě (cid:15) ¯ “ P ´ | ˆ β ´ p { µ | ě n ´ { (cid:15) ¯ ď P ´ | ˜ β ´ p { µ | ě n ´ { (cid:15) ¯ ` P ˆ | ˜ β ´ p { µ | ă max ď l ď L n | b l ´ b l ´ | ˙ , where the last inequality follows from the fact that | ˜ β ´ p { µ | ď | ˆ β ´ p { µ | ùñ | ˆ β ´ p { µ | ă max ď l ď L n | b l ´ b l ´ | .
31y the condition n { max ď l ď L n | b l ´ b l ´ | Ñ
0, we have P ˆ | ˜ β ´ p { µ | ă max ď l ď L n | b l ´ b l ´ | ˙ “ P ˆ n { | ˜ β ´ p { µ | ă n { max ď l ď L n | b l ´ b l ´ | ˙ Ñ . So ˆ β is n { -consistent. Then ˆ p ˆ β is also n { -consistent since n { p ˆ p ˆ β ´ p q “ n { p ˆ β ¯ W n ´ p p { µ q ¯ W n ` p p { µ q ¯ W n ´ p q“ n { p ˆ β ´ p { µ q ¯ W n ` n { p p { µ qp ¯ W n ´ µ q “ O p p q ..