Revelation Gap for Pricing from Samples
aa r X i v : . [ c s . G T ] F e b Revelation Gap for Pricing from Samples
Yiding Feng ∗ Jason D. Hartline † Yingkai Li ‡ Abstract
This paper considers prior-independent mechanism design, in which a single mechanismis designed to achieve approximately optimal performance on every prior distribution from agiven class. Most results in this literature focus on mechanisms with truthtelling equilibria,a.k.a., truthful mechanisms. Feng and Hartline (2018) introduce the revelation gap to quan-tify the loss of the restriction to truthful mechanisms. We solve a main open question left inFeng and Hartline (2018); namely, we identify a non-trivial revelation gap for revenue maxi-mization.Our analysis focuses on the canonical problem of selling a single item to a single agent withonly access to a single sample from the agent’s valuation distribution. We identify the sample-bidmechanism (a simple non-truthful mechanism) and upper-bound its prior-independent approx-imation ratio by 1.835 (resp. 1.296) for regular (resp. MHR) distributions. We further provethat no truthful mechanism can achieve prior-independent approximation ratio better than1.957 (resp. 1.543) for regular (resp. MHR) distributions. Thus, a non-trivial revelation gap isshown as the sample-bid mechanism outperforms the optimal prior-independent truthful mech-anism. On the hardness side, we prove that no (possibly non-truthful) mechanism can achieveprior-independent approximation ratio better than 1.073 even for uniform distributions.
One important research direction in modern computer science focuses on multi-party computation.Two fundamental concerns in this area are (i) who should be doing what part of the computation;and (i) what are their incentives to do it correctly. The second concern has been studied extensivelyin the economics field of mechanism design. For the first concern, however, the system design fieldand the mechanism design field have different high-level guidelines. The end-to-end argument (cf.Saltzer, Reed, and Clark, 1984) – a long-standing principle in system design – suggests that thecomputation should be done where the data is, i.e., in a decentralized fashion. On the other hand,due to revelation principle (see next paragraph), the mechanism design literature favors systemswhere the entire computation is done by a center with other participants truthfully reporting theirportion of the input data to the optimization. Addressing this discrepancy, in this paper, we arguethat such decentralization idea from the system design field is beneficial even in purely economicterms when robust mechanisms are desired.Revelation principle, a seminal observation in mechanism design suggests that if there is a mecha-nism with good equilibrium outcome, there is a mechanism which achieves the same outcome in atruthtelling equilibrium. This constructed mechanism asks agents to report true preferences, sim-ulates the agent strategies in the original mechanism, and outputs the outcome of the simulation. ∗ Department of Computer Science, Northwestern University. Email: [email protected] . † Department of Computer Science, Northwestern University. Email: [email protected] . ‡ Department of Computer Science, Northwestern University. Email: [email protected] . Main Results
In this paper, we focus on revenue maximization in a canonical single-item environment for a singleagent with a single sample access, i.e., the agent’s value is drawn from an unknown distribution butthe mechanism can access a single sample (independent to agent’s value) from that distribution(cf. Dhangwatnotai, Roughgarden, and Yan, 2015; Allouah and Besbes, 2019). The agent knowsher private valuation and the distribution for valuation, but she does not know the sample of themechanism. Our main theorem identifies a non-trivial revelation gap for revenue maximization inthis model. This theorem follows from three results. First, we introduce the (non-truthful) sample-bid mechanism and obtain an upper bound of its prior-independent approximation ratio. Second,we obtain a lower bound of the optimal prior-independent approximation ratio among all possi-ble mechanisms. Third, we show that any truthful mechanism is equivalent to a sampled-basedpricing mechanism introduced by Allouah and Besbes (2019) where the authors lower-bound andupper-bound the optimal prior-independent approximation ratio among all sample-based pricingmechanisms. See Table 1 for a summary of all three results. Since the prior-independent approx-imation ratio of the sample-bid mechanism is strictly better than the optimal prior-independentapproximation ratio among all truthful mechanisms, we immediately get our non-trivial revelation We impose a technical assumption (i.e. scale-invariant) to the class of truthful mechanisms, which is common inprior-independent mechanism design (Allouah and Besbes, 2018, 2019; Hartline, Johnsen, and Li, 2020).
Class of truthful mechanisms Class of all mechanismsRegular dists. MHR dists. Regular dists. MHR dists.Upper bound 1.996 ( ∗ ) ( ∗ ) ( § ) ( † ) Lower bound 1.957 ( ∗ ) ( ∗ ) ( ‡ )( ∗ ) Allouah and Besbes (2019) and Lemma 7.2; ( § ) Theorem 4.1; ( † ) Theorem 5.1; ( ‡ ) Theorem 6.1. gap for revenue maximization.In the model of a single agent with single-sample access, the class of non-truthful mechanisms isrich, which includes fairly complicated mechanisms. For example, mechanisms can ask agents toreports both her value and prior; or include multiple rounds of communication between seller andagent who sequentially reveal their private information. Nonetheless, our upper bound of theoptimal prior-independent approximation ratio is attained by a simple non-truthful mechanism – sample-bid mechanism defined as follow. • Sample-bid mechanism:
Given parameter α and sample s , the sample-bid mechanism solicitsa non-negative bid b ≥
0, charges the agent α · min { b, s } , and allocates the item to the agentif b ≥ s .From the agent’s perspective, she reports a bid to compete for the item against a random samplerealized from the same valuation distribution; and regardless of whether she wins or loses, she willalways be charged α · min { b, s } . In fact, the agent’s optimal bidding strategy could be overbiddingor underbidding, depending on the value as well as the distribution. The sample-bid mechanismhas the similar format as the Becker–DeGroot–Marschak method (Becker, DeGroot, and Marschak,1964) which has been studied and implemented in experimental economics for understanding agents’perception of the random event.In order to beat the optimal prior-independent approximation ratio among all truthful mechanisms,we need to show the approximation for the sample-bid mechanism is strictly better than 1 . < . < e / ( e − for MHR distributions. However, most approxima-tion techniques and results for non-truthful mechanisms in the literature only provide similar oror larger constants – for instance, smoothness property, permeability, and revenue covering prop-erty in price of anarchy (cf. Roughgarden, Syrgkanis, and Tardos, 2017; D¨utting and Kesselheim,2015; Hartline, 2016, see more discussion in related work). One the other hand, analyzing theapproximation of truthful mechanisms is relatively easier. In revenue maximization, one analysisapproach used extensively for truthful mechanisms is the revenue curve reduction (see next para-graph). This approach has lead to tight or nearly tight results in both prior-independent approx- Recall that the agent knows the distribution of the sample but does not know its realization. Feng and Hartline (2018) bypass this challenge in their revelation gap for welfare maximization by considering amodel where the all-pay auction (cf. Maskin, 2000) achieves prior-independent approximation ratio 1, i.e., it is indeedthe Bayesian optimal mechanism. Thecrucial observation for proving this lower bound is that for pointmass distributions, the agentperfectly knows the seller’s sample. Thus, she can strategically imitate the behavior of the values inother distributions. This restricts the seller’s ability to extract revenue from the agent, which leadsto a prior-independent approximation ratio at least 1.073 even on the restricted subclass of MHRdistributions (in fact, even on uniform distributions). Our lower bound also suggests that it willbe non-trivial to identify the non-truthful mechanism which attains the optimal prior-independentapproximation ratio.It should be noted that our better-performing non-truthful prior-independent mechanisms do notcome without drawbacks relative to truthful prior-independent mechanisms. Elegantly, truthfulprior-independent mechanisms do not require prior knowledge by any party. In contrast, non-truthful prior-independent mechanisms generally require some knowledge of the prior on the partof the agents. From this perspective, our results show that a seller is able to extract strictly higherrevenue from the agent by taking advantage of information that the agent possesses and is able tostrategize with respect to.
Important Directions
Despite the practical importance of non-truthful mechanisms, the literature on mechanism design al-most exclusively considers the design of truthful mechanisms. Thus, the most general direction fromthis paper is to systematically build a theory for the design of non-truthful mechanisms with good Such mechanisms are designed and analyzed in non-parametric implementation theory – a line of research ineconomics, see the survey of Jackson (2001) and further discussion in the related work section.
Related Work
Prior-independent mechanism design, as a standard framework for understanding the robustness ofmechanisms, has been applied to single-dimensional mechanism design (Dhangwatnotai et al., 2015,citealpRTY-12, Fu et al., 2015,Allouah and Besbes, 2018,Feng and Hartline, 2018, Hartline et al.,2020), multi-dimensional mechanism design (Devanur et al., 2011,Roughgarden et al., 2020,Goldner and Karlin,2016), makespan minimization (Chawla, Hartline, Malec, and Sivan, 2013), mechanism design forrisk-averse agents (Fu, Hartline, and Hoy, 2013), and mechanism design for agents with interdepen-dent values (Chawla, Fu, and Karlin, 2014a). Except Fu, Hartline, and Hoy (2013) and Feng and Hartline(2018), all other results focus on truthful mechanisms.There is a significant area of research studying mechanism design with sample access from the dis-tribution of agents’ preference, which has two regimes – small number of samples, and large numberof samples. In the former regime, literature studies the approximation of mechanisms with a single-sample access (Azar, Kleinberg, and Weinberg, 2014; Dhangwatnotai, Roughgarden, and Yan, 2015;Allouah and Besbes, 2019; Feng, Hartline, and Li, 2019; Correa, D¨utting, Fischer, and Schewior,2019; D¨utting, Fusco, Lazos, Leonardi, and Reiffenh¨auser, 2020; Correa, Cristi, Epstein, and Soto,2020), and mechanisms with two-sample access (Babaioff, Gonczarowski, Mansour, and Moran,2018; Daskalakis and Zampetakis, 2020). In the latter regime, the goal is to minimize the samplecomplexity, i.e., number of sample to achieve (1 − ǫ )-approximation (e.g. Cole and Roughgarden,2014; Morgenstern and Roughgarden, 2015; Huang et al., 2018; Gonczarowski and Weinberg, 2018;5uo et al., 2019; Hartline and Taggart, 2019). Except Hartline and Taggart (2019), all other re-sults focus on truthful mechanisms.Price of anarchy studies how classic non-truthful mechanisms (e.g. first-price auction, all-payauction) approximate the optimal welfare. Syrgkanis and Tardos (2013) introduce a smoothnessproperty defined on mechanisms and give an analysis framework based on this property. Withthis smoothness framework, the authors upper-bound the welfare-approximation of the first-priceauction by e / ( e − , and the welfare-approximation of the all-pay auction by 2. These two re-sults are later tightened by Christodoulou, Sgouritsa, and Tang (2015) for the all-pay auction andHoy, Taggart, and Wang (2018) for the first-price auction using some mechanism-specific argu-ments. Hartline, Hoy, and Taggart (2014) introduce a geometric framework for analyzing the priceof anarchy for both welfare and revenue. As the instantiations of the framework, authors upper-bound the revenue approximation of the first-price auction with individual monopoly reserve by e / ( e − . D¨utting and Kesselheim (2015) show that bounds from these analysis frameworks aretight up to constant factors.The literature on non-parametric implementation theory considers the same question as prior-independent mechanism design but allows mechanisms where agents cross-report their beliefs onother agents’ values (e.g., Jackson, 2001). Caillaud and Robert (2005) introduce a dynamic auc-tion for single-item multi-agent settings which is able to implement the Bayesian revenue optimalauction (Myerson, 1981) without the knowledge of agents’ distribution. Dasgupta and Maskin(2000) introduce a generalization of VCG auction for multi-agent interdependent value settings. In this auction, agents are asked to submit a function that gives a bid for every possible valu-ation of the other agents. Though this auction requires no knowledge of agents’ distributions,Dasgupta and Maskin (2000) show that it is Bayesian welfare-optimal under mild assumptions.Azar et al. (2012) study how to use scoring rules to learn agents’ distribution and implementthe auction based on this learned distribution. All results above suggest that in the multi-agentsettings, there exist complicated and arguably impractical non-truthful mechanisms whose prior-independent approximation equal or are arbitrarily close to 1. However, as we mentioned earlier,in the model of a single-agent with single-sample access, we provide a lower bound on the optimalprior-independent approximation without any restriction on mechanisms.
Model.
This paper focuses on the single-item revenue-maximization problem with a single agent.The agent has a private value v drawn from a valuation distribution (a.k.a. prior) F supported on[ v, v ]. we assume that distribution F has positive density f every where in the support. Givenallocation x and payment p , the utility of the agent is vx − p .We consider the prior-independent mechanism design with a single sample access. Namely, theseller does not know the valuation distribution F , but has a single sample s drawn from F . Theagent knows the valuation distribution F but does not observe the sample s , and the value v ofthe agent is independent of the sample s . A mechanism M = (˜ x, ˜ p ) includes an allocation rule˜ x : R × R → [0 ,
1] mapping from the agent’s bid b and the sample s to the allocation probabilityof the item; and a payment rule ˜ p : R × R → R + mapping from the agent’s bid b and the sample s to the payment charged from the agent. Let ˜ x ( b, F ) = E s ∼ F [˜ x ( b, s )], ˜ p ( b, F ) = E s ∼ F [˜ p ( b, s )] be the In general, there is no incentive compatible mechanism which outputs the welfare-optimal outcomes in interde-pendent value settings. s drawn from distribution F .The seller first announce the mechanism M = (˜ x, ˜ p ) to the buyer, and then the sample s and value v are realized from distribution F . The agent report a bid b based on her private value v , and theseller implements the mechanism M with input b and sample s . We assume that the seller has fullcommitment power on implementing the mechanism.Given a mechanism (˜ x, ˜ p ) and distribution F , the best response of the agent is b ( · , F ) : R → R which maximizes her expected utility, i.e., for every value v , b ( v, F ) ∈ argmax b v · ˜ x ( b, F ) − ˜ p ( b, F ). A mechanism (˜ x, ˜ p ) is incentive compatible (IC) if reporting the agent’s value truthfully is her bestresponse, i.e., b ( v, F ) = v for all v and F . A mechanism (˜ x, ˜ p ) is individual rational (IR) if theagent’s utility under her best response is non-negative, i.e, max b v · ˜ x ( b, F ) − ˜ p ( b, F ) ≥ v and F . For any mechanism (˜ x, ˜ p ), let x ( v, F, s ) = ˜ x ( b ( v, F ) , s ) be the interim allocation of value v given dis-tribution F and sample s when the agent follows her best response, and let p ( v, F, s ) = ˜ x ( b ( v, F ) , s )be the interim payment. Moreover, denote x ( v, F ) = E s ∼ F [ x ( v, F, s )] and p ( v, F ) = E s ∼ F [ p ( v, F, s )]as the expected interim allocation and payment. We often omit F in the notation if it is clear fromthe context.The revenue Rev F [ M ] of a mechanism M = ( x, p ) on distribution F is the expected paymentwhen the agent plays her best response, i.e., E v ∼ F [ p ( v, F )]. We evaluate mechanisms by the prior-independent approximation ratio. Definition 2.1.
The prior-independent approximation ratio of a mechanism M over a class ofdistributions DISTS is defined as Γ ( M , DISTS ) , max F ∈ DISTS
Rev F [OPT F ] Rev F [ M ] where Rev F [OPT F ] , max p (1 − F ( p )) p is the optimal revenue for distribution F (cf. Myerson,1981). Revenue Curve.
For any distribution F , let q ( v, F ) = 1 − F ( v ) be the quantile for the distri-bution, and v ( q, F ) be the value v such that q = 1 − F ( v ). Here we introduce the revenue curve inquantile space (cf. Bulow and Roberts, 1989), which is a useful tool in the revenue analysis. Definition 2.2.
For any valuation distribution F , the revenue curve R ( q, F ) of the agent is amapping from any q ∈ [0 , to the optimal revenue from an agent with value drawn from F subjectto the constraint that the item is allocated with ex ante probability q . In the later analysis in the paper, when F is clear from the context, we omit it in the notationand only use R ( q ) to represent the revenue curve and q ( v ) to represent the quantile of value v . Let φ ( v ) = v − − F ( v ) f ( v ) be the virtual value of the agent. Definition 2.3.
An valuation distribution F is regular if the virtual value of the agent is weaklyincreasing. When there are multiple bids maximizing the utility of the agent, we allow the agent to choose any bid maximizingher utility. The revenue guarantee we obtained in this paper holds even when the agent can break tie and choose thebid minimizing the revenue of the seller. Note that the utility of the agent can be negative for some realization of the sample s , but in expectation it mustbe non-negative. heorem 2.1 (Myerson, 1981) . A distribution F is regular if and only if the corresponding revenuecurve R ( q, F ) is concave. Theorem 2.2 (Myerson, 1981) . For any distribution F and any mechanism with interim allocationand payment rule x ( v ) , p ( v ) , the expected revenue of the seller equals the expected virtual value ofthe agent plus the payment of the lowest value v , i.e., E v ∼ F [ p ( v )] = E v ∼ F [ x ( v ) φ ( v )] + p ( v ) . Finally, we define the monopoly reserve and monopoly quantile of the agent given the revenuecurve R . Definition 2.4.
The monopoly quantile of the agent is q m = argmax q R ( q ) , and the monopolyreserve of the agent is v m = R ( q m ) / q m . In this section, we introduce the main mechanism considered in this paper, the sample-bid mecha-nism.
Definition 3.1 (sample-bid mechanism) . Given parameter α and sample s , the sample-bid mech-anism solicits a non-negative bid b ≥ , charges the agent α · min { b, s } , and allocates the item tothe agent if b ≥ s . In the sample-bid mechanism, the agent reports her bid without knowing the realization of thesample. From her perspective, the utility u ( v, b, F ) for her who has value v , reports bid b , andcompetes with sample s ∼ F is u ( v, b, F ) = v · F ( b ) | {z} Pr s ∼ F [ s ≤ b ] − αb · (1 − F ( b )) | {z } payment when s ≥ b − α Z max { b,v } v tdF ( t ) | {z } payment when s ≤ b Note that reporting bid equal to zero, the utility of agent is zero. Thus, sample-bid mechanismisindividually rational.
Lemma 3.1.
The sample-bid mechanism is individually rational.
On the other hand, reporting bid equal to agent’s value is not the best response in general. Weprovide a characterization of agent’s optimal bid as follows.
Lemma 3.2.
In the sample-bid mechanism, given any parameter α and distribution F , the optimalbid b ( v, F ) for the agent with value v satisfies the constraint that v = α · − F ( b ( v, F )) f ( b ( v, F )) , (1) or b ( v, F ) ∈ { , ∞} . Ties are broken according to the utility of the agent.Proof. The agent’s utility from reporting bid b is u ( v, b, F ) = v · F ( b ) − αb (1 − F ( b )) − α Z max { b,v } v tdF ( t ) In this paper, we break tie in favor of smaller quantile. Note that all the results are not affected by the tiebreaking rule. b , if the optimal bid is obtained in the interior,we have f ( b ) (cid:18) v − α · − F ( b ) f ( b ) (cid:19) = 0as a necessary condition for the optimality of the bid b . Otherwise, the optimal bid is obtained onthe boundary, where b ( v, F ) ∈ { , ∞} .Note that there might exist multiple bids b that satisfies the constraint (1) in Lemma 3.2. In thatcase, the agent chooses the bid which satisfies (1) and maximizes her utility. Another observation(Lemma 3.3) of the sample-bid mechanism is that the expected revenue of the seller scales linearlywith the valuation distribution. Since the optimal revenue scales linearly with the valuation distri-bution as well, to analyze the prior-independent approximation ratio of the sample-bid mechanism,we can focus on the valuation distributions such that the optimal revenue is normalized to 1. Lemma 3.3.
Denote by r the revenue of the sample-bid mechanism with any parameter α andany valuation distribution F † . For any ρ > and distribution F ‡ such that F ‡ is F † scaled by ρ , i.e., F † ( v ) = F ‡ ( ρv ) for all v , the revenue of the sample-bid mechanism with parameter α anddistribution F ‡ is ρr .Proof. First we show that for any value v , the bid of value v given distribution F † is equivalentto the bid of value ρv given distribution F ‡ scaled by ρ . The reason is that F † ( v ) = F ‡ ( ρv ) and f † ( v ) = ρf ‡ ( ρv ). Therefore, by Lemma 3.2, the first order condition implies that the optimal bidsatisfies b ( ρv, F ‡ ) = ρ · b ( v, F † ). Moreover, the payment satisfies˜ p ( ρb, F ‡ ) = αρb · (1 − F ‡ ( ρb )) + α Z ρb t d F ‡ ( t )= ρ ( αb · (1 − F † ( b ) + α Z b t d F † ( t )) = ρ · ˜ p ( b, F † ) . By taking expectation over the valuation, the expected revenue is scaled by ρ as well.We finish this section by providing two simple monotonicity properties of the sample-bid mechanismand defer other more complicated characterizations required in our analysis to the later sections. Lemma 3.4.
In the sample-bid mechanism, given any parameter α and distribution F , the expectedpayment for bid b is monotonically non-decreasing in b .Proof. By definition, the expected payment ˜ p ( b, F ) of bid b over the randomness of the sample s ∼ F is ˜ p ( b, F ) = αb · (1 − F ( b )) + α Z max { b,v } v t d F ( t )Taking the derivative with respect to bid b , we have ∂ ˜ p ( b, F ) ∂b = α (1 − F ( b )) − αbf ( b ) + αbf ( b ) = α (1 − F ( b )) ≥ . which finishes the proof. 9 emma 3.5. In the sample-bid mechanism, given any parameter α and distribution F , the optimalbid b ( v, F ) is monotonically non-decreasing in value v .Proof. By Myerson (1981), the equilibrium allocation of the agent is non-decreasing in value v .Moreover, given the auction format, the equilibrium allocation of the agent is increasing in the bid,and thus the optimal bid b ( v, F ) is non-decreasing in the value v . In this section, we analyze the prior-independent approximation ratio of the sample-bid mechanismover the class of MHR distributions.
Definition 4.1.
A distribution F is MHR if the hazard rate f ( v )1 − F ( v ) is monotone non-decreasingin v . Theorem 4.1.
For the sample-bid mechanism with α = 0 . , the prior-independent approximationratio over the class of MHR distributions is between [1.295, 1.296]. The lower bound in Theorem 4.1 is shown in the following example.
Example 4.2.
For the sample-bid mechanism with α = 0 . , let F be the valuation distributionsuch that F ( v ) = 1 − e − v for v ∈ [0 , . and F ( v ) = 1 for v ∈ [0 . , ∞ ) . It is easy to verifythat F is MHR. Moreover, the optimal revenue is . while the expected revenue of the sample-bid mechanism, which equals the expected revenue of posting a price equal to . fraction of theexpected welfare, is . . Thus, the prior-independent approximation ratio of the sample-bidmechanism with α = 0 . is at least . . Before the proof of the upper bound in Theorem 4.1, we first introduce a characterization of theagent’s optimal bid when the sample distribution F is MHR; and a technical property for MHRdistributions. Lemma 4.2.
In the sample-bid mechanism, given any parameter α and MHR distribution F , theoptimal bid b ( v, F ) for the agent with value v is b ( v, F ) = (cid:26) if v < α E s ∼ F [ s ] , ∞ otherwise.Proof. By the proof of Lemma 3.2, the derivative of the utility given the bid b is f ( b ) (cid:18) v − α · − F ( b ) f ( b ) (cid:19) , where the sign of the above expression flips from negative to positive only once when the bid b increases from 0 to infinity since F is MHR. Thus the utility is a quasi-convex function of the bid,which implies that the maximum utility is attained at extreme points, i.e., bid 0 or ∞ . Note thatthe utility for bidding 0 is always 0, while the utility for bidding ∞ is u ( v, ∞ , F ) = v − α E s ∼ F [ s ].Hence, the agent bid ∞ if and only her value v is at least α E s ∼ F [ s ]. Lemma 4.3 (Allouah and Besbes, 2019) . For any MHR distribution with any pair of quantile andvalues ( v , q ) , ( v , q ) such that q = q ( v ) ≤ q = q ( v ) and v ≥ v . Then for any v ≥ v , wehave q ( v ) ≥ q · e v − v v − v · ln( q q ) . emma 4.4. The expected value for any MHR distribution with monopoly quantile q m is w ≥ q m − q m · ln q m .Proof. The expected value of the agent is Z ∞ q ( v ) d v ≥ Z qm e vq m · ln q m d v = 1 q m · ln q m ( e ln q m − e ) = q m − q m · ln q m , where the inequality holds by applying Lemma 4.3 with q = q m , v = q m and q = 1 , v = 0.Now, we are ready to show Theorem 4.1. Proof of the upper bound in Theorem 4.1.
Fix any MHR distribution F . Let w , E v ∼ F [ v ]. Notethat by Lemma 4.2, our mechanism is equivalent to posting price αw to the agent. Next weanalyze the approximation ratio by considering the cases αw ≥ v m and αw < v m and optimize theparameter α such that the approximation ratio of both cases coincide. Recall that it is without lossof generality to normalize the expected revenue of the optimal mechanism to 1, i.e., q m · v m = 1.First we consider the case when αw < v m = / q m . By Lemma 4.4, we have w ≥ q m − q m · ln q m and bycombining Lemma 4.3 with ( v , q ) = ( v m , q m ) and ( v , q ) = (0 , q ( αw ) ≥ e α ( q m − .Thus, the expected revenue in this case is αw · q ( αw ) ≥ α ( q m − q m · ln q m · e α ( q m − . Then we consider the case when αw ≥ v m = / q m . In this case, combining Lemma 4.3 with( v , q ) = ( w, q w ) and ( v , q ) = ( v m , q m ), where q w ≥ / e is the quantile of the welfare (seeBarlow and Marshall, 1965), for any value v ≥ v m , we have q ( αw ) ≥ q m · e αw − vmw − vm · ln( qwqm ) . Thus theexpected revenue is αw · q ( αw ) ≥ αw · q m · e αw − vmw − vm · ln( qwqm ) ≥ αw · q m · e αw − / qm w − / qm · ln( e · qm ) . By setting α = 0 .
824 and numerically evaluating the above expressions for all possible values of w and q m with respective to the given constraints, we have that the expected revenue in both casesare at least 0 . . In this section, we analyze the prior-independent approximation of the sample-bid mechanism overthe class of regular distributions.
Theorem 5.1.
For the sample-bid mechanism with α = 0 . , the prior-independent approximationratio over the class of regular distributions is between [1 . , . . The lower bound in Theorem 5.1 is shown in the following example.
Example 5.1.
For the sample-bid mechanism with α = 0 . , let F be the valuation distributionsuch that F ( v ) = . v − . for v ∈ [1 , ∞ ) . It is easy to verify that F is regular. Moreover, theoptimal revenue is while the expected revenue of the sample-bid mechanismis . . Thus, theprior-independent approximation ratio of the sample-bid mechanism with α = 0 . is at least . .
11n Section 5.1, we introduce some technical characterizations of the sample-bid mechanism whichwill be used in the subsequent analysis. In Sections 5.2 and 5.3, we study the prior-independent ap-proximation ratio of the sample-bid mechanism over the class of regular distributions with monopolyquantile q m ≥ .
62 and q m ≤ .
62 respectively. By Lemma 3.3, without loss of generality, we re-strict our attention to the class of regular valuation distributions where the optimal revenue forthe distributions is exactly 1 (i.e., v m · q m = 1), and then lower-bound the expected revenue of thesample-bid mechanism with α = 0 . F , we define avalue threshold v ∗ ( F ) as the smallest value whose optimal bid is at least monopoly reserve v m ( F ),i.e., v ∗ ( F ) , inf { v : b ( v, F ) ≥ v m ( F ) } Denote q ( v ∗ ( F ) , F ) by q ∗ ( F ). By Lemma 3.4 and Lemma 3.5, the expected revenue Rev F (SB) ofthe sample-bid mechanism SB for valuation F can be lower-bounded as follows, Rev F (SB) = Z p ( v ( q, F ) , F ) dq ≥ p ( v ∗ ( F ) , F ) · q ∗ ( F ) + Z q ∗ ( F ) p ( v ( q, F ) , F ) dq. where p ( v, F ) is the expected payment of the agent, with value v and valuation distribution F , inthe sample-bid mechanism. We then analyze p ( v ∗ ( F ) , F ), q ∗ ( F ), and p ( v ( q, F ) , F ) for q ≥ q ∗ ( F )by providing lower bounds as the functions of q m ( F ) and other some parameters of F . Finally, bynumerically evaluating the value of lower bounds for all possible possible parameters, we concludethat the expected revenue in the sample-bid mechanism for all regular distribution (with monopolyrevenue 1) is at least 0.545, which implies the prior-independent approximation ratio / . ≈ .
835 of the sample-bid mechanism in Theorem 5.1. The details for discretizations and numericalevaluations can be found in Appendix A. Note that the bounds for the approximation ratio ofthe sample-based pricing mechanisms in Allouah and Besbes (2019) are also obtained by numericalanalysis, which requires solving a relatively more complicated dynamic program. In contrast, ournumerical analysis only requires brute force enumeration of a few parameters.As we discussed in Section 2, every valuation distribution F can be represented by its inducedrevenue curve R where R ( q ) , q F − (1 − q ) for all q ∈ [0 , In this subsection, we introduce some technical characterizations of the sample-bid mechanismwhich will be used in the later analysis.To establish a lower bound on the expected revenue of of a truthful mechanism, a classic ap-proach – revenue curve reduction – (e.g. Alaei, Hartline, Niazadeh, Pountourakis, and Yuan, 2019;Allouah and Besbes, 2018) is as follows: (i) start with an arbitrary revenue curve R , (ii) con-vert it to another revenue R with closed-form formula while the optimal revenue remains the Let R be the revenue curve induced by valuation distribution F . In Section 5.2, we lower-bound the expectedrevenue as a function of q m ( F ) and R (0). In Section 5.2, we lower-bound the expected revenue as a function of q m ( F ), q ( v m ( F ) / . , F ) and w , R q m ( F ) q ( v / . ,F ) R ( q ) q dq . R is at most the expected revenue for R whilethe optimal revenue remains the same, and finally (iv) evaluate the expected revenue for R forall possible parameters. In this section, we want to apply a similar approach to the sample-bidmechanism because it is a non-truthful mechanism. A new technical difficulty arises in step (iii).When comparing R and R , for truthful mechanisms, it is sufficient to study the change in theexpected payment (i.e. ˜ p ( b, R ) and ˜ p ( b, R )) for each bid b . However, for non-truthful mechanisms(e.g. sample-bid mechanism), the optimal bid of the agent changes when the revenue curve R isreplaced by R . In Lemma 5.2, we provide a characterization of optimal bid when we switch from R to R in a specific way (illustrated in Figure 1). We use it as a building block repeatedly inSection 5.2 and Section 5.3. Intuitively, the following lemma characterizes the phenomenon thatincreasing the revenue curve for high values does not affect the agent’s preference for low bids. Lemma 5.2.
In the sample-bid mechanism, consider any quantile q † ∈ [0 , and any pair ofrevenue curves R , R such that R ( q ) ≤ R ( q ) for any quantile q ≤ q † and R ( q † ) = R ( q † ) .Letting b † = R ( q † ) / q † . For any value v and any bid b ‡ ≥ b † , if an agent with value v and revenuecurve R prefers bid b † than b ‡ , i.e., u ( v, b † , R ) ≥ u ( v, b ‡ , R ) , then an agent with value v andrevenue curve R also prefers bid b † than b ‡ , i.e., u ( v, b † , R ) ≥ u ( v, b ‡ , R ) .Proof. By the construction of our mechanism, the utility of an agent who has value v , revenuecurve R and bids b is u ( v, b, R ) = v · (1 − q ( b, R )) − ˜ p ( b, R )and ˜ p ( b, v ) = αb · q ( b, R ) + α Z q ( b,R ) R ( q ) q dq. By the assumption that R ( q ) ≤ R ( q ) for any quantile q ≤ q † and b ‡ ≥ b † , we have q ( b ‡ , R ) ≤ q ( b ‡ , R ) ≤ q † . See Figure 1 for a graphical illustration. Thus,˜ p ( b ‡ , R ) − ˜ p ( b † , R ) = α · − b † · q † + Z q † min (cid:26) R ( q ) q , b ‡ (cid:27) dq ! ≤ α · − b † · q † + Z q † min (cid:26) R ( q ) q , b ‡ (cid:27) dq ! = ˜ p ( b ‡ , R ) − ˜ p ( b † , R ) . Thus, u ( b † , v, R ) − u ( b ‡ , v, R ) = v · (1 − q † ) − ˜ p ( b † , R ) − v · (1 − q ( b ‡ , R )) + ˜ p ( b ‡ , R ) ≤ v · (1 − q † ) − ˜ p ( b † , R ) − v · (1 − q ( b ‡ , R )) + ˜ p ( b ‡ , R ) = u ( b † , v, R ) − u ( b ‡ , v, R )and hence u ( b † , v, R ) ≥ u ( b ‡ , v, R ) implies u ( b † , v, R ) ≥ u ( b ‡ , v, R ). Lemma 5.3.
In the sample-bid mechanism with any parameter α ∈ [0 , , for an agent with concaverevenue curve R and value v greater than the monopoly reserve v m , she weakly prefers the bid v / α than any bid b † ∈ [ v m , v / α ] , i.e., u ( v, v / α , R ) ≥ u ( v, b † , R ) .
13 0 1 q † b † b ‡ q ( b ‡ , R ) q ( b ‡ , R ) R R Figure 1: Graphical illustration for Lemma 5.2. The gray dashed thick (resp. black solid) curve isrevenue curve R (resp. R ). The slopes of two dotted lines from (0 ,
0) are b ‡ and b † respectively. Proof.
Let F be a regular distribution. By the definition, the utility of the agent who has value v ,valuation distribution F and bids b is u ( v, b, F ) = v · F ( b ) − ˜ p ( b, F )By considering the first order condition as in Lemma 3.2, we have ∂u ( v, b, F ) ∂b = f ( b ) (cid:18) v − α · − F ( b ) f ( b ) (cid:19) . Thus, we can compute the difference between u ( v, v / α , F ) and u ( v, b, F ) for any value v ≥ v m andbid b ∈ [ v m , v / α ] as follows, u ( v, v / α , F ) − u ( v, b, F ) = Z vα b αf ( t ) (cid:18) vα − − F ( t ) f ( t ) (cid:19) dt ≥ Z vα b αf ( t ) (cid:18) t − − F ( t ) f ( t ) (cid:19) dt ≥ t − − F ( t ) f ( t ) ≥ t ≥ v m if F is regular. q m ≥ . In this subsection, we analyze the approximation ratio of the sample-bid mechanism over the classof regular distributions with monopoly quantile q m ≥ . Lemma 5.4.
For the sample-bid mechanism with α = 0 . , the approximation ratio over the classof regular distributions with monopoly quantile q m ≥ . is at most 1.835. R , let v ∗ ( R ) , inf { v : b ( v, R ) ≥ v m ( R ) } be the smallest value whose optimal bid b ( v, R ) for revenue curve R is at least the monopolyreserve v m ( R ). Since Lemma 3.5 guarantees that b ( v, R ) is weakly non-decreasing in v , v ∗ ( R ) iswell-defined, b ( v, R ) ≥ v m ( R ) for all v ≥ v ∗ ( R ), and b ( v, R ) < v m ( R ) for all v < v ∗ ( R ). Denote q ( v ∗ ( R ) , R ) by q ∗ ( R ). We decompose the proof of Lemma 5.4 by considering the following twosubregimes – Lemma 5.5 for revenue curve R with v ∗ ( R ) ≤ v m ( R ); and Lemma 5.7 for revenuecurve R with v ∗ ( R ) ≥ v m ( R ). Lemma 5.5.
Given any concave revenue curve R such that q m ( R ) ≥ . and v ∗ ( R ) ≤ v m ( R ) , therevenue of the sample-bid mechanism with α = 0 . is a . -approximation of the optimal revenue.Proof. Fix an arbitrary concave revenue curve R satisfying the requirement in the lemma statement,i.e., q m ( R ) ≥ .
62 and v ∗ ( R ) ≤ v m ( R ). Consider an arbitrary value v ≥ v ∗ ( R ). By Lemma 3.5, theoptimal bid of an agent with value v is at least v m ( R ). Thus, together with Lemma 3.4, her expectedpayment in sample-bid mechanism is at least the expected payment ˜ p ( v m ( R ) , R ) of bidding v m ( R ),and ˜ p ( v m ( R ) , R ) = 0 . v m ( R ) q m ( R ) + 0 . Z q m ( R ) R ( q ) q dq = 0 . . Z q m ( R ) R ( q ) q dq ≥ . . Z q m ( R ) 1 − q − q m ( R ) q dq = − . q m ( R ))1 − q m ( R ) . where the inequality uses the fact that (1) R is concave, which implies that R ( q ) ≥ − q − q m ( R ) for all q ≥ q m ( R ); and (2) v m ( R ) q m ( R ) is normalized to 1 for the revenue curve R . Since v ∗ ( R ) ≤ v m ( R ),each value with quantile smaller than q m ( R ) has ˜ p ( v m ( R ) , R ) as a lower bound of its payment inthe sample-bid mechanism. Thus, a lower bound of the expected revenue Rev R (SB) for revenuecurve R in the sample-bid mechanism is Rev R (SB) = Z p ( v ( q, F ) , F ) dq ≥ p ( v ∗ ( R ) , R ) · q ∗ ( R ) ≥ ˜ p ( v m ( R ) , R ) · q m ( R ) ≥ − . q m ( R )) q m ( R )1 − q m ( R )which is at least 0.545 for all q m ( R ) ≥ .
62. This finishes the proof, since we (without loss ofgenerality) consider revenue curve R with optimal revenue equal to 1, i.e., v m ( R ) · q m ( R ) = 1.Before diving into the subregime where v ∗ ( R ) ≥ v m ( R ), we provide a characterization (Lemma 5.6)of the optimal bid for concave revenue curves with monopoly quantile greater than 0 .
62. Specifically,Lemma 5.6 guarantees that b ( v, R ) = 0 for all value v < v ∗ ( R ). Lemma 5.6.
In the sample-bid mechanism with parameter α = 0 . , given any value v and anyconcave revenue curve R with q m ( R ) ≥ . , the optimal bid b ( v, R ) for an agent with value v andrevenue curve R satisfies b ( v, R ) ∈ { } ∪ [ v m ( R ) , ∞ ) . Proof.
We prove the lemma by contradiction. See Figure 2 for a graphical description of thefollowing construction. Suppose there exists an agent who has value v , revenue curve R s.t.15 0 1 q m ( R ) v m ( R ) b † q † ˆ qR R Figure 2: Graphical illustration for Lemma 5.6. The gray dashed (resp. black solid) curve is revenuecurve R (resp. R ). The slopes of two dotted lines from (0, 0) are v m ( R ) and b † respectively. q m ( R ) ≥ .
62 and strictly prefers a bid of b † ∈ (0 , v m ( R )) over all other bids. Denote q ( b † , R )by q † . Let ˆ q , − − q † R ( q † ) . Now consider another revenue curve R defined as follows, R ( q ) , (cid:26) q ∈ [0 , ˆ q ] , − q − ˆ q q ∈ [ˆ q, . By construction, R is a concave revenue curve s.t. (i) ˆ q ≥ .
62; (ii) b † ≤ / ˆ q ; (iii) R ( q ) ≤ R ( q )for all q ∈ [0 , q † ]; and (iv) R ( q ) ≥ R ( q ) for all q ∈ [ q † , R , R , q † , v and all b ‡ ≥ b † , we conclude that the optimal bid for an agentwith value v and revenue curve R is in [0 , b † ]. Furthermore, note that u ( v, b † , R ) ≥ u ( v, b † , R ) > R , and the second inequality holds by ourassumption that b † is strictly preferred for R . Hence, there exists an optimal bid in (0 , b † ] that isstrictly preferred to biding zero and weakly preferred to all other bids for R . Next we argue thatthis leads to a contradiction by considering v ≤ / ˆ q and v ≥ / ˆ q separately. Case (i) v ≤ / ˆ q : Note that for any bid b ∈ [0 , / ˆ q ], the utility u ( v, b, R ) has a closed-form expressionas follows, u ( v, b, R ) = v b (1 − ˆ q ) b (1 − ˆ q ) + 1 + 0 . (cid:18) b (1 − ˆ q ) + 1 (cid:19) . Considering the first order condition of u ( v, b, R ) with respect to bid b , after basic simplification,we have b = v . − − ˆ q . The allocation of bidding b † is the same for both revenue curves, while the payment of bidding b † is higher forrevenue curve R . q ∈ [0 . , and v ∈ [0 , / ˆ q ], we have v . − − ˆ q <
0, i.e.,bidding 0 is weakly preferred than any bid b ∈ (0 , b † ). Case (ii) v ≥ / ˆ q : Let b ‡ , v / . , and q ‡ , q ( b ‡ , R ) = . / v . Since v ≥ / ˆ q , the construction of R guarantees that b ‡ · q ‡ = 1 = R ( q ‡ ). Note that the utility u ( v, b ‡ , R ) has a closed-form expressionas follows, u ( v, b ‡ , R ) = v − vq ‡ + 0 . − ˆ q ) + 0 . q ) − . q − q ‡ )(1 − b ‡ q ‡ ) (cid:16) ˆ q − q ‡ − q ‡ log(ˆ q ) + q ‡ log( q ‡ ) (cid:17) . This leads to a contradiction since for all ˆ q ∈ [0 . , v ∈ [ / ˆ q , ∞ ), and (cid:16) v . − − ˆ q (cid:17) ∈ [0 , / ˆ q ], wehave u ( v, b ‡ , R ) ≥ u (cid:16) v, v . − − ˆ q , R (cid:17) , i.e., bidding 0 or v / . is weakly preferred than any bid b ∈ (0 , b † ).Now, we provide the approximation guarantee for revenue curve R with v ∗ ( R ) ≥ v m ( R ). Lemma 5.7.
Given any concave revenue curve R such that q m ( R ) ≥ . and v ∗ ( R ) ≥ v m ( R ) , therevenue of the sample-bid mechanism with α = 0 . is a . -approximation of the optimal revenue.Proof. The proof is done in four major steps:
Step 1- flattening the revenue curve for all quantile q ≥ q m ( R ) . Fix an arbitraryrevenue curve R satisfying the requirements in the lemma statement, i.e., q m ( R ) ≥ .
62 and v ∗ ( R ) ≥ v m ( R ). Consider another revenue curve R defined as follows (see Figure 3a for a graphi-cal illustration) R ( q ) , (cid:26) R ( q ) q ∈ [0 , q m ( R )] , q ∈ [ q m ( R ) , . We claim that the expected revenue of the sample-bid mechanism with α = 0 . R is at most that of revenue curve R . To see this, consider the virtual surplus for both revenuecurves. By our assumption that v ∗ ( R ) ≥ v m ( R ), every quantile q > q m ( R ) has negative virtualvalue R ′ ( q ) in R , bids zero (Lemma 5.6) and gains zero virtual surplus while their virtual value R ′ ( q ) becomes zero in R and thus gains zero virtual surplus as well. On the other side, everyquantile q ≤ q m ( R ) has identical virtual value by construction. We claim that the allocation foreach of these quantiles weakly decreases. To see this, note that the allocation of bidding any bid b ≥ v m ( R ) = v m ( R ) is the same for both revenue curves R and R , and the expected paymentincreases by a constant when the revenue curve R is replace by R . Thus the agent’s preferenceamong all bids b ≥ v m ( R ) is the same in both revenue curves R and R . However, the utilityof bidding b ≥ v m ( R ) is lower when the revenue curve is R , which implies that there may existvalue v such that the agent may prefer bidding 0 to bidding above the monopoly reserve in R ,while strictly prefer bidding above the monopoly reserve in R . By Lemma 5.6, the optimal bid for Note that q m ≥ .
62 implies that ˆ q ≥ . By first order condition, for revenue curve R , if (cid:16) v . − − ˆ q (cid:17) > / ˆ q , then bidding b ‡ already achieves higherutility for the agent compared to bidding below b ‡ . Thus it is sufficient to compare b ‡ with (cid:16) v . − − ˆ q (cid:17) in the casethat the latter is in [0 , / ˆ q ].
170 1 q m (a) The gray dashed (resp. black solid) curve isthe revenue curve R (resp. R ). By construction, q m ( R ) = q m ( R ).
10 1 v ∗ ( R ( i ) ) v ∗ ( R ( i + ) ) (b) The gray dashed (resp. black solid) curve is rev-enue curve R ( i )2 (resp. R ( i +1)2 ). The slopes of twodotted lines from (0, 0) are v ∗ ( R ( i +1)2 ) and v ∗ ( R ( i )2 )respectively. By construction, v ∗ ( R ( i +1)2 ) ≥ v ∗ ( R ( i )2 ).
10 1 v ∗ ( R ) (c) The gray dashed (resp. black solid) curve is rev-enue curve R (resp. R ). The slope of the dottedline from (0, 0) is v ∗ ( R ). r r b ‡ q ‡ (d) The gray dashed (resp. black solid, black dashed)curve is revenue curve R (resp. R (¯ r )4 , R ( r )4 ). Theslope of the dotted line from (0, 0) is b ‡ , i.e., the op-timal bid for an agent with value v ∗ ( R ) and revenuecurve R . By construction, v ∗ ( R ( r )4 ) ≤ v ∗ ( R )) ≤ v ∗ ( R (¯ r )4 ). b ‡ b † v ∗ (e) The gray dashed (resp. black solid, blackdashed) curve is revenue curve R (resp. R , R ( r )4 ).By construction, v ∗ ( R ) = v ∗ ( R ) ( , v ∗ ). The slopeof three dotted lines from (0, 0) are b ‡ , b † and v ∗ ,where b ‡ (resp. b † ) is the optimal bid for an agentwith value v ∗ and revenue curve R (resp. R ). ByLemma 3.2, b † ≤ b ‡ . Figure 3: Graphical illustration for Lemma 5.7.18ny value v is not in (0 , v m ( R )). Thus, we conclude that q ∗ ( R ) ≤ q ∗ ( R ) and (1) the optimal bid(as well as the allocation) for every quantile q ≤ q ∗ ( R ) in both R and R remains the same; and(2) for every quantile q ∈ [ q ∗ ( R ) , q ∗ ( R )), the optimal bid quantile q is 0 when the revenue curveis R . This guarantees that the virtual surplus for every quantile q ≤ q m ( R ) weakly decreasessince the virtual value is non-negative while the allocation decreases. Note that in sample-bidmechanism, the payment for lowest type is always 0, i.e., p (0) = 0. By Theorem 2.2, the expectedrevenue (a.k.a. virtual surplus) for R is at most the expected revenue (a.k.a. virtual surplus) for R . Step 2- flattening the revenue curve for all quantiles q ≥ q ∗ . In this step, we start withrevenue curve R constructed in step 1, and consider a sequence of revenue curves R (0)2 , R (1)2 , . . . where R (0)2 , R and R ( i +1)2 is recursively defined on R ( i )2 as follows, R ( i +1)2 ( q ) , R ( i )2 ( q ) q ∈ h , q ∗ ( R ( i )2 ) i ,R ( i ) ′ ( q ∗ ( R ( i )2 )) · ( q − q ∗ ( R ( i )2 )) + R ( i )2 ( q ∗ ( R ( i )2 )) q ∈ (cid:20) q ∗ ( R ( i )2 ) , − R ( i )2 ( q ∗ ( R ( i )2 )) R ( i ) ′ ( q ∗ ( R ( i )2 )) + q ∗ ( R ( i )2 ) (cid:21) , q ∈ (cid:20) − R ( i )2 ( q ∗ ( R ( i )2 )) R ( i ) ′ ( q ∗ ( R ( i )2 )) + q ∗ ( R ( i )2 ) , (cid:21) . where R ( i ) ′ ( q ∗ ( R ( i )2 )) is the right-hand derivative of R ( i )2 ( q ) at q = q ∗ ( R ( i )2 ). See Figure 3b for agraphical illustration. Invoking Lemma 5.3 and Lemma 5.6, with the same argument for valueswith positive virtual values in step 1, we can conclude that q ∗ ( R ( i )2 ) and the expected revenue for R ( i )2 in the sample-bid mechanism is weakly decreasing in i .Note that by construction, the sequence of revenue curves R (0)2 , R (1)2 , . . . converges to a revenuecurve R whose expected revenue in the sample-bid mechanism is at most the revenue for R , andsatisfying the following characterization, R ( q ) , R ( q ) q ∈ [0 , q ∗ ( R )] ,R ′ ( q ∗ ( R )) · ( q − q ∗ ( R )) + R ( q ∗ ( R )) q ∈ h q ∗ ( R ) , − R ( q ∗ ( R )) R ′ ( q ∗ ( R )) + q ∗ ( R ) i , q ∈ h − R ( q ∗ ( R )) R ′ ( q ∗ ( R )) + q ∗ ( R ) , i . See Figure 3c for a graphical illustration.
Step 3- flattening the revenue curve for all quantile q ≤ q m ( R ) . For any revenue curve R ,let p ( v ∗ ( R ) , R ) be the expected payment in the sample-bid mechanism of an agent with value v ∗ ( R )and revenue curve R . Due to Lemma 3.4 and Lemma 3.5, p ( v ∗ ( R ) , R ) · q ∗ ( R ) is a valid lower boundof the expected revenue in the sample-bid mechanism for an agent with revenue curve R . In thisstep, instead of analyzing the expected revenue, we argue that we can convert any revenue curve R (constructed in step 2) into another revenue curve R , such that (i) v ∗ ( R ) = v ∗ ( R ) ( , v ∗ ); (ii) q ∗ ( R ) ≤ q ∗ ( R ); and (iii) p ( v ∗ , R ) ≤ p ( v ∗ , R ). Finally, by showing that p ( v ∗ ( R ) , R ) · q ∗ ( R ) ≥ . R constructed in step 2, for any r ∈ [0 , ( r )4 as follows, R ( r )4 , (cid:26) r + (1 − r ) qq m ( R ) q ∈ [0 , q m ( R )] , q ∈ [ q m ( R ) , . See the black curves in Figure 3d as an example. We claim that there exists r ∗ ∈ [0 ,
1] s.t. R ( r ( , R ) satisfies properties (i) (ii) (iii) mentioned above. To see this, consider the argumentas follows.By construction, for all every value v , every bid b , the utility u ( v, b, R ( r )4 ) is decreasing continuouslyin r . Thus, v ∗ ( R ( r )4 ) is decreasing continuously in r . Let b ‡ be the optimal bid of an agentwith value v ∗ ( R ) and revenue curve R . Denote q ( b ‡ , R ) by q ‡ . Consider revenue curve R ( r )4 where r , − q m ( R ) q m ( R ) − q ‡ (1 − R ( q ‡ )). By construction, R ( r )4 ( q ) ≥ R ( q ) for all q ≤ q ‡ , and R ( r )4 ( q ) ≤ R ( q ) for all q ≥ q ‡ . See Figure 3d for a graphical illustration. Note that by construction, u ( v ∗ ( R ) , b ‡ , R ( r )4 ) = v ∗ ( R ) · (1 − q ‡ ) − αb ‡ · q ‡ − α Z q ‡ R ( r )4 ( q ) q dq ≥ v ∗ ( R ) · (1 − q ‡ ) − αb ‡ · q ‡ − α Z q ‡ R ( q ) q dq = u ( v ∗ ( R ) , b ‡ , R ) = 0Thus, v ∗ ( R ( r )4 ) ≤ v ∗ ( R ). Next, consider revenue curve R (¯ r )4 where ¯ r , − q m ( R ) q m ( R ) − q ∗ ( R ) (1 − R ( q ∗ ( R )). By construction, R (¯ r )4 ( q ) ≥ R ( q ) for all q ∈ [0 , v ∗ ( R (¯ r )4 ) ≥ v ∗ ( R ) with the similar argument for R ( r )4 Therefore, we know thatthere exists r ∗ ∈ [r , ¯ r ] such that v ∗ ( R ( r ∗ )4 ) = v ∗ ( R ). We denote R ( r ∗ )4 by R and show that R satisfies properties (ii) q ∗ ( R ) ≤ q ∗ ( R ) and (iii) p ( v ∗ , R ) ≤ p ( v ∗ , R ) with the argument below.Lemma 5.3 implies that b ‡ > v ∗ ( R ). Combining with the fact that r ∗ ≥ r , we know that property(ii) q ∗ ( R ) ≤ q ∗ ( R ) is satisfied. See Figure 3e for a graphical illustration.Combining the first order condition in Lemma 3.2 and construction of R , it is guaranteed that theoptimal bid b † of value v ∗ for revenue curve R is at most b ‡ . Furthermore, q ( b † , R ) ≥ q ( b ‡ , R ) ≥ q ( b ‡ , R ) = q ‡ by construction. By the definition, the optimal utility of value v ∗ ( R ) for any revenuecurve R is zero. Thus, p ( v ∗ , R ) = v ∗ · (1 − q ‡ ) ≥ v ∗ · (1 − q ( b † , R )) = p ( v ∗ , R ). Step 4- lower-bounding the expected revenue on R . So far, we have shown that for anarbitrary revenue curve satisfying the assumptions in lemma statement, its expected revenue inthe sample-bid mechanism is lower-bounded by p ( v ∗ ( R ) , R ) · q ∗ ( R ) for R pinned down by some( r , q m ) as follows, R , (cid:26) r + (1 − r ) qq m q ∈ [0 , q m ] , q ∈ [ q m , . By numerically verifying p ( v ∗ ( R ) , R ) · q ∗ ( R ) ≥ .
545 for all ( r , q m ) ∈ [0 , , we finish the proof.The details of this numerical evaluation is elaborated on in Appendix A. q m ≤ . In this subsection, we analyze the prior-independent approximation ratio of the sample-bid mech-anism over the class of regular distributions with monopoly quantile q m ≤ . q m q − q − q m Figure 4: Graphical illustration for Lemma 5.9. The gray dashed (resp. black solid) curve isrevenue curve R (resp. lower bound of R ). Lemma 5.8.
For the sample-bid mechanism with α = 0 . , the prior-independent approximationratio over the class of regular distributions with monopoly quantile q m ≤ . is at most . . Fix an arbitrary revenue curve R , let v ∗ ( R ) , inf { v : b ( v, R ) ≥ v m ( R ) } be the smallest value whose optimal bid b ( v, R ) for revenue curve R is at least v m ( R ). SinceLemma 3.5 guarantees that b ( v, R ) is weakly non-decreasing in v , v ∗ ( R ) is well-defined, b ( v, R ) ≥ v m ( R ) for all v ≥ v ∗ ( R ). Furthermore, by Lemma 5.3, we know that b ( v, R ) ≥ v m ( R ) / . for all v ≥ max { v ∗ ( R ) , v m ( R ) } . Denote q ( v ∗ ( R ) , R ) by q ∗ ( R ). By Lemma 3.4 and Lemma 3.5, the expectedrevenue Rev R (SB) of the sample-bid mechanism for revenue curve R can be lower-bounded asfollows, Rev R (SB) = Z p ( v ( q, R ) , R ) dq = Z min { q ∗ ( R ) ,q m ( R ) } p ( v ( q, R ) , R ) dq + Z q ∗ ( R )min { q ∗ ( R ) ,q m ( R ) } p ( v ( q, R ) , R ) dq + Z q ∗ ( R ) p ( v ( q, R ) , R ) dq ≥ ˜ p ( v m ( R ) / . , R ) · min { q ∗ ( R ) , q m ( R ) } + ˜ p ( v m ( R ) , R ) · max { , q ∗ ( R ) − q m ( R ) } + Z q ∗ ( R ) p ( v ( q, R ) , R ) dq. Denote q ( v m ( R ) / . , R ) by q ‡ ( R ), and R q m ( R ) q ‡ ( R ) R ( q ) q dq by w ( R ). In Lemma 5.9, we lower-bound theexpected payment ˜ p ( v m ( R ) / . , R ) and ˜ p ( v m ( R ) , R ) as the function of q m ( R ), q ‡ ( R ), w ( R ) and v ∗ ( R ).In Lemma 5.10, we lower-bound q ∗ ( R ) as the function of q m ( R ), q ‡ ( R ) and v ∗ ( R ). In Lemma 5.11,we upper-bound of v ∗ ( R ) as the function of q m ( R ), q ‡ ( R ) and w ( R ). In Lemma 5.12, we lower-bound p ( v ( q, R ) , R ) as a function of q m ( R ) for all quantile q ∈ [ q m ( R ) , q m ( R ), q ‡ ( R ) and w ( R ), and numerically evaluating its value for allpossible parameters. The details of the numerical evaluations in this section are similar to those ofLemma 5.7, which are elaborated on in Appendix A. Lemma 5.9.
For the sample-bid mechanism with α = 0 . , given any concave revenue curve R , the xpected payment ˜ p ( b, R ) of bidding b ∈ [0 , v m ( R )] is at least ˜ p ( b, R ) ≥ . b · (1 − q m ( R )) + 1)1 − q m ( R ) ; and the expected payment ˜ p ( v m ( R ) / . , R ) of bidding v m ( R ) / . is at least ˜ p ( v m ( R ) / . , R ) ≥ (cid:18) q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R )))1 − q m ( R ) (cid:19) . Proof.
By definition, for any b ∈ [0 , v m ( R )],˜ p ( b, R ) = 0 . b · q ( b, R ) + 0 . Z q ( b,R ) R ( q ) q dq ≥ . b · q ( b, R ) + 0 . Z q ( b,R ) 1 − q − q m ( R ) q dq = 0 . b · q ( b, R ) − . − q ( b, R )1 − q m ( R ) − . q ( b, R ))1 − q m ( R ) ≥ . b · (1 − q m ( R )) + 1)1 − q m ( R )where the first inequality uses the fact that R ( q ) ≥ − q − q m ( R ) for all q ≥ q m ( R ) from the regularity of R , and the second inequality use the fact that b · q ( b, R ) ≥ − q ( b,R )1 − q m ( R ) , and q ( b, R ) ≤ ( b · (1 − q m ( R )) +1) − from the regularity of R . See Figure 4 for a graphical illustration.Similarly, ˜ p ( v m ( R ) / . , R ) = 0 . v m ( R )0 . q ‡ ( R ) + 0 . Z q ‡ ( R ) R ( q ) q dq = q ‡ ( R ) q m ( R ) + 0 . w ( R ) + 0 . Z q m ( R ) R ( q ) q dq ≥ q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R ))1 − q m ( R ) . Lemma 5.10.
For any concave revenue curve R , the quantile q ( v, R ) for value v ≤ v m ( R ) is atleast q ( v, R ) ≥
11 + v · (1 − q m ( R )) ; and the quantile q ( v, R ) for value v ∈ [ v m ( R ) , v m ( R ) / . ] is at least q ( v, R ) ≥ q m ( R ) − q ‡ ( R ) · (1 + / . )1 + v · (1 − q m ( R )) . Proof.
Given any concave revenue curve R , consider another revenue curve R defined as follows, R ( q ) , R ( q ) q ∈ [0 , q ‡ ( R )] ,R ( q ‡ ( R )) + q − q ‡ ( R ) q m ( R ) − q ‡ ( R ) (1 − R ( q ‡ ( R ))) q ∈ [ q ‡ ( R ) , q m ( R )] , − q − q m ( R ) q ∈ [ q m ( R ) , . v m / . q m q ‡ Figure 5: Graphical illustration for Lemma 5.10. The gray dashed (resp. black solid) curve isrevenue curve R (resp. R ). The slope of the dotted line from (0, 0) is v m ( R ) / . .10 1ˆ qq m q † b † Figure 6: Graphical illustration for Lemma 5.11. The gray dashed (resp. black solid) curve isrevenue curve R (resp. R ). The slope of the dotted line from (0, 0) is b † .Since R is regular, we have R ( q ) ≤ R ( q ) for all q ∈ [0 ,
1] by construction. See Figure 5 forgraphical illustration. Thus, for any value v ≤ v m R , we have q ( v, R ) ≥ q ( v, R ) = 11 + v · (1 − q m ( R )) . Moreover, for any value v ∈ [ v m ( R ) , v m ( R ) / . ], we have q ( v, R ) ≥ q ( v, R ) = 2 q m ( R ) − q ( R ) · (1 + / . )1 + v · (1 − q m ( R )) . Lemma 5.11.
In the sample-bid mechanism with parameter α = 0 . , given any value v and anyconcave revenue curve R , the optimal bid b ( v, R ) for an agent with value v and revenue curve R isat least v m ( R ) if for all ˆ q ∈ [ q m ( R ) , , v · (1 − q ‡ ( R )) − v m ( R ) · q ‡ ( R ) − . (cid:18) w ( R ) + log (cid:18) ˆ qq m ( R ) (cid:19) − ln(ˆ q )1 − ˆ q (cid:19) ≥ v (1 − ˜ q ) + 0 . q )1 − ˆ q (2) where ˜ q , (cid:16) { / ˆ q , max { , v . − − ˆ q }} · (1 − ˆ q ) (cid:17) − . roof. Fix an arbitrary concave revenue curve R . We show that inequality (2) in the lemmastatement is a sufficient condition that bidding v m ( R ) / . is weakly preferred than bidding any bidsin [0 , v m ( R )]. The argument is similar to Lemma 5.6.We prove by contradiction, suppose there exists an revenue curve R , and value v such that in-equality (2) in the lemma statement is satisfied but the optimal bid of an agent with value v andrevenue curve R is b † ∈ [0 , v m ( R )). Denote q ( b † , R ) by q † . Let ˆ q , − − q † R ( q † ) . By construction,ˆ q ≥ q m ( R ). Now consider another revenue curve R defined as follows, R ( q ) , R ( q ) q ∈ [0 , q m ( R )] , q ∈ [ q m ( R ) , ˆ q ] , − q − ˆ q q ∈ [ˆ q, . By construction, R is a concave revenue curve s.t. (i) R ( q ) = R ( q ) for all q ∈ [0 , q m ( R )]; (ii) R ( q ) ≤ R ( q ) for all q ∈ [ q m ( R ) , q † ]; and (iii) R ( q ) ≥ R ( q ) for all q ∈ [ q † , R , R , q † , v and all b ‡ ≥ b † , we conclude that the optimal bid for an agentwith value v and revenue curve R is in [0 , b † ].Note that for any bid b ∈ [0 , / ˆ q ], the utility u ( v, b, R ) has a closed-form expression as follows, u ( v, b, R ) = v · b (1 − ˆ q ) b (1 − ˆ q ) + 1 + 0 . (cid:18) b (1 − ˆ q ) + 1 (cid:19) . Considering the first order condition of u ( v, b, R ) with respect to bid b , after basic simplification,we have b = v . − − ˆ q . Thus, the optimal bid in [0 , / ˆ q ] for revenue curve R is ˜ b , min { / ˆ q , max { , v . − − ˆ q }} . Plugging u ( v, b, R ) with b = ˜ b , we get v (1 − ˜ q ) + 0 . q )1 − ˆ q , i.e., the right hand side of inequality (2).Moreover, note that the utility u ( v, v m ( R ) / . , R ) has a closed-form expression as follows, v · (1 − q ‡ ( R )) − v m ( R ) · q ‡ ( R ) − . (cid:18) w ( R ) + log (cid:18) ˆ qq m ( R ) (cid:19) − ln(ˆ q )1 − ˆ q (cid:19) i.e., the left hand side of inequality (2). This leads to a contradiction, which finishes the proof. Definition 5.2. A pentagon revenue curve R parameterized by the quantile q k ∈ [ q m ( R ) , of kinkand the revenue r k ∈ h − q k − q m ( R ) , i on this kink is defined as follows R ( q ) , q ∈ [0 , q m ( R )] ,r k + q − q m ( R ) q k − q m ( R ) (1 − r k ) q ∈ [ q m ( R ) , q k ] , − q − q k · r k q ∈ [ q k , .
240 1 q m ( R ) q m ( R ) q k q † ˜ q b † (a) Case (i) b † ≤ ˜ v .
10 1˜ qq m ( R ) q m ( R ) q † b † (b) Case (ii) b † ≥ ˜ v . Figure 7: Graphical illustration for Lemma 5.12. The gray dashed (resp. black solid) curve isrevenue curve R (resp. R in (a) and R in (b)). The slope of the dotted line from (0, 0) is b † .An example of a pentagon revenue curve is illustrated as the solid curve in Figure 7a as the solidline. Lemma 5.12.
In the sample-bid mechanism, given any quantile ˆ q ∈ [0 , , quantile ˜ q ∈ [ˆ q, , andbid b ∈ [0 , / ˆ q ] , if for all pentagon revenue curves R P with q m ( R P ) ≥ ˆ q , the optimal bid of an agentwith value v (˜ q, R P ) and revenue curve R P is at least b ; then for all concave revenue curves R with q m ( R ) = ˆ q , the optimal bid of an agent with value v (˜ q, R ) and revenue curve R is at least b as well.Proof. Fix arbitrary ˆ q ∈ [0 , q ∈ [ˆ q, R with q m ( R ) = ˆ q . Let b † be the optimal bid for an agent with value v (˜ q, R ) ( , ˜ v ) and revenue curve R . To show thislemma, it is sufficient to assume b † ≤ / q m ( R ) . Now we consider two cases, i.e., b † ≤ ˜ v and b † ≥ ˜ v separately. Case (i) b † ≤ ˜ v : Consider the pentagon revenue curve R with q m ( R ) = ˜ q + 1 − R (˜ q ) R ′ (˜ q ) , q k = ˜ qR ′ (˜ q ) − R (˜ q ) + R ( q † )1 − q † R ′ (˜ q ) + R ( q † )1 − q † ,r k = 1 − q k − q † R ( q † ) . where R ′ (˜ q ) is the right-hand derivative of R ( q ) at q = ˜ q . By construction, we have (i) R (˜ q ) = R (˜ q ) and thus v (˜ q, R ) = v (˜ q, R ) = ˜ v ; (ii) R ( q † ) = R ( q † ); and (iii) R ( q ) ≥ R ( q ) for all q ∈ [0 , q † ]. See Figure 7a for a graphical illustration.Applying Lemma 5.2 on R , R , q † , ˜ v and all b ‡ ≥ b † , we conclude that the optimal bid for value˜ v is weakly smaller than b † . Thus, for any bid b ∈ [0 , / ˆ q ], if the optimal bid for value v (˜ q, R ) inrevenue curve R is at least b , then the optimal bid b † for value v (˜ q, R ) in revenue curve R is atleast b as well. 25 ase (ii) b † ≥ ˜ v : Consider the pentagon revenue curve R with q m ( R ) = 1 − − q † R ( q † ) , q k = q m ( R ) , r k = 1 . By construction, we have (i) q (˜ v, R ) ≤ q (˜ v, R ) and thus v (˜ q, R ) ≤ v (˜ q, R ); (ii) R ( q † ) = R ( q † );and (iii) R ( q ) ≥ R ( q ) for all q ∈ [0 , q † ]. See Figure 7b for a graphical illustration.Applying Lemma 5.2 on R , R , q † , ˜ v and all b ‡ ≥ b † , we conclude that the optimal bid for value˜ v is weakly smaller than b † . Thus, for any bid b ∈ [0 , / ˆ q ], if the optimal bid for value v (˜ q, R )in revenue curve R is at least b , then combining with Lemma 3.5, the optimal bid b † for value v (˜ q, R ) in revenue curve R is at least b as well.Now we are ready to prove Lemma 5.8. Proof of Lemma 5.8.
Fix an arbitrary concave revenue curve R with q m ( R ) ≤ .
62. We consider v ∗ ( R ) ≤ v m ( R ), v m ( R ) ≤ v ∗ ( R ) ≤ v m ( R ) / . , and v ∗ ( R ) ≥ v m ( R ) / . separately. Case (i) v ∗ ( R ) ≤ v m ( R ) : By Lemma 3.4 and Lemma 3.5, the expected revenue
Rev R (SB) of thesample-bid mechanism for revenue curve R can be lower-bounded as follows, Rev R (SB) = Z p ( v ( q, R ) , R ) dq = Z q m ( R )0 p ( v ( q, R ) , R ) dq + Z q ∗ ( R ) q m ( R ) p ( v ( q, R ) , R ) dq + Z q ∗ ( R ) p ( v ( q, R ) , R ) dq ≥ ˜ p ( v m ( R ) / . , R ) · q m ( R ) + ˜ p ( v m ( R ) , R ) · ( q ∗ ( R ) − q m ( R )) + Z q ∗ ( R ) p ( v ( q, R ) , R ) dq. Invoking Lemma 5.9 and Lemma 5.10, we can rewrite the lower bound of
Rev [ R ] as ≥ (cid:18) q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R )))1 − q m ( R ) (cid:19) · q m ( R ) − . q m ( R ))1 − q m ( R ) · (cid:18) − v ∗ ( R ) · (1 + q m ( R )) − q m ( R ) (cid:19) + Z q ∗ ( R ) p ( v ( q, R ) , R ) dq. Note that this lower bound is weakly decreasing in v ∗ ( R ) while holding everything else fixed.Let v ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) be the upper bound of v ∗ ( R ) as the function of q m ( R ) , q ‡ ( R ) , w ( R )established in Lemma 5.11. From Lemma 5.10, we can lower bound v ∗ ( q m ( R ) , q ‡ ( R ) , w ( R ))) by q ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) , (cid:0) − v ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) · (1 + q m ( R )) (cid:1) − . Let b ( q, q m ( R )) be thelower bound of the optimal bid for an agent with value v ( q, R ) and revenue curve R as the functionof q, q m ( R ) established in Lemma 5.12. Then, we can further rewrite the lower bound of Rev [ R ]as ≥ (cid:18) q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R )))1 − q m ( R ) (cid:19) · q m ( R ) − . q m ( R ))1 − q m ( R ) · (cid:16) q ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) − q m ( R ) (cid:17) + Z q ∗ ( q m ( R ) ,q ‡ ( R ) ,w ( R )) . b ( q, q m ( R )) · (1 − q m ( R )) + 1)1 − q m ( R ) dq. b ( q, q m ( R )) in the last term can be lower-bounded using Lemma 5.9.Therefore, we lower-bound Rev R (SB) as the function of q m ( R ) , q ‡ ( R ) , w ( R ). By numerically enu-merating all possible parameters, we conclude that Rev R (SB) ≥ .
545 in this case.
Case (ii) v m ( R ) ≤ v ∗ ( R ) ≤ v m ( R ) / . : The analysis is similar to case (i). By Lemma 3.4 andLemma 3.5, the expected revenue
Rev R (SB) of the sample-bid mechanism for revenue curve R canbe lower-bounded as follows, Rev R (SB) = Z p ( v ( q, R ) , R ) dq ≥ Z q ∗ ( R )0 p ( v ( q, R ) , R ) dq + Z q m ( R ) p ( v ( q, R ) , R ) dq ≥ ˜ p ( v m ( R ) / . , R ) · q ∗ ( R ) + Z q m ( R ) p ( v ( q, R ) , R ) dq. Invoking Lemma 5.9 and Lemma 5.10, we can rewrite the lower bound of
Rev [ R ] as ≥ (cid:18) q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R )))1 − q m ( R ) (cid:19) · q m ( R ) − q ‡ ( R ) · (1 + / . )1 + v ∗ ( R ) · (1 − q m ( R ))+ Z q m ( R ) p ( v ( q, R ) , R ) dq. Note that this lower bound is weakly decreasing in v ∗ ( R ) while holding everything else fixed. Let v ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) be the upper bound of v ∗ ( R ) established in Lemma 5.11. Let b ( q, q m ( R )) bethe lower bound of the optimal bid for an agent with value v ( q, R ) and revenue curve R establishedin Lemma 5.12. Then, we can further rewrite the lower bound as ≥ (cid:18) q ‡ ( R ) q m ( R ) + 0 . w ( R ) − . q m ( R )))1 − q m ( R ) (cid:19) · q m ( R ) − q ‡ ( R ) · (1 + / . )1 + v ∗ ( q m ( R ) , q ‡ ( R ) , w ( R )) · (1 − q m ( R ))+ Z q m ( R ) . b ( q, q m ( R )) · (1 − q m ( R )) + 1)1 − q m ( R ) dq. where the bid b ( q, q m ( R )) in the last term can be lower-bounded using Lemma 5.9.Therefore, we lower-bound Rev R (SB) as the function of q m ( R ) , q ‡ ( R ) , w ( R ). By numerically enu-merating all possible parameters, we conclude that Rev R (SB) ≥ .
545 in this case.
Case (iii) v ∗ ( R ) ≥ v m ( R ) / . : Lemma 5.11 upper-bounds v ∗ ( R ) as the function of q m ( R ), q ‡ ( R ) and w ( R ). By numerically enumerating all possible parameters, we conclude that v ∗ ( R ) ≥ v m ( R ) / . isnot possible for any revenue curve R with q m ( R ) ≤ . In this section, we show that no mechanism can achieve prior-independent approximation betterthan 1 .
07 even when the class of distributions are uniform distributions. Note that point massdistributions are special cases of the uniform distributions. The lower bound we will prove in this27ection holds for more general families of mechanisms than the single-round mechanisms that weintroduced in Section 2. Here we will show that even when the agent and the seller have multiplerounds of communication in general messages spaces, no mechanism can achieve prior-independentapproximation better than 1 .
07. However, since our analysis does not hinge on the exact format ofthe mechanism, we will not formally introduce the model for multi-rounds of communication.
Theorem 6.1.
For a single item, a single uniformly distributed agent, and a single valuationsample, the prior-independent approximation ratio for revenue maximization is at least . . The main idea for proving Theorem 6.1 is as follows. Consider two scenarios where the valuationdistribution of the agent is either uniform between [1 ,
2] or a pointmass with some value v ∈ [1 , , v ∈ [1 , U [1 ,
2] to win the itemand pay at most 1 in expectation. This indicates that the optimal prior-independent approximationratio is strictly above 1. By leveraging the approximation ratio in those two cases, we show thatthe optimal ratio is at least 1.07.Before the proof of Theorem 6.1, we first introduce several notations and present several propertiesfor non-truthful mechanisms M with prior-independent approximation ratio β . Lemma 6.2.
For single item, single agent, any distribution F with support [ v, v ] , for non-truthfulmechanism with prior-independent approximation ratio β , the interim allocation for agent withhighest value v is x ( v, F ) ≥ β .Proof. Suppose the interim allocation for agent with value v is x ( v, F ) < β . Since the interimallocation is monotone, the maximum expected virtual welfare for mechanism under distribution F is less than / β of the optimal expected virtual welfare, which implies the revenue is less than / β of the optimal revenue and the approximation ratio for distribution F is higher than β , acontradiction. Lemma 6.3.
For single item, single agent, and any uniform distribution F with support [ v, v ] such that v ≥ v , for a non-truthful mechanism with prior-independent approximation ratio β , theinterim utility for agent with highest value v is u ( v, F ) ≥ (cid:16) v − q v − vβ ( v − v ) (cid:17) .Proof. For uniform distribution F with support [ v, v ] such that 2 v ≥ v , the optimal mechanismOPT F is to post price v with expected revenue v . Suppose the utility for agent with value v is u ( v, F ) < (cid:16) v − q v − vβ ( v − v ) (cid:17) , the optimal mechanism subject to this constraint is to postprice v − u ( v, F ), with expected revenue u ( v,F ) v − v · ( v − u ( v, F )) < vβ , a contradiction. Lemma 6.4.
For single item, single agent, any point mass distribution F with support v , for non-truthful mechanism with prior-independent approximation ratio β , the interim utility for agent withvalue v is u ( v, F ) ≤ v (1 − / β ) .Proof. Suppose the interim utility in this case is u ( v, F ) > v (1 − / β ), the expected revenue is atmost the social welfare minus the expected utility, which is at most v − u ( v, F ) < vβ , contradictingthe fact that mechanism M achieves prior-independent approximation ratio β .28 roof of Theorem 6.1. Suppose mechanism M inducing interim allocation and payment rule x and p achieves prior-independent approximation ratio β . Consider uniform distribution F with support[1 , x (2 , F ) ≥ β , and u (2 , F ) ≥ − p − / β ). For any sample s ∈ [1 , s satisfiesthe constraint that s · x (2 , F, s ) − p (2 , F, s ) ≤ s (cid:18) − β (cid:19) (3)otherwise for distribution F s with point mass on s , an agent with value s can imitate the behavior ofan agent with value 2 in uniform distribution to achieve utility strictly higher than s (1 − / β ), andby Lemma 6.4, this contradicts to the assumption that mechanism M achieves prior-independentapproximation ratio β . Taking expectation over sample s for the left hand side of equation (3), wehave E s [ s · x (2 , F, s ) − p (2 , F, s )] ≥ E s [ s · x (2 , F, s )] − (2 − u (2 , F )) ≥ Z / β s ds − (2 − u (2 , F ))where the last inequality holds because x (2 , F ) ≥ β and the worst case happens when x (2 , F, s ) = 0for any sample s ≥ / β . Taking expectation over sample s for the right hand side of equation(3), we have E s (cid:20) s (cid:18) − β (cid:19)(cid:21) = 32 (cid:18) − β (cid:19) . Combining the inequalities, we have12 (cid:18) β (cid:19) − − (1 + p − / β ) ≤ (cid:18) − β (cid:19) . By solving the inequality, we have β ≥ . Feng and Hartline (2018) proposed the revelation gap to quantify the difference between the worstcase performance of the optimal truthful mechanism and the optimal non-truthful mechanism inprior-independent mechanism design. They showed that a non-trivial revelation gap exists for thewelfare maximization problem for agents with budgets. In this section, we show that a revelationgap also exists for the revenue maximization problem when considering the single-item single-agentsetting with single-sample access.Let M r be the family of truthful mechanisms, the family of mechanisms such that the agent maxi-mizes her utility by truthfully revealing her valuation to the seller, i.e., b ∗ ( v, F ) = v for all value v and valuation distributions F . Let M be the family of all mechanisms. We define β (MECHS , DISTS) , min M∈ MECHS Γ ( M , DISTS)as the optimal prior-independent approximation ratio among the family of mechanisms MECHS.The revelation gap for a family of distributions DISTS is then defined as the ratio β ( M r , DISTS) β ( M , DISTS) . Definition 7.1.
A mechanism is scale-invariant if the interim allocation is invariant of the scale,i.e., x ( αv, αF ) = x ( v, F ) for any distribution F , valuation v and any α > . Definition 7.2.
Given function α : R → ∆ ( R ) mapping from the sample to the randomized price,for sample s , the sample-based pricing mechanism solicits a non-negative bid b ≥ , allocates theitem to the agent if b ≥ α ( s ) , and charges the agent α ( s ) · { b ≥ α ( s ) } . It can be observed that the bid allocation rules of both sample-bid mechanism and sample-basedpricing are similar (i.e. competing against the sample), and the difference is the payment semantics.
Theorem 7.1 (Allouah and Besbes, 2019) . Under the assumption of scale-invariance, for single-item setting with regular valuation distribution, when seller has access to a single sample, theprior-independent approximation ratio of the optimal sample-based pricing mechanism is boundedin [1 . , . . Moreover, when the valuation distribution is MHR, the prior-independent approx-imation ratio is bounded in [1 . , . . Given an arbitrary valuation distribution and any mechanism that is incentive compatible only forthe given valuation distribution, the mechanism may not be equivalent to any sample-based pricingmechanism. The is because the agent only maximizes her utility by taking expectation over thesample. However, we can show that if the mechanism is incentive compatible for all possible priordistributions, then it is equivalent to consider posting a randomized price to the agent based onthe realization of the sample, i.e., a sample-based pricing mechanism.
Lemma 7.2.
For any mechanism with allocation ˜ x and payment ˜ p that is incentive compatible andindividual rational for all valuation distributions, there exists a sample-based pricing mechanismthat generates the same expected allocation and payment pointwise for any valuation of the agentand any realization of the sample.Proof. First we claim that, for any truthful mechanism with allocation ˜ x and payment ˜ p , theinduced allocation rule ˜ x ( · , s ) and payment rule ˜ p ( · , s ) are incentive compatible and individualrational given any realization of the sample s .First we prove the incentive compatibility. Suppose by contradiction, there exists constant ǫ > s and value v, v ′ such that v ˜ x ( v ′ , s ) − ˜ p ( v ′ , s ) ≥ v ˜ x ( v, s ) − ˜ p ( v, s ) + ǫ. Let F be an arbitrary distribution with positive density everywhere on the support [0 , ∞ ). Define H , u ( v, v, F ) − u ( v, v ′ , F ) as the utility loss for value v to misreport v ′ when the distribution is F .Given constant δ >
0, let F ′ be the distribution such that with probability 1 − δ , the value of theagent is s and with probability δ , the value is drawn from distribution F . It is easy to verify thatboth v and v ′ are in the support of distribution F ′ . Moreover, the utility loss for misreporting v ′ is u ( v, v, F ′ ) − u ( v, v ′ , F ′ ) ≥ (1 − δ ) ǫ + δH where (1 − δ ) ǫ + δH > δ . This implies that the mechanism is not incentivecompatible for distribution F ′ , a contradiction.Similarly, for individual rationality, if there exists constant ǫ >
0, sample s and value v, v ′ suchthat v ˜ x ( v, s ) − ˜ p ( v, s ) ≤ − ǫ, F ′ supported on [0 , ∞ ) such that agent with value v is not individualrational given distribution F ′ .Finally, since for any sample s , the induced mechanism is incentive compatible, the allocation ˜ x ( v, s )is monotone in v for any sample s . Moreover, individual rationality implies that the payment of theagent is 0 if she does not win the item. Thus the mechanism can be implemented as sample-basedpricing mechanism for any realized sample.Lemma 7.2 suggest that under the assumption of scale invariance, the bounds on prior-independentapproximation ratio of sample-based pricing in Theorem 7.1 carry over to truthful mechanisms.Then combining it with Theorem 5.1 and 6.1, we have the following corollary characterizing therevelation gap under the assumption of scale-invariance. Corollary 7.3.
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Numerical Analysis
In Section 5, we bound the prior-independent approximation ratio of the sample-bid mechanismby enumerating the possible choices of given parameters. One concern is that the parameters areselected from a continuous interval, and the revenue for valuation distributions with parametersthat are not evaluated on discretized points may be far from the revenue on discretized points. Inthis section, we formally show that this is not the case for our analysis. To provide a theoreticallower bound on all possible distributions, we will present a unified lower bound on the revenue fordistributions with parameters between discretized points. We will formalize this approach for thenumerical calculation for Lemma 5.7, and the numerical calculation for other lemmas and theoremshold similarly.By the proof of Lemma 5.7, for any revenue curve R in Figure 3e parameterized by monopolyquantile q m ∈ [¯ q m , ¯ q m ] and revenue r ∈ [¯ r , ¯ r ] for quantile 0, the revenue of the seller is lowerbounded by p ( v ∗ ( R ) , R ) · q ∗ ( R ) where v ∗ ( R ) is the critical value with bid above monopoly priceand q ∗ ( R ) is the quantile for critical value. Note that it is sufficient for us to consider revenuecurves R such that v ∗ ( R ) is at least the monopoly price. Next we show how to provide boundson parameters ¯ q m , ¯ q m , ¯ r , ¯ r , as well as lower bounds on p ( v ∗ ( R ) , R ) and q ∗ ( R ) using parameters¯ q m , ¯ q m , ¯ r , ¯ r . Lemma A.1.
There exists efficiently computed set S ⊆ R and function τ : R → R such that forany revenue curve R in Figure 3e parameterized by monopoly quantile q m ∈ [¯ q m , ¯ q m ] and revenue r ∈ [¯ r , ¯ r ] for quantile , we have1. v ∗ ( R ) ≥ v m ( R ) only if (¯ q m , ¯ q m , ¯ r , ¯ r ) ∈ S ;2. p ( v ∗ ( R ) , R ) · q ∗ ( R ) ≥ τ (¯ q m , ¯ q m , ¯ r , ¯ r ) if (¯ q m , ¯ q m , ¯ r , ¯ r ) ∈ S .Proof. First we illustrate how to find the desirable set S by numerical calculation. Note that therequirement is such that the critical value for bidding above the monopoly price is above monopolyprice, i.e., v ∗ ( R ) ≥ v m ( R ). By Lemma 5.6, it is sufficient to verify that the optimal utility of value v m ( R ) for bidding above v m ( R ) is positive. Note that by Lemma 3.2, the optimal bid above themonopoly price is b = v m α + − r − q m , with expected utility u ( v m , b ) = 1 q m · (1 − q b ) − α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) where q b = r b − − r qm . Since q m ∈ [¯ q m , ¯ q m ] and r ∈ [¯ r , ¯ r ], a sufficient condition for u ( v m , b ) > q m · (1 − ¯ q b ) − α (cid:18) ¯ b · ¯ q b + ¯ r log( ¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) > , where ¯ b = α ¯ q m + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )1 − ¯ q m + α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )1 − ¯ q m + α (1 − ¯ r ) . Note that the above inequalitycan be easily verified on discretized points.Next we construct the function τ (¯ q m , ¯ q m , ¯ r , ¯ r ) lower bound the revenue p ( v ∗ ( R ) , R ) · q ∗ ( R ). Firstnote that we can enumerate the value above monopoly price and find the minimum value that theinterim utility is strictly positive. That is, given value v ≥ v m , the optimal bid above the monopoly36rice is b = vα + − r − q m , with expected utility u ( v, b ) = v · (1 − q b ) − α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) ≥ v · (1 − ¯ q b ) − α (cid:18) ¯ b · ¯ q b + ¯ r log( ¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) > , where ¯ b = vα + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) . Let v ∗ be the minimumvalue that satisfies the above inequality. Then we have v ∗ ≥ v ∗ ( R ), and hence q ∗ ( R ) ≥ q ( v ∗ , R ) = αr q m (1 − q m ) vq m (1 − q m ) + α (1 − r ) ≥ α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m ) + α (1 − ¯ r ) . Moreover, we can similar construct an upper bound on the utility u ( v, b ) and let ¯ v ∗ be the largestvalue such that the upper bound on the utility is at most 0. Thus, we have v ∗ ( R ) ≥ ¯ v ∗ and hence p ( v ∗ ( R ) , R ) ≥ p (¯ v ∗ , R ) = α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) ≥ α (cid:18) ¯ b · ¯ q b + ¯ r log(¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) where ¯ b = ¯ v ∗ α + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )¯ v ∗ ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )¯ v ∗ ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) . By combining theinequalities, we have an lower bound on p ( v ∗ ( R ) , R ) · q ∗ ( R ) as a function of (¯ q m , ¯ q m , ¯ r0
There exists efficiently computed set S ⊆ R and function τ : R → R such that forany revenue curve R in Figure 3e parameterized by monopoly quantile q m ∈ [¯ q m , ¯ q m ] and revenue r ∈ [¯ r , ¯ r ] for quantile , we have1. v ∗ ( R ) ≥ v m ( R ) only if (¯ q m , ¯ q m , ¯ r , ¯ r ) ∈ S ;2. p ( v ∗ ( R ) , R ) · q ∗ ( R ) ≥ τ (¯ q m , ¯ q m , ¯ r , ¯ r ) if (¯ q m , ¯ q m , ¯ r , ¯ r ) ∈ S .Proof. First we illustrate how to find the desirable set S by numerical calculation. Note that therequirement is such that the critical value for bidding above the monopoly price is above monopolyprice, i.e., v ∗ ( R ) ≥ v m ( R ). By Lemma 5.6, it is sufficient to verify that the optimal utility of value v m ( R ) for bidding above v m ( R ) is positive. Note that by Lemma 3.2, the optimal bid above themonopoly price is b = v m α + − r − q m , with expected utility u ( v m , b ) = 1 q m · (1 − q b ) − α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) where q b = r b − − r qm . Since q m ∈ [¯ q m , ¯ q m ] and r ∈ [¯ r , ¯ r ], a sufficient condition for u ( v m , b ) > q m · (1 − ¯ q b ) − α (cid:18) ¯ b · ¯ q b + ¯ r log( ¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) > , where ¯ b = α ¯ q m + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )1 − ¯ q m + α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )1 − ¯ q m + α (1 − ¯ r ) . Note that the above inequalitycan be easily verified on discretized points.Next we construct the function τ (¯ q m , ¯ q m , ¯ r , ¯ r ) lower bound the revenue p ( v ∗ ( R ) , R ) · q ∗ ( R ). Firstnote that we can enumerate the value above monopoly price and find the minimum value that theinterim utility is strictly positive. That is, given value v ≥ v m , the optimal bid above the monopoly36rice is b = vα + − r − q m , with expected utility u ( v, b ) = v · (1 − q b ) − α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) ≥ v · (1 − ¯ q b ) − α (cid:18) ¯ b · ¯ q b + ¯ r log( ¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) > , where ¯ b = vα + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) . Let v ∗ be the minimumvalue that satisfies the above inequality. Then we have v ∗ ≥ v ∗ ( R ), and hence q ∗ ( R ) ≥ q ( v ∗ , R ) = αr q m (1 − q m ) vq m (1 − q m ) + α (1 − r ) ≥ α ¯ r ¯ q m (1 − ¯ q m ) v ¯ q m (1 − ¯ q m ) + α (1 − ¯ r ) . Moreover, we can similar construct an upper bound on the utility u ( v, b ) and let ¯ v ∗ be the largestvalue such that the upper bound on the utility is at most 0. Thus, we have v ∗ ( R ) ≥ ¯ v ∗ and hence p ( v ∗ ( R ) , R ) ≥ p (¯ v ∗ , R ) = α (cid:18) b · q b + r log( q m q b ) + (1 − r )( q m − q b ) q m − log q m (cid:19) ≥ α (cid:18) ¯ b · ¯ q b + ¯ r log(¯ q m ¯ q b ) + (1 − ¯ r )(¯ q m − ¯ q b )¯ q m − log ¯ q m (cid:19) where ¯ b = ¯ v ∗ α + − ¯ r − ¯ q m , ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )¯ v ∗ ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) and ¯ q b = α ¯ r ¯ q m (1 − ¯ q m )¯ v ∗ ¯ q m (1 − ¯ q m )+ α (1 − ¯ r ) . By combining theinequalities, we have an lower bound on p ( v ∗ ( R ) , R ) · q ∗ ( R ) as a function of (¯ q m , ¯ q m , ¯ r0 , ¯ r0