Distributionally Robust Pricing in Independent Private Value Auctions
aa r X i v : . [ ec on . T H ] A ug Distributionally Robust Pricing In IndependentPrivate Value Auctions ∗ Alex Suzdaltsev † August 10, 2020
Abstract
A seller chooses a reserve price in a second-price auction to maximize worst-caseexpected revenue when she knows only the mean of value distribution and an upperbound on either values themselves or variance. Values are private and iid. Usingan indirect technique, we prove that it is always optimal to set the reserve price tothe seller’s own valuation. However, the maxmin reserve price may not be unique.If the number of bidders is sufficiently high, all prices below the seller’s valuation,including zero, are also optimal. A second-price auction with the reserve equal toseller’s value (or zero) is an asymptotically optimal mechanism (among all ex postindividually rational mechanisms) as the number of bidders grows without bound.
Keywords: Robust mechanism design; Worst-case objective; Auctions; Moments prob-lemsJEL codes: D44, D82
Classic auction theory derives revenue-maximizing reserve prices under the assumption ofknown distribution of bidders’ values. One may give two interpretations to this assump-tion: (i) the probability distribution is objectively known to the seller; (ii) the distribution ∗ This paper is based on Chapter II of my PhD thesis at Stanford University and comprises a muchrevised version of an earlier working paper Suzdaltsev (2018). I would like to thank (in random order)Michael Ostrovsky, Andy Skrzypacz, Dmitry Arkhangelsky, Jeremy Bulow, Robert Wilson, Gabriel Car-roll, Ilya Segal, Evgeny Drynkin, and audience members at 2017 Conference on Economic Design, York,UK, for helpful comments. † Higher School of Economics, Saint Petersburg, Russia, [email protected] F . We then consider two specifications of the seller’s infor-mation: (1) the seller knows the mean of F and an upper bound on values; (2) she knowsthe mean of F and an upper bound on its variance. One may justify this approach invarious ways: • It may be easier for the seller to make an educated guess about two numbers thanabout a whole distribution. • It may be easier for the seller to estimate statistically a small number of parame-ters than a whole distribution. In particular, nonparametric estimators of densityfunctions converge more slowly than parametric estimators of the distribution’s mo-ments. • As shown by Wolitzky (2016), a model of a seller who knows only the mean of valuedistribution and bounds on its support can arise from the seller’s uncertainty aboutbidders’ information structures .The question of which reserve price is maxmin may be interesting not only from anormative, but from a positive perspective as well. Empirical literature on reserve pricesin auctions yields what Ostrovsky and Schwarz (2016) call a “reserve price puzzle”: thereserve prices observed in auctions are typically substantially lower than reserve pricesoptimal under the estimated distributions of values. The very formulation of the puzzlesuggests that its possible explanation involves postulating that the sellers do not possess In particular, bidder’s posterior mean can follow any distribution F with mean m and support in[0 , v ] if the prior value distribution is a binary distribution on { , v } with mean m . To apply this inour setting where values are known to be iid, we must assume that bidders information structures areidentical. See, e.g., McAfee and Vincent (1992); Paarsch (1997); McAfee et al. (2002); Haile and Tamer (2003). rR r ∗ Figure 1: A typical plot of expected revenue as a function of reserve price. The revenueis close to optimal to the left of the optimum and is far from optimal to the right of it.(The plot shown is for v i ∼ U [0 ,
1] and n = 3.)The main results of this paper state exactly this: a seller maximizing the worst-caseexpected revenue cannot do better than to set the reserve price to her own valuation,in contrast to the classical Bayesian setting where the optimal reserve is higher thanthe seller’s valuation. This is true in both settings we consider. The “low reserves underambiguity” phenomenon is an emerging theme in the literature (see section 1.1 for details);our setting is, to the best of our knowledge, the first one in which the optimal reserveprice is low for any fixed number of bidders n ≥ c is theseller’s valuation. When the number of bidders is sufficiently high, or c is sufficiently low,there is a whole interval of optimal prices that includes all prices in [0 , c ] (and possiblysome higher prices). This is also compatible with empirical evidence: the reserve pricesobserved in practice are frequently not just low, but are lower than all plausible seller’svaluations (Hasker and Sickles, 2010). Explanations for this fact proposed in the literatureinclude boundedly rational bidders who do not fully understand how participation ratedepends on the reserve price while sorting among competing sellers (Jehiel and Lamy,2015) and a combination of value interdependence and bidders’ risk aversion (Hu et al.,2019). Even though all reserve prices below c are weakly dominated by c in our model3i.e., they yield a weakly lower revenue than c for any fixed distribution of values), theirworst-case optimality provides an additional (weak) explanation for why such prices maybe observed in practice.Solving the maxmin problems posed in this paper directly by backward induction is,to the best of our knowledge, challenging. So we employ an indirect proof technique. Inthe first step, using a Lagrangian approach we identify a worst-case distribution F ∗ if thereserve price is equal to seller’s valuation, r = c (this is easier than for other prices). F ∗ is a binary distribution in the first setting (mean and upper bound on values are known)but has a continuous part in the second setting (mean and upper bound on variance areknown). We use an analogy of Nature’s problem to a textbook profit maximization prob-lem by a competitive firm with nonconvex costs to minimize the Lagrangian pointwise.Denote the worst-case revenue under r = c by R ∗ . In the second step, for each price r ≥ F r such that if the reserve price is r and the distributionis ˆ F r , the seller’s revenue is not more than R ∗ . This implies that r = c is a maxmin price.Note that the distributions ˆ F r need not be worst-case for respective prices. They are onlybad enough to discourage the seller from choosing r = c , but not necessarily the worst.We find appropriate to call them threat distributions – those Nature may threaten to useto harm the seller if she deviates from r = c . This proof technique is similar to one usedby He and Li (2020), who find a maxmin reserve price when a marginal distribution ofvalues, but not their joint distribution, is known (but, of course, the construction of ˆ F r is substantially different, as the set of possible distributions is very different from that inHe and Li (2020)).Sometimes (when the number of bidders is small or c is high) Nature’s threats arestrong enough so that r = c is the unique maxmin price; when the number of biddersis larger or c is small, other prices, including zero (as noted above), may be maxmin.This indifference occurs because in this case the lowest point in support of F ∗ happensto be strictly higher than c and the worst-case distribution is still F ∗ for all r ≤ c . Thatis, it may be so that under the worst-case distribution sale always happens, and alwayshappens at a price higher than both the seller’s valuation and the reserve price, even whenboth are positive. We find this somewhat surprising.In this paper, we address the question of optimal reserve price, but not a more generalquestion of optimal mechanism for a fixed n . A technical difficulty that does not allow touse a strong duality approach, as in, e.g., Suzdaltsev (2020) (see section 1.1), is that theset of joint distributions of values feasible for Nature is not convex due to the independenceconstraint (and these constraints are nonlinear). However, we show that the second-price4uction without a reserve (or a reserve equal to seller’s valuation) is an asymptoticallymaxmin mechanism in a large class as n grows without bound. This follows from thefact that the revenue guarantee of any individually rational mechanism cannot be higherthan the known mean of value distribution, but the revenue guarantee of the second-priceauction with a maxmin price converges to this mean as n grows. The result does not evenrequire that bidders play a Bayesian equilibrium. We compare the rates of the convergenceof the maxmin revenue to the mean across different settings to obtain insights about therelative “strength” of Nature depending on the set of distributions available to it.The result that a simple auction is asymptotically optimal must be reconciled withresults in Segal (2003) who shows that when agents’ values are independent draws from anunknown distribution, “bootstrap” schemes that estimate the distribution from bidder’sreports and set individual prices based on estimates derived from other agents’ reports, canasymptotically extract the entire full-distributional-information revenue. These schemesdiffer substantially from a classic second-price auction. The apparent discrepancy betweenthe our asymptotic result and Segal’s is due to the fact that optimization criteria aredifferent: while we employ the maxmin criterion, approximating the full-distributional-information outcome is about minimizing regret . When Nature chooses a distribution tominimize revenue itself rather than to maximize losses relative to full information, the“bootstrap” schemes cannot be significantly superior to the simple auction. This paper contributes to the growing literature on robust mechanism design. The clos-est contributions to ours are Carrasco et al. (2018a), He and Li (2020), Ko¸cyi˘git et al.(2020), Suzdaltsev (2020), Che (2019) and Neeman (2003). Carrasco et al. (2018a) studythe problem of selling the good to a single agent by a seller who maximizes worst-caseexpected revenue while knowing the first N moments of distribution. They character-ize the optimal randomized mechanism; also, they find the optimal deterministic postedprice for the single agent for settings (1) and (2) of the present paper. He and Li (2020)characterize the optimal deterministic in a second-price auction when the seller knowsthe marginal distribution of values but not their joint distribution. This setting may beseen as complementary to ours, as we assume that the marginal distribution is unknown,but a particular correlation structure (independence) is known. He and Li (2020) also The optimal deterministic posted price for the case when the seller knows mean and variance of thevalue distribution has been found earlier by Azar et al. (2013). In that paper, there is an infinite supplyof the good which essentially reduces the environment to a single-agent one. n ≥
2. In this sense, the present paper’s setting yields the most striking result among theexisting ones.Other papers seeking robustness to (payoff) type distributions include Carrasco et al.(2018b), Auster (2018), Bergemann and Schlag (2011), Bergemann and Schlag (2008).Carroll (2017), Giannakopoulos et al. (2019) and Chen et al. (2019) tackle the problemof selling multiple goods to a single agent under unknown type distribution. Bose et al.(2006) and Wolitzky (2016) study mechanism design when agents themselves are maxminwith respect to the distribution of other agents’ types. Wolitzky (2016) uses a specificationof sets of possible distributions similar to ours: bounds on support and the mean areknown. He gives a microfoundation for this specification which we mentioned earlier.A separate strand of literature studies mechanisms robust to misspecification of agents’information structures, rather than the designer’s prior. Brooks and Du (2019) identifyan optimal mechanism in the common value setting, while Du (2018) identifies a simplermechanism that asymptotically extracts full surplus. Bergemann et al. (2017) find theoptimal robust reserve price in a first-price auction under possible misspecification ofagents’ information structures. Relatedly, Chung and Ely (2007) and Chen and Li (2018)consider robustness to type distributions in a “rich” type spaces and identify conditions6nder which there exist maxmin foundations for dominant-strategy mechanisms.Others kinds of robustness explored in the literature include robustness to technol-ogy or preferences, robustness to strategic behavior and robustness to interaction amongagents and are surveyed by Carroll (2018).Finally, this paper is related to the literature seeking to explain low reserve pricesobserved in real-life auctions. Levin and Smith (1994) show that it may be optimalto use a reserve price equal to seller’s valuation under endogenous costly entry whileLevin and Smith (1996) show that unlike the textbook IPV case, the optimal reserve con-verges to the seller’s valuation when values are private but are only conditionally iid. Asmentioned above, Jehiel and Lamy (2015) and Hu et al. (2019) give explanations for whyobserved reserve prices are sometimes lower than the seller’s valuation.
In section 2, we describe the set-up. In section 3, we state and prove the results forthe case of known mean and an upper bound on values; in section 4 we do the same forthe case of known mean and an upper bound on variance. In section 5, we show thatsecond-price auction without a reserve is an asymptotically optimal mechanism among allmechanisms and compare rates of convergence of the maxmin revenue to its asymptoticvalue (which is simply the mean of value distribution) for different settings. Section 6concludes.
Consider the standard second-price auction with one object for sale and n ≥ F , but F is not fully known to the seller. We consider twospecifications of seller’s information. In the first one, the seller knows that E ( v i ) = m > v i ∈ [0 , v ], where v > m . In the second specification, the seller knows that v i ≥ E ( v i ) = m > V ar ( v i ) ≤ σ . The variance constraint is specified as inequality,rather than equality due to reasons discussed in Carrasco et al. (2018a) – with an equalityconstraint for the highest moment, the set of distributions may not be compact; also, theproof is somewhat easier to state. However, when variance is known exactly, the resultsare the same (see section 6).No further restrictions on F are made. In particular, atoms in F are allowed and7 is not necessarily regular in Myerson sense. Denote the set of feasible distributions ifonly mean and upper bound on values is known by ∆ ( m, v ) and if both mean and upperbound on variance are known by ∆ ( m, σ ).The seller’s own valuation for the object, c , may be higher than bidders’ values. Weassume that c ∈ [0 , m ). One reason for not normalizing c to zero is that we would like todistinguish between a reserve price equal to c and zero reserve price. Also, we would liketo allow distributions that put mass below c .The seller wishes to set a deterministic, public reserve price that maximizes revenue.Denote the expected revenue (including the seller’s valuation c ) if the distribution ofvalues is F and the reserve price is r by R ( F, r ). We consider the following problem:sup r ≥ inf F ∈ ∆ R ( F, r ) , (1)where either ∆ = ∆ ( v, m ) or ∆ = ∆ ( m, σ ). In other words, the seller wishes to seta price in such a way that the worst-case guarantee of revenue given her information ismaximal. Denote by R ( r ) the value of the infimum in (1), i.e. the value of this guarantee.We call any price r that solves (1) a maxmin reserve price and dthe corresponding expectedrevenue R ∗ the maxmin revenue .To present the analysis of the above problem, it will be convenient to us to phrase it asa zero-sum Stackelberg game between the seller and adversarial Nature in which the sellermoves first by setting a price r and then Nature, upon seeing r , chooses a distribution F from the choice set ∆.Denote by F ( v ) the cdf of the distribution F and by v ( i ) the i th-highest componentof the vector of valuations v . Then, assuming the bidders play dominant strategies, thefunction R ( F, r ) is given by R ( F, r ) = c · P ( v (1) ≤ r ) + r · P ( v (1) > r ∩ v (2) ≤ r ) + E F ∼ F ∼···∼ F h v (2) · { v (2) >r } i = r − ( r − c ) F n ( r ) + Z ∞ r (cid:0) − nF n − ( v ) + ( n − F n ( v ) (cid:1) dv, (2)where we used an expression relating the cdf of second-order statistic to the cdf of theparent distribution F and the identity E ( X ) = R + ∞ (1 − F ( v )) dv for a nonnegative randomvariable X with cdf F .Expressing the expected revenue in terms of the cdf F ( v ) allows to simultaneously coverall distributions regardless of presence of atoms, and also allows to reduce optimization8ver distributions to optimization over functions. Consider the problem (1) with ∆ = ∆( m, v ). Among other results, Carrasco et al. (2018a)solve this problem for n = 1, i.e., solve the monopolistic pricing problem. They show thatthere exists a unique maxmin price that exceeds seller’s costs.In contrast, a main result of this paper is when there are at least two bidders, a reserveprice equal to seller’s opportunity costs c is maxmin. Theorem 1 ( Main result I ) . Suppose the seller knows the mean of value distribution m and an upper bound on values v . Then, the set of prices r ∗ solving problem (1) includesthe seller’s valuation c . In this section, we provide the proof of theorem 1. The plan of attack, as outlinedin the introduction, consists of two steps. In the first step, we identify the worst-casedistribution F ∗ when r = c using a Lagrangian method. In the second step, for any r ≥ F r such that R ( ˆ F r , r ) ≤ R ( F ∗ , c ), without making any claim thatˆ F r is worst-case. This approach allows to circumvent the need to solve for a worst-casedistribution for each r . Identifying a worst-case distribution for r = c is much simplerthan for other prices because the term − ( r − c ) F n ( r ) in (2) disappears when r = c , sothat the expected revenue depends on F only though an integral. In the previous versionof this paper, (Suzdaltsev, 2018), we do identify worst-case distributions for each r ≥ .2.1 First step Suppose r = c . Then, Nature’s problem, as a problem of choosing a function F ( v ), maybe written as min F ( · ) (cid:18) c + Z vc (cid:0) − nF n − ( v ) + ( n − F n ( v ) (cid:1) dv (cid:19) (3)s.t. Z v (1 − F ( v )) dv = m (4) F ( v ) ∈ [0 ,
1] for all v ∈ [0 , v ] (5) F ( v ) is nondecreasing (6) F ( v ) is right-continuous (7)The constraint (4) is the mean constraint, while the constraints (5)-(7) are necessary andsufficient to ensure that the function F ( · ) chosen by Nature is a cdf. The constraint (7) isnot an issue; the monotonicity constraint may a priori be an issue, but, fortunately, turnsout not to be.To solve the problem (3)-(7), we first prove that it is without loss of generality to lookat distributions putting no mass below c . This allows to make the integration bounds inthe objective (3) and the constraint (4) the same. Lemma 1.
For every feasible cdf F in problem (3) - (7) , there exists a feasible cdf ˜ F puttingno mass below c such that the expected revenue (3) is weakly lower under ˜ F than under F . Proof of lemma 1:
Take a feasible cdf F . Define β := ( m − c ) / R ∞ c (1 − F ( v )) dv .Because F is feasible, β ∈ (0 , F ( v ) := , v < cβF ( v ) + (1 − β ) , v ≥ c. By construction, R ∞ (1 − ˜ F ( v )) dv = m so ˜ F is feasible and puts no mass below c . Therevenue is weakly lower under ˜ F than under F because ˜ F ( v ) ≥ F ( v ) for all v ≥ c and theintegrand in (3) is decreasing in F . (cid:3) The intuition behind lemma 1 is straightforward. Suppose there is some probabilitymass strictly below r = c . By transferring it all to r = c Nature will not change theexpected revenue, but will increase the mean of the distribution. Then it can restore the10ean by redistributing mass within the set { v : v ≥ c } towards lower values which willreduce the revenue.In light of lemma 1, the mean constraint can be now rewritten as Z vc (1 − F ( v )) dv = m − c (8)This allows to proceed to forming a Lagrangian.Define the Lagrangian by L ( F, λ ) := Z vc (cid:0) − nF n − ( v ) + ( n − F n ( v ) + λ (1 − F ( v )) (cid:1) dv (9)In what follows, we will minimize the Lagrangian pointwise. The validity of the La-grangian approach rests on the following lemma: Lemma 2.
Suppose F is a cdf that minimizes the Lagrangian among all cdfs for some λ ∈ R and satisfies (8) . Then, F solves the problem (3) . Proof:
Take F and any other cdf ˜ F satisfying (8) (which is without loss of generalityby lemma 1). Because F minimizes the Lagrangian, we have L ( F , λ ) ≤ L ( ˜ F , λ ) . Because both F and ˜ F satisfy (8), − λ Z vc (1 − F ( v )) dv = − λ Z vc (1 − ˜ F ( v )) dv. Summing the above relations, one gets that R ( F , c ) ≤ R ( ˜ F , c ), as desired. (cid:3)
Call the integrand in (9) H ( F, λ ) . The first-order condition for the minimization of H with respect to F is λ = n ( n − F n − ( F − . (10)If n = 2, this equation has the unique solution, so the optimal F is a constant whichcorresponds to a binary distribution on { c, v } . For n ≥
3, however, H ( F, λ ) is not convex The notation stems from the fact that the integrand is equal to the Hamiltonian of the correspondingoptimal control problem. The Minimum Principle (as applied to the relaxed problem) guarantees theexistence of the Lagrange multiplier λ such that the optimal F minimizes the Lagrangian pointwise.However, in the formal proof we construct the multiplier explicitly and therefore do not have to rely onMinimum Principle. F . In fact, for n ≥
3, Nature’s problem is isomorphic to a textbook profit maximizationproblem of a competitive firm with U-shaped marginal and average costs functions thatchooses its “output” F given the “price” λ . (The corresponding “total cost function”is T C ( F ) = ( n − F n − nF n − .) If the price is below the minimum of average costs(min AC ), the optimal output is zero; if the price is above min AC , the optimal outputis given by the minimum of the larger solution to (10) and 1; and if the price is exactlyequal to min AC , both zero output and the output minimizing the AC are optimal.Define q ∗ n := 1 − n − . This is the “output” minimizing “average costs”. Define also z ( y ) := y n − − y n − . The “supply curve” stemming from the pointwise minimization of the Lagrangian (max-imization of H ) is stated in the following lemma (we omit its proof): Lemma 3. arg max F ∈ [0 , H ( F, λ ) = { } , λ < λ ∗ ; { , q ∗ n } , λ = λ ∗ ; { min { ¯ y ( λ ) , }} , λ > λ ∗ , (11) where ¯ y ( λ ) is the larger solution to (10) and λ ∗ = min y ∈ [0 , (cid:20) ( n − y n − ny n − y (cid:21) = n ( n − z ( q ∗ n ) . (12) λ ∗ is “min AC ” (see Figure 2 for the case n = 3).] Fλ “MC” “AC” λ ∗ q ∗ = 0 . n = 3 (the argmin correspon-dence is in red). 12ortunately, a value of λ can always be found such that F ( v ) maximizes H ( F, λ ) for all v ∈ [ c, v ] and the mean constraint is satisfied. This leads to the identification of worst-casedistributions. Define v ∗ := , n = 2;max n m − v − m ( n − − , o n ≥ . Denote by δ a,b a binary distribution with support { a, b } and mean m . Note that if v ∗ >
0, the distribution δ v ∗ ,v puts a probability of q ∗ n on v ∗ . Proposition 1.
Suppose r = c . Then the distribution δ max { v ∗ ,c } ,v solves problem (3) - (7) . Proof of proposition 1:
Suppose c > v ∗ . This is equivalent to ( n − < v − cm − c .The binary distribution on { c, v } is described by a cdf F ∗ ( v ) = v − mv − c for all v ∈ [ c, v ).Denote p = v − mv − c . Consider λ = n ( n − z ( p ). Because ( n − < v − cm − c , p > q ∗ n so λ > λ ∗ .Thus, by lemma 3, F ∗ minimizes the Lagrangian for λ = λ . By lemma 2, F ∗ solves theproblem (3)-(7).Now suppose c ≤ v ∗ , i.e. ( n − ≥ v − cm − c . The binary distribution on { v ∗ , v } is suchthat its cdf F ∗ takes values 0 and q ∗ n on [ c, v ). Consider λ = λ ∗ . Thus, by lemma 3, F ∗ minimizes the Lagrangian for λ = λ . By lemma 2, F ∗ solves the problem (3)-(7). (cid:3) It is instructive to consider a specific numeric example. Note that if c = 0, Nature’sproblem for r = c is one of finding a distribution minimizing the expectation of second-order statistic in a sample given the known mean and upper bound. Example . Suppose n = 3, v = 1 and m = 1 /
2, and r = c = 0. Then, under theworst-case distribution the valuation of each bidder is equal to 1/3 with probability 3/4and is equal to 1 with probability 1/4. The worst-case expected revenue is equal to 7/16.As noted in the introduction, the fact that the support of the worst-case distributionmay be bounded away from c is a reason for why the maxmin reserve price may not beunique. In this step, for every r we construct a feasible distribution ˆ F r such that R ( ˆ F r , r ) ≤ R ( c )for all r . This implies the result of theorem 1. The distributions ˆ F r are not necessarilyworst-case given a reserve r . They may be thought of as threat distributions : distributionsthat Nature threatens to use were the seller to deviate from r = c .13 R c mv ∗ A rR c mv ∗ A Figure 3: Proof idea. The curves are graphs of R ( ˆ F r , r ). The graph of the worst-caserevenue function R ( r ) must lie everywhere weakly below the depicted curve by Step 2and must pass through point A by Step 1. n = 3; v = 1, m = 1 /
2. In the left picture, c = 0 . > / v ∗ ; c must be the unique maxmin price. On the right, c = 0 . < / v ∗ ; c may not be uniquely optimal.The construction depends on whether c > v ∗ or c ≤ v ∗ . Case 1. c > v ∗ . The construction of ˆ F r is separate for r ∈ [0 , v ∗ ), r ∈ [ v ∗ , c ), r ∈ [ c, m ), r ≥ m . Define threat distributions ˆ F r byˆ F r := δ v ∗ ,v , r ∈ [0 , v ∗ ); δ r + ,v , r ∈ [ v ∗ , c ); δ r,v , r ∈ [ c, m ); δ m , r ≥ m, where r + is a point arbitrarily close to r to the right of it. (Formally, in this case weconsider a sequence of distributions ˆ F kr , each of those binary on { r + 1 /k, v } .) Proposition 2.
Suppose c > v ∗ . Then, R ( ˆ F r , r ) < R ( c ) for all r = c and R ( ˆ F r , r ) = R ( c ) for r = c . Proof:
The fact that R ( ˆ F r , r ) = R ( c ) for r = c is obvious since ˆ F c is a worst-casedistribution for r = c , as identified by proposition 1.Consider r ∈ [ c, m ). Denote by p ( r ) = v − mv − r the probability assigned by ˆ F r to r . Then,by (2), R ( ˆ F r , r ) = r − ( r − c ) p n ( r ) + ( v − r )(1 − np n − ( r ) + ( n − p n ( r )) . dR ( ˆ F r , r ) dr = np n − − np n − np n − p ′ + ( v − r ) n ( n − z ( p ) p ′ . Since p ′ ( v − r ) = p , this simplifies to n ( n − pz ( p ) − np n − p ′ · ( r − c ) . Because p ′ > dR ( ˆ F r ,r ) dr < r = c and n = 2. Thus, R ( ˆ F r , r ) is strictlydecreasing on [ c, m ), which implies the result.For r ≥ m , R ( ˆ F r , r ) = c , but c = lim r → m − R ( ˆ F r , r ). Thus, c < R ( c ).Consider now r ∈ [ v ∗ , c ). Denote by δ r,v the binary distribution on { r, v } . R ( δ r + ,v , r )differs from R ( δ r,v , r ) only by term ( r − c ) p n , because there is a sale if v (1) = r + but notif v (1) = r . Thus, when r ∈ [ v ∗ , c ), we get dR ( ˆ F r , r ) dr = n ( n − pz ( p ) − np n − p ′ · ( r − c ) + [( r − c ) p n ] ′ = p n − (( n − p − n ( n − . For r > v ∗ , p ( r ) > q ∗ n , and for r = v ∗ , p ( r ) = q ∗ n . Thus, the above derivative is zero at r = v ∗ and positive at r ∈ ( v ∗ , c ). Thus, R ( ˆ F r , r ) is strictly increasing on [ v ∗ , c ), whichimplies the result.Finally, because R ( ˆ F r , r ) < R ( c ) for r = v ∗ , this is true for r < v ∗ as well. (cid:3) Case 2. c ≤ v ∗ . Define threat distributions ˆ F r byˆ F r := δ v ∗ ,v , r ∈ [0 , v ∗ ); δ r,v , r ∈ [ v ∗ , m ); δ m , r ≥ m, Proposition 3.
Suppose c ≤ v ∗ . Then, R ( ˆ F r , r ) ≤ R ( c ) for all r = c and R ( ˆ F r , r ) = R ( c ) for r = c . Proof:
For r ∈ [0 , v ∗ ), R ( ˆ F r , r ) = R ( c ), because ˆ F r = F ∗ and the reserve price doesnot affect the auction. For r ∈ [ v ∗ , m ), by the same reasoning as above, R ( ˆ F r , r ) is strictlydecreasing. Finally, R ( ˆ F v ∗ , v ∗ ) ≤ R ( c ) because R ( ˆ F v ∗ , v ∗ ) is the revenue under F ∗ whenthe good is not sold when all values are equal to v ∗ while R ( c ) is the revenue under F ∗ when the good is sold for v ∗ when all values are equal to v ∗ . (cid:3) R ( ˆ F r , r ) ≤ R ( c ) for all r and because distribu-tions ˆ F r are feasible to Nature, we conclude that R ( r ) ≤ R ( ˆ F r , r ) ≤ R ( c ) for all r , whichfinishes the proof of theorem 1. Theorem 1 establishes that c is a maxmin reserve price. But are there other maxminprices? Proposition 4.
Suppose the seller knows the mean of value distribution m and an upperbound on values v . Then:1. If v ∗ < c (equivalently, ( n − < v − cm − c ), c is the unique maxmin reserve price;2. If v ∗ ≥ c (equivalently, ( n − ≥ v − cm − c ), all prices r ∈ [0 , c ] are maxmin reserveprices. Proof:
Part 1 follows directly from proposition 2. To prove part 2, we show that F ∗ , the distribution identified by proposition 1, part 2, is a worst-case distribution notonly for r = c , but for all r ∈ [0 , c ) as well. Indeed, if r < c , the expected revenue may bewritten as R ( F, r ) = r − c + ( c − r ) F n ( r ) + R ( F, c ) .F ∗ minimizes R ( F, c ), but it also minimizes ( c − r ) F n ( r ) because c − r > F ∗ ( r ) = 0,as r < c ≤ v ∗ . Thus, it minimizes the sum of these two terms. (cid:3) As discussed in the introduction, proposition 4, part 2, might weakly explain whysometimes reserve prices lower than seller’s valuation are observed in real-life auctions.By using the qualifier “weakly” we emphasize the caveat that all prices below c are weaklydominated by c (yield weakly lower revenue for any fixed distribution) and thus might berefined away despite being worst-case optimal.Note, however, that proposition 4, part 2, does not say that prices r ∈ [0 , c ] are theonly maxmin prices. Indeed, in the previous version of this paper (Suzdaltsev, 2018) weshow that when the maxmin price is not unique, the set of maxmin prices may also includesome prices higher than c . As noted above, the full characterization of the set of maxminprices by backward induction requires subtler analysis that is beyond the scope of thispaper. 16 Known mean and upper bound on variance
In this section, we consider the problem (1) for ∆ = ∆ ( m, σ ), that is, consider a situationin which seller knows the mean and an upper bound for variance of value distribution.Again, it has been shown that if n = 1, there exists a unique maxmin price that exceedsseller’s costs (see Azar and Micali (2013) and Carrasco et al. (2018a)). In contrast, weshow that if n ≥
2, the seller can do no better than to set the reserve price to her ownvaluation.
Theorem 2 ( Main result II ) . Suppose the seller knows the mean of value distribution m and an upper bound on its variance σ . Then, the set of prices r ∗ solving problem (1) includes the seller’s valuation c . The plan of proof is exactly the same as in the section 3.
Suppose r = c . As compared with (3), Nature’s problem now involves one more constraint.To write it in an integral form, note that, for a nonnegative random variable v with cdf F ( · ), E ( v ) = R ∞ (1 − F ( √ s )) ds = R ∞ v (1 − F ( v )) dv. Hence, the additional constraintis R ∞ v (1 − F ( v )) dv ≤ m + σ .Thus, the new Nature’s problem is:min F ( · ) (cid:18) c + Z ∞ c (cid:0) − nF n − ( v ) + ( n − F n ( v ) (cid:1) dv (cid:19) (13)s.t. Z ∞ (1 − F ( v )) dv = m (14) Z ∞ v (1 − F ( v )) dv ≤ m + σ (15) F ( v ) ∈ [0 ,
1] for all v ∈ [0 , ∞ ) (16) F ( v ) is nondecreasing, right-continuous (17)lim v →∞ F ( v ) = 1 (18)17ogether, constraints (14) and (15) ensure that the mean of F ( · ) is equal to m , and itsvariance is no more than σ . The constraints (17), (18) ensure that F ( · ) is a cdf.Again, before proceeding to a Lagrangian, we show that Nature can restrict itself todistributions putting no mass below c . Lemma 4.
For every feasible cdf F in problem (13) - (18) , there exists a feasible cdf ˜ F putting no mass below c such that the expected revenue (13) is weakly lower under ˜ F thanunder F . Proof of lemma 4:
The proof is the same as the proof of lemma 1. The onlydifference is that one has to show that ˜ F , as constructed in the proof of lemma 1, satisfiesthe variance constraint (15). But this is true because F is a mean-preserving spread of˜ F , as R v F ( v ) dv ≥ R v ˜ F ( v ) dv and their means are the same. (cid:3) Lemma 4 allows to rewrite mean and variance constraints as (8) and Z ∞ c v (1 − F ( v )) dv ≤ m + σ − c . (19)Define the Lagrangian by L ( F, λ , λ ) = Z ∞ c (cid:0) λ + 2 tλ − nF n − ( v ) + ( n − F n ( v ) − ( λ + 2 λ v ) F ( v ) (cid:1) dv (20)The sufficiency of the pointwise minimization of the Lagrangian is now slightly subtleras now we have an inequality constraint. It can be ensured if λ has the right sign and acandidate worst-case distribution F satisfies (19) with equality. Lemma 5. If F is any cdf such that (1) F minimizes the Lagrangian among all cdfs forsome λ ≥ , λ of any sign; (2) F satisfies (8) and satisfies (19) with equality, then F solves the problem (13) - (18) . Proof:
Take any cdf ˜ F satisfying constraints (8)-(19). We shall prove that R ( F , c ) ≤ R ( ˜ F , c ) for any F satisfying conditions in the lemma. Because F minimizes the La-grangian, L ( F , λ , λ ) ≤ L ( ˜ F , λ , λ ) . Because both F and ˜ F satisfy (8), − λ Z ∞ c (1 − F ( v )) dv = − λ Z ∞ c (1 − ˜ F ( v )) dv. F satisfies (19) with equality, ˜ F satisfies (19), and λ ≥ − λ Z ∞ c v (1 − F ( v )) dv = − λ ( m + σ − c ) ≤ − λ Z ∞ c v (1 − ˜ F ( v )) dv. Summing up the above relations, one gets Z ∞ c (cid:0) − nF n − ( v ) + ( n − F n ( v ) (cid:1) dv ≤ Z ∞ c (cid:16) − n ˜ F n − ( v ) + ( n −
1) ˜ F n ( v ) (cid:17) dv, or R ( F , c ) ≤ R ( ˜ F , c ). (cid:3) Recall that z ( y ) = y n − − y n − and define φ ( q ) := R q ( z ( y ) − z ( q )) dy (cid:16)R q ( z ( y ) − z ( q )) dy (cid:17) (21)for q ∈ (0 , q ∗ n = 1 − n − . Now define v ∗∗ := max ( m − σ p φ ( q ∗ n ) − , ) . (22)Analogously to section 3, v ∗∗ will be shown to be the lowest point in support of theworst-case distribution if c = r = 0.We now introduce the family of distributions that plays a major role in both steps ofthe proof of theorem 2. The shape of the distribution is dictated by the minimization ofLagrangian when r = c . Given a parameter ρ ∈ [ v ∗∗ , m ) define a cdf G ρ ( · ) as follows: G ρ ( v ) = 0 for v < ρ ; G ρ ( ρ ) ≡ q ( ρ ) and G − ρ ( q ) = n ( n − z ( q ) − λ ( ρ )2 λ ( ρ ) (23)for q ∈ [ q ( ρ ) ,
1] where λ ( ρ ), λ ( ρ ) are parameters attuned in such a way that mean andvariance constraints hold as equalities.Equivalently, for every v ≥ ρ , G ρ ( v ) = min { ¯ y ( λ ( ρ ) + 2 λ ( ρ ) v ) , } where ¯ y ( λ ) is thelarger solution to (10) (as in (11)). For n = 2, G ρ is linear for v ≥ ρ , so the continuouspart of the distribution is uniform.To proceed, one must first check that G ρ are well-defined.19 G q ∗ = v ∗∗ Figure 4: A graph of a typical cdf from the G ρ family for n = 3. In the picture, ρ = v ∗∗ ,so the depicted cdf is worst-case if r = c < v ∗∗ . Lemma 6. G ρ is well-defined, i.e. for each ρ ∈ [ v ∗∗ , m ) , there exists a unique triple ( λ ( ρ ) , λ ( ρ ) , q ( ρ )) , λ ( ρ ) < , λ ( ρ ) > , q ( ρ ) ∈ [ q ∗ n , such that G ρ has mean m andvariance σ . The proofs of lemma 6 and subsequent lemmata is relegated to the Appendix. Theanalysis is enabled by the fact that one can write the mean and variance constraints(14),(15) as closed-form functions of λ , λ and q , even though there is generally no closed-form solution for G ρ . This is possible since the respective integrals may be rewritten asintegrals of the quantile function G − ( q ).Note that if v ∗∗ > G v ∗∗ ( v ∗∗ ) ≡ q ( v ∗∗ ) = q ∗ n . We now establish the worst-casedistribution if r = c . Proposition 5.
Suppose r = c . Then, the distribution G max { v ∗∗ ,c } solves the problem (13) - (18) . Proof of proposition 5:
By lemma 5, it suffices to prove that G max { v ∗∗ ,c } minimizesthe Lagrangian pointwise.Suppose first that c < v ∗∗ so v ∗∗ >
0. Then G v ∗∗ ( v ∗∗ ) = q ∗ n . Take λ and λ ascoming from the definition of G v ∗∗ (numbers that make G v ∗∗ satisfy the mean and varianceconstraints as equalities). We have λ + 2 λ v ∗∗ = n ( n − z ( q ∗ n ). Then it follows fromlemma 3 (with λ + 2 λ v playing the role of λ ) that G v ∗∗ minimizes the Lagrangianpointwise under the multipliers λ and λ .Now suppose c ≥ v ∗∗ . Take λ and λ as coming from the definition of G c . Thenit again follows from lemma 3 that G c minimizes the Lagrangian pointwise under themultipliers λ and λ . (cid:3) .2.2 Second step rR mcA rR v ∗∗ A mc
Figure 5: Proof idea. The curves are graphs of R ( ˆ F r , r ). The graph of the worst-caserevenue function R ( r ) must lie everywhere weakly below the depicted curve by Step 2and must pass through the point A by Step 1. m = σ = 1; c = 0; n = 2 (left), n = 3(right). If n = 2, σ/m = 1 corresponds to “high” variance, and r ∗ = 0 has to be theunique maxmin price. If n = 3, σ/m = 1 corresponds to “low” variance, and r ∗ = 0 mightnot be the unqiue maxmin price.Analogously to section 3, for every r we construct a feasible distribution ˆ F r such that R ( ˆ F r , r ) ≤ R ( c ) for all r . This implies theorem 2.As in section 3, the construction of threat distributions ˆ F r depends on whether c > v ∗∗ or c ≤ v ∗∗ ; equivalently, whether variance σ is high or low. Case 1 (High Variance). c > v ∗∗ . Define threat distributions ˆ F r byˆ F r := G v ∗∗ , r ∈ [0 , v ∗∗ ); G r + , r ∈ [ v ∗∗ , c ); G r , r ∈ [ c, m ); δ m , r ≥ m,G v ∗∗ is the same as the worst-case distribution F ∗ when r = c = 0.Recall that q ( r ) ≡ G r ( r ), the size of the atom of G r at r , and λ ( r ) , λ ( r ) are param-eters of G r (see (23)). In the next lemma, we derive closed-form expression for R ( ˆ F r , r )in terms of λ ( r ), λ ( r ) , q ( r ) is available, even though there is no closed-form solution for( λ ( r ) , λ ( r ) , q ( r )) themselves. 21 emma 7. For r ≥ v ∗∗ , R ( G r , r ) = | λ ( r ) | ( m − q ( r ) r ) − λ ( r )( m + σ − q ( r ) r ) − nz ( q ( r )) rq ( r ) + c · q n ( r ) . (24)Next, we show that the full derivative dR ( G r ,r ) dr also admits a tractable expression. (If G r were worst-case distributions, this derivative would be computable by a suitable versionof envelope theorem; but they are not.) Lemma 8.
For r ≥ v ∗∗ , dR ( G r , r ) dr = n ( n − qz − nq n − q ′ ( r − c ) . Now we are ready to state the key proposition. Recall that q ( r ) satisfies φ ( q ( r )) =1 + σ ( m − r ) . Because φ ′ ( q ) > d ( σ / ( m − r ) ) /dr > q ( r ) is a differentiable function with q ′ ( r ) > Proposition 6.
Suppose c > v ∗∗ . Then, R ( ˆ F r , r ) < R ( c ) for all r = c and R ( ˆ F r , r ) = R ( c ) for r = c . Proof:
The fact that R ( ˆ F r , r ) = R ( c ) for r = c is obvious since ˆ F c is a worst-casedistribution for r = c , as identified by proposition 5.As z < q ′ >
0, by (35) we have R ′ < r ≥ c unless n = 2 and r = c in which case R ′ = 0. Thus, R ( ˆ F r , r ) is strictly decreasing on [ c, m ) and R ( ˆ F r , r ) < R ( c ) for r ∈ ( c, m ).For r ≥ m , R ( ˆ F r , r ) = c , but c = lim r → m − R ( ˆ F r , r ). Thus, c < R ( c ).Consider now r ∈ [ v ∗∗ , c ). R ( ˆ F r , r ) = R ( G r + , r ) differs from R ( G r , r ) only by term( r − c ) q n ( r ) since there is a sale if v (1) = r + but not if v (1) = r . Thus, when r ∈ [ v ∗∗ , c ),we get dR ( ˆ F r , r ) dr = n ( n − qz − nq n − q ′ · ( r − c ) + [( r − c ) q n ] ′ = q n − (( n − q − n ( n − . For r > v ∗∗ , q ( r ) > q ∗ n , and for r = v ∗∗ , q ( r ) = q ∗ n . Thus, the above derivative is zero at r = v ∗∗ and positive at r ∈ ( v ∗∗ , c ). Thus, R ( ˆ F r , r ) is strictly increasing on [ v ∗∗ , c ), whichimplies that R ( ˆ F r , r ) < R ( c ) for r ∈ [ v ∗∗ , c ).Finally, because R ( ˆ F r , r ) < R ( c ) for r = v ∗∗ , this is true for r < v ∗∗ as well. (cid:3) ase 2 (Low Variance). c ≤ v ∗∗ . Define threat distributions ˆ F r byˆ F r := G v ∗∗ , r ∈ [0 , v ∗∗ ); G r , r ∈ [ v ∗∗ , m ); δ m , r ≥ m, The proof that R ( ˆ F r , r ) ≤ R ( c ) is exactly the same as the proof of proposition 3, with v ∗ replaced by v ∗∗ .Now, because by the above analysis R ( ˆ F r , r ) ≤ R ( c ) for all r and because distributionsˆ F r are feasible to Nature, we conclude that R ( r ) ≤ R ( ˆ F r , r ) ≤ R ( c ) for all r , which finishesthe proof of theorem 2. Analogously to proposition 4, we establish the following:
Proposition 7.
Suppose the seller knows the mean of value distribution m and an upperbound on variance σ . Then:1. (High variance case.) If v ∗∗ < c , c is the unique maxmin reserve price;2. (Low variance case.) If v ∗∗ ≥ c , all prices r ∈ [0 , c ] are maxmin reserve prices. The proof of proposition 7 is identical to the proof of proposition 4. Note that thecondition c ≤ v ∗∗ can be alternatively viewed as variance is small enough or the number ofbidders is high enough. As in section 3, part 2 of proposition 7 might weakly explain whyreserve price substantially lower than c are observed in practice. Unlike the case in section3, in the case of known bound on variance we are not aware of the full characterizationof the set of maxmin reserve prices. n = 2 In case of two bidders, closed-form expressions for threat distributions ˆ F r are available.As z ( q ) = q − n = 2, by (23), G − ( q ) is a linear function wherever it is defined andthus distributions G r are mixtures of an atom at r with a uniform distribution. The sameapplies to the worst-case distribution F ∗ when r = c .For n = 2, v ∗∗ = max { m − √ σ, } . m − √ σ is simply the lowest point in the supportof a uniform distribution with mean m and variance σ .23uppose m − √ σ ≥
0, so v ∗∗ = m − √ σ . There are two cases: c ≤ m − √ σ and c > m − √ σ . In the first case, which corresponds to either low seller’s valuation or lowvariance, the worst-case distribution for r = c is simply uniform on [ m − √ σ, m + √ σ ]and the maxmin revenue is R ∗ = m − √ σ. (25)Nature’s threats work as follows. For all r ∈ [0 , m − √ σ ], Nature can use thisuniform distribution and induce the same revenue (and this distribution is still worst-casefor r ∈ [0 , c ] as shown in the proof of proposition 7). For r ∈ ( m − √ σ, m ), Nature mayuse a distribution G r that has an atom of q ( r ) = σ − ( m − r ) / σ + ( m − r ) on r and is uniform on ( r, b ( r )] where b ( r ) = 12 (3 m − r ) + 32 σ m − r . Note that b ( r ) grows without bound when r → m ; the fact that there is no upper boundon values is important. For r ≥ m , Nature puts all mass on m .If c > m − √ σ (either high seller’s valuation or high variance), the worst-case distri-bution for r = c is G c itself. For r ∈ [0 , m − √ σ ], Nature may use uniform distributionon [ m − √ σ, m + √ σ ], for r ∈ ( m − √ σ, c ) it may use G r + , while for r ∈ ( c, m ) it againmay use G r .When n = 2, we can illustrate the fact that threat distributions ˆ F r are generally not worst-case distributions. For instance, suppose c = 0, m = σ = 1, n = 2 (Figure 5, left)and r = 0 .
5. Then ˆ F r is is such that v i is distributed uniformly on [0 . , .
25] with prob. and equal to 0 . . Expected revenue under ˆ F r is 0.32. However, if Natureuses a distribution which is a mixture of δ . and uniform distribution on [2 . , b ] (where b and the size of the atom are pinned down by moments constraints), the expected revenueis approximately 0 . < .
32. Numerically, all worst-case distributions have similargaps in support and are intractable analytically even for n = 2. It may be shown that in the low variance case the formula for maxmin revenue has thesame simple form as in (25). Indeed, in the low variance case all prices in [0 , c ] are maxmin,24nd the maxmin revenue is equal to that under the price v ∗∗ , as if there were sale when v (1) = r . Thus, replacing c with r , plugging r = v ∗∗ in (24), and then getting rid of λ and λ using (30)-(32), one gets that R ∗ n = m − γ n σ, (26)where γ n depends only on n (in the proof of proposition 8 we give a formula for γ n interms of φ ( q ∗ n )). Thus, the worst-case revenue is simply the mean minus a penalty linearin the standard deviation. However, the values of the penalties γ n are rather unexpected.For instance , R ∗ = m − √ σ ≈ m − . σR ∗ = m − √ σ ≈ m − . σR ∗ = m − √ σ ≈ m − . σ. Note that the maxmin revenue is strictly decreasing in variance. This is expected for n = 2 when the second order statistic is the minimal value, whose expectation is naturallybelow m and so higher variance reduces it. For higher n , the expectation of the secondorder statistic can be well above m and so larger variance may naturally increase ratherthan decrease it. The resolution to this paradox is that Nature chooses highly skeweddistributions such that E v (2) is below E v i = m for any n . We expect the solution to besubstantially different, with variance constraint not binding, when Nature is allowed tochoose only from symmetric distributions for n ≥ Throughout the paper so far, we have considered only the issue of optimal reserve price butnot the issue of optimal mechanism. This more general question seems to be challenging.One reason for that is that when the distribution F is unknown, one cannot assess whethera given direct mechanism is Bayesian-incentive compatible or not. (However, one can stillask what is a worst-case F for a given mechanism, having in mind the effect of F on Taking σ/m ≤ √ . c = 0. F ), Nature’s optimization problem, as seen as a problem of choosing a joint distribution ofvalues, is not a convex problem due to the stochastic independence constraint (a mixtureof two product distributions is not in general a product distributions). This precludes theuse of strong duality – a simplification trick used by, e.g., Suzdaltsev (2020).However, it is possible to establish that the second-price auction with a maxmin reserveprice (e.g. r ∗ = c ) is an asymptotically maxmin mechanism among all ex post individuallyrational mechanisms when the number of bidders is large.The argument rests only on the fact that by ex post individual rationality, the revenueof a mechanism is not higher than the social surplus, max i v i . In fact, we do not even needto assume that the bidders play a Bayesian equilibrium. All we need is that a mechanismand solution concept are such that the bidders always get a nonnegative payoff.Formally, let a mechanism be a tuple M = ( S , . . . , S n , x ( s ) , t ( s )) where, as usual, S i is the set of strategies of bidder i , and the functions x ( s ), t ( s ), with the usual codomains,specify a (possibly randomized) allocation and a vector of transfers for each strategy profile s . Let M be the set of all mechanisms. Let measurable outcome functions x ( v ) , t ( v ) mapa vector of values v to an allocation and a vector of transfers. Let O be the set of alloutcome functions. A solution concept is a correspondence SC : M × ∆ ⇒ O . That is, asolution concept maps a mechanism M ∈ M and a value distribution F ∈ ∆ to a set ofoutcome functions deemed possible. This set can depend on the distribution, as for theBayesian equilibrium solution concept. There can also be multiple “equilibrium” outcomefunctions. Given a correspondence SC , the seller restricts attention only to mechanisms M such that SC ( M, F ) = ∅ for all F ∈ ∆.We say that a mechanism M is robustly ex post individually rational under a solutionconcept SC and set of distributions ∆ iff, for all F ∈ ∆, all outcome functions ( x ( · ) , t ( · )) ∈ SC ( M , F ), all vectors of values v , and all i , v i x i ( v ) − t i ( v ) ≥
0. The qualifier “robustly”refers to the fact the inequality holds for the outcome functions, that may depend on F ,regardless of F .For example, the first-price auction is robustly ex post IR under Bayesian equilibriumas winning bidders never pay more than their values, and losers pay nothing. The Englishauction is robustly ex post IR under the weak solution concept used by Haile and Tamer(2003) who only assume that bidders never bid more than their values and never allow anopponent to win at a price they are willing to beat (and under stronger concepts as well).The schemes in Segal (2003) are robustly ex post IR under undominated strategies.26ow, let R n ( M, F, o ) be the expected revenue with n bidders under mechanism M ,distribution F and outcome functions o ∈ SC ( M, F ). Let R n ( M ) be the n -bidder revenueguarantee of a mechanism with unknown distribution F and “equilibrium” outcome o ∈ SC ( M, F ), i.e. R ( M ) := inf F ∈ ∆ inf o ∈ SC ( M,F ) R ( M, F, o ) . The worst case over “equilibrium” outcomes is taken because if the set of “equilibria”depends on F and F is unknown, the seller cannot suggest an equilibrium to play.Denote by SP A ( r ∗ ) the second-price auction with a maxmin reserve price and let( x DS ( v ) , t DS ( v )) be the usual outcome functions if bidders play dominant strategies (bidtheir values) in the SPA.Then, a simple SPA with a maxmin deterministic reserve price (e.g., r ∗ = c ) is alsoan asymptotically maxmin mechanism among all robustly ex post individually rationalmechanisms provided dominant strategies are played in the SPA. Proposition 8.
Suppose ∆ = ∆( m, v ) or ∆ = ∆( m, σ ) . Suppose the solution concept SC is such that SC ( SP A ( r ∗ ) , F ) = { ( x DS ( v ) , t DS ( v )) } for all F ∈ ∆ . Then, for anymechanism M that is robustly ex post individually rational under SC and ∆ and for any ε > , there exists N such that for all n > N , R n ( SP A ( r ∗ )) > R n ( M ) − ε . Proof:
The revenue of any robustly ex post individually rational mechanism M isnot more than the social surplus, P i t i ≤ max i v i ; thus, the worst-case expected revenueis not more than the worst-case expected surplus. However, the latter is not more than m because Nature can always choose F = δ m . (This argument is similar to the one usedin Ko¸cyi˘git et al. (2020) and He and Li (2020).) Thus, R n ( M ) ≤ m .It remains to show than R n ( SP A ( r ∗ )) converges to m as n → ∞ . For ∆ = ∆( m, v ),we have for all sufficiently high n R ∗ n = m − α n ( v − m ) , where α n = nn − (cid:18) − n − (cid:19) n − − . The fact that α n converges to zero stems from the fact that (cid:16) − n − (cid:17) n − ∼ exp( − /n )as n → ∞ . 27or ∆ = ∆( m, σ ), we have for all sufficiently high nR ∗ n = m − γ n σ, where it follows from (24) and (30) (in the Appendix) that γ n = α n p ( n − ψ ( q ∗ n ) − , (27)where ψ ( q ) = φ ( q )(1 − q ) (see the proof of lemma 6). As ψ is bounded, to prove that γ n → nα n → n → ∞ . In fact, one may show thatlim n →∞ n α n = 12 . (28)Indeed, n α n ∼ n n (exp( − /n ) −
1) + 1 n − . Then the result follows from a second-order expansion of exp( − /n ). (cid:3) Proposition 8 implies that neither a randomized reserve price nor a “bootstrap” auc-tion in which each bidder faces individual reserve computed based on an estimate of F inferred from other bidders’ reported values (a family of mechanisms investigated bySegal (2003)) is significantly better than a simple second-price auction when the selleris concerned about the worst-case performance of a mechanism, possesses only minimalstatistical information and the number of bidders is sufficiently large. As discussed inthe introduction, Segal’s schemes approximate the full-distributional-information revenuewell, but this is a criterion different from the worst-case performance.Note that if the seller knows more than the first two moments, say, three moments ofthe distribution, then the bound inf F E F ∼···∼ F max i v i ≤ m may not hold, i.e. the worst-case expected social surplus may be strictly higher than m . In other words, this prooftechnique fails. It is an open question of whether an SPA with a deterministic maxminreserve price is an asymptotically maxmin mechanism when more moments are known.Another limitation of the above result is that it effectively rules out directly elicitinginformation about F from the bidders (if they have it). More precisely, the setting abovedoes allow asking about F (strategy sets S i can be any) but the set ∆ of distributionsfeasible to Nature does not depend on bidders’ reports. An interesting direction of furtherresearch is to consider a model with a “rich” type space (i.e., types encoding both payoffand belief information) and a partially known payoff type distribution such that Nature28an choose only a payoff type distribution satisfying some prior constraints and alsobeing “close” to some types’ elicitable beliefs. See Luz (2013) and Chen and Li (2018) forrevenue maximization with rich type spaces. How does the seller’s maxmin revenue compare in the two settings we have considered inthis paper? The revenue obviously depends on the particular values of the upper bounds.However, noting that in both cases revenue converges to the mean m as the number ofbidders tends to infinity we can still get a meaningful comparison by comparing the ratesof convergence. This will compare the “strength” of Nature in two cases. We also bringinto the picture the case of correlated private values, studied by Ko¸cyi˘git et al. (2020)and Suzdaltsev (2020) (maxmin reserve price is again c in this case for all n sufficientlylarge).According to results in Ko¸cyi˘git et al. (2020) and Suzdaltsev (2020), when mean m and upper bound on values v are known, and values can be arbitrarily correlated, R n ( r ∗ ) = m − v − mn − n sufficiently large.Thus, from (27), (28) and (29) we make the following Observation. • When mean m and upper bound on values v are known, and values are iid, m − R ∗ n = Θ (cid:18) n (cid:19) as n → ∞ ; • When mean m and upper bound on variance σ are known and values are iid, m − R ∗ n = Θ (cid:18) n (cid:19) as n → ∞ ; • When mean m and upper bound on values v are known, and values can be arbitrarily29orrelated, m − R ∗ n = Θ (cid:18) n (cid:19) as n → ∞ .Thus, starting with the first setting in which mean m and upper bound on values v are known and values are iid, replacing an upper bound on values with an upper boundon variance has a similar (adverse) effect on revenue as allowing arbitrary correlation invalues. This suggests that an upper bound on values is a much more stringent constrainton Nature than an upper bound on variance. This is perhaps not surprising in hindsight:a bound on values implies a bound on variance, but not vice versa. In this paper, we showed that a seller who (1) faces multiple bidders and conducts asecond-price auction with a reserve price; (2) possesses only basic information aboutvalue distribution; (3) employs a worst-case perspective can do no better than to set thereserve price to her own valuation. This result adds to an emerging theme in the currentrobustness literature: an auctioneer’s worst-case perspective is associated with low reserveprices. The result may also help explain empirical observations of low reserve prices.One may think of several extensions of the present results. • First-price auctions.
We presented an analysis for a second-price auction. Aproblem with extending results for first-price auctions by revenue-equivalence isthat to ensure that a classic pure-strategy Bayesian equilibrium exists, one mustconstrain Nature to use atomless distributions only, but the worst-case and threatdistributions we used throughout the paper do contain atoms. However, the analysisstill goes through as these special distributions can be approximated by a sequenceof continuous distributions such that the value of revenue is the same in the limit.For example, it can be done by replacing each atom with uniform distributions onan interval below it and then tweaking the overall distribution a little in such a waythat Nature’s constraints hold. Thus, in a first-price auction the seller can still dono better than to set a reserve equal to her own valuation. • Exactly known variance.
We stated the results in section 4 for the case of aknown upper bound on variance. However, if the variance is known exactly, theresults still hold. First, even though if variance is known exactly the set of feasible30istributions is not compact (as noted by Carrasco et al. (2018a)), the infimumin Nature’s problem is well-defined. Second, for r < m all worst-case and threatdistributions we consider are such that the variance constraint binds. Third, for r ≥ m instead of m one can consider a sequence of distributions G m − /k defined in section4. Along this sequence, the probability of no sale converges to one and momentsconstraints are satisfied with equality. As for the asymptotic result (proposition 8),the key inequality necessary for the proof, inf F E F ∼···∼ F max i v i ≤ m , still holds,as under the sequence of binary distributions F k with the lowest point in support m − /k and a required mean m and variance σ , E F k ∼···∼ F k max i v i converges to m . • Higher moments.
When the seller knows F exactly (i.e. an infinite number ofmoments of F ), any optimal reserve price is strictly above c . We showed that whenthe seller knows up to two moments, an optimal price is c . It is interesting to askwhat is the minimal number K such that set of optimal prices is bounded awayfrom c if the seller knows no fewer than K moments. Is it true that K = 3? Doesthe asymptotic optimality result in section 5 carry over to the case where the sellerhas more information? • Eliciting information about F from the bidders. One may extend the modelto “rich” type spaces and constrain Nature to choose a distribution “close” to sometypes’ beliefs, as discussed in section 5.
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Appendix
Proof of lemma 6:
Fortunately, one can write constraints (14),(15) as closed-formfunctions of λ , λ and q , even though there is no closed-form solution for G ρ . This ispossible since the respective integrals may be rewritten as integrals of the quantile function G − ( q ). Indeed, one gets the following system of three equations with three unknowns( λ , λ , q ): q ( ρ ) ρ + Z q ( ρ ) n ( n − z ( q ) − λ ( ρ )2 λ ( ρ ) dq = mq ( r ) ρ + Z q ( ρ ) (cid:20) n ( n − z ( q ) − λ ( ρ )2 λ ( ρ ) (cid:21) dq = m + σ .λ ( ρ ) + 2 λ ( ρ ) ρ = n ( n − z ( q ( ρ )) . (30)34The first two equations stem from the mean and variance constraints (14),(15) whilethe third is (23), written for q = q ( ρ ).) Getting rid of λ ( ρ ), one simplifies the first twoequations to n ( n − Z q ( ρ ) ( z ( q ) − z ( q ( ρ ))) dq = 2 λ ( ρ )( m − ρ ) (31)( n ( n − Z q ( ρ ) ( z ( q ) − z ( q ( ρ ))) dq = (2 λ ( ρ )) (( m − ρ ) + σ ) . (32)Equations (31)-(32) may be collapsed to φ ( q ( ρ )) = 1 + σ ( m − ρ ) , (33)where φ ( q ) is as defined in (21). We now show that for every ρ ∈ [ v ∗∗ , m ) equation (33)admits a unique solution q ( ρ ) ∈ [ q ∗ n , φ ′ ( q ) > q ∈ [ q ∗ n , q → φ ( q ) = + ∞ .First, by direct computation, φ ′ ( q ) is proportional to z ′ ( q ) · R q ( z ( y ) − z ( q )) dy − q − R q ( z ( y ) − z ( q )) dy − q ! .z ′ ( q ) > q ≥ q ∗ n and the expression in parentheses is positive because it equals thevariance of random variable z ( Y ) − z ( q ) where random variable Y is distributed uniformlyon [ q, φ ′ ( q ) > φ ( q ) = ψ ( q ) / (1 − q ) where ψ ( q ) is the ratio of mean of square to the square ofmean of random variable z ( Y ) − z ( q ). Thus, ψ ( q ) > q <
1, so lim q → φ ( q ) = + ∞ .It follows from the definition of v ∗∗ that for every ρ ∈ [ v ∗∗ , m ), φ ( q ∗ n ) ≤ σ ( m − ρ ) ; wealso have lim q → φ ( q ) = + ∞ > σ ( m − ρ ) . Because φ ( q ) is continuous and strictly increasing,there exists a unique q ( ρ ) ∈ [ q ∗ n ,
1) solving (33).The uniqueness and signs of λ ( ρ ), λ ( ρ ) follow from (31) and (30). (cid:3) Proof of lemma 7:
Note that G r has an atom at r and a density g r for v > r .Because there is no sale when v (1) = r , R ( G r , r ) = E ( v (2) { v (2) >r } ) + r · n (1 − q ( r )) q n − ( r ) + cq n ( r ) == Z ∞ r v · n ( n − G n − r ( v ) − G n − r ( v )) g r ( v ) dv − nz ( q ( r )) rq ( r ) + cq n ( r ) . (34)35rom (23), one deduces that g r ( v ) = λ ( r ) n ( n − z ′ ( G r ( v )) . Thus, E ( v (2) { v (2) >r } ) = Z ∞ r λ ( r ) v − z ( G r ( v )) z ′ ( G r ( v )) dv. Plugging v = G − r ( q ) and using (23) again, one gets E ( v (2) { v (2) >r } ) = − Z q ( r ) n ( n − z ( q ) n ( n − z ( q ) − λ ( r )2 λ ( r ) dq. Mean and variance constraints read as q ( r ) r + Z q ( r ) n ( n − z ( q ) − λ ( r )2 λ ( r ) dq = mq ( r ) r + Z q ( r ) (cid:20) n ( n − z ( q ) − λ ( r )2 λ ( r ) (cid:21) dq = m + σ . Then, the fact that E ( v (2) { v (2) >r } ) = − λ ( r )( m − q ( r ) r ) − λ ( r )( m + σ − q ( r ) r )stems directly from the last three equations. The formula for revenue then follows. (cid:3) Proof of lemma 8:
Now consider the derivative of ˜ R ( r ) := E ( v (2) { v (2) >r } ) = − λ ( r )( m − q ( r ) r ) − λ ( r )( m + σ − q ( r ) r ). We omit arguments of functions forbrevity. From (30), we get that˜ R = 2 λ r ( r − m ) − λ (( m − r ) + σ ) − n ( n − zm + n ( n − qrz. ˜ R ′ = [2 λ ( r − m )] ′ r + 2 λ ( m − r ) − λ ′ (( m − r ) + σ ) − n ( n − z ′ q ′ m + n ( n − qrz ) ′ . Using both (31) and the differentiated version of (31), one gets˜ R ′ = n ( n − qr ) ′ z − ( m − r ) z ′ q ′ ) + n ( n − Z q ( z ( x ) − z ( q )) dx − λ ′ (( m − r ) + σ ) . However, differentiating (32) one gets λ ′ (( m − r ) + σ ) = − n ( n − q ′ z ′ R q ( z ( x ) − z ( q )) dx λ + λ ( m − r ) , λ ′ (( m − r ) + σ ) = n ( n − (cid:20) − q ′ z ′ ( m − r ) + Z q ( z ( x ) − z ( q )) dx (cid:21) . Thus, ˜ R ′ = n ( n − qr ) ′ z. Because R = ˜ R − nzqr + cq n , we get R ′ = n ( n − qr ) ′ z − nqrz ′ q ′ − n ( qr ) ′ z = n ( n − qr ) ′ z − nqrz ′ q ′ + nq n − q ′ c. Since qz ′ = ( n − z + q n − , after simplifications we finally get R ′ = n ( n − qz − nq n − q ′ ( r − c ) . (35) (cid:3)(cid:3)