IInformation Design in Optimal Auctions ∗ Yi-Chun Chen † Xiangqian Yang ‡ October 20, 2020
Abstract
We study the information design problem in a single-unit auction setting. The information designercontrols independent private signals according to which the buyers infer their binary private values.Assuming that the seller adopts the optimal auction due to Myerson (1981) in response, we characterizeboth the buyer-optimal information structure, which maximizes the buyers’ surplus, and the seller-worst information structure, which minimizes the seller’s revenue. We translate both information designproblems into finite-dimensional, constrained optimization problems in which one can explicitly solve forthe optimal information structure. In contrast to the case with one buyer ((Roesler and Szentes, 2017) and(Du, 2018)), we show that with two or more buyers, the symmetric buyer-optimal information structure isdifferent from the symmetric seller-worst information structure. The good is always sold under the seller-worst information structure but not under the buyer-optimal information structure. Nevertheless, as thenumber of buyers goes to infinity, both symmetric information structures converge to no disclosure. Wealso show that in our ex ante symmetric setting, an asymmetric information structure is never seller-worstbut can generate a strictly higher surplus for the buyers than the symmetric buyer-optimal informationstructure.
Keywords— information design; optimal auction; virtual value distribution; buyer-optimal informa-tion; seller-worst information. ∗ We are very grateful to Ben Brooks, Songzi Du, Jingfeng Lu, Satoru Takahashi, Weijie Zhong and Junjie Zhou for theirvaluable comments. Financial support from the Singapore Ministry of Education Academic Research Fund Tier 1 is gratefullyacknowledged. All errors are our own. † Department of Economics and Risk Management Institute, National University of Singapore. Email: [email protected] ‡ Department of Economics, National University of Singapore. Email: [email protected] a r X i v : . [ ec on . T H ] O c t Introduction
Consider a seller who would like to sell one object to a group of buyers. The classical optimal auction dueto Myerson (1981) assumes that each buyer knows his own valuation; moreover, each valuation followsa distribution which is common knowledge. In reality, however, the buyers may not know their ownvaluations for the good and have to assess how well the product suits their need via various informationsources. Their assessment might be based on advertisements, recommendations from their friends, orthe product description on an internet platform. In this paper, we study an information design problemin optimal auctions where each buyer learns his private valuation independently via a signal. Theseller earns the expected highest nonnegative virtual value, whereas the buyers earn the expected totalsurplus minus the seller’s revenue. We derive the buyer-optimal information structure which maximizesthe buyers’ total surplus as well as the seller-worst information structure which minimizes the seller’soptimal revenue.As the seller runs an optimal auction, providing more information to the buyers can lead to not onlya higher surplus but also a higher payment for them. Hence, the effect of a new information source onthe buyers’ welfare is not a priori clear. The buyer-optimal information structure contributes to ourunderstanding of this issue by identifying an information structure which maximizes the buyers’ surplus.In this regard, our study builds upon the prior work by Roesler and Szentes (2017) with a single buyerbut extends to an auction setup with multiple buyers. The problem is particularly relevant, when theinformation designer is a regulator who aims to promote the consumers’ welfare by requiring the sellerto disclose certain information about the product. The seller-worst information design provides a “minmax” upper bound for the revenue of an informa-tionally robust optimal auction which achieves the highest revenue guarantee (the so-called “maxmin”revenue) regardless of the equilibrium and the information structure. Indeed, a similar “minmax” upperbound is critical in establishing the strong duality results in Du (2018) for the one-buyer case and also inBergemann, Brooks, and Morris (2016) and Brooks and Du (2020a) for the multi-buyer common-valuecase. Specifically, these strong duality results show, in different contexts, that this “minmax” upperbound can indeed be achieved via an optimal informationally robust auction. In this vein, the seller-worst information provides a first step toward establishing such a strong duality result—or lack thereof,in an independent private-value setting.We assume that the seller has no value for the good and the buyers are ex ante symmetric. Inparticular, all of the buyers have an ex post valuation of the good equal to either 0 or 1 with the We assume that both the buyers’ valuations and signals are independently distributed. Terstiege and Wasser (2020) study buyer-optimal information design under monopoly pricing in which the informationdesigner may be a regulator of product information. They focus on a situation where the buyer cannot commit to ignore anyadditional information released by the seller, whereas we follow Roesler and Szentes (2017) in setting aside the issue of theseller’s disclosure. ame mean p . We follow Roesler and Szentes (2017) in assuming that each signal provides an unbiasedestimator about the buyer’s valuation. Hence, by Blackwell (1953), an information structure is feasibleif and only if it consists of a profile of independent signal distribution, all with mean p . We beginwith solving an optimal symmetric signal distribution in the two information design problems. As withderiving symmetric equilibria in symmetric auctions, it is also more tractable to derive a symmetricinformation structure in our ex ante symmetric information design problems. We show that as long as there are two or more buyers, the buyer-optimal information structure neednot be equal to the seller-worst information structure. The result sharply contrasts the results of Roeslerand Szentes (2017) and Du (2018) which show that the two information structures are equivalent whenthere is only one buyer. More precisely, Roesler and Szentes (2017) show that with one buyer, thebuyer-optimal information structure is equal to a truncated Pareto distribution with virtual value 0 forany signal less than 1, and with virtual value 1 for any signal equal to 1. Consequently, the good isalways sold. Du (2018) constructs an informationally robust mechanism which guarantees the revenueunder the buyer-optimal information structure of Roesler and Szentes (2017). Hence, the buyer-optimalinformation structure also minimizes the seller’s revenue.When there are two or more buyers, we pin down a cutoff p s which is decreasing with the number ofbuyers. If p is no more than p s , then the (symmetric) seller-worst information structure for each buyerremains the same as in the one-buyer case. If p is higher than p s , then the seller-worst signal distributionremains equal to a truncated Pareto distribution but now with virtual value k s > k s has twocountervailing effects. First, the seller’s revenue increases as the low virtual value increases. Second,to satisfy the mean constraint, increasing the low virtual value must be compensated by decreasing theprobability of having the high virtual value 1. As either the prior mean or the number of buyers grows,competition among the buyers becomes more severe and the second effect dominates the first one.The (symmetric) buyer-optimal information structure differs in a number of important ways. First,we pin down two cutoffs r b and p b which are also decreasing in the number of buyers. If p lies between r b and p b , then the buyer-optimal information structure remains the same as in the one-buyer case. When Alternatively, our analysis also applies and produces the same result if the information designer only knows that eachbuyer’s prior distribution of values has mean p and support [0 , Even with symmetric information structures, we still allow for irregular signal distributions for which the optimal auctionneed not be a second-price auction with reserve. Hence, our information design problem is not equivalent to the correspondinginformation design problem where the seller is committed to adopting a second-price auction with reserve; see Section 4.4 formore discussions and Appendix A.6 for an illustrative example. p Alwayssell? n = 1 Seller-worst Virtual value { , } for any prior mean;coincides with Roesler and Szentes (2017) YesBuyer-optimal Virtual value { , } for any prior mean Yes n ≥ p < p s p s < p ≤ { , } Virtual value { k s , } Buyer-optimal p < r b r b ≤ p ≤ p b p b < p ≤ { , } Virtual value, { , } Virtual value, { k b , } n → ∞ Seller-worst Degenerate distribution (revealing nothing) for any mean YesBuyer-optimal Degenerate distribution (revealing nothing) for any mean Yes p is less than r b , the buyer-optimal signal distribution puts positive mass on signal 0 and the remainingmass on a truncated Pareto distribution with virtual values 0 and 1. Since signal 0 induces a negativevirtual value, with positive probability the seller withholds the good. This is because to maximize thebuyers’ surplus, the information designer needs to consider not only the seller’s revenue but also the totalsurplus. The total surplus is convex in the buyers’ signal and hence favors a wider spread. When p isabove p b , the buyer-optimal signal distribution is a truncated Pareto distribution. However, also due toconvexity of total surplus, the distribution induces a low virtual value k b < k s for any signal less than 1and high virtual value 1 otherwise. We discuss the trade-off with more details in Section 4. We also show that when the number of buyers goes to infinity, the cutoffs p s , r b , and p b all tend tozero, both k b and k s monotonically increase to p , whereas the corresponding probabilities assigned tovirtual value 1 monotonically decrease. As a result, both the buyer-optimal and the seller-worst signaldistributions converge to a degenerate distribution which puts all mass on the prior mean p , i.e., nodisclosure. We summarize the results on optimal symmetric information in Table 1.We also investigate asymmetric signal distribution in both information design problems. For the We provide a definition of virtual value with arbitrary signal distribution in Section 2.2. Different from our information design problems, Yang (2019) studies the buyers’ strategic information acquisition andalso shows that the buyers’ equilibrium signal distributions are within the Pareto class. Due to the buyers’ competition ininformation acquisition, however, tie occurs with zero probability in the symmetric equilibrium signal distribution in (Yang,2019, Proposition 4), whereas tie occurs with positive probability in our buyer-optimal information. However, in a common-value model, Brooks and Du (2020a) construct a minmax/seller-worst information structure with agraded value function and the aggregate signals following an Erlang distribution. When the prior has binary support on { , } ,their information structure induces virtual values 0 and 1, regardless of the number of buyers. eller-worst problem, we show that the optimal symmetric information structure remains the uniqueoptimal solution, even if the information designer can choose different signal distributions for differentbuyers. Intuitively, averaging a profile of asymmetric virtual value distributions grants the seller lessoption values in selecting the highest virtual values and results in less revenue. This means restrictingattention to symmetric signal distributions entails no loss in minimizing the seller’s revenue.Although averaging a profile of asymmetric signal distribution entails no loss in minimizing the seller’srevenue, it may entail loss in the expected total surplus. Indeed, we demonstrate one case with two buy-ers and another case with the number of buyers approaching infinity where an asymmetric informationstructure generates strictly higher surplus for the buyers than the optimal symmetric information struc-ture. As our setup is ex ante symmetric, the result shows that asymmetric information structure canemerge endogenously as the choice of a buyer-optimal information designer.We explain how our argument differs from previous papers. In either one-buyer or common-valuemodels, the optimal information structures are constructed to make sure that under any signal realiza-tions each buyer has the same nonnegative virtual value; see Roesler and Szentes (2017), Bergemann,Brooks, and Morris (2016) and Brooks and Du (2020a). This is possible either because there is onlyone buyer or because each buyer’s common value is set to depend on all buyers’ signal realizations. Inour independent private-value model, each buyer’s interim value depends only on his signal and hencehis virtual value, being also independently distributed, must differ from other buyers’ virtual values forsome signal realizations. More importantly, when there are multiple buyers, both Bergemann, Brooks,and Morris (2016) and Brooks and Du (2020a) prove that their conjectured information structure isindeed the seller-worst/minmax information structure from their strong duality result. To the best ofour knowledge, however, such a strong duality result remains unknown in the independent private-valuesetting which we study here.To address this difficulty, we transform the control variables of the information design problems.More precisely, instead of working with signal/interim value distributions, we work with the interim virtual value distribution. After change of control variables, the information design problem becomesan isoperimetric problem in optimal control theory. The Euler-Lagrange equation in this problem canthen be invoked to argue that the virtual value distribution function is a step function with at most twosteps; see Section 4. This effectively reduces the infinite-dimensional information design problem into atractable finite-dimensional constrained optimization problem. Moreover, with the few control variablessuch as the two-step virtual value distribution functions, we are able to understand the trade-off in Since we assume independence of both valuations and signals among buyers, feasibility of a signal distribution is charac-terized by the mean constraint. While allowing the interdependence of interim valuations and signals may help us address theissue with distinct virtual values, the induced ex post valuations of different buyers from such interim valuations may end upbeing correlated and inconsistent with the independent prior; see Section 6. Since the good is always sold in their information structure (i.e., the expected total surplus is also maximized), their resultalso establishes the equivalence of the buyer-optimal and the seller-worst information structures in their common-value model. inning down their optimal choices as we elaborate above.The rest of this paper proceeds as follows. In Section 2, we describe our model and formulate theinformation design problem. Section 3 presents our main results. Section 4 demonstrates how we simplifythe control variables of the information design problems. Section 5 studies the information design problemwith asymmetric signal distributions. Section 6 discusses some issues with the extensions of our results.The appendix contains all proofs which are omitted from the main text. There is a seller who has one object to sell to a finite set N = { , , ..., n } of potential buyers. The sellerhas no value for the object. Each buyer’s prior valuation, v i , is identically and independently drawnfrom a Bernoulli distribution H on { , } . Let p = E [ v i ] = Pr ( v i = 1) denote the mean of H . To ruleout trivial cases, we assume that p ∈ (0 , x i about v i from an information designer and the distribution of v i and x i is common knowledge among the seller as well as the buyers. Following Roesler and Szentes (2017), we say a signal distribution is feasible if each signal of a buyerprovides him with an unbiased estimate about his valuation. Then, according to the characterization ofBlackwell (1953), the prior valuation distribution H is a mean-preserving spread of any feasible distribu-tion of signals. Since H is a Bernoulli distribution on { , } , the mean-preserving spread condition can bereduced to a mean constraint. Hence, a feasible symmetric information structure is a signal distribution G with G ∈ G H where G H = (cid:26) G : [0 , (cid:55)→ [0 , | (cid:90) x d G ( x ) = p and G is a CDF (cid:27) . Given a feasible signal distribution G , a revenue-maximizing mechanism is an optimal auction due toMyerson (1981). In the optimal auction, the seller’s revenue is equal to the expected highest, nonnegative,ironed virtual value max i { ˆ ϕ i ( x i | G ) , } . Formally, for any CDF G with supp ( G ) ⊂ [0 , a = inf { x ∈ [0 , | G ( x ) > } , and define Ψ( x | G ) = , if x ∈ [0 , a ); a − x (1 − G ( x )) , if x ∈ [ a, x | G ) be the convexification of Ψ under measure G . By definition, for any x ∈ [0 , ironed ) virtual valuation at x , denoted as ˆ ϕ ( x | G ), is a G -sub-gradient of Φ( x | G ); see also Yang (2019). If That is, Φ( x | G ) is the largest convex function that is everywhere weakly lower than Ψ( x | G ). ( x | G ) = Ψ( x | G ) for any x , then we say that G is a regular distribution . We use denote by ˆ ϕ an ironedvirtual value and use ϕ to denote a virtual value induced from a regular distribution. If G is regular,then the virtual value has the well known expression equal to ϕ ( x | G ) = x − − G ( x ) G (cid:48) ( x ) .We allow the information designer to choose any distribution function G , whether it is regular or irregularand whether it admits a density function or not.Let M ( x ) = { i ∈ N | ˆ ϕ ( x i | G ) ≥ max j { ˆ ϕ ( x j | G ) , }} be the set of buyers who have the largest nonneg-ative virtual value and M (cid:48) ( x ) = { i ∈ N | x i ≥ x j , ∀ j ∈ M ( x ) } be the set of buyers who not only have thehighest nonnegative virtual value but also the largest signal among those with the highest virtual value.Define an allocation rule: q i ( x i , x − i ) = | M (cid:48) ( x ) | , if i ∈ M (cid:48) ( x );0 , if i (cid:54)∈ M (cid:48) ( x ) . That is, q i ( x i , x − i ) is an optimal auction allocation rule which breaks a tie in favor of a surplus-maximizinginformation designer.We study the following information design problem parameterized by α = 0 or 1:max G ( · ) (cid:90) [0 , n n (cid:88) i =1 ( αx i − ˆ ϕ ( x i | G )) q i ( x i , x − i ) n (cid:89) i =1 (d G ( x i )) (1)s.t. (cid:90) − G ( x )d x = p. (2)If α = 1, the term (cid:80) ni =1 x i q i ( x i , x − i ) is the total surplus generated under the optimal auction allocationrule q i . Moreover, the term (cid:80) ni =1 ˆ ϕ ( x i | G ) q i ( x i , x − i ) is the seller’s revenue under the allocation rule q i ,namely the expected highest nonnegative virtual value. Hence, if α = 0, the information designer aims tominimize the seller’s revenue, and it corresponds to the seller-worst information design problem. If α = 1,the information designer aims to maximize the buyers’ surplus, and it corresponds to the buyer-optimalinformation design problem. Hereafter, we call (2) the mean constraint .Endow the space of Borel probability measures on [0 ,
1] with the weak ∗ topology. We say a signaldistribution G induces almost nonnegative virtual values if the virtual values induced by G are nonneg-ative almost everywhere on (0 , G + H ⊂ G H the feasible signal distribution with almostnonnegative virtual values. Also we say a signal distribution G is almost regular if G is regular almosteverywhere on (0 , Lemma 1.
For the problem in (1) , an optimal solution exists. roof. We first establish the seller-worst case. By Theorem 2 of Monteiro (2015), the expected revenueis a lower semicontinuous function in G . Hence, the objective function of the problem in (1) is uppersemicontinuous function in G . Third, since G H is a closed subset of the set of Borel probability measureson [0 , G H is compact. Thus, by the extreme value theorem, an optimal solution exists.For the buyer-optimal problem, the existence is more involved and we provide a formal proof inAppendix A.2. Lemma 2.
For the problem in (1) and signal distribution ˆ G ∈ G + H , if ˆ G is a (resp. strict) mean-preserving spread of signal distribution G , then ˆ G will generate (resp. strictly) more total surplus than G . Proof.
Observe that (cid:80) ni =1 x i q i ( x i , x − i ) = max { x , · · · , x n } if the good is allocated. Therefore, (cid:90) [0 , n n (cid:88) i =1 x i q i ( x i , x − i ) n (cid:89) i =1 (d G ( x i )) ≤ (cid:90) x d G n ≤ (cid:90) x d ˆ G n = (cid:90) [0 , n n (cid:88) i =1 x i q i ( x i , x − i ) n (cid:89) i =1 (cid:16) d ˆ G ( x i ) (cid:17) .The first inequality follows because the good may not be allocated under q i . The second inequalityfollows because ˆ G is a mean-preserving spread of G and max { x , · · · , x n } is convex in x . Moreover, thesecond inequality is strict if ˆ G is a strict mean-preserving spread of G . The equality follows becauseˆ G ∈ G + H , the good is always sold under q i expect when x i = 0 for every i (and in this case, the totalsurplus remains the same, whether the good is sold or not). In this section, we present our results on both information design problems. We will also sketch theirproofs in Section 4.First, the following result summarizes the seller-worst information structure:
Theorem 1.
The unique symmetric seller-worst information structure is a truncated Pareto distributions G s with G s ( x ) = − x s x − k s if x ∈ [ x s + k s , if x = 1 . where x s satisfies the mean constraint x s + k s + x s (log(1 − k s ) − log( x s )) = p . Moreover, there exists athreshold p s such that1. if p ∈ [0 , p s ] , then the optimal k s = 0 ;2. if p ∈ ( p s , , then the optimal k s > .Proof. See Section 4. The threshold p s is strictly decreasing in n and k s is strictly increasing in n . Theirexpression can be found in Appendix A.8. We say ˆ G is a (resp. strict) mean-preserving spread of signal distribution G if (cid:82) x ˆ G ( t ) − G ( t )d t ≥ x with equalityat x = 1 (resp. and with strict inequality at some x with G -positive probability). heorem 1 states that the (symmetric) seller-worst signal distribution for each buyer is equal to atruncated Pareto distribution with virtual value k s for any signal less than 1, and with virtual value 1for any signal equal to 1. Indeed, the virtual value is x − − G s ( x ) g s ( x ) = x − x s / ( x − k s ) x s / ( x − k s ) = k s when x isless than 1. Also, there exists a cut-off p s such that if p is no more than p s , then the generated virtualvalue k s = 0 remains the same as in the one-buyer case. If p > p s , the generated virtual value k s will bestrictly larger than 0.Since all virtual values are nonnegative at any signal profile, the good will always be sold. Indeed,since the seller gets no revenue so long as the virtual value is zero, it is never optimal for a seller-worstinformation designer to induce a negative virtual value. More precisely, by raising a negative virtualvalue to zero, the information designer can also decrease the probability assigned to a positive virtualvalue to balance the mean and thereby lower the seller’s revenue. The exact trade-off will become moreclear after we transform the control variable from a signal distribution to a virtual value distribution inSection 4.In contrast, raising the lower virtual value from zero to a positive number has two countervailingeffects on the seller’s revenue. First, by increasing the lower virtual value, the seller’s revenue increases.Second, to obey the mean constraint, the probability of having the high virtual value 1 must be reducedto compensate the increase in the lower virtual value. Intuitively, if either p or n is large, competitionamong the buyers becomes more severe and the second effect dominates the first one.Second, we summarize the buyer-optimal information structure: Theorem 2.
The unique symmetric buyer-optimal information structure is a truncated Pareto distribu-tions G b with G b ( x ) = θ if x ∈ [0 , x b );1 − x b (1 − θ ) x − k b if x ∈ [ x b + k b , if x = 1 . where x b satisfies the mean constraint, (1 − θ ) (cid:0) x b + k b + x b (cid:0) log(1 − k b ) − log( x b ) (cid:1)(cid:1) = p . Moreover,there exist two thresholds r b < p b such that1. if p ∈ [0 , r b ) , then θ > and k b = 0 ;2. if p ∈ [ r b , p b ) , then θ = 0 and k b = 0 ;3. if p ∈ [ p b , , then θ = 0 and k b > .Proof. See Section 4. The thresholds r b and p b are both strictly decreasing in n , whereas k b is strictlyincreasing in n . Their expression can be found in Appendix A.9.The buyer-optimal information structure looks similar to the seller-worst information structure butdiffers in several ways. First, when the prior p < r b , the buyer-optimal signal distribution puts mass θ on signal 0. That is, with probability ( θ ) n the good is not sold. In fact, to maximize the buyers’ urplus, the information designer needs to consider not only the seller’s revenue but also the expectedtotal surplus. Although putting positive mass on signal 0 will increase the seller’s revenue, it also followsfrom Lemma 2 that a mean-preserving spread of the seller-worst distribution generates more expectedtotal surplus. Theorem 2 shows that in this case, the benefit of increasing the expected total surplusdominates the cost of increasing the seller’s revenue when the mean p is lower than r b . This sharplycontrasts the seller-worst information structure in which the virtual value is always nonnegative and thegood is always sold. The comparison also helps us understand why the two information structures areequivalent and the good is always sold when there is only one buyer. With one buyer the expected totalsurplus is linear in the buyer’s signal and hence, there is no benefit in having a negative virtual value atzero.When p > p b , the lower virtual becomes positive. As we anticipate, p b is larger than p s and k b < k s . That is, the buyer-optimal information designer does not raise the lower virtual value by the same amountas he does in the seller-worst case. Again, this is because raising the lower virtual value decreases themean-preserving spread and thereby also the expected total surplus by Lemma 2.
Corollary 1.
For n → ∞ , the buyer-optimal information structure coincides with the seller-worstinformation structure in the limit. Both are given by a degenerate distribution G where G ( x ) = δ p = if x ∈ [0 , p );1 if x ∈ [ p, . Proof.
See Appendix A.10.Corollary 1 implies that when n is large, both the buyer-optimal and the seller-worst informationstructures are close to “no disclosure”. Namely that the information designer chooses the degeneratedistribution function which concentrates on p . Moreover, the seller extracts the ex ante expectation of asingle buyer’s value (i.e., p ) and leaves no surplus to the buyers. This limiting result is obtained underour assumption that the information designer has full control of the information structure among thebuyers.Corollary 1 also contrasts the result of Yang (2019). Specifically, Yang (2019) shows that when thebuyers’ information structure is a result of their strategic information acquisition, the unique symmetricequilibrium information structure converges to full information, as the number of buyers goes to infinity.Hence, the buyers retain zero surplus in the limit, whether in our buyer-optimal information structure orin the equilibrium information structure of Yang (2019). As our symmetric buyer-optimal information Indeed, suppose that p b < p s . When p ∈ [ p b , p s ], the buyer-optimal signal distribution generates positive virtual value k b , while the seller-worst distribution generates zero virtual value. The buyer-optimal information designer can replace thebuyer-optimal signal distribution by the seller-worst distribution; however, since this will generate a higher total surplus (byconvexity) and strictly lower seller revenue (by the definition of the seller-worst distribution), it contradicts the optimality ofthe buyer-optimal distribution. tructure also provides an upper bound of the buyers’ surplus under the symmetric equilibrium in Yang(2019), it follows that the gap vanishes as the number of buyers goes to infinity.In Yang (2019), the buyers’ surplus goes to zero because of the increasing competition in informationacquisition with more buyers. In our case, the limiting zero surplus is driven by the buyer-optimalinformation designer’s purposeful choice to increase the low virtual value k b in order to reduce theprobability of virtual value 1. The gain from such reduction (in the seller’s revenue) eventually dominatesthe opposite consideration to increase the spread for higher total surplus. Moreover, the total surplusis derived down to the ex ante expectation of a single buyer’s value (i.e., p ) which the seller can fullyextract. However, we will show in Section 5.2 that if we allow for the asymmetric information structure,the buyer-optimal surplus will remain strictly positive even if n → ∞ . Here we outline the steps to solve the information design problems. The key idea, as we mention inthe introduction, is to reduce the problems into tractable finite-dimensional constrained optimizationproblems.1. We first present two preliminary lemmas to restrict the class of distributions of interest to theinformation designer:(a) In solving the seller-worst information structure, we can assume without loss of generalitythat the virtual values are nonnegative. In solving the buyer-optimal information structure,similarly, we can also show that the virtual values are “almost nonnegative” in the sense thatthey are nonnegative except for some mass may be placed at signal x = 0 (Lemma 3).(b) We can show that the buyer-optimal signal distribution must be almost regular and the seller-worst signal distribution must be regular (Lemma 4).2. We change our choice variable from the distribution of signals to the distribution of virtual values(Lemma 6).
3. We show that the reformulation after change of variable leads to a tractable reduction to a finite-dimensional problem. When n = 1, the reduction yields the solution in the one-buyer case derivedby Roesler and Szentes (2017). Moreover, this approach can still be applied to derive a explicitsolution even when n ≥ We first establish the following two lemmas. The reason why we use change of variable is that it is not easy to choose the distribution and keep its almost regularity atthe same time. x θ x D i s t r i bu t i o n : G ( x ) OriginalModified x x ˆ ϕ (0)01 x V i r t u a l v a l u e : ˆ ϕ ( x ) Maybe negativeAlmost nonnegativeFigure 1: Total surplus is larger, while the seller’s revenue is the same. x x x x θ x D i s t r i bu t i o n : G ( x ) ModifiedFurther modified x x x x ϕ (0)01 x V i r t u a l v a l u e : ˆ ϕ ( x ) Almost nonnegativeNonnegativeFigure 2: Nonnegative virtual value distribution generates less the seller’s revenue
Lemma 3.
Any optimal signal distribution G which solves the information design problem in (1) mustinduce almost nonnegative virtual values (i.e., G ∈ G + H ); moreover, if G is a solution to (1) for α = 0 , itmust induce nonnegative virtual values almost everywhere on [0 , .Proof. See Appendix A.3.Figure 1 illustrates how to improve the information designer’s objective (for both the buyer-optimalcase and the seller-worst case) by modifying a signal distribution into one which admits almost non-negative virtual values. First, the red curve is a strict mean-preserving spread of the blue curve in thesub-figure on the left side; hence, by Lemma 2, the red curve can generate more expected total surplus.Second, the nonnegative ironed virtual value of the red curve coincides with that of the blue curve inthe sub-figure on the right side. As the seller will only allocate the good to a buyer with the highestnonnegative virtual value, the seller’s revenue remains the same. Therefore, the red curve generatesstrictly higher surplus for the buyers than the blue curve.Figure 2 illustrates how to further decrease the seller’s revenue by further modifying a distributioninto one which admits only nonnegative virtual values. First, the green curve generates the same meanas the red one in the sub-figure on the left side; hence the green curve is a feasible distribution. Second, x x θ − ζx − k x D i s t r i bu t i o n : G ( x ) IrregularAlmost Regular x x x ϕ (0) k x − − Gg = k Ironing x V i r t u a l v a l u e : ˆ ϕ ( x ) IrregularAlmost RegularFigure 3: An improvement by almost regular distributions the virtual values of the red curve is strictly high than those of the green cure over the interval ( x , x );hence the green curve generates strictly less revenue than the red one. As we mentioned, by raising anegative virtual value to zero, the information designer can decrease the probability assigned to a positivevirtual value and thereby lower the seller’s revenue. Lemma 4.
Any optimal signal distribution G which solves the information design problem in (1) mustbe almost regular; moreover, if G is a solution to (1) for α = 0 , it must be regular.Proof. See Appendix A.4.By Lemma 4, we will use ϕ instead of ˆ ϕ to denote the virtual value hereafter. Figure 3 illustrates howto improve the buyer-optimal information designer’s objective with almost regular distributions. First,the red curve is a strict mean-preserving spread of the blue curve in the first sub-figure. Hence, byLemma 2 and Lemma 3, the red curve can generate more expected total surplus. Second, the ironedvirtual values of the blue curve will be weakly higher than those of the red curve in the second sub-figure;hence, the seller’s revenue is weakly less. Overall, the buyer-optimal information designer’s objective valueis higher under the red curve than under the blue curve. For the seller worst case, it directly followsfrom the nonnegativity of Lemma 3 that an optimal signal distribution must be regular.Here we are able to argue that (almost) regularity entails no loss of generality, because we assumethat the seller chooses Myerson’s optimal auction. As we will argue in Section 6.2, this is no longer thecase when the seller is committed to using a second-price auction (with reserve).
We now introduce the key step of changing our control variables in the information design problems.Let F ( k ) be the distribution of virtual values given a feasible signal distribution G . Since G is almost We draw the blue curve above the red curve on the ironed interval [ x , x ] because the Pareto signal distribution withvirtual k first-order stochastically dominates the original signal distribution; see Lemma 11 in Appendix A.1. egular, F ( k ) = P rob G { x | ϕ ( x ) ≤ k } . Then, except for x = 0, the virtual value of G at signal x is ϕ ( x ) = k = x ( k ) − − G ( x ) G (cid:48) ( x ) = x ( k ) − − F ( k ) F (cid:48) ( k ) x (cid:48) ( k ) . (3)The change of variable enables us to express the true value x ( k ) in terms of the virtual value k , when ϕ is strictly increasing in x : Lemma 5.
Suppose that ϕ is strictly increasing in x . For each k with ϕ ( x ) = k , x ( k ) is the buyer’sexpected virtual value conditional on his virtual value being greater than or equal to k , i.e., x ( k ) = E [ ϕ | ϕ ≥ k ] = k + (cid:82) k (1 − F ( s ))d s − F ( k ) . (4)Since ϕ ( x ) is strictly increasing in x , we have x (cid:48) ( k ) − F (cid:48) ( k )1 − F ( k ) · x ( k ) = − kF (cid:48) ( k )1 − F ( k ) . Solving the first-order ordinary differential equation, we obtain x ( k ) as in Lemma 5.In fact, we can still make use of the expression of x ( k ) in (4), even when ϕ ( x ) is only weaklyincreasing (i.e., ϕ is almost regular). To see this, note that any weakly increasing function can beuniformly approximated by a strictly increasing function. Let { ϕ m } ∞ m =1 be a sequence of strictly in-creasing functions converging uniformly to ϕ . For each m , let G m and F m be sequences of signaldistributions and virtual value distributions corresponding to ϕ m . Specifically, by solving Equation(3), G m ( x ) = 1 − exp (cid:0)(cid:82) x ( ϕ m ( t ) − t ) − d t (cid:1) and F m is the virtual value distribution induced by G m .Since { ϕ m } converges uniformly to ϕ , we also have { G m } and { F m } uniformly converge to G and F ,respectively. In Appendix A.5, we will use the expression in (4) to establish the following two equations: (cid:90) x d G ( x ) = 1 − (cid:90) G ( x )d x = (cid:90) (1 − F ( k ))(1 − log(1 − F ( k )) − log(1 − F (0 − )))d k. (5) (cid:90) x d G n ( x ) = 1 − (cid:90) G n ( x )d x = (cid:90) n (1 − F ( k )) (cid:32) n − (cid:88) i =1 − F i ( k ) i − log(1 − F ( k )) + (cid:32) n − (cid:88) i =1 F i (0 − ) i + log(1 − F (0 − )) (cid:33)(cid:33) − F n ( k )d k + 1 . (6)where G (0) = F (cid:0) − (cid:1) . Indeed, we show in Appendix A.5 that the equations in (5) and (6) follow foreach G m and F m and then apply the bounded convergence theorem.The equation (5) is the expected mean and the equation (6) is the total surplus given the distribution G . Hence, we have the following lemma, Lemma 6.
After the change of variable, the information designer’s problem in (1) can be written as ollows: max F ( k ) (cid:90) αn (1 − F ( k )) (cid:32) n − (cid:88) i =1 − F i ( k ) i − log(1 − F ( k )) + (cid:32) n − (cid:88) i =1 F i (0 − ) i + log(1 − F (0 − )) (cid:33)(cid:33) d k (7)+ (1 − α ) (cid:90) F n ( k )d k + ( α − s.t. (cid:90) (1 − F ( k ))(1 − log(1 − F ( k )) − log(1 − F (0 − )))d k = p. Proof.
It directly follows from Equations (5) and (6). n = 1 : Roesler and Szentes (2017) revisited We are now ready to solve the information design problem for the case with n = 1, which is analyzed inRoesler and Szentes (2017). For n = 1, we can rewrite the information designer’s problem asmax F ( k ) α (cid:90) x d G (cid:124) (cid:123)(cid:122) (cid:125) total surplus − (cid:90) k d F ( k ) (cid:124) (cid:123)(cid:122) (cid:125) seller’s revenue s.t. (cid:90) (1 − F ( k ))(1 − log(1 − F ( k )) − log(1 − F (0 − )))d k = p. When n = 1, the total surplus (cid:82) x d G is linear in G ; moreover, (cid:82) x d G = p by the mean constraint. Therefore, the value of α has no effect on the optimization. This implies that the buyer-optimal infor-mation structure is equivalent to the seller-worst information structure. Since for the seller-worst case,the virtual value is always nonnegative, we have G (0) = F (0 − ) = 0, i.e., there is no mass on x = 0.The information design problem is an isoperimetric problem in optimal control theory; see Theorem4.2.1 of van Brunt (2004). To solve the optimal control problem, we can write the Lagrangian formulaas L ( F, λ ) = αp + (cid:90) ( F − λ ((1 − F )(1 − log(1 − F )) − p )) d k − . Let θ = F ( k ). Then, for each k , the Euler-Lagrange equation implies that ∂ L /∂θ = 1 − λ log(1 − θ ) = 0 . Since λ is constant for any k , there exists a unique θ ∗ ∈ (0 ,
1) such that F ( k ) should be constant andequal to θ ∗ except k = 1. That is, the support of F ( k ) has at most two points k and 1.Now, the information designer only need to choose { k, F ( k ) = θ } to maximizemax k ≥ ,θ αp − ( θ × k + (1 − θ ) × k + (1 − k )(1 − θ )(1 − log(1 − θ )) = p. The Lagrangian is L ( k, θ, λ, µ ) = αp − kθ − (1 − θ ) + λ ( p − ( k + (1 − k )(1 − θ )(1 − log(1 − θ )))) + µk. Here, we do not use the virtual value distribution to represent the total surplus when n = 1, since even if we change thevariables, the total surplus is still p by the mean constraint. ith the Euler-Lagrangian equation for θ : ∂ L ∂θ = (1 − k ) (1 − λ (log(1 − θ ))) = 0 ⇔ λ = 1 / log(1 − θ )and the Euler-Lagrangian equation for k : ∂ L ∂k = − θ − λ ( θ + (1 − θ ) log(1 − θ )) + µ = θ + log(1 − θ ) − log(1 − θ ) + µ = 0 . Since θ +log(1 − θ ) − log(1 − θ ) < θ ∈ (0 , µ >
0, and therefore, the optimal k = 0. In summary, we havereproduced the optimal signal distribution derived in Roesler and Szentes (2017), namely, G ( x ) = − − θx if x ∈ [1 − θ, x = 1 . Under the optimal signal distribution, for x ∈ [1 − θ, θ . For x = 1, the virtual value is 1 with probability 1 − θ . n ≥ Similarly to the case with n = 1, we can reduce the infinite-dimensional information design problem intoa finite-dimensional problem by the following lemma. Lemma 7.
The support of any optimal virtual value distribution F ( k ) has at most two points, say { k, } .Proof. Again, the information design problem is also an isoperimetric problem in optimal control theory.Define the following Lagrangian formula, L ( F, λ ) = (cid:90) αn (1 − F ( k )) (cid:32) n − (cid:88) i =1 − F i ( k ) i − log(1 − F ( k )) + (cid:32) n − (cid:88) i =1 F i (0 − ) i + log(1 − F (0 − )) (cid:33)(cid:33) + (1 − α ) F n ( k )d k + (cid:90) − λ (1 − F ( k ))(1 − log(1 − F ( k )) − log(1 − F (0 − ))) (cid:19) d k + pλ + ( α − θ = F (0 − ) = G (0) and θ = F ( k ). By Theorem 4.2.1 of van Brunt (2004), the Euler-Lagrangeequation for L and each state k should be satisfied as follows: I α ( θ ) ≡ nα n − (cid:88) i =1 θ i /i + n (2 − α ) θ n − + nα log(1 − θ ) − λ log(1 − θ )+ (cid:32) − nα n − (cid:88) i =1 θ i /i − ( nα − λ ) log(1 − θ ) (cid:33) = 0 . Taking derivative of I α ( θ ) with respect to θ , we have: I (cid:48) α ( θ ) = λ + nθ n − ((2 − α )( n −
1) + (2 + α ( n − − n ) θ )1 − θ . We prove the following lemma in Appendix A.7: θ θ -0.700.7 I ( θ ) (cid:37) I ( θ ) (cid:38) I ( θ ) (cid:38) θ I (cid:48) ( θ ) θ θ -0.500.5 local min local max θ I ( θ ) Figure 4: The curve of I (cid:48) ( θ ) and I ( θ ). Lemma 8.
There is at most one θ with I α ( θ ) = 0 which also satisfies the second-order condition. By Lemma 8, for any state k , F ( k ) is constant except k = 1. Moreover, F (1) = 1. Thus, the supportof virtual values has two points { k, } with F ( k ) = θ and F (1) = 1.Therefore, the information designer will choose the optimal k, θ , and θ (with θ = F ( k ) and θ = G (0) = F (0 − )) such thatmax { θ ,k,θ } (cid:32) αn (1 − k )(1 − θ ) (cid:32) n − (cid:88) i =1 − θ i i − log(1 − θ ) + (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33)(cid:33)(cid:33) + ( α −
1) + (1 − α )(1 − k ) θ n + (1 − α ) kθ n (8)s.t. (1 − k ) ((1 − θ )(1 − log(1 − θ ) + log(1 − θ )) + k (1 − θ ) = p,k ≥ , θ ≥ , and θ ≤ θ ≤ . The solution to this finite-dimensional optimization problem is standard and we present the details inAppendices A.8 and A.9. In fact, the solution to this finite-dimensional problem is unique. Since theinformation design problem has at least a solution by Lemma 1, this unique solution is globally optimaland this concludes the proof of Theorems 1 and 2.We briefly comment on how our approach differs from that of Roesler and Szentes (2017). Whenthere is only one buyer, a posted price mechanism is optimal; see Proposition 2.5 of B¨orgers and Krahmer(2015). Hence, a signal distribution matters only in determining the optimal posted price, i.e., ϕ − (0).This is how Roesler and Szentes (2017) are able to argue that it entails no loss of generality to focuson a class of Pareto distribution and the two-point Pareto distribution with virtual values 0 and 1 isthe buyer-optimal information structure. When there are multiple buyers, we may take the second-priceauction with an optimal reserve price as an extension of the posted price mechanism. Unlike Roesler We also draw the curve of I α ( θ ) and I (cid:48) α ( θ ) in Figure 4 to illustrate this lemma. In Figure 4, we choose the parameters tobe n = 3, α = 1, λ = − .
5, and θ = 0. nd Szentes (2017), however, a second-price auction with a reserve price need not be an optimal auctionagainst an irregular signal distribution.To wit, while an irregular signal distribution can be ironed into a regular signal distribution, theoptimal expected revenue under the irregular distribution is the same as the revenue of a second-priceauction with reserve under the regular distribution (obtained from ironing) rather than the irregulardistribution. We show in Section 6.2 when the seller is committed to using a second-price auctionwith no reserve price and n = 2, the seller-worst information structure is fully revealing. Fully revealingresults in the binary prior which is an irregular distribution. A recent paper by Suzdaltsev (2020) studies an optimal reserve price of a second-price auction to maximizeexpected revenue guarantee when the seller knows only the mean of buyers’ value distribution and anupper bound on values. Under the assumption that the value distribution is identically and independentlydistributed across the bidders, Suzdaltsev (2020) shows that it is optimal to set the reserve price to seller’sown valuation which is zero in our setup.In the information design problem which we analyze, since the seller moves after the informationdesigner, our seller-worst revenue is an upper bound of the revenue guarantee in Suzdaltsev (2020).Moreover, it follows from Theorem 1 that the seller-worst signal distribution is regular and has nonneg-ative virtual values. As a result, our seller will also set the reserve price equal to zero in response to theseller-worst information structure. However, our seller-worst signal distribution is derived with respectto Myerson’s optimal auction and may no longer be revenue-minimizing when the seller is committed tousing a second-price auction with an optimal reserve price. Hence, we havethe seller-worst revenue ≥ min signal max reserve Revenue ≥ max reserve min signal Revenue = Suzdaltsev’s revenue . In fact, our result shows thatthe seller-worst revenue > Suzdaltsev’s revenue.For instance, when n = 2 and p = 0 .
5, the seller-worst revenue is 2 a − a = 0 . a satisfies a − a log( a ) = p , which is obtained from truncated Pareto distribution, whereas Suzdaltsev’s revenue is p = 0 .
25 which is obtained from fully revealing; see Suzdaltsev (2020). To see the intuition, considerthe case with n = 2. Firstly, fully revealing is not optimal in our model. This is because under fullyrevealing, the seller is better off changing the reserve price from 0 to 1. Secondly, our seller-worst signaldistribution is no longer revenue-minimizing if the seller is committed to using a second-price auction For the sake of completeness, we provide an example in Appendix A.6 to illustrate this point; see also the working paperversion of Monteiro and Svaiter (2010) for a similar example. In particular, our example admits a density and nonnegative(ironed) virtual values for every signal. ith no reserve price. Since the seller obtains the minimum of the two buyers’ values which is a concavefunction, the information designer would like to maximize the spread by fully revealing; see Propositionin Section 6.2. So far we assume that the information designer chooses the same signal distribution across all buyers. Anatural question is whether the information designer can do better by choosing different signal distribu-tions for different buyers. The short answer is “No” for the seller-worst information design problem and“Yes” for the buyer-optimal information design problem. We explain further below.To allow for asymmetric signal distribution, we first need to reformulate the information designproblems. Let M ( x ) = { i ∈ N | ˆ ϕ ( x i | G i ) ≥ max j { ˆ ϕ ( x j | G j ) , }} be the set of buyers who have the largestnonnegative virtual value and M (cid:48) ( x ) = { i ∈ N | x i ≥ max j x j , ∀ j ∈ M ( x ) } be the set of buyers who notonly have the largest virtual value but also the largest signal among those with the highest virtual value.Then, the optimal auction allocation rule for buyer i when all buyers report their signals is given by, q i ( x i , x − i ) = | M (cid:48) ( x ) | , if i ∈ M (cid:48) ( x );0 , if i (cid:54)∈ M (cid:48) ( x ) . We now study the following information design problem:max { G i ( x i ) } ni =1 (cid:90) [0 , n n (cid:88) i =1 ( αx i − ˆ ϕ ( x i | G i )) q i ( x i , x − i ) n (cid:89) i =1 (d G i ( x i )) (9)s.t. (cid:90) − G i ( x i )d x i = p, ∀ i = 1 , · · · , n. In this section, we show that the optimal symmetric seller-worst information structure in Theorem1 remains the unique optimal seller-worst information structure, even if the information designer canchoose an asymmetric information structure. We first document the existence of the solution to problem(9) when α = 0. The proof is similar to the proof of Lemma 1 and omitted. Lemma 9.
For the problem in (9) with α = 0 , an optimal solution exists. The following lemma corresponds to Lemmas 3 and 4. The proof is similar and we only provide asketch in Appendix B.1. Che (2019) also studies the optimal reserve price which maximizes the expected revenue guarantee of a second-price auctionunder mean constraints. However, Che (2019) allows correlated signal distributions and hence the difference in his optimalrevenue guarantee and our seller-worst revenue is not purely driven by the order of moves between the information designerand the mechanism designer. emma 10. Any optimal signal distribution G which solves the information design problem in (9) mustbe regular and induce nonnegative virtual values almost everywhere on [0 , . Then, similarly to Lemma 6, it follows from Lemma 10 that the asymmetric information designproblem can be reformulated as:max { F i ( k ) } ni =1 (cid:90) n (cid:89) i =1 F i ( k )d k − (cid:90) (1 − F i ( k ))(1 − log(1 − F i ( k )))d k = p, ∀ i = 1 , · · · , n. We state and prove the following theorem.
Theorem 3.
The unique seller-worst information structure in Theorem 1 remains the unique seller-worstinformation structure which solves the problem in (1) with α = 0 .Proof. For any profile of virtual value distributions { F i } ni =1 , let F ( k ) ≡ n (cid:80) ni =1 F i ( k ) and denote by F a symmetric signal distribution profile where each buyer receives his signal according to F .First, the symmetric signal distribution F yields weakly less revenue. For any k , by the inequality ofarithmetic and geometric means, we have F n ( k ) = (cid:32) n n (cid:88) i =1 F i ( k ) (cid:33) n ≥ n (cid:89) i =1 F i ( k ) . (11)Moreover, equality in (11) holds if and only if F i ( k ) = F for all i . Integrating both sides yields that (cid:82) F n ( k )d k ≥ (cid:82) (cid:81) ni =1 F i ( k )d k . Hence, F yields weakly less revenue than { F i } ni =1 and strictly lessrevenue when F i (cid:54) = F for some i .Second, F generates weakly higher mean than p . Indeed, the integral term in the mean constraint isstrictly concave with respect to F ( k ). That is, I (cid:48)(cid:48) ( θ ) = − / (1 − θ ) < I ( θ ) = (1 − θ )(1 − log(1 − θ )).Moreover, since F i admits only nonnegative virtual values in [0 , F also admits only nonnegative virtualvalues in [0 , F ( k ) = F ( k − ) , if k ∈ [0 , k ) F ( k ) , if k ∈ [ k , . for some k so that ˆ F satisfies the constraint in (10). Since ˆ F ( k ) ≥ F ( k ) for any k , (cid:82) ˆ F n ( k )d k ≥ (cid:82) F n ( k )d k and ˆ F yields less revenue than F . Therefore, the improvement from { F i } ni =1 to ˆ F is strictwhen F i (cid:54) = F for some i . It follows taht the symmetric seller-worst information structure in Theorem 1remains the unique seller-worst information structure for the problem in (10). For the buyer-optimal information design problem, we demonstrate that an asymmetric signal distribu-tion can strictly improve upon the optimal symmetric signal distribution in Theorem 2. We demonstratesuch an improvement for n = 2 with p < r b and n → ∞ . roposition 1. For n = 2 with p < r b or n → ∞ , there exist asymmetric information structureswhich can strictly improve upon the optimal symmetric signal distribution in Theorem 2 and Corollary1, respectively.Proof. See Appendix B.2. To see the main idea, for n = 2 with p < r b , we focus our search of improvementon the signal distributions which put positive mass only on signal 0, signals with virtual value 0, andsignals with virtual value 1 for each buyer. Since each buyer’s virtual value distribution needs to satisfythe mean constraint, it is uniquely determined by its probability assigned to signal 0. We then optimizewithin the specific class of information structures with these two variables. For n → ∞ , we also considera specific class of information structures in which (i) the buyer i ’s signal distribution puts positive massonly on signal 0, signals with virtual value p , and signals with virtual value 1 and (ii) all the other buyers’signal distributions are a degenerate distribution which puts the entire mass on signal p .Proposition 1 shows that the buyer-optimal signal distributions (if there exists one) are asymmetricin general. Indeed, while averaging the signal distributions ( F = n F i ) can reduce the revenue, it mightreduce the total surplus. The issue is reminiscent of Bergemann and Pesendorfer (2007) which showsthat an seller-optimal/revenue-maximizing information structure is asymmetric across the buyers. Theexistence and exact shape of an asymmetric buyer-optimal information structure remains unknown tous. In this section, we discuss the issues of generalizing our results beyond the ex ante symmetric binary-value setting. In particular, we will discuss the issues with asymmetric priors, continuous priors, and aninformation structure with value interdependence. In discussing each of these three aspects, we will keepthe rest as is in order to focus on the specific issue at hand.
Suppose that each buyer i has his own prior mean p i . We only discuss the seller-worst problem for whicha solution exists. The seller-worst information problem is to maximize the same objective function in(10) with each buyer i ’s individual mean constraint now being: (cid:90) (1 − F i ( k ))(1 − log(1 − F i ( k )) − log(1 − F i (0 − )))d k = p i . In this case, our arguments for regularity and almost nonnegative virtual values are still valid. For n = 2,we can still obtain the seller-worst information structure in the following proposition. roposition 2. Suppose that n = 2 and each buyer i has prior mean p i . Then, the optimal seller-worstsignal distribution for buyer i is a truncated Pareto distribution G i ( x ) such that G i ( x ) = − x i x if x ∈ [ x i , if x = 1 , where x i is determined by buyer i ’s mean constraint.Proof. See Appendix C.1.Thus, both buyers have the virtual values { , } and the degree of asymmetry between p and p isonly reflected by x and x . For n ≥
3, the seller-worst problem remains an isoperimetric problem andwe can similarly reduce it to a finite-dimensional constrained optimization problem we do in Theorem 1.The difficulty is that the finite-dimensional problem becomes intractable and its closed-form solution(s)remain unknown to us.
We now consider continuous prior distributions. Our analysis in Sections 3–5 remains applicable, if theinformation designer only knows the mean about the prior and imposes the mean constraint. However,if the information designer has full information about the prior, then a signal distribution is feasible ifand only if the prior is a mean-preserving spread of the distribution; see (Blackwell, 1953). That is, theset of feasible signal distributions becomes G H = (cid:26) G : [0 , (cid:55)→ [0 , (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) x d G ( x ) = p, (cid:90) x G ( t )d t ≤ (cid:90) x H ( t )d t, ∀ x ∈ [0 , (cid:27) . The main issue here is to handle the mean-preserving spread constraint on the signal distributions.Change of variable is no longer useful and we need a different tool akin to the method developed inDworczak and Martini (2019). Moreover, our objective function here is also different from that inDworczak and Martini (2019). Consider, for instance, the case of a second-price auction (with noreserve). The seller-worst problem can be expressed asmax G (cid:90) (cid:0) nG n − ( x ) − ( n − G n ( x ) − (cid:1) d x (12)s.t. H is mean preserving spread of G. We obtain the following result for the case with two buyers:
Proposition 3.
For n = 2 , fully revealing (i.e., G = H ) solves the problem in (12).Proof. The objective is (cid:90) G − G − x = − (cid:90) x d (cid:0) G − G (cid:1) = (cid:90) x d G ( x ) − p. oreover, (cid:82) x d G ( x ) is maximized if G = H , since G is the CDF of the convex function max { x , x } when x and x are independently distributed according to G . Hence, fully revealing minimizes thesellers’ revenue.Proposition 3 has an implication on both the seller-worst problem and the buyer-optimal problemwith two buyers. Specifically, we can still argue that the seller-worst signal distribution must be regular,symmetric, and admit only nonnegative virtual values. Since second-price auction with reserve is optimalfor such signal distributions, identifying a seller-worst information structure amounts to solving theproblem in (12). If the prior distribution is regular and admits nonnegative virtual values, it is also afeasible choice in this problem. Hence, Proposition 3 applies and fully revealing is seller-worst. Moreover,since fully revealing maximizes the total surplus, it is also buyer-optimal. In summary, we have thefollowing corollary: Corollary 2.
For n = 2 , if the prior distribution is regular and admits nonnegative virtual values, thenfully revealing is both the unique symmetric buyer-optimal and seller-worst information structure. Note that Corollary 2 requires that the prior be regular and admit nonnegative virtual values; henceit rules out the binary prior which we analyze in Theorem 1. If n ≥ In studying a two-buyer common-value problem, Bergemann, Brooks, and Morris (2016) constructs aseller-worst information structure in which the buyers share a common interim value function dependingon the entire signal profile and each signal is independently drawn from a uniform distribution. As theyalso assume that ex post common value is either 0 or 1, we may interpret the interim valuation as theprobability that the buyers have ex post common value 1. A natural question is whether we can mimicthe construction of Bergemann, Brooks, and Morris (2016) in setting the interim value function as: v i ( s , s ) = min (cid:26) a (1 − s )(1 − s ) , (cid:27) , ∀ i = 1 , , (13)where a is determined by E s ,s [ v i ] = a (cid:0) − log( a ) + log ( a ) (cid:1) = p . However, with the interdependentvalues, we must also replace the mean constraint with the consistency condition, namely that v i ( · , · )together with the uniform distribution on S × S induces the same distribution as the prior H on { , } × { , } .In Appendix C.2, we show that the interim value function defined in (13) violates the consistencycondition and hence is not a feasible choice. The main reason is that interim value interdependence in(13) induces a correlated rather than independent distribution on { , }×{ , } . Of course, there may be ther candidate interim value functions v i ( · , · ) but their explicit form remains elusive to us. In particular,Brooks and Du (2020b) give a simulation result with interdependent values in which the seller obtainsstrictly lower revenue than our seller-worst revenue. References
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Working paper . ppendix A Proof for Section 4
We first prove a preliminary lemma which will be used in the proof for Section 4.
A.1 A Preliminary Lemma
Let x − and x +0 are the left and right hand limit of a signal x in [0 , Lemma 11.
For any distribution G , any x ∈ [0 , , and any k ∈ [ ˆ ϕ ( x − | G ) , ˆ ϕ ( x +0 | G )] , ( x − k )(1 − G ( x )) ≤ ( x − k )(1 − G ( x − )) , ∀ x ∈ [0 , . (14) Proof.
Consider a profit maximization problem of a monopolist with a constant marginal cost k . Themonopoly price ˆ ϕ − ( k ) = x is the posted price that obtains the highest expected profit; see, e.g.,Proposition 2.5 of B¨orgers and Krahmer (2015). Hence, its expected revenue is ( x − k )(1 − G ( x − )).Thus for any other posted price x ∈ [0 , x − k )(1 − G ( x )) ≤ ( x − k )(1 − G ( x − )) ≤ ( x − k )(1 − G ( x − )) . Hence, Lemma 11 holds. We can rearrange the inequality (14) to obtain G ( x ) ≥ − ( x − k )(1 − G ( x − )) x − k . In particular, the right hand side is a Pareto distribution function which generates a constant virtualvalue k . Hence, the lemma basically says that the Pareto distribution first-order stochastically dominatesany other distribution G . A.2 Proof of Lemma 1
Proof.
When α = 1, we first consider a following information design problem:max G ( · ) ∈G H (cid:32)(cid:90) [0 , n max { x , · · · , x n } n (cid:89) i =1 (d G ( x i )) − (cid:90) [0 , n n (cid:88) i =1 ( ˆ ϕ ( x i | G )) q i ( x i , x − i ) n (cid:89) i =1 (d G ( x i )) (cid:33) . (15)Denote by V ( G ) and V ( G ) the objective of problem (1) and problem (15) under the signal distri-bution G respectively.First, since max { x , · · · , x n } is continuous in x , the first term in (15) is continuous in G . Moreover,by Theorem 2 of Monteiro (2015), the expected revenue is a lower semicontinuous function in G . Hence, V ( G ) is an upper semicontinuous function in G . Also since G H is a closed subset of the set of Borelprobability measures on [0 , G H is compact. Thus, by the extreme value theorem, an optimal solutionof the problem in (15) exists. Let G (cid:63) be the optimal solution to the problem (15). econd, for any signal distribution G which induces negative virtual values with positive probability,we take the same modified distribution ˜ G θ ∈ G + H as follows,˜ G θ ( x ) = θ if x ∈ [0 , x θ );1 − x (1 − G ( x − )) x if x ∈ [ x θ , x ); G ( x ) if x ∈ [ x , , where θ denotes the mass on x = 0. Since the virtual value on x is 0 ∈ [ ˆ ϕ ( x − | G ) , ˆ ϕ ( x +0 | G )], by Lemma11, when x ∈ [ x θ , x ], we have G ( x ) ≥ ˜ G θ ( x ). If θ = 0, then ˜ G first-order stochastically dominates G ( x ), and thereby (cid:82) x d ˜ G ≥ (cid:82) x d G ( x ). If θ = G ( x − ), then x θ = x . Hence, ˜ G G ( x − ) is first-order stochastically dominated by G ( x ) and (cid:82) x d ˆ G G ( x − ) ≤ (cid:82) x d G ( x ). Since (cid:82) x d ˆ G θ ( x ) is continuousand strictly decreasing in θ , it follows from the intermediate-value theorem that there exists a unique θ ∈ [0 , G ( x − )] such that (cid:82) x d ˜ G θ = p . Moreover, by Lemma 11, since G assigns positive probabilitieson negative virtual values for signals in (0 , θ ∈ (0 , G ( x − )). Hence, ˜ G θ is a feasible signaldistribution with almost nonnegative virtual value; moreover, ˜ G θ is a strict mean-preserving spread of G . First, since the first term of the objective is convex in signal profile, the expectation of the first termunder ˜ G θ is strictly greater than that under G . Meanwhile, since ˆ ϕ (cid:16) x | ˜ G θ (cid:17) = 0 and the seller onlyallocates the good to a buyer with a non-negative virtual value, the expected virtual value is the sameunder G and ˜ G θ . Therefore, V (cid:16) ˜ G θ (cid:17) > V ( G ). Hence, G (cid:63) ∈ G + H .Third, we claim that G (cid:63) also solves the problem (1). To see this, observe that for any G and anysignal realization ( x , · · · , x n ), (cid:80) ni =1 x i q i ( x i , x − i ) ≤ max { x , · · · , x n } . If G ∈ G + H , then for any signalrealization ( x , · · · , x n ), (cid:80) ni =1 x i q i ( x i , x − i ) = max { x , · · · , x n } . Hence, V ( G ) ≤ V ( G ) for any G ∈ G H and V ( G ) = V ( G ) for any G ∈ G + H . (16)Finally, max G ∈G H V ( G ) ≤ max G ∈G H V ( G ) = V ( G (cid:63) ) = V ( G (cid:63) ) . where the first inequality and the third equality follows from (16) and the second equality follows fromthe definition of G (cid:63) . Hence, G (cid:63) solves the problem in (1). Hence an optimal solution exists in theproblem (1). A.3 Proof of Lemma 3
Proof. The case with α = 1: Let G be a signal distribution which assigns positive probability on negativevirtual values for signals in (0 , x = inf { x | ˆ ϕ ( x | G ) ≥ } . Define ˜ G θ ( x ) such that˜ G θ ( x ) = θ if x ∈ [0 , x θ );1 − x (1 − G ( x − )) x if x ∈ [ x θ , x ); G ( x ) if x ∈ [ x , , here θ denotes the mass on x = 0. Since the virtual value on x is 0 ∈ [ ˆ ϕ ( x − | G ) , ˆ ϕ ( x +0 | G )], byLemma 11, when x ∈ [ x θ , x ], we have G ( x ) ≥ ˜ G θ ( x ). If θ = 0, then ˜ G first-order stochasticallydominates G ( x ), and thereby (cid:82) x d ˜ G ≥ (cid:82) x d G ( x ). If θ = G ( x − ), then x θ = x . Hence, ˜ G G ( x − ) is first-order stochastically dominated by G ( x ) and (cid:82) x d ˆ G G ( x − ) ≤ (cid:82) x d G ( x ). Since (cid:82) x d ˆ G θ ( x ) iscontinuous and strictly decreasing in θ , it follows from the intermediate-value theorem that there existsa unique θ ∈ [0 , G ( x − )] such that (cid:82) x d ˜ G θ = p . Moreover, by Lemma 11, since G assigns positiveprobabilities on negative virtual values for signals in (0 , θ ∈ (0 , G ( x − )).Hence, ˜ G θ is a feasible signal distribution with almost nonnegative virtual value; moreover, ˜ G θ isa strict mean-preserving spread of G . Therefore, by Lemma 2, the total surplus under ˜ G θ is strictlygreater than the total surplus under G . In addition, since ˆ ϕ (cid:16) x | ˜ G θ (cid:17) = 0 and the seller only allocatesthe good to a buyer with a non-negative virtual value, the expected virtual value is the same under G and ˜ G θ . Hence, the objective value will be strictly higher as the expected total surplus becomes strictlyhigher and the seller’s revenue remains the same. Hence, any optimal signal distribution G must inducenonnegative virtual values with probability one on (0 , The case with α = 0: Suppose that ˜ G θ puts some positive mass on x = 0. For α = 0, we can furthermodify the distribution ˜ G θ to reduce the seller’s revenue as follows. Define another signal distribution,ˆ G x ( x ) = − − ˜ G θ ( x ) x if x ∈ [1 − ˆ G θ ( x ) , x ];˜ G θ ( x ) if x ∈ ( x , . For x = x , we have ˆ G x ( x ) = ˜ G ( x ) which first-order stochastically dominates ˜ G θ , then (cid:82) x d ˆ G x ≥ (cid:82) x d ˜ G θ , and for x = 1, it is a degenerate distribution with all mass on x = 0. Also since (cid:82) x d ˆ G x is continuous and strictly decreasing in x , the intermediate-value theorem implies that there exists anunique x ∈ ( x ,
1) such that (cid:82) x d ˆ G x = p . Thus, ˆ G x is a feasible information structure.Since ˆ ϕ ( x | ˆ G x ) = 0 ≤ ˆ ϕ ( x | ˜ G θ ), for each realization of signals, we have max { , ˆ ϕ ( x | ˆ G x ) } ≤ max { , ˆ ϕ ( x | ˜ G θ ) } . Moreover, the inequality is strict with positive probability. Hence, the seller’s revenueis strictly lower under ˆ G x than under ˜ G θ . A.4 Proof of Lemma 4
Proof.
For any irregular distribution G with almost nonnegative virtual values, there exists some x (cid:48) > x (cid:48) | G ) < Φ( x (cid:48) | G ). Since G ( x ) is right continuous with respect to x (cid:48) , so is Ψ( x (cid:48) | G ). Therefore,there exists an interval [ x (cid:48) , x (cid:48)(cid:48) ], such that for any x ∈ [ x (cid:48) , x (cid:48)(cid:48) ], Ψ( x | G ) < Φ( x | G ). Let [ x , x ] ⊇ [ x (cid:48) , x (cid:48)(cid:48) ]be an ironed interval such that ˆ ϕ ( x | G ) = k is constant for x ∈ [ x , x ]. And for x ∈ ( x , x ), Ψ( x | G ) ≤ Φ( x | G ). For x = x and x = x , Ψ( x | G ) = Φ( x | G ). Moreover, since Ψ( x | G ) = Φ( x | G ), G ( x ) iscontinuous at x . Then let ˆ G beˆ G = G ( x ) if x (cid:54)∈ [ x , x ];1 − (1 − G ( x ))( x − k ) x − k if x ∈ [ x , x ] . he modified distribution ˆ G has two key features: firstly, it generates the same virtual value as G forany realized signal x ; secondly, by lemma 11, since we have ˆ G ( x ) ≤ G ( x ) and ˆ G ( x ) < G ( x ) on x ∈ ( x , x )(otherwise, G ( x ) is regular), ˆ G will strictly first-order stochastically dominate G ( x ), which implies thatˆ G will generate strictly more mean than p .Hence, we can modify ˆ G again to satisfy the mean constraint. Let ˆ G θ beˆ G θ = θ if x ∈ [0 , x θ ) , ˆ G ( x ) if x ∈ [ x θ , . where θ = ˆ G ( x θ ) denotes the mass putting on signal x = 0.It is clear that (cid:90) x d ˆ G ( x ) = (cid:90) x d ˆ G ( x ) > p > (cid:90) x d ˆ G ( x ) . Since (cid:82) x d ˆ G θ ( x ) is continuous and strictly decreasing in θ , the intermediate-value theorem implies thatthere exists a unique θ ∈ (0 ,
1) such that (cid:82) x d ˆ G θ = p . Therefore, ˆ G θ is a feasible signal distribution.By construction, ˆ ϕ ( x | ˆ G θ ) = − − θ θ < , if x ∈ [0 , x θ );ˆ ϕ ( x | ˆ G ) = ˆ ϕ ( x | G ) , if x ∈ [ x θ , . Hence, max { ˆ ϕ ( x | ˆ G θ ) , } ≤ max { ˆ ϕ ( x | G ) , } .Hence, given any signal distribution G , we can modify a buyer’s signal distribution to ˆ G θ . As thearguments in Lemma 3, this modification has two effects: first, the seller’s revenue (the second termof the objective function) as a max function of nonnegative virtual values becomes weakly less; second,since the constructed distribution ˆ G θ is a strict mean-preserving spread of the original distribution G ,by Lemma 3 and Lemma 2, the total surplus is strictly higher. Therefore, with this modification, thebuyers’ total surplus is strictly higher. Hence,the buyer-optimal distribution must be almost regular.Moreover, this modification still puts positive mass on signal 0 By Lemma 3, we can further modifythis distribution into another distribution with nonnegative virtual values and generate strictly lessrevenue for the seller. Thus, the seller-worst distribution must also be regular. .5 Change of variable Denote G m (0) by F m (0), first, we have (cid:90) x d G m ( x ) = (cid:90) x ( k )d F m ( k )= (cid:90) k + (cid:82) k (1 − F m ( s ))d s (1 − F m ( k )) d F m ( k )= (cid:90) k d F m ( k ) + (cid:90) (cid:90) k − F m ( s )(1 − F m ( k )) d s d F m ( k )= (cid:90) (1 − F m ( k ))d k + (cid:90) (cid:90) s − F m ( k )) d F m ( k )(1 − F m ( s ))d s = (cid:90) (1 − F m ( k ))d k + (cid:90) (1 − F m ( s )) (cid:90) F m ( s ) F m (0 − ) − θ ) d θ d s = (cid:90) (1 − F m ( k ))d k + (cid:90) (1 − F m ( k ))( − log(1 − F m ( k )) + log(1 − F m (0 − )))d k = (cid:90) (1 − F m ( k ))(1 − log(1 − F m ( k )) + log(1 − F m (0 − )))d k. Similarly, we have (cid:90) x d G nm ( x ) = (cid:90) (cid:32) k + (cid:82) k (1 − F m ( s ))d s − F m ( k ) (cid:33) d F nm ( k )= (cid:90) k d F nm ( k ) + n (cid:90) (cid:32) (cid:82) k (1 − F m ( s ))d s − F m ( k ) (cid:33) F n − m ( k )d F m ( k )= (cid:90) k d F nm ( k ) + n (cid:90) (cid:90) k (1 − F m ( s )) F n − m ( k )1 − F m ( k ) d s d F m ( k )= (cid:90) k d F nm ( k ) + n (cid:90) (1 − F m ( s )) (cid:90) s F n − m ( k )1 − F m ( k ) d F m ( k )d s = (cid:90) k d F nm ( k ) + n (cid:90) (1 − F m ( s )) (cid:90) F m ( s ) F m (0 − ) θ n − − θ d θ d s = (cid:90) n (1 − F m ( k )) (cid:32) n − (cid:88) i =1 − F im ( k ) i − log(1 − F m ( k )) + (cid:32) n − (cid:88) i =1 F im (0 − ) i + log(1 − F m (0 − )) (cid:33)(cid:33) − F nm ( k )d k + 1 . Then by bounded convergence theorem, since G m and F m uniformly converge to G and F respectively,the equation (5) and equation (6) hold. A.6 An irregular distribution
Suppose that the support of the signal x is [1 , G ( x ) = x − , if x ∈ [1 , ); x , if x ∈ [ , . The quantile function x ( τ ) of distribution G is given by: x ( τ ) = τ + 1 , if τ ∈ [0 , );2 τ, if τ ∈ [ , . he virtual value without ironing is given by ϕ ( x ) = x − − G ( x ) g ( x ) = x − − (2 x − = 2 x − , if x ∈ [1 , ); x − − x/ / = 2 x − , if x ∈ [ , . Denote Φ( x ) = (cid:82) x ϕ ( t ) g ( t )d t . Then,Φ( x ) = (cid:82) x t − / t = ( x − x − , if x ∈ [1 , ); (cid:82) / t − / t + (cid:82) x / (2 t − t = x ( x −
2) + 1 , if x ∈ [ , . Therefore, Φ( τ ) = Φ( x ( τ )) is given byΦ( τ ) = ( x ( τ ) − x ( τ ) −
1) = τ ( τ + 1) , if τ ∈ [0 , ); x ( τ )( x ( τ ) −
2) + 1 = 1 − τ + 2 τ , if τ ∈ [ , . Denote Ψ( τ ) be the largest convex function such that Ψ( τ ) ≤ Φ( τ ). ThenΨ( τ ) = τ ( τ + 1) , if τ ∈ [0 , ); τ − , if τ ∈ [ , );1 − τ + 2 τ , if τ ∈ [ , . Therefore, the ironed virtual value ˆ ϕ ( τ ) = Ψ (cid:48) ( τ ) is given byˆ ϕ ( τ ) = τ + , if τ ∈ [0 , );1 , if τ ∈ [ , );4 τ − , if τ ∈ [ , . Replace τ by G ( x ) and the ironed virtual value ˆ ϕ ( x ) in terms of x is given byˆ ϕ ( x ) = x − , if x ∈ [1 , );1 , if x ∈ [ , );2 x − , if x ∈ [ , . Note that ˆ ϕ ( x ) ≥ ˆ ϕ (1) = 1 / ϕ ( x ) and the revenue under a second-price auction.1. First, by symmetry, the largest value x (1) induces the highest ironed virtual value and the highestvalue x (1) follows the distribution G . Then, we have E [ ˆ ϕ ( x (1) ) | G ] = (cid:90) ˆ ϕ ( x )d G ( x ) = (cid:90) ˆ ϕ ( x )2 · g ( x ) G ( x )d x = (cid:90) / (2 x − / · x − x + (cid:90) / / · · x − x + (cid:90) / / · x x + (cid:90) / (2 x − · x x = 524 + 736 + 17144 + 23 = 1916 . . Second, the lowest value x (2) follows the distribution 2 G − G , then we have E [ x (2) | G ] = (cid:90) x d(2 G − G ) = (cid:90) x · · g ( x )(1 − G ( x ))d x = (cid:90) / x · · − x )d x + (cid:90) / x · ·
12 (1 − x x = 8281 + 1481 = 9681 = 3227 = 1916 − < . Therefore, a second-price auction with an optimal reserve price 0 obtains strictly less revenue thanthe expected highest ironed virtual value. Hence, the second-price auction with an optimal reserve price0 is not an optimal auction.Note that although G ( x ) can induce the ironed virtual value ˆ ϕ ( x ), the regular distribution ˆ G whichalso induces the same virtual value ˆ ϕ ( x ) will first-order stochastically dominate G . Hence, the expectationof the lowest/second-highest value x (2) of ˆ G is strictly larger than that of G . Formally, ˆ G is given byˆ G ( x ) = x − , if x ∈ [1 , ];1 − x − , if x ∈ ( , ); x , if x ∈ [ , . For x ∈ [1 , ] ∪ [ , G ( x ) = G ( x ). For x ∈ ( , ), ˆ G ( x ) < G ( x ). Hence ˆ G first-orderstochastically dominates G . Moreover, we have E [ x (2) | ˆ G ] = = E [ ˆ ϕ ( x (1) ) | G ]. Of course, this isan example of the standard result that the optimal auction is a second-price auction with an optimalreserve price, provided that the signal distribution is regular. A.7 Proof of Lemma 8
Proof.
First, define J α ( θ ) = θ n − ((2 − α )( n −
1) + (2 + α ( n − − n ) θ ) = (1 − θ ) I (cid:48) α ( θ ) − λ Hence, I (cid:48) α ( θ ) = J α ( θ ) + λ − θ .Observe that J α ( θ ) is first increasing in θ and then decreasing in θ . When θ = 0, we have J α ( θ ) = 0.When θ = 1, we have J (1) = 0 and J (1) = −
1. We discuss the cases with α = 0 and α = 1, respectivelyas follows:1. For α = 0, we have the following four cases:(a) Case 1 . λ ≥
0: In this case, I (cid:48) ( θ ) is always positive for both α = 0. Hence, I ( θ ) is increasing.Thus, I ( θ ) will cross θ -axis from below at most once. The objective function takes a localminimal and hence we ignore the case.(b) Case 2. λ < λ ∗ where λ ∗ = − max θ J ( θ ): Then I ( θ ) is always decreasing. Thus, I ( θ ) crosses θ -axis from above at most once. c) Case 3. λ ∈ [ λ ∗ , I (cid:48) ( θ ) is first negative, then positive, and eventually becomes negative. Inaddition, I ( θ ) = 0 when θ = θ . Hence I ( θ ) will first decrease from 0, then increase, andfinally decrease. In this case, I ( θ ) will cross θ -axis at most twice. Only for the second time, I ( θ ) will cross θ -axis from above.2. For α = 1, we have another four cases:(a) Case 1 . λ ≥
1: In this case, I (cid:48) ( θ ) is always positive. Hence, I ( θ ) is increasing. Thus, I ( θ )will cross θ -axis from below at most once. The objective function takes a local minimal andhence we ignore the case.(b) Case 2. λ ∈ [0 , I (cid:48) ( θ ) is first positive and then negative, therefore, I ( θ ) will first increaseand then decrease. In this case, I ( θ ) will cross θ -axis at most twice. However, only forthe second time, I ( θ ) will cross θ -axis from the above. Hence there is only one θ such that I ( θ ) = 0 and also satisfying the second-order condition.(c) Case 3. λ < λ ∗ where λ ∗ = − max θ J ( θ ): Then I ( θ ) will be always decreasing. Thus I ( θ )will also cross θ -axis from above at most once. (d) Case 4. λ ∈ [ λ ∗ , I (cid:48) ( θ ) is first negative, then positive, and eventually becomes negative. Inaddition, I ( θ ) = 0 when θ = θ . Hence I ( θ ) will first decrease from 0, then increase, andfinally decrease. In this case, I ( θ ) will cross θ -axis at most twice. Only for the second time, I ( θ ) will cross θ -axis from above. A.8 Solving the finite-dimensional seller-worst optimization problem
For finite-dimensional seller-worst information design problem (8) ( α = 0), the Lagrangian is L ( θ, k, λ ) = (1 − k ) θ n + λ ( k + (1 − k )(1 − θ )(1 − log(1 − θ )) − p ) − . The Euler-Lagrange equation with respect to θ is: ∂ L ∂θ = (1 − k ) (cid:0) nθ n − + λ log(1 − θ ) (cid:1) = 0 . Hence, λ = nθ n − − log(1 − θ ) . Then, the Euler-Lagrange equation with respect to k implies ∂ L ∂k = − θ n + λ (1 − (1 − θ )(1 − log(1 − θ )))= θ n − − log(1 − θ ) ( θ log(1 − θ ) + n ( θ + (1 − θ ) log(1 − θ ))) (cid:44) θ n − − log(1 − θ ) I ( θ ) . Since J α ( θ ) is first increasing, and J α (0), its maximum is greater than 0; hence λ ∗ α < In this proof, we ignore the superscript of k s to make it easier to read. e have sign ∂ L ∂k = sign I ( θ ). We then calculate: I (cid:48) ( θ ) = ( n −
1) log(1 − θ ) + θ − θ , and I (cid:48)(cid:48) ( θ ) = ( n − θ − ( n − − θ ) . When n = 2, we have ∂ L ∂k < θ ∈ [0 , k = 0.When n ≥ I (cid:48)(cid:48) ( θ ) = 0 has a unique solution θ = ( n − / ( n − I (cid:48) ( θ ) is decreasing in θ if θ ∈ (0 , θ ) and increasing in θ if θ ∈ ( θ , θ → − , I (cid:48) ( θ ) → ∞ and for y = 0, I (cid:48) ( θ ) = 0. Thus,there exists θ ∈ ( θ ,
1) such that for θ ∈ (0 , θ ), I (cid:48) ( θ ) is less than 0 and for θ ∈ ( θ , I (cid:48) ( θ ) is greaterthan 0. Therefore, I ( θ ) will be decreasing when θ ∈ (0 , θ ) and increasing when θ ∈ ( θ , θ = 0, I ( θ ) = 0, for θ → − , I ( θ ) → ∞ . It follows from the intermediate-value theorem implies thatthere exists a unique θ ∗ ∈ ( θ ,
1) such that I ( θ ∗ ) = 0.The second-order condition implies that if there is a k > θ ∗ , k ) jointly satisfies the meanconstraint, then the optimal k >
0, and k is determined by the mean constraint:(1 − k )(1 − θ ∗ )(1 − log(1 − θ ∗ )) + k = p. More precisely,1. if p ∈ [0 , (1 − θ ∗ )(1 − log(1 − θ ∗ ))], then the mean constraint implies the minimal θ > θ ∗ . Hence I ( θ ) >
0, thereby ∂ L ∂k <
0, thus, the optimal k is still 0 in this case;2. if p ∈ ((1 − θ ∗ )(1 − log(1 − θ ∗ )) , k > θ ∗ , k ) jointly satisfies the meanconstraint. A.9 Solving the finite-dimensional buyer-optimal optimization problem
Consider the problem in (8) with α = 1. First, it is clear that θ should be an interior solution, so theconstraint should be θ < θ < Thus the Lagrangian of the finite-dimensional buyer-optimal problemis L ( θ, θ , k, λ, µ , µ ) = n (1 − k ) (cid:32) n − (cid:88) i =1 − θ i i − log(1 − θ ) + (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33)(cid:33) (1 − θ )+ λ ((1 − k )(1 − θ )(1 − log(1 − θ ) + log(1 − θ )) + k (1 − θ ) − p ) + µ θ + µ k .The Euler-Lagrange equations with respect to θ, θ , k are: ∂ L ∂θ =(1 − k ) (cid:32) n (cid:32) θ n − + (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33) − (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33)(cid:33) + λ (log(1 − θ ) − log(1 − θ )) (cid:33) = 0 ∂ L ∂θ = (1 − k )(1 − θ ) nθ n + λθ ((1 − θ ) + k ( θ − θ )) − (1 − θ ) θ + µ = 0 ∂ L ∂k =(1 − θ ) (cid:32) n (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33) − n (cid:32) n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:33) − λ + λ log(1 − θ ) − λ log(1 − θ )) + λ (1 − θ ) + µ = 0 . In this proof, we also ignore the superscript of k b . hus, we have λ = − n (cid:16) θ n − + (cid:16)(cid:80) n − i =1 θ i i + log(1 − θ ) (cid:17) − (cid:16)(cid:80) n − i =1 θ i i + log(1 − θ ) (cid:17)(cid:17) log(1 − θ ) − log(1 − θ ) . Then, we also have ∂ L ∂θ = (1 − k )(1 − θ ) nθ n + λθ ((1 − θ ) + k ( θ − θ )) − (1 − θ ) θ + µ = 0 ∂ L ∂k =(1 − θ ) (cid:0) − nθ n − − λ (cid:1) + λ (1 − θ ) + µ = 0 .
1. Suppose that k >
0. We have µ = 0. Thus, ∂ L ∂k = 0 and ∂ L ∂θ = 0 jointly imply that λ = n (1 − θ ) θ n − θ − θ . Therefore, ∂ L ∂θ = (1 − k )(1 − θ ) nθ n + λθ ((1 − θ ) + k ( θ − θ )) − (1 − θ ) θ + µ = n (1 − θ ) − (1 − θ ) θ (cid:18) (1 − k ) θ n − + θ n − (1 − θ ) θ − θ + kθ n − (cid:19) + µ = n (1 − θ ) − (1 − θ ) (cid:18) θ n − + θ n − (1 − θ ) θ − θ + k (cid:0) θ n − − θ n − (cid:1)(cid:19) + µ = 0 . Since θ ≥ θ , the first term of the equation above is negative. Thus, µ > θ = 0.Moreover, similar to the seller-worst case, there exists a threshold p b = (1 − θ )(1 − log(1 − θ )),where θ satisfies, (1 − θ ) θ n − log(1 − θ ) + θ n − + n − (cid:88) i =1 θ i /i + log(1 − θ ) = 0 . If p > p b , then the optimal k > θ, k ) jointly satisfies the mean constraint. Moreover, θ , k are determined by θ n − + n − (cid:88) i =1 θ i /i + log(1 − θ )(1 + (1 − θ ) θ n − ) = 0 , (1 − k )(1 − θ )(1 − log(1 − θ )) + k = p.
2. Suppose that k = 0. When p is small enough, θ may still be positive.Suppose that θ ≥ µ ≥
0. Thus, λ ≥
0. Together with ∂ L ∂θ = 0, we have λ = − n (cid:16) θ n − + (cid:16)(cid:80) n − i =1 θ i i + log(1 − θ ) (cid:17)(cid:17) log(1 − θ ) ≥ . Together with the mean constraint, we have there exists a threshold r b = (1 − θ ) (cid:16) θ n − + (cid:80) n − i =1 θ i i (cid:17) where θ satisfies − log(1 − θ ) = θ n − + (cid:80) n − i =1 θ i i , such that if p ≥ r b , then optimal θ = 0; and if p < r b , then the optimal θ > θ , θ are pinned down by n − (cid:88) i =1 θ i i + (cid:18) − p − θ (cid:19) θ n − = θ n − + n − (cid:88) i =1 θ i i + (cid:18) − p − θ (cid:19) , (1 − θ )(1 − log(1 − θ ) + log(1 − θ )) = p. . It remains to check that (i) θ is unique; (ii) θ is unique; and (iii) r b ≤ p b .(a) Let (cid:96) ( θ ) = θ n − + n − (cid:88) i =1 θ i i + log(1 − θ ) , then ∂(cid:96) ∂θ = θ n − ( nθ − ( n − − θ Thus, for θ ∈ (cid:2) , n − n (cid:1) , ∂(cid:96) ∂θ >
0, and for θ ∈ (cid:0) n − n , (cid:3) , ∂(cid:96) ∂θ <
0. In addition, (cid:96) (0) = 0 and (cid:96) (1 − ) → −∞ . Hence, (cid:96) ( θ ) will cross the θ -axis only once.(b) Let (cid:96) ( θ ) = (1 − θ ) θ n − log(1 − θ ) + θ n − + n − (cid:88) i =1 θ i /i + log(1 − θ ) , then ∂(cid:96) ∂θ = θ n − (( n − θ − ( n − − θ − (1 − θ ) log(1 − θ ))1 − θ . Since ∂∂θ ( − θ − (1 − θ ) log(1 − θ )) = log(1 − θ ) <
0, for θ ∈ (cid:104) , n − n − (cid:17) , ∂(cid:96) ∂θ >
0, and for θ ∈ (cid:16) n − n − , (cid:105) , ∂(cid:96) ∂θ <
0. In addition, (cid:96) (0) = 0 and (cid:96) (1 − ) → −∞ . Hence, (cid:96) ( θ ) will cross the θ -axis only once.(c) Since both r b and p b are in terms of θ and θ respectively via the mean constraint, p ( θ ) =(1 − θ )(1 − log(1 − θ )). In addition, since d p d θ = log(1 − θ ) <
0, it suffices to show that θ ≥ θ .Since θ and θ satisfy, θ n − + n − (cid:88) i =1 θ i /i + log(1 − θ ) = 0 ,θ n − + n − (cid:88) i =1 θ i i + log(1 − θ ) (cid:0) − θ ) θ n − (cid:1) = 0 , also (1 − θ ) θ n − >
0, this factor makes θ ≥ θ . Thus, we are done.In summary, given the threshold r b and p b ,1. If p ∈ [0 , r b ], then the optimal k = 0, the optimal θ >
0, and θ , θ are determined by n − (cid:88) i =1 θ i i + (cid:18) − p − θ (cid:19) θ n − = θ n − + n − (cid:88) i =1 θ i i + (cid:18) − p − θ (cid:19) , (1 − θ )(1 − log(1 − θ ) + log(1 − θ )) = p.
2. If p ∈ [ r b , p b ], then the optimal k = 0, the optimal θ = 0, and θ is determined by(1 − θ )(1 − log(1 − θ )) = p.
3. If p ∈ [ p b , k >
0, the optimal θ = 0, and θ , k are determined by θ n − + n − (cid:88) i =1 θ i /i + log(1 − θ )(1 + (1 − θ ) θ n − ) = 0 , (1 − k )(1 − θ )(1 − log(1 − θ )) + k = p. .10 Proof of Corollary 1 Proof.
We first show that as n → ∞ , p s , r b and p b all tend to 0.1. Recall that in the seller-worst case, the threshold is p s = (1 − θ )(1 − log(1 − θ )) where θ satisfies( n − θ log(1 − θ ) − nθ − n log(1 − θ ) = 0. As n → ∞ , by the proof of Proposition 1, ( n − θ log(1 − θ ) − nθ − n log(1 − θ ) = 0 only when θ = 0 or θ ∈ ( n − /n − , n → ∞ , we have θ → p s → r b ≤ p b , it suffices to show that p b → n → ∞ . Also by Proposition 2, the solution of θ n − + (cid:80) n − i =1 θ i /i + log(1 − θ )(1 + (1 − θ ) θ n − ) = 0can only be attained when θ = 0 or θ ∈ ( n − /n, n → ∞ , we have θ → p b → { k s , } and { k b , } , respectively. It suffices to show that both of k s and k b tendto p . And the mean constraint will further imply that the mass on signal 1 tends to 0.1. For the seller-worst case, the optimal k is determined by(1 − k )(1 − θ )(1 − log(1 − θ )) + k = p where θ satisfies ( n − θ log(1 − θ ) − nθ − n log(1 − θ ) = 0. As argued in the first part, θ → k s → p .2. For the buyer-optimal case, the optimal k is also determined by(1 − k )(1 − θ )(1 − log(1 − θ )) + k = p where θ satisfies θ n − + (cid:80) n − i =1 θ i /i + log(1 − θ )(1 + (1 − θ ) θ n − ) = 0. As we argue in the first part, θ → k b → p . B Proofs for Section 5
B.1 Proof of Lemma 10
Proof of the nonnegativity.
For each buyer i , let G i be a signal distribution which assigns positive prob-ability on negative virtual values for signals in [0 ,
1] and let x i = inf { x | ˆ ϕ ( x | G i ) ≥ } .First, we construct a distribution ˜ G θ i i similar to that in Appendix A.3:˜ G θ i i ( x ) = θ i if x ∈ [0 , x θ i );1 − x (1 − G ( x − )) x if x ∈ [ x θ i , x i ); G ( x ) if x ∈ [ x i , , By similar arguments, { ˜ G θ i i } i generates weakly less revenue than { G i } . econd, define another signal distribution ˆ G x i similar to that in Appendix A.3:ˆ G x i i ( x ) = − − ˜ G θi ( x i ) x if x ∈ [1 − ˆ G θ i i ( x i ) , x i ];˜ G θ i i ( x ) if x ∈ ( x i , . By similar arguments, { ˆ G x i i } i generates strictly less revenue than { ˜ G θ i i } i . Proof of the regularity.
For each buyer i , let G i induce ironed virtual value on [ x i , x i ]. First, weconstruct a distribution ˆ G θ i i similar to that in Appendix A.4:ˆ G θ i i ( x ) = θ i if x ∈ [0 , x θ i ) , ˆ G i ( x ) if x ∈ [ x θ i , . where ˆ G i ( x ) = G i ( x ) if x (cid:54)∈ [ x i , x i ];1 − (1 − G ( x i ))( x i − k ) x − k if x ∈ [ x i , x i ] . By similar arguments, { ˆ G θ i i } i generates weakly less revenue than { G i } .Although this modification still puts positive mass on signal 0, by the nonnegativity part of Lemma10, we can further modify this distribution into the one with nonnegative virtual values and therebyachieve the strictly less seller’s revenue. Thus, the seller-worst distribution must also be regular. B.2 Proof of Proposition 1
Proof for n = 2 . Within the class documented in the context, let θ i denote the mass on signal 0 and1 − θ i denote the mass on signal 1 for buyer i . Without loss of generality, we assume that θ ≤ θ , andthen the buyer-optimal information design problem ismax θ i ,θ i (cid:90) x d( G G ) − (1 − θ θ )s.t. (1 − θ i ) (1 − log(1 − θ i ) + log(1 − θ i )) = p ∀ i = 1 , . And the objective can be rewritten as1 − (cid:90) G G d x − (1 − θ θ )=1 − (cid:90) x θ θ d x − (cid:90) x x θ (cid:16) − a x (cid:17) d x − (cid:90) x (cid:16) − a x (cid:17) (cid:16) − a x (cid:17) d x − (1 − θ θ )=1 − θ θ x − θ ( x − x − a (log( x ) − log( x ))) − (1 − x ) − ( a + a ) log( x ) − a a (cid:18) − x (cid:19) − (1 − θ θ )=2(1 − θ )( θ − θ ) + θ (1 − θ ) (log(1 − θ ) − log(1 − θ ))+ (2 − (1 − θ ) θ − θ − θ ) (log(1 − θ ) − log(1 − θ )) . herefore, the Lagrangian is L =2(1 − θ )( θ − θ ) + θ (1 − θ ) (log(1 − θ ) − log(1 − θ )))+ (2 − (1 − θ ) θ − θ − θ ) (log(1 − θ ) − log(1 − θ ))) − (cid:88) i =1 λ i ((1 − θ i ) (1 − log(1 − θ i ) + log(1 − θ i )) − p ) . Taking the first derivative with respect to θ yields ∂ L ∂θ = − θ (1 − θ )1 − θ + λ (1 − θ )1 − θ = ( λ − θ )(1 − θ )1 − θ . Since λ is constant, ∂ L ∂θ is either always non-positive or always nonnegative. Therefore, the optimal θ is a boundary solution. That is θ = θ or θ = 0.First, if θ = θ , then the solution becomes the symmetric buyer-optimal information structuredesign problem when n = 2.Second, if θ = 0, then the information designer only needs to choose the optimal θ to maximize thebuyers’ surplus. Since p < r b , the symmetric buyer-optimal information structure puts positive mass onsignal 0. Hence, θ >
0; otherwise, by the mean constraint, both buyers distributions become identicaland place mass only on virtual values 0 and 1, which, by Theorem 2, is not optimal.However, to guarantee that the asymmetric case with θ = 0 and θ > λ and θ , for which we appeal to simulation. For instance, let p = 0 .
4, underthe symmetric buyer-optimal information, the mass on signal 0 is θ (cid:39) . θ (cid:39) . . θ = 0 and θ = 0 .
3, thenthe mean constraint implies θ (cid:39) . θ (cid:39) . . > . Proof for n → ∞ . Let θ denote the mass on signal 0 and 1 − θ denote the mass on signal 1 G i ( x ) = θ , if x ∈ [0 , x )1 − (1 − θ )(1 − p ) x − p , if x ∈ [ x , , if x = 1 . where x = p + (1 − θ )(1 − p ) / (1 − θ ). First, for x ∈ [ x , p . And since x > p , by the allocation rule, the good will be allocated to buyer i if buyer i ’s signal belongs to [ x , p which is strictly less than x when x i ∈ [ x , i must obtain somepositive information rents. Hence, the buyers’ total surplus is strictly positive. For example, let p = 0 .
4, take θ = 0 . θ = 0 . . n → ∞ . Proofs for Section 6
C.1 Proof of Proposition 2
Proof.
With change of variable, for n = 2, the seller-worst information designer’s problem can be writtenas max { F i ( k ) } i =1 (cid:90)
10 2 (cid:89) i =1 F i ( k )d k − (cid:90) (1 − F i ( k ))(1 − log(1 − F i ( k )))d k = p i , ∀ i = 1 , . Consider the following Lagrangian formula L : L ( F i ( k ) , λ i ) = (cid:90)
10 2 (cid:89) i =1 F i ( k ) − (cid:88) i =1 λ i ((1 − F i ( k ))(1 − log(1 − F i ( k )))) d k + (cid:88) i =1 λ i p i .By Theorem 4.2.1 of van Brunt (2004), for any state k , the Euler-Lagrange equations with respect to F i ( k ) are (cid:89) j (cid:54) = i F j ( k ) − λ i log(1 − F i ( k )) = 0 , ∀ i = 1 , . Denote θ i = F i ( k ) and we have θ = λ log(1 − θ ) ,θ = λ log(1 − θ ) . First, note that λ i should be negative. Then, we claim that there is a unique solution pair ( θ ∗ , θ ∗ )such that the Euler-Lagrange equations are satisfied. Hence, given any solution ( θ i , θ j ), we have θ = λ log(1 − λ log(1 − θ )) . (18)Taking the first and second derivative of the right hand side, we obtain ∂∂θ λ log(1 − λ log(1 − θ )) = λ λ (1 − θ )(1 − λ log(1 − θ )) > ,∂ ∂θ λ log(1 − λ log(1 − θ )) = λ λ (1 − λ log(1 − θ ) − λ )(1 − θ ) (1 − λ log(1 − θ )) > . Hence, the right-hand side is convex and increasing in θ . It follows that there will be only one solutionof θ such that equation (18) holds.Hence, an optimal F ∗ ( k ) and F ∗ ( k ) are both constantly equal to θ and θ for k < F ∗ ( k ) and F ∗ ( k ) have binary support { k, } . By part (1) of Lemma 9, the uniqueness of θ i implies( F ∗ , F ∗ ) with the binary support is also a global maximizer.Again, let θ i be the mass on the virtual value k for buyer i . Then the information design problem isreduced into max k,θ ,θ (1 − k ) θ θ − k + (1 − k ) ((1 − θ i )(1 − log(1 − θ i ))) = p i . he Lagrangian formula with multiplier λ i is L = (1 − k ) θ θ − − (cid:88) i =1 λ i ( k + (1 − k ) ((1 − θ i )(1 − log(1 − θ i ))) − p i )The Euler-Lagrange equation with respect to θ i is ∂ L ∂θ i = (1 − k ) ( θ − i − λ i log(1 − θ i )) = 0 = ⇒ λ i = θ − i log(1 − θ i ) . Also the Euler-Lagrange equation with respect to k is ∂ L ∂k = − θ θ − (cid:88) i =1 λ i ( θ i + (1 − θ i ) log(1 − θ i )) .Plugging λ i into the above expression, we have ∂ L ∂k = θ θ − θ (cid:18) θ log(1 − θ ) (cid:19) − θ (cid:18) θ log(1 − θ ) (cid:19) . We claim that ∂ L ∂k ≤
0. To see this, it suffices to show that 1 + θ i log(1 − θ i ) ≥ θ i . Indeed, ∂∂θ i (cid:18) − (cid:18) − θ i (cid:19) log(1 − θ i ) − θ i (cid:19) = 12 (cid:18) θ i − θ i + log(1 − θ i ) (cid:19) ≥ . Hence, − (cid:16) − θ i (cid:17) log(1 − θ i ) − θ i ≥
0. That is, 1 + θ i log(1 − θ i ) ≥ θ i . Therefore, in order the maximize theobjective, the information designer should choose k = 0. C.2 Proof of Section 6.3 If v i ( s , s ) <
1, the virtual value is ϕ i ( s , s ) = v i ( s , s ) − − G ( s i ) G (cid:48) ( s i ) ∂v i ∂s i = a (1 − s i )(1 − s j ) − (1 − s i ) a (1 − s i ) (1 − s j ) = 0 . Therefore, both buyer’s virtual value is ϕ i ( s , s ) = , if (1 − s )(1 − s ) > a ;1 , otherwise . However, the interim value v i ( s , s ) does not satisfy the consistency condition. To see this, observethat the consistency condition implies that (cid:90) s (cid:90) s Prob { v = 1 , v = 1 , s , s } d G ( s )d G ( s ) = p . Define 1 − s i = t i . Then, we have (cid:90) s (cid:90) s Prob { v = 1 , v = 1 , s , s } d G ( s )d G ( s )=Prob { t t ≤ a } + (cid:90) t t >a at t at t d t d t = a − a log( a ) + (cid:90) a (cid:90) at at t at t d t d t = a − a log( a ) + a ( − a − log( a )) = a ( a − a )) > (cid:18) a (cid:18) − log( a ) + 12 log ( a ) (cid:19)(cid:19) = p . The inequality follows because the difference between the left-hand side and the right-hand side is firstincreasing and then decreasing for a ∈ [0 ,
1] and the difference is 0 at a = 0 and a = 1.= 1.