How to Sell Hard Information
aa r X i v : . [ ec on . T H ] O c t How to Sell Hard Information ∗ Nima Haghpanah r ○ S. Nageeb Ali r ○ Xiao Lin r ○ Ron Siegel † October 19, 2020
Abstract
The seller of an asset has the option to buy hard information about the value of theasset from an intermediary. The seller can then disclose the acquired information beforeselling the asset in a competitive market. We study how the intermediary designs andsells hard information to robustly maximize her revenue across all equilibria. Even thoughthe intermediary could use an accurate test that reveals the asset’s value, we show thatrobust revenue maximization leads to a noisy test with a continuum of possible scoresthat are distributed exponentially. In addition, the intermediary always charges the sellerfor disclosing the test score to the market, but not necessarily for running the test. Thisenables the intermediary to robustly appropriate a significant share of the surplus resultingfrom the asset sale even though the information generated by the test provides no socialvalue. ∗ We thank Simone Galperti, Navin Kartik, Elliot Lipnowski, Philipp Strack, Rakesh Vohra, and variousseminar audiences for useful comments. The symbol r ○ indicates that the authors’ names are in random order. † Department of Economics, Pennsylvania State University. Emails: [email protected], [email protected],[email protected], and [email protected]. ontents
Introduction
This paper studies settings in which individuals purchase hard information from an intermediarythat they can verifiably disclose to influence the actions of others. Such settings are ubiquitous.For example, entrepreneurs often seek evidence that they can disclose to venture capitalists toobtain more funding, sellers of physical and financial assets routinely pay for evaluations thatenable them to obtain better prices, and workers commonly seek certification before applyingfor positions. The size of the market for hard information, and certification in particular, is inthe hundreds of billions. An important justification for the existence of information intermediaries in these settingsis that they generate economic value. The information they provide may facilitate assortativematching or alleviate moral hazard or adverse selection. It may also affect how the surplus isdivided between the parties (other than the intermediary). For example, a seller who knowsthat his asset is valuable may be unable to credibly convey that information to buyers on hisown, but objective evidence obtained from a trusted third-party would allow him to do so,enabling him to negotiate a better price for his asset.But would we expect to find information intermediaries where they do not provide economicvalue? Perhaps not, since purchasing information from them is voluntary and if they do notprovide economic value, they can make a profit only by reducing the surplus of the other agents.We consider such a setting with an information intermediary and show, perhaps surprisingly,that the intermediary is able to appropriate a significant part of the surplus arising from tradeby generating noisy information and charging certain fees, even when the equilibrium playedby the other parties is chosen adversarially to the intermediary’s interests.In our model, an agent owns an asset that he would like to sell in a competitive market. Boththe agent and the market have symmetric information about the asset’s market value. Beforeselling the asset, the agent can purchase additional, hard information from an intermediaryabout the asset’s value that the agent can share with the market to improve the terms of trade.The intermediary chooses a test , which stochastically maps the asset’s value to a score thatcan be verifiably disclosed, and a two-part tariff for her services. The tariff comprises a testingfee for running the test and a disclosure fee for disclosing the resulting score to the market.If the agent pays the testing fee, the test is run and the agent observes the resulting score.He then chooses whether to pay the disclosure fee to obtain hard information that enableshim to disclose the score to the market. The market cannot distinguish between the agent For a survey of the literature on the design and sale of information by intermediaries seeBergemann and Bonatti (2019). See, for example, https://tinyurl.com/y5ce9p4b. Issues of moral hazard and certification have been studied by Albano and Lizzeri (2001),Marinovic, Skrzypacz, and Varas (2018), Zapechelnyuk (2020), and Saeedi and Shourideh (2020) among oth-ers. θ ). The agent discloses the test score whenever it revealsthat the asset’s value is not the lowest possible value. The high testing fee charged by theintermediary extracts all of the agent’s expected surplus (minus θ ) from selling the asset, so hisexpected payoff is θ . This is consistent with a key intuition from standard mechanism design:since the agent’s payoff beyond what is needed to satisfy his individual rationality constraint isdue to information rents, when the agent starts with no private information the designer cankeep the agent’s payoff at his individual rationality level, extract the full surplus, and achievethis by charging an upfront fee.But the game induced by this test-fee structure has another equilibrium, in which the The intermediary can also extract all the surplus by using a binary score test, making testing free, andcharging a high disclosure fee. robustly optimal test-fee structures ,namely those that guarantee the highest revenue to the intermediary across all equilibria of theinduced game. This corresponds to the intermediary choosing the test-fee structure that max-imizes her revenue assuming that the equilibrium of the induced game is selected adversariallyto her interests. Our motivation for studying robustly optimal test-fee structures is twofold.First, the intermediary may be unable to coordinate the behavior of the agent and the marketon her most preferred equilibrium. The uncertainty about which equilibrium will be playedcould motivate her to be cautious and therefore use test-fee structures that guarantee her ahigh revenue across all equilibria. Second, for any test-fee structure, the sum of the agentand the intermediary’s revenue is constant across equilibria and equal to the asset’s ex anteexpected value, so the intermediary’s least preferred equilibrium is the agent’s most preferredequilibrium. And the agent and the market may be able to coordinate on the agent’s preferredequilibrium. As we will see, the intermediary obtains a substantial share of the surplus whenusing the robustly optimal test-fee structure. This will show that the ability of the intermediaryto secure a profit even though the information she provides has no social value and does notbenefit the agent (or the market) ex-ante does not depend on the equilibrium played in theinduced game.Finding the robustly optimal test-fee structure involves optimizing across all test-fee struc-tures and all equilibria of the induced games. Different tests induce different distributions ofprivate information about the asset’s value for the agent in the induced games, and differ-ent fees change the agent’s incentives to obtain and disclose this private information. Thus,the optimization entails comparing the equilibria of disclosure games that vary in both theamount of the agent’s private information and his disclosure costs. Despite this richness, wefind that robustly optimal test-fee structures take a relatively simple form regardless of thedistribution of the asset’s value. As we illustrate in Figure 1, robustly optimal tests are inthe “step-exponential-step” class: they feature an exponential distribution over a continuum ofscores (even if the asset’s value is drawn from a finite set), one atom below this continuum, and Our focus on adversarial equilibrium selection is shared by a rapidly growing literature in mecha-nism and information design, including Bergemann, Brooks, and Morris (2017), Du (2018), Hoshino (2019),Ziegler (2019), Inostroza and Pavan (2020), Dworczak and Pavan (2020), Halac, Kremer, and Winter (2020),Halac, Lipnowski, and Rappoport (2020), and Mathevet, Perego, and Taneva (2020). G ( s ) s Figure 1: A robustly optimal score distribution if the asset quality θ is drawn from [0 , , where G is the marginalCDF on scores. The score distribution features atoms on a low and a high score, and an exponential distributionover a continuum of intermediate scores. possibly one atom above it. The optimal disclosure fee is always positive, but the optimal test-ing fee may be positive or 0. The resulting payoff to the intermediary is positive and boundedaway from the full surplus.To derive these features of the robustly optimal test-fee structure we first observe that theintermediary does not provide any added value to the agent ex ante. This is because the marketdraws correct (Bayesian) inferences and is competitive, so for every test-fee structure and anyequilibrium, the ex ante expected market price is the ex ante expected value of the asset. Thus,for every test and for all positive fees, the agent strictly prefers an equilibrium in which themarket expects him not to have the asset tested and consequently offers him the asset’s ex anteexpected value. The agent then retains the full surplus and the intermediary’s revenue is zero.If the market were to observe whether the asset is tested, the agent could achieve this as anequilibrium outcome by not paying the testing fee. But the market does not observe whetherthe asset is tested. Therefore, the market’s expectation of whether the asset has been testedand the agent observed the resulting score must be consistent with the agent’s unobservedequilibrium choice of whether to pay the testing fee.This is how the intermediary obtains a positive payoff robustly: she uses option value asa carrot to make it non-credible for the agent not to pay for the asset to be tested. Becausethe market only learns that the test has been run if the agent pays the disclosure fee anddiscloses the test score, the intermediary creates option value for the agent by offering a testthat generates high test scores with some probability and setting the testing and disclosurefees sufficiently low. If this option value is sufficiently high, the agent cannot credibly refrainfrom paying the testing fee. The intermediary then obtains at least the testing fee in everyequilibrium. Moreover, the market then treats non-disclosure as concealing a low score, whichfurther motivates the agent to pay the disclosure fee and disclose the test score. In effect, theagent is trapped by market expectations that he has paid the testing fee and will disclose if thetest score is sufficiently high. 4ut even if the agent pays the testing fee with certainty, multiple equilibria may exist. Theseequilibria differ in the set of scores that the agent discloses, and therefore in the probability ofdisclosure and the intermediary’s revenue. The exponential test score distribution is robustlyoptimal because it eliminates potential equilibria in which the agent discloses with low proba-bility. We develop an intuition for this result by showing that the intermediary can be thoughtof as choosing an optimal “demand curve for testing,” subject to the demand curve being fea-sible and the quantity of testing demanded corresponding to the one in the equilibrium leastfavorable to the intermediary. We illustrate this approach in Section 2, and provide a generalanalysis in Sections 3–5.Section 6 describes two extensions. First, we consider a setting in which testing is costly forthe intermediary. We show that if testing costs increase in the Blackwell order, then our mainresults continue to hold, that is, there exists a robustly optimal test in the step-exponential-stepclass. Moreover, if the increase is strict, then every robustly optimal test is in this class. Second,we consider an intermediary who can sell the agent multiple pieces of evidence, and gives himthe choice of which to disclose. We show that this additional flexibility does not improve theintermediary’s revenue guarantee.The main takeaway of our paper is that even if information is neither socially valuablenor ex ante valuable to a seller, an intermediary selling hard information can profit from theseller’s inability to commit. The intermediary robustly maximizes her revenue by using test-fee structures that generate option value, use noise, and include strictly positive disclosurefees. Thus, the presence of profitable intermediaries, as in the large certification industry isnot, in itself, evidence that the provision of hard information improves the welfare of marketparticipants.Our work builds on the rich literature on verifiable disclosure and persuasion games, ini-tiated by Grossman (1981) and Milgrom (1981). Their main insight is that if a privatelyinformed agent could costlessly and verifiably disclose evidence about his type, the unique equi-librium involves full disclosure. The subsequent literature suggests a number of mechanismsthat dampen this force, including exogenously costly disclosure (Jovanovic, 1982; Verrecchia,1983) and lacking evidence with positive probability (Dye, 1985). Matthews and Postlewaite(1985) and Shavell (1994) consider an uninformed agent who decides whether to take a fullyrevealing test. Matthews and Postlewaite (1985) show that the unique equilibrium involvestesting and full disclosure of the test result when disclosure is voluntary but involves no test-ing when disclosure of the test result is mandatory. Shavell (1994) assumes the agent bears aprivately-known cost of testing, and studies how this cost dampens unraveling.The features in the models above also apply to our setting: the agent faces a cost of obtaininginformation about her type and a cost of disclosing that information in a verifiable form,and with positive probability, the agent may lack evidence. But we derive these features5ndogenously because the intermediary chooses the evidence structure as well as the cost oflearning and disclosing the evidence; the probability that the market attributes to the agenthaving evidence is determined in equilibrium. Treating these features as endogenous objectsreveals a tradeoff: all else equal, the intermediary would like the market to unravel (so that theagent discloses with maximal probability), but the instruments from which she earns revenueare exactly those that counter unraveling. This “quantity-price” tradeoff leads to the price-theoretic approach to evidence generation we develop, in which the intermediary both choosesthe optimal price and designs the optimal demand curve subject to constraints that correspondto Bayes rule and adversarial equilibrium selection.A closely related strand of the disclosure literature studies choices made by agents to influ-ence market perceptions. Ben-Porath, Dekel, and Lipman (2018) model how an agent choosesprojects when he obtains evidence of project returns with positive probability. Their analysisemphasizes option value from the possibility of disclosure as motivating the agent to chooseriskier projects. DeMarzo, Kremer, and Skrzypacz (2019) study how an agent chooses tests anddisclosures to influence the market valuation of his asset. While most of their paper concernsevidence being costlessly generated in-house by the agent, they also consider the choices thatwould be made by a monopolistic intermediary who can charge a testing fee but not a disclo-sure fee, assuming that equilibria are selected to favor the intermediary. Our analysis showsthat when the intermediary chooses to protect herself from the worst equilibrium (or assumesthat the agent and market can coordinate on the agent’s best equilibrium) and can charge adisclosure fee, she optimally designs her tests differently. In the papers discussed above, and in our work, the players are symmetrically informed atthe outset of their interaction. By contrast, Lizzeri (1999) studies signaling dynamics when theagent is perfectly and privately informed at the outset about the value of the asset and chooseswhether to have the asset tested. In his model, testing is voluntary (and observable) and ifthe agent has the asset tested, disclosure is mandatory. The intermediary’s gains come fromthe prejudicial expectation that the market forms when the privately informed agent choosesto not have the asset tested. Lizzeri shows that the intermediary can extract the full surplususing a nearly uninformative test. Were the agent instead uninformed prior to testing, theagent would retain the full surplus and the intermediary’s revenue would be 0. Our work Another strand of the literature less closely related to our work studies organizational settings inwhich an agent exerts effort to acquire hard information to persuade others; see Aghion and Tirole (1997),Dewatripont and Tirole (1999), and Che and Kartik (2009). Shishkin (2019) considers a sender who commitsto an evidence structure and then chooses whether to disclose evidence to persuade a receiver. Both the prob-ability of obtaining evidence and the form that evidence takes are determined endogenously. He finds that theoptimal evidence structure is binary certification. Harbaugh and Rasmusen (2018) study a related model in which the intermediary’s objective is to provide asmuch information as possible to the receiver. They show that the intermediary nevertheless chooses to providecoarse information to encourage the privately informed agent to have the asset tested. As an application of a novel result on Bayesian updating, Kartik, Lee, and Suen (2020) study a version of both testingand disclosure are voluntary, an intermediary can gain from using option-value to incentivizean uninformed agent to always have the asset tested. The intermediary then capitalizes onprejudicial inferences that are made if the agent chooses not to disclose the test score.
Consider an agent who will sell an asset in a competitive asset market. The market value of theasset, θ , is either 0 or 1, each with equal probability. Neither the agent nor the asset marketknow the asset’s value, but an intermediary can run a test that generates information aboutthe asset’s value. The intermediary chooses the test, T , which stochastically maps the asset’svalue to an unbiased score s in [0 , s = E [ θ | s ]. If the agent wants the asset tested, he hasto pay the intermediary a testing fee φ t . If the asset is tested, the score is reported to the agent.The agent then chooses whether to pay an additional disclosure fee φ d to disclose the score ashard information to the market; otherwise, no score is disclosed to the market. If no score isdisclosed, the market cannot distinguish between (i) the asset not being tested and (ii) the assetbeing tested but the agent not disclosing the score. The intermediary chooses both the test T and the fees (a test-fee structure), and her choice induces a disclosure game between the agentand the asset market. Some test-fee structures have multiple equilibria, and the intermediary’sobjective is to choose a test-fee structure that guarantees the highest payoff across equilibria.We use this example to illustrate several features of our analysis. We show that the interme-diary benefits from using noisy tests that pool low and high asset values as intermediate scores.We visually depict how our problem maps to setting a price on an optimally chosen demandcurve. And we show why the robustly optimal test uses an exponential score distribution. Toillustrate each of these features, it suffices to assume that testing is free and the intermediarycharges only a disclosure fee. As we later show, these features also arise when it is optimal tocharge a testing fee.We first describe the intermediary’s revenue guarantee if she used a fully revealing test, i.e.,a test in which the score is equal to the asset’s value θ . Figure 2(a) depicts our analysis ofthis case using a (inverse) demand correspondence , which for any disclosure fee in [0 ,
1] (on thevertical axis) traces the corresponding probabilities of disclosure consistent with equilibria ofthe induced game. For any such disclosure fee, there is an equilibrium in which the agent hasthe asset tested, discloses a score of 1, and conceals a score of 0; if no score is disclosed, themarket offers a price of 0 for the asset. The agent has no incentive to deviate because the payoffof 1 − φ d from disclosing a score of 1 exceeds the payoff from not disclosing, which is in turn Lizzeri’s model in which the agent is privately and imperfectly informed about the asset’s value. . p . − e (c) Figure 2: The black curve graphs the demand correspondence, and the red curve represents the robust demandcurve for (a) the fully revealing test, (b) the test with three scores, and (c) the robustly optimal test. equal to that from disclosing a score of 0. This is the unique equilibrium for any disclosure feein (0 , / For disclosure fees that weakly exceed 1 /
2, other equilibria exist. There is an equilibriumin which the agent has the asset tested but never discloses the score; if no score is disclosed,the market offers a price of 1 /
2. Given this market price, the agent prefers not to disclose ascore of 1 (or 0) because the disclosure fee is at least 1 /
2. Because the agent does not discloseany score, the intermediary’s revenue is zero in this equilibrium. There is also a mixed strategyequilibrium in which the agent discloses a score of 1 with an interior probability. For disclosurefees higher than 1 /
2, the demand correspondence in Figure 2(a) depicts the three “quantitiesof disclosure” associated with these three equilibria.Since we study the intermediary’s robust revenue across equilibria, we identify the “robustdemand curve” for disclosure, which maps each disclosure fee (price) to the lowest probabilityof disclosure (quantity) across all the equilibria associated with that fee. This is the red curvein Figure 2(a). The robust revenue for a given disclosure fee is the product of the disclosurefee and the lowest equilibrium probability of disclosure, i.e., the area of the rectangle underthe robust demand curve at the point associated with that disclosure fee. The maximal robustrevenue of ≈ / / Let us now see how the intermediary can robustly improve her revenue by introducing anintermediate score that pools the asset’s two possible values, 0 and 1. Consider a test withthree possible scores—0, 3 /
4, and 1—generated by the conditional distribution in Table 1. Thedemand correspondence and the robust demand curve for this test are shown in Figure 2(b). If the disclosure fee is 0, then for any probability p , there is an equilibrium in which the agent disclosesa score of 1 and with probability p discloses a score of 0, since the agent’s payoff is then 0 whether or not hediscloses. As is common with adversarial equilibrium selection, the intermediary cannot robustly achieve a payoff ofexactly 1 / / = 0 s = 3 / s = 1 θ = 0 1 − p p θ = 1 0 3 p − p Table 1: Distribution of score s conditional on asset value θ , where p is in (0 , / . For strictly positive disclosure fees less than 1 /
2, there is a unique equilibrium: the agent hasthe asset tested, and discloses his score if it is 3 / p ) /
2. For disclosure feeshigher than 1 /
2, other equilibria exist, and all of them feature a lower probability of disclosure.Thus, by setting a disclosure fee slightly below 1 /
2, the intermediary can obtain a revenue of(1 + p ) /
4, which is more than her revenue from the robustly optimal fully revealing test. Byintroducing the intermediate score 3 /
4, the intermediary increases the quantity of disclosuredemanded because the agent pays the disclosure fee not only when the asset’s value is 1 butalso, with some probability, when its value is 0.But the intermediary is limited in how much pooling she can introduce: a value of p that istoo high introduces a bad equilibrium in which the agent has the asset tested but only disclosesa score of 1. If no score is disclosed, the market concludes that the score is either 0 or 3 / p/ (1 + 3 p ). In this equilibrium, the probability of disclosure is (1 − p ) /
2. Toeliminate this equilibrium, the intermediary sets the value of p so that if the score is 3 /
4, theagent prefers to disclose his score rather than obtain the non-disclosure price:34 − φ d | {z } Disclose a score of 3 / > p p | {z } Nondisclosure price
With a disclosure fee slightly below 1 /
2, the above inequality holds for p less than 1 /
9. Therobust revenue is attained by setting p = 1 / / p is a “cross-equilibrium” constraint, which ensures that a low-revenue equi-librium is not created. Cross-equilibrium constraints lead to the exponential score distributionin the robustly optimal tests.To see why, let us return to the demand curve approach. These constraints lead to tests thatinduce a rectangular demand correspondence like the one in Figure 2(c): intuitively, wheneverthe demand correspondence or the robust demand curve are not a rectangle, there is someslack that the intermediary can use to change the test in a way that increases the probabilityof disclosure (by having more pooling) in the equilibrium that has the lowest probability ofdisclosure, , i.e., without violating cross-equilibrium constraints.Which tests lead to rectangular demand correspondences? Those whose marginal scoredistribution includes an exponentially distributed component. To see why, notice that with arectangular demand correspondence, there is a disclosure fee φ d for which there is an interval9 1 G ( s ) s e ( x − ) . /e (a) 1 1 G ( s | s . /e (b) 1 1 G ( s | s . Figure 3: The robustly optimal distribution of scores where (a) G ( · ) is the marginal probability, (b) G ( ·| is theconditional probability of scores for θ = 0 , and (c) G ( ·| is the conditional probability for θ = 1 . of equilibrium disclosure probabilities. In Figure 2(c), this disclosure fee is φ d = 1 /
2. Thiscontinuum of equilibria implies that there is a continuum of scores s for which s − φ d = E [ s ′ | s ′ ≤ s ] . (1)The LHS above is the market price from disclosing a score s , and the RHS is that from notdisclosing when the set of scores that do not disclose are those weakly below s . Thus, when (1)holds there is an equilibrium with a disclosure threshold of s . Using G ( s ) as the marginal CDFon scores and G ( s ′ | s ′ ≤ s ) as the conditional CDF on scores no higher than s , the RHS can bere-written as R s (1 − G ( s ′ | s ′ ≤ s )) ds ′ . It follows then that Equation 1 is equivalent to φ d = Z s G ( s ′ | s ′ ≤ s ) ds ′ = R s G ( s ′ ) ds ′ G ( s ) = (cid:18) dds (cid:16) ln( Z s G ( s ′ ) ds ′ ) (cid:17)(cid:19) − . (2)Because the above is true for an interval of scores s , it defines a differential equation whosesolution is G ( s ) = αe sφd for some constant α , , i.e., the exponential score distribution. We illustrate this solution in Figure 3, where (a) shows the marginal score distribution inthe robustly optimal test. This marginal distribution is generated by a test in which if the valueof the asset is 0, with probability 2 /e the test score is 0 and with complementary probability thetest score is distributed exponentially on [1 / , / , The exponential score distribution may bring to mind the work of Roesler and Szentes (2017) andCondorelli and Szentes (2020). In those papers and in our work, a player optimally chooses a demand curve bymanipulating an information structure or a distribution of values. In those papers, a buyer optimally choosesa unit-elastic demand curve that leads a seller to charge a low price and be indifferent between that price andhigher prices. In a similar vein, Ortner and Chassang (2018) show that a principal can reduce the cost of moni-toring by paying a monitor wages that generate a unit-elastic demand curve for bribes. In our paper, adversarialequilibrium selection leads the intermediary to choose a test that generates a rectangular demand curve, whichcorresponds to an exponential distribution of scores. The driving force is not to generate any indifference butto eliminate equilibria in which the agent does not disclose intermediate scores. /
2. This test-fee structure induces a game with a unique equilibrium:the agent has the asset tested, and discloses the score if it exceeds the disclosure fee. If noscore is disclosed the market offers a price of 0. The resulting revenue for the intermediary is ≈ (cid:0) − e (cid:1) , which is more than 1 / / A risk-neutral agent sells an asset in a competitive market comprising two risk-neutral buyerswho have a common value for the asset. The agent and both buyers are symmetrically informedabout the asset’s market value, θ , knowing only that it is drawn according to a distribution F with support Θ ⊆ [ θ, θ ], where 0 ≤ θ < θ < ∞ are the lowest and highest values in Θ, with anexpected value of µ .Prior to selling the asset, the agent can pay an intermediary to evaluate the asset, and if hedoes so, he has the option, for a fee, to disclose that evaluation to the market. The intermediarychooses how to evaluate the asset and sets the fees for her services, and we call such a schemea test-fee structure . A test-fee structure comprises:a. a test , T , where T : Θ → ∆ S is a measurable function that stochastically maps the asset’svalue to a score in some set S ,b. a testing fee , φ t ∈ R , which the agent pays for the asset to be evaluated, andc. a disclosure fee , φ d ∈ R , which the agent pays to disclose the score to the market.Because the test will only be used to determine the asset’s expected market value, wehenceforth assume without loss of generality that each test score is an unbiased estimate of thevalue. That is, S ≡ [ θ, θ ] and for every market value θ in Θ and score s in S , E [ θ | T ( θ ) = s ] = s .We denote the set of all tests by T . For each test T , let G T ( s ) denote the probability that ifthe intermediary evaluates the asset using test T , the resulting score is at most s . We refer to G T as the marginal score distribution .Turning to fees, we denote the pair of testing and disclosure fees by φ ≡ ( φ t , φ d ). The testingfee is the price the intermediary charges to evaluate and generate information about the assetand show it to the seller. The disclosure fee is the additional price to make that informationhard or verifiable so that the agent can disclose it to the buyers.We denote the test-fee structure by ( T, φ ). Each test-fee structure (
T, φ ) induces a game G ( T, φ ) between the agent and the buyers with the following timeline:1. Nature chooses the value θ of the asset according to the distribution F .11. The agent, without observing θ , decides whether to have the asset tested. If so, he paysthe testing fee φ t and observes a score s drawn according to T ( θ ) ∈ ∆ S .3. The agent decides whether to disclose the score to the market. If so, he pays the disclosurefee φ d and the buyers observe the score. If he does not disclose, or if he does not take thetest, the buyers observe a null message, N .4. The buyers bid simultaneously for the asset, and the asset is sold to the highest bidder,with ties broken uniformly.The agent’s payoff is the price at which he sells the asset minus any testing and disclosure feesthat he pays. The payoff of the buyer that buys the asset at price p is θ − p ; the other buyer’spayoff is zero. In the induced game G ( T, φ ), an agent’s (behavioral) strategy is ( σ t , σ d ), where σ t in [0 , σ d : S → [0 ,
1] is the score-contingentprobability with which he discloses the score. Let ˆ S ( T ) ≡ ∪ θ ∈ Θ Im ( T ( θ )) denote the set ofscores in the image of T . Buyer i ’s (pure) strategy σ i : ˆ S ( T ) ∪ { N } → R specifies buyer i ’s bidfollowing a disclosure of score s or the null message N . A belief system ρ i : ˆ S ( T ) ∪ { N } → ∆Θspecifies buyer i ’s posterior belief about the asset’s value following a disclosure of score s or thenull message N . We denote a strategy profile σ = ( σ t , σ d , σ , σ ) and belief system ρ = ( ρ , ρ )by ( σ, ρ ).For a game G ( T, φ ), we denote the set of Perfect Bayesian Equilibria (henceforth PBE) byΣ(
T, φ ). Because the buyers have the same beliefs and compete in a first-price auction, eachbuyer’s bid is equal to the asset’s expected value given all the available information.
We study the intermediary’s maximal revenue guarantee , namely the highest payoff that shecan guarantee herself by choosing a test-fee structure, i.e., assuming that the equilibrium in theinduced game is chosen adversarially to her interests. We have framed our analysis in terms of this market game for concreteness. But the issues and our analysisapply to other settings in which an agent obtains evidence to persuade others. For example, a principal maydecide how much to invest in a project based on her beliefs about the project’s success, and the agent mayacquire evidence to persuade the principal to invest more. Our model is isomorphic to such a setting if theprincipal’s investment increases linearly in her posterior expectation and the agent’s payoff increases linearly inthe principal’s investment. A PBE corresponds to each player behaving in a way that is sequentially rational, beliefs µ i being derivedvia Bayes Rule whenever possible, and off-path beliefs satisfying “You can’t signal what you don’t know”(Fudenberg and Tirole, 1991). The important implication for our setting is that both buyers have the samebeliefs ( ρ ( s ) = ρ ( s ) for any score s ) both on and off the equilibrium path. efinition 1. The intermediary’s maximal revenue guarantee is R M ≡ sup ( T,φ ) ∈T × R inf ( σ,ρ ) ∈ Σ( T,φ ) σ t (cid:18) φ t + φ d Z S σ d ( s ) dG T (cid:19) . (3)Recall that σ t is the probability with which the agent pays for the intermediary to test theasset and σ d ( s ) is the probability with which the agent pays to disclose a score of s . The firstterm in the parenthesis in (3) is the intermediary’s revenue from the testing fee, and the secondterm is the revenue from the disclosure fee.We compare this maximal revenue guarantee with the full informational surplus that theintermediary could (conceivably) extract in equilibrium, which is R F ≡ µ − θ , where µ is theexpected value of the asset. To see why, notice that because the market price of the asset isnever lower than θ , the agent can guarantee himself that payoff by never taking the test. Andsince the total surplus in the economy is µ , the intermediary cannot obtain more than R F inany equilibrium.Why do we study the maximal revenue guarantee? Although there are test-fee structuresthat have equilibria in which the intermediary obtains the full informational surplus, such test-fee structures have other equilibria in which the intermediary obtains 0. We alluded to this factin the introduction and in the example in Section 2. In fact, for any test-fee structure in whichthe intermediary obtains close to the full informational surplus in some equilibrium, there isanother equilibrium in which the intermediary obtains close to 0. Proposition 1.
There exists ε ∗ > such that for any ε < ε ∗ , if there exists an equilibriumof a test-fee structure ( T, φ ) in which the intermediary’s revenue is at least R F − ε , then thereexists another equilibrium of that test-fee structure in which the intermediary’s revenue is nomore than δ ( ε ) , where lim ε → δ ( ε ) = 0 . Proposition 1 formalizes the challenge that multiple equilibria present to the intermediary:choosing a test-fee structure because its most favorable equilibrium generates a high revenueleaves the intermediary vulnerable to an equilibrium that generates low or zero revenue. Animmediate implication of Proposition 1 is that the maximal revenue guarantee, R M , is boundedaway from the full information surplus, R F , by at least ε ∗ . Our main result, Proposition 2,derives a tight bound on R M that, for every µ , applies across distributions for which the asset’sexpected value is µ . Because the proof of Proposition 1 uses techniques that we develop laterin the paper, we do not develop the intuition here. The proof and a graphical intuition are inthe Appendix.A separate rationale for focusing on the maximal revenue guarantee is that for any test-feestructure, the equilibrium that minimizes the intermediary’s revenue also maximizes the agent’spayoff. Thus, if the intermediary fears that the agent and the asset market will coordinate on13he agent’s preferred equilibrium, she would choose a test-fee structure that attains her maximalrevenue guarantee. Our analysis also shows how to maximize the intermediary’s revenue across test-fee struc-tures that admit a unique equilibrium. The value of this related problem is at most R M , butbecause the robustly optimal test-fee structure that we identify in Proposition 2 has a uniqueequilibrium, it is also a solution to this related problem. To simplify the problem of solving for the maximal revenue guarantee, we take the followingpreliminary steps:1. We frame the analysis of tests purely in terms of marginal score distributions.2. We show that it suffices to consider only those test-fee structures in which the agent paysthe testing fee with probability 1 in every equilibrium.3. For every such test-fee structure, we characterize the equilibrium with the lowest revenuefor the intermediary in terms of a score threshold for disclosure.4. Because finding the maximal revenue guarantee involves optimizing with strict constraints,we formulate a relaxed problem with weak constraints that has the same solution.
For the intermediary’s revenue, all that matters about a test is its marginal score distribution:for all fees, if two tests have the same marginal score distribution, then they have the sameequilibria. Thus, henceforth, we refer to (
G, φ ) as a test-fee structure, where G is a CDFon [ θ, θ ] that corresponds to the marginal score distribution of some test T and φ is a pair oftesting and disclosure fees. We denote by s G the lowest score in the support of G .Focusing on the set of marginal score distributions induced by all possible tests is usefulbecause this set is easy to characterize. Recall that a distribution G is a mean-preservingcontraction of F if its support is in [ θ, θ ] and R s ′ θ G ( s ) ds ≤ R s ′ θ F ( s ) ds for all s ′ ∈ [ θ, ¯ θ ], withequality at s ′ = ¯ θ . We denote by Γ( F ) the set of distributions that are mean-preserving Thus, in the game induced by a test-fee structure, the agent’s preferred equilibrium is the uniquely Paretoefficient equilibrium from the perspective of players of that game. Take two test-fee structures (
T, φ ) and ( T ′ , φ ) such that T and T ′ have the same marginal score distribution.Take an equilibrium ( σ, ρ ) of G ( T, φ ). Suppose that ( σ, ρ ) were played in G ( T ′ , φ ) and observe that given ρ andthat G T = G ′ T , the strategy profile σ remains sequentially rational for the agent and the buyers. Moreover,given σ , ρ continues to satisfy Bayes’ Rule and the appropriate consistency condition. Therefore, ( σ, ρ ) is anequilibrium of G ( T ′ , φ ). F . We then have the following classical result. Lemma 0.
A marginal score distribution G is induced by an (unbiased) test if and only if G is in Γ( F ) . We use this formulation to rewrite revenue guarantees: for a test-fee structure (
G, φ ), letˆΣ(
G, φ ) be the equilibrium set of the induced game between the agent and the market. The revenue guarantee of (
G, φ ) is the lowest revenue generated in any equilibrium of this test-feestructure: R ( G, φ ) ≡ inf ( σ,ρ ) ∈ ˆΣ( G,φ ) σ t (cid:0) φ t + φ d R S σ d ( s ) dG (cid:1) . The maximal revenue guarantee isthus R M = sup ( G,φ ) ∈ Γ( F ) × R R ( G, φ ). We show that in every test-fee structure, either the asset is tested with probability 1 in everyequilibrium or there exists an equilibrium in which the asset is tested with probability 0 (andthe intermediary’s revenue is 0).
Lemma 1.
If a test-fee structure ( G, φ ) satisfies φ t < Z ¯ θµ + φ d [ s − ( µ + φ d )] dG, (P) then the asset is tested with probability in every equilibrium; otherwise, there exists an equi-librium in which the asset is tested with probability 0. The logic of Lemma 1 is that (P) is a participation constraint that must hold for the asset tobe tested with positive probability in every equilibrium. To see why, consider a test-fee structure(
G, φ ) and an equilibrium in which the asset is tested with probability 0. Because the marketexpects non-disclosure with probability 1, the price of the asset conditional on non-disclosureis µ .If the agent deviates and has the asset tested, then he optimally pays φ d and discloses thescore whenever it is higher than µ + φ d . This is strictly profitable if µ < − φ t + Z θθ max { µ, s − φ d } | {z } Option Value dG, which, by re-arranging, yields (P). Thus, if (P) holds, the asset is tested with positive proba-bility in every equilibrium. The proof of Lemma 1 shows if (P) holds, the asset is in fact testedwith probability 1 in every equilibrium. This result is in Rothschild and Stiglitz (1970) and Blackwell and Girshick (1979), and features in recentwork on information design (e.g. Gentzkow and Kamenica, 2016; Roesler and Szentes, 2017).
Lemma 2.
For any test-fee structure ( G, φ ) with φ d < or φ t < , there exists a test-feestructure ( G ′ , φ ′ ) with non-negative fees such that R ( G, φ ) is strictly less than R ( G ′ , φ ′ ) . For any test-fee structure that satisfies (P), an adversarial equilibrium is one that minimizes thedisclosure probability, but in which the asset is tested with probability 1. We show that suchequilibria are threshold equilibria in which the agent does not disclose his score if he obtains ascore at the threshold.With Lemmas 1 and 2 in hand, we restrict attention to test-fee structures that satisfy (P)and have non-negative disclosure fees. Given a test-fee structure (
G, φ ), consider thresholds τ that weakly exceed s G , where s G is the lowest score in the support of G . We say that such athreshold τ is an equilibrium threshold for a test-fee structure ( G, φ ) if τ − φ d = E G [ s | s ≤ τ ] . (4)If τ satisfies (4), then there is an equilibrium in which the asset is tested with probability 1and the agent discloses the score if and only if it strictly exceeds τ . To see why, suppose thatthe agent behaves in this way and (4) holds. Then the LHS of (4) is the difference betweenthe market price of the asset when a score of τ is disclosed and the disclosure fee, and theRHS is the market price for the asset when no score is disclosed. Thus, the agent is indifferentbetween disclosing and not disclosing a score of τ . Because the market price following disclosureincreases in the score but the market price following non-disclosure is constant in the score, theagent strictly prefers not to disclose scores lower than τ and to disclose scores higher than τ .A test-fee structure may have multiple equilibrium thresholds. It may also have otherequilibria, in which if the agent obtains a threshold score, he chooses to disclose his score withstrictly positive probability rather than probability 0. The mixed strategy equilibria for thefully revealing and the 3-score tests in Section 2 are examples of such equilibria. We provein Lemma 3 below that from the standpoint of adversarial equilibrium selection, it suffices tofocus on equilibria in which the agent discloses his score only if strictly exceeds the threshold.We proceed as follows. For each test-fee structure, we show that a highest equilibriumthreshold exists and provide a characterization of the highest equilibrium threshold that welater use to find the robustly optimal test-fee structure. Finally, we show that this highestequilibrium threshold corresponds to an adversarial equilibrium in which the agent discloses his16core if and only if it strictly exceeds the highest equilibrium threshold. Lemma 3.
If a test-fee structure ( G, φ ) satisfies (P) and φ d ≥ , then the following are true: (a) A highest equilibrium threshold τ exists. (b) A score threshold τ ≥ s G is the highest equilibrium threshold if and only if τ − φ d = E G [ s | s ≤ τ ] ,τ ′ − φ d > E G [ s | s ≤ τ ′ ] ∀ τ ′ > τ. (HE)(c) There exists an adversarial equilibrium in which the agent discloses score s if and only if s > τ where τ is the highest equilibrium threshold. Figure 4 illustrates our characterization of the highest equilibrium threshold. τ − E [ s | s ≤ τ ]Fee θ Threshold Score τ τ τ θθ − µφ d Figure 4: The worst equilibrium at a disclosure fee is that with the highest equilibrium threshold. For example,at a disclosure fee of φ d , the highest equilibrium threshold is τ . We have seen that if a test-fee structure satisfies (P), then the agent takes the test with prob-ability 1 and the adversarial equilibrium is characterized by the highest equilibrium threshold.The revenue guarantee in the robustly optimal test-fee structure is therefore R M = sup ( G,φ,τ ) ∈ Γ( F ) × R φ t + φ d (1 − G ( τ )) s.t. (P) and (HE) . The intermediary’s maximization problem has constraints with strict inequalities, but we showthat the problem’s value R M is unchanged if those inequalities are made weak. These are the weak participation constraint φ t ≤ Z ¯ θµ + φ d [ s − ( µ + φ d )] dG (w-P)17nd the weak-highest equilibrium constraint (which defines the weak-highest equilibrium thresh-old), τ − φ d = E G [ s | s ≤ τ ] τ ′ − φ d ≥ E G [ s | s ≤ τ ′ ] ∀ τ ′ > τ. (w-HE)We write the relaxed problem asmax ( G,φ,τ ) ∈ Γ( F ) × R φ t + φ d (1 − G ( τ )) s.t. (w-P) and (w-HE) . In addition to having a solution and the same value as the original problem, the relaxedproblem has several attractive features. First, for any test-fee structure that solves the relaxedproblem, there is a “nearby” test-fee structure whose revenue guarantee is close to the maximalrevenue guarantee R M . Second, for any convergent sequence of test-fee structures that achievesthe maximal revenue guarantee, the limiting test-fee structure is a solution to the relaxedproblem. Thus, solutions to the relaxed problem identify necessary and sufficient features oftest-fee structures whose revenue guarantees approximate R M . We call these solutions robustlyoptimal test-fee structures .Let us formalize this discussion. A sequence of test-fee structures { ( G n , φ n ) } n =1 , ,... convergesto a test-fee structure ( G, φ ) if G n converges weakly to G and φ n converges to φ . For a test-fee structure ( G, φ ) and threshold τ , we denote the associated equilibrium revenue for theintermediary by ˆ R ( G, φ, τ ). We use this notation to link the relaxed and original problems.
Lemma 4. (a) An optimal solution ( G, φ, τ ) to the relaxed problem exists and R M = ˆ R ( G, φ, τ ) .(b) Consider any optimal solution ( G, φ, τ ) to the relaxed problem. Then there exists a se-quence of test-fee structures and thresholds { ( G n , φ n , τ n ) } n =1 , ,... such that (i) for each n , ( G n , φ n , τ n ) satisfy (P) and (HE) , (ii) ( G n , φ n ) converges to ( G, φ ) , and (iii) R M =lim n →∞ ˆ R ( G n , φ n , τ n ) .(c) Consider any sequence of test-fee structures and thresholds { ( G n , φ n , τ n ) } n =1 , ,... such that(i) for each n , ( G n , φ n , τ n ) satisfy (P) and (HE) , (ii) ( G n , φ n ) converges to a test-feestructure ( G, φ ) , and (iii) R M = lim n →∞ ˆ R ( G n , φ n , τ n ) . Then there exists τ such that ( G, φ, τ ) satisfy (w-P) and (w-HE) , and R M = ˆ R ( G, φ, τ ) . Robustly Optimal Test-fee Structures
This section contains our main result. We show that robustly optimal test-fee structures usetests that have a “step-exponential-step” form; i.e., the distributions over scores are exponentialover an interval, have up to two mass points, one above and one below the interval, and havezero density everywhere else. We also show that the optimal disclosure fees are strictly positive,and we derive a tight bound on the intermediary’s maximal revenue guarantee.1 θG ( s ) sθ τ τ τ τ g Figure 5: A step-exponential-step distribution G . As illustrated in Figure 5, a test-fee structure (
G, φ ) is in the step-exponential-step class ifthere exists g ∈ [0 ,
1] and thresholds τ < τ < τ ≤ τ such that G ( s ) = g if s ∈ [ τ , τ ) ge ( s − τ ) / ( τ − τ ) if s ∈ [ τ , τ )1 if s ≥ τ , (5) G assigns probability 0 to [0 , τ ) ∪ ( τ , τ ), and the fees are φ d = τ − τ , (6) φ t = (1 − ge ( τ − τ ) /φ d )( τ − ( µ + φ d )) . (7) Proposition 2.
For any distribution F of the asset’s value, the following hold: (a) There exists a test-fee structure in the step-exponential-step class that is robustly optimal. (b)
Every robustly optimal test-fee structure has a strictly positive disclosure fee. (c) If ( G, φ ) is a robustly optimal test-fee structure, then the testing fee φ t is strictly positiveif and only if scores in ( µ + φ d , θ ] arise with positive probability. (d) The maximal revenue guarantee is at most ( θ − µ )(1 − e ( θ − µ ) / ( θ − µ ) ) , and this bound isattained when the support of F is binary, i.e., { θ, θ } . d q Fee0 Prob. ofDisclosure1 θ − µ (a) CDF θ Score µ + φ d θ τ (b) Figure 6: How an intermediary gains from “flattening” the robust demand curve. (a) depicts a demand curvein blue, and by flattening the demand curve the intermediary can induce a higher probability of disclosure atdisclosure fee φ d . (b) shows that this can be done without changing the score distribution above µ + φ d , whichguarantees that (P) continues to be satisfied without changing the testing fee. Proposition 2 partially characterizes the robustly optimal test-fee structures, and providesa tight upper bound on the maximal revenue guarantee. There is always a robust solution inthe step-exponential-step class, disclosure fees are strictly positive in any robust solution, andtesting fees are strictly positive only if the robustly optimal score distribution takes a particularform. We show in the proof of Proposition 2(a) that if the intermediary uses the robustlyoptimal test in the step-exponential-step class, charges the testing fee in (7) and a disclosurefee slightly below those in (6), then that test-fee structure has a unique equilibrium. In thatequilibrium, the asset is tested with probability 1 and the agent discloses his score whenever itexceeds τ . Our previous result, Proposition 1, already established that the maximal revenueguarantee is bounded away from full surplus, but the bound provided was not tight; (d) specifiesa tight bound.Proposition 2 clarifies the interaction between the testing fee, the disclosure fee, and thetest. Lemma 10 in the Appendix shows that if the intermediary could not charge a disclosurefee, she would choose a fully revealing test and a testing fee that the agent pays with probability1. In the robustly optimal solution, the intermediary uses noisy tests because she can chargea disclosure fee. If she could not charge a testing fee, there would be a solution in the step-exponential-step class that would not have a step at the top (above the exponential part of thecurve). Her ability to charge a testing fee in addition to a disclosure fee leads to a solution inthe step-exponential-step class that may have a step at the top.The intuition for Proposition 2(a) builds on the robust demand curve approach describedin Section 2. Consider a test-fee structure with disclosure fee φ d in which the probability ofdisclosure in an adversarial equilibrium is q . Suppose, as is shown in Figure 6(a), that thedemand correspondence is not flat to the left of the point ( q, φ d ). Then we can modify thetest-fee structure and improve the intermediary’s robust revenue as follows: flatten the demand20orrespondence to the left of ( q, φ d ) and push it to the right of ( q, φ d ). Doing this increasesthe disclosure probability in the adversarial equilibrium for any disclosure fee slightly lowerthan φ d . Moreover, as seen in Figure 6(b), this modification can be done without changingthe score distribution on scores above µ + φ d , so the option value of a test, given by the RHSof (P), remains unchanged. Therefore, this modification does not generate a new adversarialequilibrium in which the agent doesn’t pay the testing fee. Thus, the intermediary can increasethe probability of disclosure by decreasing the disclosure fee arbitrarily slightly and withoutchanging the testing fee, which improves her revenue guarantee. A similar logic shows that every robustly optimal test-fee structure has a score distributionthat is exponential over a non-degenerate interval (Lemma 12 in the Appendix). Outside ofthat interval, there is some flexibility because scores below τ are those that the agent strictlyprefers not to disclose, and scores above µ + φ d , which exceeds τ , are those for which (HE)is slack. In the step-exponential-step distribution scores below τ are pooled as a single scoreand scores above µ + φ d are pooled as a single score. This creates a score distribution thatis a mean-preserving contraction of the distribution F of the asset value. But there may beother robustly optimal distributions that are the same on the interval [ τ , τ ] and are alsomean-preserving contractions of F .Turning to disclosure fees, we prove Proposition 2(b) by showing that for the intermediaryto charge a testing fee, the test must induce a high option value by having high scores in itssupport. But then the gain of adding a small disclosure fee outweighs the resulting reductionin the option value and the associated reduction in the testing fee.Proposition 2(c) follows from the participation constraint (P). If scores in ( µ + φ d , θ ] arisewith positive probability., then even if the market expects the asset to not be tested, the agentstrictly prefers to have the asset tested at a testing fee of 0 and a disclosure fee of φ d . Increasingthe testing fee slightly while keeping the disclosure fee unchanged increases the intermediary’srevenue in the adversarial equilibrium and does not generate a new equilibrium in which theasset is not tested. We use this logic in Proposition 4 below to provide conditions on primitivesthat guarantee a strictly positive testing fee.We prove Proposition 2(d) by considering a relaxed problem in which the score distribu-tion need not be a mean-preserving contraction of distribution F , while still having the sameexpectation, i.e., E G [ s ] = E F [ θ ]. We solve this relaxed problem completely, and its value thenprovides an upper bound on the maximal revenue guarantee. This bound is tight when thesupport of F is binary because the mean-preserving contraction condition is then equivalent to E G [ s ] = E F [ θ ]. The following result describes the complete solution for this case. A reader may wonder how this is possible while the total surplus is constant and (w-P) (defined on p. 17)binds. This is because (w-P) is a non-standard participation constraint. The modification to the test decreasesthe agent’s payoff in the adversarial equilibrium without affecting his indifference between paying the testingfee and not paying it if the market expects him not to pay it, which is what (w-P) represents. roposition 3. Suppose that the support of distribution F is { θ, θ } . The unique robustlyoptimal test-fee structure involves: (a) No testing fee but a strictly positive disclosure fee: φ ∗ t = 0 and φ ∗ d = θ − µ . (b) The following marginal score distribution that has an atom at θ , a gap above it, and thenan exponential form (with no atom at the top): G ∗ ( s ) = e θ − µθ − µ if s ∈ [ θ, θ + θ − µ ) e s − θθ − µ if s ∈ [ θ + θ − µ, θ ] . (8)We know from Proposition 2 that the robustly optimal disclosure fee is always strictlypositive; Proposition 3 shows that the robustly optimal testing fee may be 0. We now providea sufficient condition on the distribution F for the optimal testing fee to be strictly positive. Proposition 4.
If the distribution F of the asset’s value is log-concave, then any robustlyoptimal test-fee structure includes a strictly positive testing fee. Several commonly studied distributions such as the uniform distribution over [ θ, θ ] andthe truncated Normal and Pareto distributions are log-concave (Bagnoli and Bergstrom, 2005).Log-concavity requires the tail of the distribution to be less heavy than the tail of the exponentialdistribution.
This section shows that our results also hold when testing is costly for the intermediary andwhen the intermediary can offer the agent multiple pieces of evidence that he can disclose.
Our analysis assumed that testing is costless, but in reality testing is often costly. Suppose thatthe cost to the intermediary of running a test with marginal score distribution G is c ( G ). Weassume that c is lower semi-continuous in the weak* topology and monotone in the Blackwellorder, that is, garbling a test weakly reduces its cost; in other words, whenever G ′ is a mean-preserving contraction of G , c ( G ′ ) ≤ c ( G ). This condition is standard when informationacquisition is costly, and corresponds to a less informative test being less costly to generate. A special case is when the intermediary starts with a finite set of initial tests and can garblethose tests to obtain additional tests. Lower semi-continuity guarantees that solutions to the relevant maximization problems exist. Recent examples of analyses that assume monotonicity in the Blackwell order areDe Oliveira, Denti, Mihm, and Ozbek (2017) and Pomatto, Strack, and Tamuz (2019).
Proposition 5.
If costs are lower semi-continuous and monotone in the Blackwell order, thereexists a robustly optimal test-fee structure in the step-exponential-step class, and every robustlyoptimal test-fee structure uses a strictly positive disclosure fee.
Proposition 5 is a corollary of Lemma 12 in the Appendix, which is one of the steps in theproof of Proposition 2. Lemma 12 shows that for any test-fee structure (
G, φ ), there exists atest-fee structure in the step-exponential-step class that uses a mean-preserving contraction of G and has a weakly higher revenue guarantee. Because such a test-fee structure is a garbling of G , it has a weakly lower cost. Therefore, there must exist a robustly optimal test-fee structurein this class. The argument used to prove Proposition 2 proves that the optimal disclosurefee is strictly positive.
Our analysis assumed that the intermediary provided the agent with a single piece of evidencethat he could verifiably disclose. One could envision the intermediary providing the agent withmultiple pieces of evidence and a choice of which to disclose. We show that this additionalgenerality does not change the robustly optimal test-fee structures.To see this, suppose that the intermediary designs, along with the test-fee structure, anarbitrary evidence structure, which specifies a message space M and the set of messages avail-able for each score, described by M : S ⇒ M . We call this an evidence-test-fee structure . Ourbaseline model corresponds to the special case in which M ( s ) = { s } for every score s . Theagent first decides whether to have the asset tested and pay the testing fee φ t . If he pays thetesting fee, then he observes the test score s and then chooses whether to disclose each message m in M ( s ) to the buyers. To disclose any of these messages, the agent pays the disclosure fee φ d to the intermediary. If the agent does not have the asset tested or does not disclose anymessage, then the buyers observe the null message N . The following result shows that usingevidence-test-fees structures does not improve the maximal revenue guarantee. Proposition 6.
For every adversarial equilibrium in the game induced by an evidence-test-feestructure, there exists a test-fee structure that has an adversarial equilibrium with the samerevenue for the intermediary.
If the intermediary chose the equilibrium in addition to the test-fee structure, Proposition 6would follow from the logic of the revelation principle. But adversarial equilibrium selection If costs are strictly monotone in the Blackwell order, then every robustly optimal test-fee structure is in thestep-exponential-step class.
When assets are traded, it is common for sellers to disclose to buyers information obtainedfrom a third party about the value of the assets. One rationale for the existence of suchinformation intermediaries is that their presence generates economic value by, for example,mitigating moral hazard or facilitating assortative matching. A less obvious rationale for thepresence of such intermediaries, even if they provide no economic value, is that once sellers ofassets have the option to obtain hard information from an intermediary, potential buyers mayhave an unfavorable view of assets whose sellers do not disclose favorable information.Our paper investigates the scope of this second rationale. In a setting in which informationhas no social value, we study how an intermediary designs and prices evidence for a disclosuregame between an asset owner and the asset market. We show that even if the equilibria ofthe disclosure game are chosen to minimize the intermediary’s revenue, she can still guaranteeherself a high revenue across equilibria. We study how she accomplishes this.First, she uses option value as a carrot. Because the agent prefers not to have the assettested and the market to anticipate this, the intermediary chooses a test and fees so that theagent cannot credibly refrain from having the asset tested. We show that this correspondsto the intermediary creating option value by including high score realizations in the test andcharging sufficiently low testing and disclosure fees. The market then correctly expect the agentto have the asset tested, and treats non-disclosure with prejudice. In this way, the intermediaryexploits the agent’s commitment problem to guarantee herself revenue.Second, the intermediary uses positive disclosure fees and noisy tests. Disclosure fees andnoisy tests are closely related; were the intermediary constrained to charging only testing fees,she could optimally use tests that are fully revealing. To develop an intuition for the optimalcombination of test and fees we propose a demand curve approach in which every test cor-responds to a robust demand curve, and the intermediary can be thought of as choosing anoptimal price on an optimal robust demand curve. The optimal disclosure fee maximizes thearea of a rectangle under the robust demand curve, so the optimal robust demand curve has24 rectangular component, which leads to an exponential distribution of scores. Finally, whilewe have focused on two-part tariffs, which are often used in practice, it would be interestingto study a broader range of pricing structures, including disclosure fees that depend on therealized test score and fees paid by prospective buyers of the asset.
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Appendix
A.1 Proofs for Section 3
A.1.1 Proof of Proposition 1 on p. 13
Proof.
Consider a test-fee structure (
G, φ ). If (w-P) is violated, then Lemma 1 shows that thereis an equilibrium with zero revenue, and the proposition follows. Suppose that (w-P) holds.Using integration by parts, we can rewrite (w-P) as φ t ≤ R θµ + φ d [1 − G ( s )] ds . This expressionimplies thatany testing fee that satisfies (w-P) is at most the area above the score distribution G from µ + φ d to θ , shaded dark in Figure 7. The revenue from disclosure is at most φ d Pr[ s ≥ θ + φ d ],shaded light in Figure 7. This is because in any equilibrium, a score strictly less than θ + φ d strictly prefers to conceal. So the total revenue is at most the shaded area above G . Since G isa mean-preserving contraction of the prior distribution, µ = Z θθ sdG ( s ) = Z θθ [1 − G ( s )] ds + θ and therefore the area above G is equal to R F = µ − θ .CDF Score θ Gµ + φ d θ θ + φ d Figure 7: The revenue from testing fee fee is shaded dark. The revenue from disclosure fee is at most the areashaded light.
Now suppose that there exists an equilibrium in which revenue is R F − ε . Since revenue isat most the shaded area above G and the total area above G is R F , then the un-shaded areaabove G must be at most ε . In particular, the area above G from θ + φ d to µ + φ d is at most ε . Since G is monotone, φ t ≤ Z θµ + φ d [1 − G ( s )] ds ≤ ( θ − ( µ + φ d ))(1 − G ( µ + φ d )) ≤ (cid:18) θ − ( µ + φ d ) µ − θ (cid:19) Z µ + φ d θ + φ d [1 − G ( s )] ds ≤ (cid:18) θ − µµ − θ (cid:19) ε ε goes to zero, the revenue from testing goes to zero as well. Thus to completethe proof we only need to show that the revenue from disclosure also goes to zero.For ε small enough so that µ + φ d ≥ θ + φ d + √ ε , we have ε ≥ Z µ + φ d θ + φ d [1 − G ( s )] ds ≥ Z θ + φ d + √ εθ + φ d [1 − G ( s )] ds ≥ √ ε (1 − G ( θ + φ d + √ ε )) , where the third inequality follows since G is monotone. That is, the probability that the scoreis more than τ := θ + φ d + √ ε is at most √ ε . Thus if there exists an equilibrium thresholdabove τ , the disclosure probability in that equilibrium is at most √ ε . To show that there existsan equilibrium threshold above τ , we apply Lemma 3 by showing that E [ s | s ≤ τ ] > τ − φ d .The expectation of G can be written as µ = G ( τ ) E [ s | s ≤ τ ] + (1 − G ( τ )) E [ s | s > τ ] ≤ G ( τ ) E [ s | s ≤ τ ] + (1 − G ( τ )) θ. Rearranging terms yields θ − E [ s | s ≤ τ ] ≤ θ − µG ( τ ) ≤ θ − µ − √ ε . Therefore since τ = θ + φ d + √ ε and θ < µ , for small enough ε we have τ − φ d = θ + √ ε < µ − θ √ ε − √ ε = θ − θ − µ − √ ε ≤ E [ s | s ≤ τ ] , and therefore by Lemma 3 there must exist an equilibrium threshold higher than τ .To complete the proof, recall that φ d < θ − µ , so the revenue from disclosure fee is no morethan √ ε ( θ − µ ) . Thus the total revenue is bounded by δ ( ε ) ≡ ( θ − µµ − θ ) ε + √ ε ( θ − µ ) . (cid:3) A.2 Proofs for Section 4
A.2.1 Proof of Lemma 1 on p. 15.
Proof.
Consider a test-fee structure (
G, φ ) that satisfies (P). Assume towards a contradictionthat there exists an equilibrium in which the agent has the asset tested with probability strictlyless than 1. In this equilibrium, let p N be the price that the agent obtains when he does notdisclose his score. In any equilibrium, a agent discloses the score if s − φ d > p N and does notdisclose his score if s − φ d < p N . Therefore, the expected value conditional on non-disclosure isno higher than the prior expected value: p N ≤ µ .Because the agent has the asset tested with probability strictly less than 1, his equilibriumpayoff equals p N . Consider the following deviation: having the asset tested with probability 1, Precisely, for ε ≤ ε ∗ ≡ ( µ − θ θ ) . s satisfies s > p N + φ d . The payoff from this deviationis − φ t + R θp N + φ d ( s − φ d ) dG + R p N + φ d θ p N dG where the first term is the testing fee, the second isthe payoff from disclosing a high score, and the third is the payoff from concealing a low score.Taking the difference between this deviation payoff and the equilibrium payoff of p N yields − φ t + Z θp N + φ d ( s − φ d ) dG + Z p N + φ d θ p N dG − p N = − φ t + Z θp N + φ d ( s − p N − φ d ) dG ≥ − φ t + Z θµ + φ d ( s − µ − φ d ) dG> . where the equality follows from algebra, the first inequality follows from p N ≤ µ , and thesecond inequality follows from (P). Therefore, the deviation is strictly profitable, and thisstrategy profile is not an equilibrium.Now consider a test-fee structure ( G, φ ) that violates (P), and where φ t ≥ Z θµ + φ d [ s − ( µ + φ d )] dG. (9)We show that now there exists an equilibrium where the agent has the asset tested with prob-ability 0. Consider a strategy profile in which the agent’s strategy involves having the assettested with probability 0, and therefore, he has no opportunity to disclose on the path of play.Off-path, the agent discloses when obtaining a score s ≥ µ + φ d .In this strategy profile, the market price from non-disclosure is µ . We claim that thisstrategy profile is an equilibrium. First, the agent’s behavior is sequentially rational off-path.Second, by deviating to take the test, the agent receives an expected payoff lower than µ : Z µ + φ d θ µdG + Z θµ + φ d [ s − φ d ] dG − φ t = µ + Z θµ + φ d [ s − ( µ + φ d )] dG − φ t ≤ µ where the equality is algebra, and the inequality follows from (9). So the agent has no profitabledeviation, and the strategy profile is an equilibrium. (cid:3) A.2.2 Proof of Lemma 2 on p. 16
Proof.
Consider a test-fee structure (
G, φ ) where at least one of the fees is negative.If φ t − R θµ + φ d [ s − ( µ + φ d )] dG <
0, from Lemma 1, the asset is tested with probability1 in all equilibria. Suppose φ d <
0, then in the unique equilibrium all scores disclose, so30 ( G, φ ) = φ t + φ d . Let φ ′ d = 0 and φ ′ t = φ t + (1 − G ( µ + φ d )) φ d . Then φ ′ t − R θµ + φ ′ d [ s − ( µ + φ ′ d )] dG ≤ φ t − R θµ + φ d [ s − ( µ + φ d )] dG <
0, so we can find φ ′′ t > φ ′ t and φ ′′ d = φ ′ d = 0 such that φ ′′ t − R θµ + φ ′′ d [ s − ( µ + φ ′′ d )] dG <
0. Then Lemma 1 implies that under (
G, φ ′′ ) the asset is testedwith probability 1 and R ( G, φ ′′ ) = φ ′′ t > φ ′ t = φ t + (1 − G ( µ + φ d )) φ d ≥ φ t + φ d = R ( G, φ ).Now if φ ′′ t ≥
0, we have constructed a test-fee structure with non-negative fees that generatesa strictly higher revenue. If φ ′′ t <
0, since φ ′′ d = 0, it implies R ( G, φ ′′ ) ≤
0. But then wecan easily construct a test-fee strcuture ( G ′′′ , φ ′′′ ) that generates a strictly positive revenue: let G ′′′ = F and φ ′′′ d = 0. By assumption F is not degenerated (i.e., θ < θ ), so R θµ ( s − µ ) dF > . Let φ ′′′ t ∈ (0 , R θµ ( s − µ ) dF ), and from Lemma 1 in all equilibria the asset is tested withprobability 1 under ( G ′′′ , φ ′′′ ), and R ( G ′′′ , φ ′′′ ) = φ ′′′ t >
0. Now suppose φ t < φ d ≥ φ ′′′′ t = 0 and φ ′′′′ d = φ d , then ( G, φ ′′′′ ) satisfies (P) unless R θµ + φ d [ s − ( µ + φ d )] dG = 0. So if R θµ + φ d [ s − ( µ + φ d )] dG >
0, a higher testing fee is charged while the revenue from disclosure feeis unchanged, because the disclosure decision doesn’t depend on the testing fee (this can alsobe seen from φ t does not appear in (HE)). If R θµ + φ d [ s − ( µ + φ d )] dG = 0, then G ( µ + φ d ) = 1,which implies τ = µ + φ d is the threshold satisfying (HE), and R ( G, φ ) = φ t + 0 <
0. But thenthe ( G ′′′ , φ ′′′ ) constructed above gives a strictly higher revenue with non-negative fees.If φ t − R θµ + φ d [ s − ( µ + φ d )] dG ≥
0, from Lemma 1, R ( G, φ d ) ≤
0. Then the ( G ′′′ , φ ′′′ )constructed above gives a strictly higher revenue with non-negative fees. (cid:3) A.2.3 Proof of Lemma 3 on p. 17
The following lemma is used in our proof.
Lemma 5.
Suppose f is an increasing function defined on [ a, b ] ⊂ R , and g is a continuousfunction defined on [ a, b ] . If f ( a ) > g ( a ) and f ( b ) < g ( b ) , there exists x ∗ ∈ ( a, b ) such that f ( x ∗ ) = g ( x ∗ ) . Moreover, ¯ x = max { x | f ( x ) = g ( x ) } exists and ¯ x = sup { x ∈ [ a, b ] | f ( x ) ≥ g ( x ) } .Proof. Define S = { x ∈ [ a, b ] | f ( x ) ≥ g ( x ) } . Becase S is non-empty and bounded above by b , there exists a unique supremum ¯ x = sup S . Since f is increasing, f ( a ) > g ( a ), and g iscontinuous, it follows that for sufficiently small ε > f ( a + ε ) > f ( a ) > g ( a + ε ), which implies¯ x > a . Consider an increasing sequence x n ∈ S such that x n → ¯ x . Since f is an increasingfunction, f (¯ x ) ≥ lim sup n →∞ f ( x n ). Recall that f ( x n ) ≥ g ( x n ), so taking limsup for both sidesyields f (¯ x ) ≥ lim sup n →∞ f ( x n ) ≥ lim sup n →∞ g ( x n ) = g (¯ x )where the equality holds because g is continuous. So f (¯ x ) ≥ g (¯ x ), which also implies ¯ x = b .31ow we prove that f (¯ x ) = g (¯ x ). Suppose towards a contradiction that f (¯ x ) > g (¯ x ). Because g is continuous, there exists ε > f (¯ x ) > g (¯ x + ε ). Since f is increasing, it thenfollows that f (¯ x + ε ) ≥ f (¯ x ) > g (¯ x + ε )which means ¯ x + ε ∈ S , contradicting ¯ x = sup S . Therefore, it follows that ¯ x = max { x | f ( x ) = g ( x ) } . (cid:3) Proof of Lemma 3 (a) . A threshold τ is an equilibrium if and only if E G [ s | s ≤ τ ] = τ − φ d . Define a ( τ ) = E G [ s | s ≤ τ ] and b ( τ ) = τ − φ d . Since a ( τ ) is bounded by µ and b ( τ ) → ∞ as τ → ∞ , there exists ¯ τ large enough such that a ( τ ) < b ( τ ) for all τ ≥ ¯ τ . So if an equilibriumexists, the threshold τ must be in [ s G , ¯ τ ].We have a ( s G ) = s G ≥ b ( s G ) and a (¯ τ ) ≤ b (¯ τ ), a is increasing and b is continuous. FromLemma 5, there exists τ ∈ [ s G , ¯ τ ] such that a ( τ ) = b ( τ ), and max { τ | a ( τ ) = b ( τ ) } exists. Sothe set of equilibrium thresholds is non-empty and a highest equilibrium threshold exists. (cid:3) Proof of Lemma 3 (b) . The “if” part follows directly from definition, so we prove the “only if”part. Suppose that τ is the highest threshold of ( G, φ ), so τ − φ d = E [ s | s ≤ τ ] and for all τ ′ > τ , τ ′ − φ d = E [ s | s ≤ τ ′ ]. We show that indeed we must have τ ′ − φ d > E [ s | s ≤ τ ′ ].Suppose for contradiction that there exists τ ′ > τ such that τ ′ − φ d < E [ s | s ≤ τ ′ ]. Since a ( τ ′ ) > b ( τ ′ ), a (¯ τ ) ≤ b (¯ τ ), a is increasing and b is continuous, from Lemma 5, there exists τ ∗ ∈ ( τ ′ , ¯ τ ] ⊂ ( τ, ¯ τ ] such that a ( τ ∗ ) = b ( τ ∗ ). It implies threshold τ ∗ > τ is an equilibrium,which leads to a contradiction. Therefore, for all τ ′ > τ , we must have τ ′ − φ d > E [ s | s ≤ τ ′ ] . (cid:3) Proof of Lemma 3 (c) . We show that the agent discloses and only discloses scores greater thanthe highest threshold τ is an adversarial equilibrium. Suppose there exists another equilibriumthat gives the intermediary a lower revenue. In this equilibrium, let ˜ τ = p N + φ d , where p N isthe price of no disclosure. Since all scores strictly greater than ˜ τ must disclose and all scoresstrictly lower than ˜ τ must conceal in equilibrium, the equilibrium price satisfies E [ s < ˜ τ ] ≤ p N ≤ E [ s | s ≤ ˜ τ ] . The intermediary’s revenue from this equilibrium is φ t + φ d [1 − G (˜ τ )+ λ ( G (˜ τ ) − sup s<τ G ( s ))],for some λ ∈ [0 , s = τ . Sincethis equilibrium gives the intermediary a lower revenue,1 − G (˜ τ ) + λ ( G (˜ τ ) − sup s<τ G ( s )) < − G ( τ ) , τ > τ . But then from part (b) the characterization of τ and the fact that ˜ τ > τ ,˜ τ − φ d > E [ s | s ≤ ˜ τ ] ≥ p N which contradicts ˜ τ = p N + φ d . (cid:3) A.2.4 Proof of Lemma 4 on p. 18
We prove Lemma 4 by first proving part (c), then (a), and then (b).We use the following lemma to prove Lemma 4(c).
Lemma 6. If ( G n , τ n ) → ( G, τ ) , and lim n →∞ G n ( τ n ) exists, then lim n →∞ G n ( τ n ) ≤ G ( τ ) .Proof. It suffices to show that for any c > G ( τ ), there exists N such that for n ≥ N , G n ( τ n ) < c .Because G is right-continuous, it follows that for any c > G ( τ ), there exists ¯ ε > c > G ( τ + 2¯ ε ) + ¯ ε (10)Notice that τ n → τ , so there exists N such that for n > N , τ n ≤ τ + ¯ ε , which implies G n ( τ n ) ≤ G n ( τ + ¯ ε ) (11)Because weak convergence is metrized by the Levy metric, L ( G, F ) = inf { ε > | F ( x − ε ) − ε ≤ G ( x ) ≤ F ( x + ε ) + ε for all x } . Thus, if G n converges weakly to G , there exists N such that for all n > N , L ( G, G n ) < ¯ ε , andhence G ( τ + 2¯ ε ) + ¯ ε ≥ G n ( τ + ¯ ε ) (12)Combining (10), (11) and (12), it follows that for every n ≥ max { N , N } , c > G ( τ + 2¯ ε ) + ¯ ε ≥ G n ( τ + ¯ ε ) ≥ G n ( τ n ) . (cid:3) Proof of Lemma 4(c).
Let us begin by verifying (w-P). We have φ t = lim n →∞ φ nt ≤ lim n →∞ Z θµ + φ nd [ s − ( µ + φ nd )] dG n ( s ) = Z θµ + φ d [ s − ( µ + φ d )] dG ( s )where the equalities follow from taking limits and the inequality follows since ( G n , φ n , τ n )satisfies (P). So ( G, φ ) satisfies (w-P). 33ow we turn to (HE). Since τ n − φ nd = E [ s | s ≤ τ n ] ≤ µ and φ nd → φ d , we have τ n beingbounded above. On the other hand τ n ≥ s G ≥ θ so τ n is bounded below. From Bolzano-Weierstrass theorem, there exists a subsequence n k and τ such that lim k →∞ τ n k = τ .For any τ ′ > τ , we show that φ d G ( τ ′ ) ≤ R τ ′ θ G ( s ) ds . If φ d G ( τ ′ ) > R τ ′ θ G ( s ) ds , there existssmall enough ε such that φ d G ( τ ′ + ε ) > R τ ′ + εθ G ( s ) ds and G is continuous at τ ′ + ε . Solim n →∞ φ nd G n ( τ ′ + ε ) = φ d G ( τ ′ + ε ). But then we have a contradiction: φ d G ( τ ′ + ε ) > Z τ ′ + εθ G ( s ) ds = lim k →∞ Z τ ′ + εθ G n k ( s ) ds ≥ lim k →∞ φ n k d G k n ( τ ′ + ε ) = φ d G ( τ ′ + ε )where the second inequality holds because τ ′ > τ implies for large enough k , τ ′ + ε > τ ′ > τ n k and thus φ n k d G n k ( τ ′ + ε ) ≤ R τ ′ + εθ G n k ( s ) ds .Next we show that φ d G ( τ ) = R τθ G ( s ) ds . Since G is right continuous and for all τ ′ > τ , φ d G ( τ ′ ) ≤ R τ ′ θ G ( s ) ds , we must have φ d G ( τ ) ≤ R τθ G ( s ) ds . Since φ n k d G n k ( τ n k ) = R τ nk θ G n k ( s ) ds for all k , lim k →∞ φ n k d G n k ( τ n k ) exists and equals lim k →∞ R τ nk θ G n k ( s ) ds = R τθ G ( s ) ds . Therefore φ d G ( τ ) ≥ lim k →∞ φ n k d G n k ( τ n k ) = Z τθ G ( s ) ds where the inequality holds from Lemma 6. Notice that the argument above also implies φ d G ( τ ) = lim k →∞ φ n k d G n k ( τ n k ).Now we have shown ( G, ( φ d , φ t )) and τ satisfies (w-P) and (w-HE), and next we considerthe revenues comparison: V = lim n →∞ φ nt + φ nd (1 − G n ( τ n )) = φ t + lim k →∞ φ n k d (1 − G n k ( τ n k )) = φ t + φ d (1 − G ( τ )) . So ( G, ( φ d , φ t )) and τ indeed generate a revenue as the limit of the revenues generated by( G n , ( φ nd , φ nt )) and τ n . (cid:3) We use the following lemma to prove Lemma 4(a) and Lemma 4(b).
Lemma 7.
For every test-fee structure and threshold ( G, φ, τ ) that satisfy (w-P) and (w-HE) ,there exists a sequence of test-fee structures and thresholds { ( G n , φ n , τ n ) } n =1 , ,... such that (i)for each n , ( G n , φ n , τ n ) satisfy (P) and (HE) , (ii) ( G n , φ n ) converges to ( G, φ ) , and (iii) ˆ R ( G, φ, τ ) ≤ lim n →∞ ˆ R ( G n , φ n , τ n ) .Proof. Consider a test-fee structure ( φ t , φ d , G ) and τ satisfying (w-P) and (w-HE). Therefore, φ t ≤ Z θ max { µ + φ d ,θ } ( s − µ − φ d ) dG ( s ) and τ − φ d = E [ s | s ≤ τ ] 34 ′ − φ d ≥ E [ s | s ≤ τ ′ ] for all τ ′ > τ. Let φ nd = φ d − n and φ nt = φ t − n , then φ nt < φ t ≤ Z θµ + φ d ( s − µ − φ d ) dG ( s ) ≤ Z θµ + φ nd ( s − µ − φ nd ) dG ( s ) , so (P) is satisfied.From Lemma 3, we know that under ( φ nt , φ nd , G ), the highest equilibrium threshold ˆ τ n exists.Moreover, since τ − φ nd > τ − φ d = E [ s | s ≤ τ ] τ ′ − φ nd > τ ′ − φ d ≥ E [ s | s ≤ τ ′ ] for all τ ′ > τ. It implies the highest threshold ˆ τ n < τ . So the revenue under ( φ nt , φ nd , G ) is φ nt + φ nd (1 − G (ˆ τ n )) ≥ φ nt + φ nd (1 − G ( τ )) ≥ φ t + φ d (1 − G ( τ )) − n where the last term goes to ˆ R ( G, φ, τ ) as n → ∞ . (cid:3) Proof of Lemma 4(a).
Consider any sequence ( G n , ( φ nt , φ nd )) and τ n satisfying (P) and (HE)and generating value V n → R M .The disclosure fee being non-negative ( φ nd ≥
0) implies φ nt ≤ R θµ ( s − µ ) dG n ( s ) ≤ θ − µ . Forany φ nd > θ − µ , (P) and (HE) implies φ nt < G ( τ n ) = 1, and thus V n <
0. So withoutloss of generality, we can consider φ nd ≤ θ − µ . Moreover, for any φ nt < −
1, by replacing it with φ nt = −
1, (P) and (HE) are still satisfied, and V n increases. So without loss of generality, wecan restrict attention to φ nt ≥ − G n , ( φ nt , φ nd )) such that φ n j t ∈ [ − , θ − µ ], φ n j d ∈ [0 , θ − µ ], G n ∈ Γ( F ).Proposition 1 of Kleiner, Moldovanu, and Strack (2020) proves that Γ( F ) is compact, whichimplies that there exists a converging subsequence ( G n k , ( φ n k t , φ n k d )) → ( G, ( φ t , φ d )). FromLemma 4(c), we can find τ such that ( G, ( φ t , φ d )) and τ satisfies the constraints (w-P) and(w-HE) in the relaxed problem, and generates a revenue R M . Moreover, from Lemma 7, anytest-fee structure and threshold satisfying (w-P) and (w-HE) must at most generate a revenueof R M , so ( G, ( φ t , φ d )) and τ is an optimal solution to the relaxed problem. (cid:3) Proof of Lemma 4(b).
By Lemma 7, there exists a sequence of test-fee structures and thresholds { ( G n , φ n , τ n ) } n =1 , ,... such that (i) for each n , ( G n , φ n , τ n ) satisfy (P) and (HE), (ii) ( G n , φ n )converges to ( G, φ ), and (iii) lim n →∞ ˆ R ( G n , φ n , τ n ) ≥ ˆ R ( G, φ, τ ) = R M . By definition of R M ,we also have lim n →∞ ˆ R ( G n , φ n , τ n ) ≤ R M . Therefore, lim n →∞ ˆ R ( G n , φ n , τ n ) = R M . (cid:3) .3 Proofs for Section 5 This section contains the proofs of Propositions 2–4. The three statements of Proposition 2build on each other in reverse order. We therefore prove Proposition 2 in reverse order, begin-ning with (d) and ending with (a). Proposition 3 is a corollary of one of the steps (Lemma 9)used in Proposition 2(d).
A.3.1 Proof of Proposition 2 (d) on p. 19
We prove Proposition 2(d) by solving a relaxed problem where the mean-preserving contractionconstraints are relaxed to requiring the score distribution to have same expectation as the priormean: max ( G,φ,τ ) ∈ ∆[ θ,θ ] × R ˆ R ( G, φ, τ ) s.t. (w-P) , (w-HE) , and E G [ s ] = E F [ s ] (RE)Recall that ˆ R ( G, φ, τ ) ≡ φ t + φ d (1 − G ( τ )) is the revenue corresponding to a test-fee structure( G, φ ) and a threshold τ . This is a relaxed problem because G ∈ Γ( F ) implies G ∈ ∆[ θ, θ ] and E G [ s ] = E F [ s ].We use the following lemma to solve the relaxed problem. The lemma shows that in anyoptimal testing fee structure, the only possible score above µ + φ d is θ . In other words, the scoredistribution G is flat from µ + φ d to θ , with possibly a discrete jump at θ so that G ( θ ) = 1. Lemma 8.
If a test-fee structure ( G, φ ) with a weak-highest equilibrium threshold τ is anoptimal solution to the relaxed problem (RE) , then G ( s ) = G ( µ + φ d ) for all s ∈ [ µ + φ d , θ ) .Proof. Using integration by parts, the testing fee of an optimal test-fee structure is φ t = Z θµ + φ d [ s − ( µ + φ d )] dG ( s ) = − Z θµ + φ d G ( s ) ds + ( θ − ( µ + φ d ))Thus, revenue is ˆ R ( G, φ, τ ) = − φ d G ( τ ) − Z θµ + φ d G ( s ) ds + ( θ − µ ) . Since τ is a weak-highest equilibrium threshold, φ d G ( τ ) = R τθ G ( s ) ds . Substituting this equalityinto revenue, we haveˆ R ( G, φ, τ ) = − Z τθ G ( s ) ds − Z θµ + φ d G ( s ) ds + ( θ − µ ) = Z µ + φ d τ G ( s ) ds, (13)where the second equality followed from the constraint that the integral of G is 1 − µ .36ow consider an optimal test-fee structure ( G, φ ) with a weak-highest equilibrium threshold τ , and suppose for contradiction that G ( s ) > G ( µ + φ d ) for some s ∈ [ µ + φ d , θ ). Constructa distribution G ′ as follows. Let G ′ ( s ) = αG ( s ) for some α and all s ≤ µ + φ d , and G ′ ( s ) = G ′ ( µ + φ d ) for all s ∈ [ µ + φ d , θ ). By the assumption that G ( s ) > G ( µ + φ d ) for some s ∈ [ µ + φ d , θ ),there exists α > G and G ′ are equal. Define φ ′ d = φ d , and φ ′ t sothat the upper bound on the testing fee (w-P) holds with equality for distribution G ′ .We now show that τ is a weak-highest equilibrium threshold of the test-fee structure ( G ′ , φ ′ ),and gives higher revenue than ( G, φ ). Indeed, for any τ ′ such that τ ≤ τ ′ ≤ µ + φ d , since G ′ isequal to G multiplied by α , we have R τ ′ θ G ( s ) dsG ( τ ′ ) = R τθ G ( s ) dsG ( τ ) . Thus, from integration by parts, τ ′ − E G ′ [ s | s ≤ τ ′ ] = τ ′ − E G [ s | s ≤ τ ′ ] ≥ φ d , with equality at τ ′ = τ . In addition, for all τ ′ ≥ µ + φ d , since G ′ is flat we have R τ ′ θ G ′ ( s ) dsG ′ ( τ ′ ) ≥ R µ + φ d θ G ( s ) dsG ( µ + φ d ) ≥ φ d . It remains to show that the revenue of ( G ′ , φ ′ ) is higher than that of ( G, φ ). This fact followsdirectly from (13), since G ′ > G below µ + φ d . (cid:3) Equipped with Lemma 8, we solve the relaxed problem (RE).
Lemma 9.
The value of (RE) is ( θ − µ )(1 − e θ − µθ − µ ) , and it has a unique solution φ ∗ t = 0 , φ ∗ d = θ − µ , and G ∗ ( s ) = e θ − µθ − µ if s ∈ [ θ, θ + θ − µ ) e s − θθ − µ if s ∈ [ θ + θ − µ, θ ] . Proof.
We first argue that the test-fee structure ( G ∗ , φ ∗ ) achieves the revenue bound ( θ − µ )(1 − e θ − µθ − µ ). We will later show that ( θ − µ )(1 − e θ − µθ − µ ) is an upper bound on revenue and therefore( G ∗ , φ ∗ ) is optimal. We start by verifying that ( G ∗ , φ ∗ ) is a feasible solution. Weak participation constraint is satisfied
Since µ + φ ∗ d = θ , the weak participationconstraint (w-P) holds with equality, i.e., Z θµ + φ ∗ d [ s − ( µ + φ ∗ d )] dG ∗ ( s ) = 0 = φ ∗ t . If τ = µ + φ d , ˆ R ( G, φ, τ ) = 0. As we have shown in the proof of Lemma 2, the optimal revenue is strictlypositive. So in an optimal test-fee structure (
G, φ ) with threshold τ , τ < µ + φ d . he constraint E G [ s ] = E F [ θ ] is satisfied Directly from the definition of G ∗ , for any τ ′ ≥ θ + θ − µ we have Z τ ′ θ G ∗ ( s ) ds = ( θ − µ ) e θ − µθ − µ + ( θ − µ ) e s − θθ − µ (cid:12)(cid:12)(cid:12)(cid:12) τ ′ θ + θ − µ = ( θ − µ ) e τ ′− θθ − µ . Thus, in particular, R θθ G ∗ ( s ) = θ − µ and hence, by integration by parts, E G ∗ [ s ] = Z θθ sdG ∗ ( s ) = θ − Z θθ G ∗ ( s ) ds = µ. Therefore, E G [ s ] = E F [ θ ]. Weak-highest equilibrium constraint is satisfied
We show that τ = θ + θ − µ is aweak-highest equilibrium threshold, i.e., it satisfies (w-HE). For any τ ′ ≥ θ + θ − µ , E [ s | s ≤ τ ′ ] = R τ ′ θ sdG ∗ ( s ) G ∗ ( τ ′ ) = τ ′ G ( τ ′ ) − R τ ′ θ G ∗ ( s ) dsG ∗ ( τ ′ ) = τ ′ − ( θ − µ ) e τ ′− θθ − µ e τ ′− θθ − µ = τ ′ − φ ∗ d . Therefore, τ = θ + θ − µ satisfies (w-HE) and is a weak-highest equilibrium threshold.The lemma below shows that E [ s ] = µ and τ = θ + θ − µ is a weak-highest equilibrium threshold.Thus, the revenue of the test fee structure ( G ∗ , φ ∗ ) with weak-highest equilibrium threshold τ = θ + θ − µ is φ ∗ t + φ ∗ d (1 − G ( θ + θ − µ )) = 0 + ( θ − µ )(1 − e θ − µθ − µ ) . Upper bound on revenue
It remains to show that the revenue of any test-fee structure isat most ( θ − µ )(1 − e θ − µθ − µ ).For any τ ′ such that τ ≤ τ ′ ≤ µ + φ d , the total area under G is θ − µ = Z τ ′ θ G ( s ) ds + Z θτ ′ G ( s ) ds ≤ φ d e φd ( τ ′ − τ ) G ( τ ) + Z θτ ′ G ( s ) ds ≤ φ d e φd ( τ ′ − τ ) G ( τ ) + ( θ − τ ′ ) G ( µ + φ d ) , G ( s ) ≤ G ( µ + φ d ) for all s ≤ µ + φ d , and G ( s ) = G ( µ + φ d ) for all s ≥ µ + φ d by Lemma 8.Let τ ′ = φ d + µ − θ (1 − G ( µ + φ d )) G ( µ + φ d ) . Notice that τ ′ ∈ [ τ, µ + φ d ] because from the definition of τ , τ = φ d + E [ s | s ≤ τ ] ≤ φ d + E [ s | s ≤ µ + φ d ] = τ ′ , and µ < θ implies µ − θ (1 − G ( µ + φ d )) G ( µ + φ d ) < µ .Therefore, by the above discussion we have G ( τ ) ≥ φ d e − φd ( τ ′ − τ ) (( θ − µ ) − ( θ − τ ′ ) G ( µ + φ d ))= 1 φ d e − φd ( τ ′ − τ ) (( θ − µ ) − ( θ − µG ( µ + φ d ) − φ d ) G ( µ + φ d ))= e − φd ( τ ′ − τ ) G ( µ + φ d ) ≥ e − φd ( θ − θ − θ − µG ( µ + φd ) ) G ( µ + φ d ) , (14)where the last inequality followed since θ + φ d ≤ τ .We now use (14) and Lemma 8 to bound revenue. Since by Lemma 8, the distribution isflat above µ + φ d , the testing fee is φ t = Z θµ + φ d [ s − ( µ + φ d )] dG ( s ) = (1 − G ( µ + φ d ))( θ − ( µ + φ d ))Thus revenue isˆ R ( G, φ, τ ) ≤ φ d (cid:20) − e − φd ( θ − θ − θ − µG ( µ + φd ) ) G ( µ + φ d ) (cid:21) + (1 − G ( µ + φ d ))( θ − ( µ + φ d )) . The above expression is non-decreasing in φ d . Since φ d ≤ θ − µ , an upper bound on revenue isobtained by substituting φ d = θ − µ into the above expression, which yields( θ − µ )(1 − e − θ − θθ − µ + G ( µ + φd ) G ( µ + φ d )) . This expression is non-decreasing in G ( µ + φ d ). Since G ( µ + φ d ) ≤
1, an upper bound onrevenue is obtained by substituting G ( µ + φ d ) = 1 into the above expression, which yields( θ − µ )(1 − e θ − µθ − µ ), completing the proof. (cid:3) Proof of Proposition 2 (d) . The solution to the relaxed problem in Lemma 9 is ( θ − µ )(1 − e θ − µθ − µ ).Therefore ( θ − µ )(1 − e θ − µθ − µ ) is an upper bound on the revenue of any test-fee structure. Moreover,if the support of the prior is binary (i.e. { θ, θ } ), G ∈ Γ( F ) is equivalent to G ∈ ∆[ θ, θ ] and E G [ s ] = E F [ s ], so the bound is attained in this case. (cid:3) .3.2 Proof of Proposition 2 (c) on p. 19 Proof of Proposition 2 (c) . In a robustly optimal test-fee structure, the constraint (w-P) mustbind, otherwise the intermediary can strictly increases the revenue by increasing φ t , so φ t = R ¯ θµ + φ d [ s − ( µ + φ d )] dG . Notice that R ¯ θµ + φ d [ s − ( µ + φ d )] dG > G ( µ + φ d ) <
1, which thenimplies the result. (cid:3)
A.3.3 Proof of Proposition 2 (b) on p. 19
We prove Proposition 2(b) in two steps. We first identify test-fee structures that are robustlyoptimal among ones with zero disclosure fees. We show that it is in fact robustly optimal touse a fully revealing test. We will then show that such a test-fee structure can be improvedupon using a small disclosure fee.
Lemma 10.
The test-fee structure ( F, φ ) with φ t = R θµ [ s − µ ] dF ( s ) and φ d = 0 is robustlyoptimal among all test-fee structures with zero disclosure fee.Proof. Consider a test-fee structure with zero disclosure fee. The revenue is Z θµ ( s − µ ) dG ( s ) = ( θ − µ ) − Z θµ G ( s ) ds. Thus revenue is maximized by minimizing R θµ G ( s ) ds .The MPC constraints require that R θτ G ( s ) ds ≥ R θτ F ( s ) ds for all τ ∈ [ θ, θ ]. Specifically at τ = µ , we have Z θµ G ( s ) ds ≥ Z θµ F ( s ) ds. Thus the optimal revenue is at most ( θ − µ ) − Z θµ F ( s ) ds which is obtained by setting G = F . (cid:3) Equipped with Lemma 10, we prove Proposition 2(b).
Proof of Proposition 2 (b) . Consider the revenue from charging a positive disclosure fee φ ′ d , andreducing the testing fee to φ ′ t so that the upper bound on testing fee (w-P) binds, ensuringparticipation with probability one. This change involves a loss and a gain in revenue. The lossis from lower testing fee, which is the difference between θ − µ − R θµ G ( s ) ds and θ − ( µ + φ ′ d ) − R θµ + φ ′ d G ( s ) ds . By algebra, the loss is at most φ ′ d (1 − G ( µ )).40he gain from charging a positive disclosure fee is φ ′ d (1 − G ( τ )), where τ is a weak-highestequilibrium threshold. Notice that since θ is in the support of the distribution, we have G ( τ ′ ) > τ ′ > θ . Thus, by integration by parts we have τ ′ − E [ s | s ≤ τ ′ ] = R τ ′ θ G ( s ) dsG ( τ ′ ) > . for all τ ′ > θ . Therefore, for any τ ′ > θ , there exists a small enough disclosure fee such that anyweak-highest equilibrium threshold is below τ ′ . In particular, for a small enough disclosure fee,any weak-highest equilibrium threshold τ is below µ and satisfies G ( τ ) ≤ G ( µ ). As a result,for small enough disclosure fee, the gain is at least the loss. Further, if G ( s ) > G ( µ ) for some s ∈ [ µ, θ ), then there exists φ d such that the loss is strictly less than φ ′ d (1 − G ( µ )). This is thecase if G is non-binary. As a result, for a non-binary G , there exists φ ′ , φ ′ d >
0, such that therevenue of (
G, φ ′ ) is strictly larger than the revenue of ( G, φ ).Suppose first that the prior F is non-binary. By Lemma 10, the test-fee structure ( F, φ )where φ d = 0 is optimal among all test-fee structures with zero disclosure fee. The argumentabove shows that there exists φ ′ with positive disclosure fee and such that the revenue of ( F, φ ′ )is strictly higher than that of ( F, φ ), and hence of any test-fee structure with zero disclosurefee.Now suppose that the prior F is binary. With binary support, the mean-preserving con-traction constraints become E G [ s ] = E F [ θ ]. Therefore, by Lemma 9, the optimal revenue is( θ − µ )(1 − e θ − µθ − µ ). By Lemma 10, the optimal revenue among test-fee structures with zerodisclosure fee is obtained by full revelation. The revenue is the testing fee θ − µ − Z θµ F ( s ) ds = ( θ − µ ) − ( θ − µ ) θ − θ = ( θ − µ )( µ − θ ) θ − θ , which is strictly less than the optimal revenue ( θ − µ )(1 − e θ − µθ − µ ) for all µ ∈ ( θ, θ ). Since θ is inthe support of the distribution, it ensures that µ ∈ ( θ, θ ). (cid:3) A.3.4 Proof of Proposition 2 (a) on p. 19
We use two lemmas in the proof of Proposition 2(a). The first lemma bounds the rate at whichthe integral of a score distribution can grow given the (w-HE) constraint.
Lemma 11.
Suppose that φ d > . Let τ be a weak-highest equilibrium threshold. Then for any τ a and τ b where τ ≤ τ a ≤ τ b , Z τ b θ G ( s ) ds ≤ e φd ( τ b − τ a ) Z τ a θ G ( s ) ds, ith equality if and only if (w-HE) holds with equality for all thresholds in [ τ a , τ b ] .Proof. Using integration by parts, the weak-highest equilibrium threshold constraint (w-HE)can be written as φ d ≤ τ ′ − E [ s | s ≤ τ ′ ] = R τ ′ θ G ( s ) dsG ( τ ′ ) , for all τ ′ ≥ τ , with equality at τ ′ = τ . The right hand side is the inverse of the derivative ofln( R τ ′ θ G ( s ) ds ) with respect to τ ′ . Thus, ddτ ′ (ln( Z τ ′ θ G ( s ) ds )) ≤ φ d . (15)Integrating from τ a to τ b ,ln( Z τ b θ G ( s ) ds ) − ln( Z τ a θ G ( s ) ds ) ≤ φ d ( τ b − τ a ) . Raising both sides to the power of e , R τ b θ G ( s ) ds R τ a θ G ( s ) ds ≤ e φd ( τ b − τ a ) , with equality if and only if (w-HE) holds with equality for all thresholds in [ τ a , τ b ]. (cid:3) The following lemma is the main step in the proof of Proposition 2(a). It shows that forany test-fee structure, there exists a test-fee structure that is in the step-exponential-step classand has a weakly higher revenue guarantee. The lemma further establishes that any robustlyoptimal test must be exponential over an interval.
Lemma 12.
For any test-fee structure ( G, φ ) with φ d > and a corresponding weak-highestthreshold τ , there exists a mean-preserving contraction G ′ of G , a fee structure φ ′ , and athreshold τ ′ such that the test-fee structure ( G ′ , φ ′ ) is in the step-exponential-step class, τ ′ isa weak-highest threshold and the robust revenue of ( G ′ , φ ′ , τ ′ ) is at least the robust revenue of ( G, φ, τ ) . Further, if ( G, φ ) is robustly optimal and has weak-highest equilibrium threshold τ ,then φ d > and there exists a threshold τ ∈ [ τ , µ + φ d ] , such that G is exponential from τ to τ , i.e., G ( s ) = G ( τ ) e φd ( s − τ ) , ∀ s ∈ [ τ , τ ] , nd is flat from τ to µ + φ d , i.e., G ( s ) = G ( µ + φ d ) , ∀ s ∈ [ τ , µ + φ d ] . Proof.
Given a test-fee structure (
G, φ ) with a weak-highest threshold τ , clearly by definitionof τ , we have τ ≤ µ + φ d . Moreover, if τ = µ + φ d , the intermediary receives zero revenue,which is never optimal. So we consider τ < µ + φ d .We first consider the case G ( µ + φ d ) = G ( τ ), where τ is a weak-highest equilibriumthreshold under ( G, φ ). Define τ a = E G [ s | s ≤ τ ], and τ b = E G [ s | s > µ + φ d ]. From thedefinition of τ , we have τ a = E G [ s | s ≤ τ ] = τ − φ d . The mean constraint requires G ( τ ) τ a +(1 − G ( τ )) τ b = µ , which implies G ( τ ) = τ b − µτ b − τ a .Given the test G , and to satisfy (w-P), the highest testing fee the intermediary can chargeis φ t = R µ + φ d [ s − ( µ + φ d )] dG ( s ). The disclosure fee being charged under equilibrium threshold τ is φ d (1 − G ( τ )). So the intermediary’s robust revenue bound under test G is Z θµ + φ d [ s − ( µ + φ d )] dG ( s ) + φ d (1 − G ( τ )) = θ − µ − φ d − Z θµ + φ d G ( s ) ds + φ d − φ d G ( τ )= Z µ + φ d θ G ( s ) ds − Z τ θ G ( s ) ds = Z µ + φ d τ G ( s ) ds = G ( τ )( µ + φ d − τ )= ( τ b − µ )( µ − τ a ) τ b − τ a . Now we construct the following G ′ , which is in the class of step-exponential-step with thelast step being degenerated: G ′ ( s ) = s ∈ [ θ, τ a ) e τa − µτb − µ if s ∈ [ τ a , τ a + τ b − µ ) e s − τbτb − µ if s ∈ [ τ a + τ b − µ, τ b ]1 if s ∈ [ τ b , θ ] . It can be verified that G ′ is a mean-preserving contraction of G , and τ ′ = τ a + τ b − µ is aweak-highest equilibrium threshold under test-fee structure ( G ′ , φ ′ d , φ ′ t ) with φ ′ d = τ b − µ and φ ′ t = 0. In this equilibrium, the intermediary’s revenue is ( τ b − µ )(1 − e τa − µτb − µ ) > ( τ b − µ )( µ − τ a ) τ b − τ a .Now consider the other case G ( µ + φ d ) > G ( τ ). We construct a class of distributions G α parametrized by α ∈ [0 , α ( s ) = αG ( s ) if s ≤ τ min { αG ( τ ) e φd ( s − τ ) , G ( µ + φ d ) } if τ < s < µ + φ d G ( s ) if s ≥ µ + φ d . (16)Notice that G α is well-defined since φ d >
0, and is monotone non-decreasing and is between 0and 1. Therefore, G α is a distribution.We first show that there exists α ≤ G α and G are equal. Wedo so using a continuity argument. Let τ ( α ) be the lowest score τ ′ > τ such that G α ( τ ′ ) = G α ( µ + φ d ). Consider α = 1. We have Z τ ( α ) θ G ( s ) ds = e φd ( τ ( α ) − τ ) Z τ θ G ( s ) ds = e φd ( τ ( α ) − τ ) Z τ θ G ( s ) ds ≥ Z τ ( α ) θ G ( s ) ds, where the inequality followed from Lemma 11. As a result, since G is weakly higher than G for all scores above τ , we have R θθ G ( s ) ds ≥ R θθ G ( s ) ds . Further, Z µ + φ d θ G ( s ) ds = 0 ≤ Z µ + φ d θ G ( s ) ds. As a result, since G and G are equal above µ + φ d , we have R θθ G ( s ) ds ≤ R θθ G ( s ) ds . The areaunder G α increases continuously in α . Therefore, there exists α ≤ G α and G are equal. For the rest of the proof fix such an α .We now show that G α is a mean-preserving contraction of G . Since α ≤
1, the area under G α up to any threshold τ ′ ≤ τ is weakly less than that for G . For τ ′ ≥ τ ( α ), since the totalarea under G and G α are equal and G α is weakly higher than G above µ + φ d , we must have R τ ′ θ G α ( s ) ds ≤ R τ ′ θ G ( s ) ds . Finally, for any τ ′ such that τ ≤ τ ′ ≤ τ ( α ) we have Z τ ′ θ G α ( s ) ds = e − φd ( τ − τ ′ ) Z τ θ G α ( s ) ds ≤ e − φd ( τ − τ ′ ) Z τ θ G ( s ) ds ≤ Z τ ′ θ G ( s ) ds, where the inequality followed from Lemma 11. Thus, G α is a mean-preserving contraction of G . Test-fee structure ( G α , φ ) has weakly higher revenue than ( G, φ ). Since the two distributionsare equal above µ + φ d , the (w-P) constraint is satisfied for ( G α , φ ). Further, τ is a weak-highest44quilibrium threshold for ( G α , φ ). Since α ≤
1, the disclosure probability in ( G α , φ ) is weaklyhigher than ( G, φ ). In fact, if α <
1, the robust revenue of ( G α , φ ) is more than that of ( G, φ ).To see the first statement of the lemma, consider a score distribution G ′ that is equal to G α except that it pools the scores below τ and also pools the scores above τ . Formally, G ′ ( s ) = G α ( τ ) if s ∈ [ τ , τ ] ,G α ( s ) if s ∈ [ τ , τ ] , s ∈ [ τ , θ ] , where τ and τ are such that the areas under G ′ and G α are equal. Notice that the test-feestructure ( G ′ , φ ) is in the step-exponential-step class, with a non-degenerated exponential partbecause τ > τ . Further distribution G ′ is a mean-preserving contraction of G α and therefore G . Finally, the robust revenue of the test-fee structure ( G ′ , φ ) with weak-highest equilibriumthreshold τ is equal to that of ( G α , φ ), and therefore at least that of ( G, φ ). This establishesthe first statement of the lemma.To see the second statement, notice that if (
G, φ ) is optimal, then by Proposition 2(b), φ d >
0. Further, it must be that α = 1, as otherwise the revenue of ( G α , φ ) is strictly higher.Therefore, G = G , which means that G is exponential from τ to τ , and flat from τ to µ + φ d ,as claimed. (cid:3) Proof of Proposition 2 (a) . Consider any robustly optimal test-fee structure (
G, φ ). By Proposition 2(b), φ d >
0. By Lemma 12, the robust revenue of (
G, φ ) is at most the robust revenue of somestep-exponential-step test-fee structure ( G ′ , φ ) where G ′ is a mean-preserving contraction of G .Therefore, ( G ′ , φ ) is robustly optimal.Next we show that in fact there is a unique equilibrium under any robustly optimal step-exponential-step test-fee structure with a slightly lower disclosure fee. Let ( G, φ ) be a robustlyoptimal step-exponential-step test-fee structure: G ( s ) = g if s ∈ [ τ , τ ) ge ( s − τ ) / ( τ − τ ) if s ∈ [ τ , τ )1 if s ≥ τ , , the disclosure fee is φ d = τ − τ , and the testing fee is φ t = (1 − ge ( τ − τ ) / ( τ − τ ) )( τ − ( µ + φ d )).Notice that for any τ ∈ [ τ , τ ], E G [ s | s ≤ τ ] = τ − R τθ G ( s ) dsG ( τ ) = τ − g ( τ − τ ) e τ − τ τ − τ ge τ − τ τ − τ = τ − τ − τ = τ − φ d . τ > τ , E G [ s | s ≤ τ ] ≤ τ − φ d . For τ ∈ [ τ , τ ), E G [ s | s ≤ τ ] = τ < τ − φ d .Consider a test-fee structure ( G, φ ′ ) with φ ′ t = φ t and φ ′ d = φ d − ε for ε < τ − τ , then thereexists a unique threshold τ ′ = τ − ε satisfying (HE). Moreover, since ( G, φ ) satisfies (w-P),(
G, φ ′ ) satisfies (P) because R ¯ θµ + φ d [ s − ( µ + φ d )] dG < R ¯ θµ + φ d − ε [ s − ( µ + φ d − ε )] dG . Therefore,in the unique equilibrium under ( G, φ ′ ), the asset is tested with probability 1 and the agentdiscloses all the scores above τ . (cid:3) A.3.5 Proof of Proposition 3 on p. 22
Proof.
With binary support, the mean-preserving contraction constraints become E G [ s ] = E F [ θ ]. Therefore, the proposition follows from Lemma 9. (cid:3) A.3.6 Proof of Proposition 4 on p. 22
We first simplify the step-exponential-step distribution identified in Section 5 in the case wherethe testing fee is zero.
Lemma 13.
Consider an optimal step-exponential-step test-fee structure defined in (5) , (6) ,and (7) . If φ ∗ t = 0 , then τ = µ + φ ∗ d and G ∗ ( τ ) = 1 . Further, if the prior distribution islog-concave, then the mean-preserving constraints are slack, R τ ′ θ G ∗ ( s ) ds < R τ ′ θ F ( s ) ds , for all τ ′ from τ to θ .Proof. For the optimal testing fee to be zero, the area above G ∗ from µ + φ ∗ d to θ must be zero,and hence G ∗ ( µ + φ d ) = 1. From Lemma 12, G ∗ ( τ ) = 1.The constraint that the area under the score distribution G ∗ is θ − µ can be written as θ − µ = Z τ θ G ∗ ( s ) ds + Z τ τ ge s − τ τ − τ ds + ( θ − τ )= g ( τ − τ ) + ( τ − τ ) ge s − τ τ − τ | τ τ + ( θ − τ )= ( τ − τ ) + θ − τ . We must therefore have τ = µ + ( τ − τ ) = µ + φ ∗ d .We now show that the MPC constraints must be slack on interval [ τ , τ , θ ] since G is equal to 1 but F ( τ ′ ) is strictly less than one for all τ ′ <
1. It thus remains to show that the constraints are slack on some interval [ τ ′ , τ ]. Noticethat log-concavity implies continuity in the interior, which will be used in the later arguments.We first claim that F ( τ ) ≥ G ( τ ) = e − µ − τ τ − τ . Suppose F ( τ ) < G ( τ ), then F ( x ) < G ( x ) forany x ∈ [ τ , τ ), which implies the constraint must be slack on the interval because R xθ F ( s ) ds = R τ θ F ( s ) ds − R τ x F ( s ) ds > R τ θ G ( s ) ds − R τ x G ( s ) ds = R xθ G ( s ) ds . From the right-continuity of F , the constraint is also slack at τ . If this is the case, we can construct a new distribution46arameterized by τ ′ = τ − ε and τ ′ = τ − ε , the distribution on [ τ , θ ] doesn’t change so all theconstraints on [ τ , θ ] are still satisfied. Also the original constraints on [ θ, τ ] are slack, they arestill satisfied for small ε . By charging the same disclosure fee φ d = τ − τ , this new distributioncan induce a higher disclosure probability, which contradicts to G being optimal.If F ( τ ) = G ( τ ), the same argument goes through if the constructed distribution doesn’tviolate the mean-preserving constraints. If the constraint is slack at τ , then all the constraintsto the left are slack, so the constructed distribution is still a profitable deviation. If the con-straint binds at τ , the right derivative of F at τ must be greater than the right derivative of G . Also the left derivative of F must be greater than G from log-concavity. So the local changeof the distribution doesn’t violate any constraint either, which leads to a profitable deviation.So we must have F ( τ ) > G ( τ ), and the constraint at τ is slack at τ due to the con-tinuity of F . Moreover, from the log-concavity of F , and F ( τ ) > G ( τ ), F ( τ ) < G ( τ ), F crosses the exponential part of G from above once and only once. To see this, notice thatlog( F ) is concave and log( G ) is linear on [ τ , τ ], and a concave function can only cross a linearfunction from above once. Let x ∗ denote the intersection point, we have F ( x ) > G ( x ) for x ∈ [ τ , x ∗ ) and F ( x ) < G ( x ) for x ∈ ( x ∗ , τ ]. Notice that the constraint is slack at τ , whichmeans R τ θ G ( s ) ds ≤ R τ θ F ( s ) ds , then for any x ∈ [ x ∗ , τ ), the constraint is also slack because R xθ G ( s ) ds = R τ θ G ( s ) ds − R τ x G ( s ) ds < R τ θ GF ds − R τ x F ( s ) ds = R xθ F ( s ) ds . Similarly theconstraint is slack at τ implies the constraint is also slack at any x ∈ ( τ , x ∗ ].Therefore, all MPC constraints are slack on [ τ , θ ). (cid:3) Given Lemma 13 we now prove Proposition 4.
Proof of Proposition 4.
Consider the class of step-exponential-step distributions defined in (5)where τ = θ . That is, G ( s ) = g if s = τ ge ( s − τ ) / ( τ − τ ) if s ∈ [ τ , τ ]1 if s = θ. A degree of freedom is removed given the constraint that the area under G is θ − µ . Noticethat the distribution with zero testing fee identified in Lemma 13 is a special case where τ = µ + ( τ − τ ). Consider such a distribution. We show that reducing τ by a small amountincreases the revenue, and does not violate any of the constraints.The intermediary’s revenue from charging disclosure fee φ d = τ − τ and the highest possibletesting fee φ t = R θµ + φ d ( s − µ − c ) dG ( s ) is φ t + φ d (1 − G ( τ )) = Z θµ + x − x [ s − µ − ( τ − τ )] dG ( s ) + ( τ − τ )(1 − G ( τ ))47 θ − µ − ( τ − τ ) − Z θµ + τ − τ G ( s ) ds + τ − τ − Z τ θ G ( s ) ds = Z µ + τ − τ τ G ( s ) ds = ge τ − τ τ − τ (2( τ − τ ) + µ − τ ) − ( τ − τ ) g. Consider the change in revenue from decreasing τ , evaluated at τ = µ + ( τ − τ ) and g = e − µ − τ τ − τ , ddτ ( φ t + φ d (1 − G ( τ ))) =( τ − τ ) ddτ (cid:18) τ − τ τ − τ (cid:19) + 2 − dτ dτ − g = (cid:18) dτ dτ − (cid:19) − τ − τ τ − τ + 2 − dτ dτ − g =1 − g − τ − τ τ − τ =1 + τ − µτ − τ − e τ − µτ − τ < . The inequality follows from e x > x for x >
0. Thus decreasing τ strictly increasesrevenue. (cid:3) A.4 Proofs for Section 6
A.4.1 Proof of Proposition 6 on p. 23
Proof.
Consider an evidence-test-fee structure, denoted by fees ( φ t , φ d ), an unbiased test T :Θ → ∆ S , and an evidence structure M : S ⇒ M such that for each s , M ( s ) is a Borel space.A strategy profile ( σ, p ) consists of the agent’s strategy σ = ( σ T , σ D ), where σ T ∈ [0 ,
1] and σ D maps s ∈ S to ∆( M ( s ) ∪ { N } ), and the market price p : M → [ θ, ¯ θ ]. Let ( σ, p ) be anadversarial equilibrium. First we consider the case in which the agent has the asset tested withprobability 1, that is σ T = 1.Consider the disclosure stage. Let G ( σ,p ) be the induced distribution of prices, i.e., G ( σ,p ) ( x ) = P r [ p ( σ D ( s )) ≤ x ] for any x , taking into account both the randomization over the score andthe agent’s strategy. Let τ ≡ p ( N ) + φ d . We show that the following holds, mirroring ourcharacterization of highest equilibria (HE): τ − φ d ≤ E G ( σ,p ) [ x | x ≤ τ ] , (17) τ ′ − φ d > E G ( σ,p ) [ x | x ≤ τ ′ ] , ∀ τ ′ > τ. (18)48et us argue why (17) holds. Since τ − φ d = p ( N ), it suffices to show that p ( N ) is weakly lessthan E G ( σ,p ) [ x | x ≤ τ ]. Observe that p ( σ D ( s )) is at least p ( N ) with probability 1: if p ( σ D ( s ))were strictly less than p ( N ), the agent could profitably deviate to sending message N andobtain a strictly higher price. But then this implies that p ( N ) ≤ E [ p ( σ D ( s )) | p ( σ D ( s )) ≤ τ ] = E G ( σ,p ) [ x | x ≤ τ ].To see that (18) holds, suppose for contradiction that τ ′′ − φ d ≤ E G ( σ,p ) [ p | p ≤ τ ′′ ] for some τ ′′ > τ . By Lemma 5 there exists τ ′ > τ such that τ ′ − φ d = E G ( σ,p ) [ p | p ≤ τ ′ ]. Consider thestrategy profile ( σ ′ , p ′ ) defined as follows. The agent’s strategy σ ′ is the same as σ except thatthe agent conceals a score s if p ( σ ( s )) ≤ τ ′ . For m = N , p ′ ( m ) = p ( m ), and for m = N , theprice is p ′ ( m ) = E G ( σ,p ) [ x | x ≤ τ ′ ] ≥ E G ( σ,p ) [ x | x ≤ τ ] ≥ p ( m ). Notice that since any message m that is disclosed in ( σ ′ , p ′ ) is also disclosed in ( σ, p ), the prices p ′ are defined on path via theBayes rule.To see that ( σ ′ , p ′ ) is an equilibrium, consider any score s such that p ( σ ( s )) > τ withpositive probability. Since s can send a message m = N such that p ( m ) > τ , it optimallyrandomizes over messages other than N that lead to the same (and maximum) price, which,abusing notation, we denote p ( σ ( s )). Since p ( m ) = p ′ ( m ) for all m = N , σ ( s ) is optimal amongall strategies that send N with probability 0 given prices p ′ . Therefore, for such a score, itis optimal to follow σ ( s ) if p ′ ( σ ( s )) > τ ′ = p ′ ( N ) + φ d , and to conceal if p ′ ( σ ( s )) ≤ τ ′ , asprescribed by σ ′ . Now consider a score s such that p ( σ ( s )) ≤ τ with probability 1. For sucha score, it is optimal given prices p to conceal, i.e., for any message m = N that s can send, p ( m ) − φ d ≤ p ( N ). Since p ′ ( m ) = p ( m ) and p ′ ( N ) ≥ p ( N ), it is also optimal to conceal givenprices p ′ , as prescribed by σ ′ .Now consider the testing stage. If φ t ≥ Z θµ + φ d [ x − ( µ + φ d )] dG ( σ,p ) . (19)then there exists an equilibrium in the evidence-test-fee structure where the agent has the assettested with probability 0. The argument parallels that of Lemma 1. Consider a strategy profile( σ ′ , p ′ ) such that that σ ′ T = 0, off-path the agent follows σ ( s ) if p ′ ( σ ( s )) > µ + φ d ( p ′ ( σ ( s ))is well-defined as argued above) and otherwise conceals, and the prices are p ′ ( N ) = µ , and p ( m ) = p ′ ( m ) for all m = N . Since the set of disclosed messages in ( σ ′ , p ′ ) is a subset of that in( σ, p ), an argument similar to above shows that the agent’s disclosure strategy is sequentiallyrational. Also, by deviating to take the test, the agent receives an expected payoff lower than µ , Z µ + φ d θ µdG ( σ,p ) + Z ¯ θµ + φ d [ x − φ d ] dG ( σ,p ) − φ t = µ + Z ¯ θµ + φ d [ x − ( µ + φ d )] dG ( σ,p ) − φ t ≤ µ G ( σ,p ) , φ ). By Lemma 3, τ is a weak-highest equilibriumthreshold of the test-fee structure. Also since (19) is violated, by Lemma 1, the test is takenwith probability 1 in all equilibria. Therefore, adversarial revenue of the test-fee structure isequal to the adversarial revenue of the evidence-test-fee environment.Now we consider the case σ T ∈ [0 , φ t ≥ Z θp ( N )+ φ d [ x − ( p ( N ) + φ d )] dG ( σ,p ) ≥ Z θµ + φ d [ x − ( µ + φ d )] dG ( σ,p ) so (19) holds. Form the same argument above we know that the adversarial revenue is at mostzero, which can be obtained in a test-fee structure with any test and zero fees. (cid:3)(cid:3)