Cowles Foundation for Research in Economics at Yale University

Transcription

1 Cowles Foundation for Research in Economics at Yale University Cowles Foundation Discussion Paer No. 1743RR SELLING INFORMATION Johannes Hörner and Andrzej Skrzyacz December 2009 Revised November 2012 An author index to the working aers in the Cowles Foundation Discussion Paer Series is located at: htt://cowles.econ.yale.edu/p/au/index.htm This aer can be downloaded without charge from the Social Science Research Network Electronic Paer Collection: htt://ssrn.com/abstract= Electronic coy available at: htt://ssrn.com/abstract=

2 Selling Information Johannes Hörner and Andrzej Skrzyacz November 11, 2012 Abstract An Agent who owns information that is otentially valuable to a Firm bargains for its sale, without commitment and certification ossibilities, short of disclosing it. We roose a model of gradual ersuasion and show how gradualism hels mitigate the hold-u roblem (that the Firm would not ay once it learns the information). An examle illustrates how it is otimal to give away art of the information at the beginning of the bargaining, and sell the remainder in dribs and drabs. The Agent can only aroriate art of the value of information. Introducing a third-arty allows her to extract the maximum surlus. Keywords: value of information, dynamic game. JEL codes: C72, D82, D83 We thank Daron Acemoglu, Robert Aumann, Hector Chade, Alex Frankel, Drew Fudenberg, David Kres, Max Kwiek, R.Vijay Krishna, Sofia Moroni, Romans Pancs, Arthur Robson, Rann Smorodinsky and seminar articiants at the Barcelona JOCS, the Collegio Carlo Alberto, Turin, Essex University, the Euroean University Institute, Harvard-MIT, University of Illinois at Urbana-Chamaign, Northwestern University, Oxford University, Simon Fraser University, Stanford University, University of British Columbia, University of Western Ontario, UC San Diego, Yale, X-HEC Paris, Paris Sciences Economiques, Toulouse School of Economics, the Euroean Summer Symosium in Economic Theory at Gerzensee 2012, the SED 2009 meeting, the Stony Brook 2010 game theory conference, and the 2011 Asian meeting of the Econometric Society for useful comments and suggestions. Yale University, 30 Hillhouse Ave., New Haven, CT 06520, USA. johannes.horner@yale.edu. Stanford University, Graduate School of Business, 518 Memorial Way, Stanford, CA 94305, USA. andy@gsb.stanford.edu. 1 Electronic coy available at: htt://ssrn.com/abstract=

3 1 Introduction In the late 1960s, Italian inventor Aldo Bonassoli and Count Alain de Villegas contacted Elf, the French ublic oil comany, to sell their discovery: a device that could detect oil fields from the air. Early tests were sectacularly successful. Contracts worth 200m Swiss Francs (in 1976), and 250m Swiss Francs (in 1978) were signed as tests roceeded, with the agreement of both French resident Valery Giscard d Estaing and Prime Minister Raymond Barre. Unfortunately, the device was a hoax, exosed in The story of the Great Oil Sniffer Hoax was made ublic in December Elf never comleted aying for the final contract, but nevertheless had sent over $150m. But one cannot blame the inventors to be careful. Who has not heard of Robert Kearns, the inventor of the intermittent windshield wier systems, who was a little too forthcoming with information about his invention with engineers of Ford? This dilemma, known as the Arrow information aradox (1959) is the subject of this aer. The otential buyer of information needs to know its value before urchasing it since otherwise she may end u aying for a hoax. But often the only way to verify the information is to transmit it fully and once the seller has this detailed knowledge, she has no incentives to ay for it. This roblem has obvious economic imlications, as rewarding innovative activity is key in encouraging it. It is a roblem not just for inventors, but owners of information in general: hedge funds claim to have secial investment techniques which, of course, they cannot disclose; similarly, exerts have confidential sources of information. Scientists and engineers claim to have suerior and valuable knowledge again, which cannot be disclosed. We rovide a game-theoretic analysis of the interaction between a buyer (Firm) and a seller (Agent) and examine when and how information should be transmitted, and ayments made. In doing so, we determine how much of the information value can be aroriated by the seller, and how this roblem is mitigated if sufficiently elaborate ways of transmitting information can 1 The early tests were successful because the inventors had an inside contact at Elf, who gave them mas of the oil fields that were being surveyed. The alleged ma dislayed by the device s screen was simly a hotograh. This was also how the hoax was uncovered: the two inventors routinely hotocoied objects that were then laced behind a wall. Magically, the image of the object behind the wall aeared on the screen. Until an engineer bent the object, a ruler, right before it was laced behind the wall. 2 Electronic coy available at: htt://ssrn.com/abstract=

4 be alied, or a third arty, such as a trusted intermediary, can be involved. The game of information sale that we consider is dynamic. Within a round, the layers make voluntary ayments and then the Agent can disclose some information to the Firm. We assume that information is verifiable and divisible. In articular, in our model the Agent has one of two tyes (i.e., she is knowledgeable or not) and the information transmission is modeled as tests to verify the Agent s information. Verifiability of information means that each test has a known difficulty: the cometent Agent can always ass it (in the baseline model), but the incometent Agent asses it with a robability commensurate to its difficulty. Easier tests have a higher robability of being assed by an incometent Agent. Divisibility of information means that there is a rich set of tests with varying difficulties. There is no commitment on either side. Weconstruct tightboundsonthelimitsofthecometent Agent sayoffasthenumber ofossible communication rounds grows to infinity (they also establish bound on the difference of the cometent and incometent ayoffs, which might be the relevant measure for those alications in which incentives to acquire information are exlicitly taken into account). We characterize three such bounds: when we consider only ure-strategy equilibria (in which a cometent Agent always asses the test), when we allow for mixed-strategy equilibria, and finally when we allow for tests so hard that even a cometent Agent may occasionally fail. The latter case is equivalent to allowing for a trusted intermediary. Lack of commitment creates a hold-u roblem: since the Agent is selling information, once the Firm learns it, it has no reason to ay for it. Therefore, it seems at first difficult to make the Firm ay different amounts to different tyes, since such screening would inform the Firm about the Agent s tye and lead it to renege on ayments. Although we can make the Firm ay for a iece of information, it is necessary that it ays before it learns it. That leads to our first main result that slitting information generally increases the cometent Agent s ayoff. That is, the cometent Agent s ayoff is higher in equilibria in which she takes two tests in a sequence (and is aid for each searately) than if she takes both of them at once (which is equivalent to taking one harder test). That intuition underlies the structure of the best equilibrium in ure strategies in our leading examle: first, an initial chunk of information is given away for free that makes the Firm very uncertain about the Agent s cometence. Then 3

5 the Agent sells information in dribs and drabs and gets aid a little for each bit. Although the exression for the limit ayoff deends on the assumtion that there is a rich set of tests and arbitrarily many rounds of communication, the benefit of slitting does not deend on either assumtion. Second, we show how randomization can hel imrove the erformance of the contract. In the best ure-strategy equilibrium the incometent Agent collects on average a non-trivial amount of ayments, which leaves room for imrovement. We first show that using (non-observable) mixed strategies can hel by taking advantage of the fact that the cometent Agent and the Firm may have different (endogenous) risk attitudes (more recisely, that the sum of their continuation ayoffs need not be concave). An imortant ractical imlication is that the cometent Agent can gain from the ossibility that the Firm s belief that she is cometent might go down during the bargaining rocess. Performance can be further imroved if the layers have access to tests that both Agents tyes can fail with ositive robability, or alternatively, if arties have access to a trusted intermediary that can noise u the test s outcome. In fact, we rove that, with the hel of such a third-arty, the Agent can aroriate the maximum surlus (Theorem 3). Our finding that selling information gradually is beneficial to the seller should (in terms of roviding the highest incentives to acquire information) come as no surrise to anyone who was ever involved in consulting. The free first consultation is also reminiscent of standard business ractice. The further benefits of intermediation might be more surrising. Yet it is indeed common ractice to hire third arties to evaluate the value of information. This third-arty structure is used as a buffer to ensure that the buyer does not have access to any unnecessary confidential information about the seller at any oint during the sales rocess. 2 Most of the aer analyzes the information sales roblem for the secific ayoff structure that arises in a simle examle in which the Agent s information is decisive for a Firm s otimal decision that is exlicitly modeled. However, there is nothing articular about this examle. We generalize our results to arbitrary secifications of how the Firm s ayoff varies with its 2 We thank Rann Smorodinsky for sharing his exerience in this resect. As a seller of software, the sale involved no less than three third arties secialized in this kind of intermediation Johnson-Laird, Inc., Construx Software and NextGeneration Software Ltd. 4

6 belief about the Agent s cometence. This secification could arise from decision roblems that are more comlicated than the binary one considered in the examle. We rove that selling information in small bits is rofitable as long as this ayoff function is star-shaed, that is, as long as its average is increasing in the belief. Moreover, we show that, with rich enough tests, the tye-1 Agent can extract the entire exected value quite generally. The aer is related to the literature on hold-u, for examle Gul (2001) and Che and Sákovics (2004). One difference is that in our game what is being sold is information and hence the value of ast ieces sold deends on the realization of value of additional ieces. Moreover, we assume that there is no hysical cost of selling a iece of information and hence the Agent does not care er se about how much information the Firm gets or what action it takes. In contrast, in Che and Sákovics (2004) each iece of the roject is costly to the Agent and the roblem is how to rovide incentives for this observable effort rather than unobservable effort in our model. Finally, our focus is on the different ways of information transmission, which is not resent in any of these aers. The formal maximization roblem, and in articular the structural constraints on information revelation, are reminiscent of the literature on long chea talk. See, in articular, Forges (1990) and Aumann and Hart (2003), and, more generally, Aumann and Maschler (1995). As is the case here, the roblem is how to slit a martingale otimally over time. That is, the Firm s belief is a martingale, and the otimal strategy secifies its distribution over time. There are imortant differences between our aer and the motivation of these aers, however. In articular, unlike in that literature, ayoff-relevant actions are taken before information disclosure is over, since the Firm ays the Agent as information gets revealed over time. In fact, with a mediator, the Agent also makes ayments to the Firm during the communication hase. As in Forges and Koessler (2008), messages are tye-deendent, as the Agent is constrained in the messages she can send by the information she actually owns. Chea-talk (i.e. the ossibility to send messages from sets that are tye-indeendent) is of no hel in our model. Rosenberg, Solan and Vieille (2009) consider the roblem of information exchange between two informed arties in a reeated game without transfers, and establish a folk theorem. In all these aers, the focus is on identifying the best equilibrium from the Agent s ersective in the ex ante sense, before her tye is known. 5

7 In our case, this is trivial and does not deliver differential ayoffs to the Agent s tyes (i.e., a higher ayoff to the cometent tye). The martingale roerty is distinctive of information, and this is a key difference between our set-u and other models in which gradualism aears. In articular, the benefits of gradualism are well known in games of ublic goods rovision (see Admati and Perry, 1991, Comte and Jehiel, 2004 and Marx and Matthews, 2000). Contributions are costly in these games, whereas information disclosure is not costly er se. In fact, costlessness is a second hallmark of information disclosure that lays an imortant role in the analysis. (On the other hand, the secific order of moves is irrelevant for the results, unlike in contribution games.) The oortunity cost of giving information away is a function of the equilibrium to be layed. So, unlike in ublic goods game, the marginal (oortunity) cost of information is endogenous. Relative to sales of rivate goods, the marginal value of information cannot be ascertained without considering the information as a whole, very much as for ublic goods. Our focus (roving one s knowledge) and instrument (tests that imerfectly discriminate for it) are reminiscent of the literature on zero-knowledge roofs, which also stresses the benefits of reeating such tests. This literature that starts with the aer of Goldwasser, Micali and Rackoff (1985) is too large to survey here. A key difference is that, in that literature, assing a test conveys information about the tye without revealing anything valuable (factoring large numbers into rimes does not hel the tester factoring numbers himself). In many economic alications, however, it is hard to convince the buyer that the seller has information without giving away some of it, which is costly as it is in our model. Indeed, unlike in ublic goods games, or zero-knowledge roofs, slitting information is not always otimal. As mentioned, this hinges on a (commonly satisfied) roerty of the Firm s ayoff, as a function of its belief about the Agent s tye. Less related are some aers in industrial organization. Our aer is comlementary to Anton and Yao (1994 and 2002) in which an inventor tries to obtain a return to his information in the absence of roerty rights. In Anton and Yao (1994) the inventor has the threat of revealing information to cometitors of the Firm and it allows him to receive ayments even after she gives the Firm all information. In Anton and Yao (2002) some contingent ayments are allowed 6

8 and the inventor can use them together with cometition among firms to obtain ositive return to her information. In contrast, in our model, there are no contingent ayments and we assume that only one Firm can use the information. Finally, there is a vast literature directly related to the value of information. See, among others, Admati and Pfleiderer (1988 and 1990). Eső and Szentes (2007) take a mechanism design aroach to this roblem, while Gentzkow and Kamenica (2011) aly ideas similar to Aumann and Maschler (1995) to study otimal information disclosure olicy when the Agent does not have rivate information about the state of the world, but cares about the Firm s action. 2 The Main Examle We start with a simle examle in which we exlicitly model how the Firm s value for the Agent s information arises from a decision roblem. Later sections take this value as exogenous data. 2.1 Set-U There is one Agent (she) and a Firm (it). The Agent is of one of two ossible tyes: ω Ω := {0,1}, she is either cometent (1) or not (0). We also refer to these as tye-1 and tye-0. The Agent s tye is rivate information. The Firm s rior belief that ω = 1 is 0 (0,1). The game is divided into K rounds of communication (we focus on the limit as K grows large), followed by an action stage. In the action stage the Firm must choose either action I ( Invest ) or N ( Not Invest ). Not Investing yields a ayoff of 0 indeendently of the Agent s tye. Investing yields a ayoff of 1 if ω = 1 and γ < 0 if ω = 0. That is, investing is otimal if the Agent is cometent, as such an Agent has the know-how (or information) to make the investment thrive; however, if the Agent is incometent, it is safer to abstain from investing. In each of the K rounds of communication timing is as follows. First, the Firm and the Agent choose a monetary transfer to the other layer, t A k and tf k, resectively.3 Second, after these 3 The Reader might wonder why we allow the Agent to ay the Firm. After all, it is the Agent who owns the unique valuable good, information. Such ayments will turn out to be irrelevant when only ure strategies are considered, but will lay a role in the second art of the aer with more comlex communication. 7

9 simultaneous transfers, the Agent chooses whether to reveal some information by undergoing a test. 4 We roose the following concrete model of gradual ersuasion/communication using tests. We assume that for every m [0,1], there exists a test that the cometent Agent asses for sure, but that the incometent Agent asses with robability m. The level of difficulty, m, is chosen by the Agent and observed by the Firm. 5 If the Firm s rior belief about the Agent being cometent is and a test of difficulty m is chosen and assed, the osterior belief is = +(1 )m. Thus, the range of ossible osterior beliefs as m varies is [,1] (if the test is assed). An uninformative test corresonds to the case m = 1. If the Agent fails the test, then the Firm correctly udates its belief to zero. To allow for rich communication, tests of any desired recision m are available at each of the K rounds, and their outcomes are conditionally indeendent. In words, by disclosing information, the Agent affects the Firm s belief that she is cometent. Persuasion can be a gradual rocess: after the Agent discloses some information, the Firm s osteriorbelief canbearbitrary, rovidedtheriorbeliefisnotdegenerate. ButtheFirmuses Bayesian udating. Viewed as a stochastic rocess whose realization deends on the disclosed information, the sequence of osterior beliefs is a martingale from the Firm s oint of view. 6 Fornow, wedonotallowthecometent Agenttoflunkthetestonurose, nordoweconsider tests so difficult that even the cometent Agent might fail them. Describing strategies in terms of martingales (see footnote 6), this means that we restrict attention to rocesses whose samle aths are either non-decreasing, or absorbed at zero. We discuss these richer communication 4 Nothing hinges on this timing. Payments could be made sequentially rather than simultaneously, and occur after rather than before the test is taken. 5 Because the level of difficulty is determined in equilibrium, it does not matter that the Agent chooses it rather than the Firm. 6 More abstractly, as in the literature on reeated games with incomlete information, we can think of an Agent s strategy as a choice of a martingale the Firm s beliefs given the consistency requirements imosed by Bayes rule. For concreteness, we model this as the outcome of a series of tests whose difficulty can be varied. Alternatively, we may think of information as being divisible, and the Agent choosing how much information to disclose at each round; the incometent Agent might be a charlatan who might be lucky or skilled enough to roduce some ersuasive evidence, as in the Great Oil Sniffer Hoax. While the articular imlementation of information transmission is irrelevant for the urose of equilibrium analysis, it is somewhat simler to describe the game using tests, as it does not require a continuum of tyes charlatans of varying skill levels. 8

10 ossibilities in the second art of the aer. 2.2 Histories, Strategies and Payoffs More formally, a (ublic) history of length k is a sequence h k = {(t A k,tf k,m k,r k )}k 1 k =0, where (t A k,tf k,m k,r k ) R2 + [0,1] {0,1}. Here, m k is the difficulty of the test chosen by the Agent in stage k and r k is the outcome of that test (which is either ositive, 1, or negative, 0). The set of all such histories is denoted H k (set H 0 := ). A (behavior) strategy σ F for the Firm is a collection ({τ F k }K 1 k=0,αf ), where (i) τ F k is a robability transition τ F k : H k R +, secifying a transfer t F k := τf (h k ) as a function of the (ublic) history so far, as well as (ii) an action (a robability transition as well), α F : H K {I,N} after the K-th round. A (behavior) strategy σ A for the Agent is a collection {τ A k,µa k }K 1 k=0, where (i) τ A k : Ω H k R + is a robability transition secifying the transfer t A k := τa (h k ) in round k given the history so far and given the information she has, (ii) µ A k : Ω H k R 2 + [0,1] is a robability transition secifying the difficulty of the test (i.e., the value of m), as a function of the Agent s tye, the history u to the current round, and the transfers that were made in the round. All choices are ossibly randomized. These definitions imly that there is no commitment on either side: the Firm(and the Agent) can sto making ayments at any time, and nothing comels the Agent to disclose information if she refers not to. In terms of ayoffs, we assume there is neither discounting nor any other tye of frictions during the K rounds (for examle, taking the tests is free). Absent any additional information revelation, the Firm s otimal action is to invest if and only if its belief that her tye is 1 satisfies := γ 1+γ. 9

11 Hence, its ayoff from the otimal action is given by w() := ( (1 )γ) +, where x + := max{0,x}. While our analysis covers both the case in which the rior belief 0 is below or above, we have in mind the more interesting case in which 0 is smaller than. The ayoff w() is the Firm s outside otion. Here, given the motivating investment decision, the secific outside otion reduces to a call otion. This is the distinguishing feature of our main examle. In our general results we cover a richer class of outside otion secifications. The Agent does not value the knowledge that she otentially holds, nor does she care about the Firm s investment decision. All she cares about is getting aid. The Firm cares about the ayoff from the investment decision, net of any ayments to the Agent. Given some final history h K (which does not include the Firm s final action to invest or not), tye-ω Agent s realized ayoff is the sum of all net transfers over all rounds, indeendently of her tye: K 1 V ω (h K ) = (τ F k τ A k). k=0 Given tye ω, the Firm s overall ayoff results from its action, as well as from the sum of net transfers. If the Firm chooses the safe action, it gets K 1 W(ω,h K,N) = (τ A k τ F k). If instead the Firm decides to invest, it receives K 1 W(ω,h K,I) = (τ A k τf k )+1 1 ω=1 γ 1 ω=0, k=0 where 1 A denotes the indicator function of the event A. Ariorbelief 0 andastrategyrofileσ := (σ F,σ A )defineadistributionoverω H K {I,N}, and we let V(σ),W(σ), or simly V,W, denote the exected ayoffs of the Agent and the Firm, k=0 10

12 resectively, with resect to this distribution. When the strategy rofile is understood, we also write V(h k ),W(h k ) for the layers continuation ayoffs, given history h k. We further write V 0,V 1, for the ayoff to the Agent, when we condition on the tye ω = 0,1. The solution concet is erfect Bayesian equilibrium, as defined in Fudenberg and Tirole (1991, Definition 8.2). 7 We assume that layers have access to a ublic randomization device at the beginning of each round (a draw from a uniform distribution), as this facilitates an argument in a roof. The best equilibrium that we identify (whether in ure or mixed strategies, or with a mediator) turns out not to take advantage of this device, so that results do not deend on it. A central assumtion of our model is that information revealed by successful tests is valuable: if the Firm decides to sto making ayments once its belief reaches some level, its exected ayoff is given by w() and that is increasing (at least over some range) in. There are cases in which tests can be conducted whose outcome reveals nothing valuable. (As is the case with zero-knowledge roofs which could be catured in our model by taking w() equal to 0 for all < 1 and w(1) = 1.) In ractice, however, it is difficult to think of demonstrations (bluerints, rototyes, etc.) that do not involve some valuable information leakage. A cometent Agent might still be useful to the Firm after she roduces sufficient evidence to convince the Firm that she is cometent. There is no roblem for her in getting comensated for the value that she might retain in this way; our interest is in the value that she might give away in the rocess of convincing the Firm. For examle, while roving that the Agent knows how to solve a articular roblem may not give the Firm the full solution, it would give the Firm confidence that the roblem is solvable and even some hints about the direction it would need to follow to find the solution on its own. 2.3 Preliminary Remarks This game admits a lethora of equilibria, but our focus is on identifying the best equilibrium for the cometent Agent. It is not difficult to motivate our interest in this equilibrium. After all, rewarding agents for their exertise is socially desirable if acquiring this knowledge is costly. 7 Fudenberg and Tiroledefine erfectbayesianequilibria forfinite multistage gameswith observedactionsonly. Here instead, both the tye sace and the action sets are infinite. The natural generalization of their definition is straightforward and omitted. 11

13 Clearly, there are many ways to model this information acquisition stage rior to the game. Our working aer develos a articular examle. As such, the equilibria we construct can be interreted as the best relational contracts for the cometent Agent subject to different restrictions on the communication technology and self-enforcing constraints. As usual, how good ayoffs can be sustained on the equilibrium ath deends on the worst unishment ayoffs that are consistent with a continuation equilibrium. In our game, after every history there is a babbling equilibrium in which the Agent never undergoes a test (i.e., chooses m = 1 in each eriod), and neither the Agent nor the Firm make ayments. This gives the Agent a ayoff of 0, and the Firm a ayoff of w(), its outside otion. This equilibrium achieves the lower bound on the ayoffs of all the articiants simultaneously, so it is the most otent unishment available. This imlies that it is without loss of generality that we can restrict attention to equilibria in which any observable deviation triggers reversion to this equilibrium (the Firm getting then its outside otion w() given its belief once the deviation occurs). To induce comliance, it suffices to make sure that all layers receive at least their minmax ayoff (0 and w()) at any time. IftheFirmassignsrobabilitytothetye-1Agent, then, fromitsointofview, theexected total surlus is at most 1 + (1 ) 0 = (this is in the best ossible scenario in which it eventually takes the right investment decision). Hence, given some equilibrium, any history h k and resulting belief, continuation ayoffs must satisfy V 1 (h k )+(1 )V 0 (h k )+W(h k ). (1) With only one round of communication, K = 1, both tyes of the Agent have to receive the same ayoff in any equilibrium so V 1 w() in this case. By (1), w() is also the uer bound on the average, or ex ante ayoff of the Agent. How much can gradual communication imrove V 1? By (1), given that W(h k ) w() and V 0 (h k ) 0, the tye-1 Agent cannot receive more than 1 w()/. Clearly w() < 1 w()/ whenever w() <, so the uer bound is strictly larger than the maximum ex ante ayoff. Can we imrove on the latter? Before we resent the analysis that answers this question, we make two observations. 12

14 First, it is worth ointing out that, in some cases, maximizing the incentives to acquire information is not about maximizing the tye-1 Agent s ayoff V 1, but the difference V 1 V 0. But the two objectives coincide. This can be seen in three stes: first, in terms of the Agent s equilibrium ayoffs (V 0,V 1 ), there is no loss of generality in assuming that the equilibrium achieving this ayoff is efficient, i.e., that it satisfies (1) with equality: disclosing the tye in the last eriod on the equilibrium ath does not affect the Agent s ayoff and only makes comliance with the equilibrium strategy more attractive to the Firm. Second, if (1) holds as an equality, then V 1 V 0 = V 1 +W Hence, maximizing the difference in the tyes ayoffs amounts to maximizing the sum V 1 +W. Third, maximizing V 1 is equivalent to maximizing V 1 +W. This is because ayoffs between the rincial and the Agent can be transferred one-to-one via the first ayment that the Firm makes: if W > w( 0 ), we can decrease W and increase V 1 by the same amount by requiring the Firm to make a larger ayment ufront. Hence, in maximizing V 1 +W over all equilibria, there is no loss in assuming that W = w( 0 ), a fixed quantity, and so in maximizing V 1 only. Second, while we characterize the equilibrium maximizing V 1 (or V 1 V 0 ), one may be interested in other equilibrium ayoffs. A artial characterization is as follows. The Firm cannot hoe for more than, the entire surlus, and there is a trivial equilibrium that achieves this uer bound: on ath, no ayment is ever made, and the Agent reveals her tye in the last eriod (m = 0). There is an equally simle equilibrium that achieves the maximum ex ante ayoff of the Agent, V 1 + (1 )V 0 : the Firm is exected to ay the difference 0 w( 0 ) in the initial eriod, and the agent reveals her tye (m = 0) if and only if the Firm makes this ayment. In this game, the strategy that maximizes this ex ante ayoff of the informed layer is trivial, unlike in standard games with incomlete information (see Aumann and Maschler, 1995). Because the tye-1 Agent can always mimic the tye-0 Agent in the choice of m (and once a test is failed Agent s ayoff is 0), the cometent tye ayoff must be at least as large as the incometent s. This imlies that the best equilibrium for the tye-0 Agent maximizes the Agent s ex ante ayoff. 13

15 Pure Strategies Figure 1: A feasible action We now turn to the focusof the analysis: what equilibrium maximizes theayoff of the tye-1 Agent, and how much of the surlus can she aroriate? Note that this maximum ayoff is non-decreasing in K, the number of rounds: layers can always choose not to make transfers or disclose any information in the first round. Hence, for any 0, the highest equilibrium ayoff for the tye-1 Agent has a well-defined limit as K that we seek to identify. In this section we assume that the tye-1 Agent always asses any test she takes. She is not allowed to flunk the test on urose, a ossibility that we will allow in Section 4: there we enrich the descrition of the game to allow the agent to choose whether to ass the test after she chooses the difficulty m. In that richer game the analysis in this section is equivalent to restricting attention to ure strategies. From the Firm s oint of view, its osterior will take one of two values: either it jums from 0 u to some, if the test is successful. Or it jums down to zero. This is illustrated in Figure 1. The two arrows indicate the two ossible osterior beliefs. As mentioned, viewed from the Firm s ersective, this belief must follow a martingale: the Firm s exectation of its osterior belief must be equal to its rior belief. This is not the case from the Agent s oint of view. Given her knowledge of the tye, she assigns different robabilities to these osterior beliefs than the Firm. If she is cometent, she knows for sure that the belief will not decrease over time. If she is incometent, the exectation of the osterior belief is below 0, as she does not know whether she will be lucky in taking the test (the rocess is then a suermartingale). Instead of describing the information art of an equilibrium outcome by the tests taken so far 14

16 {m k } and their results, we can equivalently describe it by martingale slitting, i.e. the sequence of Firm s beliefs that the tye is 1, conditional on all tests so far. As long as the Agent asses the tests, the Firm s equilibrium beliefs follow a non-decreasing sequence { 0,..., K+1 } starting at the Firm s rior belief, 0, and ending at K+1 = 1 (assuming, without loss, that the equilibrium is efficient). If the Agent fails a test, the osterior dros to zero. An equilibrium must also secify ayments. It turns out that the tye-1 Agent s ayoff decreases if the Firm is ever granted any ayoff in excess of its outside otion. On the one hand, the Agent could demand more in earlier rounds by romising to leave some surlus to the Firm in later rounds. On the other hand, the willingness-to-ay of the Firm for this future surlus is lower than the cost of such a romise to the tye-1 Agent. The reason is that the Firm assigns a lower robability than the tye-1 Agent to the osterior increasing (and ayments once the osterior dros to zero are not individually rational). Finally, it is not hard to see that there is no oint here in having the Agent make any ayments. To sum u, if the Firm s belief in the next round is either k+1 or 0, given the current belief k, then the equilibrium secifies that the Firm ays her willingness-to-ay E F [w( )] w( k ), where is the (random) belief in the next round, with ossible values 0 and k+1, and E F [ ] is the exectation oerator for the Firm. This leaves us with the determination of the sequence of osterior beliefs The Main Examle: A Geometric Analysis We already know that it is ossible for the Agent to aroriate some of the value of her information, but the question is whether she can get more than 0 w( 0 ), which is just as much as the tye-0 Agent gets in the equilibrium we constructed so far. Consider first the case K = 1. In this case, the highest equilibrium ayoff to the tye-1 Agent is indeed equal to 0 w( 0 ). Suose that a successful test takes the osterior to 1 0. Using 8 In addition, an equilibrium must also secify how layers behave off the equilibrium ath. As discussed, the most effective unishment for deviations is reversion to the babbling equilibrium, and this is assumed throughout. 15

17 the martingale roerty, it must be that the robability that the osterior is 1 is 0 / 1, because hence, the Firm is willing to ay 0 = E F [w( )] w( 0 ) = 0 1 w( 1 ) w( 0 ) 0 w( 0 ), where the inequality follows from w( 1 ) 1. Setting 1 to 1 is best, as it makes the inequality tight: with one round, revealing all information is otimal. Note that, when 0, w( 0 ) = 0, and the highest ayoff in one round that the tye-1 Agent can achieve is the rior 0, which is increasing in 0. This suggests one way to imrove on the ayoff with two rounds. In the first round, the Agent takes a test whose success leads to a osterior of for free. Indeed, the Firm is not willing to ay for a test that does not affect its outside otion. In the second round, the equilibrium of the one-round game is layed, given the belief. This second and only ayment yields w( ) = > 0. This is illustrated in the right anel of Figure 2. The lower kinked line is the outside otion w, the uer straight line is total surlus,. Hence, the ayment in the second round is given by the length of the vertical segment at in the right anel, which is larger than the ayment with only one round, given by the length of the vertical segment at 0. To sum u: the Agent gives away a chunk of information for free, making the Firm really unsure whether investing is a good idea. Then she charges as much as she can for the disclosure of all her information. Is the slitting that we described otimal with two eriods to go? As it turns out, it is so if and only if 0 < ( ) 2. But there are many other ways of slitting information with two eriods to go that imrove uon the one-round equilibrium, and among them, slits that also imrove over the one-eriod equilibrium when 0 >. The otimal strategy is given at the end of this 16

18 $ Figure 2: Revealing information in two stes subsection. Can we do better with more rounds? Consider Figure 3. As shown on the left anel, information is revealed in three stages. First, the belief is slit into 0 and. Second, at (assuming this belief is reached), it is slit in 0 and. Finally, at, it is slit in 0 and 1. The right anel shows how to determine the tye-1 Agent s ayoff. The two solid (red) segments reresent the maximum ayments that can be demanded at the second and third stage. (No ayment is made in the first, as the slitting does not affect the Firm s outside otion.) Their added lengths measures the tye-1 Agent s ayoff. Comare with our two-stage equilibrium, in which all information is disclosed once the belief reaches : the ayment is equal to the length of the vertical segment between the outside otion w at and the chord connecting (0,0) and (1,1) evaluated at (i.e., the lower red segment, lus the dotted segment). The ayoff with three stages is larger, as the chords from the origin to the oint (,w()) become steeer as increases. We could go on: information slitting is beneficial. Figure 4 illustrates the total ayoff that results from a slitting that involves many small stes (which is the sum of all vertical segments). Does it follow that the cometent Agent extracts the maximum value of information as K? Unfortunately, no: see the right anel of Figure 4. As the Firm s belief goes from d to, the Firm must ay (using the martingale roerty, the test must be assed with 17

19 $ Figure 3: Revealing information in three stes: evolution (left) and ayoff (right) robability d ) d w() w( d), yet its outside increases by w() w( d). The tye-1 Agent gives u the difference w()d/ in the rocess. This foregone ayoff need not be large when the ste size d is small, but then again, the smaller the ste size, the larger the number of stes involved. Note that this foregone ayoff does not benefit the Firm, which is always charged its full willingness-to-ay. The tye-0 Agent reas this ayoff. As a result, her ayoff does not vanish, even as K. What does the maximum ayoff converge to as K? Plugging in the secific form of w from our leading examle, the ayment for a slitting of into {0,+d} is +d w(+d) w() = ((+d) γ(1 d))) ( γ(1 )) = γd +d +O(d2 ), where O(x) < M x for some constant M and all x. If the entire interval [,1] is divided in this 18

20 $ $ 1 1 w() loss d 1 Figure 4: Revealing information in many stes (left); Foregone rofit at each ste (right) fashion into smaller and smaller intervals, the resulting ayoff to the cometent Agent tends to 1 γ d = γ(ln1 ln ) = γln. This suggests that the limiting ayoff is indeendent of the exact way in which information (above ) is divided u over time, as long as the mesh of the artition tends to zero. V V V10 V3 0.5 V V Figure 5: Revealing information in many stes (left); Payoff as a function of K (right). 19

21 Lemma 1 As K, the maximum ayoff to the tye-1 Agent in ure strategies tends to, for 0 <, V 1 ( 0 ) := γln. This lemma follows as immediate corollary from the next one. Note that this ayoff is indeendent of 0 (for 0 < ). Indeed, the first chunk of information, leading to a osterior belief of if K is large enough, is given away for free. It does not affect the Firm s outside otion, but it makes the Firm as unsure as can be about the right decision. From that oint on, the Agent starts selling information in excruciatingly small bits, leaving no surlus whatsoever to the Firm, as in the left anel of Figure 5. We conclude this subsection by the exlicit descrition of the equilibrium that achieves the maximum ayoff of the tye-1 Agent, as a function of the number of rounds and the rior belief 0. Here, (x) := min{0,x} 0. Lemma 2 The maximum equilibrium ayoff of the tye-1 Agent in a game with K rounds is recursively given by Kγ(1 1/K 0 ) ( 0 γ(1 0 )) if 0 ( ) K K 1, V 1,K ( 0 ) = V 1,K 1 ( ) if 0 < ( ) K K 1, for K > 1, with V 1,1 ( 0 ) = γ(1 0 ) ( 0 γ(1 0 )). On the equilibrium ath, in the initial round, the tye-1 Agent reveals a iece of information leading to a osterior belief of K 1 K 0 if 0 ( ) K K 1, 1 = if 0 < ( ) K K 1, after which the lay roceeds as in the best equilibrium with K 1 rounds, given rior 1. Note that, fixing 0 <, and letting K, it holds that 0 < ( ) K K 1 for all K large enough, so that, with enough rounds ahead, it is otimal to set 1 = in the first, and then to follow the sequence of osterior beliefs ( ) K 1 K,( ) K 2 K,...,1, and the sequence of osteriors 20

22 successively used becomes dense in [,1]. Therefore, with sufficiently many rounds, the equilibrium involves rogressive disclosure of information, with a first big ste leading to the osterior belief, given the rior belief 0 <, followed by a succession of very small disclosures, leading the Firm s belief gradually u all the way to one. The right anel of Figure 5 shows how the ayoff varies with K. Note also that, for any K and any equilibrium, if and > are beliefs on the equilibrium ath, then V 0 ( ) V 1 ( ) V 0 () V 1 (), as long as only the Firm makes ayments. Indeed, going from to, the tye-1 Agent forfeits the ayments that the Firm might have made over this range of beliefs (hence V 1 ( ) < V 1 ()), while the tye-0 Agent only forfeits them in the event that she is able to ass the test: hence she loses less, and might even gain (for instance, she might not have been able to ass the first free test at < ). As a result, the tye-1 Agent has a reference for lower beliefs, relative to the tye-0 Agent. Having to give away information is more costly to an Agent who knows that she owns it. This lays an imortant role once mixed strategies are considered. 3.2 General Outside Otions Assuming that the outside otion is given by a call otion, as in our main examle, leads to closed-form exressions. However, the analysis can be generalized. Suose that the ayoff of the Firm (gross of any transfers) as a function of its osterior belief after the K rounds is a non-decreasing continuous function w(), and normalize w(0) = 0, w(1) = 1. We further assume that w(), for all [0,1], for otherwise full information disclosure is not socially desirable. This ayoff can be thought as the reduced-form of some decision roblem that the Firm faces, as in our baseline model. In that case, w must be convex, but since it is a rimitive here, we do not assume so. Recall that the best equilibrium with many rounds called for a first burst of information released for free (assuming < ), after which information is disclosed in dribs and drabs. One might wonder whether this is a general henomenon. The answer, as it turns out, deends on the shae of the outside otion. It is in the interest of the tye-1 Agent to slit information as finely as ossible for any rior belief 0 if and only 21

23 if the function w is (strictly) star-shaed, i.e., if and only if the average, w()/, is a strictly increasing function of. 9 More generally, if a function is star-shaed on some intervals of beliefs, but not on others, then information will be sold in small bits at a ositive rice for beliefs in the former tye of interval, and given away for free as a chunk in the latter. In our main examle, w is not star-shaed on [0, ], as the average value w()/ is constant (and equal to zero) over this interval. However, it is star-shaed on [,1]. Hence our finding. Let us first consider a star-shaed outside otion. If in a given round the Firm s belief goes from to either (+d) or 0, the Agent can charge u to +d w(+d) w() = (w () w()/)d+o(d 2 ) for it. 10 Given the Firm s rior belief 0, the tye-1 Agent s ayoff becomes then (in the limit, as the number of rounds K goes to infinity) 1 0 [w () w()/]d = w(1) w( 0 ) 1 0 w()d/, which generalizes the formula that we have seen for the secial case w() = ( (1 )γ) That is, the tye-1 Agent s ayoff is the area between the marginal ayoff of the Firm and its average ayoff. To see that slitting information as finely as ossible is best in that case, fix some arbitrary interval of beliefs [, ], and consider the alternative strategy under which the osterior belief of the Firm jum from to, the ayment from the Firm to the Agent in that round is given by w( ) w(). 9 This condition alreadyaearsin the economicsliterature in the study of risk (see Landsbergerand Meilijson, 1990). It is weaker than convexity: the function α is star-shaed for α > 1, but only convex for α In case w() is not differentiable, then w () is the right-derivative, which is well-defined in case w is starshaed. 11 In our main examle, w is (globally) weakly star-shaed: that is, the function w()/ is only weakly increasing. The formula for the maximum ayoff in the limit K is the same whether there is a jum in the first eriod or not. But for any finite K, slitting information disclosures over the range [ 0, ] is subotimal, as it is a wasted eriod, whose cost only vanishes in the limit. 22

24 If instead this interval of beliefs is slit as finely as is ossible, the ayoff over this range is Hence, slitting is better if and only if w( ) w() 1 which is satisfied if the average w()/ is increasing. w() d. w() w( ) d, (2) Equation (2) also exlains why slitting information finely is not a good idea if the average outside otion is strictly decreasing over some range [, ], as the inequality is reversed in that case. What determines the jum? As mentioned, the ayoff from a jum is w( )/ w(), while the marginal benefit from finely slitting information disclosures at any given belief (in articular, at and ) is w () w()/. Setting the marginal benefits equal at and, resectively, yields that w( ) = w() and w ( ) = w( ). See Figure 6. The left anel illustrates how having two rounds imroves on one round. Starting with a rior belief 0, the highest equilibrium ayoff the tye-1 Agent can receive in one round is given by the dotted black segment. If instead information is disclosed in two stes, with an intermediate belief 1, the tye-1 Agent s ayoff becomes thesum of thetwo solid(red) segments, which is strictly more, since w()/ is strictly increasing. The right anel illustrates the jum in beliefs that occurs over the relevant interval when w()/ is not strictly increasing, as occurs in our leading examle for <. There is a simle way to describe the maximum resulting ayoff. Given a non-negative function f on [0,1], let sha f denote the largest weakly star-shaed function that is smaller than f. In light of the revious discussion (see right anel of Figure 6), the following result should not be too unexected. 23

25 $ $ w() w() Figure 6: Slitting information with an arbitrary outside otion Theorem 1 The maximum equilibrium ayoff to the tye-1 Agent in ure strategies tends to, as K, 1 V 1 ( 0 ) = 1 sha w(ˆ 0 ) where ˆ 0 := min{ [ 0,1] : w() = sha w()}. ˆ 0 sha w()d/, Thatis, thesameformulaasinthecaseofastar-shaedfunctionalies, rovidedonealies it to the largest weakly star-shaed function that is smaller than w. In words, the maximum ayoff to the tye-1 Agent is the area between the marginal and the average outside otion of the Firm, after regularizing this outside otion by considering the largest weakly star-shaed function below it. 12 The roof also elucidates the structure of the otimal information disclosure olicy, at least in the limit. Let I w := cl { [0,1] : sha w() = w() and w()/ is strictly increasing at }. 12 There is an obvious analogy with results in the literature in marginal ricing, in which, not surrisingly, star-shaed functions lay an imortant role as well. 24

26 In our main examle, sha w() = w() for all, but I w = [1/2,1]. Then the set of on-ath beliefs as K held by the firm is contained, and dense, in I w if I w. If I w =, any olicy is otimal. Note that this result immediately imlies that the highest ayoff to the tye-1 Agent is higher, the lower the outside otion w. That is, if we consider two functions w, w such that w w, then the corresonding ayoffs satisfy V 1 Ṽ 1. The favorite outside otion for the Agent is w() = 0 for all < 1, and w(1) = 1 (though this does not quite satisfy our maintained continuity assumtions). In that case, the tye-1 Agent aroriates the entire surlus. This is the case considered in the literature on zero-knowledge roofs: the revision in the Firm s belief that successive information disclosures entail does not affect its willingness-to-ay. 4 Mixed Strategies and Mediation So far, we have assumed that the cometent Agent always asses the test, which imlies that the Firm s osterior belief is either non-decreasing, or absorbed at zero. There are two reasons why even the cometent Agent may fail. First, she may be able to choose to flunk the test (it turns out that such otion may imrove uon the equilibria considered so far). In ractice, it is hard to see what revents an Agent from failing intentionally a given test: software can be criled or slowed down, rototyes can be damaged or imaired, imrecise or even incorrect answers can begiven. To model this ossibility, we addathirddimension to the Agent s strategy; namely, in every round, after a test has been rivately erformed, the Agent has the choice, in case of a success, to reort a failure. As further notation is not needed, we refer the interested Reader to the working aer for a formal definition. Because the model considered in Section 3 corresonds to the secial case in which the cometent Agent always asses the test the only interesting ure strategy in the extended model we refer to this version as the model with mixed strategies. A second reason for why a cometent Agent might fail a test is simly that the test might be noisy, or very hard. One might devise rocedures that are so difficult that even knowledgeable agents might be occasionally unsuccessful; not many recognized exerts rovide correct 25