Selling Information. May 26, Abstract

Transcription

1 Selling Information Johannes Hörner and Andrzej Skrzyacz May 26, 2011 Abstract We characterize otimal selling rotocols/equilibria of a game in which an Agent first uts hidden effort to acquire information and then transacts with a Firm that uses this information to take a decision. We determine the equilibrium ayoffs that maximize incentives to acquire information. Our analysis is similar to finding ex ante otimal self-enforcing contracts since information sharing, outcomes and transfers cannot be contracted uon. We show when and how selling and transmitting information gradually hels. We also show how mixing/side bets increases the Agent s ayoff. In fact, she can get the entire surlus with rich enough tests. 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, Romans Pancs, Arthur Robson and seminar articiants at the Barcelona JOCS, the Collegio Carlo Alberto, Turin, Essex University, the Euroean University Institute, Harvard-MIT, Illinois at Urbana-Chamaign, Oxford, Simon Fraser University, Stanford, University of British Columbia, University of Western Ontario, UC San Diego, Yale, X-HEC Paris and the SED 2009 meeting 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

2 1 Introduction In this aer, we study a dynamic game of selling information in which layers cannot use external enforcement of contracts. Motivated by a moral hazard roblem of acquiring information, we describe equilibria that maximize ex ante efficiency of a decision roblem in which an Agent needs to acquire information that a Firm can later use to make a decision. The selling information game is divided into rounds of communication. Within a round, the layers first can transfer ayments and then the Agent can send 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 has a binary information about the otimal Firm decision) 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 so that the tye-1 Agent can always ass it but the tye-0 Agent can ass it only with some known robability (so it is not a chea talk). Easy tests have a high robability of being assed by the tye-0. Divisibility of information means that there is a rich set of tests with varying difficulties. In this game we construct tight bounds on the limits of the difference between tye-1 and tye-0 Agent ayoffs as the number of ossible communication rounds grows to infinity (we show an examle where maximizing this difference is necessary for otimal incentive rovision to acquire information). We characterize three such bounds: when we consider only ure-strategy equilibria (in which tye-1 always asses the test), when we allow for mixed-strategy equilibria (when tye-1 may be mixing between assing and failing a test) and finally if we allow for noisy tests that even the tye-1 may not be able to ass (in the absence of noisy test, the same outcome can be achieved with the hel of a trusted intermediary, for instance). Since we assume that the agents cannot commit to ayments or information disclosure, our equilibria can be viewed as the best self-enforcing contracts that the layers would like to coordinate on ex ante. Alternatively, these equilibria describe the maximum ayoffs achievable in any equilibrium without external enforcement (as a function of the communication rotocol) and hence allow us to divide the value of exlicit contracts into the coordination art and the enforcement art. Lack of commitment creates a hold-u roblem: since the Agent is selling information, once 2

3 the Firm learns it, it has no reason to ay for it (see Arrow, 1959). 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 difference in ayoffs. That is, it is usually better for incentives if the Agent 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: first, an initial chunk of information is given away for free that leads the Firm to be indifferent between both decisions. Then 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 very rich set of tests and arbitrarily many rounds of communication, the benefit of slitting does not deend on either assumtion. Second, we show how mixed strategies can hel imrove erformance of the contract. In the best ure-strategy equilibrium the tye-0 Agent collects (in exectation) 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 tye-1 Agent and the Firm may have different (endogenous) risk attitudes (more recisely, if the sum of their continuation ayoffs is not concave). Mixed strategies can be further imroved uon if the layers have access to tests that both tyes can fail with ositive robability, or alternatively, by assuming that layers have access to a trusted intermediary that can noise u the tests (i.e., the intermediator s role is to allow the Agent to commit to a mixed strategy). Such tests allow us to exloit also the difference in beliefs between the Firm and the two tyes of the Agent, regarding the very own evolution of the Firm s belief about the Agent s tye. This form of communication makes it ossible for tye-1 of the Agent to use side-bets (in which the Agent ays the Firm uon a failure of the test) to extract the entire surlus of the information. Most of the aer (Sections 2 and 3) analyzes the information sales roblem for the secific ayoff structure that is inherited from the motivating game of information acquisition. However, 3

4 there is nothing articular about this motivating examle. In Section 4, we generalize our results to arbitrary secifications of how the Firm s ayoff varies with its belief about the Agent s information. 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 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 4

5 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, i.e. before her tye is known. In our case, this is trivial and does not deliver differential ayoffs to the Agent s tyes. Therefore, such an equilibrium does not rovide the Agent with incentives to engage in inventive activities in the first lace (which determine the robability with which an Agent is informed). To do so requires identifying the best equilibrium from the oint of view of a articular tye of the Agent. 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. But it is imortant to stress that by information, we mean here information that is relevant for commonly known choice, such as an investment oortunity. The object of this information is not unknown er se. Our focus (roving one owns information) and instrument (tests that imerfectly discriminate between an Agent that holds information or not) are reminiscent of the literature on zeroknowledge 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 5

6 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 the gives the Firm all information. In Anton and Yao (2002) some contingent ayments are allowed and the inventor can use them together with cometition among firms to obtain ositive return to his 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 (2009) aly ideas similar to Aumann and Maschler (1995) to study otimal information disclosure in the standard (twoeriod) signaling model. 2 The Main Examle We shall motivate this examle by considering the following decision roblem. 2.1 The Decision Problem Consider the roblem of an Agent who must decide whether to acquire information or not. This information will then be sold to a Firm in a second stage. This second stage is the ultimate focus of our analysis. Here we describe this first stage in which information is acquired. There are two states of Nature, s N {L, H}, with rior P[s N = H] = ρ (0, 1). The Agent rivately chooses an effort level e [0, ē] at cost c (e). The differentiable function c is strictly increasing and convex, with c(0) = c (0) = 0, and c (ē) > ρ. 6

7 Given e, the Agent rivately observes a signal, the rivate state ω Ω := {0, 1}, with P [ω = 0 s N = L] = 1, P [ω = 1 s N = H] = e. Hence, conditional on rivate state 1, the robability of s N = H is 1, while P [s N = H ω = 0] = ρ 1 e 1 eρ < ρ. Given effort e, the unconditional robability of the Agent being in the rivate state 1 is 0 := ρ e. Information is useful because an investment decision must be taken, whose return deends on the state of nature. Not investing yields a safe (i.e., state-indeendent) ayoff normalized to 0. Investing yields a ayoff 1 when s N = H and ˆγ < 0 when s N = L. Hence, conditional on the Agent s rivate state, this imlies that investing yields an exected return of γ := ρ 1 e ( 1 eρ ( ˆγ) + 1 ρ 1 e ) 1. 1 eρ if ω = 0, and 1 if ω = 1. We assume that γ > 0 (for all feasible e), i.e., investing is otimal if and only if the rivate state is ω = 1. Hence, the exected surlus is ρ e c (e), so that the first-best effort solves c (e FB ) = ρ. However, it is not the Agent, but the Firm who takes the investment decision and reas its benefits. The Firm observes neither the exerted effort level nor the resulting rivate state. Based on its exectation e about the Agent s effort level, the Firm forms a osterior belief 0 that ω = 1. This belief affects how the surlus will be slit in the second stage. Anticiating revenues V 1 ( 0 ) and V 0 ( 0 ) from the second stage, if the rivate state is ω = 1 or 0, resectively, the Agent s effort e must maximize ρev 1 ( 0 ) + (1 ρe) V 0 ( 0 ) c (e), 7

8 and so, assuming that V 1 ( 0 ) > V 0 ( 0 ), e solves c (e ) = ρ(v 1 ( 0 ) V 0 ( 0 )). (1) Unless V 1 ( 0 ) = 1 and V 0 ( 0 ) = 0, equilibrium effort is below first-best. 1 This is a standard hold-u roblem, although investment is unobservable here. Because c is convex, social welfare is then maximized when the difference V 1 ( 0 ) V 0 ( 0 ) is highest. This gives us the motivation and the objective function for the game layed in the second stage that determines the slit of the surlus, to which we now turn. 2.2 The Game of Selling Information Some basic ingredients of this game are inherited from the decision roblem. As this game can be understood indeendently from the decision roblem, we reeat them here, so that the exosition be self-contained. There are two risk-neutral layers: an Agent (she) and a Firm (it). There are two states of the world, ω Ω := {0, 1}. The Agent is rivately informed of the state of the world at the beginning of the game, but the Firm is not. The Firm s rior belief that the state is 1 is 0, which is common knowledge. The fact that the Agent is erfectly informed is a normalization. 2 The game lasts K rounds, but our focus will be on what haens as K grows arbitrarily large. After the K rounds have elased, the Firm must take a binary action a {I, N}. Either the Firm chooses to Invest (I) or to Not Invest (N). Not investing yields a safe ayoff normalized to 0. Investing yields a ayoff 1 if ω = 1 and γ < 0 if ω = 0. That is, the Investing action is risky: it can ay more than the safe action, but only in one state. The arameter γ measures the cost of taking this action, if it is inaroriate. Because the Agent knows the state, call her the tye-1 Agent if ω = 1, and the tye-0 Agent otherwise. Note that, absent any information revelation, the Firm s otimal action is to invest if and 1 Shares of surlus V 0, V 1 will always be in [0, 1]. 2 Here, a state of the world is what we called a rivate state in the decision roblem. With that interretation of the information that is available to the Agent, the fact that she is erfectly informed is somewhat tautological. 8

9 only if and obtain thereby a ayoff of := γ 1 + γ, w() := ( (1 )γ) +, where x + := max{0, x}. While our analysis will cover both the case in which the rior belief 0 is below or above, we shall often focus on the more interesting case in which 0 is smaller than, unless stated otherwise. The ayoff w() is the Firm s outside otion, and we shall generalize the analysis to outside otions with rather arbitrary secifications in Section 4. In each of the K rounds before the action is taken, the Firm and Agent can make a monetary transfer, and the Agent can reveal some information if she wishes to. More recisely, the strategy has two arts. In round k = 1,..., K, as a function of the history of transfers and information disclosures u to that oint, the Agent and the Firm can simultaneously make a non-negative transfer t A k and tf k, resectively, to the other arty.3 Second, once these transfers are made and observed, the Agent may disclose some verifiable information. 4 Information disclosure is modeled as follows. The Agent chooses a number m [0, 1]. This choice is observed by the Firm. This number corresonds to the difficulty of the test that the Agent icks: The tye-1 Agent can always ass the test (though she can choose to fail it), while the tye-0 Agent can only ass it with robability m. The realizations of tests are indeendent across eriods (and values of m), conditional on the state. Note that the Agent can always choose an uninformative test if she wishes to, by icking m = 1. This is interreted as not revealing any information. If m < 1 and the Agent asses the test, we say that information gets disclosed. Note that, given any belief (0, 1) that the Firm might assign to state 1 and for any, there exists a test that leads the Firm to udate its belief to, if the Agent asses it. Indeed, 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. Indeed, as we shall see, such ayments are irrelevant when only ure strategies are considered. But they lay a critical role once more general strategies are considered. 4 It is easy to see that nothing hinges on this timing. Payments could be made sequentially rather than simultaneously, and could occur after rather than before information disclosure. 9

10 if the Agent icks the value m = 1 1. indeendently of her tye, and does not flunk it on urose, it follows from Bayes rule that the osterior belief assigned to ω = 1 is equal to + (1 )m =. 5 If the Agent fails the test, then the Firm correctly udates its belief to zero. 6 The set of ossible tests that we assume is rich, and imlies that information is erfectly divisible. 7 This allows us to conduct the analysis entirely in terms of beliefs, and to make abstraction from issues relative to the tye of information that is being disclosed, leaving oen some fascinating questions (for instance, in which order should information be released?). Tests only serve the urose of modeling how beliefs can evolve gradually, and could be relaced with any other formalism achieving the same end. But richer sets of tests could be conceived of and will be considered in the analysis: for instance, we might wish to consider tests that even the tye-1 Agent could fail with some ositive robability, so that the Firm s belief that the Agent is of tye 1 can go down just as gradually as it can go u. The Agent does not care about the Firm s decision er se. All she seeks to do is to maximize the sum of the net transfers she receives during the K rounds. The Firm seeks to maximize the ayoff from its decision after the K rounds, net of the ayments that it has made. There is neither discounting, nor any other tye of frictions during the K rounds. In articular, there is no cost to disclosing information. 5 A modeling issue arises for = 0. What if the Firm, after some history of transfers and disclosures, assigns robability 0 to ω = 1, but the Agent then asses a test with m = 0? But our urose is to identify the best equilibrium, not to characterize the set of all equilibria, and so this issue is irrelevant: the equilibria we shall describe remain equilibria if it is required that layers cannot switch away from robability 1 beliefs, and remain the best equilibria if this requirement (not imosed by erfect Bayesian equilibrium) is droed. 6 That is, unless the tye-1 Agent is exected to flunk it on urose with ositive robability. 7 Yet our result that it is better to slit information by using two easier tests instead of a difficult one also holds when the set of tests is coarse. 10

11 2.2.1 Histories and Payoffs 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 Firm in stage k and r k is the result of that test (which is either ositive, 1, or negative, 0). The set of all such histories is denoted H k (set H 0 := ). Given some final history h K (this does not include the Firm s final action to invest or not), the Agent s realized ayoff is simly 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 state 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 ) ω=1 γ 1 ω=0, k=0 k=0 where 1 A denotes the indicator function of the event A Strategies and Equilibrium 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. 11

12 A (behavior) strategy σ A for the Agent is a collection {τ A k, µa k, ρa k }K 1 k=0, where (i) τa 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 information that is released in round k (i.e., the value of m), as a function of the state, the history u to the current round, and the transfers that were made in the round, and (iii) ρ A k : Ω H k R 2 + {0, 1} is the decision to flunk the test on urose, given the outstanding test. 8 A rior belief 0 and a strategy rofile σ := (σ F, σ A ) define a distribution over Ω 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, 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 state ω = 0, 1. The solution concet is erfect Bayesian equilibrium, as defined in Fudenberg and Tirole (1991, Definition 8.2). 9 This game admits a lethora of equilibria. Our focus is to identify the equilibrium that maximizes the sread V 1 V 0. Given the decision roblem of Subsection 2.1, the motivation is twofold. First, if the Firm and the Agent could coordinate ex ante (i.e. before the decision roblem) and make side-ayments, then clearly it would be in their interest to choose an equilibrium that maximizes the ex ante ayoffs, as a form of a self-enforcing contract (or relational contract). Second, we are interested in the uer bound on efficiency that can be achieved without roerty rights, through such self-enforcing contracts, to better understand the agency costs, and how they deend on the coordination failures vs. on the constraints from the way information is sold and acquired (that is, in the sirit of mechanism design, we searate the question of what is the most any equilibrium can achieve from the question of how to coordinate on that equilibrium). To reca, we are interested in the limit of the difference in the Agents ayoffs as the number 8 Note that, for notational simlicity, we assume that the Agent s rivate strategy does not deend on her ast rivate information whether she has flunked the test on urose in the ast aside from the state of the world. Nothing can be gained by considering such strategies. Further, this allows us to view this game as a multistage game, and so to aly Fudenberg and Tirole s definition of erfect Bayesian equilibrium. 9 Fudenberg and Tirole define erfect Bayesian equilibria for finite multistage games with observed actions only. Here instead, both the tye sace and the action sets are infinite. The natural generalization of their definition is straightforward and omitted. 12

13 of rounds becomes arbitrarily large. 10 To do so, we shall relax the roblem by assuming that layers have access to a ublic randomization device at the beginning of every round (a draw from a uniform distribution), as this will facilitate one argument. The resulting equilibria that we consider (whether we consider ure or mixed strategies, or allow a mediator) turn out not to take advantage of this device, so that the findings hold for the model without it. 2.3 Preliminary Remarks If the robability of state 1 is, given the history h k, then the exected surlus (assuming that the Firm takes an otimal eventual decision) is 1 + (1 ) 0 =. This means that continuation ayoffs must satisfy V 1 (h k ) + (1 ) V 0 (h k ) + W(h k ). (2) From any history onward, the Agent can secure a ayoff of zero, indeendently of her tye: V 1 (h k ) 0, V 0 (h k ) 0. (3) The Firm, on the hand, can secure a higher continuation ayoff. If it receives no further information, it receives its outside otion w() = ( γ (1 )) +. (4) Since additional information cannot hurt the Firm, this is a lower bound on W(h k ). It is easy to see that (2) (with = 0 ), (3) and (4) define the set of feasible and individually rational (continuation) ayoffs. We conclude this section with a series of observations about the selling information game. Fix K and 0 throughout. - There exists an equilibrium (the worst equilibrium) that minmaxes both layers simulta- 10 The equilibrium we shall obtain is also an equilibrium of the infinite-horizon, undiscounted game, but taking limits allows us to uniquely in down the limiting strategy rofile. 13

14 neously: Making no transfers (execting none) and releasing no information is an equilibrium, with ayoffs W = w( 0 ), V 1 = V 0 = Although there are many ways for the Agent to signal her information through transfers or deviations in terms of the test difficulty that she icks, and therefore, many out-of-equilibrium beliefs to worry about, such beliefs lay no role: observable deviations by the Firm do not affect its beliefs (this is the no signalling what you don t know ingredient of erfect Bayesian equilibrium), and observable deviations by the Agent can be deterred through the threat of reverting to this worst equilibrium, indeendently of how this affects the Firm s belief. 12 equilibrium is efficient if the constraint (2) is binding, that is, if the tye-1 Agent discloses all her information eventually, on the equilibrium ath. - If an equilibrium gives (V 0, V 1 ) to the Agent, there is an efficient equilibrium that does so: Indeed, the Agent can always disclose the state in the last eriod on the equilibrium ath. This cannot weaken the incentives for the layers to carry out the lanned transfers (because it can only increase the ayoff from following the secified equilibrium actions), but it guarantees that the correct action is taken. 13 Some efficient equilibrium ayoffs giving all the surlus to one of the arties are easy to describe. - There is an equilibrium in which the Firm receives W 0 = 0 : no transfers are ever made, and the tye-1 Agent reveals the state in the last eriod, so that the osterior belief is 1 with robability 0, and 0 otherwise. - There is an equilibrium in which the Agent s exected ayoff is 0 V 1 +(1 0 )V 0 = 0 w( 0 ): 11 For brevity, we often write V ω for V ω (h k ), but because the maximum equilibrium ayoff only deends on the Firm s belief (and the number of eriods left), we also sometimes write V ω (), where is this belief. Finally, we also write V ω for the resulting function of the belief. Hoefully, no confusion will arise. 12 This also imlies that the equilibrium ayoffs that we shall determine can easily be obtained as well for alternative orders of moves within in eriod, such as disclosure before transfer, etc. 13 This is clear if only at most the Agent randomizes on ath, as her ayoff from releasing additional information at the end does not affect her incentives. If the Firm is suosed to randomize on ath, there exists an equivalent equilibrium that takes advantage of the ublic randomization device in which, conditional on the device s outcome, its action is ure, and so its incentives from following the equilibrium action are reinforced by this additional disclosure. This is the only oint in the analysis in which the relaxation (to a game with a ublic randomization device) is used. An 14

15 the Firm ays this amount in the first eriod, and the Agent reveals the state. If the Firm fails to ay, lay reverts to the worst equilibrium. This shows that attaining the maximum exected ayoff of the informed layer is trivial in our game, unlike in many games with incomlete information (see Aumann and Maschler, 1995). Note also that, since the tye-1 Agent can always mimic the tye-0 Agent, her ayoff must be at least as high as the tye-0 s ayoff. This imlies that the maximal equilibrium ayoff for the tye-0 s Agent is the one that maximizes the Agent s ex ante ayoff, as described above. However, all these equilibria are terrible for roviding incentives to acquire information: in all of them the two tyes of the Agent earn the same ayoff and hence there is no return to the effort. What is non-trivial is to identify an equilibrium that maximizes V 1 V 0, the difference in ayoffs of the two tyes. We now argue that maximizing V 1 among all equilibria is equivalent to maximizing V 1 +W, the sum of the Firm s and tye-1 Agent s ayoff, as well as to maximizing V 1 V 0 : - An equilibrium that maximizes V 1 also maximizes V 1 + W over all equilibria: Given some equilibrium yielding ayoffs (V 0, V 1, W), note that V 1 + W V 1 + w ( 0 ). Otherwise, by simly starting from the equilibrium that yields V 1 to the tye-1 Agent and W to the Firm, and by increasing the initial transfer that the Firm is asked to make by an amount W w ( 0 ), we would obtain another equilibrium in which the tye-1 Agent gets a ayoff strictly above V 1 a contradiction. Given that w ( 0 ) is fixed, the conclusion follows. Therefore, the equilibrium that maximizes the tye-1 Agent s ayoff cannot leave any surlus to the Firm. - An equilibrium that maximizes V 1 also maximizes V 1 V 0 : Efficient equilibria, to which attention can be restricted to, satisfy 0 V 1 + (1 0 )V 0 + W = 0, (5) 15

16 so that V 1 V 0 = V 1 + W Thus, given the rior belief 0, maximizing the ayoff difference V 1 V 0 is equivalent to maximizing the sum V 1 + W, but as we have already remarked, this is in turn imlied by maximizing V 1 only. Therefore, we can simlify further and focus on maximizing V 1, the tye-1 Agent s ayoff. - The set of equilibrium ayoffs 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, the highest equilibrium ayoff for the tye-1 Agent has a well-defined limit given 0 that we shall seek to identify. The functions V 1, V m 1, V int 1 are the (ointwise) limit ayoffs (as 0 varies) in ure, mixed unobservable, and mixed observable strategies, that we shall consider in turn. 3 Equilibrium Analysis We now turn to the focus of the analysis: what equilibrium maximizes the ayoff of the tye-1 Agent, and how much of the surlus can she aroriate? 3.1 Pure Strategies We start by considering ure strategies by the Agent. A ure strategy calls for the Agent to disclose a secific iece of information at each round (revealing nothing being a secial case). Of course, this is only ossible if the Agent owns this iece of information. If she is of tye 1, she does. But this is not necessarily the case if she is of tye 0. This imlies that, from the Firm s oint of view, and ignoring the uninteresting case in which the Agent is suosed to reveal nothing, its osterior will take one of two values: either it will jum from 0 u to some, if the iece of information is revealed. Or it will jum down to zero. This is illustrated in Figure 1 below. The two arrows indicate the two ossible osterior beliefs. Note that, as a stochastic rocess, and 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, however, from the Agent s oint of view. Given her knowledge of 16

17 0 0 1 Figure 1: A feasible action the state, she assigns different robabilities to these osterior beliefs than the Firm. If she is the tye-1 Agent, she knows for sure that the belief will not decrease over time, because she owns all ieces of information (the belief rocess is then a submartingale, relative to her information). Conditioning only on the state being 0, but not on the secific information owned by the tye-0 Agent, the exectation of the osterior belief is below 0 (the rocess is then a suermartingale). But deending uon whether the tye-0 Agent haens to own or not the secific iece of information whose disclosure the equilibrium calls for, she knows for sure whether the osterior will be or 0. More generally, an equilibrium outcome secifies a martingale slitting, summarized by the sequence of Firm s beliefs that the state is 1, conditional on all ieces of information having been exhibited u that round. We let K+1 denote the final osterior belief (after all ieces to be disclosed in equilibrium have been). This non-decreasing sequence { 0,..., K+1 } starts at the Firm s rior belief, 0, and ends u at K+1 = 1 if the equilibrium is efficient, which we can assume. If a iece of information fails to be disclosed, the osterior immediately dros to zero. Of course, an equilibrium must also secify transfers, as well as how layers behave off the equilibrium ath. The most effective unishment for deviations (whether in terms of information disclosure or ayment) is reversion to the worst equilibrium, and this is assumed throughout. On-ath equilibrium transfers are easily determined by backward induction. Leaving any surlus to the Firm in the last round brings no benefit to the Agent at that stage. Of course, the Agent could demand more in earlier rounds by leaving surlus to the Firm in later rounds, but the willingness-to-ay of the Firm for these later surluses must be discounted by the robability 17

18 assigned by the Firm that the Agent will fail to be able to disclose the iece of information leading to this later round. (If the osterior dros to zero, clearly all ayments sto). Because the tye-1 Agent knows that this information will be disclosed, she gains from backloading these transfers as much as ossible. Therefore, if the Firm s belief in the next round is either k+1, or 0, given the current belief k, then the Firm is willing 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. The Agent does not make any transfers. In other words, the Agent extracts the maximal ayment she can hoe for from the Firm at every round. This sounds intuitive, but as we shall see, this will no longer be otimal when a more general class of mechanism is considered. This leaves us with the determination of the sequence of osterior beliefs. 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. Unless the Agent can reveal the information slowly, we show that the answer is negative: If K = 1, the highest equilibrium ayoff to the tye-1 Agent is equal to 0 w( 0 ). With one round of communication, the ayoff of the Agent can come only from the ayment in the first (and only) round. Therefore, the ayoffs of the two tyes of Agents have to be the same in all equilibria (for K = 1). To identify the best for the tye-1 Agent, recall that we can focus on efficient equilibria, in which the osterior is either 0 or 1 = 1. Because beliefs must follow a martingale from the Firm s oint of view, it must be that the robability that the osterior is 1 is 0 / 1, because 0 = This means that the additional value from this information, relative to what the Firm can secure, 18

19 $ Figure 2: Revealing information in two stes is E F [w( )] w( 0 ) = 0 1 w ( 1 ) w ( 0 ) = 0 w ( 0 ). Note that, when 0, the highest ayoff in one round that the tye-1 Agent can get in equilibrium is simly the rior 0. Note also that this ayoff is increasing in 0. This immediately suggests one way to imrove on the ayoff with as little as two rounds. In the first ste, the Agent discloses for free the iece of information leading to a osterior belief of (or 0, if she fails to do so). In the second round, the equilibrium of the one-round game is layed, given the belief. This second and only ayment yields w( ) = > 0. The right anel of Figure 2 illustrates. 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 clearly larger than the ayment with only one round, given by the length of the vertical segment at 0. 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 will be given at the end of this subsection. 19

20 $ Figure 3: Revealing information in three stes: evolution (left) and ayoff (right) Allowing additional rounds will further imrove what the tye-1 Agent can achieve. This can be understood grahically. Consider Figure 3. As shown on the left anel, information is revealed in three stes. 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 grahically. The two solid (red) segments reresent the maximal ayments that the tye-1 Agent can demand at each round for the information that is being released in the second and third round. (In the first round, no ayment can be demanded, because if future ayments drive down the Firm s continuation ayoff from the second round onward to its outside otion, its continuation ayoff is zero whether its osterior goes u or down). Thus, the sum of their lengths is the ayoff of the tye-1 Agent. In contrast, in the equilibrium involving two rounds only, in which information is fully disclosed once the belief reaches, the ayment to the Agent is only equal to the distance 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 segment, lus the dotted segment). It is clear that the rofit with three rounds exceeds the rofit with only two, as the chords from the origin to the oint (, w()) become steeer as increases. It is intuitively clear that further slitting information is beneficial, if ossible. Figure 4 20

21 $ $ w() loss d 1 Figure 4: Revealing information in many stes (left); Foregone rofit at each ste (right) illustrates the total ayoff that results from a slitting that involves many small stes (which is the sum of all vertical segments). The Reader might be temted to conject that, in the limit as K, the tye-1 Agent will be able to extract the full value of the information. The right anel exlains why this conjecture is incorrect. As the Firm s belief goes from d to, its outside otion increases from w( d) to w(), yet the tye-1 Agent only charges a fraction of this, giving u w()d/ in this rocess. This loss, or foregone rofit, 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 that the disclosure olicy involves. As a result, the tye-1 Agent cannot avoid but to give u a fraction of the value of the information. Note that this sacrificed rofit does not benefit the Firm, which is always charged its full willingness-to-ay. Therefore, it benefits the tye-0 Agent, whose rofit does not tend to zero, even as the number of rounds goes to infinity. What does the maximum ayoff converge to as the number of rounds increase? Here is a heuristic derivation of the solution. Note that, for, the ayment that the tye-1 Agent 21

22 can extract from the Firm if the following osterior belief is {0, + d} is w ( + d) w () = + d (( + d) γ(1 d))) ( γ(1 )) = γd + d + O(d2 ), where O(x) < K x for some constant K and all x. If the entire interval [, 1] is divided in this fashion in smaller and smaller intervals, the resulting ayoff tends to 1 γ d = γ(ln1 ln ) = γ ln. 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). Lemma 1 As K, the maximum ayoff to the tye-1 Agent in ure strategies tends to, for 0 <, V 1 ( 0) := γ ln. This lemma will follow 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, is given away for free. It does not affect the Firm s outside otion, but it makes the 22

23 Firm as unsure as can be about what it is 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 maximal equilibrium ayoff of the tye-1 Agent with K rounds, given the Firm s rior belief 0, 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 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, 23

24 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 roduce the relevant information: hence she loses less, and might even gain (for instance, she might not have been able to roduce the first iece of evidence that is given away at < ). As a result, and quite generally, 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 in the analysis of mixed strategies that we do next. An imlication of this analysis is that, with ure strategies, there is no role for ayments going from the Agent to the Firm. We believe, but have not shown, that the converse also holds, and that, without ayments from the Agent to the Firm, one cannot imrove on the equilibrium in ure strategies. 3.2 Mixed Strategies We now consider mixed strategies by the Agent. Secifically, consider the following scenario. The tye-1 Agent asses the test with ositive but non-unitary robability; that is, she flunks on urose some of the time. The tye-0 Agent asses the test whenever she is able to. This requires (i) the tye-1 Agent to be indifferent between the two resulting continuations, and (ii) the tye-0 Agent to (weakly) refer not flunking the test. In this case, failure to exhibit information does not lead to a osterior of zero. Indeed, the tye-1 Agent might conceivably mix in such a way that exhibiting information leads to a lower osterior (though this won t occur in the analysis). Whether one views mixed strategies as lausible in their own right or not, such dynamics of beliefs would also result from ure strategies with an aroriately extended set of actions: if the Agent can commit to run a test which is noisy (e.g., alying her exertise to a articular task, or letting the Firm exeriment with, or make measurements of, her invention), the osterior belief will not necessarily dro to zero after a failure (oerating systems do crash occasionally). In fact, such tests endow the Agent with even more commitment than mixed strategies as considered here, as they do not require the Agent to be indifferent over the resulting outcomes. The imortance 24

25 of such commitment will be evaluated in the next subsection. One might wonder what the tye-1 Agent could gain from using mixed strategies. The rationale is actually well-known. The tye-1 Agent and the Firm have both differences in references over osterior beliefs, and differences in beliefs about the event that these osterior beliefs materialize. The next subsection will show how to take advantage of the heterogeneity in beliefs. Mixed strategies take advantage of the heterogeneity in references. To illustrate this, consider the case in which γ = 1, so that = 1/2, and consider the limiting case K =, for simlicity. Using the best equilibrium (for the tye-1 Agent) as a benchmark, the references of the Firm are iecewise affine in (w() = (2 1) + ). Meanwhile, the tye-1 Agent has a ayoff function that is convex in (over [1, 2/1]), given by ln. We shall use side bets to take advantage of this differential attitude towards the resolution of uncertainty. Of course, if the tye-1 Agent gains from such side bets, and the Firm does not lose from them (as her ayoff is already down to her outside otion), it must be that the tye-0 Agent loses. Her ayoff function is given by V 0 () = 1 + ( ln)/(1 ), and w() = 2 1 for 1/2. See the left anel of Figure 6. Side bets, however, require ayments to go back and forth between the Firm and the Agent. 14 If the Firm ays more than the fundamental exected value of the information disclosed, in anticiation of the returns of such side bets, it had better be that the Agent has incentives to honor such ayments if necessary. If the osterior belief droed to zero, even the threat of reversion to the worst equilibrium could not disciline the Agent into aying back. Therefore, the stakes of such bets are limited on two accounts: the tye-0 Agent should be willing to make the requisite ayments if the case occurs, and the tye-1 Agent must be indifferent between the two continuation equilibria. Note that, if the tye-1 Agent is indifferent between the two continuations, the tye-0 Agent refers the one starting with the higher osterior belief, given their relative references over starting beliefs, so that the tye-0 Agent will disclose the requisite information, whenever she is able to (as we argued at the end of the revious subsection). The left anel of Figure 6 illustrates how the mixing works, starting from a given belief 14 As mentioned, we do not know what ayoffs can be obtained if mixed strategies are allowed, but ayments from the Agent to the Firm are not. We cannot rule out that some of the back ayments of the bets could take the form of deductions on later ayments by the Firm, but is unclear how far such deductions could substitute for exlicit ayments by the Agent. 25