Selling Information. September 5, Abstract

Transcription

1 Selling Information Johannes Hörner and Andrzej Skrzyacz Setember 5, 2014 Abstract A Firm considers hiring an Agent who may be cometent for a otential roject or not. The Agent can rove her cometence, but faces a hold-u roblem. We roose a model of ersuasion and show how gradualism hels mitigate the hold-u roblem. We show when 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

2 1 Introduction A firm contemlates hiring an exert for a otential roject. For examle, the firm considers hiring a consulting comany to lead a otential re-organization of its division; the firm considers hiring an advisor to hel with a otential acquisition, or an inventor to hel build a new roduct feature, or a roduct itself, in the case of an agency engaged in a rocurement rocess. The agent has rivate information whether she is cometent (tye 1) or incometent (tye 0). Hiring an incometent agent leads to a negative exected net resent value (ENPV) from the roject. Hiring a cometent agent yields a ositive ENPV. While the firm could hire the exert/agent by offering her a ayment equal to her outside otion, the firm s roblem is that it is uncertain whether hiring the agent and going forward with the roject is the best course of action. The firm could take its decision based on the rior belief, but taking into account the agent s information would result in efficiency gains. How could the agent ersuade the firm that she is cometent and at the same time be rewarded for cometence? How should the agent sell that information otimally so that the firm would make an informed hiring decision and the agent would be utmost motivated to acquire cometence? To answer these questions, we roose a game-theoretic model of gradual ersuasion between the Agent and the Firm. Persuasion can be thought of as reliminary rojects, or temorary emloyment sells (that either side can revoke at any time). During the trial eriod the Agent is aid and gradually reveals information. In articular, the Agent erforms tasks (that we call tests ) that are informative about her tye. In every eriod, the Firm observes the task chosen by the Agent, its outcomes and then decides whether to continue the trial eriod, fire the Agent, or hire her and start the main roject. Neither the Firm nor the Agent can commit to future ayments, hiring decisions, test-taking (choice of tasks) or information sharing. There is neither commitment nor contractibility. Most imortantly, the Firm cannot commit to ay the Agent contingent on the information she reveals. In ractice, there are obvious ways in which the two sides can commit, more or less formally. The Agent might be concerned about her reutation for cometence, for instance. The legal structure might rovide some rotection to intellectual roerty rights (as atents do). 2

3 Motivated by incentives to acquire cometence, we characterize equilibria that maximize the difference in exected ayoffs of the Agent of tye 1 and tye 0 (as we show, this is equivalent to solving for equilibria that maximize the ayoffs of the cometent agent). With one-time information transmission (whether information disclosure is full or artial) both tyes of the Agent would get the same ayment. Maximizing incentives to acquire cometence calls for multi-stage communication. The game we consider is therefore dynamic: there is a multi-stage information selling stage (which we call the game of information sale) followed by the final decision of the Firm, that deends on the information the Firm has acquired. Within a round of communication, the two arties make voluntary monetary transfers and then the Agent can disclose some information to the Firm. We assume that information is verifiable and divisible. In articular, the information transmission is modeled as tests to verify the Agent s tye. 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 available tests of varying difficulties. Our main result is that, very generally, slitting information over time increases the cometent Agent s ayoff. That is, the cometent Agent s ayoff is higher in equilibria in which she takes two tests (and is aid for them) in a sequence, than if she takes both of them at once (which is equivalent to taking one harder test). The intuition is that when we slit the tests, the Firm ays still the same amount on average, but the incometent Agent may fail the first test, so that tye gets a smaller fraction of the exected ayment. That effect exlains the structure of the best equilibrium in our first model: first, an initial chunk of information is given away for free that leads the Firm to utmost confusion regarding whether hiring the agent is the correct decision. If the Agent asses this first test, she is then hired on a temorary basis, and during the trial, as long as the Agent erforms, she sells information in dribs and drabs and gets aid a little for each bit (and a failure of a test leads to a termination). These features seem consistent with the ractice of reliminary reorts/trial eriods before firms make large financial commitments to rojects with the hel of external 3

4 exerts. Our finding that selling information gradually is beneficial to reward cometence of the seller should come as no surrise to anyone who was ever involved in consulting. The free first consultation is also reminiscent of standard business ractice. We derive a tight bound on the cometent Agent s equilibrium ayoff as the number of ossible communication rounds grows to infinity. Although the closed-form exression for the limit ayoff relies on the divisibility of information and the arbitrary number of rounds that we allow, there is no discontinuity: the benefit of slitting information does not deend on either assumtion. While the acquisition of cometence is our leading interretation for the model, we also develo a more general version that could be useful for other alications. For examle, the Firm may be buying the Agent s advice (not needing the Agent s hel to execute the roject) and the Agent could be rivately informed about a state of the world, which in turn affects the ENPV of the roject. The Firm s action set does not need to be binary either (for examle, the Firm may be choosing a size of its investment in the roject). To model such situations, we allow for more general ayoffs if the Firm decides to make decisions without further information from the Agent. In articular, these ayoffs cature the ossibility that revealing information the Agent has about the state makes it easier for the Firm to make good decisions without further Agent s inut. We show that the benefits of slitting information aly to this more general model and characterize the best equilibrium ayoffs for the tye-1. We rove that selling information in small bits is rofitable as long as the Firm s ayoff function (maing the current osterior belief to Firm s ENPV if it was to make a decision without any further information from the Agent) is star-shaed, that is, as long as its average value is increasing in the belief. Slitting information might hel the Agent in extracting more surlus from the relationshi, but it does not suffice to extract the entire surlus. In the last section of the aer we show how enlarging the class of test technologies to include some that involve noisy communication between the Agent and the Firm can hel. In articular, we show that, even with our extreme assumtions of non-contractibility and non-commitment, with rich enough tests, the cometent Agent can extract the entire exected value quite generally. 4

5 Related Literature: The aer is related to the literature on hold-u, for examle Gul (2001) andche andsákovics (2004). Onedifference isthat inour gamewhat isbeing sold isinformation and hence the value of ast ieces sold can deend on the future ieces that are disclosed (or failed to be). An agent that eventually reveals himself to be incometent destroys by the same token the value of the consideration that he had established. This roerty of beliefs that they can go u or down is an essential ingredient for our results. 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. 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. Ely, Frankel and Kamenica (2013) is another analysis of otimal martingale slitting, although information has instrumental value there. There are imortant differences between our aer and the motivation of these aers, however. First and foremost, 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, so communication and ayments are concurrent here. In fact, with a mediator, the Agent also makes ayments to the Firm during the communication hase. As in Matthews and Postlewaite (1995) or 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. Pure 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 (2013) 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. In our case, this is trivial and does not deliver differential 5

6 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. Our leading motivating examle of an agent that tries to convince a firm to hire her is closely related to that of the standard model of signaling, as in Sence (1973). There are three differences. First, in terms of technology: In job market signaling, all messages can be sent by all tyes, but at different costs. Here instead, messages are free, but the message sace is tye-deendent. Second, in terms of market structure: in standard job market signaling, it is usually assumed that firms comete for the worker, who reas the entire surlus, so the issue 6

7 of how much surlus signaling allows her to aroriate does not really come u. Third, in terms of objective function: our goal is to understand what equilibrium maximizes the ayoff difference between the cometent and incometent tye, a question that isn t usually central to the analysis of signaling. These are not major differences, however: tye-deendent messages can be viewed as actions that are rohibitively costly to some tyes; 1 erfect cometition among firms does not seem like an essential ingredient of Sence s analysis, and likewise for the fact that we ignore here any outside otion for the agent. The major difference, in our view, is that we are interested in how ersuasion should be dynamically structured, to rovide maximum rewards for (and so incentives to acquire) cometence. With the excetion of Forges (1990), who analyzes a very different game, we are not aware of an analysis of longer communication or signaling in a job market related setting. 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 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 Kamenica and Gentzkow (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. 1 Although this interretation of our tests does not quite work, because the incometent agent does not know whether she will be lucky or not in assing a given test. 7

8 test. 3 We roose the following concrete model of gradual ersuasion/communication using tests. 2 Hiring Cometent Exerts and Motivating Cometence We start with a simle model in which we exlicitly derive how the Firm s ayoff of hiring the Agent as a function of the Firm s beliefs about her cometence arises from a decision roblem. 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). In Section 2.6, we discuss how this belief might come about. 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 (hire the agent and Invest in the roject) or N (not hire the agent and Not Invest in the roject). Not Investing yields a ayoff of 0 indeendently of the Agent s tye. Investing yields an exected net ayoff of 1 if ω = 1 and γ < 0 if ω = 0 (net of the costs of hiring the Agent). That is, investing is otimal if the Agent is cometent, as such an Agent has the skill, 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.2 Second, after these simultaneous transfers, the Agent chooses whether to reveal some information by undergoing a 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 2 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 given the testing technology in this section, but will lay a role in the second art of the aer with more comlex communication. 3 Nothing hinges on this timing. Payments could be made sequentially rather than simultaneously, and occur after rather than before the test is taken. 8

9 chosen by the Agent and observed by the Firm. 4 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 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). 4 Because the level of difficulty is determined in equilibrium, it does not matter that the Agent chooses it rather than the Firm. 5 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. For now, we do not allow the cometent Agent to flunk the test on urose, nor do we consider tests so difficult that even the cometent Agent might fail them. This means that the samle ath of the belief martingale is either non-decreasing, or absorbed at zero. We discuss richer communication ossibilities in Section 4. 9

10 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, but in this and next section we restrict attention to ure-strategy equilibria. 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). We discuss frictions in the next section. 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. 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. Since we assumed that the firm makes a binary investment decision, the secific outside otion reduces to a call otion. We consider a richer 10

11 class of outside otion secifications in the next section. To focus attention on the communication stage of the game, we assume for now that if the Firm decides to invest, it hires the Agent at a salary equal to her outside otion (we consider the erhas more realistic case in which the Agent strictly refers to be hired in section 2.5). As a result, the Agent does not care about the Firm s investment decision or about its own cometence er se. All she cares about is getting aid as much as ossible over the K rounds of communication. The Firm cares about the ayoff from the investment decision, net of any ayments to the Agent during the communication stage (recall that the investment ayoffs already include the cost of hiring 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 τf 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, 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 k=0 11

12 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). 6 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 (in this and later sections) turns out not to take advantage of this device, so that results do not deend on it. 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 it is costly. Clearly, there are many ways to model acquisition of cometence - Aendix B rovides a articular examle. 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 6 Fudenberg and Tiroledefine erfect Bayesianequilibria 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. 12

13 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? It is worth ointing out that, in some cases, maximizing the incentives to acquire cometence 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. 13

14 0 0 1 Figure 1: A feasible action 2.4 The Best Equilibrium for the Cometent Agent We now turn to the focus of the analysis: what equilibrium maximizes the ayoff of the cometent 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 and the next section we consider equilibria in which the cometent 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 (a restriction that we recall in formal statements). 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). 14

15 Instead of describing the information art of an equilibrium outcome by the tests taken so far {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 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 7 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 unless mentioned otherwise. 15

16 is indeed equal to 0 w( 0 ). Suose that a successful test takes the osterior to 1 0. Using 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 16

17 $ Figure 2: Revealing information in two stes over the one-eriod equilibrium when 0 >. The otimal strategy is given at the end of this 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 (vertical) 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.) To understand these ayments, note that in each round the Agent is aid the difference between the exectation of the value of the outside otion tomorrow and the current outside otion. The exectation is weighted by the robabilities of each osterior belief: geometrically, it is the value, at the current belief, of the line that connects the outside otion at the two ossible osterior beliefs tomorrow (so, 0 and for the first ositive ayment, and 0 and 1 for the second). Thus, the added lengths of the solid vertical segments measure the tye-1 Agent s ayoff. Comare with our two-stage equilibrium, in which all information is disclosed once the belief reaches. The ayment for our three-stage equilibrium is measured the same way: it is the length ofthe vertical segment connecting theoutside otionat to theoint onthe lineconnect- 17

18 $ Figure 3: Revealing information in three stes: evolution (left) and ayoff (right) ing the outside otions at 0 and 1. So it is the sum of the left vertical segment, and the dotted segment above it. As is clear, the right solid segment exceeds the left dotted segment: the ayoff with three stages must be larger, as the chords from the origin to the oint (,w()) become steeer as increases. With three stes as deicted, the tye-1 Agent reas two ayments that add to more than the one ayment with two stes. Intuitively, the Firm is willing to ay more with three stes because it is less likely to have to have to make all these ayments if it is dealing with the tye-0 Agent. By the time the osterior belief reaches, by definition, there is a chance that the tye-0 Agent has been found out; hence, the second ayment is a conditional ayment, and because the conditioning event is suggestive that the Agent is indeed cometent, the Firm is willing to ay more in total when some of the ayments are conditional. 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 18

19 $ $ 1 1 w() loss d 1 Figure 4: Revealing information in many stes (left); Foregone rofit at each ste (right) robability d ) d w() w( d), yet its outside otion 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 19

20 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). Lemma 1 Consider the binary-hiring model: w() = [ (1 )γ] +. As K, the maximum ayoff to the tye-1 Agent in ure strategies tends to, for 0 <, If 0, V 1 ( 0 ) = γln 0. 8 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. 8 The suerscrit refers to the restriction to ure strategies. 20

21 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 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, 21

22 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 noise (mixed strategies) are considered. 2.5 Agent Prefers to be Hired So far, we have assumed that the Firm can hire the Agent at her outside otion, so that the Agent is indifferent whether the Firm hires her or not. A erhas more realistic assumtion is that the Agent s comensation is strictly higher than her outside otion, so that she strictly refers being hired. How does that affect our analysis? Assume that the Agent s surlus from emloyment is smaller than the exected net losses the firm would incur if it hired an incometent exert, so that not investing remains the efficient action if the Agent is incometent (otherwise, investing would be otimal in both states and communicating cometence would not be imortant). In this case the equilibrium that maximizes V 1 (or V 1 V 0 ) has the same equilibrium ath as described above. The only difference is in the off-ath behavior suorting it. First, in our construction above deviations are unished by a babbling equilibrium. That means that if and the Agent deviates, the Firm still hires her. When the Agent strictly refers being hired, this no longer suffices. Once is sufficiently high, the incometent tye would refer not to take any more tests (even if she gets comensated for them) to avoid the risk of losing emloyment. In articular, she would not take the last test in round K that is fully revealing. To sustain our equilibrium outcome we can use the following continuation equilibrium: if the Agent ever fails to take the test she is exected to take, the Firm s belief about her cometence dros to 0, and the babbling equilibrium is then layed. This clearly rovides incentives for the Agent to take the rescribed tests. Second, one may worry that the Firm would not make any ayments to the Agent knowing that she wants to be emloyed and hence has strict incentives in the last eriod to reveal enough information to get emloyed. This creates no difficulty: our equilibrium calls for the Agent to take for free a test in the first eriod so that Firm s beliefs increase to. After that, it is always 22

23 a best resonse for the Firm to hire her. So if the Firm deviates to not aying for future tests, the deviation to a babbling equilibrium (in which the Agent reveals no more information and is still hired at the end) remains incentive comatible. Positive emloyment surlus has two additional effects on the equilibrium that are worth ointing out. First, it increases V 1 V 0, because only the cometent Agent catures the emloyment surlus. Since our equilibrium is searating, leaving emloyment surlus to the cometent Agent strengthens incentives for acquisition of cometence. Second, one may be worried that in our original construction the Agent is indifferent between revealing and not revealing the last iece of information in round K, because some considerations left out of the model might break this indifference, leading to unraveling. With a ositive emloyment surlus the Agent has strict incentives to take the last, fully-revealing test, because otherwise the Firm would think that she must be hiding something, and not hire her. At the same time, it does not lead to unraveling because the strict incentives hold only on the equilibrium ath. If the Firm deviates by not aying for some of the tests, the Agent would still be hired, so it would be rational not take the last test (the cometent Agent would be indifferent and the incometent Agent strictly refers to follow the unishment). For further discussion of robustness see Section Free Entry The rior belief 0 lays an imortant role in the analysis, so it is worth discussing how it might come about. One natural way of endogenizing it is to exlicitly model a rior stage in which agents must decide on whether to invest in cometence, so that the benefit of doing so, which deends on 0, must equal its cost. As a result, the fraction of agents doing the necessary investment gives rise to the value of 0 that makes recisely this fraction willing to do so. More directly, we may assume that there is a unit mass of agents (on the short side of the market), whose choice is to enter either as incometent for free, or as cometent ones at a cost of c (0, 1). There are lenty of equilibria including one where only incometent agents enter, and 0 = 0. 9 The best equilibrium from a social oint of view is the one maximizing 0 ; indeed, each cometent agent generates a surlus of 1 at a cost of c. But note that it is not an equilibrium 9 A small cost of entering as incometent would eliminate this equilibrium. 23

24 for every agent to enter as a cometent one, as the resulting belief 0 = 1 would stri them from any additional rewards from further ersuading the Firm of their cometence. In the best equilibrium, both tyes of agents must enter, and so the constraint V 1 ( 0 ) V 0 ( 0 ) = c must hold. Maximizing 0 thus means selecting the equilibrium that maximizes the difference V 1 V 0, as this is the one for which it is ossible to ick the highest 0 satisfying the constraint. This is recisely the equilibrium that we have characterized. Hence, the best equilibrium yields a rior belief 0 that solves V 1 ( 0 ) V 0 ( 0 ) = c, 0 V 1 ( 0 )+(1 0 )V 0 ( 0 )+w( 0 ) = 0. If c ( γ(1+(1+γ)ln ), γln ), the unique solution has 0 = c+γln (0, ). 10 If on 1 c the other hand c γ(1+(1+γ)ln ), the unique solution lies in (,1). As one would exect, a lower cost leads to a higher rior belief that an agent is cometent, as it is then cheaer for them to enter. 3 General Outside Otions Assuming that the outside otion is given by a call otion, as in our main examle, rovides a simle illustration of the benefits of slitting, as well as closed-form exressions. However, the analysis can be generalized. Such a generalization has two benefits. First, it clarifies what drives the benefits of slitting information. Second, it encomasses a broader class of alications. Plainly, there is no reason to confine ourselves to binary decisions by the Firm. For instance, the Firm might choose between hiring the agent if it deems it rofitable; remaining in an Arm s length relationshi with the Agent; or stoing the relationshi altogether. 11 Cometence is not a one-dimensional attribute. Furthermore, the value of hiring a cometent exert might rely on more than the cometence itself: it might deend on the feasibility of the firm s roject, on the match between this roject 10 If c γln, there is no equilibrium with cometent entrants. 11 We thank a referee for suggesting such a ternary decision, as well as useful variations. 24

25 and the agent s exertise. Also, 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. Hence, it makes sense to assume that the firm s outside otion deends on its belief in the agent s cometence in arbitrary ways. Convexity seems to be a natural roerty to imose, but as we show an even weaker condition suffices to generalize our results. 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-negative continuous function w(), 12 andnormalize 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 if the function w is (strictly) star-shaed, i.e., if and only if the average, w()/, is a strictly increasing function of. 13 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 12 We may without loss assume w to be non-decreasing. 13 This condition alreadyaearsin the economicsliterature in the study of risk (see Landsbergerand Meilijson, 1990). It is weaker than convexity. 25