Inside and outside information

Transcription

1 Inside and outside information Daniel Quigley University of Oxford Ansgar Walther University of Warwick This version: May 4, 2017 Abstract We analyze a game of communication where uninformed parties receive verifiable inside information, which is disclosed strategically by self-interested parties, and outside information, which is beyond the control of insiders. For a range of parameters, the classic unraveling result reverses. The highest-quality insiders remain quiet, silence feeds on itself, and only mediocre insiders disclose. Better outside information can trigger discontinuous drops in inside disclosures, and leave outsiders less informed overall. Finally we characterize the optimal design of outside information, which is the solution to a constrained Bayesian persuasion problem. We derive new implications for buyer-seller interactions, political contests and financial markets. JEL Classification: D82, D83, D72, G14. Keywords: Communication, persuasion, evidence, unraveling. We thank Vince Crawford, Martin Cripps, Peter Eso, Ian Jewitt, Emir Kamenica, Navin Kartik, Paul Klemperer, Tim McQuade, Meg Meyer, Eduardo Perez-Richet, Larry Samuelson, Adrien Vigier, Selma Walther, Lucy White, Mungo Wilson, Peyton Young, and audiences at Northwestern, Oxford, Imperial College, Warwick, the ECB, the Federal Reserve (New York and Board of Governors), Frankfurt School of Management, Cass Business School, Einaudi Institute for Finance and Economics, and Illinois (Urbana- Champaign) for their comments. A previous version of this paper was circulated under the title Crowding out disclosure.

2 1 Introduction We have made an effort to provide credible evidence of this paper s quality. We have searched and cited several strands of related literature, illustrated our intuitions with diagrams, and included detailed proofs and extensions in the Appendix. The editor has the option to consider the information that we have provided in addition to the assessment of independent referees. In this navel-gazing example, the editor bases her decision on two varieties of information: Inside information is provided by self-interested agents (authors) who have an interest in persuading her to take a favorable action (accept). Outside information is generated by third parties (referees) and is beyond insiders direct control. The interaction of inside and outside information is important in many transactions. Firms disclose inside information about the quality of their products and publish financial accounts. Their customers and investors also observe outside information such as online product reviews, analyst opinions and credit ratings. Job market candidates list evidence of their achievements on their CVs, and recruiters rely on these lists in conjunction with recommendations from past employers. Some politicians publish their tax returns to prove that they are not crooks, 1 while voters obtain additional information from investigative journalism. How are insiders incentives to disclose information affected by the presence of outside information? Can the role of outside information help us to understand empirical patterns of insiders disclosures? How should outside information be designed when inside disclosures respond endogenously? In this paper, we address these questions in a standard Sender-Receiver model of strategic communication. An informed Sender can convey a verifiable but costly message about his quality, or stay quiet. 2 An uninformed Receiver bases her decisions on this message and the realization of an outside signal that is beyond Sender s control. Sender wants Receiver to believe that his quality is high. Motivated by the above examples, we focus on the case where inside information is pre-emptive to some extent: When Sender decides whether to send a verifiable message, he cannot perfectly predict the realization of outside information. 3 A natural backdrop for our analysis is the classic unraveling result: In the absence of outside information, insiders tend to make full disclosures (Grossman and Hart, 1980; 1 As we discuss in detail, insiders typically have control over how much to reveal, and may not reveal much in equilibrium. President Trump chose not to publish his tax returns in the 2016 election, for example, while a suite of top British politicians did publish them following the Mossack-Fonseca scandal (see 2 On the costs of verifiable disclosure, Lewis (2011) demonstrates the relevance of technological disclosure costs for online auctions. Leuz and Wysocki (2016) survey a large body of accounting research that shows that disclosures are costly, both for technological reasons and because of concerns about releasing proprietary data to competitors. 3 In Section 3, we discuss the motivation and possible foundations of this assumption in detail. 2

3 Milgrom, 1981; Hagenbach et al., 2014). Consider the concrete example of a market where sellers (Senders) want to persuade buyers (Receivers) that the quality of their goods is high. When disclosure costs are small enough, the highest-quality sellers will happily disclose that they are the best. Buyers now expect a disclosure from the best, and infer that no news is bad news. The second-highest-quality sellers also disclose to avoid this pessimistic assessment, no news becomes worse news still, inducing the third-best sellers to make a disclosure, and so forth. Disclosures by high quality types feed on themselves, and all but the worst insiders reveal their quality in equilibrium. This unraveling mechanism is mitigated if is unclear whether insiders know anything (Dye, 1985; Shin, 2003), or when disclosure costs are large (Verrecchia, 1983), 4 but a robust implication is that those with high quality have the strongest incentive to reveal inside information. We argue that unraveling can reverse in the presence of outside information. In Section 2, we demonstrate this in a model where Receiver takes a binary action, as in the case of a buyer who decides whether or not to purchase a good at a pre-determined price. If outside information is sufficiently precise, the highest-quality sellers expect a favorable signal, and predict that they are likely to make a sale even if they do not disclose anything. Thus, they rationally stay quiet to save on the costs of disclosure. Buyers now expect silence from the best, and no news is ambiguous news. Silence then becomes more attractive for the secondbest sellers. If they too stay quiet, no news becomes better news still, the third-best sellers are tempted to stay quiet, and so forth. Silence by high types feeds on itself. In equilibrium, disclosures are rare and tend to come only from mediocre sellers. In Sections 3 to 5, we generalize the positive insights from the binary example, and address the normative question of how outside signals should be designed. Our main results are as follows: First, optimal disclosure strategies are not generally monotone. Disclosures need not come predominantly from high-quality Senders. Monotonicity breaks when the best Senders have weaker incentives to disclose than others, as is the case in the binary-action example. In a general model, this occurs when outside signals are sufficiently precise relative to the marginal benefit of being perceived as the best, and we propose a simple algorithm that can find out whether equilibrium strategies are monotone in most games. In a natural class of models, which includes market games between sellers and buyers with endogenous prices, we can relate this idea to the shape of payoffs: If Sender s payoffs are sufficiently concave as a function of his perceived quality, then disclosure strategies are not monotone. The marginal benefit of being thought of as the best is relatively low in this case, and the best types are 4 Mathios (2000) and Jin and Leslie (2003) demonstrate empirically that, in line with incomplete unraveling, insiders do not make full voluntary disclosures. 3

4 happy to wait for outside information to approximately reveal their quality. If payoffs are sufficiently convex, by contrast, we obtain monotone equilibria and disclosures by the best types, because the marginal benefit of being thought of as the best is dominant. Second, the reverse unraveling mechanism amplifies silence. Most strikingly, outside information introduces discontinuities around full disclosure. An infinitesimal improvement in the quality of outside information can take us from a situation where almost all Senders have strict incentives to disclose, to a situation where no inside disclosures are made at all. Third, outside information has a strong tendency to crowd out inside information. Indeed, for any game where Sender makes disclosures with positive probability in equilibrium, an improvement in the quality of outside information (in the sense of the Blackwell (1953) order 5 ) can leave Receiver worse informed (also in the Blackwell sense) in equilibrium. Fourth, we study the optimal design of outside information. We consider the problem of a Designer who chooses the distribution of outside information ex ante, in order to maximize Sender s expected utility. We show that this problem reduces to a constrained version of the Bayesian persuasion problem in Kamenica and Gentzkow (2011). The constraints arise because inside disclosures generate a lack of commitment. We derive necessary and sufficient conditions under which the constraint is slack. Under these conditions, the Kamenica-Gentzkow scheme is also the optimal signal in the presence of inside information, and therefore robust to relaxing the commitment assumption. We also take first steps towards the second-best solution when inside information imposes binding constraints on outside signal design: In a model with two types of Sender, the second-best signal is always more informative than the Kamenica-Gentzkow scheme. Intuitively, the second-best signal reveals additional outside information in order to crowd out any inside disclosures that might disturb the information structure. Our positive results speak to recent empirical work, where non-monotone disclosure strategies are a common finding (Luca and Smith, 2015; Bederson et al., 2016). 6 This is usually rationalized by the too cool for school effect (Feltovich et al., 2002; Daley and Green, 2014): High-quality insiders do not pay for costly signals if their quality is likely to be revealed anyway, so that the absence of a signal becomes a counter-signal of quality. results underline the empirical relevance of this idea: The reverse unraveling mechanism amplifies non-disclosure, and we do not require very precise outside information for all types 5 An experiment τ is more informative than τ in the Blackwell order if τ can be expressed as a garbled version of τ. Equivalently, every Bayesian decision-maker weakly prefers to observe realizations of τ instead of τ. A precise definition is in Section Feltovich et al. (2002) provide experimental evidence of non-monotonicity. Luca and Smith (2015) show that mid-ranked business schools are most likely to disclose their rankings, while Bederson et al. (2016) find that restaurants with top hygiene ratings did not post these ratings online. Moreover, Edelman (2011) finds that websites displaying trust certificates are less likely to be trustworthy on average. Our 4

5 of Sender to generate substantial opacity. Moreover, our result on concave payoffs lead us to a new testable prediction: Non-monotone strategies should be observed in markets where there are diminishing marginal returns to perceived quality, for example because high perceived quality attracts imitators and heightens competition. Our result on crowding out is also related to the analysis of bank stress test by Shahhosseini (2016). She finds that the announcement of Federal Reserve stress tests (which provide outside information) led tested banks to reduce loan charge-offs and to keep problematic loans on their books. To the extent that these actions represent a reduction in banks voluntary transparency, her results are in line with our prediction that outside information crowds out inside information. In Section 6, we develop additional applications. The first relates to the literature on corporate disclosures. We consider a firm who sells either bonds or shares to investors. The firm s managers can disclose inside information about future profits, and investors observe outside signals such as credit ratings or analyst reports. We show that disclosures in bond markets are likely to come from mediocre firms with mediocre credit ratings, because the market price of bonds is a concave function of perceived quality. Conversely, the market price of shares is convex, and disclosures in equity markets are likely to come from the best firms. In the existing literature, Verrecchia (1983) and Acharya et al. (2011) emphasize that corporate disclosures are driven by mangers desire to keep stock prices high (or their cost of capital low), while Diamond and Verrecchia (1991) show that managers make disclosures to improve market liquidity. Our results suggest that, in addition to these factors, corporate capital structure and compensation plans play a key role in determining disclosure strategies, by shaping managers marginal values of perceived quality. Our second application considers political communication during election campaigns. In recent work, Kartik and McAfee (2007) show that candidates can use policy positions to signal their inherent quality or character. We hold policy positions fixed, but allow candidates to verifiably disclose their relative quality at a cost. Voters also observe outside information such as media coverage. This model predicts that (i) candidates have the strongest incentives to release inside information during close elections; and that (ii) precise news media coverage can reduce incentives to provide inside information. Finally, we revisit recent applications of optimal signal design and Bayesian persuasion. In Kamenica and Gentzkow s leading example, a prosecutor designs signals to maximize the chances of a guilty verdict, while in Alonso and Camara (2016b), a political communicator designs signals to maximize the probability that a population of voters ratifies a new policy. We show that in both cases, the optimal signal design remains implementable in the presence of inside information. In Goldstein and Leitner (2015), and optimally designed financial stress tests pools strong banks with weak ones in order to insure the latter against under-funding 5

6 and bank runs. We show that this stress test is not always implementable when inside information is available, because good banks may find it profitable to undo the pooling scheme by disclosing their quality. Our results relate to several further strands of literature. First, we contribute to the literature on verifiable communication. 7 Grossman and Hart (1980), Grossman (1981), Milgrom (1981) and Milgrom and Roberts (1986) point out the unraveling of information in various settings. Hagenbach et al. (2014) derive conditions for full disclosure in a wider class of games with pre-play certifiable communication. Jovanovic (1982) and Verrecchia (1983) emphasize that disclosure costs mitigate unraveling, while Dye (1985) and Shin (1994; 2003) show that uncertainty about Sender s information can dampen incentives to be truthful. Kartik (2009) shows that disclosures only partially separate Senders when lying about quality is possible but costly. Our contribution is to study outside information as a first-order determinant of incentives to communicate. In particular, our work complements that of Acharya et al. (2011), who study corporate disclosures in a dynamic game, and show that the anticipation of (outside) public revelations can lead to clustered announcements of bad (inside) news. Second, we emphasize that better public (outside) information can reduce total amount of available information. Related results emerge in macroeconomic models with dispersed information: Vives (1997) and Amador and Weill (2010) show that public information can reduce the weight that agents place on their private signals, thus inhibiting the informativeness of the price system. Morris and Shin (2002) and Angeletos and Pavan (2007) show that public information induces potentially wasteful coordination motives. Our results imply that, in addition to these effects, public information can reduce overall informativeness because it crowds out the release of inside information. Third, our analysis of optimal information design relates to the literature on Bayesian persuasion following Rayo and Segal (2010) and Kamenica and Gentzkow (2011). Perez-Richet (2014) characterizes persuasion by a privately informed Sender, who is subject to incentive constraints. Kolotilin et al. (2016) study persuasion when Receiver is privately informed, and Alonso and Camara (2016a) introduce heterogeneous priors. In these papers, persuasion is subject to informational constraints, and the optimal design frequently changes as a result. Our contribution is to take first steps in analyzing Bayesian persuasion subject to a different set of constraints, which arise from the strategic disclosure of inside information. This point also connects our work to the literature on mechanism design with verifiable evidence (Green and Laffont, 1986; Bull and Watson, 2004; Kartik and Tercieux, 2012; Ben-Porath et al., 2016), which studies the constraints that players ability to disclose verifiable evidence places on the implementability of social choice functions. We analyze the implementability 7 See Milgrom (2008) and Dranove and Jin (2010) for surveys. 6

7 of information structures in particular, of optimal Bayesian persuasion when evidence is available to insiders. 2 A simple model of inside and outside information A seller (Sender) offers an indivisible good at an exogenously determined market price p, and a buyer (Receiver) is deciding whether to purchase it. The value of the good to the buyer is its quality θ, which is drawn from a commonly known distribution with smooth density f(θ) and full support on [ θ, θ ]. Quality is privately observed by the firm and we also refer to it as the seller s type. We consider the following game: 1. The seller privately observes θ, and can send an message m {θ, } that represents inside information. The message m = θ verifiably reveals θ, since it is not feasible for other quality types. The null message m = is costless but reveals no verifiable information. Revealing verifiable information is costly: Sending m = θ reduces the seller s profits by c > The consumer observes m, as well as an exogenous signal s of θ that represents outside information. The signal is s = θ + kɛ, where k is a parameter measuring noise, and the error ɛ is drawn from a distribution with smooth, log-concave density g(ɛ) and full support on [ 1, 1] The buyer decides whether or not to buy the good. If she buys, her payoff is θ p and the firm s payoff is p c 1 m=θ. If she does not buy, payoffs are zero for both players. We assume that E[θ] < p, so that the average good is not worth buying without further information, and that p > c, so that profits from a sale are sufficient to recoup the costs of disclosure. As a benchmark, assume that there is no public signal s (or equivalently, that the noise k = ). Since the average good is not worth buying, good firms with quality θ > p have a strong incentive to disclose their quality in order to guarantee a sale. Bad firms with quality θ < p have a dominant strategy to remain quiet. However, given the strategy of good firms, no news is bad news: Failure to make a disclosure is interpreted as evidence that θ < p. In the unique Perfect Bayesian equilibrium of this game, the seller discloses and a sale occurs if and only if θ p, meaning that equilibrium play is as if consumers had perfect information. This 8 Log-concavity guarantees that a high realization of s is good news about θ in the sense of the Monotone Likelihood Ratio Property (Milgrom, 1981). 7

8 is an instance of unraveling in the sense of Grossman and Hart (1980): Strong incentives for disclosure among good types renders asymmetric information irrelevant. When outside signals are informative, incentives change, and unraveling can turn into reverse unraveling. This can be seen most clearly in two steps that rely on iterated deletion of dominated strategies. First, note that unraveling is not necessarily a equilibrium. Suppose that all sellers θ > p are expected to make disclosures, as in the unraveling equilibrium above. In addition, buyers observe the informative public signal s. There is a critical outside signal s 0 = p + k such that s > s 0 reveals beyond doubt that the good is worth buying. 9 Now high-quality sellers may deviate from full disclosure because they are likely to draw a good enough signal to ensure a sale, even if they do not disclose anything. Indeed, a high quality seller θ > p finds it optimal to deviate by staying quiet if p P r [s > s 0 θ] > p c P r[s < s 0 θ] < c p, (1) that is, if the expected profit from sales after a sufficiently impressive public signal outweighs the profit from a guaranteed sale, net of disclosure costs. The equivalent way to write the condition is that the probability of drawing a signal below s 0, and thereby missing out on a sale, is less than the cost-benefit ratio of disclosure c/p. This condition is more likely to hold for high values of θ, since better sellers draw better signals on average. Thus, the best sellers have the strongest incentives to deviate from full disclosure. Figure 1 illustrates this in panel (a). When signals are sufficiently precise, e.g. when k is small, then the probability that the best type θ draws an outside signal below s 0 (the blue shaded area) is smaller than the cost-benefit ratio of disclosure, and the best type θ prefers to stay quiet. Condition (1) will hold for a set of the best sellers θ (θ 0, θ], where θ 0 is the critical type solving (1) with equality (the red shaded area equals c for this type). As a result, all types above θ 0 would deviate from an unraveling equilibrium by staying quiet. Intuitively, precise signals increase the confidence of high quality sellers, and therefore lead them to deviate from full disclosure, in line with the too cool for school effect in Feltovich et al. (2002). Second, the silence of the best sellers sets off a process of reverse unraveling which encourages silence among other types. If buyers believe that high quality sellers with θ > θ 0 stay quiet, they view silence more favorably. Therefore, the outside signal that guarantees a sale in the absence of inside disclosure falls to s 1 < s 0, where the new critical signal s 1 solves the condition E[θ s 1, θ / [p, θ 0 )] = p and keeps the buyer indifferent. Since the critical signal falls, the probability of a guaranteed sale in the absence of disclosure increases, which means 9 We impose a natural restriction on out-of-equilibrium beliefs here: If she observes a signal s / [θ k, θ+k], which could not have come from type θ, then she attaches zero probability to this type. 8

9 P r[s s 0 θ 0] = c p s 0 (a) Deviations from full disclosure. The blue (solid) curve is the density of outside signals drawn by the best type of seller θ. For this type, the probability of drawing a signal below s 0 (the blue shaded area) is smaller than the cost-benefit ratio c/p. The red (dashed) curve is the density of signals for the critical type θ 0 for whom the probability of drawing a signal below s 0 (the red shaded area) equals the cost-benefit ratio. All types above θ 0 can profit by deviating from full disclosure. θ 0 θ s P r[s s 1 θ 1] = c p s 1 s 0 θ 0 θ s (b) Reverse unraveling. When it is common knowledge that types θ > θ 0 stay quiet, the critical signal outside signal that ensures a sale falls to s 1. The probability that type θ 0 draws a signal below s 1 (the red shaded area) is less than the cost-benefit ratio c/p. The thick black curve is the density of signals for the new critical type θ 1 for whom this probability (the grey shaded area) equals the cost-benefit ratio. All types above θ 1 deviate from disclosure. Figure 1: Reverse unraveling in the binary example. that a wider set of high quality sellers θ (θ 1, θ 0 ] will find it profitable to deviate by staying quiet. Figure 1 illustrates this idea in panel (b). Intuitively, the fact that types above θ 0 stay quiet makes silence more attractive, since it offers the opportunity to pool with the best. This process continues: Silence becomes an even better signal because more above-average sellers θ (θ 1, θ 0 ] have been added to the pool of quiet types. Consequently, the critical outside signal falls further to s 2 < s 1, and even more high-quality sellers become confident enough to stay quiet. An equilibrium is reached when this process converges to a fixed point. In equilibrium, there is a threshold θ such that the seller stays quiet if θ < p or θ > θ, while mediocre types θ (p, θ ) make disclosures. When signals are precise enough, we will have θ = p and there are no disclosures at all. 9

10 Panel (a) in Figure 2 shows how θ can be found by examining best responses. The best response function B(θ ) is defined as the highest type θ who prefers to disclose when buyers believe that types above θ and below p stay quiet. This function is upward-sloping due to strategic complementarities: When θ increases, the buyer believes that fewer types above the threshold p stay quiet, and views silence less favorably. Consequently, more types above the threshold elect to make a disclosure as a best response. Intersections of B(θ ) with the 45-degree (dashed) line are equilibria with B(θ ) = θ. Improvements in the precision of signals reduce incentives for high types to disclose, and consequently shift B(θ ) down. Because of strategic complementarities, B(θ ) is upward-sloping, and there can be multiple equilibria when signals are of intermediate precision. This model yields three central insights, which we derive formally in Appendix A: First, if signals are precise enough in the sense that (1) holds for some θ, then the equilibrium of the Sender-Receiver game is either fully opaque, with no disclosure at all, or non-monotone, with disclosures only by mediocre sellers. Second, reverse unraveling amplifies silence. Most strikingly, it generates discontinuities in equilibrium play. Full disclosure by good types (unraveling) is an equilibrium if no high quality seller would deviate by staying quiet, that is, if (1) fails to hold for θ = θ. But whenever this is not the case, we can show that the highest disclosing type θ in any equilibrium is bounded strictly away from θ. Even when the best seller θ = θ is given just an infinitesimal incentive to stay quiet, for example via a small improvement in the informativeness of the outside signal, there is a discrete mass of types θ (θ, θ] who follow suit and stay quiet in equilibrium. This is driven by the reverse unraveling loop described above, that is, by the fact that the silence of the highest types discourages disclosures further down the distribution. Panel (b) in Figure 2 illustrates this result. Third, an increase in the precision of outside information can leave the buyer strictly worse informed in equilibrium. For example, assume that we start in a situation where the signal s noise k =. Then unraveling is the only equilibrium, and the buyer has perfect equilibrium information about sellers with θ > p. Suppose that outside information becomes more precise, so that the noise parameter k falls, to the point where a discrete set of types (θ, θ] remain quiet in equilibrium. After this change, the buyer has less-than-perfect information about this subset of sellers, and is therefore strictly less informed (in the sense of the Blackwell (1953) criterion) than before the change, even though the quality of outside information has increased in the Blackwell sense. The intuition we have presented depends on the assumption that signals do not have full support. Signals s > p+k cannot possibly be drawn by types whose good is not worth buying. We believe that deviations from full support are empirically realistic, since very favorable 10

11 Best response B(θ ) θ B A C p D 0 p (a) Best responses and equilibria. The blue (upper) curve shows the best response when outside signals are imprecise; the unique equilibrium is full disclosure at point A. The black (middle) curve is drawn for intermediate signal precision; there are multiple equilibria at A, B and C due to strategic complements. The red (lower) curve is drawn for low signal precision; the only equilibrium in this case is full opacity at D. θ θ Best response B(θ ) θ A p ɛ B 0 p (b) Discontinuity around full disclosure. The blue (upper) curve shows the best response before a marginal increase in the precision of signals; while there are multiple equilibria, the full disclosure equilibrium at point A exists. The red (lower) curve shows the best response after a marginal increase in precision; since the best response function is infinitely steep near the top type θ, a small shift of order ɛ leads to a discrete shift in equilibrium play. The most informative equilibrium is now at point B. Figure 2: Equilibrium in the binary example. θ θ 11

12 news tend to rule out very bad outcomes and vice versa: Casual empiricism suggests that most small businesses and developing countries have no chance of obtaining an AAA credit rating from the major agencies; a worker who has been fired from his previous job for stealing would not expect a glowing reference with positive probability, and an innocent defendant in a trial might rationally attach zero probability to having her fingerprints found on the murder weapon. However, many of our general results in the next Section go through even when we allow outside signals to have full support, and we will state explicitly when it is necessary to allow for deviations from it. 3 A general Sender-Receiver game We study a game between a Sender, who has the opportunity to disclose verifiable inside information, and a Receiver, who decides on an action based on these disclosures and outside information. Receiver needs to choose an action a A. Payoffs depend on this action and on the state of the world θ Θ = {θ 1,..., θ N }, where θ N > θ N 1 >... > θ 1. A and Θ are subsets of R. We focus on the case of pure persuasion, where Sender s payoff v(a) depends only on the action taken and is strictly increasing in a. Receiver s payoff u(a, θ) is log-supermodular in a and θ, so that she 10 optimally chooses higher actions when optimistic about θ. 11 We assume that, when Receiver knows the type with certainty, she has a unique best response denoted a (θ) = arg max a A u(a, θ). Sender privately observes the state θ, which we refer to as his type, and sends a message m {θ, }. The message m = θ is available only to type θ, and therefore amounts to full and verifiable disclosure of θ, but it reduces Sender s utility by c > 0. The null message m =, which we refer to as staying quiet, is costless but reveals no verifiable information. In the Online Appendix, we show that similar results obtain in more general message spaces, as long as verifiable messages are more costly than cheap talk. In addition, Receiver privately observes an outside signal s S, where S is a finite subset of R. The players have a common prior belief about the joint distribution of types θ and outside signals s. We write µ 0 (θ) for the prior distribution of θ and π(s θ) for the conditional distribution of s given θ, θ Θ µ 0 (θ) = 1 and s S π(s θ) = 1 for all θ. We assume without loss that µ 0 (θ) > 0 for all θ. 10 Throughout the paper, we use male pronouns for Sender and female ones for Receiver. 11 More precisely, the Receiver s optimal action increases whenever her beliefs about θ become more optimistic in the sense of the Monotone Likelihood Ratio Property (Milgrom, 1981; Athey, 2002). Seidmann and Winter (1997) and Giovannoni and Seidmann (2007) study verifiable message games where these conditions on preferences are relaxed. 12

13 The timing is as follows: First, Sender observes θ and chooses m. Second, Receiver observes inside information m and outside information s before choosing an action a. 12 We consider Perfect Bayesian Equilibria: Sender and Receiver choose messages and actions to maximize expected payoffs, and the Receiver s posterior beliefs about θ are derived using Bayes law on the equilibrium path. Off the equilibrium path, we require that Receiver places zero probability on type θ if she observes a signal such that π(s θ ) = 0. The assumption of pre-emptive disclosure In our model, Sender commits to a disclosure strategy m before he knows the realization s of outside information. This is important for our results: One of our key intuitions is that the best types have weaker incentives to disclose if their decision is pre-emptive to some extent. However, if verifiable messages can be sent between the realization of s and Receiver s action a, a similar logic to unraveling dictates that the best types of Sender will have strong incentives to make such a disclosure in equilibrium. We focus on the case of pre-emptive disclosures because it captures frictions that arise in most of the applications that motivate our paper. In many applications, Receivers react very quickly to outside news, and irreparable damage to Sender s prospects may be done if he waits until after this event to prove his quality. For example, financial markets respond very quickly to bad news or credit downgrades and managers may lose their job or reputation before they have a chance to respond. The release of bad news on the eve of an election may damage a candidate s chances regardless of her subsequent communication. This is especially relevant in situations if verifiable information takes time to prepare and circulate. In financial markets, financial reports need to be prepared and externally audited in advance of their release. 13 In product markets, there can be considerable delays in circulating information regarding the quality of a new product, for example via advertising campaigns, to a dispersed audience of potential customers. More generally, if economic agents have limited capacity for processing information as in Sims (2003), Receiver may be unable to (or rationally choose not to) process further communications by Sender once the outside 12 This action is not contractible: Sender and Receiver cannot commit to a contract specifying a as a function of m and s. Hart et al. (2017) derive a class of verifiable message games in which equilibria with and without commitment are identical.. 13 A potential variation on our model is a setting where the verifiable report m = θ takes time to prepare, but where Sender can prepare it in advance and decide whether to release it once s has been observed. In this environment, Sender has stronger incentives to prepare the report than in our model, because he retains the option to keep it to himself in case s turns out to be better news than the truth. However, similar arguments to our main results are likely to go through: The best types of Sender have a relatively weak incentive to prepare a verifiable report in the first place, because they anticipate that the outside signal s will be good enough to secure a favorable action. 13

14 signal s has resolved a significant portion of the uncertainty. In other applications, there are alternative frictions that lead to a formally equivalent game: In the case of research papers and job applications, the outside information (i.e. referee reports or past employers recommendations) remains private information for the Receiver until after the decision of interest is made, so that the Sender must decide on a communication strategy before he knows outside information perfectly. Additional notation We write V (θ) = v(a (θ)) for Sender s payoff when he is taken to be type θ for certain. When Sender stays quiet and outside information is s, we write α(s) arg max E µ [u(a, θ) s, m = ] for Receiver s (potentially random) best response. This action depends on equilibrium beliefs about θ given s. In particular, the interpretation of m = by a Bayesian Receiver depends on which types play this strategy. We define the net payoff from a verifiable disclosure as N (θ) V (θ) E[v(α(s)) θ]. (2) Sender prefers to disclose if N (θ) c. As pointed out by Milgrom and Roberts (1986) and others, skeptical beliefs, in which Receiver assumes that Sender is of the worst type θ(s) = min{θ π(s θ) > 0} that is consistent with her outside information s, can be useful for analyzing equilibrium. We define the maximal punishment that Sender can suffer by staying quiet as the difference between the payoff he obtains under full disclosure and the payoff he obtains by staying quiet and facing a skeptical Receiver: M(θ) = V (θ) E[V (θ(s)) θ]. The maximal punishment is an upper bound on the net payoff from disclosure; we have N (θ) M(θ) for all feasible equilibrium beliefs. Indeed, type θ has a dominant strategy to stay quiet if M(θ) < c. Note that M(θ 1 ) = 0, so that the worst type θ 1 always has a dominant strategy to stay quiet. 4 Inside and outside information in equilibrium 4.1 Monotone and non-monotone equilibria We call an equilibrium monotone increasing if the probability of disclosure P r[m = θ θ] is increasing in the type θ, and strictly increasing for some pair of types. We call an equilibrium 14

15 opaque if nobody discloses and P r[m = θ θ] = 0. Finally, we call an equilibrium nonmonotone if P r[m = θ θ] is strictly increasing for some pair of types and strictly decreasing for another. The fact that the worst type θ 1 has a dominant strategy to stay quiet guarantees that these are the only possibilities. A focal point in the literature on inside information is the unraveling equilibrium: Sender discloses unless he is the worst type θ 1, and Receiver responds to silence by adopting skeptical beliefs. This is a special case of monotone equilibrium and fully eliminates asymmetric information. In our setting an unraveling equilibrium exists if and only if c min θ>θ 1 M(θ) c 0, (3) where M(θ) denotes type θ s maximal punishment. 14 In the absence of outside information, disclosure strategies remain monotone increasing, and thus qualitatively similar to unraveling, even when (3) does not hold. If Sender stays quiet and Receiver has no outside information, he responds with a fixed optimal action α. Thus Sender s payoff from staying quiet is independent of θ and given by E[v(α)]. 15 The net payoff N (θ) = V (θ) E[v(α)] is now increasing in θ. Since all types are valued equally if quiet, high types have the most to gain from disclosure. It is immediate that all equilibria without outside information are monotone increasing, unless costs c are prohibitively high. We now argue, in line with our intuition in Section 2, that outside information introduces non-monotonicity. When outside information is precise relative to the slope of Sender s payoff v(a), it generates strong incentives for the best types to stay quiet. We are interested particularly in cases where (3) does not hold, so that there is no unraveling equilibrium. We begin by focusing on a class of models for which we can directly relate disclosure strategies to the shape of payoff functions. Then, we use iterated deletion of dominated strategies to derive a characterization of equilibrium disclosures in the general model. Concavity and convexity in a model with virtual types Assume that Receiver s optimal action is the expected value of a function of θ: arg max E µ [u(a, θ)] = E µ [X (θ)], 14 The argument behind (3) is simple. Positing an unraveling equilibrium, we need to check that no type θ > θ 1 wishes to deviate and stay quiet, which is the case if the net payoff from disclosure N (θ) c. With skeptical beliefs, the net payoff equals the maximal punishment N (θ) = M(θ), so that (3) is sufficient to rule out profitable deviations. It is easy to check that (3) is also necessary: If maximal punishments cannot entice Sender to disclose, he prefers to stay quiet regardless of Receiver s beliefs. 15 Receiver may randomize α, but if he does, then this randomization must be conditionally independent of θ in the standard definition of Perfect Bayesian Equilibrium. 15

16 for some increasing function X(θ), and that Sender s utility is simply v (a) = a. This arises in standard seller-buyer interactions where a denotes the willingness-to-pay of a buyer, or indeed of a mass of buyers in a competitive market, for an indivisible item that gives her utility X(θ). This case permits a useful re-interpretation of payoffs in terms of virtual types. If Sender stays quiet and has true type θ = θ i, his expected payoff is E[α(s) θ i ] = N π(s θ i )E[X(θ) s, ] = q ij X(θ j ), s S j=1 where q ij = E[P r[θ j s, ] θ i ] is the expected probability mass that Receiver places on type θ j. Payoffs in the absence of disclosure are therefore equivalent to a game in which Sender draws a virtual type θ j according to the conditional distribution q ij. Note that this is indeed a probability distribution since j q ij = 1 for all i. We write Q ij = k j q ik for the cumulative distribution of virtual types. Virtual types are useful because they inherit this ordering of signals: In the Appendix, we show that type θ n+1 draws better virtual types in the sense of first-order stochastic dominance than type θ n. To clarify the exposition in the virtual type model, we assume that outside signals have full support and satisfy the strict Monotone Likelihood Ratio Property (MLRP). This property needs to be defined carefully since we want to allow for outside signals that do not have full support. An appropriate notion of strict MLRP is that s > s implies π(s θ )π(s θ) > π(s θ)π(s θ ), unless all of the probabilities π(..) in this expression are zero. We further assume that neighboring types share signals: For each i, there is an s such that π(s θ i ) > 0 and π(s θ i 1 ) > 0 (clearly, any signal distribution with full support satisfies this restriction). We write X i = X(θ i+1 ) X(θ i ) for the increment in Receiver s action if she learns that Sender s type increases from θ i to the next-best type θ i+1. re-write the net payoff from disclosure in Equation (2) as: i 1 N (θ i ) = X j Q ij j=1 N 1 j=i Integrating by parts, we can X j (1 Q ij ). (4) Equation (4) expresses the net payoff from disclosure as the sum of two components. The first term is the downside risk that Sender takes by staying quiet; with probability Q ij his virtual type is below θ j for j < i, and the associated incremental loss is X j. The second term is the upside risk; with probability 1 Q ij, Sender s virtual type is above θ j for j > i, and the associated incremental gain is again X j. If the downside exceeds the upside by more than c, Sender prefers to disclose. 16

17 To compare upside and downside risk, we first define the measure of concavity χ min i X i X i+1. When χ > 1, payoffs are concave in the sense that the marginal value of being perceived as a better type diminishes as Sender s type improves. Sufficient concavity rules out monotone increasing disclosure strategies in equilibrium: Proposition 1. If the concavity χ of payoffs is sufficiently large, then for a range of disclosure costs c, there are no monotone increasing equilibria. To understand this result, take a candidate monotone equilibrium where Sender discloses if and only if θ > θ n for some n. In terms of Equation (4), the highest quiet type θ n perceives no upside. 16 As we move from θ n to the lowest disclosing type θ n+1, the upside from staying quiet therefore weakly increases, in proportion to payoff increments X n+1,..., X N The downside from staying quiet falls in proportion to the left-tail increments X 1,..., X n 1, because type θ n+1 has a lower probability of drawing left-tail virtual types than θ n, but it increases in proportion to X n, because type θ n+1 can secure this incremental payoff by making a full disclosure in states where he draws virtual type θ n. If payoffs are sufficiently concave, then the difference in incentives between types θ n and θ n+1 is dominated by the downside, and in particular, by the left-tail increments X 1,..., X n 1. This implies that the net incentive to disclose gets weaker as we move from type θ n to θ n+1, which contradicts equilibrium. The formal proof constructs a uniform bound χ 0 (0, ), which depends on the prior and signal distribution but not on equilibrium play, so that χ > χ 0 implies leads to this type of contradiction for all possible (pure or mixed) monotone increasing strategies and associated beliefs. A simple example clarifies the logic of Proposition 1. The example also highlights that the concavity of payoffs needs to be large relative to the mass placed on left-tail realizations of outside information. Example with three types: Consider three types θ {θ 1, θ 2, θ 3 } and five outside signals s {s 0,..., s 4 }. The prior is uniform with µ 0 (θ i ) = 1/3. Each type draws the outside signal to the left of his type with probability π(s i 1 θ i ) = p, that to the right with probability π(s i+1 θ i ) = r, and the signal matching his type with the remaining probability π(s i θ i ) = q = 16 In order to be perceived as a virtual type θ > θ n, he would need to draw an outside signal that could not have come from the set of types θ θ n who stay quiet in equilibrium. This is impossible. 17 The precise weights on these increments depend on how off-path beliefs are specified for signals that types below θ n cannot send. 17

18 1 p r. Assume that Receiver s optimal action is E µ [X(θ)] and Sender s utility is v(a) = a. The measure χ of concavity in this example is simply χ = X 1 X 2. The maximal punishment for silence is M(θ 2 ) = X 1 (p + q) for the middle type and M(θ 3 ) = X 1 p + X 2 (p + q) for the top type. We have M(θ 3 ) < M(θ 2 ) when χ > 1 + p, that is, when the concavity q of payoffs is large relative to the likelihood ratio of left-tail outside signals to intermediate ones. Under this condition, an unraveling equilibrium exists if and only if c M(θ 3 ) c 0. Whenever c > c 0, the top type must stay quiet, and since the bottom type also has a dominant strategy to stay quiet, there can be no monotone increasing equilibrium. As we will see below, strictly non-monotone equilibria, where only the middle type discloses, exist for a range of parameters. A partial converse to Proposition 1 is that, when payoffs are convex instead of concave, non-monotone equilibria cannot exist. To establish this, we use the measure of convexity ξ = min i X i+1 X i. When ξ > 1, payoffs are convex in the sense that the marginal value of being perceived as a better type increases as Sender s type improves. Proposition 2. If the convexity ξ of payoffs is sufficiently large, then there are no nonmonotone equilibria. Consider a hypothetical non-monotone equilibrium. Let the highest type who stays quiet in this equilibrium be θ n. Since the equilibrium is non-monotone, we can also find some type θ d < θ n who makes a disclosure (the proof deals with mixed strategies). As before, we can argue that θ n perceives no upside from staying quiet. As we move from θ d to θ n, the upside from staying quiet therefore weakly decreases, which gives θ n a stronger incentive to disclose. The downside falls in proportion to the increments X 1,..., X d 1 because type θ n draws better virtual types than θ d, but increases in proportion to X d,..., X n 1, because type θ n can secure these incremental payoffs by making a full disclosure in states where his virtual type is between θ d and θ n. The latter effect, that is, the benefits of full disclosure for the higher type, dominate when payoffs are sufficiently convex. For this reason, the downside of staying quiet is higher for type θ n, who therefore has a strictly stronger incentive to disclose than θ d. This contradicts equilibrium. We return to the three-type example to illustrate Proposition 2. The example also reveals properties of the lower bound on convexity that is needed to rule out non-monotone disclosure strategies. The bound is tighter whenever top types have a significant advantage 18

19 over mediocre types, in the sense that downside risk is significantly smaller for the top types of Sender. Example with three types (cont.): In the example above, the convexity measure is ξ = χ 1 = X 2 X 1. Consider a non-monotone equilibrium in pure strategies, where only the middle type θ 2 discloses. In this equilibrium, Receiver is certain that θ = θ 1 when the outside signal is s s 1, and equally certain that θ = θ 3 when s s 3. When s = s 2, she places probability r on type θ p+r 1 and complementary probability p on type θ p+r 3. The implied distribution of virtual types has Q 21 = p + q r = Q p+r 22 and Q 31 = p r = Q p+r 32. For optimality, the middle type must prefer to disclose and the top type must prefer to stay quiet: N (θ 2 ) c N (θ 3 ). Thus a non-monotone equilibrium exists for some c if and only if N (θ 2 ) N (θ 3 ). Substituting into (4) and rearranging, this is equivalent to ξ λ, where 1 λ λ = Q 21 Q 31 is the downside risk perceived by the top type relative to the middle type. Conversely, when the convexity ξ > Proposition 2. λ, there is no non-monotone equilibrium in line with 1 λ Propositions 1 and 2 establish useful bounds: Given sufficient concavity or convexity of payoffs, we can characterize equilibrium disclosure strategies. In between these bounds, for example when payoffs are linear, the nature of equilibrium disclosures is very sensitive to the distribution of outside information. The three-type example illustrates this point further. In the case of linear payoffs (ξ = 1) non-monotone equilibria exist in the example if and only if λ = p(1 r ) + q r 1. With a symmetric outside signal distribution (p = r) this p+r p+r 2 is impossible unless signals are perfectly revealing. When outside signals are precise in the sense that q > 1, we have a non-monotone equilibrium if and only if outside information is 2 right-skewed, with r sufficiently large. Intuitively, a right skew increases the advantage of p top types over mediocre types, since the outside signals drawn by mediocre types are now interpreted chiefly as having come from low types. As a result, payoffs must now be strictly convex to rule out non-monotone disclosures. Iterated deletion of dominated, non-monotone strategies We return to our general model and characterize non-monotonicity of equilibrium in terms of the maximal punishment M(θ), using an adapted notion of iterated deletion of dominated strategies. We first define a procedure for iterated deletion of strictly dominated, non-monotone disclosures (DNMD): Let Θ n be the set of types who stay quiet after n iterations of the procedure. At stage n of the procedure, we first delete from Sender s strategy any disclosure by a type θ i for whom staying quiet is strictly dominant, assuming that Receiver 19