Abstract Sender, who is either good or bad, wishes to look good at an exogenous deadline. Sender privately observes if and when she can release a public flow of information about her private type. Releasing information earlier exposes to greater scrutiny, but signals credibility. In equilibrium bad Sender releases information later than good Sender. We find empirical support for the dynamic predictions of our model using data on the timing of U.S. presidential scandals and U.S. initial public offerings. In the context of elections, our results suggest that October Surprises are driven by the strategic behaviour of bad Sender. 1. Introduction Election campaigns consist of promises, allegations, and scandals. While most of them are inconsequential, some are pivotal events that can sway elections. Rather than settling existing issues, these bombshells typically start new debates that, in time, provide voters with new information. When bombshells are dropped, their timing is hotly debated. Was the bombshell intentionally timed to sway the election? What else did media and politicians know, when they dropped the bombshell, that voters might only discover after the election? The 2016 U.S. presidential campaign between Democrat Hillary Clinton and Republican Donald Trump provides several examples. Just eleven days before the election, FBI director James Comey announced that his agency was reopening its investigation into Secretary Clinton’s emails. The announcement reignited claims that Clinton was not fit to be commander in chief because of her mishandling of classified information. Paul Ryan, the Republican Speaker of the House, went as far as to demand an end to classified intelligence briefings to Clinton. Some commentators maintain that Comey’s announcement cost Clinton the election.1 While the announcement conveyed the impression of an emerging scandal, Clinton was confident that no actual wrongdoing would be revealed by the new investigation —there would be no real scandal. Comey’s letter to Congress stated that “the FBI cannot yet assess whether or not this material may be significant, and I cannot predict how long it will take us to complete this additional work”. The Clinton campaign—and Democrats generally—were furious, accusing Comey of interfering with the election. Comey wrote that he was briefed on the new material only the day before the announcement. But his critics maintained that the FBI had accessed the new emails weeks before the announcement and speculated about how long Comey sat on the new material and what he knew about it.2 Similarly, one month before the election, the Washington Post released a video featuring Donald Trump talking disparagingly about women.3 The video triggered a heated public debate about whether Trump was fit to be president. It revived allegations that he had assaulted women and even prominent Republicans called for Trump to end his campaign.4 Within a week of the video’s release, five women came forward accusing Trump of sexual assault. Trump himself denied all accusations and dismissed the video as “locker-room banter”, and “nothing more than a distraction from the important issues we are facing today.”5 Others echoed his statement that real scandals are about “actions” and not “words”, and took the media coverage of the video as proof of a conspiracy against Trump.6 The concentration of scandals in the last months of the 2016 campaign is far from an exception. Such October surprises are commonplace in U.S. presidential elections, as shown in Figure 1. Political commentators argue that such bombshells may be strategically dropped close to elections so that voters have not enough time to tell real from fake news. Yet, if all fake news were released just before an election, then voters may rationally discount October surprises as fake. Voters may not do so fully, however, since while some bombshells may be strategically timed, others are simply discovered close to the election. Figure 1 View largeDownload slide Distribution of scandals implicating U.S. presidents running up for reelection, from 1977 to 2008. Data from Nyhan (2015). Figure 1 View largeDownload slide Distribution of scandals implicating U.S. presidents running up for reelection, from 1977 to 2008. Data from Nyhan (2015). Therefore, the strategic decision of when to drop a bombshell is driven by a trade-off between credibility and scrutiny. On the one hand, dropping the bombshell earlier is more credible, in that it signals that its sender has nothing to hide. On the other hand, it exposes the bombshell to scrutiny for a longer period of time—possibly revealing that the bombshell is a fake. This credibility-scrutiny trade-off also drives the timing of announcements about candidacy, running mates, cabinet members, and details of policy platforms. An early announcement exposes the background of the candidate or her team to more scrutiny, but boosts credibility. The same trade-off is likely to drive the timing of information release in other contexts outside the political sphere. For instance, a firm going public can provide a longer or shorter time for the market to evaluate its prospectus before the firm’s shares are traded. This time can be dictated by the firm’s liquidity needs and development plans, but can also be chosen strategically to influence the market. A longer time allows the market to learn more about the firm’s prospective performance. Therefore, the market perceives a longer time as a signal of the firm’s credibility, increasing the share price. But a longer time also exposes the firm to more scrutiny, possibly revealing that the firm’s future profitability is low. In all these situations, (1) an interested party has private information and (2) she cares about the public opinion at a given date. Crucially, (3) she can partially control how much time the public has to learn about her information. In this article we introduce a Sender-Receiver model of these dynamic information release problems. In our benchmark model of Section 2, (1) Sender privately knows her binary type, good or bad, and (2) wants Receiver to believe that she is good at an exogenous deadline; (3) Sender privately observes whether and when an opportunity to start a public flow of information about her type arrives and chooses when to exercise this opportunity. We call this opportunity an arm and say that Sender chooses when to pull the arm.7 In Section 3.1, we characterize the set of perfect Bayesian equilibria. Intuitively, bad Sender is willing to endure more scrutiny only if pulling the arm earlier boosts her credibility in the sense that Receiver holds a higher belief that Sender is good if the arm is pulled earlier. Therefore, bad Sender withholds the arm with strictly positive probability. Our main result is that, in all equilibria, bad Sender pulls the arm later than good Sender in the likelihood ratio order. We prove that there exists an essentially unique divine equilibrium (Cho and Kreps, 1987).8 In this equilibrium, good Sender immediately pulls the arm when it arrives and bad Sender is indifferent between pulling the arm at any time and not pulling it at all. Uniqueness allows us to analyse comparative statics in a tractable way in a special case of our model where the arm arrives according to a Poisson process and pulling the arm starts an exponential learning process in the sense of Keller et al. (2005). We do this in Section 4 and show that the comparative static properties of this equilibrium are very intuitive. Both good and bad Sender gain from a higher Receiver’s prior belief that Sender is good. Instead, whereas good Sender gains from a faster learning process and a faster arrival of the arm, bad Sender loses from these. When learning is faster and when the arm arrives more slowly, bad Sender delays pulling the arm for longer and pulls it with lower probability. In this case, the total probability that (good and bad) Sender pulls the arm is also lower. When Receiver’s prior belief is higher, withholding information is less damning, so bad Sender strategically pulls the arm with lower probability, but the probability that good Sender pulls the arm is mechanically higher. We show that the strategic effect dominates the mechanical effect if and only if Receiver’s prior belief is sufficiently low. We show that the probability density with which bad Sender pulls the arm is single-peaked in time, and derive the conditions under which it monotonically increases with time. We also characterize the shape of the probability density with which (good and bad) Sender pulls the arm, and show it has at most two peaks—an earlier peak driven by good Sender and a later peak driven by bad Sender. In Section 5, we apply our model to the strategic release of political scandals in U.S. presidential campaigns. In equilibrium, while real scandals are released as they are discovered, fake scandals are strategically delayed and concentrated towards the end of the campaign. In other words, our credibility-scrutiny trade-off predicts that the October surprise phenomenon is driven by fake scandals. Using data from Nyhan (2015), we find empirical support for this prediction. To the best of our knowledge, this is the first empirical evidence about the strategic timing of political scandals relative to the date of elections and the first direct evidence of an October Surprise effect. Finally, we apply our model to the timing of U.S. initial public offerings (IPOs). Our model links a stock’s long-run performance to the time gap between the announcement of an IPO and the initial trade date. Firms with higher long-run returns should choose longer time gaps in the likelihood ratio order. Using an approach developed by Dardanoni and Forcina (1998), we find empirical support for this prediction. Related Literature.Grossman and Hart (1980), Grossman (1981), and Milgrom (1981) pioneered the study of verifiable information disclosure and established the unraveling result: if Sender’s preferences are common knowledge and monotonic in Receiver’s action (for all types of Sender) then Receiver learns Sender’s type in any sequential equilibrium. Dye (1985) first pointed out that the unraveling result fails if Receiver is uncertain about Sender’s information endowment.9 When Sender does not disclose information, Receiver is unsure as to why, and thus cannot conclude that the non-disclosure was strategic, and hence does not “assume the worst” about Sender’s type. Acharya et al. (2011) and Guttman et al. (2013) explore the strategic timing of information disclosure in a dynamic version of Dye (1985).10Acharya et al. (2011) focus on the interaction between the timing of disclosure of private information relative to the arrival of external news, and clustering of the timing of announcements across firms. Guttman et al. (2013) analyse a setting with two periods and two signals and show that, in equilibrium, both what is disclosed and when it is disclosed matters. Strikingly, the authors show that later disclosures are received more positively. All these models are unsuited to study either the credibility or the scrutiny sides of our trade-off, because information in these models is verified instantly and with certainty once disclosed. In our motivating examples, information is not immediately verifiable: when Sender releases the information, Receiver only knows that “time will tell” whether the information released is reliable. To capture this notion of partial verifiability, we model information as being verified stochastically over time in the sense that releasing information starts a learning process for Receiver akin to processes in Bolton and Harris (1999), Keller et al. (2005). In Brocas and Carrillo (2007), an uninformed Sender, wishing to influence Receiver’s beliefs, chooses when to stop a public learning process.11 In contrast, in our model Sender is privately informed and she chooses when to start rather than stop the process.12 Our application to U.S. presidential scandals also contributes to the literature on the effect of biased media and campaigns on voters’ behaviour (e.g. Mullainathan and Shleifer, 2005; Gentzkow and Shapiro, 2006; Duggan and Martinelli, 2011; Li and Li, 2013).13DellaVigna and Kaplan (2007) provide evidence that biased media have a significant effect on the vote share in U.S. presidential elections. We focus on when a biased source chooses to release information and show that voters respond differently to information released at different times in the election campaign. 2. The Model In our model, Sender’s payoff depends on Receiver’s posterior belief about Sender’s type at a deadline. We begin with a benchmark model in which (1) Sender’s payoff is equal to Receiver’s posterior belief, (2) Sender is perfectly informed, (3) Sender’s type does not affect when the arm arrives, and (4) the deadline is deterministic. Section 3.2 relaxes each of these assumptions and shows that our main results continue to hold. 2.1. Benchmark model There are two players: Sender (she) and Receiver (he). Sender is one of two types $$\theta\in\left\{ G,B\right\} $$: good ($$\theta=G$$) or bad ($$\theta=B$$). Let $$\pi\in\left(0,1\right)$$ be the common prior belief that Sender is good. Time is discrete and indexed by $$t\in\left\{ 1,2,\ldots,T+1\right\} $$. Sender is concerned about being perceived as good at a deadline $$t=T$$. In particular, the expected payoff of type $$\theta\in\left\{ G,B\right\} $$ is given by $$v_{\theta}\left(s\right)=s$$, where $$s$$ is Receiver’s posterior belief at $$t=T$$ that $$\theta=G$$. Time $$T+1$$ combines all future dates after the deadline, including never. An arm arrives to Sender at a random time according to distribution $$F$$ with support $$\left\{ 1,2,\ldots,T+1\right\} $$. If the arm has arrived, Sender privately observes her type and can pull the arm immediately or at any time after its arrival, including time $$T+1$$. Because Sender moves only after the arrival of the arm, it is immaterial for the analysis whether Sender learns her type when the arm arrives or when the game starts. Pulling the arm starts a learning process for Receiver. Specifically, if the arm is pulled at a time $$\tau$$ before the deadline ($$\tau\leq T$$), Receiver observes realizations of a stochastic process \[ L=\left\{ L_{\theta}\left(t;\tau\right),\tau\leq t\leq T\right\} . \] The process $$L$$ can be viewed as a sequence of signals, one per each time from $$\tau$$ to $$T$$ with the precision of the signal at time $$t$$ possibly depending on $$\tau$$, $$t$$, and all previous signals. Notice that if the arm is pulled at $$\tau=T$$, Receiver observes the realization $$L_{\theta}\left(T,T\right)$$ before taking his action. For notational convenience, we assume that $$L$$ is either discrete or atomless. It is more convenient to work directly with the distribution of beliefs induced by the process $$L$$ rather than with the process itself. Recall that $$s$$ is Receiver’s posterior belief that Sender is good after observing all realizations of the process from $$\tau$$ to $$T$$. Let $$m$$ denote Receiver’s interim belief that Sender is good upon observing that she pulls the arm at time $$\tau$$ and before observing any realization of $$L$$. Given $$\tau$$ and $$m$$, the process $$L$$ generates a distribution $$H\left(.\mid\tau,m\right)$$ over Receiver’s posterior beliefs $$s$$; given $$\tau$$, $$m$$, and $$\theta$$, the process $$L$$ generates a distribution $$H_{\theta}\left(.\mid\tau,m\right)$$ over $$s$$. Notice that if the arm is pulled after the deadline ($$\tau=T+1$$), then the distributions $$H_{\theta}\left(.\mid\tau,m\right)$$ and $$H\left(.\mid\tau,m\right)$$ assign probability one to $$s=m$$. Assumption 1 says that (1) pulling the arm later reveals strictly less information about Sender’s type in Blackwell (1953)’s sense and (2) the learning process never fully reveals Sender’s type. Assumption 1. (1) For all $$\tau,\tau^{\prime}\in\left\{ 1,2,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$, $$H\left(.\mid\tau,\pi\right)$$ is a strict mean-preserving spread of $$H\left(.\mid\tau^{\prime},\pi\right)$$. (2) The support of $$H\left(.\mid1,\pi\right)$$ is a subset of $$\left(0,1\right)$$. For example, consider a set of (imperfectly informative) signals $$\mathcal{S}$$ with some joint distribution and suppose that pulling the arm at $$\tau$$ reveals to Receiver a set of signals $$\mathcal{S}_{\tau}\subset\mathcal{S}$$. Assumption 1 holds whenever $$\mathcal{S}_{\tau^{\prime}}$$ is a proper subset of $$\mathcal{S}_{\tau}$$ for all $$\tau<\tau^{\prime}$$. We characterize the set of perfect Bayesian equilibria, henceforth equilibria. Let $$\mu\left(\tau\right)$$ be Receiver’s equilibrium interim belief that Sender is good given that Sender pulls the arm at time $$\tau\in\left\{ 1,2,\dots,T+1\right\} $$. Also, let $$P_{\theta}$$ denote an equilibrium distribution of pulling time $$\tau$$ given Sender’s type $$\theta$$ (with the convention that $$P_{\theta}\left(0\right)=0$$). 2.2. Discussion We now pause to interpret key ingredients of our model using our main application—the timing of U.S. presidential scandals in the lead-up to elections. Receiver is the median voter and Sender is an opposition member or organization wishing to reduce a candidate’s chances of being elected. The candidate is either fit ($$\theta=B$$) or unfit ($$\theta=G$$) to run the country. The prior belief that the candidate is unfit is $$\pi$$. At a random time, the opposition may privately receive scandalous material against the candidate (arrival of the arm). The opposition can choose when and whether to release the material (pull the arm). After it is released, the material is subject to scrutiny, and the median voter gradually learns about the candidate’s type. Crucially, the opposition has private information about what the expected outcome of scrutiny is. We say that the scandal is real (fake) if further scrutiny is likely to reveal that the candidate is unfit (fit) to run the country. If, at the time of the election (deadline), the median voter believes that the candidate is likely to be unfit to run the country, the candidate’s chances of being elected are weak. Notice that releasing a scandal might backfire. For example, before the FBI reopened its investigation over Secretary Clinton’s emails, the median U.S. voter had some belief $$\pi$$ that Secretary Clinton had grossly mishandled classified information and was therefore unfit to be commander in chief. Further investigations could have revealed that her conduct was more than a mere procedural mistake. In this case, the median voter’s posterior belief $$s$$ would have been higher than $$\pi$$. On the contrary, the FBI might not have found any evidence of misconduct, despite investigating yet more emails. In this case, the median voter’s posterior belief $$s$$ would have been lower than $$\pi$$. In this application, Sender’s payoff depends on Receiver’s belief at the deadline because this belief affects the probability that the median voter elects the candidate. Specifically, suppose that the opposition is uncertain about the ideological position $$r$$ of the median voter, which is uniformly distributed on the unit interval. If the candidate is not elected, the median voter’s payoff is normalized to $$0$$. If the incumbent is elected, the median voter with position $$r$$ gets payoff $$r-1$$ if the candidate is unfit and payoff $$r$$ otherwise. The opposition gets payoff $$0$$ if the candidate is elected and $$1$$ otherwise. Therefore, Sender’s expected payoff is given by \[ v_{\theta}\left(s\right)=\Pr\left(r\leq s\right)=s\text{ for }\theta\in\left\{ G,B\right\} . \] Furthermore, Receiver’s expected payoff $$u\left(s\right)$$ is given by \[ u\left(s\right)=\int_{s}^{1}\left[s\left(r-1\right)+\left(1-s\right)r\right]dr=\frac{\left(1-s\right)^{2}}{2}. \] The Receiver’s ex-ante expected payoff is therefore given by \begin{align} \mathbb{E}\left[u\left(s\right)\right] & =\frac{\left(1-\mathbb{E}\left[s\right]\right)^{2}+\mathbb{\mathbb{E}}\left[\left(s-\mathbb{E}\left[s\right]\right)^{2}\right]}{2}=\frac{\left(1-\pi\right)^{2}+\mathrm{Var\left[s\right]}}{2}.\label{eq:ele} \end{align} (1) 3. Analysis 3.1. Equilibrium We begin our analysis by deriving statistical properties of the model that rely only on players being Bayesian. These properties link the pulling time and Receiver’s interim belief to the expectation of Receiver’s posterior belief. First, from (good and bad) Sender’s perspective, keeping the pulling time constant, a higher interim belief results in a higher expected posterior belief. Furthermore, pulling the arm earlier reveals more information about Sender’s type. Therefore, from bad (good) Sender’s perspective, pulling the arm earlier decreases (increases) the expected posterior belief that Sender is good. In short, Lemma 1 says that credibility is beneficial for both types of Sender, whereas scrutiny is detrimental for bad Sender but beneficial for good Sender. Lemma 1. (Statistical Properties).Let $$\mathbb{E}\left[s\mid\tau,m,\theta\right]$$ be the expectation of Receiver’s posterior belief $$s$$ conditional on the pulling time $$\tau$$, Receiver’s interim belief $$m$$, and Sender’s type $$\theta$$. For all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$, and all $$m,m^{\prime}\in\left(0,1\right]$$ such that $$m<m^{\prime}$$, (1)$$\mathbb{E}\left[s\mid\tau,m^{\prime},\theta\right]>\mathbb{E}\left[s\mid\tau,m,\theta\right]$$ for $$\theta\in\left\{ G,B\right\} $$; (2)$$\mathbb{E}\left[s\mid\tau^{\prime},m,B\right]>\mathbb{E}\left[s\mid\tau,m,B\right]$$; (3)$$\mathbb{E}\left[s\mid\tau,m,G\right]>\mathbb{E}\left[s\mid\tau^{\prime},m,G\right]$$. Proof. In Appendix A. ǁ We now show that in any equilibrium, (1) good Sender strictly prefers to pull the arm whenever bad Sender weakly prefers to do so, and therefore (2) if the arm has arrived, good Sender pulls it with certainty whenever bad Sender pulls it with positive probability. Lemma 2. (Good Sender’s Behaviour).In any equilibrium: (1)For all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$ and $$\mu\left(\tau\right),\mu\left(\tau^{\prime}\right)\in\left(0,1\right)$$, if bad Sender weakly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$, then $$\mu\left(\tau\right)>\mu\left(\tau^{\prime}\right)$$ and good Sender strictly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$; (2)For all $$\tau\in\left\{ 1,\ldots,T\right\} $$ in the support of $$P_{B}$$, we have $$P_{G}\left(\tau\right)=F\left(\tau\right)$$. Proof. In Appendix B. ǁ The proof relies on the three statistical properties from Lemma 1. The key to Lemma 2 is that if bad Sender weakly prefers to pull the arm at some time $$\tau$$ than at $$\tau^{\prime}>\tau$$, then Receiver’s interim belief $$\mu\left(\tau\right)$$ must be greater than $$\mu$$$$\left(\tau^{\prime}\right)$$. Intuitively, bad Sender is willing to endure more scrutiny only if pulling the arm earlier boosts her credibility. Since $$\mu\left(\tau\right)>\mu\left(\tau^{\prime}\right)$$, good Sender strictly prefers to pull the arm at the earlier time $$\tau$$, as she benefits from both scrutiny and credibility. Next, we show that bad Sender pulls the arm with positive probability whenever good Sender does, but bad Sender pulls the arm later than good Sender in the first-order stochastic dominance sense. Moreover, bad sender pulls the arm strictly later unless no type pulls the arm. An immediate implication is that bad Sender always withholds the arm with positive probability. Lemma 3. (Bad Sender’s Behaviour).In any equilibrium, $$P_{G}$$ and $$P_{B}$$ have the same supports and, for all $$\tau\in\left\{ 1,\ldots,T\right\} $$ with $$P_{G}\left(\tau\right)>0$$, we have $$P_{B}\left(\tau\right)<P_{G}\left(\tau\right)$$. Therefore, in any equilibrium, $$P_{B}\left(T\right)<F\left(T\right)$$. Proof. In Appendix B. ǁ Intuitively, if there were a time $$\tau\in\left\{ 1,\dots,T\right\} $$ at which only good Sender pulled the arm with positive probability, then, upon observing that the arm was pulled at $$\tau$$, Receiver would conclude that Sender was good. But then, to achieve this perfect credibility,14 bad Sender would want to mimic good Sender and therefore strictly prefer to pull the arm at $$\tau$$, contradicting that only good Sender pulled the arm at $$\tau$$. Nevertheless, bad Sender always delays relative to good Sender. Indeed, if bad and good Sender were to pull the arm at the same time, then Sender’s credibility would not depend on the pulling time. But with constant credibility, bad Sender would never pull the arm to avoid scrutiny. Therefore, good Sender must necessarily pull the arm earlier than bad Sender. We now show that, at any time when good Sender pulls the arm, bad Sender is indifferent between pulling and not pulling the arm. That is, in equilibrium, pulling the arm earlier boosts Sender’s credibility as much as to exactly offset the expected cost of longer scrutiny for bad Sender. Thus, Receiver’s interim beliefs are determined by bad Sender’s indifference condition (2) and the consistency condition (3). The consistency condition follows from Receiver’s interim beliefs being determined by Bayes’s rule and Sender’s equilibrium strategy. Roughly, it says that a weighted average of interim beliefs is equal to the prior belief. Lemma 4. (Receiver’s Beliefs).In any equilibrium, \begin{equation} \int v_{B}\left(s\right)dH_{B}\left(s|\tau,\mu\left(\tau\right)\right)=v_{B}\left(\mu\left(T+1\right)\right)\it{ for\, all }\,\tau\it{ in\, the\, support\, of }\,P_{G},\label{pit} \end{equation} (2) \begin{equation} \sum_{\tau\in {\rm supp}\left(P_{G}\right)}\frac{1-\mu\left(\tau\right)}{\mu\left(\tau\right)}\left(P_{G}\left(\tau\right)-P_{G}\left(\tau-1\right)\right)=\frac{1-\pi}{\pi}.\label{piT} \end{equation} (3) Proof. In Appendix B. ǁ We now characterize the set of equilibria. Part 1 of Proposition 1 states that, for any set of times, there exists an equilibrium in which good Sender pulls the arm only at times in this set. Moreover, in any equilibrium, at any time when good Sender pulls the arm, she pulls it with probability $$1$$ and bad Sender pulls it with strictly positive probability. The probability with which bad Sender pulls the arm at any time is determined by the condition that the induced interim beliefs keep bad Sender exactly indifferent between pulling the arm then and not pulling it at all. Part 2 of Proposition 1 characterizes the set of divine equilibria of Banks and Sobel (1987) and Cho and Kreps (1987).15 In such equilibria, good Sender pulls the arm as soon as it arrives. Proposition 1. (Equilibrium). (1)For any $$\mathcal{T}\subseteq\left\{ 1,\dots,T+1\right\} $$ with $$T+1\in\mathcal{T}$$, there exists an equilibrium in which the support of $$P_{G}$$ is $$\mathcal{T}$$. In any equilibrium, $$P_{G}$$ and $$P_{B}$$ have the same supports, and for all $$\tau$$ in the support of $$P_{G}$$, $$P_{G}\left(\tau\right)=F\left(\tau\right)$$ and \begin{equation} P_{B}\left(\tau\right)=\frac{\pi}{1-\pi}\sum_{t\in supp\left(P_{G}\right)\text{ s.t. }t\leq\tau}\frac{1-\mu\left(t\right)}{\mu\left(t\right)}\left(P_{G}\left(t\right)-P_{G}\left(t-1\right)\right),\label{B} \end{equation} (4)where $$\mu\left(\tau\right)\in\left(0,1\right)$$ is uniquely determined by (2) and (3). (2)There exists a divine equilibrium. In any divine equilibrium, for all $$\tau\in\left\{ 1,\ldots,T+1\right\} $$, $$P_{G}\left(\tau\right)=F\left(\tau\right)$$. Proof. In Appendix B. ǁ Although there exist a plethora of divine equilibria, in all such equilibria, pulling probabilities of good and bad Sender, as well as Receiver’s beliefs, are uniquely determined by $$P_{G}=F$$ and (2)-(4). In this sense, there exists an essentially unique divine equilibrium. Our main testable prediction is that bad Sender pulls the arm strictly later than good Sender in the likelihood ratio order sense. Corollary 1. (Equilibrium Dynamics).In the divine equilibrium, \[ \frac{P_{B}\left(\tau\right)-P_{B}\left(\tau-1\right)}{P_{G}\left(\tau\right)-P_{G}\left(\tau-1\right)}<\frac{P_{B}\left(\tau+1\right)-P_{B}\left(\tau\right)}{P_{G}\left(\tau+1\right)-P_{G}\left(\tau\right)}\,\it{ for\, all }\,\tau\in\left\{ 1,\dots,T\right\} . \] Proof. In Appendix B. ǁ Corollary 1 implies that, conditional on pulling time $$\tau$$ being between any two times $$\tau^{\prime}$$ and $$\tau^{\prime\prime}$$, bad Sender pulls the arm strictly later than good Sender in the first-order stochastic dominance sense (Theorem 1.C.5, Shaked and Shanthikumar, 2007): \[ \frac{P_{B}\left(\tau\right)-P_{B}\left(\tau^{\prime}\right)}{P_{B}\left(\tau^{\prime\prime}\right)-P_{B}\left(\tau^{\prime}\right)}<\frac{P_{G}\left(\tau\right)-P_{G}\left(\tau^{\prime}\right)}{P_{G}\left(\tau^{\prime\prime}\right)-P_{G}\left(\tau^{\prime}\right)}\text{ for all }\tau^{\prime}<\tau<\tau^{\prime\prime}. \] Our model also gives predictions about the evolution of Receiver’s beliefs. Pulling the arm earlier is more credible as Receiver’s interim beliefs $$\mu\left(\tau\right)$$ decrease over time. Moreover, pulling the arm instantaneously boosts credibility in the sense that Receiver’s belief at any time $$\tau$$ about Sender’s type is higher if Sender pulls the arm than if she does not. Corollary 2. (Belief Dynamics).Let $$\tilde{\mu}\left(\tau\right)$$ denote Receiver’s interim belief that Sender is good given that she has not pulled the arm before or at $$\tau$$. In the divine equilibrium, \[ \mu\left(\tau-1\right)>\mu\left(\tau\right)>\tilde{\mu}\left(\tau-1\right)>\tilde{\mu}\left(\tau\right)\,\it{ for\, all }\,\tau\in\left\{ 2,\dots,T\right\} . \] Proof. In Appendix B. ǁ 3.2. Discussion of model assumptions We now discuss how our results change (or do not change) if we relax several of the assumptions made in our benchmark model. We discuss each assumption in a separate subsection. The reader may skip this section without any loss of understanding of subsequent sections. 3.2.1. Nonlinear Sender’s payoff In the benchmark model, we assume that Sender’s payoff is linear in Receiver’s posterior belief: $$v_{G}\left(s\right)=v_{B}\left(s\right)=s$$ for all $$s$$. In our motivating example, this linearity arises because the opposition is uncertain about the ideological position $$r$$ of the median voter. If there is no such uncertainty, then the median voter reelects the incumbent whenever $$s$$ is below $$r$$, where $$r\in\left(0,1\right)$$ is a constant. Therefore, Sender’s payoff is a step function: \begin{equation} v_{\theta}\left(s\right)=v\left(s\right)=\begin{cases} 0 & \text{if }s<r;\\ 1 & \text{if }s>r. \end{cases} \end{equation} (5) We now allow for Sender’s payoff to be nonlinear in Receiver’s posterior belief and even type dependent. To understand how the shapes of the payoff functions $$v_{G}$$ and $$v_{B}$$ affect our analysis, we extend the statistical properties of Lemma 1, which describe the evolution of Receiver’s posterior belief from Sender’s perspective. First and not surprisingly, a more favourable interim belief results in more favourable posterior beliefs for all types of Sender and for all realizations of the process. Moreover, Receiver’s posterior belief follows a supermartingale (submartingale) process from bad (good) Sender’s perspective. Lemma 1′ formalizes these statistical properties, using standard stochastic orders (see, e.g., Shaked and Shanthikumar, 2007). Distribution $$Z_{2}$$ strictly dominates distribution $$Z_{1}$$ in the increasing convex (concave) order if there exists a distribution $$Z$$ such that $$Z_{2}$$ strictly first-order stochastically dominates $$Z$$ and $$Z$$ is a mean-preserving spread (reduction) of $$Z_{1}$$. Lemma 1′. (Statistical Properties).For all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$, and all $$m,m^{\prime}\in\left(0,1\right]$$ such that $$m<m^{\prime}$$, (1)$$H_{\theta}\left(.\mid\tau,m^{\prime}\right)$$ strictly first-order stochastically dominates $$H_{\theta}\left(.\mid\tau,m\right)$$ for $$\theta\in\left\{ G,B\right\} $$; (2)$$H_{B}\left(.\mid\tau^{\prime},m\right)$$ strictly dominates $$H_{B}\left(.\mid\tau,m\right)$$ in the increasing concave order; (3)$$H_{G}\left(.\mid\tau,m\right)$$ strictly dominates $$H_{G}\left(.\mid\tau^{\prime},m\right)$$ in the increasing convex order. Proof. In Appendix A. ǁ To interpret Lemma 1′, we assume that the payoff of both types of Sender is a continuous strictly increasing function of Receiver’s posterior belief, so that both types of Sender want to look good.16 Part 1 says that credibility is beneficial for both types of Sender, regardless of the shape of their payoff functions. Part 2 (part 3) says that from bad (good) Sender’s perspective, pulling the arm earlier results in more spread out and less (more) favourable posteriors provided that the interim belief does not depend on the pulling time. So scrutiny is detrimental for bad Sender if her payoff is not too convex but beneficial for good Sender if her payoff is not too concave. Therefore, for a given process satisfying Assumption 1, Proposition 1 continues to hold if bad Sender is not too risk-loving and good Sender is not too risk-averse. In fact, Proposition 1 continues to hold verbatim if bad Sender’s payoff is weakly concave and good Sender’s payoff is weakly convex (the proof in Appendix B explicitly allows for this possibility).17 Much less can be said in general if the payoff functions $$v_{G}$$ and $$v_{B}$$ have an arbitrary shape. For example, if $$v_{G}$$ is sufficiently concave, then good Sender can prefer to delay pulling the arm to reduce the spread in posterior beliefs. Likewise, if $$v_{B}$$ is sufficiently convex, then bad Sender can prefer to pull the arm earlier than good Sender to increase the spread in posterior beliefs. These effects work against our credibility-scrutiny trade-off and Proposition 1 no longer holds.18 Nevertheless, bad Sender weakly delays pulling the arm relative to good Sender under the following single crossing assumption. Assumption 2. For all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$ and $$\mu\left(\tau\right),\mu\left(\tau^{\prime}\right)\in\left(0,1\right)$$, if bad Sender weakly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$, then good Sender strictly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$. This assumption holds in the benchmark model by Lemma 2. This assumption also holds if Sender’s payoff is the step function in (5) whenever pulling the arm later reveals strictly less useful information about Sender’s type, in the sense that Receiver is strictly worse off.19 Lemma 2′. (Good Sender’s Behaviour).Let $$v_{\theta}$$ be given by (5). If for all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$ \begin{equation} \int_{r}^{1}H\left(s\mid\tau^{\prime},m\right)ds>\int_{r}^{1}H\left(s\mid\tau,m\right)ds\,\it{ for\, all }\,m\in\left(0,1\right),\label{eq:anton} \end{equation} (6)then Assumption 2 holds. Proof. In Appendix B. ǁ If Assumption 2 holds and $$v_{\theta}$$ is strictly increasing, then in the unique divine equilibrium, good Sender pulls the arm as soon as it arrives and bad Sender pulls the arm weakly later than good Sender—there may exist an equilibrium in which both good and bad Sender pull the arm as soon as it arrives.20 3.2.2. Imperfectly informed Sender In many applications, Sender does not know with certainty whether pulling the arm would start a good or bad learning process for Receiver. For example, when announcing the reopening of the Clinton investigation, Director Comey could not know for certain what the results of the investigation would eventually be. We generalize our model to allow for Sender to only observe a signal $$\sigma\in\left\{ \sigma_{B},\sigma_{G}\right\} $$ about an underlying binary state $$\theta$$, with normalization \[ \sigma_{G}=\Pr\left(\theta=G\mid\sigma_{G}\right)>\pi>\Pr\left(\theta=G\mid\sigma_{B}\right)=\sigma_{B}. \] The statistical properties of Lemma 1 still hold. Lemma 1″. (Statistical Properties).Let $$\mathbb{E}\left[s\mid\tau,m,{{\sigma}}\right]$$ be the expectation of Receiver’s posterior belief $$s$$ conditional on the pulling time $$\tau$$, Receiver’s interim belief $$m$$, and Sender’s signal $$\sigma$$. For all $$\tau,\tau^{\prime}\in\left\{ 1,\ldots,T+1\right\} $$ such that $$\tau<\tau^{\prime}$$, and all $$m,m^{\prime}\in\left(0,1\right]$$ such that $$m<m^{\prime}$$, (1)$$\mathbb{E}\left[s\mid\tau,m^{\prime},\sigma\right]>\mathbb{E}\left[s\mid\tau,m,\sigma\right]$$; (2)$$\mathbb{E}\left[s\mid\tau^{\prime},m,\sigma_{B}\right]>\mathbb{E}\left[s\mid\tau,m,\sigma_{B}\right]$$; (3)$$\mathbb{E}\left[s\mid\tau,m,\sigma_{G}\right]>\mathbb{E}\left[s\mid\tau^{\prime},m,\sigma_{G}\right]$$. Proof. In Appendix A. ǁ These statistical results ensure that credibility is always beneficial for Sender, whereas scrutiny is detrimental for Sender with signal $$\sigma_{B}$$ but beneficial for Sender with signal $$\sigma_{G}$$. Therefore, all our results carry over. Moreover, we can extend our analysis to allow for signal $$\sigma$$ to be continuously distributed on the interval $$\left[\underline{{\sigma}},\bar{\sigma}\right)$$, with normalization $$\sigma=\Pr\left(\theta=G\mid\sigma\right)$$. In particular, in this case, there exists a partition equilibrium with $$\bar{\sigma}=\sigma_{0}>\sigma_{1}>\dots>\sigma_{T+1}=\underline{{\sigma}}$$ such that Sender $$\sigma\in\left[\sigma_{t},\sigma_{t-1}\right)$$ pulls the arm as soon as it arrives unless it arrives before time $$t\in\left\{ 1,\dots,T+1\right\} $$ (and pulls the arm at time $$t$$ if it arrives before $$t$$). 3.2.3. Type-dependent arrival of the arm In many applications, it is more reasonable to assume that the distribution of the arrival of the arm differs for good and bad Sender. For example, fake scandals may be easy to fabricate, whereas real scandals need time to be discovered. We generalize the model to allow for different distributions of the arrival of the arm for good and bad Sender. In particular, the arm arrives at a random time according to distributions $$F_{G}=F$$ for good Sender and $$F_{B}$$ for bad Sender. The proof of Proposition 1 (in Appendix B) explicitly allows for the arm to arrive (weakly) earlier to bad Sender than to good Sender in the first-order stochastic dominance sense: $$F_{B}\left(t\right)\geq F_{G}\left(t\right)$$ for all $$t$$. This assumption is clearly satisfied if bad Sender has the arm from the outset or if bad and good Sender receive the arm at the same time. More generally, Proposition 1 continues to hold verbatim unless the arm arrives sufficiently later to bad Sender than to good Sender such that $$F_{B}\left(t\right)<P_{B}\left(t\right)$$ for some $$t$$, where $$P_{B}\left(t\right)$$ is given by (4). But even then, Corollary 1 still holds. That is, bad Sender pulls the arm strictly later than good Sender. Yet, bad Sender may do so for the simple mechanical (rather than strategic) reason that the arm arrives to her later than to good Sender.21 3.2.4. Stochastic deadline In the benchmark model, we assume that the deadline $$T$$ is fixed and common knowledge. In some applications, the deadline $$T$$ may be stochastic. In particular, suppose that $$T$$ is a random variable distributed on $$\left\{ 1,\dots,\bar{T}\right\} $$ where time runs from $$1$$ to $$\bar{T}+1$$. Now the process $$L$$ has $$T$$ as a random variable rather than a constant. For this process, we can define the ex-ante distribution $$H$$ of posteriors at $$T$$, where $$H$$ depends only on pulling time $$\tau$$ and interim belief $$m$$. Notice that Assumption 1 still holds for this ex-ante distribution of posteriors for any $$\tau,\tau^{\prime}\in\left\{ 1,\dots,\bar{T}+1\right\} $$. Therefore, from the ex-ante perspective, Sender’s problem is identical to the problem with a deterministic deadline and all results carry over. 4. Poisson Model To get more precise predictions about the strategic timing of information release, we now assume that the arrival of the arm and Receiver’s learning follow Poisson processes. In this Poisson model, time is continuous $$t\in\left[0,T\right]$$.22 The arm arrives to Sender at Poisson rate $$\alpha$$, so that $$F\left(t\right)=1-e^{-\alpha t}$$. Once Sender pulls the arm, a breakdown occurs at Poisson rate $$\lambda$$ if Sender is bad, but never occurs if Sender is good, so that $$H\left(.\mid\tau,m\right)$$ puts probability $$\left(1-m\right)\left(1-e^{-\lambda\left(T-\tau\right)}\right)$$ on $$s=0$$ and the complementary probability on \[ s=\frac{m}{m+\left(1-m\right)e^{-\lambda\left(T-\tau\right)}}. \] Returning to our main application, the Poisson model assumes that scandals can be conclusively debunked, but cannot be proven real. It also assumes that the opposition receives the scandalous material against the president at a constant rate, independent of whether they are real or fake. As discussed in Section 3.2.3, the results would not change if real documents take more time to be discovered than fake document take to be fabricated. Our benchmark model does not completely nest the Poisson model. In fact, part (ii) of Assumption 1, that the learning process never fully reveals Sender’s type, fails in the Poisson model because a breakdown fully reveals that Sender is bad. Nevertheless, if only part (i) of Assumption 1 is satisfied, a version of Proposition 1 continues to hold, with the difference that bad Sender never pulls the arm before some time $$\bar{t}$$. Specifically, Proposition 1 holds for all $$\tau\geq\bar{t}$$, whereas $$\mu\left(\tau\right)=1$$ and $$P_{B}\left(\tau\right)=0$$ for all $$\tau<\bar{t}$$. Intuitively, even if Receiver believes that only good Sender pulls the arm before $$\bar{t}$$, bad Sender strictly prefers to pull the arm after $$\bar{t}$$ to reduce the risk that Receiver fully learns that Sender is bad. We can, therefore, explicitly characterize the divine equilibrium of the Poisson model. First, good Sender pulls the arm as soon as it arrives.23 Second, bad Sender is indifferent between pulling the arm at any time $$t\geq\bar{t}\geq0$$ and not pulling it at all. Third, bad Sender strictly prefers to delay pulling the arm if $$t<\bar{t}$$. In the divine equilibrium of the Poisson model, $$\mu\left(t\right)=1$$ for all $$t<\bar{t}$$, and equations (2) and (3) become \begin{eqnarray*} \frac{\mu\left(t\right)e^{-\lambda\left(T-t\right)}}{\mu\left(t\right)+\left(1-\mu\left(t\right)\right)e^{-\lambda\left(T-t\right)}} & = & \mu\left(T\right)\text{ for all }t\ge\bar{t},\\ \int_{0}^{T}\alpha\frac{1-\mu\left(t\right)}{\mu\left(t\right)}e^{-\alpha t}dt+\frac{1-\mu\left(T\right)}{\mu\left(T\right)}e^{-\alpha T} & = & \frac{1-\pi}{\pi}. \end{eqnarray*} Adding the boundary condition $$\lim_{t\downarrow\bar{t}}\mu\left(t\right)=1$$ yields the explicit solution $$\mu\left(t\right)$$ and uniquely determines $$\bar{t}$$. Proposition 2. In the divine equilibrium, good Sender pulls the arm as soon as it arrives and Receiver’s interim belief that Sender is good given pulling time $$t$$ is: \[ \mu\left(t\right)=\begin{cases} \frac{\mu\left(T\right)}{1-\mu\left(T\right)\left(e^{\lambda\left(T-t\right)}-1\right)} & {\it{if }}\,t\geq\bar{t};\\ 1 & {\it{otherwise,}} \end{cases} \] where $$\mu\left(T\right)$$ is Receiver’s posterior belief if the arm is never pulled and \[ \bar{t}=\begin{cases} 0 & {\it{if }}\,\,\pi<\bar{\pi};\\ T-\frac{1}{\lambda}\ln\frac{1}{\mu\left(T\right)} & {\it{otherwise,}} \end{cases} \] \begin{eqnarray*} \mu\left(T\right) & = & \left\{ \begin{array}{l} \left[\frac{\alpha e^{\lambda T}+\lambda e^{-\alpha T}}{\alpha+\lambda}+\frac{1-\pi}{\pi}\right]^{-1}\,{\it{ if }}\,\,\pi<\bar{\pi};\\ \left[\frac{\left(\alpha+\lambda\right)\left(1-\pi\right)}{\lambda\pi}e^{\alpha T}+1\right]^{-\frac{\lambda}{\alpha+\lambda}}\,\it{ otherwise,} \end{array}\right.\\ \bar{\pi} & = & \left[1+\frac{\lambda}{\alpha+\lambda}\left(e^{\lambda T}-e^{-\alpha T}\right)\right]^{-1}. \end{eqnarray*} The parameters of the model affect welfare directly and through Sender’s equilibrium behavior. Proposition 3 says that, in the divine equilibrium, direct effects dominate. Specifically, a higher prior belief $$\pi$$ results in higher posterior beliefs, which increases both bad and good Sender’s welfare. Moreover, a higher breakdown rate $$\lambda$$ or a higher arrival rate $$\alpha$$ allows Receiver to learn more about Sender, which decreases (increases) bad (good) Sender’s welfare. Proposition 3 also derives comparative statics on Receiver’s welfare given by (1).24 Proposition 3. In the divine equilibrium, (1)the expected payoff of bad Sender increases with $$\pi$$ but decreases with $$\lambda$$ and $$\alpha$$; (2)the expected payoff of good Sender increases with $$\pi$$, $$\lambda$$, and $$\alpha$$; (3)the expected payoff of Receiver decreases with $$\pi$$ but increases with $$\lambda$$ and $$\alpha$$. Proof. In Appendix C. ǁ 4.1. Static analysis We now explore how the parameters of the model affect the probability that Sender releases information. The probability that bad Sender pulls the arm is \begin{equation} P_{B}\left(T\right)=1-\frac{\pi}{1-\pi}\frac{1-\mu\left(T\right)}{\mu\left(T\right)}e^{-\alpha T},\label{q} \end{equation} (7) which follows from \begin{equation} \mu\left(T\right)=\frac{\pi e^{-\alpha T}}{\pi e^{-\alpha T}+\left(1-\pi\right)\left(1-P_{B}\left(T\right)\right)}.\label{eq:pi(T)} \end{equation} (8) Proposition 4 says that bad Sender pulls the arm with a higher probability if the prior belief $$\pi$$ is lower, if the breakdown rate $$\lambda$$ is lower, or if the arrival rate $$\alpha$$ is higher. Proposition 4. In the divine equilibrium, the probability that bad Sender pulls the arm decreases with $$\pi$$ and $$\lambda$$ but increases with $$\alpha$$. Proof. In Appendix C. ǁ Intuitively, if the prior belief $$\pi$$ is higher, bad Sender has more to lose in case of a breakdown. Similarly, if the breakdown rate $$\lambda$$ is higher, pulling the arm is more likely to reveal that Sender is bad. In both cases, bad Sender is more reluctant to pull the arm. In contrast, if the arrival rate $$\alpha$$ is higher, good Sender is more likely to pull the arm and Receiver will believe that Sender is bad with higher probability if she does not pull the arm. In this case, bad Sender is more willing to pull the arm. The total probability that Sender pulls the arm is given by the weighted sum of the probabilities $$P_{B}\left(T\right)$$ and $$P_{G}\left(T\right)$$ that bad Sender and good Sender pull the arm: \begin{equation} P_{B}\left(T\right)=\pi P_{G}\left(T\right)+\left(1-\pi\right)P_{B}\left(T\right)=1-\frac{\pi e^{-\alpha T}}{\mu\left(T\right)}.\label{P} \end{equation} (9) A change in $$\lambda$$ affects $$P_{B}\left(T\right)$$, but not $$P_{G}\left(T\right)$$; a change in $$\alpha$$ affects both $$P_{B}\left(T\right)$$ and $$P_{G}\left(T\right)$$ in the same direction. Therefore, Sender pulls the arm with a higher total probability if the breakdown rate is lower or if the arrival rate is higher. The prior belief $$\pi$$ has a direct and an indirect effect on the total probability that Sender pulls the arm. On the one hand, holding $$P_{B}\left(T\right)$$ constant, $$P$$$$\left(T\right)$$ directly increases with $$\pi$$, because $$P_{G}\left(T\right)>P_{B}\left(T\right)$$. On the other hand, $$P\left(T\right)$$ indirectly decreases with $$\pi$$, because $$P_{B}\left(T\right)$$ decreases with $$\pi$$. Proposition 5 says that the indirect effect dominates the direct effect when $$\pi$$ is sufficiently low. Proposition 5. In the divine equilibrium, the total probability that Sender pulls the arm decreases with $$\lambda$$, increases with $$\alpha$$, and is quasiconvex in $$\pi$$: decreases with $$\pi$$ if \[ \pi<\frac{\alpha e^{\alpha T}}{\alpha e^{\alpha T}+\lambda\left(e^{\alpha T}-1\right)}\in\left(0,1\right) \] and increases with $$\pi$$ otherwise. Proof. In Appendix C. ǁ The probabilities $$P_{G}\left(T\right)=1-e^{-\alpha T}$$ and $$P_{B}\left(T\right)$$ that good Sender and bad Sender pull the arm also determine Receiver’s posterior belief $$\mu\left(T\right)$$. By (8), $$\mu\left(T\right)$$ decreases with the breakdown rate $$\lambda$$, because $$P_{B}\left(T\right)$$ decreases with $$\lambda$$. Equation (8) also suggests that there are direct and indirect effects of the prior belief $$\pi$$ and the arrival rate $$\alpha$$ on $$\mu\left(T\right)$$. On the one hand, holding $$P_{B}\left(T\right)$$ constant, $$\mu\left(T\right)$$ directly increases with $$\pi$$ and decreases with $$\alpha$$. On the other hand, $$\mu\left(T\right)$$ indirectly decreases with $$\pi$$ and increases with $$\alpha$$, because $$P_{B}\left(T\right)$$ decreases with $$\pi$$ and increases with $$\alpha$$. Proposition 6 says that the direct effect always dominates the indirect effect in the Poisson model. Proposition 6. In the divine equilibrium, Receiver’s posterior belief if the arm is never pulled increases with $$\pi$$ but decreases with $$\lambda$$ and $$\alpha$$. Proof. In Appendix C. ǁ 4.2. Dynamic analysis The Poisson model also allows for a more detailed analysis of the strategic timing of information release. By Proposition 2, bad Sender begins to pull the arm at time $$\bar{t}$$. In the spirit of Proposition 4, bad Sender begins to pull the arm later if the prior belief $$\pi$$ is higher, if the breakdown rate $$\lambda$$ is higher, or if the arrival rate $$\alpha$$ is lower. Proposition 7. In the divine equilibrium, $$\bar{t}$$ increases with $$\pi$$ and $$\lambda$$ but decreases with $$\alpha$$. Proof. In Appendix C. ǁ At each time $$t$$ after $$\bar{t}$$, bad Sender pulls the arm with a strictly positive probability density $$p_{B}\left(t\right)$$ (Figure 2a). Proposition 8 says that $$p_{B}\left(t\right)$$ first increases and then decreases with time. Figure 2 View largeDownload slide Pulling density and breakdown probability; $$\alpha=1$$, $$\lambda=2$$, $$\pi=.5$$, $$T=1$$. (a) dotted: $$p_{G}\left(t\right)$$; solid: $$p_{B}\left(t\right)$$; dashed: $$p\left(t\right)$$; (b) dotted: $$\lambda\left(T-t\right)$$; solid: $${p_{B}\left(t\right)}/{p_{G}\left(t\right)}$$; dashed: $$Q\left(t\right)$$. Figure 2 View largeDownload slide Pulling density and breakdown probability; $$\alpha=1$$, $$\lambda=2$$, $$\pi=.5$$, $$T=1$$. (a) dotted: $$p_{G}\left(t\right)$$; solid: $$p_{B}\left(t\right)$$; dashed: $$p\left(t\right)$$; (b) dotted: $$\lambda\left(T-t\right)$$; solid: $${p_{B}\left(t\right)}/{p_{G}\left(t\right)}$$; dashed: $$Q\left(t\right)$$. Proposition 8. In the divine equilibrium, the probability density that bad Sender pulls the arm at time $$t$$ is quasiconcave: increases with $$t$$ if \[ t<t_{b}\equiv T-\frac{1}{\lambda}\ln\left(\frac{\alpha}{\alpha+\lambda}\frac{1}{\mu\left(T\right)}\right) \] and decreases with $$t$$ otherwise. Proof. In Appendix C. ǁ Intuitively, the dynamics of $$p_{B}\left(t\right)$$ are driven by a strategic and a mechanic force. Strategically, as in Corollary 1, bad Sender delays pulling the arm with respect to good Sender, so that the likelihood ratio $$p_{B}\left(t\right)/p_{G}\left(t\right)$$ increases with time, where $$p_{G}\left(t\right)=\alpha e^{-\alpha t}$$ is the probability density that good Sender pulls the arm at time $$t$$. Mechanically, $$p_{B}\left(t\right)$$ roughly follows the dynamics of $$p_{G}\left(t\right)$$. If the arrival rate $$\alpha$$ is sufficiently small, so that the density $$p_{G}\left(t\right)$$ barely changes over time, the strategic force dominates and the probability that bad sender pulls the arm monotonically increases with time ($$t_{b}>T$$). Instead, if the arrival rate $$\alpha$$ is sufficiently large, so that $$p_{G}\left(t\right)$$ rapidly decreases over time, the mechanic force dominates and the probability that bad sender pulls the arm monotonically decreases with time ($$t_{b}<T$$). The total probability density $$p\left(t\right)$$ that Sender pulls the arm is a weighted sum of $$p_{G} \left(t\right)$$ and $$p_{B}\left(t\right)$$, so that $$p\left(t\right)=\pi p_{G}\left(t\right)+\left(1-\pi\right)p_{B}\left(t\right)$$ (Figure 2a). Therefore, until $$\bar{t}$$, $$p\left(t\right)=\pi p_{G}\left(t\right)$$, and thereafter, as in Proposition 8, $$p\left(t\right)$$ first increases and then decreases with time. Proposition 9. In the divine equilibrium, the total probability density that Sender pulls the arm at time $$t$$ decreases with $$t$$ from $$0$$ to $$\bar{t}$$ and is quasiconcave in $$t$$ on the interval $$\left[\bar{t},T\right]$$: increases with $$t$$ if \[ \bar{t}<t<t_{s}\equiv T-\frac{1}{\lambda}\ln\left(\frac{\alpha}{\alpha+\lambda}\frac{1+\mu\left(T\right)}{\mu\left(T\right)}\right) \] and decreases with $$t$$ if $$t>t_{s}$$. Proof. In Appendix C. ǁ Let the breakdown probability$$Q\left(t\right)$$ be the probability that a breakdown occurs before the deadline given that the arm is pulled at time $$t$$ (Figure 2b). Proposition 10 says that, as time passes, the breakdown probability first increases and then decreases.25 Proposition 10. In the divine equilibrium, the breakdown probability is quasiconcave: increases with $$t$$ if \[ t<t_{b}\equiv T-\frac{1}{\lambda}\ln\left(\frac{1+\mu\left(T\right)}{2\mu\left(T\right)}\right)<T \] and decreases with $$t$$ otherwise. Proof. In Appendix C. ǁ Intuitively, the breakdown probability increases with the amount of scrutiny and with the likelihood ratio $$p_{B}\left(t\right)/p_{G}\left(t\right)$$. Obviously, Sender is exposed to more scrutiny if she pulls the arm earlier. But the likelihood ratio $$p_{B}\left(t\right)/p_{G}\left(t\right)$$ is lower earlier, because bad Sender strategically delays pulling the arm. Proposition 10 says that this strategic effect dominates for earlier times. 5. Applications 5.1. U.S. presidential scandals Returning to our presidential scandals example, the main prediction of our model is that fake scandals are released later than real scandals. We explore this prediction using Nyhan’s (2015) data on U.S. presidential scandals from 1977 to 2008. For each week, the data report whether a new scandal involving the current U.S. president was first mentioned in the Washington Post. Although scandals might have first appeared on other outlets, we agree with Nyhan that the Washington Post is likely to have mentioned such scandals immediately thereafter. As our model concerns scandals involving the incumbent in view of his possible reelection, we focus on all the presidential elections in which the incumbent was a candidate. Therefore we consider only the first term of each president from 1977 to 2008, beginning on the first week of January after the president’s election.26,27 In all cases, the election was held on the 201st week after this date. We construct the variable weeks to election as the difference between the election week at the end of the term and the release week of the scandal. For each scandal,28 we locate the original Washington Post article as well as other contemporary articles on The New York Times and the Los Angeles Times. We then search for subsequent articles on the same scandal in following years until 2016, as well as court decisions and scholarly books when possible. We check whether factual evidence of wrongdoing or otherwise reputationally damaging conduct was conclusively verified at a later time. If so, we check whether the evidence involved the president directly or close family members or political collaborators chosen or appointed by the president or his administration. We code these scandals as real. For the remaining scandals, we check whether a case for libel was successful or all political actors linked to the scandal were cleared of wrongdoings. We code these scandals as fake. The only scandal we were not able to code by this procedure is the “Banca Nazionale del Lavoro” scandal (also known as “Iraq-gate”). We code this scandal as real, but we check in Online Appendix A that all our qualitative results are robust to coding it as fake. In Online Appendix A, we report the complete list of scandals and a summary motivation of our coding decisions. Figure 3 shows the empirical distributions of the first mention of real and fake presidential scandals in the Washington Post as a function of weeks to election. Although we do not observe scandals released after the election ($$t=T+1$$ in our model) and cannot pinpoint the date at which the campaign begins ($$t=1$$ in our model), Corollary 1 implies that fake scandals are released later than real scandals conditional on any given time interval. The left panel covers the whole presidential term; the right panel focuses on the election campaign period only, which we identify with the last 60 weeks before the election. Both figures suggest that fake scandals are released later than real scandals. Because of the small sample size (only 15 scandals), formal tests have low power. Nevertheless, using the Dardanoni and Forcina (1998) test for the likelihood ratio order (which implies first-order stochastic dominance), we almost reject the hypothesis that the two distributions are equal in favour of the alternative hypothesis that fake scandals are released later ($$p$$-value: $$0.114$$); we cannot reject the hypothesis that fake scandals are released later in favour of the unrestricted hypothesis at all standard statistical significance levels ($$p$$-value: $$0.834$$).29 Figure 3 View largeDownload slide US presidential scandals and weeks to election. Distribution of real and fake scandals. (a) whole term; (b) last 60 weeks only. Figure 3 View largeDownload slide US presidential scandals and weeks to election. Distribution of real and fake scandals. (a) whole term; (b) last 60 weeks only. Our Poisson model offers a novel perspective over the October surprise concentration of scandals towards the end of the presidential election campaign (Figure 1). In equilibrium, real scandals are released as they are discovered by the media. Unless real scandals are more likely to be discovered towards the end of the first term of a president, then we should not expect their release to be concentrated towards the end of the campaign (see $$p_{G}\left(t\right)$$ in Figure 2a). Instead, fake scandals are strategically delayed, and so they should be concentrated towards the end of the first term of the president and just before the election (see $$p_{B}\left(t\right)$$ in Figure 2a). In other words, our model predicts that the October surprise effect is driven by fake scandals. In contrast, were the October surprise effect driven by the desire to release scandals when they are most salient, then the timing of release of real and fake scandals would be similar. Figure 4 is a replica of Figure 1, but with scandals coded as real and fake. Fake scandals are concentrated close to the election, with a majority of them released in the last quarter before the election. In contrast, real scandals appear to be scattered across the entire presidential term. Figure 4 View largeDownload slide Distribution of real and fake scandals. Figure 4 View largeDownload slide Distribution of real and fake scandals. Our Poisson model also predicts how different parameters affect the release of a U.S. presidential scandal. We now illustrate how Nyhan’s (2015) empirical findings may be interpreted using our model. Nyhan (2015) finds that scandals are more likely to appear when the president’s opposition approval rate is low. In our model, the approval rate is most naturally captured by the prior belief $$1-\pi$$ (the belief that the president is fit to run the country). In our Poisson model, a higher $$\pi$$ has a direct and an indirect effect on the probability of release of a scandal. On the one hand, a higher $$\pi$$ means that the president is more likely to be involved in a real scandal, thus directly increasing the probability that such a scandal is released. On the other hand, the opposition optimally resorts to fake scandals more when the president is so popular that only a scandal could prevent the president’s reelection. Therefore, a higher $$\pi$$ reduces the incentive for the opposition to release fake scandals, indirectly reducing the probability that a scandal is released. We can then interpret Nyhan’s finding as suggesting that the direct effect on average dominates the indirect effect. But the president’s opposition approval rate also measures opposition voters’ hostility towards the president, which might be captured by the rate $$\lambda$$ at which voters learn that a scandal is fake.30 Indeed, Nyhan conjectures that when opposition voters are more hostile to the president, then they are “supportive of scandal allegations against the president and less sensitive to the evidentiary basis for these claims [and] opposition elites will be more likely to pursue scandal allegations” (p. 6). Consistently, in our Poisson model, when voters take more time to tell real and fake scandals apart, the opposition optimally resorts to fake scandals. Nyhan (2015) also finds that fewer scandals involving the president are released when the news agenda is more congested. Such media congestion may have the following two effects. First, when the news agenda is congested, the opposition media has less time to devote to investigate the president. In our Poisson model, this is captured by a lower arrival rate $$\alpha$$, which in turn reduces the probability that a scandal is released. Second, when the news agenda is congested, public scrutiny of the scandal is slower as the attention of media, politicians, and voters are captured by other events. In our Poisson model, this is captured by a lower breakdown rate $$\lambda$$, which in turn increases the probability that a scandal is released. We can interpret Nyhan’s finding as suggesting that the media congestion effect through the arrival rate $$\alpha$$ dominates the effect through the breakdown rate $$\lambda$$.31 5.2. Initial public offerings We now apply our model to the timing of IPOs. Sender is a firm that needs liquidity in a particular time frame, and this time frame is private information of the firm. The need for liquidity could arise from the desire to grow the firm, expand into new products or markets, or because of operating expenses outstripping revenues. It could also arise because of investors having so-called “drag-along rights”, where they can force founders and other shareholders to vote in favour of a liquidity event. When announcing the IPO, firms have private information regarding their prospective long-run performance. Good firms expect their business to out-perform the market’s prior expectation; bad firms expect their business to under-perform the market’s prior expectation. After a firm announces an IPO, the market scrutinizes the firm’s prospectus, and learns about the firm’s prospective performance. The initial trade closing price of the stock is determined by the market’s posterior belief at the initial trade date. Therefore, after the initial trade date, as the firms’ potential is gradually revealed to the market, good firms’ stocks out-perform bad firms’ stocks. Since the true time frame is private information, the firm can “pretend” to need liquidity faster than it actually does, and it has significant control over the time gap between the announcement of the IPO and the initial trade date. A shorter time gap decreases the amount of scrutiny the firm undergoes before going public, but also reduces credibility. Therefore, our model predicts that bad firms should choose a shorter time gap than good firms.32 We explore this prediction using data on U.S. IPOs from 1983 to 2016. For each IPO, we record the time gap and calculate the cumulative return of the stock, starting from the initial trade date. We measure the stock’s performance as its return relative to the market return over the same period. Following Loughran and Ritter (1995), we evaluate IPOs’ long-run performance $$y\in\left\{ 3,5\right\} $$ years after the initial trade date. For each value of $$y$$, we code as good (bad) those IPOs that performed above (below) market.33 Figure 5 shows the empirical distributions of time gap for good and bad IPOs evaluated at 3 and 5 years after the initial trade date. Both figures suggest that bad firms choose a shorter time gap in the first-order stochastic dominance sense, but with the effect being more clearly visible after 5 years. This pattern is consistent with our idea that firms’ private information is only gradually (and slowly) revealed to the market once the period of intense scrutiny of the IPO ends. Figure 5 View largeDownload slide US IPOs and time gap. Distributions for good and bad IPOs. (a) 3 years; (5) years. Figure 5 View largeDownload slide US IPOs and time gap. Distributions for good and bad IPOs. (a) 3 years; (5) years. Our main prediction in Corollary 1 is that the distribution of time gap for good IPOs dominates the distribution of time gap for bad IPOs in the likelihood ratio order.34 We evaluate this prediction using an approach developed by Dardanoni and Forcina (1998). This approach tests (1) the hypothesis $$H_{0}$$ that the distributions are identical against the alternative $$H_{1}$$ that the distributions are ordered in the likelihood ratio order; as well as (2) the hypothesis $$H_{1}$$ against an unrestricted alternative $$H_{2}$$. The hypothesis of interest $$H_{1}$$ is accepted if the first test rejects $$H_{0}$$ and the second test fails to reject $$H_{1}$$. Following Roosen and Hennessy (2004), we partition the variable time gap into $$k$$ intervals that are equiprobable according to the empirical distribution of time gap. We report in Table 1 the $$p$$-values of the two statistics for the case of $$k=7$$. Table 1 Dardanoni and Forcina test for likelihood ratio order (p-values) 3 years 5 years H0 versus H1 0.001 0.000 H1 versus H2 0.000 0.361 Obs. 529 403 3 years 5 years H0 versus H1 0.001 0.000 H1 versus H2 0.000 0.361 Obs. 529 403 Table 1 Dardanoni and Forcina test for likelihood ratio order (p-values) 3 years 5 years H0 versus H1 0.001 0.000 H1 versus H2 0.000 0.361 Obs. 529 403 3 years 5 years H0 versus H1 0.001 0.000 H1 versus H2 0.000 0.361 Obs. 529 403 For both 3 and 5 years performance, we reject the hypothesis $$H_{0}$$ in favour of $$H_{1}$$ at the 1% significance level. Furthermore, for 5 years performance, we cannot reject the hypothesis $$H_{1}$$ in favour of $$H_{2}$$ at all standard significance levels. In Online Appendix B we give some further details about our data and the test, and we explore how the results of the test may change under alternative specifications. 6. Concluding Remarks We have analysed a model in which the strategic timing of information release is driven by the trade-off between credibility and scrutiny. The analysis yields novel predictions about the dynamics of information release. We also offered supporting evidence for these predictions using data on the timing of U.S. presidential scandals and the announcement of IPOs. Our model can also be used to deliver normative implications for the design of a variety of institutions. In the context of election campaigns, our results could be employed to evaluate laws that limit the period in which candidates can announce new policies in their platforms or media can cover candidates. For example, more than a third of the world’s countries mandate a blackout period before elections: a ban on political campaigns or, in some cases, on any mention of a candidate’s name, for one or more days immediately preceding elections.35 The framework we have developed has further potential applications. For instance, the relationship between a firm’s management team and its board of directors often exhibits the core features of our model: management has private information and potentially different preferences than the board; the board’s view about a project or investment determines whether it is undertaken; and management can provide more or less time to the board in evaluating the project or investment. The comparative statics of our model may speak to how this aspect of the management–board relationship may vary across industries and countries. Similarly, in various legal settings an interested party with private information may come forward sooner or later, notwithstanding an essentially fixed deadline for the legal decision-maker (due to institutional or resource constraints). A natural example is witnesses in a criminal investigation, but the same issues often arise in civil matters or even parliamentary inquiries. In each of these applications, the credibility-scrutiny trade-off plays an important role, and we hope our model, characterization of equilibrium, and comparative statics will serve as a useful framework for studying them in the future. The editor in charge of this paper was Nicola Gennaioli. APPENDIX A. Statistical Properties Proof of Lemma 1. Follows from Lemma 1$$'$$. ∥ Proof of Lemma 1$$'$$. Part 1. By Blackwell (1953), Assumption 1 with $$\tau^{\prime}=T+1$$ implies that pulling the arm at $$\tau$$ is the same as releasing an informative signal $$y$$. By Bayes’s rule, posterior $$s$$ is given by: \[ s=\frac{mq\left(y\mid G\right)}{mq\left(y\mid G\right)+\left(1-m\right)q\left(y\mid B\right)}, \] where $$q\left(y\mid\theta\right)$$ is the density of $$y$$ given $$\theta$$. (If $$L$$ is discrete, then $$q\left(y\mid\theta\right)$$ is the discrete density of $$y$$ given $$\theta$$.) Therefore, \begin{equation} \frac{q\left(y\mid G\right)}{q\left(y\mid B\right)}=\frac{1-m}{m}\frac{s}{1-s}. \end{equation} (A10) Writing (A10) for interim beliefs $$m$$ and $$m^{\prime}$$, we obtain the following relation for corresponding posterior beliefs $$s$$ and $$s^{\prime}$$: \[ \frac{1-m^{\prime}}{m^{\prime}}\frac{s^{\prime}}{1-s^{\prime}}=\frac{1-m}{m}\frac{s}{1-s}, \] which implies that \begin{equation} s^{\prime}=\frac{\frac{m^{\prime}s}{m}}{\frac{m^{\prime}s}{m}+\frac{\left(1-m^{\prime}\right)\left(1-s\right)}{1-m}}. \end{equation} (A11) Therefore, $$s^{\prime}>s$$ for $$m^{\prime}>m$$; so part 1 follows. Part 2. By Blackwell (1953), Assumption 1 implies that pulling the arm at $$\tau$$ is the same as pulling the arm at $$\tau^{\prime}$$ and then releasing an additional informative signal $$y$$ with conditional density $$q\left(y\mid\theta\right)$$. Part 2 holds because for any strictly increasing concave $$v_{B}$$, we have \begin{eqnarray*} \mathbb{E}\left[v_{B}\left(s\right)\mid\tau,m,B\right] & = & \mathbb{E}\left[v_{B}\left(\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\right)\mid\tau^{\prime},m,B\right]\\ & = & \mathbb{E}\left[\mathbb{E}\left[v_{B}\left(\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\right)\mid\tau^{\prime},s,B\right]\mid\tau^{\prime},m,B\right]\\ & \leq & \mathbb{E}\left[v_{B}\left(\mathbb{E}\left[\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,B\right]\right)\mid\tau^{\prime},m,B\right]\\ & < & \mathbb{E}\left[v_{B}\left(\frac{s\mathbb{E} \left[\displaystyle\frac{q\left(y\mid G\right)}{q\left(y\mid B\right)}\mid\tau^{\prime},s,B\right]}{s\mathbb{E}\left[\displaystyle\frac{q\left(y\mid G\right)}{q\left(y\mid B\right)}\mid\tau^{\prime},s,B\right]+\left(1-s\right)}\right)\mid\tau^{\prime},m,B\right]\\ & = & \mathbb{E}\left[v_{B}\left(s\right)\mid\tau^{\prime},m,B\right], \end{eqnarray*} where the first line holds by Bayes’s rule, the second by the law of iterated expectations, the third by Jensen’s inequality applied to concave $$v_{B}$$, the fourth by strict monotonicity of $$v_{B}$$ and Jensen’s inequality applied to function $$sz/\left(sz+1-s\right)$$ which is strictly concave in $$z$$, and the last by definition of expectations. Part 3. Analogously to Part 2, Part 3 holds because for any strictly increasing convex $$v_{G}$$, we have \begin{eqnarray*} \mathbb{E}\left[v_{G}\left(s\right)\mid\tau,m,G\right] & = & \mathbb{E}\left[v_{G}\left(\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\right)\mid\tau^{\prime},m,G\right]\\ & = & \mathbb{E}\left[\mathbb{E}\left[v_{G}\left(\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\right)\mid\tau^{\prime},s,G\right]\mid\tau^{\prime},m,G\right]\\ & \geq & \mathbb{E}\left[v_{G}\left(\mathbb{E}\left[\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,G\right]\right)\mid\tau^{\prime},m,G\right]\\ & > & \mathbb{E}\left[v_{G}\left(\frac{s}{s+\left(1-s\right)\mathbb{E}\left[\displaystyle\frac{q\left(y\mid B\right)}{q\left(y\mid G\right)}\mid\tau^{\prime},s,G\right]}\right)\mid\tau^{\prime},m,G\right]\\ & = & \mathbb{E}\left[v_{G}\left(s\right)\mid\tau^{\prime},m,G\right]. \end{eqnarray*} ∥ Proof of Lemma 1$$''$$. The proof of part 1 is the same as in Lemma 1$$'$$. As noted before, pulling the arm at $$\tau$$ is the same as pulling the arm at $$\tau^{\prime}$$ and then releasing an additional informative signal $$y$$ with conditional density $$q\left(y\mid\theta\right)$$. Let $$s_{\sigma}$$ be the probability that Sender is good given that Receiver’s posterior is $$s$$ and Sender’s signal is $$\sigma$$. By (A11), \[ s_{\sigma}=\frac{\frac{\sigma s}{\pi}}{\frac{\sigma s}{m}+\frac{\left(1-\sigma\right)\left(1-s\right)}{1-m}}. \] We have, \begin{eqnarray*} \mathbb{E}\left[s\mid\tau,m,\sigma\right] & = & \mathbb{E}\left[\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},m,\sigma\right]\\ & = & \mathbb{E}\left[\mathbb{E}\left[\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,\sigma\right]\mid\tau^{\prime},m,\sigma\right]\\ & = & \mathbb{E}\left[\left.\begin{array}{c} s_{\sigma}\mathbb{E}\left[\displaystyle\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,G\right]+\\ +\left(1-s_{\sigma}\right)\mathbb{E}\left[\displaystyle\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,B\right] \end{array}\right|\tau^{\prime},m,\sigma\right]\\ & = & \mathbb{E}\left[\left.\begin{array}{c} s_{\sigma}\mathbb{E}\left[\displaystyle\frac{sq\left(y\mid G\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,G\right]+\\ +\left(1-s_{\sigma}\right)\mathbb{E}\left[\displaystyle\frac{sq\left(y\mid B\right)}{sq\left(y\mid G\right)+\left(1-s\right)q\left(y\mid B\right)}\mid\tau^{\prime},s,G\right] \end{array}\right|\tau^{\prime},m,\sigma\right]\\ & = & \mathbb{E}\left[s\mathbb{E}\left[\displaystyle\frac{s_{\sigma}+\left(1-s_{\sigma}\right)\displaystyle\frac{q\left(y\mid B\right)} {q\left(y\mid G\right)}}{s+\left(1-s\right)\displaystyle\frac{q\left(y\mid B\right)}{q\left(y\mid G\right)}}\mid\tau^{\prime},s,G\right]\mid\tau^{\prime},m,\sigma\right]\\ & \gtrless & \mathbb{E}\left[s\mid\tau^{\prime},m,\sigma\right]\mbox{ whenever }s_{\sigma}\gtrless s, \end{eqnarray*} where the last line holds by Jensen’s inequality applied to function $$\left(s_{\sigma}+\left(1-s_{\sigma}\right)z\right)/\left(s+\left(1-s\right)z\right)$$, which is strictly convex (concave) in $$z$$ whenever $$s_{\sigma}>s$$ ($$s_{\sigma}<s$$). Because $$\sigma_{G}>\pi>\sigma_{B}$$, we have $$s_{\sigma_{G}}>s>s_{\sigma_{B}}$$, so parts 2 and 3 follow. ∥ B. Benchmark Model To facilitate our discussion in Section 3.2, we prove our results under more general assumptions than in our benchmark model. First, we assume that $$v_{G}\left(s\right)$$ is continuous, strictly increasing, and (weakly) convex, and $$v_{B}\left(s\right)$$ is continuous, strictly increasing, and (weakly) concave. Second, we assume that the arm arrives at a random time according to distributions $$F_{G}=F$$ for good Sender and $$F_{B}$$ for bad Sender, where $$F_{B}\left(t\right)\geq F_{G}\left(t\right)$$ for all $$t$$. Proof of Lemma 2. Part 1. Suppose, on the contrary, that $$\mu\left(\tau\right)\leq\mu\left(\tau^{\prime}\right)$$. Then \begin{eqnarray*} \int v_{B}\left(s\right)dH_{B}\left(s|\tau,\mu\left(\tau\right)\right) & < & \int v_{B}\left(s\right)dH_{B}\left(s|\tau^{\prime},\mu\left(\tau\right)\right)\\ & \leq & \int v_{B}\left(s\right)dH_{B}\left(s|\tau^{\prime},\mu\left(\tau^{\prime}\right)\right), \end{eqnarray*} where the first inequality holds by part 2 of Lemma 1$$'$$ and the second by part 1 of Lemma 1$$'$$. Therefore, bad Sender strictly prefers to pull the arm at $$\tau^{\prime}$$ than at $$\tau$$. A contradiction. Good Sender strictly prefers to pull the arm at $$\tau$$ because \begin{eqnarray*} \int v_{G}\left(s\right)dH_{G}\left(s|\tau,\mu\left(\tau\right)\right) & > & \int v_{G}\left(s\right)dH_{G}\left(s|\tau^{\prime},\mu\left(\tau\right)\right)\\ & > & \int v_{G}\left(s\right)dH_{G}\left(s|\tau^{\prime},\mu\left(\tau^{\prime}\right)\right), \end{eqnarray*} where the first inequality holds by part 3 of Lemma 1$$'$$ and the second by $$\mu\left(\tau\right)>\mu\left(\tau^{\prime}\right)$$ and part 1 of Lemma 1$$'$$. Part 2. By part 1 of this lemma applied to $$\tau$$ and $$\tau^{\prime}=T+1$$, it suffices to show that $$\mu\left(\tau\right),\mu\left(T+1\right)\in\left(0,1\right)$$ for all $$\tau$$ in the support of $$P_{B}$$. First, by Bayes’s rule, $$\tau$$ being in the support of $$P_{B}$$ implies $$\mu\left(\tau\right)<1$$. Second, by Bayes’s rule, $$F_{G}\left(T\right)<1$$ implies $$\mu\left(T+1\right)>0$$. Third, $$\mu\left(T+1\right)>0$$ implies $$\mu\left(\tau\right)>0$$, otherwise $$\tau$$ could not be in the support of $$P_{B}$$ because $$v_{B}\left(\mu\left(T+1\right)\right)>v_{B}\left(0\right)=\mathbb{E}\left[v_{B}\left(s\right)\mid\tau,0,B\right]$$. Finally, $$\mu\left(\tau\right)<1$$ implies $$\mu\left(T+1\right)<1$$, otherwise $$\tau$$ could not be in the support of $$P_{B}$$ because $$v_{B}\left(1\right)>\mathbb{E}\left[v_{B}\left(s\right)\mid\tau,\mu\left(\tau\right),B\right]$$. ∥ Proof of Lemma 3. By part 2 of Lemma 2, each $$t^{\prime}$$ in the support of $$P_{B}$$ is also in the support of $$P_{G}$$. We show that each $$t^{\prime}$$ in the support of $$P_{G}$$ is also in the support of $$P_{B}$$ by contradiction. Suppose that there exists $$t^{\prime}$$ in the support of $$P_{G}$$ but not in the support of $$P_{B}$$. Then, by Bayes’s rule $$\mu\left(t^{\prime}\right)=1$$; so bad Sender who receives the arm at $$t\leq t^{\prime}$$ gets the highest possible equilibrium payoff $$v_{B}\left(1\right)$$, because she can pull the arm at time $$t^{\prime}$$ and get payoff $$v_{B}\left(1\right)$$ (recall that, for all $$t$$, the support of $$H\left(.|t,\pi\right)$$ does not contain $$s=0$$ by part (ii) of Assumption (1)). Because bad Sender receives the arm at or before $$t^{\prime}$$ with a positive probability (recall that, for all $$t$$, $$F_{B}\left(t\right)\geq F_{G}\left(t\right)>0$$ by assumption), there exists time $$\tau$$ at which bad Sender pulls the arm with a positive probability and gets payoff $$v_{B}\left(1\right)$$. But then $$\mu\left(\tau\right)=1$$, contradicting that bad Sender pulls the arm at $$\tau$$ with a positive probability. Suppose, on the contrary, that there exists $$\tau$$ such that $$P_{G}\left(\tau\right)>0$$ and $$P_{B}\left(\tau\right)\geq P_{G}\left(\tau\right)$$. Because $$P_{\theta}\left(\tau\right)=\sum_{t=1}^{\tau}\left(P_{\theta}\left(t\right)-P_{\theta}\left(t-1\right)\right)$$, there exists $$\tau^{\prime}\leq\tau$$ in the support of $$P_{B}$$ such that $$P_{B}\left(\tau^{\prime}\right)-P_{B}\left(\tau^{\prime}-1\right)\geq P_{G}\left(\tau^{\prime}\right)-P_{G}\left(\tau^{\prime}-1\right)$$. Similarly, because $$1-P_{\theta}\left(\tau\right)=\sum_{t=\tau+1}^{T+1}\left(P_{\theta}\left(t\right)-P_{\theta}\left(t-1\right)\right)$$ and $$1-P_{G}\left(\tau\right)>0$$ (recall that $$P_{G}\left(T\right)\leq F_{G}\left(T\right)<1$$), there exists $$\tau^{\prime\prime}>\tau$$ in the support of $$P_{G}$$ such that $$P_{G}\left(\tau^{\prime\prime}\right)-P_{G}\left(\tau^{\prime\prime}-1\right)\geq P_{B}\left(\tau^{\prime\prime}\right)-P_{B}\left(\tau^{\prime\prime}-1\right)$$. By Bayes’s rule, \begin{eqnarray*} \mu\left(\tau^{\prime}\right) & = & \frac{\pi\left(P_{G}\left(\tau^{\prime}\right)-P_{G}\left(\tau^{\prime}-1\right)\right)}{\pi\left(P_{G}\left(\tau^{\prime}\right)-P_{G}\left(\tau^{\prime}-1\right)\right)+\left(1-\pi\right)\left(P_{B}\left(\tau^{\prime}\right)-P_{B}\left(\tau^{\prime}-1\right)\right)}\leq\pi\\ & \leq & \frac{\pi\left(P_{G}\left(\tau^{\prime\prime}\right)-P_{G}\left(\tau^{\prime\prime}-1\right)\right)}{\pi\left(P_{G}\left(\tau^{\prime\prime}\right)-P_{G}\left(\tau^{\prime\prime}-1\right)\right)+\left(1-\pi\right)\left(P_{B}\left(\tau^{\prime\prime}\right)-P_{B}\left(\tau^{\prime\prime}-1\right)\right)}=\mu\left(\tau^{\prime\prime}\right)\text{.} \end{eqnarray*} Therefore, by Lemma 2, bad Sender strictly prefers to pull the arm at $$\tau^{\prime\prime}$$ than at $$\tau^{\prime}$$, contradicting that $$\tau^{\prime}$$ is in the support of $$P_{B}$$. ∥ Proof of Lemma 4. By Lemma 3, $$P_{G}$$ and $$P_{B}$$ have the same supports and therefore $$\mu\left(\tau\right)\in\left(0,1\right)$$. Let the support of $$P_{G}$$ be $$\left\{ \tau_{1},...,\tau_{n}\right\} $$. Notice that $$\tau_{n}=T+1$$ because $$P_{G}\left(T\right)\leq F_{G}\left(T\right)<1$$. Since $$\tau_{n-1}$$ is in the support of $$P_{B}$$ and \[ P_{B}\left(\tau_{n-1}\right)<P_{G}\left(\tau_{n-1}\right)=F_{G}\left(\tau_{n-1}\right)\leq F_{B}\left(\tau_{n-1}\right)\text{,} \] where the first inequality holds by Lemma 3, the equality by part 2 of Lemma 2, and the last inequality by assumption $$F_{B}\left(t\right)\geq F_{G}\left(t\right)$$. Therefore, bad Sender who receives the arm at $$\tau_{n-1}$$ must be indifferent between pulling the arm at $$\tau_{n-1}$$ or at $$\tau_{n}$$. Analogously, bad Sender who receives the arm at $$\tau_{n-k-1}$$ must be indifferent between pulling it at $$\tau_{n-k-1}$$ and at some $$\tau\in\left\{ \tau_{n-k},\ldots,\tau_{n}\right\} $$. Thus, by mathematical induction on $$k$$, bad Sender is indifferent between pulling the arm at any $$\tau$$ in the support of $$P_{G}$$ and at $$T+1$$, which proves (2). By Bayes’s rule, for all $$\tau$$ in the support of $$P_{G}$$, \begin{equation} \frac{1-\pi}{\pi}\left(P_{B}\left(\tau\right)-P_{B}\left(\tau-1\right)\right)=\frac{1-\mu\left(\tau\right)}{\mu\left(\tau\right)}\left(P_{G}\left(\tau\right)-P_{G}\left(\tau-1\right)\right)\text{.} \end{equation} (B12) Summing up over $$\tau$$ yields (3). ∥ Proof of Proposition 1. Part 1. We first show that, for each $$\mathcal{T}\subseteq\left\{ 1,\dots,T+1\right\} $$ with $$T+1\in\mathcal{T}$$and each $$\tau\in\mathcal{T}$$, there exist unique $$P_{G} \left(\tau\right)$$, $$P_{B} \left(\tau\right)$$, and $$\mu\left(\tau\right)$$ given by part 1 of this proposition. It suffices to show that there exists a unique $$\left\{ \mu\left(\tau\right)\right\} _{\tau\in\mathcal{T}}\in\left[0,1\right]^{\left|\mathcal{T}\right|}$$ that solves (2) and (3). Using (A11) with $$m=\pi$$ and $$m^{\prime}=\mu\left(\tau\right)$$, the left hand side of (2) can be rewritten as \[ V_{B}\left(\mu\left(\tau\right)\right)\equiv\int v_{B}\left(\frac{\frac{\mu\left(\tau\right)s}{\pi}}{\frac{\mu\left(\tau\right)s}{\pi}+\frac{\left(1-\mu\left(\tau\right)\right)\left(1-s\right)}{1-\pi}}\right)dH_{B}\left(s|\tau,\pi\right). \] Because $$v_{B}$$ is continuous and strictly increasing, $$V_{B}$$ is also continuous and strictly increasing. Furthermore, $$V_{B}\left(0\right)=v_{B}\left(0\right)$$ and $$V_{B}\left(1\right)=v_{B}\left(1\right)$$. Therefore, for all $$\mu\left(T+1\right)\in\left[0,1\right]$$ and all $$\tau\in\mathcal{T}$$, there exists a unique $$\mu\left(\tau\right)$$ that solves (2). Moreover, for all $$\tau\in\mathcal{T}$$, $$\mu\left(\tau\right)$$ is continuous and strictly increasing in $$\mu\left(T+1\right)$$, is equal to $$0$$ if $$\mu\left(T+1\right)=0$$, and is equal to $$1$$ if $$\mu\left(T+1\right)=1$$. The left-hand side of (3) is continuous and strictly decreasing in $$\mu\left(\tau\right)$$ for all $$\tau\in\mathcal{T}$$. Moreover, the left-hand side of (3) is $$0$$ when $$\mu\left(\tau\right)=1$$ for all $$\tau\in\mathcal{T}$$, and it approaches infinity when $$\mu\left(\tau\right)$$ approaches $$0$$ for all $$\tau\in\mathcal{T}$$. Therefore, substituting each $$\mu \left(\tau\right)$$ in (3) with a function of $$\mu\left(T+1\right)$$ obtained from (2), we conclude that there exists a unique $$\mu\left(T+1\right)$$ that solves (3). We now construct an equilibrium for each $$\mathcal{T}\subseteq\left\{ 1,\dots,T+1\right\} $$ with $$T+1\in\mathcal{T}$$. Let $$P_{G}\left(\tau\right)$$ and $$P_{B}\left(\tau\right)$$ be given by part 1 of this proposition for all $$\tau\in\left\{ 1,\dots,T+1\right\} $$. Let $$\mu\left(\tau\right)$$ be given by part 1 of this proposition for all $$\tau\in\mathcal{T}$$ and $$\mu\left(\tau\right)=0$$ otherwise. Notice that, so constructed, $$P_{G}$$, $$P_{B}$$, and $$\mu$$ exist and are unique. $$P_{G}$$ is clearly a distribution. $$P_{B}$$ is also a distribution, because $$P_{B}\left(\tau\right)$$ increases with $$\tau$$ by (4) and $$P_{B}\left(T+1\right)=1$$ by (3) and (4). Furthermore, $$\mu$$ is a consistent belief because (B12) holds for all $$\tau\in\mathcal{T}$$ by (4). It remains to show that there exists an optimal strategy for Sender such that good and bad Sender’s distributions of pulling time are given by $$P_{G}$$ and $$P_{B}$$. First, both good and bad Sender strictly prefer not to pull the arm at any time $$\tau\notin\mathcal{T}$$, because, by part (ii) of Assumption (1) , pulling the arm at $$\tau$$ gives Sender a payoff of $$v_{\theta}\left(0\right)<v_{\theta}\left(\mu\left(T+1\right)\right)$$. Second, by (2), pulling the arm at any time $$\tau\in\mathcal{T}$$ gives bad Sender the same expected payoff $$v_{\theta}\left(\mu\left(T+1\right)\right)$$. Finally, by part 1 of Lemma 2, good Sender strictly prefers to pull the arm at time $$\tau\in\mathcal{T}$$ than at any other time $$\tau^{\prime}>\tau$$. Finally, in any equilibrium, $$P_{G}$$ and $$P_{B}$$ have the same supports by Lemma 3. Moreover, for all $$\tau$$ in the support of $$P_{G}$$, $$P_{G}\left(\tau\right)=F\left(\tau\right)$$ by part 2 of Lemma 2, $$P_{B}\left(\tau\right)$$ satisfies (4) by (B12), $$\mu\left(\tau\right)\in\left(0,1\right)$$ by Lemma 3, and $$\mu\left(\tau\right)$$ satisfies (3) and (4) by Lemma 4. Part 2. First, we notice that, by part 1 of Proposition 1, there exists an equilibrium with $$\mathcal{T}=\left\{ 1,\dots,T+1\right\} $$. In this equilibrium, there are no out of equilibrium events and therefore it is divine. Adopting Cho and Kreps (1987)’s definition to our setting (see, $$e.g.$$, Maskin and Tirole, 1992), we say that an equilibrium is divine if $$\mu\left(\tau\right)=1$$ for any $$\tau\notin {\rm supp}\left(P_{G}\right)$$ at which condition D1 holds. D1 holds at $$\tau$$ if for all $$m\in\left[0,1\right]$$ that satisfy \begin{equation} \int v_{B}\left(s\right)dH_{B}\left(s|\tau,m\right)\geq\max_{t\in {\rm supp}\left(P_{G}\right),t>\tau}\int v_{B}\left(s\right)dH_{B}\left(s|t,\mu\left(t\right)\right) \end{equation} (B13) the following inequality holds: \begin{equation} \int v_{G}\left(s\right)dH_{G}\left(s|\tau,m\right)>\max_{t\in {\rm supp}\left(P_{G}\right),t>\tau}\int v_{G}\left(s\right)dH_{G}\left(s|t,\mu\left(t\right)\right). \end{equation} (B14) Suppose, on the contrary, that there exists a divine equilibrium in which $$P_{G}\left(\tau\right)<F_{G}\left(\tau\right)$$ for some $$\tau\in\left\{ 1,\ldots,T\right\} $$. By part 1 of Proposition 1, $$\tau\notin {\rm supp}\left(P_{G}\right)$$. Let $$t^{*}$$ denote $$t$$ that maximizes the right-hand side of (B14). By Lemma 3, $$\mu\left(t^{*}\right)<1$$ , and, by Lemma 4, $$t^{*}$$ maximizes the right-hand side of (B13). Therefore, by part 1 of Lemma 2, D1 holds at $$\tau$$; so $$\mu\left(\tau\right)=1$$. But then $$\tau\notin {\rm supp}\left(P_{G}\right)$$ cannot hold, because \[ \int v_{G}\left(s\right)dH_{G}\left(s|\tau,1\right)=v_{G}\left(1\right)>\max_{t\in {\rm supp}\left(P_{G}\right)}\int v_{G}\left(s\right)dH_{G}\left(s|t,\mu\left(t\right)\right). \] ∥ Proof of Corollary 2. By Lemma 4 and part 2 of Proposition 1, bad Sender is indifferent between pulling the arm at any time before the deadline and not pulling the arm at all. Then, by Lemma 2, $$\mu\left(\tau-1\right)>\mu\left(\tau\right)$$ for all $$\tau$$. Using (4) with $$P_{G}=F_{G}$$, we have that for all $$\tau<T$$, \begin{eqnarray} \frac{1-\tilde{\mu}\left(\tau\right)}{\tilde{\mu}\left(\tau\right)} & = & \frac{1-\pi}{\pi}\frac{1-P_{B}\left(\tau\right)}{1-P_{G}\left(\tau\right)}\nonumber \\ & = & \frac{\sum_{t=\tau+1}^{T+1}\frac{1-\mu\left(t\right)}{\mu\left(t\right)}\left(F_{G}\left(t\right)-F_{G}\left(t-1\right)\right)}{1-F_{G}\left(\tau\right)}\\ & = & \mathbb{E}_{F}\left[\frac{1-\mu\left(t\right)}{\mu\left(t\right)}\mid t\ge\tau+1\right].\nonumber \end{eqnarray} (B15) Since $$\mu\left(\tau-1\right)>\mu\left(\tau\right)$$ for all $$\tau$$, (B15) implies that $$\tilde{\mu}\left(\tau-1\right)>\tilde{\mu}\left(\tau\right)$$ and $$\mu\left(\tau\right)>\tilde{\mu}\left(\tau-1\right)$$ for all $$\tau$$. ∥ Proof of Corollary 1. Using (4) with $$P_{G}=F_{G}$$, we have \[ \frac{1-\mu\left(\tau\right)}{\mu\left(\tau\right)}=\frac{1-\pi}{\pi}\frac{P_{B}\left(\tau\right)-P_{B}\left(\tau-1\right)}{P_{G}\left(\tau\right)-P_{G}\left(\tau-1\right)}. \] To complete the proof, notice that, by Corollary 2, $$\mu\left(\tau\right)>\mu\left(\tau^{\prime}\right)$$ whenever $$\tau<\tau^{\prime}$$. ∥ Proof of Lemma 2$$'$$. Given Receiver’s interim belief $$m$$ and pulling times $$\tau$$ and $$\tau^{\prime}$$, we write $$H$$ and $$H^{\prime}$$ for distributions $$H\left(.\mid\tau,m\right)$$ and $$H\left(.\mid\tau^{\prime},m\right)$$ of Receiver’s posterior belief $$s$$ from Receiver’s perspective, and we write $$H_{\theta}$$ and $$H_{\theta}^{\prime}$$ for distributions $$H_{\theta}\left(.\mid\tau,m\right)$$ and $$H_{\theta}\left(.\mid\tau^{\prime},m\right)$$ of Receiver’s posterior belief $$s$$ from type-$$\theta$$ Sender’s perspective. For any interim belief $$m\in\left(0,1\right)$$ and pulling times $$\tau,\tau^{\prime}$$, by Bayes’s rule, we have \begin{eqnarray*} dH\left(s\right) & = & mdH_{G}\left(s\right)+\left(1-m\right)dH_{B}\left(s\right),\\ s & = & \frac{mdH_{G}\left(s\right)}{mdH_{G}\left(s\right)+\left(1-m\right)dH_{B}\left(s\right)}, \end{eqnarray*} so that $$dH_{G}\left(s\right)=\frac{s}{m}dH\left(s\right)$$ and $$dH_{B}\left(s\right)=\frac{1-s}{1-m}dH\left(s\right)$$. Likewise, $$dH_{G}^{\prime}\left(s\right)=\frac{s}{m}dH^{\prime}\left(s\right)$$ and $$dH_{B}^{\prime}\left(s\right)=\frac{1-s}{1-m}dH^{\prime}\left(s\right)$$. For any pulling time $$\tau$$ and interim beliefs $$m,m^{\prime}\in\left(0,1\right)$$, each posterior belief $$s$$ under interim belief $$m$$ transforms into the posterior belief $$s^{\prime}$$ given by (A11) under interim belief $$m^{\prime}$$. Let $$m=\mu\left(\tau\right)$$ and $$m^{\prime}=\mu\left(\tau^{\prime}\right)$$. Bad Sender weakly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$ if and only if \[ \int_{0}^{1}v_{B}\left(s\right)dH_{B}\left(s\right)\geq\int_{0}^{1}v_{B}\left(\frac{\frac{m^{\prime}s}{m}}{\frac{m^{\prime}s}{m}+\frac{\left(1-m^{\prime}\right)\left(1-s\right)}{1-m}}\right)dH_{B}^{\prime}\left(s\right), \] which is equivalent to \begin{equation} \int_{0}^{1}v_{B}\left(s\right)\left(1-s\right)dH\left(s\right)\geq\int_{0}^{1}v_{B}\left(\frac{\frac{m^{\prime}s}{m}}{\frac{m^{\prime}s}{m}+\frac{\left(1-m^{\prime}\right)\left(1-s\right)}{1-m}}\right)\left(1-s\right)dH^{\prime}\left(s\right). \end{equation} (B16) Similarly, good Sender strictly prefers to pull the arm at $$\tau$$ than at $$\tau^{\prime}$$ if and only if \begin{equation} \int_{0}^{1}v_{G}\left(s\right)sdH\left(s\right)>\int_{0}^{1}v_{G}\left(\frac{\frac{m^{\prime}s}{m}}{\frac{m^{\prime}s}{m}+\frac{\left(1-m^{\prime}\right)\left(1-s\right)}{1-m}}\right)sdH^{\prime}\left(s\right). \end{equation} (B17) Because $$s$$ and $$s^{\prime}$$ are in $$\left(0,1\right)$$, and $$s^{\prime}$$ is strictly increasing in $$s$$, for any $$r\in\left(0,1\right)$$, we have that $$s^{\prime}>r$$ if and only if $$s>r^{\prime}$$ for some $$r^{\prime}\in\left(0,1\right)$$, which depends on $$m$$ and $$m^{\prime}$$. Thus, for $$v_{\theta}$$ given by (5), the inequalities (B16) and (B17) can be rewritten as \begin{eqnarray} \int_{r}^{1}\left(1-s\right)dH\left(s\right) & \geq & \int_{r^{\prime}}^{1}\left(1-s\right)dH^{\prime}\left(s\right),\\ \end{eqnarray} (B18) \begin{eqnarray} \int_{r}^{1}sdH\left(s\right) & > & \int_{r^{\prime}}^{1}sdH^{\prime}\left(s\right), \end{eqnarray} (B19) where$$\int_{r}^{1}$$ and $$\int_{r^{\prime}}^{1}$$ stand for the Lebesgue integrals over the sets $$\left(r,1\right]$$ and $$\left(r^{\prime},1\right]$$. Notice also that we are using a selection from Receiver’s best response correspondence for which $$v\left(r\right)=0$$. The proof goes through for other selections, after adding appropriate terms on both sides of (B18) and (B19). Integrating by parts, we can rewrite (B18) and (B19) as \begin{eqnarray} -\left(1-r\right)H\left(r\right)+\int_{r}^{1}H\left(s\right)ds & \geq & -\left(1-r^{\prime}\right)H^{\prime}\left(r^{\prime}\right)+\int_{r^{\prime}}^{1}H^{\prime}\left(s\right)ds,\\ \end{eqnarray} (B20) \begin{eqnarray} -rH\left(r\right)-\int_{r}^{1}H\left(s\right)ds & > & -r^{\prime}H^{\prime}\left(r^{\prime}\right)-\int_{r^{\prime}}^{1}H^{\prime}\left(s\right)ds. \end{eqnarray} (B21) Suppose that (B20) and (6) hold and let us show that (B21) holds. We have $$H^{\prime}\left(r^{\prime}\right)>H\left(r\right)$$, because \begin{eqnarray*} \left(1-r\right)\left(H^{\prime}\left(r^{\prime}\right)-H\left(r\right)\right) & = & \left(1-r^{\prime}\right)H^{\prime}\left(r^{\prime}\right)+\left(r^{\prime}-r\right)H^{\prime}\left(r^{\prime}\right)-\left(1-r\right)H\left(r\right)\\ & \geq & \left(1-r^{\prime}\right)H^{\prime}\left(r^{\prime}\right)+\int_{r}^{r^{\prime}}H^{\prime}\left(s\right)ds-\left(1-r\right)H\left(r\right)\\ & \geq & \int_{r}^{1}H^{\prime}\left(s\right)ds-\int_{r}^{1}H\left(s\right)ds>0, \end{eqnarray*} where the equality holds by rearrangement, the first inequality holds by monotonicity of $$H$$, the second by (B20), and the last by (6). The inequality (B21) then holds because \begin{eqnarray*} r^{\prime}H^{\prime}\left(r^{\prime}\right)-rH\left(r\right)+\int_{r^{\prime}}^{1}H^{\prime}\left(s\right)ds-\int_{r}^{1}H\left(s\right)ds & > & r^{\prime}H^{\prime}\left(r^{\prime}\right)-rH\left(r\right)-\int_{r}^{r^{\prime}}H^{\prime}\left(s\right)ds\\ & \geq & r^{\prime}H^{\prime}\left(r^{\prime}\right)-rH\left(r\right)-H^{\prime}\left(r^{\prime}\right)\left(r^{\prime}-r\right)\\ & = & r\left(H^{\prime}\left(r^{\prime}\right)-H\left(r\right)\right)>0, \end{eqnarray*} where the first inequality holds by (6), the second by monotonicity of $$H$$, and the last by the established inequality $$H^{\prime}\left(r^{\prime}\right)>H\left(r\right)$$. ∥ C. Poisson Model Proof of Proposition 6. $$\underline{{\text{For}}\,\pi:}$$ Differentiating $$\mu\left(T\right)$$ in Proposition 2 with respect to $$\pi$$, we have \[ \frac{d\mu\left(T\right)}{d\pi}=\left\{ \begin{array}{l} \frac{1}{\pi^{2}}\mu\left(T\right)^{2}\text{ if }\pi<\bar{\pi},\\ \frac{e^{\alpha T}}{\pi^{2}}\mu\left(T\right)^{2+\frac{\alpha}{\lambda}}\text{ otherwise,} \end{array}\right\} >0. \] For $$\underline{{\text{For}}\,\lambda:}$$ First, when $$\pi<\bar{\pi}$$, $$\frac{d\mu\left(T\right)}{d\lambda}<0$$ since $$e^{-\left(\alpha+\lambda\right)T}>1-\left(\alpha+\lambda\right)T$$ for all $$\alpha,\lambda,T>0$$. Second, when $$\pi>\bar{\pi},$$ \begin{eqnarray*} \frac{d\mu\left(T\right)}{d\lambda} & = & \frac{d}{d\lambda}e^{-\frac{\lambda}{\alpha+\lambda}\ln\left(1+\phi\left(\lambda\right)\right)},\\ \phi & \equiv & \frac{\alpha+\lambda}{\lambda}\frac{1-\pi}{\pi}e^{\alpha T}>0. \end{eqnarray*} Thus, $$\frac{d\mu\left(T\right)}{d\lambda}<0$$, because \[ \frac{d}{d\lambda}\frac{\lambda}{\alpha+\lambda}\ln\left(1+\phi\right)=\frac{\alpha}{\alpha+\lambda}\left[\frac{\ln\left(1+\phi\right)}{\alpha+\lambda}-\frac{1}{\lambda}\frac{1-\pi}{\pi}\frac{1}{1+\phi}\right]>0, \] where the inequality follows from $$\left(1+\phi\right)\ln\left(1+\phi\right)>\phi$$. $$\underline{{\text{For}}\,\alpha:}$$ First, when $$\pi<\bar{\pi}$$, \begin{eqnarray*} \frac{d\mu\left(T\right)}{d\alpha} & = & -\left(\mu\left(T\right)\right)^{2}\frac{\chi}{\left(\alpha+\lambda\right)^{2}}<0,\\ \chi & \equiv & \lambda\left\{ e^{\lambda T}-\left[1+\left(\alpha+\lambda\right)T\right]e^{-\alpha T}\right\} >0, \end{eqnarray*} where the last passage follows from $$e^{\left(\alpha+\lambda\right)T}>1+\left(\alpha+\lambda\right)T$$ for all $$\alpha,\lambda,T>0$$. Second, when $$\pi\geq\bar{\pi}$$, by log-differentiation, \[ \frac{d\mu\left(T\right)}{d\alpha}=\mu\left(T\right)\frac{\lambda}{\alpha+\lambda}\left[\frac{\ln\left(1+\phi\right)}{\alpha+\lambda}-\frac{1}{1+\phi}\frac{d\phi}{d\alpha}\right]. \] Thus, \begin{equation} \frac{d\mu\left(T\right)}{d\alpha}<0\iff\frac{\left(1+\phi\right)\ln\left(1+\phi\right)}{\phi}<1+T\left(\alpha+\lambda\right). \end{equation} (C22) For $$\pi=\bar{\pi}$$, $$\phi=e^{\left(\alpha+\lambda\right)T}-1>0$$; so \[ \frac{d\mu\left(T\right)}{d\alpha}<0\iff\ln\left(1+\phi\right)<\phi, \] which is true for all $$\phi>0$$. Then $$\frac{d\mu\left(T\right)}{d\alpha}<0$$ for $$\pi\geq\bar{\pi}$$ follows because $$\phi<e^{\left(\alpha+\lambda\right)T}-1$$ for $$\pi>\bar{\pi}$$ and the left hand side of (C22) increases with $$\phi$$ for $$\phi>0$$. ∥ Proof of Proposition 3. Part 1. Recall that (1) Sender’s payoff equals Receiver’s posterior belief about Sender at $$t=T$$ and (2) in equilibrium, bad Sender (weakly) prefers not to pull the arm at all than pulling it at any time $$t\in\left[0,T\right]$$. Therefore, bad Sender’s expected payoff equals Receiver’s belief about Sender at $$t=T$$ if the arm has not been pulled: \begin{equation} \mathbb{E}\left[v_{B}\right]=\mu\left(T\right). \end{equation} (C23) Part $$1$$ then follows from Proposition 6. Part 2. By the law of iterated expectations, \begin{eqnarray} \mathbb{E}\left[s\right] & = & \pi\mathbb{E}\left[v_{G}\right]+\left(1-\pi\right)\mathbb{E}\left[v_{B}\right]=\pi\nonumber \\ \Rightarrow\mathbb{E}\left[v_{G}\right] & = & 1-\frac{1-\pi}{\pi}\mu\left(T\right), \end{eqnarray} (C24) where $$s$$ is Receiver’s posterior belief about Sender at $$t=T$$ and we use (C23) in the last passage. Thus, good Sender’s expected payoff increases with $$\alpha$$ and $$\lambda$$ by Proposition 6. Finally, it is easy to see that $$\mathbb{E}\left[v_{G}\right]$$ increases in $$\pi$$ after substituting $$\mu\left(T\right)$$ in $$\mathbb{E}\left[v_{G}\right]$$. Part 3. We shall show that in the divine equilibrium \begin{eqnarray} \mathbb{E}\left[u\right] & = & \frac{\left(1-\pi\right)\left(1-\mu\left(T\right)\right)}{2}. \end{eqnarray} (C25) Part 3 then follows from Proposition 6. Since $$\mathbb{E}\left[s\right]=\pi$$, by (1) and (C24), it is sufficient to prove that $$\mathbb{E}\left[s^{2}\right]=\pi\mathbb{E}\left[v_{G}\right]$$. We divide the proof in two cases: $$\pi\leq\bar{\pi}$$ and $$\pi>\bar{\pi}$$. If $$\pi\leq\bar{\pi},$$ Receiver’s expected payoff is given by the sum of four terms: (1) Sender is good and the arm does not arrive; (2) Sender is good and the arm arrives; (3) Sender is bad and she does not pull the arm; and (4) Sender is bad and she pulls the arm. Thus, \begin{eqnarray*} \mathbb{E}\left[s^{2}\right] & = & \pi e^{-\alpha T}\left(\mu\left(T\right)\right)^{2}\\ & & +\pi\int_{0}^{T}\left(e^{\lambda\left(T-t\right)}\mu\left(T\right)\right)^{2}\alpha e^{-\alpha t}dt\\ & & +\left(1-\pi\right)\left(1-P_{B}\left(T\right)\right)\left(\mu\left(T\right)\right)^{2}\\ & & +\left(1-\pi\right)\int_{0}^{T}e^{-\lambda\left(T-t\right)}\left(e^{\lambda\left(T-t\right)}\mu\left(T\right)\right)^{2}\frac{\pi}{1-\pi}\left(\frac{1-\mu\left(t\right)}{\mu\left(t\right)}\right)\alpha e^{-\alpha t}dt. \end{eqnarray*} Solving all integrals and rearranging all common terms we get \begin{eqnarray*} \mathbb{E}\left[s^{2}\right] & = & \pi\mathbb{E}\left[v_{G}\right]. \end{eqnarray*} If $$\pi>\bar{\pi}$$, Receiver’s expected payoff is given by the sum of five terms: (1) Sender is good and the arm does not arrive; (2) Sender is good and the arm arrives before $$\bar{t}$$; (3) Sender is good and the arm arrives between $$\bar{t}$$ and $$T$$; (4) Sender is bad and she does not pull the arm; (5) Sender is bad and she pulls the arm. Thus, \begin{eqnarray*} \mathbb{E}\left[s^{2}\right] & = & \pi e^{-\alpha T}\left(\mu\left(T\right)\right)^{2}\\ & & +\pi\left(1-e^{-\alpha\bar{t}}\right)\\ & & +\pi\int_{\bar{t}}^{T}\left(e^{\lambda\left(T-t\right)}\mu\left(T\right)\right)^{2}\alpha e^{-\alpha t}dt\\ & & +\left(1-\pi\right)\left(1-P_{B}\left(T\right)\right)\left(\mu\left(T\right)\right)^{2}\\ & & +\left(1-\pi\right)\int_{\bar{t}}^{T}e^{-\lambda\left(T-t\right)}\left(e^{\lambda\left(T-t\right)}\mu\left(T\right)\right)^{2}\frac{\pi}{1-\pi}\left(\frac{1-\mu\left(t\right)}{\mu\left(t\right)}\right)\alpha e^{-\alpha t}dt. \end{eqnarray*} Solving all integrals and rearranging all common terms we again get \begin{eqnarray*} \mathbb{E}\left[s^{2}\right] & = & \pi\mathbb{E}\left[v_{G}\right]. \end{eqnarray*} ∥ Proof of Proposition 4. $$\underline{{\text{For}}\,\pi:}$$ Differentiating $$P_{B}\left(T\right)$$ in (7) with respect to $$\pi$$, we have \begin{align*} \frac{dP_{B}\left(T\right)}{d\pi} & =\frac{e^{-\alpha T}}{\mu\left(T\right)\left(1-\pi\right)}\times\left[\frac{\pi}{\mu\left(T\right)}\frac{d\mu\left(T\right)}{d\pi}-\frac{1-\mu\left(T\right)}{1-\pi}\right]\\ & =\frac{e^{-\alpha T}}{\mu\left(T\right)\left(1-\pi\right)}\times\left\{ \begin{array}{l} \left[\frac{\mu\left(T\right)}{\pi}-\frac{1-\mu\left(T\right)}{1-\pi}\right]\text{ if }\pi<\bar{\pi},\\ \left[e^{\alpha T}\frac{\mu\left(T\right)^{1+\frac{\alpha}{\lambda}}}{\pi}-\frac{1-\mu\left(T\right)}{1-\pi}\right]\text{ otherwise.} \end{array}\right\} \end{align*} First, when $$\pi<\bar{\pi}$$, $$\frac{dP_{B}\left(T\right)}{d\pi}<0$$ because $$\mu\left(T\right)<\pi$$. Second, when$$\pi\geq\bar{\pi}$$, $$\frac{dP_{B}\left(T\right)}{d\pi}<0$$ if and only if \begin{align*} 1+\frac{\alpha}{\alpha+\lambda}\phi & >\left(1+\phi\right)^{\frac{\alpha}{\alpha+\lambda}},\\ \phi & \equiv\frac{\alpha+\lambda}{\lambda}\frac{1-\pi}{\pi}e^{\alpha T}>0. \end{align*} Thus, $$\frac{dP_{B}\left(T\right)}{d\pi}<0$$, because $$1+x\phi>\left(1+\phi\right)^{x}$$ for all $$\phi>0$$ and $$x\in\left(0,1\right)$$. $$\underline{{\text{For}}\,\lambda:}$$ Differentiating $$P_{B}\left(T\right)$$ in (7) with respect to $$\lambda$$, we have \begin{align*} \frac{dP_{B}\left(T\right)}{d\lambda} & =\frac{\pi}{1-\pi}\frac{e^{-\alpha T}}{\mu\left(T\right)^{2}}\frac{d\mu\left(T\right)}{d\lambda}<0, \end{align*} where the inequality follows from Proposition 6. $$\underline{{\text{For}}\,\alpha:}$$ Without loss of generality we can set $$T=1$$. Differentiating $$P_{B}\left(T\right)$$ in (7) with respect to $$\alpha$$, we have \[ \frac{dP_{B}\left(T\right)}{d\alpha}=\frac{\pi}{1-\pi}e^{-\alpha}\left[\frac{1-\mu\left(T\right)}{\mu\left(T\right)}+\frac{1}{\left(\mu\left(T\right)\right)^{2}}\frac{d\mu\left(T\right)}{d\alpha}\right]. \] First, when $$\pi<\bar{\pi}$$, \begin{eqnarray*} \frac{1-\pi}{\pi}e^{2\alpha}\frac{dP_{B}\left(T\right)}{d\alpha} & = & \left(\frac{1}{\pi}-2\right)e^{\alpha}+\frac{\left(\alpha\left(\alpha+\lambda\right)-\lambda\right)e^{\alpha+\lambda}+\lambda\left(1+2\left(\alpha+\lambda\right)\right)}{\left(\alpha+\lambda\right)^{2}}\\ & > & \left(\frac{1}{\bar{\pi}}-2\right)+\frac{\left(\alpha\left(\alpha+\lambda\right)-\lambda\right)e^{\alpha+\lambda}+\lambda\left(1+2\left(\alpha+\lambda\right)\right)}{\left(\alpha+\lambda\right)^{2}}\\ & = & \frac{1}{\left(\alpha+\lambda\right)^{2}}\left(\lambda\left(1+\left(\alpha+\lambda\right)\right)+\left(\left(\alpha+\lambda\right)^{2}-\lambda\right)e^{\left(\alpha+\lambda\right)}-\left(\alpha+\lambda\right)^{2}e^{\alpha}\right)\\ & = & \sum_{k=3}^{\infty}\left[\frac{\left(\alpha+\lambda\right)^{k}}{\left(k-2\right)!}-\lambda\frac{\left(\alpha+\lambda\right)^{k-1}}{\left(k-1\right)!}-\left(\alpha+\lambda\right)^{2}\frac{\alpha^{k-2}}{\left(k-2\right)!}\right]\equiv\sum_{k=3}^{\infty}c_{k}>0, \end{eqnarray*} where the inequality holds because each term $$c_{k}$$ in the sum is positive: \begin{eqnarray*} c_{k} & = & \frac{\left(\alpha+\lambda\right)^{2}\left(\left(\alpha+\lambda\right)^{k-2}-\alpha^{k-2}\right)}{\left(k-2\right)!}-\frac{\left(\alpha+\lambda\right)^{2}\lambda\left(\alpha+\lambda\right)^{k-3}}{\left(k-1\right)!}\\ & = & \frac{\left(\alpha+\lambda\right)^{2}\lambda\left(\sum_{n=0}^{k-3}\left(\alpha+\lambda\right)^{k-3-n}\alpha^{n}\right)}{\left(k-2\right)!}-\frac{\left(\alpha+\lambda\right)^{2}\lambda\left(\alpha+\lambda\right)^{k-3}}{\left(k-1\right)!}\\ & > & \frac{\left(\alpha+\lambda\right)^{2}\lambda\left(\alpha+\lambda\right)^{k-3}}{\left(k-2\right)!}-\frac{\left(\alpha+\lambda\right)^{2}\lambda\left(\alpha+\lambda\right)^{k-3}}{\left(k-1\right)!}>0. \end{eqnarray*} Second, when $$\pi\ge\bar{\pi}$$, $$\frac{dP_{B}\left(T\right)}{d\alpha}>0$$ if and only if \[ \frac{1+\phi}{\phi}\left[\ln\left(1+\phi\right)+\frac{\left(\alpha+\lambda\right)^{2}}{\lambda}\left(1-\left(1+\phi\right)^{-\frac{\alpha+\lambda}{\lambda}}\right)\right]-1-\alpha-\lambda>0 \] \[ \phi\equiv\frac{\alpha+\lambda}{\lambda}\frac{1-\pi}{\pi}e^{\alpha T}. \] The left-hand side increases with $$\alpha$$, treating $$\phi$$ as a constant. Then the inequality holds because it holds for $$\alpha\rightarrow0:$$ \begin{eqnarray*} \frac{1+\phi}{\phi}\left[\ln\left(1+\phi\right)+\lambda\left(1-\left(1+\phi\right)^{-1}\right)\right]-1-\lambda & > & 0\\ \iff\frac{1+\phi}{\phi}\ln\left(1+\phi\right) & > & 1. \end{eqnarray*} ∥ Proof of Proposition 5. $$\underline{{\text{For}}\,\lambda:}$$ Differentiating $$P\left(T\right)$$ in (9) with respect to $$\lambda$$, we have \begin{eqnarray*} \frac{dP\left(T\right)}{d\lambda} & = & \left(1-\pi\right)\frac{dP_{B}\left(T\right)}{d\lambda}<0, \end{eqnarray*} where the inequality follows from Proposition 4. $$\underline{{\text{For}}\,\alpha:}$$ Differentiating $$P\left(T\right)$$ in (9) with respect to $$\alpha$$, we have \begin{eqnarray*} \frac{dP\left(T\right)}{d\alpha} & > & \left(1-\pi\right)\frac{dP_{B}\left(T\right)}{d\alpha}>0, \end{eqnarray*} where the last inequality follows from Proposition 4. $$\underline{{\text{For}}\,\pi:}$$ Differentiating $$P\left(T\right)$$ in (9) with respect to $$\pi$$, we have \begin{eqnarray*} \frac{dP\left(T\right)}{d\pi} & = & \frac{\pi e^{-\alpha T}}{\mu\left(T\right)^{2}}\left(\frac{d\mu\left(T\right)}{d\pi}-\frac{\mu\left(T\right)}{\pi}\right). \end{eqnarray*} We now show that \[ \frac{dP\left(T\right)}{d\pi}\geq0\iff\pi\geq\frac{\alpha e^{\alpha T}}{\left(\alpha+\lambda\right)e^{\alpha T}-1}. \] First, when $$\pi<\bar{\pi}$$, we have $$dP\left(T\right)/d\pi<0$$ because $$\mu\left(T\right)<\pi$$ and \[ \frac{d\mu\left(T\right)}{d\pi}=\frac{\mu\left(T\right)^{2}}{\pi^{2}}<\frac{\mu\left(T\right)}{\pi}. \] Second, when $$\pi\ge\bar{\pi}$$, we have $$dP\left(T\right)/d\pi<0$$ if and only if \[ \frac{d\mu\left(T\right)}{d\pi}=e^{\alpha T}\frac{\mu\left(T\right){}^{2+\frac{\alpha}{\lambda}}}{\pi^{2}}<\frac{\mu\left(T\right)}{\pi}. \] Substituting $$\mu\left(T\right)$$, we get that this inequality is equivalent to \[ \pi<\frac{\alpha e^{\alpha T}}{\alpha e^{\alpha T}+\lambda\left(e^{\alpha T}-1\right)}. \] It remains to show that \[ \frac{\alpha e^{\alpha T}}{\alpha e^{\alpha T}+\lambda\left(e^{\alpha T}-1\right)}>\bar{\pi}. \] Substituting $$\bar{\pi}$$, we get that this inequality is equivalent to \[ \frac{e^{\left(\alpha+\lambda\right)T}-1}{\alpha+\lambda}>\frac{e^{\alpha T}-1}{\alpha}, \] which is satisfied because function $$\left(e^{x}-1\right)/x$$ increases with $$x$$. ∥ Proof of Proposition 7. First, for $$\pi<\bar{\pi}$$, $$\bar{t}=0$$. Second, for $$\pi\ge\bar{\pi}$$, $$\bar{t}$$ increases with $$\pi$$ and decreases with $$\alpha$$ because $$\mu\left(T\right)$$ increases with $$\pi$$ and decreasing with $$\alpha$$. Furthermore, \begin{align*} \frac{d\bar{t}}{d\lambda} & =\frac{1}{\alpha+\lambda}\left(\frac{1}{\alpha+\lambda}\ln\left(1+\phi\right)+\frac{\alpha}{\lambda^{2}}\frac{1-\pi}{\pi}e^{\alpha T}\frac{1}{1+\phi}\right)>0,\\ \phi & \equiv\frac{\alpha+\lambda}{\lambda}\frac{1-\pi}{\pi}e^{\alpha T}. \end{align*} ∥ Proof of Proposition 8. The density $$p_{B}\left(t\right)$$ is equal to $$0$$ for $$t\leq\bar{t}$$ and is given by \[ p_{B}\left(t\right)\equiv\frac{dP_{B}\left(t\right)}{dt}=\frac{\pi}{1-\pi}\frac{\alpha e^{-\alpha t}\left(1-\mu\left(T\right)e^{\lambda\left(T-t\right)}\right)}{\mu\left(T\right)} \] for $$t>\bar{t}$$. Differentiating $$p_{B}\left(t\right)$$ with respect to $$t$$ for $$t>\bar{t}$$, we get \[ \frac{dp_{B}\left(t\right)}{dt}=\frac{\pi}{1-\pi}\frac{\alpha e^{-\alpha t}}{\mu\left(T\right)}\left[\left(\alpha+\lambda\right)\mu\left(T\right)e^{\lambda\left(T-t\right)}-\alpha\right]>0 \] if and only if \[ t<T-\frac{1}{\lambda}\ln\left(\frac{\alpha}{\alpha+\lambda}\frac{1}{\mu\left(T\right)}\right). \] We can therefore conclude that $$p_{B}\left(t\right)$$ is quasiconcave on the interval $$\left[\bar{t},T\right]$$. ∥ Proof of Proposition 9. The density $$p\left(t\right)$$ is given by \begin{align*} p\left(t\right) & =\begin{cases} \pi\alpha e^{-\alpha t} & \mbox{if }t<\bar{t}\\ \pi\alpha e^{-\alpha t}+\pi\alpha e^{-\alpha t}\frac{1-\mu\left(T\right)e^{\lambda\left(T-t\right)}}{\mu\left(T\right)} & \mbox{if }t\geq\bar{t}. \end{cases} \end{align*} Obviously, for $$t\leq\bar{t}$$, $$p\left(t\right)$$ is decreasing in $$t$$. For $$t>\bar{t}$$, differentiating $$p\left(t\right)$$ with respect to $$t$$, we get \[ \frac{dp\left(t\right)}{dt}=\pi\alpha e^{-\alpha t}\left[\left(\alpha+\lambda\right)e^{\lambda\left(T-t\right)}-\alpha\frac{1+\mu\left(T\right)}{\mu\left(T\right)}\right]>0 \] if and only if \[ t<T-\frac{1}{\lambda}\ln\left(\frac{\alpha}{\alpha+\lambda}\frac{1+\mu\left(T\right)}{\mu\left(T\right)}\right). \] ∥ Proof of Proposition 10. The breakdown probability at $$t$$ is given by \begin{eqnarray*} Q\left(t\right) & \equiv & \left(1-e^{-\lambda\left(T-t\right)}\right)\left[1-\mu\left(t\right)\right]. \end{eqnarray*} Notice that $$Q\left(t\right)$$ is continuous in $$t$$ because $$\mu\left(t\right)$$ is continuous in $$t$$. Also, $$Q\left(t\right)$$ equals $$0$$ for $$t\leq\bar{t}$$, is strictly positive for all $$t\in\left(\bar{t},T\right)$$, and equals $$0$$ for $$t=T$$. Substituting $$\mu\left(t\right)$$ and differentiating $$Q\left(t\right)$$ with respect to $$t$$ for $$t\geq\bar{t},$$ we get \[ \frac{dQ\left(t\right)}{dt}=-\lambda\frac{e^{-\lambda\left(T-t\right)}\left(1+\mu\left(T\right)\right)-2\mu\left(T\right)}{\left[1-\mu\left(T\right)\left(1-e^{\lambda\left(T-t\right)}\right)\right]^{2}}>0 \] if and only if \[ t<T-\frac{1}{\lambda}\ln\left(\frac{1+\mu\left(T\right)}{2\mu\left(T\right)}\right). \] ∥ Acknowledgments. We are grateful to the editor, four anonymous referees, Alessandro Bonatti, Steven Callander, Yeon-Koo Che, Wouter Dessein, William Fuchs, Drew Fudenberg, Robert Gibbons, Navin Kartik, Keiichi Kawai, Hongyi Li, Jin Li, Brendan Nyhan, Carlos Pimienta, Andrea Prat, and participants at various seminars and conferences for helpful comments and suggestions. Valentino Dardanoni, Antonio Forcina, and Jutta Roosen kindly shared their codes for the likelihood ratio order test. We especially thank Aleksandra Balyanova and Barton Lee for excellent research assistance. We acknowledge support from the Australian Research Council (ARC) including the ARC Discovery Project DP140102426 (Gratton), the ARC Future Fellowship FT130101159 (Holden), and the ARC Discovery Early Career Research Award DE160100964 (Kolotilin). Footnotes 1. For example, Paul Krugman wrote that the announcement “very probably installed Donald Trump in the White House” (New York Times, January 13, 2017). 2. Matt Zapotosky, Ellen Nakashima, and Rosalind S. Helderman, Washington Post, October 30, 2016. 3. Although the video was filmed eleven years prior to the release, raising the question of whether it was strategically timed, the Washington Post maintains it obtained the unedited video only a few hours before its online release (Farhi, Paul, Washington Post, October 7, 2016). 4. Aaron Blake, Washington Post, October 9, 2016. 5. Los Angeles Times, transcript of Trump’s video statement, October 7, 2016. 6. Susan Page and Karina Shedrofsky, USA TODAY, October 26, 2016 7. In Section 3.2, we generalize the model in several directions allowing for more general utility functions, for Sender to be imperfectly informed, for Sender’s type to affect when the arm arrives, and for the deadline to be stochastic. 8. The equilibrium is essentially unique in the sense that the probability with which each type of Sender pulls the arm at any time is uniquely determined. 9. See also Jung and Kwon (1988), Shin (1994), and Dziuda (2011). The unraveling result might also fail if disclosure is costly (Jovanovic, 1982) or information acquisition is costly (Shavell, 1994). 10. Shin (2003, 2006) also studies dynamic verifiable information disclosure, but he does not allow Sender to choose when to disclose. A series of recent papers consider dynamic information disclosure with different focuses to us, including: Ely et al. (2015); Ely (2016); Grenadier et al. (2016); Hörner and Skrzypacz (2016); Bizzotto et al. (2017); Che and Hörner (2017); Orlov et al. (2017). 11. Brocas and Carrillo (2007) also show that if the learning process is privately observed by Sender but the stopping time is observed by Receiver, then in equilibrium Receiver learns Sender’s information (akin to the unraveling result), as if the learning process was public. Gentzkow and Kamenica (2017) generalize this result. 12. In our model Sender can influence only the starting time of the experimentation process, but not the design of the process itself. Instead, in the Bayesian persuasion literature (e.g. Rayo and Segal, 2010; Kamenica and Gentzkow, 2011), Sender fully controls the design of the experimentation process. 13. See also Prat and Stromberg (2013) for a review of this literature in the broader context of the relationship between media and politics. 14. By part (ii) of Assumption 1, such perfect credibility can never be dented: $$H_{\theta}\left(.\mid\tau,1\right)$$ assigns probability $$1$$ to $$s=1$$ for all $$\theta$$ and $$\tau$$. 15. Divinity is a standard refinement used by the signalling literature. It requires Receiver to attribute a deviation to those types of Sender who would choose it for the widest range of Receiver’s interim beliefs. In our setting, the set of divine equilibria coincides with the set of monotone equilibria in which Receiver’s interim belief about Sender is non-increasing in the pulling time. Specifically, divinity rules out all equilibria in which both types of Sender do not pull the arm at some times, because Receiver’s out-of-equilibrium beliefs for those times are sufficiently unfavourable. 16. It is sufficient for our results to assume that Sender’s payoff is an upper hemicontinuous correspondence (rather than a continuous function) of Receiver’s posterior belief. For example, this is the case if Sender’s and Receiver’s payoffs depend on Receiver’s action and Sender’s type, and Receiver’s action set is finite. In the above example with constant ideological position, Sender’s payoff in (5) is a correspondence with $$v\left(r\right)=\left[0,1\right]$$, because it is optimal for Receiver to randomize between the two actions when $$s=r$$. 17. More generally, Proposition 1 holds whenever $$sv_{G}\left(s\right)$$ is strictly convex and $$\left(1-s\right)v_{B}\left(s\right)$$ is strictly concave, that is, for all $$s$$, Sender’s Arrow-Prat coefficient of absolute risk aversion $$-v_{\theta}^{\prime\prime}\left(s\right)/v_{\theta}^{\prime}\left(s\right)$$ is less than $$2/s$$ for good Sender and more than $$-2/\left(1-s\right)$$ for bad Sender. For the Poisson model of Section 4, Proposition 1 continues to hold for any risk attitude of good Sender and only relies on bad Sender being not too risk-loving. 18. These effects are common in the Bayesian persuasion literature (Kamenica and Gentzkow, 2011). In this literature, Sender is uninformed. Therefore, from her perspective, Receiver’s beliefs follow a martingale process (Ely et al., 2015), so only convexity properties of Sender’s payoff affect the time at which she pulls the arm. 19. The inequality (6) holds if and only if $$\int_{x}^{1}H\left(s\mid\tau^{\prime},\pi\right)ds>\int_{x}^{1}H\left(s\mid\tau,\pi\right)ds$$ for all $$x\in\left(0,1\right)$$. In comparison, part (i) of Assumption 1 holds if and only if $$\int_{x}^{1}H\left(s\mid\tau^{\prime},\pi\right)ds\geq\int_{x}^{1}H\left(s\mid\tau,\pi\right)ds$$ for all $$x\in\left(0,1\right)$$ with strict inequality for some $$x\in\left(0,1\right)$$. 20. For $$v_{\theta}$$ given by (5), we can show that bad Sender withholds the arm with strictly positive probability, $$P_{B}\left(T\right)<F\left(T\right)$$, in all divine equilibria, if $$\pi>r$$. In this case, however, $$v_{\theta}$$ is not strictly increasing, and there exist divine equilibria in which good Sender does not always pull the arm as soon as it arrives. For example, there exists a divine equilibrium in which bad and good Sender never pull the arm by the deadline: $$P_{G}\left(T\right)=P_{B}\left(T\right)=0$$. In this equilibrium, both bad and good Sender enjoy the highest possible payoff, $$1$$. 21. In this case, there exist some time $$\tau$$ at which bad Sender strictly prefers to pull the arm and (2) no longer holds for $$\tau$$. 22. Technically, we use the results from Section 3.1 by treating continuous time as an appropriate limit of discrete time. 23. In every divine equilibrium, $$P_{G}\left(t\right)=F\left(t\right)$$ for all $$t\in\left[\bar{t},T\right]$$ and $$P_{B}\left(t\right)=0$$ for all $$t\in\left[0,\bar{t}\right]$$. But for each distribution $$\hat{P}$$ such that $$\hat{P}\left(t\right)\leq F\left(t\right)$$ for all $$t\in\left[0,\bar{t}\right)$$ and $$\hat{P}\left(t\right)=F\left(t\right)$$ for all $$t\in\left[\bar{t},T\right]$$, there exists a divine equilibrium with $$P_{G}=\hat{P}$$. For ease of exposition, we focus on the divine equilibrium in which $$P_{G}\left(t\right)=F\left(t\right)$$ for all $$t\in\left[0,T\right]$$. 24. For an arbitrary Receiver’s Bernoulli payoff function, which depends on Receiver’s action and Sender’s type, the second-order Taylor approximation of Receiver’s expected payoff increases with the variance of his posterior belief, and therefore with $$\lambda$$ and $$\alpha$$. In contrast, the comparative statics with respect to $$\pi$$ are less robust. For example, if Receiver’s Bernoulli payoff function is $$-\left(a-\theta\right)^{2}$$, where $$a\in\mathbb{R}$$ is Receiver’s action and $$\theta\in\left\{ 0,1\right\} $$ is Sender’s type, then Receiver’s expected payoff decreases with $$\pi$$ for $$\pi<1/2$$ and increases with $$\pi$$ for $$\pi>1/2$$. 25. If the arrival rate $$\alpha$$ is sufficiently small, then $$t_{b}$$ is negative and hence the breakdown probability monotonically decreases with time. 26. This corresponds to the first terms of five presidents: Jimmy Carter (1976–80), Ronald Reagan (1980–84), George H. W. Bush (1988–92), Bill Clinton (1992–96), and George W. Bush (2000–04). Each president run for reelection and three (Reagan, Clinton, and Bush) served two full terms. 27. Nyhan (2015) does not provide data on scandals involving the president-elect between Election Day and the first week of January of the following year, but it contains data on scandals involving the president-elect between the first week of January and the date of his inauguration: there are no such scandals. 28. We omit from our sample the “GSA corruption” scandal during Jimmy Carter’s presidency as the allegations, explicit and implicit, of the scandal, while involving the federal administration, did not involve any of the member of Carter’s administration or their collaborators (if anything, as Carter run with the promise to end corruption in the GSA, the scandal might have actually reinforced his position). In any case, we check in Online Appendix A that our qualitative results are robust to the inclusion of this scandal. 29. We discuss this test in greater detail in the context of the next application. For this application, we use $$k=3$$ equiprobable time intervals. For the election campaign period only (10 scandals), the $$p$$-values are $$0.003$$ and $$0.839$$, respectively. 30. The rate of learning $$\lambda$$ might also be related to the verifiability of information, which may depend on the scandal’s type ($$e.g.,$$ infidelity versus corruption). 31. Another possible explanation (not captured by our model) for Nyhan’s finding is that media organizations strategically avoid releasing scandals when voters’ attention is captured by other media events and scandals may be less effective (see Durante and Zhuravskaya, 2017). 32. One way to map this application into our model is as follows. Suppose that a firm learns at date $$t_{\ell}$$ that it needs liquidity in a time frame $$\Delta_{F}$$, meaning that the latest possible initial trade date is $$t_{\ell}+\Delta_{F}$$. Both $$t_{\ell}$$ and $$\Delta_{F}$$ are privately known by the firm. Date $$t_{\ell}$$ is drawn according to the (improper) uniform distribution on the set of integers $$\mathbb{Z}$$. The time frame is $$\Delta_{F}\equiv T-t$$, where $$t$$ has a distribution $$F$$ on $$\left\{ 1,\dots,T+1\right\} $$. The firm chooses a time gap $$\Delta_{G}\equiv T-\tau$$ subject to $$\tau\in\left\{ t,\dots,T+1\right\} $$, meaning that it announces an IPO at a date $$t_{a}\in\left\{ t_{\ell},\dots,t_{\ell}+\left(\Delta_{F}-\Delta_{G}\right)\right\} $$ with the initial trade date at $$t_{a}+\Delta_{G}\leq t_{\ell}+\Delta_{F}$$. Announcing an IPO at date $$t_{a}$$ with the initial trade date at $$t_{a}-1$$ means that the firm accesses liquidity through other channels than an IPO. With this mapping, all our results hold exactly with $$P_{G}$$ and $$P_{B}$$ being the distributions of $$\tau=T-\Delta_{G}$$ for good and bad firms, respectively. 33. 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Published by Oxford University Press on behalf of The Review of Economic Studies Limited. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
The Review of Economic Studies – Oxford University Press
Published: Oct 1, 2018
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