Keeping the Listener Engaged: a Dynamic Model of Bayesian Persuasion
aa r X i v : . [ ec on . T H ] M a r Keeping the Listener Engaged: a DynamicModel of Bayesian Persuasion ∗ Yeon-Koo Che Kyungmin Kim Konrad Mierendorff † March 17, 2020
Abstract
We consider a dynamic model of Bayesian persuasion. Over time, a sender per-forms a series of experiments to persuade a receiver to take a desired action. Dueto constraints on the information flow, the sender must take real time to persuade,and the receiver may stop listening and take a final action at any time. In ad-dition, persuasion is costly for both players. To incentivize the receiver to listen,the sender must leave rents that compensate his listening costs, but neither playercan commit to her/his future actions. Persuasion may totally collapse in Markovperfect equilibrium (MPE) of this game. However, for persuasion costs sufficientlysmall, a version of a folk theorem holds: outcomes that approximate Kamenicaand Gentzkow (2011)’s sender-optimal persuasion as well as full revelation (whichis most preferred by the receiver) and everything in between are obtained in MPE,as the cost vanishes.
Keywords : Bayesian persuasion, general Poisson experiments, Markov perfectequilibria, folk theorem.
JEL Classification Numbers:
C72, C73, D83 ∗ We thank Martin Cripps, Faruk Gul, Emir Kamenica, Stephan Lauermann, George Mailath, MegMeyer, Sven Rady, S¨onje Reiche, Nikita Roketskiy, Hamid Sabourian, Larry Samuelson, and audiencesin various seminars and conferences for helpful comments and discussions. † Che: Department of Economics, Columbia University (email: [email protected]); Kim: De-partment of Economics, Emory University (email: [email protected]); Mierendorff: Departmentof Economics, University College London (email: k.mierendorff@ucl.ac.uk) Introduction
Persuasion is a quintessential form of communication in which one individual (the sender)pitches an idea, a product, a political candidate, or a point of view, to another indi-vidual (the receiver). Whether the receiver ultimately accepts that pitch—or is “per-suaded”—depends on the underlying truth (the state of world) but also importantly onthe information the sender manages to communicate. In remarkable elegance and general-ity, Kamenica and Gentzkow (2011) show how the sender should communicate informationin such a setting, when she can perform any (Blackwell) experiment instantaneously , with-out any cost incurred by her or by the receiver. This frictionlessness gives full commitmentpower to the sender, as she can publicly choose any experiment and reveal its outcome,all before the receiver can act.In practice, however, persuasion is rarely frictionless. Imagine a salesperson pitchinga product to a potential buyer. The buyer has some interest in buying the product butrequires some evidence that it matches his needs. To convince the buyer, the salespersonmight demonstrate certain features of the product or marshal customer testimonies orsales records, any of which takes real time and effort. Likewise, to process information,the buyer must pay attention, which is costly.In this paper, we study the implications of these realistic frictions. Importantly, withthe friction that real information takes time to generate, the sender no longer automat-ically enjoys full commitment power. Plainly, she cannot promise to the receiver whatexperiments she will perform in the future, so her commitment power is reduced to hercurrent “flow” experiment. Given the lack of commitment by the sender, the receivermay stop listening and take an action at any time if he does not believe that the sender’sfuture experiments are worth waiting for; the buyer in the example above may walk awayat any time when he becomes pessimistic about the product or about the prospect of thesalesperson eventually persuading him. We will examine how well and in what manner the sender can persuade the receiver in this limited commitment environment. As willbecome clear, the key challenge facing the sender is to instill the belief that she will beworth listening to, namely, to keep the receiver engaged .We develop a dynamic version of the canonical persuasion model: the state is binary, L or R , and the receiver can take a binary action, ℓ or r . The receiver prefers to matchthe state, by taking action ℓ in state L and r in state R , while the sender prefers thereceiver to choose r regardless of the state. Time is continuous. At each point in time,the sender may perform some “flow” experiment (unless the game has ended). In response,the receiver may take an action and end the game, or he could wait, in which case thegame continues. Both the sender’s choice of experiment and its outcome are publiclyobservable. Therefore, the two players always keep the same belief about the state.For information generated by the sender, we consider a rich class of Poisson exper-2ments. Specifically, we assume that at each instant in time the sender can generate acollection of Poisson signals that arrive at rates depending on their accuracy. The possiblesignals are flexible in their directionalities : a signal can be either good-news (inducing ahigher posterior belief than the current belief), or bad-news (inducing a lower posteriorthan the current belief), and the news can be of arbitrary accuracy : the sender can chooseany target posterior, although more accurate news (with targets closer to either 0 or 1)takes longer to arrive. Our model generalizes the existing Poisson models in the literaturewhich considered either a good-news or bad-news Poisson experiment of a given accuracy(see, e.g., Keller, Rady, and Cripps, 2005; Keller and Rady, 2015; Che and Mierendorff,2019).Any real experiment, regardless of its accuracy, requires a fixed cost c > Markov perfect equilibria (MPE) of this game—namely, subgame per-fect equilibrium strategy profiles prescribing the flow experiment chosen by the senderand the action ( ℓ, r, “wait”) taken by the receiver as a function of the belief p that thestate is R . We are particularly interested in the equilibrium outcomes when the frictionsare sufficiently small (i.e., in the limit as the flow cost c converges to zero). In addition,we investigate the persuasion dynamics or the “type of pitch” the sender uses to persuadethe receiver in equilibrium of our game. Is persuasion possible? If so, to what extent?
How well the sender can persuadedepends on, among other things, whether the receiver finds her worth listening to, ormore precisely on his belief about the sender providing enough information to justifyhis listening costs. That belief depends on the sender’s future experimentation strategy,which in turn rests on what the receiver will do if the sender betrays her trust and renegeson her information provision. The multitude of ways in which the players can coordinateon these choices yield a version of a folk theorem. There is an MPE in which no persuasionoccurs. However, we also obtain a set of MPEs that range from ones that approximateKamenica and Gentzkow (2011)’s sender-optimal persuasion to ones that approximate fullrevelation, and covers everything in between, when the cost c becomes arbitrarily small.In the “persuasion failure” equilibrium, the receiver is pessimistic about the sender gen-erating sufficient information, so he simply takes an action immediately without waitingfor information. Up against that pessimism, the sender becomes desperate and maximizesher chance of once-and-for-all persuasion, which turns out to be the sort of strategy thatthe receiver fears the sender would employ, justifying her pessimism.In a “persuasion” equilibrium, the receiver expects the sender to deliver sufficient3nformation that would compensate his listening costs. This optimism in turn motivatesthe sender to deliver on her “promise” of informative experimentation; if she reneges onher experimentation, the ever optimistic receiver would simply wait for resumption ofexperimentation an instant later, instead of taking the action that the sender would hopeshe takes. In short, the receiver’s optimism begets the sender’s generosity in informationprovision, which in turn justifies that optimism. As will be shown, an equilibrium ofthis “virtuous cycle” of beliefs can support outcomes that approximate KG’s optimalpersuasion, full-revelation and anything in between, as the flow cost c tends to 0. Persuasion dynamics.
Our model informs us what kind of pitch the sender shouldmake at each point in time, how long it takes for the sender to persuade, if ever, andhow long the receiver listens to the sender before taking his action. The dynamics of thepersuasion strategy adopted in equilibrium unpacks rich behavioral implications that areabsent in the static persuasion model.In our MPEs, the sender optimally makes use of the following three strategies: (i) confidence-building , (ii) confidence-spending , and (iii) confidence-preserving . The confidence-building strategy involves a bad-news Poisson experiment that induces the receiver’s belief(that the state is R ) to either drift upward or jump to zero. This strategy triggers upwardmovement of the belief when the state is R , but quite likely even when it is L ; in fact,it minimizes the probability of bad-news, by insisting that the news be conclusive. Inthis sense, the sender can be seen as “ R -biasing” or “overselling” the desired action. Thesender finds it optimal to use this strategy when the receiver’s belief is already close tothe target belief that would lead the receiver to choose r .The confidence-spending strategy involves a good-news Poisson experiment that gen-erates an upward jump to some target belief, either one inducing the receiver to choose r , or at least one inducing him to listen to the sender. Such a jump arises rarely, how-ever, and absent that jump, the receiver’s belief drifts downward. In that sense, thisstrategy is a risky one that “spends” the receiver’s confidence over time. This strategy isused in general when the receiver is already quite pessimistic about R , so that either theconfidence-building strategy would take too long or the receiver would simply not listen.In particular, it is used as a “last ditch” effort, when the sender is close to giving up onpersuasion or when the receiver is about to choose ℓ .The confidence-preserving strategy combines the above two strategies—namely, agood-news Poisson experiment inducing the belief to jump to a persuasion target, and abad-news Poisson experiment inducing the belief to jump to zero. This strategy is effectiveif the receiver is sufficiently skeptical relative to the persuasion target (i.e., the belief thatwill trigger him to choose r ) so that the confidence-building strategy will take too long.Confidence spending could accomplish persuasion fast and thus can be used for a rangeof beliefs, but the sender would be running down the receiver’s confidence in the process.4ence, at some point the sender finds it optimal to switch to the confidence-preservingstrategy, which prevents the receiver’s belief from deteriorating further. Technically, thebelief where the sender switches to this strategy constitutes an absorbing point of the be-lief dynamics; from then on, the belief does not move, unless either a sudden persuasionbreakthrough or persuasion breakdown occurs.The equilibrium strategy of the sender combines these three strategies in differentways under different economic conditions, thereby exhibiting rich and novel persuasiondynamics. Our equilibrium characterization in Section 5 describes precisely how thesender does this. Related literature.
This paper relates to several strands of literature. First, it con-tributes to the Bayesian persuasion literature that began with Kamenica and Gentzkow(2011) and Aumann and Maschler (1995), by studying the problem in a dynamic envi-ronment. Several recent papers also consider dynamic models (e.g., Brocas and Carrillo,2007; Kremer, Mansour, and Perry, 2014; Au, 2015; Ely, 2017; Renault, Solan, and Vieille,2017; Bizzotto, Rudiger, and Vigier, 2018; Che and H¨orner, 2018; Henry and Ottaviani,2019; Ely and Szydlowski, 2020; Orlov, Skrzypacz, and Zryumov, forthcoming). In most ofthese papers, there are no restrictions on the set of feasible experiments, and full commit-ment is assumed outright. When there are restrictions on the set of feasible experiments(e.g., Brocas and Carrillo, 2007; Henry and Ottaviani, 2019), the receiver cannot chooseto stop listening. Hence, our central issues—namely, a lack of commitment by the senderto persuade and by the receiver to listen—do not arise in those papers. Second, the receiver’s problem in our paper involves a stopping problem, which hasbeen studied extensively in the single agent context, beginning with Wald (1947) andArrow, Blackwell, and Girshick (1949). In particular, Nikandrova and Pancs (2018), Cheand Mierendorff (2019) and Mayskaya (2016) study an agent’s stopping problem whenshe acquires information through Poisson experiments. Che and Mierendorff (2019)introduced the general class of Poisson experiments adopted in this paper. However, thegenerality is irrelevant in their model, because the decision-maker optimally chooses onlybetween two conclusive experiments (i.e., never chooses a non-conclusive experiment).Finally, the current paper is closely related to repeated/dynamic communication mod-els. Margaria and Smolin (2018), Best and Quigley (2017), and Mathevet, Pearce, and Henry and Ottaviani (2019) consider a version of non-commitment problem but one in which thereceiver has a stronger commitment power than in our model: the receiver in their model (e.g., a drugapprover) can effectively force the sender (e.g., a drug company) to experiment by not approving thesender’s application (e.g., for a new drug). The sender’s desire for the receiver to “wait” arises in Orlov,Skrzypacz, and Zryumov (forthcoming), but “waiting” is a payoff-relevant action in their context ofexercising a real option; that is, it is desired in its own merit and not as a means for persuasion as in thecurrent paper. The Wald stopping problem has also been studied with drift-diffusion learning (e.g., Moscarini andSmith, 2001; Ke and Villas-Boas, 2016; Fudenberg, Strack, and Strzalecki, 2017), and in a model thatallows for general endogenous experimentation (see Zhong, 2019).
We consider a game in which a
Sender (“she”) wishes to persuade a
Receiver (“he”).There is an unknown state ω which can be either L (“left”) or R (“right”). The receiverultimately takes a binary action ℓ or r , which yields the following payoffs for the senderand the receiver: Payoffs for the sender and the receiver state/actions ℓ rL (0 , u Lℓ ) ( v, u Lr ) R (0 , u Rℓ ) ( v, u Rr )The receiver gets u ωa if he takes action a ∈ { ℓ, r } when the state is ω ∈ { L, R } . Thesender’s payoff depends only on the receiver’s action: she gets v > r and zero otherwise. We assume u Lℓ > max { u Lr , } and u Rr > max { u Rℓ , } , so that thereceiver prefers to match the action with the state, and also v >
0, so that the senderprefers action r to action ℓ . Notice that this payoff structure corresponds to the leadingexamples considered by Kamenica and Gentzkow (2011) (KG, hereafter) and Bergemannand Morris (2019), where a prosecutor seeks to persuade a judge to convict a defendant,or a regulator seeks to dissuade a depositor from running on a bank. Both players beginwith a common prior p that the state is R , and use Bayes rule to update their beliefs. KG Benchmark.
By now, it is well understood how the sender may optimally persuadethe receiver if she can commit to an experiment without any restrictions. For each a ∈{ ℓ, r } , let U a ( p ) denote the receiver’s expected payoff when he takes action a with belief p . In addition, let ˆ p denote the belief at which the receiver is indifferent between actions ℓ and r , that is, U ℓ (ˆ p ) = U r (ˆ p ). Specifically, for each p ∈ [0 , U ℓ ( p ) := pu Rℓ + (1 − p ) u Lℓ and U r ( p ) := pu Rr + (1 − p ) u Lr . Therefore,ˆ p = (cid:0) u Lℓ − u Lr (cid:1) / (cid:0) u Rr − u Rℓ + u Lℓ − u Lr (cid:1) , which is well-defined in (0 ,
1) under our assumptions on thereceiver’s payoffs. p ˆ p v Sender Receiver p ˆ p u Lℓ u Lr u Rℓ u Rr Figure 1: Payoffs from static persuasion. Solid red curves: payoffs without persuasion(information). Dashed blue curve: the sender’s expected payoff in the KG solution. Bluedots: payoffs in the KG solution at prior p . Dash-dotted green curves: payoffs under afully revealing experiment.If the sender provides no information, then the receiver takes action r if and onlyif p ≥ ˆ p . Therefore, persuasion is necessary only when p < ˆ p . In this case, the KGsolution prescribes an experiment that induces only two posterior beliefs, q − = 0 and q + = ˆ p . The former leads to action ℓ , while the latter results in action r . This experimentis optimal for the sender, because ˆ p is the minimum belief necessary to trigger action r ,and setting q − = 0 maximizes the probability of generating ˆ p , and thus action r . Theresulting payoff for the sender is p v/ ˆ p , as given by the dashed blue line in the left panelof Figure 1. The flip side is that the receiver enjoys no rents from persuasion; his payoffis U ( p ) := max { U ℓ ( p ) , U r ( p ) } , the same as if no information were provided, as depicted inthe right panel of Figure 1. Dynamic model.
We consider a dynamic version of the above Bayesian persuasionproblem. Time flows continuously starting from 0. Unless the game has ended, at eachpoint in time t ≥
0, either the sender performs an informative experiment from a feasibleset at the flow cost of c >
0, or she simply “passes,” in which case she incurs no cost.Both the feasible experiments and the nature of flow costs will be made precise below. Ifthe sender experiments, then the receiver also pays the same flow cost c and observes theexperiment and its outcome. If the sender passes, then the receiver incurs no flow costand obtains no information. Then, the receiver decides whether to take an irreversibleaction a ∈ { ℓ, r } or to “wait.” The former ends the game, while the latter lets the gamecontinue to the next instant.There are two notable modeling assumptions. First, the receiver can take a game- In this sense, flow cost c is interpreted as a “listening cost” rather than a waiting cost. This distinctiondoes not matter in the continuous time game; our analysis below will not change even if the receiver incurscost c , regardless of whether the sender passes or not. However, it is relevant in the discrete-time versionof our model. Feasible experiments.
We endow the sender with a class of Poisson experiments.Specifically, at each point in time, the sender may expend one unit of a resource (attention)across different experiments that generate Poisson signals. The experiments are indexedby i ∈ N . Each Poisson experiment i ∈ N generates breakthrough news that moves thebelief to a target posterior q i ∈ [0 ,
1] of the sender’s choosing. The sender also choosesthe share α i ∈ [0 ,
1] of her resources allocated to experiment i , subject to the (budget)constraint that P ∞ i =1 α i ≤
1. We call a collection of experiments ( α i , q i ) i ∈ N an informationstructure .Given an information structure ( α i , q i ) i ∈ N , a Poisson jump to posterior to q i = p occursat the arrival rate of α i λ p (1 − q i ) | q i − p | if ω = L, and α i λ q i (1 − p ) | q i − p | if ω = R. The unconditional arrival rate is then given by(1 − p ) · α i λ p (1 − q i ) | q i − p | + p · α i λ q i (1 − p ) | q i − p | = α i λ p (1 − p ) | q i − p | . (1)These arrival rates are micro-founded via a class of binary experiments in a discretetime model, as we show in Section 6.1. Further, they provide a natural generalization ofthe Poisson models considered in the existing literature. To see this, suppose that the Suppose that the sender’s cost is given by c s , while that of the receiver is c r . Such a model isequivalent to our normalized one in which c ′ r = c ′ s = c r and v ′ = v ( c r /c s ). When solving the model for afixed set of parameters ( u ωa , v, c, λ ), this normalization does not affect the results. If we let c tend to 0,we are implicitly assuming that the sender’s and receiver’s (unnormalized) costs, c s and c r , converge tozero at the same rate. One can extend this to an uncountable set of experiments. However, in our model, the sender nevermixes over an infinite number of experiments, and thus such extra generality is unnecessary. For q i = p the experiment is uninformative and we set the arrival rate to zero in both states. Thishas the same effect on information as setting α i = 0. The class of feasible information structures is formulated in terms of the current belief p and jump-target beliefs q i . We emphasize, however, that there is an underlying class of information structuresthat is independent of beliefs. Hence, which experiments are feasible does not depend on the currentbelief of the players. Section 6.1 makes this clear in a discrete time foundation, and Appendix A.2 statesthe class of feasible information structures in continuous time without reference to beliefs. This featuredistinguishes our approach from the rational inattention model (Sims, 2003; Matejka and McKay, 2015),in which costs or constraints are based on a measure of information that quantifies the uncertainty in theposterior beliefs induced by an information structure (see also Frankel and Kamenica, forthcoming). q . The jump toposterior q then occurs at the rate of λp (1 − p ) / | q − p | . Conclusive R -evidence ( q = 1)is obtained at the rate of λp , as is assumed in “good” news models (see, e.g., Keller,Rady, and Cripps, 2005). Likewise, conclusive L -evidence ( q = 0) is obtained at therate of λ (1 − p ), as is assumed in “bad” news models (see, e.g., Keller and Rady, 2015).Our model allows for such conclusive news, but it also allows for arbitrary non-conclusivenews with q ∈ (0 , q > p , the arrival rate increases as the news becomes lessprecise ( q decreases), and it approaches infinity as the news becomes totally uninformative(i.e., in the limit as q tends to p ). Lastly, limited arrival rates, together with the budgetconstraint P i α i ≤
1, capture the important feature of our model that any meaningfulpersuasion takes time and requires delay.For our purpose, it suffices to consider either informative information structures wherethe constraint is binding P i α i = 1, or “passing” which corresponds to α i = 0 for all i ∈ N .If the sender uses an informative information structure, both players incur a flow cost of c . If the sender passes, neither player incurs any flow cost. If no Poisson jump arrives when the sender uses the information structure ( α i , q i ) i ∈ N ,the belief drifts according to the following law of motion: ˙ p = − X i : q i >p α i − X i : q i
0, we obtain the updating formula. -drifting, targeting 0: 0 1 p t L -drifting, targeting q : 0 1 p t q Stationary, targeting 0 or q : 0 1 p t q Figure 2: Three prominent feasible experiments.prove particularly relevant for our purpose and will be frequently referred to. Theyformalize the three modes of persuasion discussed in the introduction: • R -drifting experiment (confidence building): α = 1 with q = 0. The senderdevotes all resources to a Poisson experiment with the (posterior) jump target q = 0.In the absence of a jump, the posterior drifts to the right, at rate ˙ p = λp (1 − p ) . • L -drifting experiment (confidence spending): α = 1 with q = q for some q > p .The sender devotes all resources to a Poisson experiment with jumps targetingsome posterior q > p . The precise jump target q will be specified in our equilibriumconstruction. In the absence of a jump, the posterior drifts to the left, at rate˙ p = − λp (1 − p ) . • Stationary experiment (confidence preserving): α = α = 1 / q = 0and q = q for some q > p . The sender assigns equal resources to an experimenttargeting q = 0 and one targeting q = q . Absent jumps, the posterior remainsunchanged. Solution concept.
We study (pure-strategy) Markov Perfect equilibria (MPE, here-after) of this dynamic game in which both players’ strategies depend only on the currentbelief p . Formally, a profile of Markov strategies specifies for each belief p ∈ [0 , α i , q i ) i ∈ N chosen by the sender, and an action a ∈ { ℓ, r, wait } chosenby the receiver. An MPE is a strategy profile that, starting from any belief p ∈ [0 , Naturally, this solution concept limits the useof (punishment) strategies depending solely on the payoff-irrelevant part of the histories,and serves to discipline the strategies off the equilibrium path.We impose a restriction that captures the spirit of “perfection” in our continuous timeframework. Suppose that at some p , the receiver would choose action ℓ immediately,unless a Poisson signal causes a discrete jump in beliefs. In continuous time, the latterevent occurs with probability 0, and therefore, the sender’s strategy at p is inconsequential There are well known technical issues in defining a game in continuous time (see Simon and Stinch-combe, 1989; Bergin and MacLeod, 1993). In Appendix A.1, we formally define admissible strategyprofiles that guarantee a well defined outcome of the game and define Markov perfect equilibria. Lℓ u Lr u Rℓ u Rr p ˆ p U ℓ ( p ) U dKG ( p )Figure 3: Replicating the KG outcome through R -drifting experiments.for the players’ payoffs. We require the sender to choose a strategy that maximizes her flow payoff in such a situation. This can be seen as selecting an MPE that is robust to adiscrete-time approximation: in discrete time, a Poisson jump would occur with a positiveprobability, and thus the sender’s strategy would have non-trivial payoff consequences.See Appendix A.1 for a formal definition. Persuading the Receiver to Listen
We begin by illustrating the key issue facing the sender: persuading the receiver to listen .To this end, consider any prior p < ˆ p , so that persuasion is not trivial, and suppose thatthe sender repeatedly chooses R -drifting experiments with jumps targeting q = 0 untilthe posterior either jumps to 0 or drifts to ˆ p , as depicted on the horizontal axis in Figure3. This strategy exactly replicates the KG solution (in the sense that it yields the sameprobabilities of reaching the two posteriors, 0 and ˆ p , as the KG solution), provided thatthe receiver listens to the sender for a sufficiently long time. But will the receiver wait until a target belief of ˆ p or is reached? The answer is no.The KG experiment leaves no rents for the receiver even without listening costs, and thuslistening will make the receiver strictly worse off compared with choosing ℓ immediately:in Figure 3, the receiver’s expected gross payoff from the static KG experiment is U ℓ ( p ).Due to listening costs, the receiver’s expected payoff under the dynamic KG strategy,denoted here by U dKG ( p ), is strictly smaller than U ℓ ( p ). In other words, the dynamicstrategy implementing the KG solution cannot persuade the receiver to wait and listen,so it does not permit any persuasion. Indeed, this problem leads to the existence of a The KG outcome can also be replicated by other dynamic strategies. For instance, the sender could c > Theorem 1 (Persuasion Failure) . For any c > , there exists an MPE in which nopersuasion occurs, that is, for any p , the receiver immediately takes either action ℓ or r .Proof. Consider the following strategy profile: the receiver chooses ℓ for p < ˆ p and r for p ≥ ˆ p ; and the sender chooses the L -drifting experiment with jump target ˆ p for all p ∈ (ˆ π ℓL , ˆ p ) and passes for all p / ∈ (ˆ π ℓL , ˆ p ), where the cutoff ˆ π ℓL is the belief at which thesender is indifferent between the L -drifting experiment and stopping (so that the receiverchooses ℓ ). In order to show that this strategy profile is indeed an equilibrium, first consider thereceiver’s incentives given the sender’s strategy. If p (ˆ π ℓL , ˆ p ), then the sender neverprovides information, so the receiver has no incentive to wait, and will take an actionimmediately. If p ∈ (ˆ π ℓL , ˆ p ), then the sender never moves the belief into the region wherethe receiver strictly prefers to take action r (i.e., strictly above ˆ p ). This implies thatthe receiver’s expected payoff is equal to U ℓ ( p ) minus any listening cost she may incur.Therefore, again, it is optimal for the receiver to take an action immediately.Now consider the sender’s incentives given the receiver’s strategy. If p ≥ ˆ p , thenit is trivially optimal for the sender to pass. Now suppose that p < ˆ p . Our refinement,discussed at the end of Section 2, requires that the sender choose an information structurethat maximizes her flow payoff, which is given by max ( α i ,q i ) i ∈ N X q i = p α i λ p (1 − p ) | q i − p | { q i ≥ ˆ p } v − c subject to X α i α i = 1 . If the sender chooses any nontrivial experiment, its jump target must be q i = ˆ p . Hencethe optimal information structure is either ( α = 1 , q = ˆ p ) or α i = 0 for all i . The formeris optimal if and only if λp (1 − p )ˆ p − p v ≥ c , or equivalently p ≥ ˆ π ℓL .The no-persuasion equilibrium constructed in the proof showcases a total collapse oftrust between the two players. The receiver does not trust the sender to convey valu-able information (e.g., an experiment targeting q > ˆ p ), so she refuses to listen to her.This attitude makes the sender desperate for a quick breakthrough; she tries to achieve repeatedly choose a stationary strategy with jumps targeting q = ˆ p and q = 0 until either jump occurs.However, this (and in fact, any other) strategy would not incentivize the receiver to listen, for the samereason as in the case of repeating R -drifting experiments. Specifically, ˆ π ℓL satisfies c = λ ˆ π ℓL (1 − ˆ π ℓL )ˆ p − ˆ π ℓL v ⇐⇒ ˆ π ℓL = 12 + c λv − s(cid:18)
12 + c λv (cid:19) − c ˆ pλv . The equation follows from the fact that under the given strategy profile, the sender’s value functionis V ( p ) = v if p ≥ ˆ p and V ( p ) = 0 otherwise; and when the target posterior is q i , a Poisson jump occursat rate λp (1 − p ) / | q i − p | . p , which is indeed not enough for the receiver to be willing towait. Can trust be restored? In other words, can the sender ever persuade the receiver tolisten to her?
She certainly can, if she can commit to a dynamic strategy, that is, ifshe can credibly promise to provide more information in the future. Consider the follow-ing modification of the dynamic KG experiment discussed above: the sender repeatedlychooses R -drifting experiments with jumps targeting zero, until either the jump occurs orthe belief reaches p ∗ > ˆ p . If the receiver waits until p either jumps to 0 (in which case shetakes action ℓ ) or reaches p ∗ (in which case she takes action r ), then her expected payoffis equal to U R ( p ) = p ∗ − p p ∗ u Lℓ + p p ∗ U r ( p ∗ ) − (cid:18) p log (cid:18) p ∗ − p ∗ − p p (cid:19) + 1 − p p ∗ (cid:19) cλ . Importantly, if p ∗ is sufficiently larger than ˆ p (and c is sufficiently small), then U R ( p ) (thedotted curve in Figure 4) stays above max { U ℓ ( p ) , U r ( p ) } (the black kinked curve) while p drifts toward p ∗ , so the receiver prefers to wait. Intuitively, unlike in the KG solution,this “more generous” persuasion scheme promises the receiver enough rents that make itworth listening to.If c is sufficiently small, the required belief target p ∗ need not exceed ˆ p by much. Infact, p ∗ can be chosen to converge to ˆ p as c →
0. In this fashion, a dynamic persuasionstrategy can be constructed to virtually implement the KG solution when c is sufficientlysmall.At first glance, this strategy seems unlikely to work without the sender’s commitmentpower. How can she credibly continue her experiment even after the posterior has risenpast ˆ p ? Why not simply stop at the posterior ˆ p —the belief that should have convinced thereceiver to choose r ? Surprisingly, however, the strategy works even without commit-ment. The reason lies with the fact that the equilibrium beliefs generated by the Markovstrategies themselves can provide a sufficient incentive for the sender to go above ˆ p . Wealready argued that, with a suitably chosen p ∗ > ˆ p , the receiver is incentivized to wait pastˆ p , due to the “optimistic” equilibrium belief that the sender will continue to experiment To understand this explicit solution, first notice that under the prescribed strategy profile, the receivertakes action ℓ when p jumps to 0, which occurs with probability ( p ∗ − p ) /p ∗ , and action r when p reachesto p ∗ , which occurs with probability p /p ∗ . The last term captures the total expected listening cost. Thelength of time τ it takes for p to reach p ∗ absent jumps is derived as follows: p ∗ = p p + (1 − p ) e − λτ ⇔ τ = 1 λ log (cid:18) p ∗ − p ∗ − p p (cid:19) . Hence, the total listening cost is equal to(1 − p ) Z τ ctd (cid:0) − e − λt (cid:1) + (cid:0) p + (1 − p ) e − λτ (cid:1) cτ = (cid:18) p log (cid:18) p ∗ − p ∗ − p p (cid:19) + 1 − p p ∗ (cid:19) cλ . Lℓ u Lr u Rℓ u Rr p ˆ p p ∗ U ℓ ( p ) = U R ( p )Figure 4: Persuasive R -drifting experimentsuntil a much higher belief p ∗ is reached. Crucially, this optimism in turn incentivizes thesender to carry out her strategy: were she to deviate and, say, “pass” at q = ˆ p , thereceiver would simply wait (instead of choosing r ), believing that the sender will shortlyresume her R -drifting experiments after the “unexpected” pass. Given this response,the sender cannot gain from deviating: she cannot convince the receiver to “prematurely”choose r . To sum up, the sender’s strategy instills optimism in the receiver to wait andlisten to the sender, and this optimism, or the power of beliefs , in turn incentivizes thesender to carry out the strategy. The equilibrium logic outlined in the previous section applies not just to strategy profilesthat approximate the KG solution, but also to other strategy profiles with a target belief p ∗ ∈ (ˆ p, anypayoff for the sender between the KG solution and her payoff from full revelation can be We will show in Section 5.2 that under certain conditions, using R -drifting experiments is not justbetter than passing but also the optimal experiment (best response), given that the receiver waits. Here,we illustrate the possibility of persuasion for this case. The logic extends to other cases where the senderoptimally uses different experiments to persuade the receiver. To be formal, suppose that the current belief is slightly below p ∗ (say, p = p ∗ − λdt ), and the senderdeviates and passes. Given the belief that she will continue her R -drifting strategy in the next instant,the receiver prefers waiting to taking action r immediately if and only if c ≤ λ (1 − p ∗ )( u Lℓ − U r ( p ∗ )) + λp ∗ (1 − p ∗ ) U ′ r ( p ∗ ) ⇔ p ∗ ≤ p := (cid:0) u Lℓ − u Lr (cid:1) λ − c (cid:0) u Lℓ − u Lr (cid:1) λ . In other words, even if p is close to p ∗ (and strictly above ˆ p ), the receiver is willing to wait as long as p ∗ is not too high. Importantly, p approaches 1 as c tends to 0. p ˆ p p ∗ v p ˆ p vp v Sender Receiver p ˆ p p ∗ U ( p ) u Lℓ u Lr u Rℓ u Rr Figure 5: Implementable payoff set for each player at each p . virtually supported as an MPE payoff. Theorem 2 (Folk theorem) . Fix any prior p ∈ (0 , . For any sender payoff V ∈ ( p v, min { p / ˆ p, } v ) , if c is sufficiently small, there exists an MPE in which the senderobtains V ; likewise, for any receiver payoff U ∈ (cid:0) U ( p ) , p u Rr + (1 − p ) u Lℓ (cid:1) , if c is suffi-ciently small, there exists an MPE in which the receiver achieves U . Figure 5 depicts how the set of implementable payoffs for each player varies accordingto p in the limit as c tends to 0. Theorem 2 shows that any payoffs in the green andred shaded areas can be implemented in an MPE, provided that c is sufficiently small. Inthe left panel, the upper bound for the sender’s payoff is given by the KG-optimal payoffmin { p / ˆ p, } v , and the lower bound of the green shaded area is given by the sender’spayoff from full revelation p v . For the receiver, by contrast, full revelation defines theupper bound p u Rr + (1 − p ) u Lℓ , whereas the KG-payoff, which leaves no rent for thereceiver, is given by U ( p ). In both panels, the thick blue lines correspond to the players’payoffs in the no-persuasion equilibria of Theorem 1.Note that Theorem 2 is silent about payoffs in the gray shaded region. In the staticKG environment, these payoffs can be achieved by the (sender-pessimal) experiment thatsplits the prior p into two posteriors, 1 and q ∈ (0 , ˆ p ). The following theorem shows thatthe sender’s payoffs in this region cannot be supported as an MPE payoff, for a sufficientlysmall c > Theorem 3. If p < ˆ p , then the sender’s MPE payoff is either equal to or at least p v − c/λ . If p ≥ ˆ p , then the sender’s MPE payoff cannot be smaller than p v − c/λ .Proof. Fix p < ˆ p , and consider any MPE. If the receiver’s strategy is to wait at p , thenthe sender can always adopt the stationary strategy with jump-targets 0 and 1, whichwill guarantee her the payoff of p v − c/λ . If the receiver’s strategy is to stop at p , In order to understand this payoff, notice that the strategy fully reveals the state, and thus thesender gets v only in state R . In addition, in each state, a Poisson jump occurs at rate λ/
2, and thus theexpected waiting time equals 2 /λ , which is multiplied by c to obtain the expected cost. ℓ immediately, in which case the sender’s payoff is equal to0. Therefore, the sender’s expected payoff is either equal to 0 or above p v − c/λ .Now suppose p ≥ ˆ p , and consider any MPE. As above, if p belongs to the waitingregion, then the sender’s payoff must exceed at least p v − c/λ . If p belongs to thestopping region, then the sender’s payoff is equal to v . In either case, the sender’spayoff is at least as much as p v − c/λ .We prove the folk theorem by constructing MPEs with a particularly simple structure: Definition 1.
A Markov perfect equilibrium is a simple MPE (henceforth, SMPE) ifthere exist p ∗ ∈ (0 , ˆ p ) and p ∗ ∈ (ˆ p,
1) such that the receiver chooses action ℓ if p < p ∗ ,waits if p ∈ ( p ∗ , p ∗ ), and chooses action r if p ≥ p ∗ . In other words, in an SMPE, the receiver waits for more information if p ∈ W andtakes an action, ℓ or r , otherwise, where W = ( p ∗ , p ∗ ) or W = [ p ∗ , p ∗ ) denotes the waitingregion: | p =0 ℓ z }| { ———————— p ∗ “wait” z }| { ———————— p ∗ r z }| { —————— | While this is the most natural equilibrium structure, we do not exclude the possibility ofMPEs that violate this structure. Our folk theorem, as well as Theorem 3, is independentof whether such non-simple equilibria exist or not.To prove the folk theorem, we begin by fixing p ∗ ∈ (ˆ p, c sufficientlysmall, we identify a unique value of p ∗ that yields an SMPE. We then show that as c tends to 0, p ∗ approaches 0 as well. This implies that given p ∗ , the limit SMPE spansthe sender’s payoffs on the line that connects (0 ,
0) and ( p ∗ , v ) (the dashed line in theleft panel of Figure 5) and the receiver’s payoffs on the line that connects (0 , u Lℓ ) and( p ∗ , U r ( p ∗ )) (the dashed line in the right panel). By varying p ∗ from ˆ p to 1, we can coverthe entire shaded areas in Figure 5. Note that with this construction, we also obtain acharacterization of feasible payoff vectors ( V, U ) for the sender and receiver that can arisein an SMPE in the limit as c tends to 0. We state this in the following corollary. Corollary 1.
For any prior p ∈ [0 , , in the limit as c tends to , the set of SMPEpayoff vectors ( V, U ) is given by (cid:26) ( V, U ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ p ∗ ∈ [max { p , ˆ p } ,
1] : V = p p ∗ v, U = p p ∗ U r ( p ∗ ) + p ∗ − p p ∗ u Lℓ (cid:27) , At ˆ p , the receiver is indifferent between ℓ and r . In any MPE , however, she must take action r ifˆ p is not in the waiting region. This is necessary for the existence of a best response of the sender. Forexample, in the no-persuasion equilibrium, if the receiver takes action ℓ at ˆ p , then the sender has nooptimal jump-target for p < ˆ p . We do not restrict the receiver’s decision at the lower bound p ∗ , so that the waiting region can beeither W = ( p ∗ , p ∗ ) or W = [ p ∗ , p ∗ ). Requiring W = ( p ∗ , p ∗ ) can lead to non-existence of an SMPE(see Proposition 2 below, as well as the discussion in Footnote 19). Requiring W = [ p ∗ , p ∗ ) can leadto non-admissibility of the sender’s best response in Proposition 3 (see the discussion of admissibility inAppendix A.1). ith the addition of the no-persuasion payoff vector (0 , U ( p )) for p < ˆ p . While the folk theorem in Section 4 is of clear interest, it is equally interesting to teaseout the behavioral implications from our dynamic persuasion model. In this section, weprovide a full description of SMPE strategy profiles and illustrate the resulting equilibriumpersuasion dynamics. We first explain why the sender optimally uses the three modes ofpersuasion discussed in the Introduction and Section 2. Then, using them as buildingblocks, we construct full SMPE strategy profiles.
Suppose that the sender runs a flow experiment that targets q i when the current beliefis p . Then, the belief jumps to q i at rate λp (1 − p ) / | q i − p | . Absent jumps, it movescontinuously according to (2). Therefore, given the sender’s value function V ( · ), her flowbenefit is given by v ( p ; q i ) := λ p (1 − p ) | q i − p | ( V ( q i ) − V ( p )) − sgn( q i − p ) λp (1 − p ) V ′ ( p ) . At each point in time, the sender can choose any countable mixture over differentexperiments. Therefore, at each p , her flow benefit from optimal persuasion is equal to v ( p ) := max ( α i ,q i ) i X q i = p α i v ( p ; q i ) subject to X i ∈ N α i = 1 . (3)The function v ( p ) represents the gross flow value from experimentation. If p / ∈ W thenthe sender simply compares this to the flow cost c . Specifically, if p > p ∗ , then thereceiver takes action r immediately, and thus V ( p ) = v for all p > p ∗ . It follows that v ( p ) = 0 < c , so it is optimal for the sender to pass, which is intuitive. If p < p ∗ then thesender has only one instant to persuade the receiver, and therefore experiments only when v ( p ) ≥ c : if v ( p ) < c then persuasion is so unlikely that she prefers to pass. If p ∈ W , thenin equilibrium, the sender must continue to experiment, which suggests that v ( p ) ≥ c .When the sender’s optimal solution involves experimentation, her value function V ( · ) isadjusted so that her flow benefit v ( p ) coincides with the corresponding flow cost c . Moreformally, v ( p ) = c is the HJB equation that the sender’s value function must satisfy. sgn( x ) denotes the sign function that assigns 1 if x >
0, 0 if x = 0, and − x <
0. Note that thesender’s value function is not everywhere differentiable in all equilibria. Here, we ignore this to give asimplified argument illustrating the properties of the optimal strategy for the sender. The formal proofscan be found in Appendix B. p ∗ p ∗ v p p p Figure 6: Optimal Poisson jump targets for different values of p . The solid curve representsthe sender’s value function in an SMPE with p ∗ and p ∗ .The following proposition shows that the potentially daunting task of characterizingthe sender’s equilibrium strategy reduces to searching among a small set of experiments,instead of all feasible experiments, at each belief. Proposition 1.
Consider an SMPE where the receiver’s strategy is given by p ∗ < ˆ p < p ∗ .(a) For all p ∈ (0 , , there exists a best response that uses at most two experiments, ( α , q ) and ( α , q ) .(b) Suppose that V ( · ) is non-negative, increasing, and strictly convex over ( p ∗ , p ∗ ] , and V ( p ∗ ) /p ∗ ≤ V ′ ( p ∗ ) .(i) If p ∈ ( p ∗ , p ∗ ) , then α + α = 1 , q = p ∗ , and q = 0 .(ii) If p < p ∗ , then either α = α = 0 (i.e., the sender passes), or α = 1 and q = p ∗ or q = p ∗ . For part (a) of Proposition 1, notice that the right-hand side in equation (3) is linearin each α i and the constraint P i ∈ N α i ≤ p and the other above p . This result implies that v ( p ) can be written as v ( p ) = max ( α ,q ) , ( α ,q ) λp (1 − p ) (cid:20) α V ( q ) − V ( p ) q − p + α V ( q ) − V ( p ) p − q − ( α − α ) V ′ ( p ) (cid:21) , subject to α + α = 1 and q < p < q .Part (b) of Proposition 1 states that if V ( · ) satisfies certain properties, which willbe shown to hold in equilibrium later, then there are only three candidates for optimalPoisson jump targets, 0, p ∗ , and p ∗ , regardless of p ∈ (0 , p ∗ ). To see this, observe that One may wonder why we allow for two experiments. In fact, linearity implies that there existsa maximizer that puts all weight on a single experiment. Mixing, however, is necessary to obtain an admissible
Markov strategy. For example, if p is an absorbing belief, then admissibility requires that thestationary strategy be used at that belief, requiring two experiments. See Section A.1 for details. maximizes ( V ( q ) − V ( p )) / ( q − p ) (the slope of V between p and q ) subject to q > p ,while q minimizes ( V ( q ) − V ( p )) / ( p − q ) (the slope between q and p ) subject to q < p .As shown in Figure 6, under the given assumptions on V ( · ), if p ∈ ( p ∗ , p ∗ ) then q = p ∗ and q = 0 are optimal (see p and the dotted lines). Similarly, if p < p ∗ then q = 0 isoptimal and q is either p ∗ (see p and the dotted line) or p ∗ (see p and the dash-dottedline).Proposition 1 implies that the sender makes use of the following three modes of per-suasion while p < p ∗ . R -drifting experiment: This corresponds to choosing α = 1 and q = 0, that is, thesender uses an experiment that generates Poisson jumps to 0 at (unconditional) rate λ (1 − p ), upon which the receiver immediately takes action ℓ . In the absence of Poisson jumps, p continuously drifts rightward at rate ˙ p = λp (1 − p ). Targeting q = 0 is explained by thesame logic as selecting zero as the belief that induces action ℓ in the static model: the jumpto zero is less likely than jump to any other q < p , as the arrival rate λp (1 − p ) / ( p − q ) isincreasing in q . Intuitively, this experiment can be interpreted as the strategy of buildingthe receiver’s confidence slowly but steadily . This strategy has low risks of losing thereceiver’s attention and, therefore, is particularly useful when the current belief is alreadyclose to p ∗ , in which case the sender can achieve persuasion relatively quickly and at lowcost. L -drifting experiment: This corresponds to choosing α = 1 and q = p ∗ or possibly q = p ∗ if p < p ∗ . In other words, the sender generates rightward Poisson jumps that leadto either p ∗ or p ∗ . In the absence of Poisson jumps, the belief continuously drifts leftwardat rate ˙ p = − λp (1 − p ). This strategy is the polar opposite of the R -drifting experiment.It can yield fast success, but the success is unlikely to happen. In addition, when thereis no success, the receiver’s confidence diminishes. As is intuitive, this strategy is usefulwhen the current belief is significantly away from the target belief and, therefore, thesender has a strong incentive to take risks. Stationary experiment:
This arises when the sender targets two beliefs, q = p ∗ and q = 0, with equal weights ( α = α ). In this case, unless the belief jumps to0 or p ∗ , it stays constant (thus, “stationary”). This can be interpreted as a mixturebetween R -drifting and L -drifting experiments and, therefore, combines their strengthsand weaknesses. Naturally, this strategy is useful when p is not so close to p ∗ or p ∗ . We will show that any other mixture (in which α = α ) never arises in equilibrium. .2 Equilibrium Characterization We now explain how the sender optimally combines the three modes of persuasion intro-duced in Section 5.1, and provide a full description of the unique SMPE strategy profilefor each set of parameter values.The structure of the sender’s equilibrium strategy depends on two conditions. Thefirst condition concerns how demanding the persuasion target p ∗ is: p ∗ ≤ η ≈ . . (C1)As explained later, this condition determines whether the sender always prefers the R -drifting strategy to the stationary strategy or not; η is the largest value of p ∗ such that thesender prefers the former strategy to the latter for all p < p ∗ . Notice that this conditionholds for p ∗ close to ˆ p , as with a strategy that approximates the KG solution (as long asˆ p ≤ η ).The structure of the sender’s equilibrium strategy also depends on the following con-dition: v > U r ( p ∗ ) − U ℓ ( p ∗ ) . (C2)The left-hand side quantifies the sender’s gains when she successfully persuades the re-ceiver and induces action r , while the right-hand side represents the corresponding gainsby the receiver. If (C2) holds, then the sender has a stronger incentive to experimentthan the receiver, and thus p ∗ (the belief below which some player wishes to stop) isdetermined by the receiver’s incentives. Conversely, if (C2) fails, then the sender is lesseager to experiment, and thus p ∗ is determined by the sender’s incentives.We first provide an equilibrium characterization for the case where (C2) is satisfied. Proposition 2.
Fix p ∗ ∈ (ˆ p, and suppose that v > U r ( p ∗ ) − U ℓ ( p ∗ ) . For each c suffi-ciently small, there exists a unique SMPE such that the waiting region has upper bound p ∗ . The waiting region is W = [ p ∗ , p ∗ ) for some p ∗ < ˆ p , and the sender’s equilibriumstrategy is as follows: (a.i) In the waiting region when p ∗ ∈ (ˆ p, η ) : the sender plays the R -drifting strategy withleft-jumps to for all p ∈ [ p ∗ , p ∗ ) .(a.ii) In the waiting region when p ∗ ∈ ( η, : there exist cutoffs p ∗ < ξ < π LR < p ∗ such that for p ∈ [ p ∗ , ξ ) ∪ ( π LR , p ∗ ) , the sender plays the R -drifting strategy with As explained in Section 2, the payoffs of the two players are directly comparable, because theirinformation cost c is normalized to be the same. When they have different information costs, it sufficesto interpret the payoffs in (C2) as relative to each player’s individual information cost. It is important for existence of the sender’s best response that the waiting region is W = [ p ∗ , p ∗ ).See Lemma 15 in Appendix B. Notice that in the knife-edge case when p ∗ = η , there are two SMPEs, one as in (a.i) and another asin (a.ii). In the latter, however, π LR = ξ and the L -drifting strategy is not used in the waiting region.The two equilibria are payoff-equivalent but exhibit very different dynamic behavior when p ∈ [ p ∗ , ξ ]. ∗ ∈ (ˆ p, η ) : | —— | {z } pass π ℓL ←−←− | {z } jump: p ∗ π ←−←− | {z } jump: p ∗ p ∗ −→−→−→−→−→−→−→−→−→ | {z } R -drifting, jump to:0 p ∗ —————————— | {z } pass | p ∗ ∈ ( η,
1) : | —— | {z } pass π ℓL ←−←− | {z } jump: p ∗ π ←−←− | {z } jump: p ∗ p ∗ −→−→−→−→ | {z } jump:0 ξ |{z} stationary ←−←−←− | {z } jump: p ∗ π LR −→−→−→−→ | {z } jump:0 p ∗ ——— | {z } pass | Figure 7: The sender’s SMPE strategies in Proposition 2, that is, when v > U r ( p ∗ ) − U ℓ ( p ∗ ). left-jumps to ; for p = ξ , she uses the stationary strategy with jumps to or p ∗ ;and for p ∈ ( ξ, π LR ] , she adopts the L -drifting strategy with right-jumps to p ∗ .(b) Outside the waiting region: there exist cutoffs < π ℓL < π < p ∗ such that for p < π ℓL , the sender passes; for p ∈ ( π ℓL , π ) , she uses the L -drifting strategy withjumps to q = p ∗ ; and for p ∈ [ π , p ∗ ) , she uses the L -drifting strategy with jumps to q = p ∗ .The lower bound of the waiting region p ∗ converges to zero as c → . Figure 7 summarizes the sender’s SMPE strategy in Proposition 2, depending onwhether p ∗ < η or not. If p ∗ ∈ (ˆ p, η ), then the sender uses only R -drifting experiments inthe waiting region [ p ∗ , p ∗ ). If p ∗ > η , then she also employs L -drifting experiments andthe stationary experiment for a range of beliefs. In order to understand this difference,recall from Section 5.1 that R -drifting experiments are particularly useful when p is closeto p ∗ . If p ∗ < η , then the sender would not want to deviate from R -drifting at any p < p ∗ . If p ∗ > η , however, R -drifting is no longer uniformly optimal for the sender: if p is sufficiently small, then it would take too long for the sender to gradually move p to p ∗ through R -drifting. In that case, other experiments that are more risky but can generatefaster success than R -drifting experiments can be preferred. The structure of our modelthen implies that there exists a point ξ ∈ ( p ∗ , p ∗ ) around which L -drifting is optimalif p > ξ , while R -drifting is optimal if p < ξ . In other words, ξ is an absorbing pointtowards which the belief converges from both sides in the absence of jumps. Consequently,at p = ξ , the sender plays the stationary strategy and the belief p does not drift any longer.For an economic intuition, consider a salesperson courting a potentially interestedbuyer. If the buyer needs only a bit more reassurance to buy the product, then thesalesperson should adopt a conservative low-risk pitch that slowly “works up” the buyer.The salesperson may still “slip off” and lose the buyer (i.e., p jumps down to 0). But mostlikely, the salesperson “weathers” that risk and gets the buyer over the last hurdle (i.e., q = p ∗ is reached). This is exactly what our equilibrium persuasion dynamics describeswhen p is close to p ∗ . If p ∗ < η , this holds for all p ∈ [ p ∗ , p ∗ ) since the buyer does not Although the sender plays the stationary strategy only at one belief ξ , in the absence of Poissonjumps, p reaches ξ in finite time whenever it starts from p ∈ [ p ∗ , π LR ]. Therefore, the sender’s strategyat ξ has a significant impact on the players’ expected payoffs. p ∗ > η , the buyer requires a lotof convincing and there are beliefs where the buyer is rather uninterested (as in a “cold”call). Then the salesperson must offer a big pitch that can quickly overcome the buyer’sskepticism. It is likely to fail, but may generate quick, unexpected success, which is betterthan to spend a significant amount of time in order to slowly convince a skeptical buyer.Condition (C2) implies that p ∗ is determined by the receiver’s incentive: p ∗ is the pointat which the receiver is indifferent between taking action ℓ immediately and waiting (i.e., U ℓ ( p ∗ ) = U ( p ∗ )). Intuitively, (C2) suggests that the receiver gains less from experimenta-tion, and is thus less willing to continue, than the sender. Therefore, at the lower bound p ∗ , the receiver wants to stop, even though the sender wants to continue persuading thereceiver (i.e., V ( p ∗ ) > p < p ∗ , the sender plays only L -drifting experiments, unless she prefers to pass(below π ℓL ). This is intuitive, because the receiver would take action ℓ immediately unlessthe sender generates an instantaneous jump. It is intriguing, though, that the sender’starget posterior can be either p ∗ or p ∗ , depending on how close p is to p ∗ : in the salescontext used above, if the buyer is fairly skeptical, then the salesperson needs to use a bigpitch. But, depending on how skeptical the buyer is, she may try to get enough attentiononly for the buyer to stay engaged (targeting q = p ∗ ) or use an even bigger pitch toconvince the prospect to buy outright (targeting q = p ∗ ). If p is just below p ∗ (see p inFigure 6), then the sender can jump into the waiting region at a high rate: note that thearrival rate approaches ∞ as p tends to p ∗ . In this case, it is optimal to target p ∗ , therebymaximizing the arrival rate of Poisson jumps: the salesperson is sufficiently optimisticabout her chance of grabbing the buyer’s attention, so she only aims to make the buyerstay. If p is rather far away from p ∗ (see p in Figure 6), then the sender does not enjoya high arrival rate. In this case, it is optimal to maximize the sender’s payoff conditionalon Poisson jumps, which she gets by targeting p ∗ : the salesperson tries to sell her productright away and if it does not succeed, then she just lets it go.Next, we provide an equilibrium characterization for the case when (C2) is violated. Proposition 3.
Fix p ∗ ∈ (ˆ p, and assume that v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ) . For each c suffi-ciently small, there exists a unique SMPE such that the waiting region has upper bound p ∗ . The waiting region is W = ( p ∗ , p ∗ ) for some p ∗ < ˆ p , and the sender’s equilibriumstrategy is as follows: (a.i) In the waiting region when p ∗ ∈ (ˆ p, η ) : there exists a cutoff π LR ∈ W such that for p ∈ ( π LR , p ∗ ) , the sender uses the R -drifting strategy with left-jumps to ; and for p ∈ ( p ∗ , π LR ) , she uses the L -drifting strategy with right-jumps to p ∗ .(a.ii) In the waiting region when p ∗ ∈ ( η, : there exist cutoffs p ∗ < π LR < ξ < π LR < p ∗ ,such that for p ∈ [ π LR , ξ ) ∪ [ π LR , p ∗ ) , the sender plays the R -drifting strategy with It is important that the waiting region is an open interval. Otherwise the strategy profile violatesadmissibility at p ∗ . See Appendix A.1. ∗ ∈ (ˆ p, η ) : | ——— | {z } pass p ∗ ←−←−←−←−←− | {z } L -drifting, jump: p ∗ π LR −→−→−→−→−→−→−→−→ | {z } R -drifting, jump:0 p ∗ —————————— | {z } pass | p ∗ ∈ ( η,
1) : | ——— | {z } pass p ∗ ←−←−←− | {z } jump: p ∗ π LR −→−→−→−→ | {z } jump:0 ξ |{z} stationary ←−←−←−←− | {z } jump: p ∗ π LR −→−→−→−→ | {z } jump: 0 p ∗ ——— | {z } pass | Figure 8: The sender’s SMPE strategy in Proposition 3, that is, when v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ). left-jumps to ; for p = ξ , she adopts the stationary strategy with jumps to or p ∗ ;and for p ∈ ( p ∗ , π LR ) ∪ ( ξ, π LR ) , she uses the L -drifting strategy with right-jumps to p ∗ .(b) Outside the waiting region the sender passes.The lower bound of the waiting region p ∗ converges to zero as c tends to . Figure 8 describes the persuasion dynamics in Proposition 3. There are two maindifferences from Proposition 2. First, if p < p ∗ then the sender simply passes: recallthat in Proposition 2, there exists π ℓL ∈ (0 , p ∗ ) such that the sender plays L -driftingexperiments whenever p ∈ ( π ℓL , p ∗ ). Second, when p is just above p ∗ , the sender adopts L -drifting experiments, and thus the game may stop at p ∗ : recall that in Proposition 2,the sender always plays R -drifting experiments just above p ∗ , and the game never endsby the belief reaching p ∗ . Both these difference are precisely due to the failure of (C2): if v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ) then the sender is less willing to continue than the receiver, and thus p ∗ is determined by the sender’s participation constraint (i.e., V ( p ∗ ) = 0). This impliesthat the sender has no incentive to experiment once p falls below p ∗ . In addition, when p is just above p ∗ , the sender goes for a big pitch by targeting p ∗ and, therefore, play L -drifting experiments. Invoking the salesperson’s problem again, (C2) fails (i.e., v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ) holds)when either the salesperson is not so motivated, perhaps because of her compensationstructure, or the stake to the buyer is sufficiently large. In this case, the salespersonprefers to make big sales pitches that can sell the product quickly (i.e., targeting p ∗ ) untilshe runs down the buyer’s confidence to p ∗ . To provide further intuition for the difference, we note that in Proposition 2, p ∗ is determined by thereceiver’s participation constraint. In other words, the receiver does not let the sender experiment forall beliefs where the sender would like to. When using the confidence spending L -drifting strategy, thisimplies that the sender is stopped earlier than she would like, which diminishes its value. By contrast,the R -drifting strategy’s value is unaffected by a high p ∗ , since it builds confidence and moves the beliefaway from p ∗ . Hence, the sender does not use the L -drifting strategy stopping at p ∗ in Proposition 2. Concluding Discussions
We conclude by discussing two important modeling assumptions and suggesting somedirections for future research.
Our timing structure as well as feasible information structures can be micro-founded bythe following discrete time model. Time consists of discrete periods k = 0 , , ..., withperiod length ∆. We consider only short period lengths ∆ ∈ (0 , /λ ). At the beginning ofperiod k = 0 , , ... , (unless the game has ended before), the sender performs experimentswhose outcomes are realized after the elapse of time ∆. At the end of that period, afterobserving the experiments and their outcomes, the receiver takes an action a ∈ { ℓ, r, w } .Either of the first two actions ends the game, whereas w moves the game to the nextperiod k + 1.In any period k , the sender allocates a share α i of resources to experiment i , whichhas the following binary form: Binary-signal experiment i state/signal L -signal R -signal L x i − x i R − y i y i An information structure in discrete time is given by a countable set of experiments { ( α i , x i , y i ) } i ∈ N with weights α i , as in the continuous time formulation. We impose theconstraints ( α i , x i , y i ) ∈ [0 , and 1 ≤ x i + y i ≤ α i λ ∆ for all i , and P i ∈ N α i ≤ The shares could be interpreted as the intensities of messages sent simultaneously.Observe that as ∆ → x i + y i → i (i.e., the experiments become unin-formative), which again captures the idea that information takes time to generate andcommunicate. It is routine to show that a mixture of Poisson experiments ( α i , q i ) i ∈ N sat-isfying the arrival rates (1) and the drift rates (2) can be obtained in the limit as ∆ → The players in our model incur flow costs but they do not discount their future payoffs.This assumption simplifies the analysis significantly. In particular, it allows us to derivethe value functions associated with the three main modes of persuasion in closed form. Theassumption, however, has no qualitative impact on our main results. Specifically, consider This discrete-time foundation for our class of Poisson experiments appeared in Che and Mierendorff(2019). Arguably, the richness of the experiments did not play an important role in that paper, sinceconclusive experiments with q i = 0 or q i = 1 always prove optimal for a single decision maker.
24n extension of our model in which there are both flow costs and discounting. SMPEsin such a model would converge to those in our model as the discount rate converges tozero. Therefore, all of our results are robust to the introduction of a small amount ofdiscounting.If one considers a model with only discounting (i.e., without flow cost c ), however,the persuasion dynamics presented in Section 5 needs some modification. Among otherthings, the sender will never voluntarily stop experimentation: recall that in our flow-costmodel, the sender never experiments if the belief p is sufficiently close to 0 (below π ℓL ).When there is only discounting, the opportunity cost of experimentation is 0, no matterhow small p is (i.e., no matter how unlikely persuasion is to succeed). This implies that thelower bound p ∗ of the waiting region is always determined by the receiver’s participationconstraint, as in our Proposition 2. In addition, at the lower bound p ∗ , the sender willplay either R -drifting experiments or the stationary strategy, because she always prefersplaying the stationary strategy (which ensures the belief to stay within the waiting region)to L -drifting experiments at p ∗ (which may move the belief out of the waiting region).Nevertheless, the main economic lessons from our flow-cost model are likely to applyto the discounting version. Specifically, all three theorems in Section 4 would continue tohold. Furthermore, the advantages of the three main modes of persuasion remain un-changed. Therefore, the persuasion dynamics is also likely to be similar to those describedin Propositions 2 and 3, provided that the belief is not so close to p ∗ . In particular, if p is rather close to p ∗ , then the sender will play only R -drifting experiments. But, if p issufficiently far away from p ∗ , then the sender combines the three modes of persuasion asin the case of p ∗ > η in Propositions 2 and 3. We studied a dynamic persuasion model in which real information takes time to generateand neither the sender nor the receiver has commitment power over future actions. Thereare several variations of our model that could be worth investigating. For example, onemay consider a model in which the sender faces the same flow information constraint as inour model but has full commitment power over her dynamic strategy: given our discussionin Section 3, it is straightforward that the sender can approximately implement the KGoutcome. However, it is non-trivial to characterize the sender’s optimal dynamic strategy. The proofs of Theorems 1 and 3 can be readily modified. For Theorem 2, one can show that themain economic logic behind it (namely, “the power of beliefs” explained at the end of Section 3) holdsalso with discounting. To be formal, let ρ denote the common discount rate. Then, it suffices to modifythe argument in footnote 17 as follows: ρU r ( p ∗ ) ≤ λ (1 − p ∗ )( u Lℓ − U r ( p ∗ )) + λp ∗ (1 − p ∗ ) U ′ r ( p ∗ ) ⇔ p ∗ ≤ p := (cid:0) u Lℓ − u Lr (cid:1) λ − u Lr ρ (cid:0) u Lℓ − u Lr (cid:1) λ + ( u Rr − u Lr ) ρ . Notice that p approaches 1 as ρ tends to 0. at each point in time mightappear extremely complex to analyze, and a clear analysis might seem unlikely. Yet, themodel produced a remarkably precise characterization of the sender’s optimal choice ofinformation—namely, not just when to stop acquiring information but more importantly what type of information to search for. This modeling innovation may fruitfully apply toother dynamic settings. 26 Continuous Time Formulation
A.1 Markov Strategies in Continuous time
An information structure was defined as a collection ( α i , q i ) i ∈ N of experiments, where each( α i , q i ) specifies a Poisson experiment with jump-targets q i and associated weight α i . Theset of feasible information structures is thus given by I = ( ( α i , q i ) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i ≥ ∞ X i =1 α i ≤ q i ∈ [0 , ) . We define a game in Markov strategies. The sender’s strategy is a measurable function σ S : [0 , → I that maps the belief p to an information structure σ S ( p ). The receiver’sstrategy is a measurable function σ R : [0 , → A , that maps the belief p to an action σ R ( p ) ∈ A := { ℓ, r, w } . We impose the following admissibility restrictions in order toensure that a strategy profile σ = ( σ S , σ R ) yields a well defined outcome. Admissible Strategies for the Sender.
Our first restriction ensures that σ S givesrise to a well-defined evolution of the (common) belief about the state. For a Markovstrategy σ S ( p ), with experiments ( α i ( p ; σ S ) , q i ( p ; σ S )) i ∈ N , Bayesian updating leads to thefollowing integral equation for the belief p t : p t = p e − λ R t ( α + s − α − s ) ds p e − λ R t ( α + s − α − s ) ds + (1 − p ) , (4)where α + t = P i : q i ( p t ; σ S ) >p t α i ( p t ; σ S ) is the total weight on upward jumps at time t , and α − t = P i : q i ( p t ; σ S )
0, let ˜ p ( k +1)∆ = ˜ p k e − λ ∆( α + k ∆ − α − k ∆ ) ˜ p k e − λ ∆( α + k ∆ − α − k ∆ ) + (1 − ˜ p k ) . This can be used to define ˜ p k ∆ recursively for each ˜ p = p , and yields a step-function p ∆ t := ˜ p ⌊ t/ ∆ ⌋ ∆ . Definition 2.
A measurable function σ S : [0 , → I is an admissible strategy for the We can take I to be a subset of R N , the set of sequences ( α , q ) , ( α , q ) , . . . in R , with the product σ -algebra B ( R N ) = B ( R ) ⊗ B ( R ) ⊗ . . . , where B ( R ) is the Borel σ -algebra on R . In this part, we follow Klein and Rady (2011), with the difference that in their model, the evolutionof beliefs is jointly controlled by two players. Given that in our model, only the sender controls theinformation structures, we can dispense with their assumption that Markov strategies are constant onthe elements of a finite interval partition of the state space. The corresponding differential equation is given by ˙ p t = − (cid:0) α + t − α − t (cid:1) λp t (1 − p t ) . ender if for all p ∈ [0 , ∆ → p ∆ t exists andsolves (4).This definition imposes two restrictions on Markov strategies. First, there must be asolution to (4). Indeed, there are Markov strategies for which no solution exists. Consider,for example, a strategy of the following form: σ S ( p ) = ( α = 1 , q = 1) , if p ≥ p ′ , ( α = 1 , q = 0) , if p < p ′ . This strategy does not lead to a well-defined evolution of the belief if p is given by the“absorbing belief” p = p ′ . To satisfy admissibility, we can set σ S ( p ) = (( α = 1 / , q =1) , ( α = 1 / , q = 0)), while keeping the strategy otherwise unchanged.The second restriction guarantees that if there are multiple solutions, we can selectone of them by taking the pointwise limit of the discrete time approximation. Consider,for example, the following strategy: σ S ( p ) = ( α = 1 , q = 0) , if p ≥ p ′ , ( α = 1 , q = 1) , if p < p ′ . (5)If p = p ′ , then there is an “obvious” solution p t = p ′ e λt p ′ e λt +(1 − p ′ ) > p ′ for t >
0. However,there exists another solution p t = p ′ e − λt p ′ e − λt +(1 − p ′ ) consistent with p = p ′ . But, in discretetime, ˜ p ∆ > p ′ for any ∆ >
0, and thus lim ∆ → p ∆ t = p t . This means that the strategyin (5) is admissible, while the latter strategy with p t is not. In general, when there aremultiple solutions, admissibility enables us to select the “obvious” one that would beobtained from the discrete time approximation. With this selection, admissibility of thesender’s strategy guarantees a well defined belief for all t > p . Admissible Strategy Profiles.
In addition to a well defined evolution of beliefs, weneed to ensure that a strategy profile σ = ( σ S , σ R ) leads to a well defined stopping timefor any initial belief p . Consider for example the function σ R ( p ) = w if p ≤ p ′ ,r if p > p ′ . If the sender uses the (admissible) Markov strategy given by σ S ( p ) = ( α = 1 , q = 0) for all p , and the prior belief is p < p ′ , then the function σ R ( p ) does not lead to a well-definedstopping time. To be concrete, suppose that the true state is ω = R . In this case, no28oisson jumps occur, and the belief drifts upwards. Let t ′ denote the time at which thebelief reaches p ′ . The receiver’s strategy implies that for any t ≤ t ′ , the receiver plays w and for any t > t ′ , the receiver has stopped before t . Hence, the stopping time is not welldefined. Clearly, the following modified strategy fixes the problem:ˆ σ R ′ ( p ) = w if p > p ′ ,r if p ≤ p ′ . This example demonstrates that we need a joint restriction on the sender’s and the re-ceiver’s strategies to ensure a well defined outcome.To formally define admissibility, we need the following notation: for a given strategyof the receiver σ R , let W = (cid:8) p ∈ [0 , (cid:12)(cid:12) σ R ( p ) = w (cid:9) and S = [0 , \ W be the receiver’s waiting region and stopping region , respectively, and denote the closures of these sets by W and S . Definition 3.
A strategy profile σ = ( σ S , σ R ) is admissible if (i) σ S is an admissiblestrategy for the sender, and (ii) for each p ∈ W ∩ S , either p ∈ S , or if p / ∈ S , then thereexits ε > p t ( p ) ∈ W for all t < ε , where p t ( p ) = lim ∆ → p ∆ t is the selectedsolution to (4) with p = p .Requirement (i) guarantees that the sender’s strategy gives rise to a well defined beliefat all t > p ∈ W , the belief evolution is such that absent jumps the beliefremains in the waiting region.One may wonder why we do not simply require that the stopping region is a closedset. This is stronger than requirement (ii) and it turns out that in some cases it can leadto non-existence of an equilibrium. Payoffs and Equilibrium.
Let σ = ( σ S , σ R ) be a profile of strategies. If σ is not admis-sible, then both players receive −∞ from playing the strategy profile. If σ is admissible,then for each prior belief p , both players’ expected payoffs are well-defined: V σ ( p ) = v P (cid:2) σ R ( p τ ) = r (cid:12)(cid:12) p (cid:3) − c E (cid:20)Z τ { P α i ( p t ) =0 } dt (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21) for the sender, and U σ ( p ) = E (cid:20) U σ R ( p τ ) ( p τ ) − c Z τ { P α i ( p t ) =0 } dt (cid:12)(cid:12)(cid:12)(cid:12) p (cid:21) For example, in the equilibrium characterized in Proposition 2, we have S = [0 , p ∗ ) ∪ [ p ∗ , S to be closed and set S = [0 , p ∗ ] ∪ [ p ∗ ,
1] instead, the sender does not have a best response for p ∈ ( π , p ∗ ) since v ( p ; q i ) fails to be upper semi-continuous in q i at q i = p ∗ . τ is the stopping time defined by the strategy profile and p τ is thebelief when the receiver stops. Definition 4 (Markov Perfect Equilibrium) . An admissible strategy profile σ = ( σ S , σ R )is a Markov perfect equilibrium (MPE), if(i) for any p ∈ [0 ,
1] and any admissible strategy profile ˆ σ = (ˆ σ S , σ R ), V ˆ σ ( p ) ≤ V σ ( p ),(ii) for any p ∈ [0 ,
1] and any admissible strategy profile ˆ σ = ( σ S , ˆ σ R ), U ˆ σ ( p ) ≤ U σ ( p ),and(iii) for any p ∈ S : (refinement) σ S ( p ) ∈ arg max ( α i ,q i ) ∈I X i : q i = p α i λp (1 − p ) | q i − p | (cid:0) V σ ( q i ) − { σ R ( p )= r } v (cid:1) − { P α i =0 } c. Parts (i) and (ii) in this definition require that no player have a profitable deviationto a Markov strategy that, together with the opponent’s strategy, forms an admissiblestrategy profile. Part (iii) formalizes our refinement. We do not explicitly require thatdeviations to non-Markov strategies should not be profitable. This requirement is in facthard to formulate since we do not define a game that allows for non-Markov strategies.However, given the opponent’s strategy, each player faces a Markov decision problem.Therefore, if there is a policy in this decision problem that yields a higher payoff than thecandidate equilibrium strategy, then there is also a profitable deviation that is Markov.
A.2 Belief-free formulation of feasible information structures incontinuous time.
In this section, we formulate the class of feasible information in continuous time withoutreference to beliefs. Denote by ˜ I the set of information structures available to the sender.We have ˜ I = ( ( α i , γ i , ω i ) i ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α i ≥ ∞ X i =1 α i ≤ γ i ∈ [ λ, ∞ ); ω i ∈ { L, R } ) . Let ˜ I = ( α i , γ i , ω i ) i ∈ N ∈ ˜ I be a typical information structure. As discussed in section2, α i is the share of resources allocated to experiment i . An information structure with P ∞ i =1 α i = 0 corresponds to “passing”. For an experiment with α i >
0, the parameter γ i controls the arrival rate of jumps that the experiment generates, and the experiment i generates Poisson jumps at rate α i γ i if the true state is ω = ω i , and at rate α i ( γ i − λ ) ifthe true state is ω = ω i .This specification implies that if the current belief is p t , a jump from an experiment30ith ω i = R leads to a posterior q ( p t , γ i , R ) = p t γ i p t γ i + (1 − p t ) ( γ i − λ ) . Hence the sender can choose any jump target q > p t by setting (where we solved theprevious formula for γ i ): γ i = λ (1 − p t ) qq − p t . Note that this expression for γ i (multiplied by α i ) is precisely the conditional arrival ratein state R of the experiment ( α i , q i ) for q i > p in Section 2. Similarly the arrival rate instate L is γ i − λ = λ (1 − p t ) q − λ ( q − p t ) q − p t = λp t (1 − q ) q − p t . Hence, the experiments ( α i , γ i , R ) ∈ [0 , × [ λ, ∞ ) × { R } specify the Poisson experimentswith upward jumps that were described with reference to current beliefs and posteriorsin Section 2. Similarly, experiments ( α i , γ i , L ) ∈ [0 , × [ λ, ∞ ) × { L } specify experimentswith downward jumps. Therefore, the set of information structures ˜ I describes preciselythe feasible information structures available to the sender as defined in the main text (orthe set I in Appendix A.1). 31 Proofs of Propositions 2 and 3
This appendix provides formal proofs for Propositions 2 and 3, from which Theorem 2and Corollary 1 are also straightforward to obtain. In Section B.1, we derive the generalvalue functions that correspond to the three policies, stationary, R -drifting, and L -driftingexperiments. In Section B.2, we define the various cutoffs used in Propositions 2 and 3.In Section B.3, we show that the strategy profiles given in Propositions 2 and 3 (andsupplemented by the cutoffs defined in Section B.2) indeed constitute equilibria. Finally,in Section B.4, we show that for c sufficiently small, the strategy profiles in Propositions2 and 3 are a unique SMPE in each case. To avoid breaking the flow in the presentation,proofs of most Lemmas are relegated to Appendix C.The main challenge in the proofs is to characterize the best response of the senderif the receiver uses a strategy with waiting region ( p ∗ , p ∗ ), or [ p ∗ , p ∗ ). Inside the waitingregion, the value function of the sender’s best response must be non-negative and satisfythe HJB equation c = max ( α i ,q i ) i ∈ N ∈I X q i = p α i v ( p ; q i ) , (HJB)where I denotes the set of feasible information structures (see Appendix A.1). If thevalue function satisfies the conditions of Proposition 1.b, then the maximization problemon the right-hand side can be restricted to information structures ( α i , q i ) i ∈ N ∈ I s , where I s is the set of simple information structures I s = { ( α i , q i ) i ∈ N | q = p ∗ , q = 0 , α = 1 − α , and α i = 0 for i > } , so that the sender’s problem is to choose a single variable α = α . This simplifies theHJB equation to c = λp (1 − p ) max α (cid:20) α v − V ( p ) p ∗ − p − (1 − α ) V ( p ) p − (2 α − V ′ ( p ) (cid:21) . (HJB-S)Characterizing solutions to (HJB-S) involves the three policies discussed in Section B.1and the cutoffs in Section B.2. In Lemma 15 in Section B.3, we characterize the sender’s(unrestricted) best response and show that it satisfies the restriction ( α i , q i ) i ∈ N ∈ I s .To prove this, we first construct the best response and the corresponding value functionunder the restriction ( α i , q i ) i ∈ N ∈ I s . We then show that the value function satisfies theconditions of Proposition 1.b. This allows us to appeal to Lemmas 13 and 14 (also inSubsection B.3), to show that the candidate value function satisfies (HJB), which impliesthat the sender’s best response under the restriction ( α i , q i ) i ∈ N ∈ I s is also an unrestrictedbest response. This indirect way of characterizing the sender’s unrestricted best response If the value function is not everywhere differentiable, it is a viscosity solution of the HJB equation.
32s necessary since we do not have a direct proof that the value function satisfies theconditions of Proposition 1.b. The rest of Section B.3 verifies that the strategy profiles given in Propositions 2 and3 specify a best response for the sender in the stopping region (Lemma 16), and likewiseverifies that the receiver plays a best response to the sender’s strategy (Sections B.3.4and B.3.5). In Section B.4, we show uniqueness. Given the general observation thatfor any waiting region W = [ p ∗ , p ∗ ) or W = ( p ∗ , p ∗ ), the sender’s best response uses onlyexperiments in I s , we can use the characterization of the sender’s best response in Lemma15 in the uniqueness proof. B.1 Value Functions
In this section, we derive value functions for the stationary strategy, and the R - and L -drifting policies, and prove crucial properties for these that will be used in later proofs.As discussed in the introductory paragraph to this appendix, if the sender is restricted toinformation structures in I s , the sender’s value function V ( p ) satisfies (HJB-S). Similarly,the receiver’s value function satisfies c = λp (1 − p ) (cid:20) α U r ( p ∗ ) − U ( p ) p ∗ − p − (1 − α ) u Lℓ − U ( p ) p − (2 α − U ′ ( p ) (cid:21) . (HJB-R)Note that this is not the HJB equation for the receiver’s problem, but it can be used toderive the receiver’s payoff for any given information acquisition policy of the sender. B.1.1 Stationary Strategy
Let V S ( p ) denote the sender’s expected payoff from playing the stationary strategy atbelief p (assuming that the receiver waits at belief p ). In the stationary strategy, thesender devotes equal attention to jumps to 0 and p ∗ . Setting α = 1 / V S ( p ) = λp (1 − p ) v − c ( p ∗ − p ) λp ∗ (1 − p ) = pp ∗ v − C S ( p ) , (6)where C S ( p ) = 2 c ( p ∗ − p ) λp ∗ (1 − p ) . (7) Consider for example convexity: Standard arguments to not apply since the sender’s payoff dependson the receiver’s strategy so that the sender’s value cannot be written as the envelope of linear functions. Intuitively, under the stationary strategy, the belief stays constant in the absence of a jump. Inaddition, if a jump occurs, either to 0 or to p ∗ , then the game ends immediately. Therefore, the sender’sbenefit of playing the strategy is equal to the probability of jumping to p ∗ times v , while the associatedcost is equal to the expected time of the first jump, either to 0 or p ∗ , times c . For the receiver’s valuefunction (below), it suffices to take into account that his payoff is equal to u Lℓ if p jumps to 0 and equalto U r ( p ∗ ) if p jumps to p ∗ . α = 1 / U S ( p ) = p ∗ − pp ∗ u Lℓ + pp ∗ U r ( p ∗ ) − C S ( p ) . (8)Note that C S ( p ) is concave, and thus both V S ( p ) and U S ( p ) are convex over (0 , p ∗ ): V ′′ S ( p ) = U ′′ S ( p ) = − C ′′ ( p ) = − c (1 − p ∗ ) λp ∗ (1 − p ) > . (9) B.1.2 R -drifting StrategiesGeneric value functions: If the sender uses an R -drifting experiment with jump-target zero for an interval of beliefs, then the sender’s value function satisfies (HJB-S)with α = 0 on that interval: c = − λ (1 − p ) V + ( p ) + λp (1 − p ) V ′ + ( p ) . (10)Here, V + ( p ) denotes a generic function that satisfies (10); “+” signifies the upward driftof the belief. If the sender uses the R -drifting strategy until the belief reaches a stoppingbound q ( > p ), and the sender’s value at q is given by the boundary condition V ( q ) = X ,we can solve the ODE (10) to obtain the particular solution V + ( p ; q, X ) = pq X − C + ( p ; q ) , where C + ( p ; q ) = (cid:18) p log (cid:18) q − q − pp (cid:19) + 1 − pq (cid:19) cλ . Similarly, let U + ( p ) denote the receiver’s generic value function from the R -driftingstrategy with jumps to zero. It satisfies (HJB-R) with α = 0: c = λ (1 − p ) (cid:0) u Lℓ − U + ( p ) (cid:1) + λp (1 − p ) U ′ + ( p ) . (11)For a boundary condition U ( q ) = X , we obtain the particular solution U + ( p ; q, X ) = q − pq u Lℓ + pq X − C + ( p ; q ) . Note that, as for V S ( p ) and U S ( p ), V + ( p ) and U − ( p ) are convex: V ′′ + ( p ; q, X ) = U ′′ + ( p ; q, X ) = − C ′′ + ( p ; q ) = cλp (1 − p ) > . (12)Here and in the following, we are slightly abusing notation for the partial derivatives of34hese functions with respect to p , e.g., V ′′ + ( p ; q, X ) = ∂ V + ( p ; q, X ) /∂p . R -drifting with stopping bound p ∗ : In Propositions 2 and 3, for p close to p ∗ thesender uses the R -drifting strategy with a stopping bound p ∗ , and at p ∗ the receivertakes action r immediately. Hence, we have the boundary condition V ( p ∗ ) = v for thesender and U ( p ∗ ) = U r ( p ∗ ) for the receiver. To avoid cluttering notation, we denote thecorresponding value functions by V R ( p ) and U R ( p ), suppressing the dependence on p ∗ .Then, we have V R ( p ) := V + ( p ; p ∗ , v ) = pp ∗ v − C + ( p ; p ∗ ) , (13) U R ( p ) := U + ( p ; p ∗ , U r ( p ∗ )) = p ∗ − pp ∗ u Lℓ + pp ∗ U r ( p ∗ ) − C + ( p ; p ∗ ) . (14) Comparison of V R ( p ) and V S ( p ) : Clearly, the sender will not use R -drifting withstopping bound p ∗ at beliefs where V R ( p ) < V S ( p ). We now derive conditions under which V R ( p ) < V S ( p ) can arise for some beliefs p ∈ (0 , p ∗ ).From (6) and (8), we have V S ( p ∗ ) = v = V R ( p ∗ ) , (15) V ′ S ( p ∗ ) = vp ∗ + 2 cλp ∗ (1 − p ∗ ) > vp ∗ + cλp ∗ (1 − p ∗ ) = V ′ R ( p ∗ ) , (16) V S (0) = − cλ < − cλ = lim p → V R ( p ) . (17)Hence, V S ( p ) is dominated by V R ( p ) for p close to 0 and p ∗ . This implies that if V S ( p ) >V R ( p ∗ ) for some p ∈ (0 , p ∗ ), then there must be at least two intersection points.We begin by characterizing intersections between V S ( p ) and V + ( p ) (the latter being ageneric solution to (10)). For each p ∗ ≥ /
9, define the following two cutoffs: ξ ( p ∗ ) := 3 p ∗ − s(cid:18) p ∗ (cid:19) − p ∗ ∈ (cid:18) , p ∗ (cid:19) ,ξ ( p ∗ ) := 3 p ∗ s(cid:18) p ∗ (cid:19) − p ∗ ∈ (cid:18) p ∗ , p ∗ (cid:19) .ξ in Propositions 2 and 3 corresponds to ξ ( p ∗ ). In what follows, we will use ξ instead of ξ . Lemma 1.
Fix p ∗ ∈ (ˆ p, , let V + ( p ) be a solution to (10) , and suppose that V + ( p ) = V S ( p ) for some p ∈ [0 , p ∗ ] .(a) If p ∗ < / , then V ′ + ( p ) < V ′ S ( p ) . b) If p ∗ ≥ / , then V ′ + ( p ) < V ′ S ( p ) if p / ∈ [ ξ ( p ∗ ) , ξ ( p ∗ )] , = V ′ S ( p ) if p ∈ { ξ ( p ∗ ) , ξ ( p ∗ ) } ,> V ′ S ( p ) if p ∈ ( ξ ( p ∗ ) , ξ ( p ∗ )) . Lemma 1.a implies that if p ∗ < / V + can cross V S only from above. Since thisrules out the possibility of at least two intersections, it follows that V S ( p ) < V R ( p ) for all p ∈ (0 , p ∗ ) if p ∗ < / V S and V R can intersect eachother. (17) implies that at the lowest intersection, which we call π SR , V R ( p ) must cross V S ( p ) from above. At the highest intersection which we call π SR , (15)–(16) imply that V R ( p ) must cross V S ( p ) from below. The only possibility for this pattern of intersections toarise is that π SR ∈ (0 , ξ ), π SR ∈ ( ξ , ξ ), and there are no other intersections. This impliesthat there are two intersections if V S ( ξ ( p ∗ )) > V R ( ξ ( p ∗ )), and there is no intersection if V S ( ξ ( p ∗ )) < V R ( ξ ( p ∗ )). If V S ( ξ ( p ∗ )) = V R ( ξ ( p ∗ )), then the two value functions do notintersect but touch each other at ξ ( p ∗ ). We define the constant η as the value for p ∗ forwhich this knife-edge case obtains: V S ( ξ ( η )) = V R ( ξ ( η )) . An explicit solution for η is not available, but numerically we obtain η = . > / V S ( p ) and V R ( p ), the following lemma shows that thetwo functions never intersect if p ∗ < η , and intersect exactly twice if p ∗ > η . Lemma 2. If p ∗ < η , then V R ( p ) > V S ( p ) for all p ∈ [0 , p ∗ ) . If p ∗ = η , then V R ( p ) ≥ V S ( p ) for all p < p ∗ , with equality holding only when p = ξ . Finally, if p ∗ > η , then there aretwo points of intersection π SR ∈ (0 , ξ ) and π SR ∈ ( ξ , ξ ) . R -drifting followed by stationary strategy: We have shown that at ξ , the senderstrictly prefers the stationary strategy to R -drifting with stopping bound p ∗ , if and onlyif p ∗ > η . In this case, Propositions 2 and 3 prescribe that for some p < ξ , the sender usethe R -drifting experiment until the belief drifts to ξ , where she switches to the stationarystrategy. Let V RS ( p ) and U RS ( p ) denote the value functions that derive from this dynamic Note that in this condition η enters both through ξ ( η ), and as an omitted argument p ∗ = η of V S and V R . That is, we have ξ ( η ) η v − c ( η − ξ ( η )) λη (1 − ξ ( η )) = ξ ( η ) η v − (cid:18) ξ ( η ) log (cid:18) η − η − ξ ( η ) ξ ( η ) (cid:19) + 1 − ξ ( η ) η (cid:19) cλ . p < ξ ) V RS ( p ) = V + ( p ; ξ , V S ( ξ )) = pp ∗ v − C + ( p, ξ ) − pξ C S ( ξ ) , (18) U RS ( p ) = U + ( p ; ξ , U S ( ξ )) = p ∗ − pp ∗ u Lℓ + pp ∗ U r ( p ∗ ) − C + ( p, ξ ) − pξ C S ( ξ ) . (19) B.1.3 L -drifting strategiesGeneric Value Functions: If the sender uses the L -drifting experiment with jumps to p ∗ , then her value function satisfies (HJB-S) with α = 1: c = λp (1 − p ) (cid:18) v − V − ( p ) p ∗ − p − V ′− ( p ) (cid:19) , (20)where we use V − ( p ) to denote a generic function that satisfies (20). For a stopping bound q < p , and a boundary condition V ( q ) = X , we obtain the particular solution V − ( p ; q, X ) := p − qp ∗ − q v + p ∗ − pp ∗ − q X − C − ( p ; q ) , (21)where C − ( p ; q ) = − p ∗ − pp ∗ (1 − p ∗ ) (cid:18) p ∗ log 1 − q − p + (1 − p ∗ ) log qp − log p ∗ − qp ∗ − p (cid:19) cλ . (22)Similarly, When α = 1, (HJB-R) simplifies to c = λp (1 − p ) (cid:18) U r ( p ∗ ) − U − ( p ) p ∗ − p − U ′− ( p ) (cid:19) . (23)For a boundary condition U ( q ) = X , we obtain the following generic solution: U − ( p ; q, X ) = p − qp ∗ − q U r ( p ∗ ) + p ∗ − pp ∗ − q X − C − ( p ; q ) . Again, the value functions V − ( p ; q, X ) and U − ( p ; q, X ) are convex in p : V ′′− ( p ; q, X ) = U ′′− ( p ; q, X ) = − C ′′− ( p ; q ) = ( p ∗ − p ) + p ∗ (1 − p ∗ ) p (1 − p ) ( p ∗ − p ) cλ > . (24) L -drifting with stopping bound p ∗ : In Proposition 3, the sender uses the L -driftingexperiment with a stopping bound equal to p ∗ . At p ∗ , the receiver takes action ℓ , whichyields the boundary conditions V ( p ∗ ) = 0 for the sender, and U ( p ∗ ) = U ℓ ( p ∗ ) for thereceiver. Denoting the associated value functions by V L ( p ) and U R ( p ), we get V L ( p ) = V − ( p ; p ∗ ,
0) = p − p ∗ p ∗ − p ∗ v − C − ( p ; p ∗ ) , (25)37 L ( p ) = U − ( p ; p ∗ , U ℓ ( p ∗ )) = p − p ∗ p ∗ − p ∗ U r ( p ∗ ) + p ∗ − pp ∗ − p ∗ U ℓ ( p ∗ ) − C − ( p ; p ∗ ) . (26) L -drifting followed by stationary strategy: If p ∗ > η , then Propositions 2 and 3also prescribe another L -drifting strategy: for an interval above ξ , the sender uses the L -drifting strategy with jumps to p ∗ until belief reaches ξ , at which point she switchesto the stationary strategy. We denote the values of this strategy by V LS ( p ) and U RS ( p ),respectively: V LS ( p ) = V − ( p ; ξ , V S ( ξ )) = pp ∗ v − C − ( p ; ξ ) − p ∗ − pp ∗ − ξ C S ( ξ ) , (27) U LS ( p ) = U − ( p ; ξ , U S ( ξ )) = p ∗ − pp ∗ u Lℓ + pp ∗ U r ( p ∗ ) − C − ( p ; ξ ) − p ∗ − pp ∗ − ξ C S ( ξ ) . (28) B.1.4 The Crossing Lemma
The following lemma provides a crossing condition for intersections of generic functions V + ( p ) and V − ( p ). Lemma 3. (Crossing Lemma) Let V + ( p ) be a solution to (10) , and V − ( p ) a solution to (20) . If V − ( p ) = V + ( p ) for some p ∈ (0 , p ∗ ) , thensign (cid:0) V ′ + ( p ) − V ′− ( p ) (cid:1) = sign ( V − ( p ) − V S ( p )) . Lemma 3 suggests that the crossing patterns between V + ( p ) and V − ( p ) are fully deter-mined by the relationship between V + ( p ) = V − ( p ) and V S ( p ). This leads to the followingcrucial observations. Lemma 4.
Suppose p ∗ ∈ [ η, . Then(a) V ′ S ( ξ ) = V ′ LS ( ξ ) = V ′ RS ( ξ ) , and(b) V RS ( p ) > V S ( p ) for all p ∈ [0 , ξ ) and V LS ( p ) > V S ( p ) for all p ∈ ( ξ , ξ ) . Moreover, the Crossing Lemma 3 will be crucial to show that the cutoffs used inPropositions 2 and 3 are well defined (see Section B.2), and that kinks in the valuefunctions of the strategies specified in Propositions 2 and 3 are convex (see the proof ofLemma 15).
B.1.5 Additional Notation for the Value Functions
For some proofs and derivations that follow, it is convenient to have a unified notationfor V R ( p ) and V RS ( p ), as many arguments apply to both in the same way. To this end,note that V S ( p ∗ ) = V R ( p ∗ ) = v , and thus V R ( p ) can be written as V R ( p ) = V + ( p ; p ∗ , V S ( p ∗ )) .
38n the other hand, we have V RS ( p ) = V + ( p ; ξ , V S ( ξ )). In other words, the two functionshave an identical structure except for the point at which the value of stationary strategyis used in the boundary condition. We therefore define q R := ( p ∗ if p ∗ ≤ η,ξ if p ∗ > η, and V RS ( p ; q R ) := V + ( p ; q R , V S ( q R )) . Note that this implies that V RS ( p ; q R ) = V R ( p ) if p ∗ ≤ η and V RS ( p ; q R ) = V RS ( p ) if p ∗ > η .We will also use a similar notation for the receiver: U RS ( p ; q R ) := U + ( p ; q R , U S ( q R )). B.2 Cutoffs
We proceed to formally define various cutoffs used in Propositions 2 and 3. As a generalrule for notation, we use φ • to denote cutoffs stemming indifference conditions of thereceiver, and π • to denote those stemming from indifference conditions of the sender. Thesubscripts refer to the strategies or actions between which the player is indifferent at therespective cutoff. For example, at π ℓL the sender is indifferent between action ℓ and the L -drifting strategy. B.2.1 p : Upper Bound of p ∗ In Propositions 2 and 3, the sender always plays R -drifting experiments if the belief p isclose to p ∗ . A necessary condition for this to be possible in equilibrium is that the receiverprefers waiting to taking action r , that is, U R ( p ) ≥ U r ( p ) for p ∈ ( p ∗ − ε, p ∗ ) for some ε >
0. The following lemma shows that this is satisfied if and only if p ∗ ≤ p , where p isdefined as p := 1 − u Lℓ − u Lr cλ . Lemma 5 (Crossing of U r and U R ) . For any p ∗ ∈ (ˆ p, , U ′ R ( p ∗ ) S U ′ r ( p ∗ ) , if and only if p ∗ S p . B.2.2 p ∗ in Proposition 2 In Proposition 2, the sender uses R -drifting experiments if p is slightly above p ∗ . Thelower bound p ∗ is then given by the belief φ ℓR at which the receiver is indifferent between R -drifting and taking action ℓ : U ℓ ( φ ℓR ) = U RS ( φ ℓR ; q R ) = ( U R ( φ ℓR ) if p ∗ ≤ η,U RS ( φ ℓR ) if p ∗ > η. (29)39lthough φ ℓR is not available in close form, the following lemma shows that the cutoff φ ℓR is well defined, and converges to zero as c tends to 0. In the second case ( p ∗ > η )the cutoff is well defined if U S ( ξ ) exceeds the stopping payoff, which is the case for c sufficiently small. Lemma 6.
Let p ∗ ∈ (ˆ p, .(a) If p ∗ ≤ η , then there exists a unique belief φ ℓR ∈ (0 , ˆ p ) that satisfies (29) , and φ ℓR → as c → .(b) If p ∗ > η , then for c sufficiently small, U S ( ξ ) > max { U ℓ ( ξ ) , U r ( ξ ) } and thereexists a unique belief φ ℓR ∈ (0 , ˆ p ) that satisfies (29) . In addition, φ ℓR → as c → . Let π ℓR denote a similar cutoff for the sender. This cutoff does not appear in theequilibrium characterization, but it will be useful when we verify the sender’s incentives. π ℓR is given by the sender’s indifference between action ℓ and the R -drifting strategy:0 = V RS ( π ℓR ; q R ) = ( V R ( φ ℓR ) if p ∗ ≤ η,V RS ( φ ℓR ) if p ∗ > η. (30)Convexity in p of V RS ( p ; q R ) and lim p → V RS ( p ; q R ) = − c/λ < π ℓR is welldefined for c sufficiently small. The next lemma shows that the relationship between φ ℓR and π ℓR is fully determinedby Condition C2. Intuitively, the player who gains more form persuasion has the lowerindifference belief. This also explains why in Proposition 2, where Condition C2 holds,we defined p ∗ = φ ℓR > π ℓR . Lemma 7. π ℓR S φ ℓR if and only if v T U r ( p ∗ ) − U ℓ ( p ∗ ) . B.2.3 p ∗ in Proposition 3 Proposition 3 specifies that for p slightly above p ∗ , the sender uses L -drifting experimentswith stopping bound p ∗ given by the belief π ℓL , at which the sender is indifferent betweenstopping with action ℓ and playing the L -drifting experiment for one more instant: − c dt + λπ ℓL (1 − π ℓL ) p ∗ − π ℓL v dt = 0 ⇐⇒ π ℓL λv − ( λv + c ) π ℓL + p ∗ c = 0 . (31)This quadratic equation has a unique solution in (0 , p ∗ ) given by π ℓL = 12 + c λv − s(cid:18)
12 + c λv (cid:19) − cp ∗ λv . (32) If p ∗ > η , then for c sufficiently small, V RS ( ξ ) = V S ( ξ ) >
0, so that the intermediate value theoremimplies existence of the cutoff.
40e show that the sender prefers a short period of L -drifting experiment followed by action ℓ , over immediate stopping with action ℓ , if and only if p ∗ ≥ π ℓL . Formally, we insert V ( p ∗ ) = 0 in (20) and use simple algebra to obtain V ′ L ( p ∗ ) T ⇐⇒ v T p ∗ − p ∗ p ∗ (1 − p ∗ ) cλ ⇐⇒ p ∗ T π ℓL . (33)Let φ ℓL denote a similar cutoff for the receiver, that is, the belief at which the receiveris indifferent between stopping with action ℓ and allowing the L -drifting experiment forone more instant: − c + λφ ℓL (1 − φ ℓL ) p ∗ − φ ℓL ( U r ( p ∗ ) − U ℓ ( φ ℓL )) − λφ ℓL (1 − φ ℓL ) U ′ ℓ ( φ ℓL ) = 0 . (34)The unique solution on (0 ,
1) is given by φ ℓL = c + λ ∆ U ( p ∗ ) − q ( c + λ ∆ U ( p ∗ )) − λc ∆ U ( p ∗ ) p ∗ λ ∆ U ( p ∗ ) , where ∆ U ( p ∗ ) = U r ( p ∗ ) − U ℓ ( p ∗ ). Analogously to π ℓL , the receiver prefers waiting totaking action ℓ immediately, if and only if p ≥ φ ℓL . Formally we have U ′ L ( p ∗ ) T U ′ ℓ ( p ∗ ) ⇐⇒ p ∗ T φ ℓL . (35)As for π ℓR and φ ℓR , Condition C2 determines the relationship between π ℓL and φ ℓL .As the following lemma shows, the player who benefits more from persuasion has thelower cutoff. This gives an intuition why for Proposition 3, where C2 is violated, we set p ∗ = π ℓL > φ ℓL . Lemma 8. π ℓL S φ ℓL if and only if v T U r ( p ∗ ) − U ℓ ( p ∗ ) . The following Lemma shows that for any belief p < p ∗ , the R -drifting strategy dom-inates the L -drifting strategy with stopping bound p ∗ = π ℓL if the cost is sufficientlylow. Lemma 9.
For any p ∈ (0 , p ∗ ) , there exists c ( p ) > such that for all c < c ( p ) , we have π ℓL < p and V L ( p ) < V R ( p ) when the stopping bound for V L is p ∗ = π ℓL . We will later use the following corollary of Lemma 9, which holds since by Lemma 2, V R ( ξ ) < V S ( ξ ) if p ∗ > η . The careful reader will wonder why we do not set p ∗ > π ℓL since the sender’s incentives only rule out p ∗ < π ℓL . The argument is more subtle and will become clear in Subsection B.3.5 as well as SubsectionB.4. Note however, that π ℓL → c →
0. Therefore Lemma 9 does not imply that the R -driftingstrategy dominates the L -drifting strategy on the whole waiting region. In the lemma, for c < c ( p ), wehave p ∗ = π ℓL < p , leaving room for an interval where the L -drifting strategy is not dominated by the R -drifting strategy. orollary 2. Suppose p ∗ > η . If c is sufficiently small and p ∗ = π ℓL , then V L ( ξ ) So far, we have considered cutoff beliefs at which the players are indifferent betweenexperimenting and stopping with some action. Now we define the cutoffs that arise fromthe sender’s indifference conditions between different experiments. π LR in Proposition 3: If p ∗ ≤ η , then we define the cutoff π LR as the point of indiffer-ence between the L -drifting strategy with stopping bound p ∗ and the R -drifting strategywith stopping bound p ∗ , that is, V L ( π LR ) = V R ( π LR ) , if p ∗ ≤ π ℓR . (36)If p ∗ > π ℓR , then V R ( p ∗ ) > V L ( p ∗ ), and we set π LR = p ∗ . The following lemma showsthat the cutoff is well defined. Lemma 10. Suppose p ∗ ∈ (ˆ p, η ] .(a) If c is sufficiently small and p ∗ ≤ π ℓR , then there exists a unique π LR ∈ [ p ∗ , p ∗ ) that solves (36) . Moreover, V L ( p ) > V R ( p ) if p ∈ [ p ∗ , π LR ) and V L ( p ) < V R ( p ) if p ∈ ( π LR , p ∗ ) . If p ∗ > π ℓR , then V L ( p ) < V R ( p ) for all p ∈ ( p ∗ , p ∗ ) .(b) If p ∗ = π ℓL , then π LR → as c → . π LR in Propositions 2 and 3: For p ∗ > η , we define π LR as the point of indifferencebetween R -drifting with stopping bound p ∗ and L -drifting followed by the stationarystrategy at ξ : V R ( π LR ) = V LS ( π LR ) . (37)The following lemma shows that the cutoff is well defined. Lemma 11. Suppose p ∗ ∈ ( η, . Then there is a unique π LR ∈ ( ξ , p ∗ ) that solves (37) .Moreover, V LS ( p ) > V R ( p ) if p ∈ [ ξ , π LR ) and V LS ( p ) < V R ( p ) if p ∈ ( π LR , p ∗ ) . π LR in Proposition 3: Proposition 3 uses the additional cutoff π LR . We define it as thepoint of indifference between L -drifting with stopping bound p ∗ and R -drifting followedby the stationary strategy at ξ : V L ( π LR ) = V RS ( π LR ) if p ∗ ≤ π ℓR . (38)42f p ∗ > π ℓR , then V RS ( p ∗ ) > π LR = p ∗ . The following Lemma shows thatthis cutoff is well defined if p ∗ < π ℓR and V L ( ξ ) < V S ( ξ ). Lemma 12. Suppose that p ∗ ∈ ( η, . If p ∗ < π ℓR and V L ( ξ ) < V S ( ξ ) , then thereexists a unique π LR ∈ ( p ∗ , ξ ) that solves (38) , V L ( p ) > V RS ( p ) if p ∈ [ p ∗ , π LR ) , and V L ( p ) < V RS ( p ) if p ∈ ( π LR , ξ ) . If p ∗ ≥ π ℓR , then V L ( p ) < V RS ( p ) for all p ∈ ( p ∗ , ξ ) . B.3 Equilibrium In this section we show that the strategy profiles specified in Proposition 2 and 3 areindeed equilibria when c is sufficiently small. B.3.1 Verifying Sender Optimality in the Waiting Region We first consider the sender’s strategies in the waiting region ( p ∗ , p ∗ ). The following lem-mas provide conditions that can be used to verify the optimality of the sender’s strategy.Note that they apply unchanged even if the waiting region is the half-open interval [ p ∗ , p ∗ ). Lemma 13. (Unimprovability) Let V ( p ) ≥ be a candidate value function for the senderthat is continuous on [ p ∗ , , is strictly convex on ( p ∗ , p ∗ ) , and satisfies V ( p ) = 0 for p < p ∗ , V ( p ) = v for p ≥ p ∗ , p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ + ) , and V ( p ) ≥ V S ( p ) for all p ∈ ( p ∗ , p ∗ ) .(a) If V ( p ) is differentiable and satisfies (20) at p ∈ ( p ∗ , p ∗ ) , then V ( p ) satisfies (HJB) at p . If V ( p ) > V S ( p ) , then L -drifting with jumps to p ∗ is the unique optimal policyat p .(b) If V ( p ) is differentiable and satisfies (10) at p ∈ ( p ∗ , p ∗ ) , then V ( p ) satisfies (HJB) at p . If V ( p ) > V S ( p ) , then R -drifting with jumps to is the unique optimal policyat p . If the sender’s value function is differentiable for all p ∈ ( p ∗ , p ∗ ), optimality of acandidate solution whose value function V ( p ) satisfies (20) or (10), is shown in two steps.First, if V ( p ) ≥ V S ( p ) for all p ∈ ( p ∗ , p ∗ ), then the strategy is optimal on the restrictedset of information structures I s , i.e., V ( p ) satisfies (HJB-S). Second, if it also satisfiesthe conditions of Proposition 1.b, then V ( p ) satisfies (HJB) and the candidate solutionis optimal on the unrestricted set of feasible information structures I . The next lemmaextends this to the case where the value function has convex kinks. In this case, we showthat the value function is a viscosity solution of (HJB). A standard verification argumentthen implies that the candidate solution is optimal. Lemma 14. (Unimprovability with Kinks) Let V ( p ) be a candidate value function thatsatisfies the conditions of Lemma 13. Suppose further that for any p ∈ ( p ∗ , p ∗ ) where Corollary 2 shows that V L ( ξ ) < V S ( ξ ) holds for c sufficiently small if p ∗ ∈ ( η, We define V ( p − ) := lim x ր p V ( x ), V ′ ( p − ) := lim x ր p V ′ ( x ), V ( p + ) := lim x ց p V ( x ), and V ′ ( p + ) :=lim x ց p V ′ ( x ). ( p ) is not differentiable, we have V ′ ( p − ) < V ′ ( p + ) , and there exists ε > such that V ( p ) satisfies (20) for p ′ ∈ ( p − ε, p ) ; V ( p ) satisfies (10) for p ′ ∈ ( p, p + ε ) . Then V ( p ) is a viscosity solution of (HJB) on ( p ∗ , p ∗ ) . Moreover, if p ∗ V ′ ( p ∗ + ) > V ( p ∗ + ) and V ( p ) > V S ( p ) , then the optimal policy is unique at p if V ( p ) is differentiable; and theoptimal policy at p can only be R -drifting with jumps to or L -drifting with jumps to p ∗ if V ( p ) is not differentiable at p . B.3.2 The Sender’s Best Response in the Waiting Region The following lemma characterizes the sender’s best response given the waiting region.Note that p ∗ > π ℓL in Proposition 2 and p ∗ = π ℓL in Proposition 3. Therefore, it suffices toconsider the case where p ∗ ≥ π ℓL . To verify that the strategy profiles in the Propositionsare equilibria, it suffices to consider p ∗ ≤ φ ℓR (cases (a)–(d) in the lemma). For theuniqueness proof, however, we need to consider all possible p ∗ ∈ [ π ℓL , ˆ p ). Lemma 15. Suppose that the receiver plays a strategy with waiting region W = [ p ∗ , p ∗ ) or W = ( p ∗ , p ∗ ) , where π ℓL ≤ p ∗ < ˆ p < p ∗ ≤ p . Then, for c sufficiently small, the sender’sbest response and the associated value function V ( p ) have the following properties in W :(a) If p ∗ ≤ η and p ∗ < π ℓR , then the sender’s best response is given by L -drifting with jump to p ∗ if p < π LR ,R -drifting with jump to if p ≥ π LR . Admissibility requires W = ( p ∗ , p ∗ ) in this case. The best response is unique for all p < p ∗ if p ∗ < η , and for all p = ξ if p = η . The corresponding value functionsatisfies V ( p ) = max { V L ( p ) , V R ( p ) } ≥ V S ( p ) , with strict inequality for all p < p ∗ if p ∗ < η , and for all p = ξ if p = η .(b) If p ∗ ≤ η and p ∗ ≥ π ℓR , then the sender’s best response is given by “ R -drifting withjumps to ” for all p ∈ [ p ∗ , p ∗ ) . It is unique for all p < p ∗ if p ∗ < η , and for all p = ξ if p = η . The value function satisfies V ( p ) = V R ( p ) ≥ V S ( p ) , with strictinequality for all p < p ∗ (and p = ξ ) if p ∗ < η ( p = η ). This means α ∈ { , } and the policy is unique up to tie-breaking. If p = η , then it is also a best response that the sender plays the stationary strategy at ξ . c) If p ∗ > η and p ∗ < π ℓR , then the sender’s best response is given by L -drifting with jump to p ∗ if p ≤ π LR ,R -drifting with jump to if p ∈ [ π LR , ξ ) , stationary if p = ξ ,L -drifting with jump to p ∗ if p ∈ ( ξ , π LR ] ,R -drifting with jump to if p ≥ π LR . Admissibility requires W = ( p ∗ , p ∗ ) in this case. The best response is unique up totie-breaking at π LR and π LR . The associated value function satisfies V ( p ) = max { V L ( p ) , V RS ( p ) } > V S ( p ) , if p < ξ ,V S ( ξ ) if p = ξ , max { V LS ( p ) , V R ( p ) } > V S ( p ) , if p ∈ ( ξ , p ∗ ) . (d) If p ∗ > η and p ∗ ∈ [ π ℓR , ξ ) , then the sender’s best response is given by R -drifting with jump to if p ∈ [ p ∗ , ξ ) , stationary if p = ξ ,L -drifting with jump to p ∗ if p ∈ ( ξ , π LR ) ,R -drifting with jump to if p ∈ [ π LR , p ∗ ) . The best response is unique up to tie-breaking at π LR . The associated value functionsatisfies V ( p ) = V RS ( p ) > V S ( p ) , if p < ξ ,V S ( ξ ) if p = ξ , max { V LS ( p ) , V R ( p ) } > V S ( p ) , if p ∈ ( ξ , p ∗ ) . (e) If p ∗ > η and p ∗ ∈ [ ξ , π SR ) , existence of a best response requires W = [ p ∗ , p ∗ ) . With W = [ p ∗ , p ∗ ) , the sender’s best response is given by stationary if p = p ∗ ,L -drifting with jump to p ∗ if p ∈ ( p ∗ , π LR ) ,R -drifting with jump to if p ∈ [ π LR , p ∗ ) . The best response is unique up to tie-breaking at π LR . The associated value function atisfies V ( p ) = V S ( ξ ) if p = p ∗ , max { V LS ( p ) , V R ( p ) } > V S ( p ) , if p ∈ ( p ∗ , p ∗ ) . (f ) If p ∗ > η and p ∗ ≥ π SR , then the sender’s best response and value function are as incase (b). The best response is unique and if p = π SR . The value function satisfies V ( p ) = V R ( p ) ≥ V S ( p ) , with strict inequality for p = π SR .(g) In all cases, the value function associated with the sender’s best response is strictlyconvex on [ p ∗ , p ∗ ) , and satisfies V ( p ∗− ) = v , V ( p ) > for p > p ∗ , and p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ + ) . B.3.3 The Sender’s Best Response outside the Waiting RegionLemma 16. Suppose that the receiver uses a strategy with waiting region W = [ p ∗ , p ∗ ) or W = ( p ∗ , p ∗ ) , where π ℓL ≤ p ∗ < ˆ p < p ∗ ≤ p , and suppose that c is sufficiently small sothat π ℓR is well defined. Then the sender’s best response has the following properties for p < p ∗ :(a) The sender passes whenever p < π ℓL .(b) If π ℓL ≤ p ∗ ≤ π ℓR , then the sender chooses the L -drifting experiment with jumps to p ∗ for all p ∈ ( π ℓL , p ∗ ) .(c) If p ∗ > π ℓR and W = [ p ∗ , p ∗ ) , then the sender chooses the L -drifting experiment withjumps to p ∗ for all p ∈ [ π ℓL , π ) and the L -drifting experiment with jumps to p ∗ forall p ∈ ( π , p ∗ ) , where π ∈ ( π ℓL , π ℓR ) is defined by V ( p ∗ ) p ∗ − π = vp ∗ − π ⇐⇒ π = p ∗ v − p ∗ V ( p ∗ ) v − V ( p ∗ ) . (d) If p ∗ > π ℓR , W = ( p ∗ , p ∗ ) , and V ( p ∗ ) > , then for p ∈ ( π , p ∗ ) the sender’s bestreply does not exists. The non-existence in part (d) is the reason why we must allow for both W = ( p ∗ , p ∗ ),and W = [ p ∗ , p ∗ ). The former is required by admissibility to obtain equilibria in whichthe sender uses the L -drifting experiment with stopping bound p ∗ . The latter is requiredto ensure the existence of a best response for the sender in equilibria in which V ( p ∗ ) > Lemma 17. φ ℓL < φ ℓR and π ℓL < π ℓR . B.3.4 Equilibrium Verification in Proposition 2 Suppose that Condition C2 holds. Lemmas 15 and 16 imply that the sender’s strategy isa best response in each case. For the receiver, if p ≥ p ∗ , then it is clearly optimal to stop46nd take action r , because the sender will no longer experiment. We consider the othercases in more detail below. For the following, recall from Section B.2.2 that p ∗ = φ ℓR inProposition 2. Receiver optimality when p < p ∗ : If p ∈ ( π , p ∗ ), then the sender plays L -driftingexperiments with jumps to p ∗ . After observing the signals form this experiment, takingaction ℓ is the receiver’s best response whether a jump to p ∗ has occurred or not. Therefore,it is optimal for the receiver to take action ℓ immediately, without incurring more listeningcosts. If p < π ℓL , then the sender simply passes and provides no additional information.Therefore, it is obviously optimal for the receiver to stop and take action ℓ .It remains to consider beliefs p ∈ [ π ℓL , π ]. Here, the sender uses L -drifting experimentswith jumps to p ∗ . By (35), given the sender’s strategy, the receiver’s best response is tostop if p < φ ℓL . We complete the proof by showing that for c sufficiently small, φ ℓL > π . Lemma 18. Suppose that Condition C2 holds. Then there exists c ( p ∗ ) such that φ ℓL > π if c < c ( p ∗ ) . Receiver optimality when p ∈ [ p ∗ , p ∗ ) : First, consider the case where p ∗ ≤ η (so thatthe sender plays only R -drifting experiments over ( p ∗ , p ∗ )). Waiting yields a payoff of U R ( p ). Lemma 5 implies that U R ( p ) > U r ( p ) for all p < p ∗ , as long as p ∗ < p . Since p approaches 1 as c tends to 0, p ∗ < p is guaranteed for c sufficiently small. In addition,uniqueness of φ ℓR (see Lemma 6) implies that U R ( p ) > U ℓ ( p ) for all p > p ∗ . It followsthat U R ( p ) ≥ max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ p ∗ , p ∗ ], with strict inequalities for p ∈ ( p ∗ , p ∗ ).This implies that waiting is a best response for p ∈ [ p ∗ , p ∗ ].Now consider the case where p ∗ > η . First, suppose that p ≥ ξ . As for the casewhere p ∗ ≤ η , U R ( p ) > U r ( p ) for p < p ∗ . Note that U R ( p ) → p ∗ − p ∗ p ∗ u Lℓ + p ∗ p ∗ U r ( p ∗ ) > U ℓ ( p )as c → 0. This implies that U R ( p ) ≥ max { U ℓ ( p ) , U r ( p ) } for all p ≥ ξ if c is sufficientlysmall. Next, note that U LS ( p ) > U R ( p ) if and only if V LS ( p ) > V R ( p ) since the senderand the receiver incur the same cost. Therefore, whenever the LS -strategy is used on[ ξ , p ∗ ], it increases the receiver’s value compared to U R ( p ). Hence the receiver’s value isgreater or equal U R ( p ) ≥ max { U ℓ ( p ) , U r ( p ) } for p ≥ ξ , if c is sufficiently small. Turningto p < ξ , we note that U RS ( p ) > max { U ℓ ( p ) , U r ( p ) } for p ∈ [ p ∗ , ξ ]: U RS ( p ) > U ℓ ( p ) isproven in the same way as U R ( p ) > U ℓ ( p ) in the case where p ∗ ≤ η ; and U RS ( p ) > U r ( p )follows from U RS ( p ) > U R ( p ) > U r ( p ) (which holds since U RS ( ξ ) = U S ( ξ ) > U R ( ξ ) byLemma 2). Hence the receiver’s value exceeds max { U ℓ ( p ) , U r ( p ) } also for p ∈ [ p ∗ , ξ ] if c is sufficiently small. This completes the verification of the receiver’s incentives in theequilibrium specified by Proposition 2 when p ∗ > η .47 .3.5 Equilibrium Verification in Proposition 3 Suppose Condition C2 fails. As for Proposition 2, Lemmas 15 and 16 imply that thesender’s strategy is a best response in each case. The optimality of the receiver’s strategyoutside the waiting region is straightforward, because the sender passes for all p / ∈ ( p ∗ , p ∗ ).For the following, recall from Section B.2.3 that p ∗ = π ℓL in Proposition 3. Receiver optimality when p ∈ ( p ∗ , p ∗ ) and p ∗ ≤ η : First, consider p > π LR . If c issufficiently small, then p ∗ < p , and thus U R ( p ) > U r ( p ) for all p < p ∗ (see Lemma 5).Moreover, by Lemma 7, φ ℓR < π ℓR because Condition C2 does not hold. Since V R ( p ) < p < π ℓR we have π LR ≥ π ℓR > φ ℓR . This implies that U R ( p ) > U ℓ ( p ) for p > π LR .(This follows from convexity of U R ( p ) since U R ( φ ℓR ) = U ℓ ( φ ℓR ) and U R ( p ∗ ) > U ℓ ( p ∗ ).)Hence we have U ( p ) = U R ( p ) ≥ max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ π LR , p ∗ ].For p ∈ [ p ∗ , π LR ], note that p ∗ = π ℓL > φ ℓL , where the inequality follows from Lemma8 if Condition C2 is violated. Condition (35) and convexity of U L ( p ) then imply that U L ( p ) > U ℓ ( p ) for all p > p ∗ = π ℓL . To show that U L ( p ) > U r ( p ) for p ∈ [ p ∗ , π LR ] weshow that for c sufficiently small, π LR < ˆ p so that U r ( p ) < U ℓ ( p ). To see this note thatfor p ∗ = π ℓL , Lemma 10.b implies that π LR → c → π LR < ˆ p for c sufficiently small. Hence we have U L ( p ) > max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ p ∗ , π LR ]. Thiscompletes the proof that waiting is a best response of the receiver for all p ∈ ( p ∗ , p ∗ ). Receiver optimality when p ∈ ( p ∗ , p ∗ ) and p ∗ > η : For the case p ∗ ≤ η , we showedthat U ( p ) = U R ( p ) ≥ max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ π LR , p ∗ ]. The proof works virtuallyunchanged for the current case and shows that U ( p ) = U R ( p ) ≥ max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ φ ℓR , p ∗ ], and hence for all p ∈ [ π LR , p ∗ ]. Moreover by Lemma 6.b, for c sufficientlylow, U RS ( ξ ) = U S ( ξ ) ≥ max { U ℓ ( ξ ) , U r ( ξ ) } . Noting that U RS ( p ) > U R ( p ) for all p ≤ ξ ,this implies that U RS ( p ) > U r ( p ) for all p ≤ ξ . Moreover, since Condition C2 is violated,we have π LR > π ℓR > φ ℓR which implies that U RS ( p ) > U ℓ ( p ) for all p ∈ [ π LR , ξ ]. Thisshows that U ( p ) = U R ( p ) ≥ max { U r ( p ) , U ℓ ( p ) } for all p ∈ [ π LR , ξ ].Next consider the beliefs where the sender uses an L -drifting strategy. For p ∈ [ ξ , π LR ],we have to show U LS ( p ) > max { U r ( p ) , U ℓ ( p ) } . Note that U LS ( p ) − U R ( p ) = C + ( p ; p ∗ ) − C − ( p ; ξ ) − p ∗ − pp ∗ − ξ C S ( ξ ) = V LS ( p ) − V R ( p ) . Since V LS ( p ) > V R ( p ) for p ∈ [ ξ , π LR ], we therefore have U LS ( p ) > U R ( p ) > max { U r ( p ) , U ℓ ( p ) } for p ∈ [ ξ , π LR ]. We have already shown that U R ( p ) > max { U r ( p ) , U ℓ ( p ) } for p ∈ ( φ ℓR , p ∗ ) and since φ ℓR < ξ for c sufficiently small we therefore have U ( p ) = U LS ( p ) > max { U r ( p ) , U ℓ ( p ) } for p ∈ [ ξ , π LR ].It remains to consider p ∈ [ p ∗ , π LR ]. As in the case where p ∗ ≤ η , we have U L ( p ) >U ℓ ( p ) for p ∈ [ p ∗ , π LR ]. Again we show that U L ( p ) > U r ( p ) for p ∈ [ p ∗ , π LR ] if c is48ufficiently small. To show this, note that π LR < π LR since V RS ( p ) > V R ( p ) for all p < ξ . Hence π LR → π LR → U ( p ) = U L ( p ) > max { U r ( p ) , U ℓ ( p ) } for p ∈ [ p ∗ , π LR ]. B.4 Uniqueness We complete the proofs of Propositions 2 and 3 by proving that the given strategy profilesare the unique SMPE in each case. B.4.1 Lower Bound of p ∗ We begin with two useful observations, formally reported in the following lemma. Lemma 19. Fix p ∗ ∈ (ˆ p, p ) . In any SMPE, p ∗ ≥ max { φ ℓL , π ℓL } . In addition, if p ∗ > max { φ ℓL , π ℓL } , then V ( p ∗ ) > and U ℓ ( p ∗ ) = U ( p ∗ ) . In the proof, the crucial step is to show that for any strategy profile, the sender’s payoffis negative if p < π ℓL , and the receiver’s payoff is less than U ℓ ( p ) if p < φ ℓL . To show this,we consider a hypothetical strategy in which the sender can choose α = α = 1, violatingthe constraint α + α ≤ 1. The value of this strategy, which is an upper bound for anyfeasible strategy, is negative for the sender if p < π ℓL , and below U ℓ ( p ) for the receiver if p < φ ℓL . Therefore, the lower bound of the waiting region must be greater than or equalto max { φ ℓL , π ℓL } .To understand the second result of Lemma 19, notice first that if V ( p ∗ ) = 0, thenthe optimal jump-target for p < p ∗ is p ∗ . This implies that for p ∗ > max { φ ℓL , π ℓL } , thereceiver would prefer to wait when p ∈ ( φ ℓL , p ∗ ), in contrast to the conjectured equilibrium.Therefore p ∗ > max { φ ℓL , π ℓL } requires V ( p ∗ ) > 0. If V ( p ∗ ) > 0, however, the sender willtarget p ∗ for beliefs in the stopping region close to p ∗ . U ℓ ( p ∗ ) = U ( p ∗ ) is needed toguarantee that such jumps do not give the receiver an incentive to wait.Lemma 19 implies that there are only two cases to consider: (i) p ∗ = max { φ ℓL , π ℓL } ;or (ii) p ∗ > max { φ ℓL , π ℓL } and p ∗ is determined by the receiver’s incentives (i.e., U ℓ ( p ∗ ) = U ( p ∗ )). Propositions 2 and 3 correspond to each of these two cases. B.4.2 Proof of Uniqueness in Proposition 2 In Proposition 2, Condition C2 holds, that is, v > U r ( p ∗ ) − U ℓ ( p ∗ ). In this case, byLemma 8, φ ℓL > π ℓL ⇔ max { φ ℓL , π ℓL } = φ ℓL : the receiver is less willing to continue and,therefore, stops at a higher belief than the sender. In addition, we have the followingimportant observation: Lemma 20. If v > U r ( p ∗ ) − U ℓ ( p ∗ ) and c is sufficiently small, then p ∗ > φ ℓL = max { φ ℓL , π ℓL } in any SMPE. c sufficiently small, V RS ( p ∗ ; q R ) > p ∗ = φ ℓL . This implies that the sender uses the R -drifting strategy close to p ∗ whichyields negative utility for the receiver for p ∈ [ φ ℓL , φ ℓR ). Hence, p ∗ = φ ℓL implies that thereceiver’s incentives are violated. Therefore, we must have p ∗ > φ ℓL in equilibrium.Since p ∗ > φ ℓL = max { φ ℓL , π ℓL } by Lemma 20, Lemma 19 requires that the receiver’svalue at p ∗ is equal to U ℓ ( p ∗ ). To determine the receiver’s value at p ∗ , note that Lemma15 implies that the sender either uses R -drifting experiments at p ∗ (Parts (b), (d) or (f)of the lemma) or she uses the the stationary strategy (Part (e) of the lemma). For c → U S ( p ) , U RS ( p ; q R ) → (( p ∗ − p ) /p ∗ ) U ℓ (0) + ( p/p ∗ ) U r ( p ∗ ). Since the receiver’s valueat p ∗ must be equal to U ℓ ( p ∗ ), we must therefore have p ∗ → c → 0. Therefore, for c sufficiently small Lemma 15.(e) never applies, and the sender uses R -drifting experimentsat p ∗ . Hence, the receiver’s indifference condition implies p ∗ = φ ℓR . It then follows fromLemma 15 that the strategy profile stated in Proposition 2 is the only one that can arisein equilibrium, establishing uniqueness of the equilibrium in Proposition 2. B.4.3 Proof of Uniqueness in Proposition 3 In Proposition 3, v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ), and thus π ℓL = max { φ ℓL , π ℓL } . Using Lemma 19,we show that p ∗ = π ℓL if Condition C2 fails: Lemma 21. If v ≤ U r ( p ∗ ) − U ℓ ( p ∗ ) , then p ∗ = π ℓL in any SMPE. With p ∗ = π ℓL , uniqueness of the equilibrium in Proposition 3 then follows immediatelyfrom the characterization of the sender’s best response in Lemma 15.50 Remaining Proofs C.1 Proof of Proposition 1 Proof of Proposition 1. Recall that v ( p ) is defined as follows: v ( p ) := max ( α i ,q i ) i X q i = p α i v ( p ; q i ) subject to X i ∈ N α i ≤ , where v ( p ; q i ) := λ p (1 − p ) | q i − p | ( V ( q i ) − V ( p )) − sgn( q i − p ) λp (1 − p ) V ′ ( p ) . As explained in Section 5.1, the first result of Proposition 1 (namely that for each p ∈ (0 , α , q ) and ( α , q ), where 0 ≤ q < p 1, and assume that V ( · ) is non-negative, increasing and convex over( p ∗ , p ∗ ] and V ( p ∗ ) /p ∗ < V ′ ( p ∗ ). Clearly, the optimal q maximizes ( V ( q ) − V ( p )) / ( q − p ),while the optimal q maximizes ( V ( q ) − V ( p )) / ( p − q ).Suppose that p ∈ ( p ∗ , p ∗ ). The optimality of q = p ∗ follows immediately from con-vexity of V ( · ). A similar argument shows that the optimal q is either 0 or p ∗ . In orderto show that 0 is optimal, notice that V ′ ( p ∗ ) > V ( p ∗ ) /p ∗ implies that V ( p ) − V ( p ∗ ) p − p ∗ = R pp ∗ V ′ ( x ) dxp − p ∗ ≥ R pp ∗ V ′ ( p ∗ ) dxp − p ∗ = V ′ ( p ∗ ) > V ( p ∗ ) p ∗ , which is equivalent to V ( p ) p > V ( p ∗ ) p ∗ ⇔ V ( p ) − V ( p ∗ ) p − p ∗ > V ( p ) p = V ( p ) − V (0) p ⇔ V ( p ∗ ) − V ( p ) p − p ∗ < V (0) − V ( p ) p − . Now suppose that p < p ∗ . Again, the optimality of q = p ∗ or q = p ∗ follows fromconvexity of V ( · ) over ( p ∗ , p ∗ ]. The optimality of q = 0 simply comes from the fact that V ( p ) = 0 for all p < p ∗ . 51 .2 Proof of Lemma 1 Proof of Lemma 1. Substituting V + ( p ) = V S ( p ) in (10) we obtain V ′ + ( p ), and differentiat-ing (6) we obtain V ′ S ( p ): V ′ + ( p ) = vp ∗ + 2 p − p ∗ p ∗ p (1 − p ) cλ and V ′ S ( p ) = vp ∗ + 1 − p ∗ (1 − p ) cλp ∗ . Straightforward algebra yields V ′ + ( p ) S V ′ S ( p ) ⇐⇒ pp ∗ − p ∗ − p S . The quadratic expression 3 pp ∗ − p ∗ − p is negative for all p ∈ [0 , p ∗ ] if p ∗ < / p ∗ < / V ′ + ( p ) < V ′ S ( p ) for all p ∈ (0 , p ∗ ). This proves part (a).If p ∗ ≥ / 9, the quadratic expression 3 pp ∗ − p ∗ − p has the real roots ξ ( p ∗ ) and ξ ( p ∗ ), which are distinct if p ∗ > / 9. Hence V ′ + ( q ) = V ′ S ( p ) if p ∈ { ξ ( p ∗ ) , ξ ( p ∗ ) } .Since the quadratic expression is concave in p , V ′ + ( q ) > V ′ S ( p ) if p ∈ ( ξ ( p ∗ ) , ξ ( p ∗ )), and V ′ + ( q ) < V ′ S ( p ) if p / ∈ [ ξ ( p ∗ ) , ξ ( p ∗ )]. This proves part (b). C.3 Proof of Lemma 2 Proof of Lemma 2. We have already argued that for p ∗ < / V S ( p ) < V R ( p ) for all p ∈ (0 , p ∗ ). Now suppose that p ∗ ≥ / 9, so that ξ ( p ∗ ) and ξ ( p ∗ ) are well defined.Our notation for V S ( p ) does not explicitly note the dependence on p ∗ . In this proof,to avoid confusion, we explicitly note this dependence and write V S ( p ; p ∗ ). The equationdefining η is therefore given by V S ( ξ ( η ); η ) = V + ( ξ ( η ); η, v ) , where the right-hand side makes explicit the dependence of V R ( p ) on p ∗ = η . We havealready argued in the text before the statement of the lemma that that V S ( p ; p ∗ ) 4, and the second since p ∗ ≥ / 9. Thiscompletes the proof. C.4 Proof of Lemma 3 Proof of Lemma 3. From (10) we get V ′ + ( p ) = V + ( p ) p + cλp (1 − p ) , and from (20) we get V ′− ( p ) = v − V − ( p ) p ∗ − p − cλp (1 − p ) . Given V + ( p ) = V − ( p ), the difference is equal to V ′ + ( p ) − V ′− ( p ) = p ∗ p ( p ∗ − p ) V − ( p ) + 2 cλp (1 − p ) − vp ∗ − p = p ∗ p ( p ∗ − p ) ( V − ( p ) − V S ( p )) . Since p ∗ / ( p ( p ∗ − p )) is positive for all p < p ∗ , this proves the Lemma. C.5 Proof of Lemma 4 Proof of Lemma 4. By construction, V S ( ξ ) = V + ( ξ ) = V RS ( ξ ) and V ′ S ( ξ ) = V ′ + ( ξ ) = V ′ RS ( ξ ). In addition, V LS ( ξ ) = V RS ( ξ ). Then, by Lemma 3, V ′ LS ( ξ ) = V ′ RS ( ξ ) = V ′ S ( ξ ).For p = ξ , consider V RS ( p ) first. We will show that V ′′ RS ( ξ ) > V ′′ S ( ξ ). Since V ′ RS ( ξ ) = V ′ S ( ξ ) by Lemma 4.(a), this implies that V RS ( p ) > V S ( p ) for p ∈ ( ξ − ε, ξ ) for some ε > 0. Lemma 1 then implies that V RS ( p ) > V S ( p ) for all p ∈ [0 , ξ ).To complete the proof it remains to show V ′′ RS ( ξ ) > V ′′ S ( ξ ). Direct calculation yields V ′′ RS ( ξ ) > V ′′ S ( ξ ) ⇔ − − ξ ) ξ > − − p ∗ )(1 − ξ ) p ∗ ⇔ ξ < p ∗ − p ∗ . ξ ≤ p ∗ / p ∗ > / 9, the last inequality is satisfied.Next consider V LS ( p ). The proof works virtually identically except that we have: V ′′ LS ( ξ ) > V ′′ S ( ξ ) ⇔ p ∗ − pp ∗ + p ( ξ ) ( p ∗ − ξ ) > − − p ∗ )(1 − ξ ) p ∗ ⇔ (5 p ∗ − ξ ) + (3 − p ∗ ) p ∗ ( ξ ) + 3( p ∗ ) ξ − ( p ∗ ) < . Since ξ ≤ p ∗ / p ∗ > / 9, the last inequality is satisfied. This implies that V LS ( p ) >V S ( p ) for p ∈ ( ξ , ξ + ε ) for some ξ > 0. Part (a) implies that Lemma 1 can also be appliedto V − ( p ). Therefore, if V LS ( p ) and V S ( p ) intersect at p ∈ ( ξ , ξ ), then V LS ( p ) crosses V S ( p )from below. Since V LS ( p ) > V S ( p ) for p ∈ ( ξ , ξ + ε ), this implies V LS ( p ) > V S ( p ) for all p ∈ ( ξ , ξ ). C.6 Proof of Lemma 5 Proof of Lemma 5. Substituting U R ( p ∗ ) = U r ( p ∗ ) in (11), we get U ′ R ( p ∗ ) = − p ∗ (cid:0) u Lℓ − p ∗ u Rr − (1 − p ∗ ) u Lr (cid:1) + cλp ∗ (1 − p ∗ ) = U ′ r ( p ∗ ) − u Lℓ − u Lr p ∗ + cλp ∗ (1 − p ∗ ) . Simple algebra then shows that U ′ R ( p ∗ ) S U ′ r ( p ∗ ) is equivalent to p ∗ S p. C.7 Proof of Lemma 6 Proof of Lemma 6. (a) If p ∗ ≤ η , φ ℓR is defined by U ℓ ( φ ℓR ) = U R ( φ ℓR ). Since p ∗ > ˆ p , U R ( p ∗ ) = U r ( p ∗ ) > U ℓ ( p ∗ ), and from (14) we have lim p → U R ( p ) = U ℓ (0) − c/λ . Thereforean intersection φ ℓR exists and since U ℓ ( p ) is linear and U R ( p ) is convex, the intersection isunique. Finally, if p ∗ ≤ p , U ′ R ( p ∗ ) ≤ U ′ r ( p ∗ ) and, since U r ( p ) is linear and U R ( p ) is convex, U R ( p ) > U r ( p ) for all p < p ∗ . In particular this implies that U ℓ ( φ ℓR ) = U R ( φ ℓR ) > U r ( φ ℓR )which shows that φ ℓR < ˆ p .(b) We first show that U S ( ξ ) > max { U ℓ ( ξ ) , U r ( ξ ) } for c sufficiently small. Thisfollows directly from U S ( ξ ) → p ∗ − ξ p ∗ u Lℓ + ξ p ∗ U r ( p ∗ ) > max { U ℓ ( ξ ) , U r ( ξ ) } , where thestrict inequality follows from p ∗ > ˆ p and the fact that max { U ℓ ( ξ ) , U r ( ξ ) } is piecewiselinear and V -shaped.Hence, U RS ( ξ ) = U S ( ξ ) > max { U ℓ ( ξ ) , U r ( ξ ) } for c sufficiently small. From (19) wehave lim p → U RS ( p ) = U ℓ (0) − c/λ . Therefore an intersection φ ℓR ∈ (0 , ξ ) exists and since U ℓ ( p ) is linear and by (12), U R ( p ) is convex, the intersection is unique. Finally, U RS ( p ) >U R ( p ) for all p < ξ and therefore U ℓ ( φ ℓR ) = U RS ( φ ℓR ) > U R ( φ ℓR ) > U r ( φ ℓR ), where wehave used that U R ( p ) > U r ( p ) for all p < p ∗ as in the case p ∗ ≤ η . U ℓ ( φ ℓR ) > U r ( φ ℓR )shows that φ ℓR < ˆ p .For both (a) and (b), the convergence φ ℓR → U R ( p ) , U RS ( p ) → (( p ∗ − p ) /p ∗ ) u Lℓ +54 p/p ∗ ) U r ( p ∗ ) > U ℓ ( p ) for all p > C.8 Proof of Lemma 7 Proof of Lemma 7. In this proof we use the simplified notation U RS ( p ; q R ) that was in-troduced in section B.1.5. With this notation (29) and (30) can be written as follows: U ℓ ( φ ℓR ) = U RS ( φ ℓR , q R ) and 0 = V RS ( π ℓR , q R ) . Substituting U ℓ ( φ ℓR ) and U RS ( φ ℓR , q R ) in the first condition we obtain φ ℓR u Rℓ + (1 − φ ℓR ) u Lℓ = p ∗ − φ ℓR p ∗ u Lℓ + φ ℓR p ∗ U r ( p ∗ ) − C + ( φ ℓR ; q R ) − φ ℓR q R C S ( q R ) , which reduces to 1 p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) = 1 φ ℓR C + ( φ ℓR ; q R ) + 1 q R C S ( q R ) . Similarly, from the second condition we obtain1 p ∗ v = 1 π ℓR C + ( π ℓR ; q R ) + 1 q R C S ( q R ) . This implies that v > U r ( p ∗ ) − U ℓ ( p ∗ ) ⇔ π ℓR C + ( π ℓR ; q R ) > φ ℓR C + ( φ ℓR ; q R ) ⇔ π ℓR < φ ℓR , where the last equivalence is due to the fact that ddp (cid:18) p C + ( p ; q R ) (cid:19) = − p C + ( p ; q R ) + 1 p C ′ + ( p ; q R )= − p (cid:18) p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R − p (cid:18) log (cid:18) q R − q R − pp (cid:19) − − p − q R (cid:19)(cid:19) cλ = − p (1 − p ) cλ < . C.9 Proof of Lemma 8 Proof of Lemma 8. Equation (31) is equivalent to v = p ∗ − π ℓL π ℓL (1 − π ℓL ) cλ U r ( p ∗ ) − U ℓ ( p ∗ ) = p ∗ − φ ℓL φ ℓL (1 − φ ℓL ) cλ . The desired result follows from the fact that ( d/dp ) (( p ∗ − p ) / ( p (1 − p ))) < C.10 Proof of Lemma 9 Proof of Lemma 9. We first show thatlim c → V L ( p ) = pp ∗ v = lim c → V R ( p ) . For V L ( p ) we derive a lower bound that converges to pv/p ∗ . Since V ′ L ( π ℓL ) = 0 and V L ( p ) isconvex, we have V L ( p ) > p > π ℓL . Therefore V L ( p ) > V − ( p ; q, 0) for any q ∈ ( π ℓL , p ).From (21) and (22), we obtain lim c → V − ( p ; q, 0) = p − qp ∗ − q v. Since π ℓL → c → V L ( p ) ≥ lim q → lim c → V − ( p ; q, 0) = pp ∗ v. Moreover, since C − ( p ; π ℓL ) > V L ( p ) < pv/p ∗ and therefore lim c → V L ( p ) = pv/p ∗ . For V R ( p ) the limit follows from lim c → C + ( p ; p ∗ ) = 0.To show that V L ( p ) < V R ( p ) for c sufficiently small, we compare the derivatives withrespect to c of both functions in a neighborhood of c = 0. For V R ( p ), we have dV R ( p ) dc = − ∂C + ( p ; p ∗ ) ∂c = − (cid:18) p log (cid:18) p ∗ − p ∗ − pp (cid:19) + p ∗ − pp ∗ (cid:19) < . This is equal to zero for p = 0 and p = p ∗ and continuous in p . Therefore, ∂V R ( p ) /∂c isuniformly bounded in ( p, c ) on [0 , p ∗ ] × R + .For V L ( p ) we have dV L ( p ) dc = ∂V L ( p ) ∂π ℓL ∂π ℓL ∂c + ∂V L ( p ) ∂c . The first term on the RHS is zero since ∂V L ( p ) ∂π ℓL = ( p ∗ − p ) [ c ( p ∗ − π ℓL ) − (1 − π ℓL ) π ℓL vλ ](1 − π ℓL ) π ℓL ( p ∗ − π ℓL ) λ = 0 . ∂V L ( p ) ∂c = − C − ( p ; π ℓL ) c = p ∗ − pp ∗ (1 − p ∗ ) (cid:18) p ∗ log 1 − π ℓL − p + (1 − p ∗ ) log π ℓL p − log p ∗ − π ℓL p ∗ − p (cid:19) λ . Taking the limit and using π ℓL → 0, we getlim c → dV L ( p ) dc = lim c → ∂V L ( p ) ∂c = −∞ . for all p ∈ (0 , p ∗ ) . This shows that for any p ∈ (0 , p ∗ ) there exists c ( p ) > V L ( p ) < V R ( p ) for c ∈ (0 , c ( p )). C.11 Proof of Lemma 10 Proof of Lemma 10. (a) For c sufficiently small, Lemma 9 implies that V L ( p ) < V R ( p )for all p ∈ [ ξ , p ∗ ), so that any intersection must be at p < ξ . Therefore, by Lemma 2, V L ( p ) = V R ( p ) > V S ( p ) at any intersection. Then, by the Crossing Lemma 3, V R ( p ) canintersect V L ( p ) only from below which implies that there is at most one intersection of V R ( p ) and V L ( p ).If p ∗ = π ℓR , then an intersection exists at π LR = p ∗ since V R ( p ∗ ) = 0 = V L ( p ∗ ). If p ∗ < π ℓR , then V R ( p ∗ ) < V L ( p ∗ ) and since V L ( ξ ) < V R ( ξ ) for c sufficiently small, theintermediate value theorem implies that there exists an intersection of V R ( p ) and V L ( p )in ( p ∗ , p ∗ ). In both cases the intersection is unique by the same argument as above. Theremaining claim of the Lemma holds since V R ( p ) crosses V L ( p ) from below.If p ∗ > π ℓR , then V R ( p ∗ ) > V L ( p ∗ ) so that the number of intersections must beeven which implies that there is no intersection and V L ( p ) < V R ( p ) for all p ∈ ( p ∗ , p ∗ ).(b) From Lemma 9 if follows directly that π LR → c → p ∗ = π ℓL (this meansif we adjust p ∗ to maintain p ∗ = π ℓL as c → C.12 Proof of Lemma 11 Proof of Lemma 11. By Lemma 2, V LS ( ξ ) = V S ( ξ ) > V R ( ξ ) if p ∗ > η . From (22), (13)and (27), we obtain C ′− ( p ; q ) = − C − ( p ; q ) p ∗ − p − p ∗ − pp ∗ (1 − p ∗ ) (cid:18) p ∗ − p − − p ∗ p − p ∗ − p (cid:19) cλ andlim p → p ∗ V ′ LS ( p ∗ ) = − lim p → p ∗ C ′− ( p ; q ) = ∞ . Therefore, V LS ( p ) < V R ( p ) for p ∈ ( p ∗ − ε, p ∗ ) for some ε > . This shows that there is atleast one intersection of V LS ( p ) and V R ( p ) in the interval ( ξ , p ∗ ).Next we show that the intersection is unique. We have V LS ( ξ ) = V S ( ξ ) and V LS ( p ∗ ) =57 S ( p ∗ ). By the Crossing Lemma 3, the conditions characterizing intersections of V S and V + in Lemma 1 also apply to V − . These crossing conditions imply V LS ( p ) > V S ( p ) forall p ∈ ( ξ , p ∗ ) . This implies that we can apply the Crossing Lemma 3 to intersections of V LS and V R . Since V LS ( p ) > V S ( p ), the Crossing Lemma 3 implies that V ′ LS ( p ) < V ′ R ( p )at any intersection p ∈ ( ξ , p ∗ ). Therefore there is a unique intersection, and this definesthe cutoff π LR . Clearly, for p ∈ [ ξ , π LR ), V LS ( p ) > V R ( p ), and for p ∈ ( π LR , p ∗ ), V LS ( p ) Proof of Lemma 12. We have V RS ( ξ ) = V S ( ξ ) and from (6) and (18) we have V RS (0) >V S (0). Lemma 1 implies that if V RS ( p ) = V S ( p ) for p < ξ then V ′ RS ( p ) < V ′ S ( p ). Therefore,there cannot be any intersection for p < ξ and we have V RS ( p ) > V S ( p ) for p < ξ .Since V RS ( p ) > V S ( p ) for all p < ξ , the Crossing Lemma 3 implies that there canbe at most one intersection of V RS ( p ) and V L ( p ) for p ∈ ( p ∗ , ξ ) and V L ( p ) must cross V RS ( p ) from above. If p ∗ ≥ π ℓR , then V L ( p ∗ ) = 0 ≤ V RS ( p ∗ ), hence there cannot be anyintersection which implies that V L ( p ) < V RS ( p ) for all p ∈ ( p ∗ , ξ ). Next suppose that p ∗ < π ℓR . This implies that V RS ( p ∗ ) < V L ( p ∗ ), therefore by the intermediate valuetheorem, an intersection exists if V L ( ξ ) < V S ( ξ ), and we have already shown that theintersection is unique. Clearly, V L ( p ) > V RS ( p ) if p ∈ [ p ∗ , π LR ) and V L ( p ) < V RS ( p ) if p ∈ ( π LR , ξ ). C.14 Proof of Lemma 13 Proof of Lemma 13. Let p ∈ ( p ∗ , p ∗ ). Strict convexity of V ( p ) on ( p ∗ , p ∗ ) and p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ ) imply that it suffices to consider jumps to zero and p ∗ in (HJB). If p ∗ V ′ ( p ∗ + ) >V ( p ∗ ), then 0 or p ∗ is the unique optimal jump target. Hence (HJB) simplifies to (HJB-S)which can be written as cλp (1 − p ) = − V ( p ) p + V ′ ( p ) + max α (cid:20) V ( p ) p + v − V ( p ) p ∗ − p − V ′ ( p ) (cid:21) α. For part (a) we substitute V ′ ( p ) using (20) in the coefficient of α and rearrange as follows: V ( p ) p + v − V ( p ) p ∗ − p − V ′ ( p ) = 2 cλp (1 − p ) + V ( p ) p − v − V ( p ) p ∗ − p = 2 cλp (1 − p ) − vp ∗ − p + V ( p ) p ∗ − p + V ( p ) p = p ∗ p ( p ∗ − p ) V ( p ) − λp (1 − p ) v − c ( p ∗ − p ) λp (1 − p )( p ∗ − p )= p ∗ p ( p ∗ − p ) ( V ( p ) − V S ( p )) . α = 1 is a maximizer in (HJB-S) if V ( p ) ≥ V S ( p ) and (HJB-S) holds if cλp (1 − p ) = v − V ( p ) p ∗ − p − V ′ ( p ) , which is equivalent to (20). If V ( p ) > V S ( p ), α = 1 is the unique maximizer.For part (b), we substitute V ′ ( p ) from (10) instead to obtain V ( p ) p + v − V ( p ) p ∗ − p − V ′ ( p ) = v − V ( p ) p ∗ − p − V ( p ) p − cλp (1 − p )= − p ∗ p ( p ∗ − p ) ( V ( p ) − V S ( p ))Hence α = 0 is a maximizer in the HJB equation if V ( p ) ≥ V S ( p ) and (HJB-S) holds if cλp (1 − p ) = − V ( p ) p + V ′ ( p ) , which is equivalent to (10). If V ( p ) > V S ( p ), α = 0 is the unique maximizer. C.15 Proof of Lemma 14 Proof of Lemma 14. By Lemma 13, V ( p ) satisfies (HJB) for all points p ∈ ( p ∗ , p ∗ ) whereit is differentiable. To show that it is a viscosity solution, we have to show that for allpoints p ′ ∈ ( p ∗ , p ∗ ) where V ( p ) is not differentiable,max α ∈ [0 , λp ′ (1 − p ′ ) (cid:20) α w − V ( p ′ ) p ∗ − p ′ − (1 − α ) V ( p ′ ) p ′ − (2 α − z (cid:21) ≤ c for all z ∈ [ V ′ ( p ′− ) , V ′ ( p ′ + )]. We have simplified the condition for a viscosity solution usingthe fact that V ( p ) ≥ 0, and V ( p ) satisfies (20) below the kink and (10) above the kink,the kink is convex (i.e., V ′ ( p ′− ) < V ′ ( p ′ + )), and that V ( p ) is strictly convex and satisfies p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ + ).Since the term in the square bracket is linear in α , it suffices to check this conditionfor α ∈ { , } . For α = 1 we have − (2 α − z < − (2 α − V ′ ( p ′ + ) and for α = 0 wehave − (2 α − z < − (2 α − V ′ ( p ′− ). Hence it is sufficient to check λp ′ (1 − p ′ ) max (cid:26) w − V ( p ′ ) p ∗ − p ′ − V ′ ( p ′ + ) , − V ( p ′ ) p ′ + V ′ ( p ′− ) (cid:27) ≤ c. Since V ( p ) is continuous and satisfies (20) for p ′′ ∈ ( p ′ − ε, p ′ ) and satisfies (10) for p ′′ ∈ ( p ′ , p ′ + ε ), the last condition holds with equality. Therefore, V ( p ) is a viscositysolution of (HJB). Uniqueness is shown as in the proof of Lemma 13.59 .16 Proof of Lemma 15 Proof of Lemma 15. For the proof of this Lemma, we verify for each case (a)–(f) thatthe stated value functions verify the conditions of Lemmas 13 and 14. This implies thatthe candidate value function is a viscosity solution of (HJB). The value function mustnecessarily be a viscosity solution of (HJB) (see, e.g., Theorem 10.8 in Oksendal andSulem, 2009). While we are not aware of a statement of sufficiency that covers preciselyour model, the arguments in Soner (1986) can be easily extended to show uniqueness ofthe viscosity solution to (HJB). This proves that the candidate value function V ( p ) isthe value function of the sender’s problem in the waiting region. Uniqueness of the bestresponse follows from Lemmas 13 and 14. At the end of the proof we also address theexistence issue for part (e).We now verify the conditions of Lemmas 13 and 14. Outside the waiting region thesender’s value function satisfies V ( p ) = 0 for p < p ∗ and V ( p ) = v for p > p ∗ . The otherproperties are verified one by one: V ( p ) > Note first that for c sufficiently small, π ℓR < ξ , which we use in some of thecases.(a) Since p ∗ ≥ π ℓL , we have V L ( p ) > p ∈ ( p ∗ , p ∗ ] and therefore V ( p ) > p ∗ ≥ π ℓR , we have V R ( p ) > p ∈ ( p ∗ , p ∗ ] and therefore V ( p ) > V ( p ) > p ∈ ( p ∗ , ξ ), and the argumentfor case (b) implies that V ( p ) > p ∈ ( ξ , p ∗ ] since π ℓR < ξ . Finally, for c sufficiently small, V S ( ξ ) > p ∗ ≥ π ℓR , we have V RS ( p ) > p ∈ ( p ∗ , ξ ], and therefore V ( p ) > p ∈ ( p ∗ , ξ ]. The argument for case (b) implies that V ( p ) > p ∈ ( ξ , p ∗ ] since π ℓR < ξ . Finally, for c sufficiently small, V S ( ξ ) > p ∗ ≥ ξ > π ℓR , as in (b), we have V R ( p ) > p ∈ ( p ∗ , p ∗ ] and therefore V ( p ) ≥ 0. For c sufficiently small V S ( ξ ) > p ∗ π SR > ξ > π ℓR , as in (b) we have V R ( p ) > p ∈ ( p ∗ , p ∗ ] and therefore V ( p ) ≥ Continuity. We only need to verify continuity at the cutoffs since elsewhere the candi-date value functions are solutions to the ODEs (10) and (20): • at p ∗ if the waiting region is [ p ∗ , p ∗ ). By admissibility, this rules out cases (a) and(c).(e) Since V LS ( p ∗ ) = V S ( p ∗ ) is used as the boundary condition for V LS ( p ), thecandidate value function is continuous at p ∗ .(b,d,f) In all other cases V R ( p ) is continuous at p ∗ . • at ξ : Since V RS ( ξ ) = V LS ( ξ ) = V S ( ξ ) from the boundary condition used to define V LS and V RS , the candidate value functions in (c)–(d) are continuous at ξ .60 All other cutoffs ( π LR , π LR , π LR ) are given by indifference conditions between thevalue functions in the adjacent regions of beliefs. Therefore, the candidate valuefunction is continuous at theses cutoffs. V ( p ) ≥ V S ( p ). (a,b) By Lemma 2, V R ( p ) ≥ V S ( p ) for all p < p ∗ since p ∗ ≤ η . The inequality is strict if p ∗ < η or p ∗ = η and p = η . In (a), when V ( p ) = V L ( p ) we have V L ( p ) ≥ V R ( p ) sothat V L ( p ) ≥ V S ( p ) (or V L ( p ) > V S ( p )) as well.(c,d) By Lemma 12, V ( p ) ≥ V RS ( p ) for p ∈ ( p ∗ , ξ ), and by Lemma 4.(b) V RS ( p ) > V S ( p )for p ∈ ( p ∗ , ξ ). Hence V ( p ) > V S ( p ) for p ∈ ( p ∗ , ξ ). By Lemma 2, V R ( p ) > V S ( p ) for p ∈ [ ξ , p ∗ ) and by Lemma 4.(b) V LS ( p ) > V S ( p ) for p ∈ ( ξ , ξ ). Hence V ( p ) > V S ( p )for p ∈ ( ξ , p ∗ ).(e) By Lemma 2, V R ( p ) > V S ( p ) for p ∈ [ ξ , p ∗ ). Since p ∗ < π RS in case (e), Lemma 2implies that p ∗ ∈ ( ξ , ξ ). The proof of Lemma 4.(b) can be extended to show that V LS ( p ) > V S ( p ) for p ∈ ( ξ , ξ ) when V LS ( p ) = V − ( p ; p ∗ , V S ( p ∗ )) for p ∗ ∈ ( ξ , ξ ).Hence V ( p ) > V S ( p ) for p ∈ ( p ∗ , p ∗ ).(f) Since p ∗ ≥ π RS , Lemma 2 implies that V R ( p ) > V S ( p ) for all p ∈ ( p ∗ , p ∗ ). Strict convexity. The candidate value functions are defined piece-wise using the func-tions V R , V RS , V L , and V LS , which are all strictly convex (see (12) and (24), respectively).Lemma 4 implies that V ′ LS ( ξ ) = V ′ RS ( ξ ) in (c) and (d). Therefore, it only remains toverify that at the remaining cutoffs the value function has convex kinks. To do this weemploy the Crossing Lemma 3. Note that at any cutoff x ∈ { π LR , π LR , π LR } , V ( p ) satis-fies (10) for p ∈ ( x, x + ε ), and (20) for p ∈ ( x − ε, x ) for some ε > 0. Since V ( p ) ≥ V S ( p )(see above), the Crossing Lemma 3 imples that V ′ ( x − ) ≤ V ′ ( x + )—that is V ( p ) is eithercontinuously differentiable at x or has a convex kink. Therefore V ( p ) is strictly convex. p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ ). (a,c) In these two cases we have V ( p ∗ + ) = V ( p ∗ ) = 0 and by (33), V ′ ( p ∗ + ) ≥ V ( p ) = V ( p ) if p > p ∗ ,V R ( p ) if p ≤ p ∗ in cases (b) and (f), V RS ( p ) if p ≤ p ∗ in case (d) . Since V R ( p ) is strictly convex on [0 , p ∗ ] and V RS ( p ) strictly convex on [0 , ξ ], ˜ V ( p )is a strictly convex function on [0 , p ∗ ]. Moreover lim p → ˜ V ( p ) = lim p → V R ( p ) =lim p → V RS ( p ) = − c/λ < 0. This implies that p ∗ ˜ V ′ ( p ∗ ) > ˜ V ( p ∗ ) − lim p → ˜ V ( p ) > ˜ V ( p ∗ ) and therefore p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ ).61e) In this case we follow a similar argument as in cases (b,d,f) and define˜ V ( p ) = V ( p ) if p ≥ p ∗ ,V + ( p ; p ∗ , V S ( p ∗ )) if p < p ∗ . By the Crossing Lemma (3), we have V ′ L ( p ∗ ) = V ′ + ( p ; p ∗ , V S ( p ∗ )) so that ˜ V ( p ) isconvex on [0 , p ∗ ) and as in cases (b,d,f), lim p → ˜ V ( p ) = − c/λ < 0. This implies that p ∗ ˜ V ′ ( p ∗ ) ≥ ˜ V ( p ∗ ) and therefore p ∗ V ′ ( p ∗ + ) ≥ V ( p ∗ ). Existence in case (e). Suppose p ∗ ∈ [ ξ , π SR ). Let V [ p ∗ ,p ∗ ) ( p ) be the value function ofthe best response if W = [ p ∗ , p ∗ ) (stated in part (e)). Clearly, if W = ( p ∗ , p ∗ ) the sendercannot achieve V [ p ∗ ,p ∗ ) ( p ) for p < π LR , since this would require using V LS ( p ), but thesender cannot switch to the stationary strategy at p ∗ if the receiver stops at p ∗ . However,the sender can achieve a value arbitrarily close to V [ p ∗ ,p ∗ ) ( p ) by using the strategy R -drifting with jump to 0 if p ∈ ( p ∗ , p ∗ + ε ) , stationary if p = p ∗ + ε,L -drifting with jump to p ∗ if p ∈ ( p ∗ + ε, π LR ) ,R -drifting with jump to 0 if p ∈ [ π LR , p ∗ ) . For fixed ε > V ε ( p ) = V + ( p ; p ∗ + ε, V S ( p ∗ + ε )) if p ∈ ( p ∗ , p ∗ + ε ) ,V S ( p ∗ + ε ) if p = p ∗ + ε,V − ( p ; p ∗ + ε, V S ( p ∗ + ε )) if p ∈ ( p ∗ + ε, π LR ) ,V R ( p ) if p ∈ [ π LR , p ∗ ) . Since the ODE (20) is Lipschitz continuous, lim ε → V − ( p ; p ∗ + ε, V S ( p ∗ + ε )) = V LS ( p ) forall p ∈ ( p ∗ , π LR ). Hence, lim ε → V ε ( p ) = V [ p ∗ ,p ∗ ) ( p ) for all p ∈ W . Since for p < π LR , thelimit value cannot be achieved by any strategy if W = ( p ∗ , p ∗ ), the sender has no bestresponse. C.17 Proof of Lemma 16 Proof of Lemma 16. Let V ( p ) denote the value function associated with the sender’s bestresponse. For p < p ∗ we have V ( p ) = 0 since the receiver stops immediately and for p ∈ W , V ( p ) is as characterized in Lemma 15.For p < p ∗ , our refinement requires that the sender chooses the experiment that yieldsthe highest flow payoff if this flow payoff is positive (see Appendix (A.1)). The flow payoff62s given by − c + max ≤ q ≤ p ≤ q ≤ p ∗ , α ∈ [0 , λp (1 − p ) (cid:20) α V ( q ) − V ( p ) q − p + (1 − α ) V ( q ) − V ( p ) p − q (cid:21) . Since V ( x ) = 0 for all x < p ∗ and V ( q ) > q > p ∗ , we must have α = 1. Theoptimal value for q maximizes q − V ( p ) q − p . Since V ( p ) is convex for p ∈ [ p ∗ , p ∗ ] by Lemma 15.e, we must have q ∈ { p ∗ , p ∗ } .Consider first the case that p ∗ ≤ π ℓR . In this case, Lemma 15 implies V ( p ∗ ) = 0.Therefore the optimal jump-target is q = p ∗ . By (33), the flow payoff from the L -driftingexperiment with jump-target p ∗ is negative for p < π ℓL . This proves part (b) and part(a) for the case p ∗ ≤ π ℓR .Next consider the case that p ∗ > π ℓR . In this case, Lemma 15 implies V ( p ∗ ) > q = p ∗ if V ( p ∗ ) p ∗ − p > vp ∗ − p ⇐⇒ p > π , and q = p ∗ otherwise, with indifference at p = π . Hence q = p ∗ if and only if p ≥ π .Convexity of V ( p ) on [ p ∗ , p ∗ ], together with convexity of V RS ( p ; q R ), implies that π ℓR < π and Lemma 17 implies π ℓL < π ℓR < π . Hence for p < π ℓL the optimal target is q = p ∗ which yields a negative flow payoff (by (33)) so that passing is optimal.For p ∈ [ π ℓL , π ) the optimal target is q = p ∗ , which yields a positive flow payoff (alsoby (33)). For p ∈ [ π , p ∗ ] the optimal target is q = p ∗ . Here, the flow payoff is positivesince it is greater than the flow payoff if the target is q = p ∗ , and the latter leads to apositive flow payoff (again by (33)). This shows part (c) and part (a) for p ∗ > π ℓR .Finally, for part (d), note that the flow payoff fails upper semi-continuity in q at q = p ∗ . Hence there exists no best response in p ∈ ( π , p ∗ ). For p < π ℓL , the argument in(c) remains valid which proves the statement of part (a) under the assumptions of part(d). C.18 Proof of Lemma 17 Proof of Lemma 17. Since π ℓR is the lowest value of p such that V RS ( p ; q R ) ≥ itsuffices to show that V RS ( π ℓL ; q R ) < 0. From (31) we have c = λπ ℓL (1 − π ℓL ) p ∗ − π ℓL v. (39) See Section B.1.5 for the definition of V RS ( p ; q R ). φ ℓL < φ , it suffices to show that U ℓ ( φ ℓL ) > U RS ( φ ℓL ; q R ). Rearranging (34)we have. c = λφ ℓL (1 − φ ℓL ) p ∗ − φ ℓL ( U r ( p ∗ ) − U ℓ ( p ∗ )) . (40)Recall that V RS ( p ; q R ) = pq R V S ( q R ) − (cid:18) p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R (cid:19) cλ = pp ∗ v − (cid:18) p ( p ∗ − q R ) p ∗ q R (1 − q R ) + p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R (cid:19) cλ , and U RS ( p ; q R ) = pp ∗ U r ( p ∗ ) + p ∗ − pp ∗ u Lℓ − (cid:18) p ( p ∗ − q R ) p ∗ q R (1 − q R ) + p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R (cid:19) cλ . Rearranging the terms, and using (39) and (40), we get that both V R ( π ℓL ; q R ) < U ℓ ( φ ℓL ) > U RS ( φ ℓL ; q R ) reduce to the following inequality:2 p ( p ∗ − q R ) p ∗ q R (1 − q R ) + p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R > p ∗ − pp ∗ (1 − p ) . First, consider the case where q R = p ∗ (i.e., p ∗ ≤ η ). In this case, the necessaryinequality simplifies to p log (cid:18) p ∗ − p ∗ − pp (cid:19) + 1 − pp ∗ > p ∗ − pp ∗ (1 − p ) ⇔ log (cid:18) p ∗ − p ∗ − pp (cid:19) > p ∗ − pp ∗ (1 − p ) . This holds for any p ∈ (0 , p ∗ ), because the two sides are identical if p = p ∗ , and ddp (cid:18) log (cid:18) p ∗ − p ∗ − pp (cid:19) − p ∗ − pp ∗ (1 − p ) (cid:19) = − − p ) (cid:18) − pp − − p ∗ p ∗ (cid:19) < . Now consider the case where q R = ξ (i.e., p ∗ > η ). For this case, we show that thefollowing function is always positive: f ( p ) = 2 p ( p ∗ − ξ ) p ∗ ξ (1 − ξ ) + p log (cid:18) ξ − ξ − pp (cid:19) + 1 − pξ − p ∗ − pp ∗ (1 − p ) . It is straightforward that f (0) = 0 and f ( ξ ) = ( p ∗ − ξ ) / ( p ∗ (1 − ξ )) > 0. We obtain thedesired result by showing that f is strictly concave over [0 , ξ ]. To that end, observe that f ′ ( p ) = 2( p ∗ − ξ ) p ∗ ξ (1 − ξ ) + log (cid:18) ξ − ξ − pp (cid:19) − − p − ξ + 1 − p ∗ p ∗ (1 − p ) , f ′′ ( p ) = − p (1 − p ) + 2(1 − p ∗ ) p ∗ (1 − p ) = − − p ) (cid:18) − pp − − p ∗ ) p ∗ (cid:19) . For any p ∈ (0 , ξ ], f ′′ ( p ) < 0, because1 − pp ≥ − ξ ξ = 4 − p ∗ + p p ∗ (9 p ∗ − p ∗ − p p ∗ (9 p ∗ − > − p ∗ ) p ∗ whenever p ∗ > η. C.19 Proof of Lemma 18 Proof of Lemma 18. Since φ ℓL is the lowest value of p such that (cf. (35)) p (1 − p ) p ∗ − p λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) ≥ c, it suffices to show that π (1 − π ) p ∗ − π λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) < c. (41)We show that (41) holds if c is sufficiently small and Condition C2 holds. Step 1: φ ℓR as a function of c , and lim c → φ ℓR = 0 . Adopting the notation introduced in in Section B.1.5, the receiver’s value functionassociated with R -drifting experiment below q R can be written as U RS ( p ; q R ) = pp ∗ U r ( p ∗ ) + (cid:18) − pp ∗ (cid:19) u Lℓ − C RS ( p ; q R )where C RS ( p ; q R ) = C R ( p ; q R )+ pq R C S ( q R ) = (cid:18) p log (cid:18) q R − q R − pp (cid:19) + 1 − pq R + pq R p ∗ − q R ) p ∗ (1 − q R ) (cid:19) cλ . Recall that φ ℓR is defined to be the value such that U ℓ ( φ ℓR ) = U RS ( φ ℓR ; q R ) ⇔ C RS ( φ ℓR ; q R ) = φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . (42)As c tends to 0, C RS ( p ; q R ) approaches 0 for any p > 0. Therefore, the right-hand side ofthis equation must also converge to 0, which implies that φ ℓR converges 0. Step 2: π as a function of φ ℓR , and lim c → π = 0 . π is defined to be the value such that V RS ( φ ℓR ; q R ) φ ℓR − π = vp ∗ − π ⇔ π = φ ℓR v − p ∗ V RS ( φ ℓR ; q R ) v − V RS ( φ ℓR ; q R ) , where V RS ( φ ℓR ; q R ) = φ ℓR p ∗ v − C RS ( φ ℓR ; q R ) . Replacing C RS ( φ ℓR ; q R ) with equation (42), we get V RS ( φ ℓR ; q R ) = φ ℓR p ∗ v − φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . Plugging this into the equation for π , we obtain π = p ∗ φ ℓR ( U r ( p ∗ ) − U ℓ ( p ∗ ))( p ∗ − φ ℓR ) v + φ ℓR ( U r ( p ∗ ) − U ℓ ( p ∗ )) . (43)Since lim c → φ ℓR = 0, we also have lim c → π = 0. Step 3: deriving an equivalent inequality to (41) . Since lim c → π = 0, inequality (41) holds with equality in the limit as c tends to 0.We show that it holds with strictly inequality when c is sufficiently small (but strictlypositive) by showing that the derivative of the left-hand side at 0 is strictly less than 1(which is the derivative of the right-hand side), that is,lim c → ddc (cid:18) π (1 − π ) p ∗ − π λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) (cid:19) < . Notice that φ ℓR is a function of c (via equation (42)), and π can be expressed as a functionof φ ℓR (via equation (43)). Therefore, the above inequality is equivalent tolim c → ddπ (cid:18) π (1 − π ) p ∗ − π λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) (cid:19) dπ dφ ℓR dφ ℓR dc < . Since lim c → π = 0 and ddπ (cid:18) π (1 − π ) p ∗ − π λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) (cid:19) = p ∗ − p ∗ π + π ( p ∗ − π ) λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) , we have lim c → ddπ (cid:18) π (1 − π ) p ∗ − π λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) (cid:19) = λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) p ∗ . c sufficiently small:lim c → dπ dφ ℓR dφ ℓR dc < p ∗ λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . (44) Step 4: Proving (44) . We complete the proof by showing thatlim c → dπ dφ ℓR = U r ( p ∗ ) − U ℓ ( p ∗ ) v and lim c → dφ ℓR dc = p ∗ λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . Since Condition C2 ensures that lim c → dπ /dφ ℓR < 1, these are (exactly) sufficient forthe desired inequality.From equation (43), we get dπ dφ ℓR = p ∗ v (( p ∗ − φ ℓR ) v + φ ℓR ( U r ( p ∗ ) − U ℓ ( p ∗ ))) p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . Combining this with lim c → φ ℓR = 0 (from Step 1) immediately yieldslim c → dπ dφ ℓR = U r ( p ∗ ) − U ℓ ( p ∗ ) v . For lim c → dφ ℓR /dc , define F ( φ ℓR , c ) := φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) − C RS ( φ ℓR ; q R )= φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) − (cid:18) φ ℓR log (cid:18) q R − q R − φ ℓR φ ℓR (cid:19) + 1 − φ ℓR q R + φ ℓR q R p ∗ − q R ) p ∗ (1 − q R ) (cid:19) cλ . Note that φ ℓR is implicitly defined by F ( φ ℓR , c ) = 0. We have ∂F∂c = − (cid:18) φ ℓR log (cid:18) q R − q R − φ ℓR φ ℓR (cid:19) + 1 − φ ℓR q R + φ ℓR q R p ∗ − q R ) p ∗ (1 − q R ) (cid:19) λ , = − C RS ( φ ℓR ; q R ) c = − c (cid:18) φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) − F ( φ ℓR , c ) (cid:19) , = − c φ ℓR p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) , where we have used (42) in the second line. Next, we have ∂F∂φ ℓR = U r ( p ∗ ) − U ℓ ( p ∗ ) p ∗ − (cid:18) log (cid:18) q R − q R − φ ℓR φ ℓR (cid:19) − − φ ℓR − q R + 2( p ∗ − q ) p ∗ q R (1 − q R ) (cid:19) cλ = U r ( p ∗ ) − U ℓ ( p ∗ ) p ∗ − φ ℓR C RS ( φ ℓR ; q R ) + 1 φ ℓR (1 − φ ℓR ) cλ = 1 φ ℓR (1 − φ ℓR ) cλ dφ ℓR dc = φ ℓR (1 − φ ℓR ) p ∗ c λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . Recall that it suffices to show thatlim c → dφ ℓR dc = p ∗ λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . To obtain this result, notice that the differential equation for φ ℓR ( c ) can be expressed asfollows: dφ ℓR dc φ ℓR (1 − φ ℓR ) = − ddc (cid:18) φ ℓR + log (cid:18) − φ ℓR φ ℓR (cid:19)(cid:19) = K c = − K (cid:18) c (cid:19) ′ . where K := λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) /p ∗ . Therefore, we have1 φ ℓR ( c ) + log (cid:18) − φ ℓR ( c ) φ ℓR ( c ) (cid:19) = Kc + χ, (45)where χ is the constant of integration. Multiplying both sides by φ ℓR ( c ) and letting c tend to 0, we have1 + lim c → φ ℓR ( c ) log (cid:18) − φ ℓR ( c ) φ ℓR ( c ) (cid:19) = K lim c → φ ℓR ( c ) c + lim c → φ ℓR ( c ) χ. Using that lim c → φ ℓR ( c ) = 0 (by Step 1) and lim x → x log x = 0, it follows that lim c → φ ℓR ( c ) /c =1 /K . Applying these results to the original differential equation, we conclude thatlim c → φ ′ ℓR ( c ) = K (cid:18) lim c → φ ℓR ( c ) c (cid:19) (1 − lim c → φ ℓR ( c )) = K K = 1 K = p ∗ λ ( U r ( p ∗ ) − U ℓ ( p ∗ )) . C.20 Proof of Lemma 19 Proof of Lemma 19. We first show that in any SMPE, p ∗ ≥ max { φ ℓL , π ℓL } . To thisend, we consider the following “hypothetical” environment in which the sender is lessconstrained than in our model: she may choose two experiments, one generating upwardjumps to q ( > p ) and the other generating downward jumps to q ( < p ). However, thesender is not constrained to split her attention between the two experiments. Instead,she can devote “full attention” to both. Specifically, she now has access to informationstructures in which α = α = 1, q < p < q , and α i = 0 for i > 2. Assuming that the Note that we cannot apply the implicit function theorem to determine dφ ℓR (0) /dc since F ( φ ℓR , c ) isnot continuously differentiable at ( φ ℓR , c ) = (0 , r whenever p ≥ p ∗ , the sender’s flow value of experimentationis given by ˜ v ( p ) = λp (1 − p ) max ≤ q V ′′ F A ( p ) = 2(1 − p ) − p ∗ p ∗ cλ > . Therefore, if we set V F A ( p ) = 0 whenever the value V F A ( p ) derived above is negative, thenthe function V F A ( p ) is convex on [0 , p ∗ ].For the same reason as in Section 5.1, the optimal q maximizes the slope V ( q ) − V ( p )) / ( q − p ), while the optimal q minimizes the slope ( V ( p ) − V ( q )) / ( p − q ). Convexityof V F A ( p ) over [0 , p ∗ ] implies that if V ( p ) = V F A ( p ) then q = p ∗ and q = 0 are optimalfor the sender. Conversely, since V F A ( p ) is the value function that arises with the optimaljump targets q = p ∗ and q = 0, it must be indeed the optimal value function. Finally,since V F A ( p ) is the value function for a problem in which the sender is less constrained,it forms an upper bound of the sender’s value function in the original problem (given thewaiting region [ p ∗ , p ∗ ]).The corresponding value function of the receiver can be derived in a similar fashion: U F A ( p ) = p ∗ − pp ∗ u Lℓ + pp ∗ U r ( p ∗ ) − C F A ( p ) . Specifically, this can be interpreted as the solution to the problem in which the receiverchooses two jump targets in [0 , p ∗ ] and can devote full attention to each target.In order to conclude that p ∗ ≥ max { φ ℓL , π ℓL } , first observe that V F A ( π ℓL ) = 0 and U F A ( φ ℓL ) = U ℓ ( φ ℓL ), which can be directly shown by plugging the values of π ℓL and φ ℓL in Section B.2 into each function. Combining this with the fact that V ( p ) ≤ V F A ( p ) and U ( p ) ≤ U F A ( p ) for all p ≤ p ∗ , but V ( π ℓL ) = V F A ( π ℓL ) and U ( φ ℓL ) ≤ U F A ( φ ℓL ) leads tothe desired result.For the second result, suppose by contradiction that p ∗ > max { φ ℓL , π ℓL } and V ( p ∗ ) =69. If p ∗ > π ℓL , then for all p ∈ ( π ℓL , p ∗ ), the sender uses the L -drifting experiment withjumps to p ∗ . This follows from the same arguments as in the proof of Lemma 16.(b). Thereceiver’s strategy in the candidate equilibrium prescribes to stop for p < p ∗ . However,this is not a best response for p ∈ ( φ ℓL , p ∗ ), which is a non-empty interval if φ ℓL < p ∗ .Therefore we must have V ( p ∗ ) > p ∗ > max { φ ℓL , π ℓL } .Now suppose by contradiction that U ( p ∗ ) > U ℓ ( p ∗ ). We show that this implies thatthere exists ε > 0, such that it is optimal for the receiver to wait if p ∈ [ p ∗ − ε, p ∗ ),contradicting the conjectured equilibrium. If V ( p ∗ ) > 0, the same argument as in theproof of Lemma 16.(c) implies that the optimal experiment for the sender is “ L -driftingwith jumps to p ∗ ” for p ∈ [ π , p ∗ ). The (flow) benefit of this experiment for the receiver is λp (1 − p ) p ∗ − p ( U ( p ∗ ) − U ℓ ( p )) − λp (1 − p ) U ′ ℓ ( p ) = λp (1 − p ) p ∗ − p ( U ( p ∗ ) − U ℓ ( p ∗ )) . If U ( p ∗ ) > U ℓ ( p ∗ ), this exceeds the cost c for p sufficiently close to p ∗ . Hence, the receiveris willing to stop for all p < p ∗ only if U ( p ∗ ) = U ℓ ( p ∗ ). C.21 Proof of Lemma 20 Proof of Lemma 20. Note first that both φ ℓL and π ℓL approach 0 as c tends to 0. More-over, both U S ( p ) and U RS ( p ; q R ) (whether q R = p ∗ or q R = ξ ) converge to pU r ( p ∗ ) /p ∗ +( p ∗ − p ) u Lℓ /p ∗ . Therefore, the characterization of p ∗ from Lemma 19 implies that p ∗ → c → p ∗ is given by the indifference condition for the receiver which implies p ∗ = φ ℓR → 0, or p ∗ = max { φ ℓL , π ℓL } → 0. Hence, p ∗ < ξ if c is sufficiently low. Wetherefore assume that c is small enough so that p ∗ < ξ .Suppose p ∗ ≤ η . If p ∗ = max { φ ℓL , π ℓL } , Lemma 8 and Condition C2 imply that p ∗ = φ ℓL . Lemma 18 shows that there exits c ( p ∗ ) > c < c ( p ∗ ), π < φ ℓL .Note that convexity of V R ( p ) implies that V R ( p ) > p > π . But this implies that p ∗ = φ ℓL cannot be the lower bound in an equilibrium. If it was, the sender would preferto use R -drifting for p close to φ ℓL which yields V R ( φ ℓL ) > V − ( φ ℓL ; φ ℓL , φ ℓL < φ ℓR which implies that U R ( p ) < U ℓ ( p ) for p ∈ ( φ ℓL , φ ℓR ). Therefore, it is not abest response for the receiver to wait for p close to φ ℓL . This contradicts the fact that thelower bound of the waiting region is p ∗ = φ ℓL .The proof for the case p ∗ > η is similar. The only necessary modification is thatwe have to show that V RS ( p ) > p > π , which again follows from convexity of thesender’s value function. C.22 Proof of Lemma 21 Proof of Lemma 21. Suppose by contradiction that p ∗ > π ℓL . Then, by Lemma 19, U ( p ∗ ) = U ℓ ( p ∗ ) and V ( p ∗ ) > 0. According to Lemma 15, V ( p ∗ ) > L -drifting at p ∗ , and the game ends only when p reaches 0 or p ∗ .This implies that the players’ expected values at p ∗ can be written as V ( p ∗ ) = p ∗ p ∗ v − C ( p ∗ ) and U ( p ∗ ) = p ∗ p ∗ U r ( p ∗ ) + p ∗ − p ∗ p ∗ u Lℓ − C ( p ∗ ) , where C ( p ) represents the total cost of delay common for both players. By straightforwardalgebra, U ( p ∗ ) − U ℓ ( p ∗ ) = p ∗ p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) − C ( p ∗ ) . If U ( p ∗ ) = U ℓ ( p ∗ ) and V ( p ∗ ) > 0, but Condition C2 fails, then we arrive at the followingcontradiction: C ( p ∗ ) < p ∗ p ∗ v ≤ p ∗ p ∗ ( U r ( p ∗ ) − U ℓ ( p ∗ )) = C ( p ∗ ) . eferences Arrow, K. J., D. Blackwell, and M. A. Girshick (1949): “Bayes and MinimaxSolutions of Sequential Decision Problems,” Econometrica , pp. 213–244. 5 Au, P. H. 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