aa r X i v : . [ ec on . T H ] M a r Persuading part of an audience
Bruno Salcedo ∗† March 4, 2019
Abstract
I propose a cheap-talk model in which the sender can useprivate messages and only cares about persuading a subset of her au-dience. For example, a candidate only needs to persuade a majorityof the electorate in order to win an election. I find that senders cangain credibility by speaking truthfully to some receivers while lyingto others. In general settings, the model admits information transmis-sion in equilibrium for some prior beliefs. The sender can approximateher preferred outcome when the fraction of the audience she needs topersuade is sufficiently small. I characterize the sender-optimal equi-librium and the benefit of not having to persuade your whole audiencein separable environments. I also analyze different applications andverify that the results are robust to some perturbations of the model,including non-transparent motives as in Crawford and Sobel (1982),and full commitment as in Kamenica and Gentzkow (2011).
Keywords
Cheap talk · Information transmission · Persuassion
JEL classification
D83 · C72 · D72 · L15 ∗ Department of Economics, Western University, brunosalcedo.com, [email protected]. † I am grateful to Yi Chen for introducing me to the work of Lipnowski and Ravid (2018),and to the Economics faculty at Western University for their support and guidance. some but not all of them. Mymain finding is that having to persuade only part of an audience significantlyfacilitates information transmission and increases persuasion power.Let us examine the first example in more detail. Suppose a politician (thesender, she) is running for office. All voters (the receivers, he) share the samepreferences. The unknown state of the world equals either 0 or 1. Each voter willvote for the politician if his expectation about the state of the world is greaterthan 1 /
2. The voters share a common prior expectation in the interval (1 / , / /
3. In that case, every voter who receives thismessage prefers to vote for the politician.I study a general cheap-talk model with many ex-ante homogeneous receiversand both public and private communication. I depart from the literature byassuming that there are n receivers, but the sender only cares about the highest n < n actions taken. In such cases, the utility of the sender can be completelydetermined by strict subsets of the receivers. Thus, she only needs to persuadepart of her audience in order to maximize her utility. I call the gap between n and n an excess audience .I find that the sender can influence the behavior of receivers in equilibrium in a2ery wide class of environments as long as there is an excess audience (Proposition4). In some environments, effective information transmission is only possible ifthere is an excess audience (Proposition 6). When the fraction of the audiencethat the sender cares about is small enough, she can achieve her preferred outcomein equilibrium (Proposition 3).I characterize the sender’s benefit from having an excess audience under aseparability assumption (Theorem 5). Lipnowski and Ravid (2018) characterizethe sender’s maximum equilibrium payoff when the sender cares about her entireaudience in terms of her value function. The value function is the highest payoffthe sender can obtain when all the receivers behave optimally given their poste-rior beliefs. Under Lipnowski and Ravids’ assumptions, the sender’s maximumequilibrium payoff equals the quasiconcave envelope of her value function. I findthat an additional step is needed in the presence of an excess audience.This step involves a generalization of the politician’s communication strategydescribed above. The sender starts by randomly and privately splitting her au-dience into a target audience that she wants to persuade, and the rest of thereceivers. Receivers in the target audience always receive whichever message in-duces the behavior most favorable to the sender. The communication strategyfor the rest of the receivers is chosen to maximize the credibility of the messagesent to the target group. This message conveys information because individualreceivers are not told whether they were assigned to the target audience.This kind of strategy allows the sender to implant a fixed posterior belief ina fixed proportion of the audience regardless of the state. I say that such beliefsare attainable . The set of attainable beliefs admits a simple and computationallytractable characterization (Lemmas 1 and 2). When the sender wishes to persuadeher entire audience, the only attainable belief is the prior. However, the set ofattainable beliefs is strictly increasing with the size of the excess audience. Themissing step to characterize the sender’s maximum equilibrium payoff is to replacethe original value with the maximum value over the set of attainable beliefs.The benefit from having an excess audience is always non-negative. It is strictlypositive for some prior beliefs. And it is monotone in the fraction of the audiencethat the sender wishes to persuade. Moreover, as the fraction of the audiencethat the sender cares about converges to zero, the set of attainable beliefs totallycovers the interior of the simplex. Consequently, the maximum equilibrium payoffapproaches the best feasible payoff for the sender (Proposition 7).Section 4 applies the techniques developed in the paper to analyze the election3xample, financial advice, and the role of advertisement for crowdfunding. Sec-tion 5 considers two extensions of the model. First, I analyze a model with fullcommitment as in Kamenica and Gentzkow (2011) and the information design lit-erature (Bergemann and Morris, 2016, Taneva, 2019). I find a characterization ofthe maximum sender equilibrium payoff in the full-commitment game, assumingthat the state space is finite. The characterization is similar to the one for thecheap-talk game. The only difference is that it uses the concave envelope of thevalue function instead of the quasiconcave envelope.I also analyze an example with the classic quadratic loss functions from Craw-ford and Sobel (1982). This example does not satisfy all the assumptions requiredfor my characterization. However, it is still possible to use strategies with a ran-dom target audience in order to transmit information. When the fraction of theaudience that the sender cares about is small, the sender can approximate herpreferred outcome and transmit large amounts of information to most of her au-dience in equilibrium. Unlike the case without an excess audience, informationtransmission is possible for any degree of bias.Since the seminal work of Vincent Crawford and Joel Sobel, different authorshave found different mechanisms for an expert to gain credibility. Informationtransmission is possible via cheap talk when incentives are not too misaligned,or there are multiple senders (Battaglini, 2002), or multiple dimensions of infor-mation (Chakraborty and Harbaugh, 2010), or strategic complementarities (Levyand Razin, 2004, Baliga and Sjöström, 2012), or the sender has transparent mo-tives (Lipnowski and Ravid, 2018), among other reasons. An excess audience isa novel mechanism which allows for information transmission in some settings inwhich none of the aforementioned mechanisms operate.Some authors have studied cheap-talk communication with multiple audiences.However, this literature has focused on situations when the sender cares about theactions of all receivers, either directly or indirectly. Farrell and Gibbons (1989)showed that senders with multiple audiences sometimes prefer public communica-tion and sometimes private communication. Goltsman and Pavlov (2011) showthat the sender might be strictly better off by combining both types of messages.Hence, I allow the sender to use both private and public messages.Basu et al. (2018) study the problem of an informed sender who employs cheaptalk to try to prevent an ethnic conflict. Their model has a large audience, andthe sender is allowed to use private messages. However, they restrict attentionto strategies that are anonymous, conditional on observed heterogeneity. This4estriction precludes the strategies with random target audiences that I analyze.Instead, they exploit preference complementarities in order to find an equilibriumwith effective information transmission.There is a large body of literature using cheap talk to study information trans-mission between politicians and electorates, dating at least as far back as Harring-ton (1992). Some recent work in this area includes Schnakenberg (2015), Panova(2017), Jeong (2019), and Kartik and Van Weelden (2017). Other recent papersanalyze the problem from the information design perspective, including Alonsoand Câmara (2016), and Chan et al. (2019). Within this literature, the papersthat assume talk is cheap focus on public messages. My contributions highlightthe importance of private anonymous communication (e.g., through social media).
1. Model
There is one sender s , and a set of receivers r ∈ R = { , . . . , n } . The senderand receivers share a common prior belief π about the true state θ ∈ Θ. Thesender learns the state. She then sends a private message m pr ∈ M to eachreceiver r , and a public message m ∈ M to all receivers. Each receiver observesthe compound message m r = ( m , m pr ), but observes neither the state nor otherreceivers’ private messages. Then, all receivers observe an uninformative publicsunspot ω distributed uniformly on [0 , r chooses anaction a r ∈ A . All receivers have identical preferences. The utility of r dependsonly on his own action and the state. It is given by u R ( a r , θ ). The sender’s utility u S ( a , . . . , a n ) does not depend on the state.I impose some technical restrictions. Each of A , Θ and M is a compactseparable metric space containing at least two elements. M and M are rich I adopt the following notational conventions throughout the paper. For each separablemetric space Y , B Y denotes the Borel σ -algebra on Y , and ∆ Y the set of probability measures on( Y, B Y ) endowed with the w eak* topology. Given measures π, τ ∈ ∆ Y , π ≪ τ denotes absolutecontinuity, π ∼ τ denotes equivalence, and dπ/dτ denotes the Radon-Nikodym derivative of π with respect to τ . The support of π is denoted by supp π . Given a set X , a function g : X → ∆ Y ,a point x ∈ X and an event Y ′ ∈ B Y , let g ( Y ′ | x ) := [ g ( x )]( Y ′ ) . Profiles of elements of X aredenoted by x = ( x , . . . , x n ) ∈ X n . Throughout the paper, “a.s.” means “almost surely withrespect to π ”. not restrict the set of equilibrium outcomes. The utility functionsare continuous.I study the perfect Bayesian equilibria of this game.
Communication strategies , receiver strategies , and updating rules are measurable maps µ : Θ → ∆( M × M n ), α r : M × M × [0 , → A , and β r : M × M → ∆Θ, respectively. Let µ r ( · | θ ) denotethe marginal distribution over m r induced by µ ( θ ). Let BR( π ) be the set of actionsthat maximize R Θ u R ( a, θ ) dπ ( θ ). An equilibrium is a tuple ( α , β , µ ) consisting aprofile of receiver strategies, a profile of updating rules, and a communicationstrategy, such that (i) For every receiver r , β r is consistent with Bayes’ rule given µ r and π . (ii) α r ( m, ω ) ∈ A ∗ ( β r ( m r )) for every receiver r and compound message m r . (iii) For every message profile m ∈ M × M n , if there exists a state θ such that µ ( m | θ ) >
0, then m maximizes R u S ( α ( m , ω ) , . . . , α n ( m n , ω )) dω . Throughout most of the paper, I maintain the assumption that the sender onlycares about the tail of the empirical distribution of actions taken by the receivers.
Assumption 1 A ⊆ R and there exists an integer n ′ such that for every pair ofaction profiles a and ˜ a , if a ( i ) = ˜ a ( i ) for every i ≥ n + 1 − n ′ , then u S ( a ) = u S (˜ a ),where a ( i ) denotes the i -th order statistic of a .Define the pivotal number of receivers to be the smallest integer n ∈ { , . . . , n } that satisfies the condition from Assumption 1. The pivotal fraction of the audi-ence is γ = n /n . The sender’s utility only depends on the highest n actionstaken by the receivers. If n = 0, then the sender is indifferent between all out-comes. If n = n , then the sender cares about the entire empirical distribution ofreceiver actions, but not about the identity of the receivers taking each action. If0 < n < n then the sender cares about persuading fewer receivers than she cantalk to. In that case, I say that there is an excess audience . A sufficient condition is k M k , k M k ≥ k ( A × Θ × [0 , n k . xample 1 Recall the election example from the introduction. Letting a r = 1denote a vote for the sender and a r = 0 a vote against her, the outcome of theelection is determined by the median action. If n is odd, then n = ( n + 1) / γ = ( n + 1) / n . See Section 4.1 for more on this example. Example 2
Suppose that the sender is the owner of a coffee shop from a localfranchise T . The shop is located at a tourist destination with several shops fromthe same franchise. Most of the potential customers are travelers who are unfa-miliar with the franchise and will buy coffee from it at most once. The utilityof a potential customer equals θ if they buy a coffee at one of the coffee shops( a r = 1), and 0 otherwise ( a r = 0). The sender’s profit is normalized to equal thenumber of customers she serves. She can talk with n passing-by receivers, butshe can serve at most n of them. Receivers who choose a r = 1 but are beyondthe capacity of the sender, will buy their coffee at a different shop from the samefranchise. Therefore, u S ( a ) = P n i =1 a ( n +1 − i ) .
2. Attainable posteriors
This section discusses two technical lemmas that drive the rest of the results.Readers interested in the main results can skip to Section 3. A key step in myanalysis is to determine the maximum influence that the sender can exert overthe beliefs of part of her audience. If the sender wants to guarantee that thereare always n receivers having certain posterior beliefs, what values can theseposteriors take? Definition 1
For γ ∈ [0 , π ′ ∈ ∆Θ is γ - attained by a communicationstrategy µ and a profile of updating rules β if (i) For every receiver r , β r is consistent with Bayes’ rule given µ . (ii) For every state θ and every message profile m in the support of µ ( θ ), thereexists a set T ⊆ R such that k T k ≥ nγ and β r ( m r ) = π for all r ∈ T .Say that π is γ -attainable if there exist µ and β that γ -attain it. Let Π( γ )7e the set of γ -attainable beliefs. The following lemma characterizes the set of γ -attainable beliefs using a set of linear restrictions on likelihood ratios. Lemma 1
For all γ ∈ (0 , and π ∈ ∆Θ , the following statements are equivalent (i) π is γ -attainable. (ii) π ( E ) π ( E ′ ) ≥ γπ ( E ′ ) π ( E ) for any two events E, E ′ ∈ B Θ . (iii) π ≪ π and dπ/dπ ∈ [ γc , c ] a.s. for some c > . Lemma 1 implies that Π( γ ) is a nonempty closed and convex polytope. Notethat all possible posterior beliefs are 0-attainable, and only π is 1-attainable. Allthe proofs are in the appendix. The following example shows one way to reachthe bounds from ( ii ). Example 3
Suppose that Θ = { , } and fix some γ = k/n with k ∈ { , . . . , n } .Identify each π ∈ ∆Θ with the probability p := π (1). The only two non-trivialevents to consider are { } and { } . Hence, part ( ii ) of Lemma 1 implies that abelief is γ -attainable if and only if γ p − p ! ≤ p − p ≤ γ p − p ! . (1)After some simple algebra, this implies that the set of γ -attainable beliefs corre-sponds to " p p + (1 − p ) /γ , p p + (1 − p ) γ . (2)The upper bound in (2) can be attained by the following communication strat-egy. The sender first chooses a random target audience T ⊂ R consisting ofexactly k receivers. Each receiver r only observes one of two possible (compound)messages m r = m ′ or m r = m ′′ . The sender always sends message m ′ to all thereceivers in T . Receivers not in T receive message m ′ if and only if θ = 1. Thisstrategy results in the conditional probabilities µ r ( m ′ |
1) = 1 and µ r ( m ′′ |
0) = γ . Condition ( iii ) is a stronger version of the condition from Theorem 2.1 in Diaconis andZabell (1982). They find that a posterior belief π is consistent with a prior π and Bayes’ Ruleif and only if π ≪ π and dπ/dπ ≤ c a.s. for some c > . β r (1 | m ′ ) equals the upper bound of (2).Since at least k receive message m ′ , this belief is γ -attained.Lemma 2 below asserts that each extreme point of Π( γ ) corresponds to a par-tition of states into only two blocks. States in one block have increased likelihoodsrelative to the prior, and states in the other block have decreased likelihoods. Thischaracterization makes Π( γ ) computationally tractable. It is particularly advanta-geous in monotone environments when the sender would always prefer to increasethe receivers’ beliefs about the state. Lemma 2
For any γ ∈ (0 , , a belief π ∈ ∆Θ is an extreme point of Π( γ ) , if andonly if there exists an event E + ∈ B Θ such that for every event E ∈ B Θ π ( E ) = π ( E ∩ E + ) + γπ ( E \ E + ) π ( E + ) + γπ (Θ \ E + ) . (3) Example 4
Suppose that Θ = { , , } and π = (1 / , / , / π (1) /π (2) = 3 / π (1) /π (
3) = 3, and π ( /π (
3) = 2. Itfollows from part ( ii ) of Lemma 1 that a posterior π is ⁄ -attainable if and only if π (1) /π (2) ∈ [3 / , π (1) /π (3) ∈ [3 / , π (2) /π (3) ∈ [1 , /
2) is the shaded irregular hexagon surrounding π . The vertex π ′ = (1 / , / , /
9) is given by (3) with E + = { θ , θ } . It maximizes R Θ θ dπ ( θ )subject to π ∈ Π(1 / Example 5
Suppose θ is distributed uniformly on [0 , γ -attainablebelief that maximizes the expectation of the state? Theorem 32.3 in Rockafellar(1970) implies that the maximum is attained at an extreme point of Π( γ ). Lemma2 thus implies that the maximizer π ∗ takes the form (3), and it must be the casethat E + = [ θ ,
1] for some θ ∈ (0 , . It follows that Z θ dπ ∗ ( θ ) = Z θ γθγθ + 1 − θ dθ + Z θ θγθ + 1 − θ dθ = γθ + 1 − θ γθ + 1 − θ ) . (4)This expression is maximized when θ = 1 / (1 + √ γ ), and the maximum is R Θ θ dπ ∗ ( θ ) = 1 / (1 + √ γ ). Note that this maximum equals 1 / γ = 1,equals 1 when γ = 0, and is strictly decreasing in γ .9 b b bb b p ′ π √ / √ / θ = 1 θ = 2 θ = 3 Figure 1 – ⁄ -attainable beliefs for Example 4.
3. Information transmission and persuasion
It is possible to exert great influence over very small fractions of an audi-ence. Suppose that the sender’s utility is maximized by a constant action profile.Further suppose that this action profile is a best response to a belief π ∗ with aRadon-Nikodym derivative bounded both above and away from zero. Lemma 1implies that π ∗ would be γ -attainable for sufficiently low values of γ . Hence, whenthe sender only cares about small fractions of her audience, she would be able toreach her preferred outcome in equilibrium. Formally, Assumption 2
There exist an action a ∗ and a belief π ∗ such that a ∗ ∈ BR( π ∗ )and u S ( a ∗ , . . . , a ∗ ) ≥ u S ( a ) for every action profile a .10 ssumption 3 π ∗ ≪ π , and there exist two numbers 0 < ν ≤ ¯ ν < ∞ such that dπ ∗ /dπ ∈ [ ν, ¯ ν ] a.s.. Proposition 3
Under assumptions 1–3 there exists ¯ γ = ¯ γ ( π , π ∗ ) ∈ (0 , suchthat if γ ≤ ¯ γ , then the game admits an equilibrium in which the sender obtainsher preferred outcome. I have assumed that the sender has transparent motives , in that her preferencesdo not depend on the state and are thus common knowledge. Cheap-talk modelswith transparent motives often allow for some information transmission in equi-librium. See, for instance, Theorem 1 in Chakraborty and Harbaugh (2010) andProposition 1 in Lipnowski and Ravid (2018). The question I address is whetherthe sender can transmit sufficient information in order to influence the behaviorof her audience to her benefit.Let v denote the sender’s value function . That is, v ( π ) specifies the maximumutility that the sender could obtain if all receivers shared a posterior π and actedoptimally, v ( π ) = max n u S ( a ) (cid:12)(cid:12)(cid:12) a r ∈ BR( π ) for all receivers r o . (5)Say that an equilibrium exhibits effective information transmission if the sender’sexpected equilibrium payoff is strictly greater than v ( π ). An excess audienceallows for effective information transmission for some prior beliefs under two mildsensitivity assumptions that rule out trivial cases. Assumption 4 u S ( a ∗ , . . . , a ∗ ) > u S ( a ′ , . . . , a ′ ) for every action a ′ = a ∗ , and thereexists π ′ ∈ ∆Θ such that a ∗ BR( π ′ ), π ′ ≪ π ∗ , and there exist two numbers0 < ν ′ ≤ ¯ ν ′ < ∞ such that dπ ′ /dπ ∗ ∈ [ ν ′ , ¯ ν ′ ] a.s.. Proposition 4
Under assumptions 1, 2, and 4, if there is an excess audience, thenthere exists a nonempty set P = P ( π ∗ , π ′ , γ ) ⊆ ∆Θ such that the game has an If k Θ k < + ∞ , Assumption 3 holds if and only if supp π = supp π ∗ . If k Θ k < + ∞ , the last condition is satisfied whenever supp π ′ = supp π ∗ . quilibrium with effective information transmission as long as π ∈ P . Moreover,if k Θ k < + ∞ , then P has a nonempty interior. The sender’s maximum equilibrium payoff v ∗ is the maximum utility that thesender can obtain in any equilibrium. This section characterizes v ∗ under thefollowing separability assumption. Assumption 5
There exist a strictly increasing function U S : A → R such that u S ( a ) = 1 n n X i =1 U S (cid:16) a ( n +1 − i ) (cid:17) . (6)The characterization relies on two operators defined on the set of upper-semicontinuous functions from beliefs to sender payoffs. First, env q v is the qua-siconcave envelope of v . That is, env q v is the pointwise-minimum, quasiconcaveand upper semicontinuous function that majorizes v . Second, att v gives the max-imum of v arising from γ -attainable beliefs, i.e.,att v ( π ) = max n v ( π ′ ) (cid:12)(cid:12)(cid:12) π ∈ Π( γ , π ) o , (7)where Π( γ , π ) is the set of beliefs that would be γ -attainable if π = π . Intu-itively, env q v operates by “flooding the valleys” while att v is obtained by “widen-ing the hills.” See the left and center panels of Figure 2 in Section 4.2 for anexample. Theorem 5
Under assumptions 1 and 5, v ∗ = att env q v ( π ) . Lemma 1 implies that att env q v = env q v when γ = 1. Hence, Theorem 5reduces to Theorem 2 in Lipnowski and Ravid (2018) in that case. However, thetwo results differ whenever there is an excess audience and the assumptions fromProposition 4 hold. The difference between the results corresponds to the benefitfrom an excess audience defined in the following subsection.12 .4. The benefit from an excess audience and private communication What happens to the sender’s payoff when she has to persuade a larger orsmaller fraction of her audience? Proposition 6 below gives sufficient conditionsunder which an excess audience is necessary for effective information transmission.These conditions are satisfied by the election, excess capacity, and labor marketapplications in Section 4.
Proposition 6
Under assumptions 1 and 5, if v is quasi-concave and there isan equilibrium with effective information transmission, then there is an excessaudience. Define the benefit from excess audience to be the difference between v ∗ andthe maximum equilibrium payoff to the sender in an alternative environment with n = n . It follows from Theorem 5 and Lemma 1 that this benefit equals thedifference between v ∗ and env q v ( π ). Moreover, Assumption 5 guarantees thatboth v ∗ and env q v ( π ) are measured in the same units. Proposition 7
Under assumptions 1–5, there exist ¯ γ = ¯ γ ( π , π ∗ ) > and anonempty set P = P ( π ∗ , π ′ , γ ) ⊆ ∆Θ such that the benefit from excess audience (i) is non-negative and non-increasing in γ , (ii) is strictly positive whenever π ∈ P and γ < , and (iii) equals ˜ u S ( a ∗ ) − v ( π ) whenever γ ≤ ¯ γ . Consider an alternative model in which the sender can use only public messages.She still cares only about the actions of part of her audience. However, publicmessages only allow her to persuade either all of the receivers or none of them. The benefit from private communication is the gap between v ∗ and the maximum senderequilibrium value in this alternative model. Without private messages, the only γ -attainable belief is the prior. Hence, the benefit from private communicationcoincides with the benefit from excess audience under Assumption 5.13 . Examples Social media allows politicians to personalize campaign advertising at a lowcost. Suppose that the sender is a politician running for office. Each receiver r will either vote for the sender ( a r = 1) or against her ( a r = 0). The state is either0 or 1, and voters share a common prior belief with p := π (1) ∈ (0 , γ ∈ (0 , n = min { n ′ | n ′ > nγ } . All receivers share the samepreferences. Receiver r prefers a r = 1 if and only if his posterior beliefs satisfy p r := π r (1) ≥ η , where η ∈ (0 ,
1) is a fixed parameter.When is victory attainable for the politician? She wins the election when theposterior beliefs of at least n voters satisfy π r > η . From (1), there exists a γ -attainable belief satisfying this condition if and only if η − η < γ p − p ! . (8)In such cases, there exists an equilibrium in which the sender always wins theelection regardless of the state. The condition is satisfied whenever: ( i ) the votersprior attitude towards the sender is positive ( p is high), ( ii ) the voters have a lowbar for the sender ( η is small), or ( iii ) the sender only needs a small fraction ofvotes in order to win the election ( γ is small). For the case γ = 1 / η = 1 / π > / The state θ ∈ { , } indicates the winner of a rigged boxing match. Thesender is an informed bookie who knows the state and would like to persuade thereceivers to place large bets. However, she is time constrained. She can talk with n receivers, but she can handle at most n ≤ n bets. The rest of the bets will be14andled by other bookies. The total utility of the sender equals V + ηV , where V θ is the total volume of bets on θ that she handles, and η > w > a r ∈ [ − w, w ]. A positive bet represents a bet on θ = 1 whereas a negative betrepresents a bet on θ = 0. Bets on different states have different exogenous netreturns ρ , ρ >
0, with ρ ρ <
1. Receivers have logarithmic Bernoulli utilityfunctions. For example, a receiver with beliefs π that places a bet on θ = 1maximizes Z Θ u R ( a r , θ ) dπ ( θ ) = p log( w + a r ) + (1 − p ) log( w − a r ) , (9)subject to a r ∈ (0 , w ], where p = π (1).This example deviates slightly from our environment because the sender caresabout both tails of the distribution of actions. However, the conclusion of Theorem5 still applies. The receivers’ best response correspondence is given byBR( π ) = wρ [(1 + ρ ) p −
1] if p ≥
11 + ρ − wρ [(1 + ρ )(1 − p ) −
1] if p ≤ ρ ρ . (10)The function att v can be computed by substituting the bounds from (2) into u S (BR( p )). Figure 2 illustrates env q v (left), att v for γ ∈ { / , / } (middle),and v ∗ for γ = 1 / The sender owns a start-up company financed via an online crowdfundingplatform. The receivers are potential backers. Each receiver pledges an investmentlevel a r ≥
0. Say that the company is backed if P r ∈ R a r ≥ η , where η > u S πu S πu S Figure 2 – Value functions for the bookie example with w = 0 . , η = 2 , ρ = 1 / ,and ρ = 1 / . Left panel: env q v . Center panel: att v with γ = 1 / (red) and γ = 1 / (blue). Right panel: v ∗ with γ = 1 / . fixed parameter. The sender gets a payoff of 1 if the company is backed and apayoff of 0 otherwise.A backed company might be a success or a failure. The company succeedswith probability θ ∈ [0 ,
1] if it is backed, and it fails for sure otherwise. Theprior belief about θ is uniform on [0 , θ and cancommunicate with the receivers via private cheap talk.If the company is not backed, the pledged investments are refunded. If thecompany is backed and fails, all the investments are lost. Otherwise, the investorsreceive a net return ρ >
0. The receivers have logarithmic Bernoulli utility func-tions. Hence, u R ( a r , θ ) = − θe − ( w + ρa r ) − (1 − θ ) e − ( w − a r ) (11)if the company is backed, and u R ( a r , θ ) = − e − w otherwise.Lemmas 1 and 2 can be used to construct equilibria in which the sender benefitsfrom having an excess audience. Note that the company will be backed if at least n ′ receivers to pledge at least η/n ′ . If η/n ′ <
1, a receiver who is optimisticenough about θ would be willing to pledge that amount. Optimistic beliefs canbe ( n ′ /n )-attained as long as n is sufficiently larger than n ′ . Proposition 8 If n ≥ η/wρ , then the crowdfunding game has a perfect Baye-sian equilibrium in which the company is backed for sure regardless of the state. . Extensions Suppose now that the sender chooses and commits to a communication strategy before learning the state of nature. This timing corresponds to the informationdesign paradigm used by Kamenica and Gentzkow (2011). See also Bergemannand Morris (2016). Define a commitment protocol to be a tuple ( α , β , µ ) satisfyingconditions ( i ) and ( ii ) in the definition of an equilibrium. The sender’s maximumcommitment payoff v ∗∗ ( p ) is the maximum utility that the sender can obtain inany commitment protocol.Since every equilibrium is a commitment protocol, Propositions 3 and 4 con-tinue to hold in the game with commitment. Also, it is possible to obtain ageometric characterzation of v ∗∗ . Let env v denote the concave envelope of v ,that is, the pointwise-minimum, concave function that majorizes v . Theorem 9 If k Θ k < + ∞ and Assumption 5 holds, then v ∗∗ = att env v ( π ) . Figure 3 illustrates this result for the election example with γ = η = 1 / v , which corresponds to the maximumcommitment value without an excess audience (Kamenica and Gentzkow, 2011).This would also be the maximum commitment value for the sender if she wasrestricted to use only public messages. The middle panel shows v ∗ = att v . Theright panel shows v ∗∗ . The benefit from commitment is thus given by the gapbetween the right and the middle panel. The benefit from an excess audienceunder commitment is given by the gap between the right and the left panel. Assuming that the sender does not care about the state simplifies the analysisand plays a crucial role in the proof Theorem 5. However, an excess audience can The proof is based on Proposition 1 in Kamenica and Gentzkow (2011), which assumes afinite state space. The discussion in Section 3 of their online appendix suggests that it mightbe possible to extend the result to compact separable state spaces. u S πu S πu S Figure 3 – Commitment value for election example with γ = η = 1 / . still be beneficial even when the sender’s motives are not transparent. Considerthe classic quadratic-loss game from Crawford and Sobel (1982), with the twistthat the sender faces an excess audience.The state is distributed uniformly on [0 , A = [0 , u R ( a r , θ ) = − ( θ − a r ) . The sender’s utility isgiven by u S ( a , θ ) = − n n X i =1 (cid:16) θ + b − a ( n +1 − i ) (cid:17) , (12)where b > / bias of the sender relative tothe receivers. Suppose that the sender is only allowed to use private messages.When n is equal to the total number of receivers, this is a particular instanceof the environment studied by Goltsman and Pavlov (2011). Since all the receiversare biased in the same direction, there are only babbling equilibira. In contrast,when n is much larger than n , there can be effective information transmission. Proposition 10
For all ǫ > , there exists n < ∞ such that whenever n ≥ n themaximum sender equilibrium value in the quadratic-loss game is greater than − ǫ if θ + b ≤ , and greater than − b − ǫ otherwise. The proof is constructive. The sender uses a strategy based on a finite partitionof [0 , n − n and n . She then reveals truthfully which block of the partition contains θ to themembers of the first group. She misleads the members of the second group sothat they choose her preferred action. When n is very large, it is possible to18onstruct incentive compatible equilibria of this sort with very fine partitions.
6. Closing remarks
When talk is cheap, information transmission requires the sender to be indiffer-ent between all messages she uses. The present work identifies a novel mechanismthat can create indifference. When the sender only cares about persuading a strictsubset of her audience, she is indifferent between the messages she sends to therest of the receivers. It is possible for her to gain credibility by being truthfulwith some receivers while lying to others. This mechanism can greatly facilitateinformation transmission and increase the sender’s power to persuade.The present work provides a full characterization of the sender-optimal equilib-rium assuming that the receivers do not care about each other actions, the senderhas transparent motives, and her preferences are monotone and satisfy a separa-bility condition. These restrictions greatly simplify the analysis. They make itpossible to characterize the set of equilibria combining the idea of γ -attainabilitywith the techniques from Lipnowski and Ravid (2018). However, they appear tobe inessential for many of the results. The value of having a large audience ingeneral settings is left as an open problem. References
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A. Proofs
A.1. Attainable posteriors
Proof of Lemma 1. If γ = 1, then the only belief that satisfies either ( i ), ( ii ) or( iii ) is π , and it satisfies all three conditions. Hence, for the rest of the proof,suppose that γ ∈ (0 , i ) ⇒ ( ii )) Suppose that π is γ -attained by some β and µ , and fix any twoevents E ′ and E ′′ . For each receiver r , let M r be the set of compound messages m such that β r ( m ) = π , and let χ r be the indicator that m r ∈ M r . Also let χ = P r ∈ R χ r be the number of receivers whose posterior equals π . The prior beliefs π , communication strategy µ , and updating rule β induce a joint probabilitymeasure Pr over θ , m r , χ r and χ .First, suppose that π ( E ′ ) π ( E ′′ ) = 0. Since γ >
0, there exists at least onereceiver r such that Pr( M r ) >
0. This implies that π is obtained from π usingBayes’ rule. Therefore, π ≪ π and ( iii ) holds with equality.Now consider the case π ( E ′ ) π ( E ′′ ) >
0. Note that E [ χ | E ′ ] = X r ∈ R E [ χ r | E ′ ] = X r ∈ R Pr( M r | E ′ ) , (13)where I am using short notation for the events θ ∈ E ′ and m r ∈ M r . Since p ′ is21 -attained by x and β , it follows thatPr( χ ≥ γn | E ′ ) = 1 ⇒ E [ χ | E ′ ] ≥ γn. (14)Combining (13) and (14) it follows that1 n X r ∈ R Pr( M r | E ′ ) ≥ γ. (15)Therefore there exists receiver r ′ such that Pr( M r ′ | E ′ ) ≥ γ and Pr( M r ′ ) > π ( E ′ ) = π ( E ′ ) Pr( M r ′ | E ′ )Pr( M r ′ ) and π ( E ′′ ) = π ( E ′′ ) Pr( M r ′ | E ′′ )Pr( M r ′ ) . (16)Taking the ratio of these equations yields: π ( E ′ ) π ( E ′′ ) = π ( E ′ ) π ( E ′′ ) · Pr( M r ′ | E ′ )Pr( M r ′ | E ′′ ) ≥ π ( E ′ ) π ( E ′′ ) · γ , (17)which is equivalent to ( ii ) after rearranging terms.(( ii ) ⇒ ( iii )) Now suppose that ( ii ) and take any event E such that π ( E ) = 0,and thus π (Ω \ E ) = 1. It follows that γπ ( E ) π (Ω \ E ) ≤ π (Ω \ E ) π ( E ) = 0.Since γ >
0, this implies that π ( E ) = 0. Since E was arbitrary, this implies that π ≪ π . Hence, dπ/dπ : Θ → [0 , ∞ ) exists by the Radon-Nikodym theorem.Let ¯ ν, ν ∈ R + ∪ {∞} denote the supremum and infimum of the support of dp ′ /dπ ( θ ), respectively, i.e.,¯ ν = sup ( ν (cid:12)(cid:12)(cid:12) π dπdπ ( θ ) ≥ ν ! > ) (18)and ν = inf ( ν (cid:12)(cid:12)(cid:12) π dπdπ ( θ ) ≤ ν ! > ) . (19)By construction, 0 ≤ ν < ∞ , ν ≤ ¯ ν , and dπ/dπ ∈ [ ν, ¯ ν ] a.s.. If ¯ ν = ν , then ( iii ) issatisfied by c = ¯ ν . Otherwise, there exists ν , ν ∈ R such that ν < ν < ν < ¯ ν .I will show that if ( ii ) holds, then γν ≤ ν for any such pair. Therefore, γ ¯ ν ≤ ν and ( iii ) is satisfied by c = ¯ ν . 22et E = { θ | dπ/dπ ( θ ) ≤ ν } and E = { θ | dπ/dπ ( θ ) ≥ ν } . Since dπ/dπ is measurable, so are E and E . The way ¯ ν and ν were defined implies that π ( E ) > π ( E ) >
0. It follows from the Radon-Nikodym theorem that π ( E ) = Z E dπdπ ( θ ) dπ ( θ ) ≤ Z E ν dπ ( θ | E ) = ν π ( E ) . (20)By a similar argument, it follows that π ( E ) ≥ ν π ( E ) . (21)The fact that ν < ¯ ν implies that π ( E ) >
0. Hence, we can divide (20) by (22).Doing so yields ν ν π ( E ) π ( E ) ≥ π ( E ) π ( E ) ≥ γ π ( E ) π ( E ) , (22)where the second inequality follows from condition ( ii ). Therefore, γν ≤ ν .(( iii ) ⇒ ( i )) The proof is constructive. Suppose that condition ( iii ) holds andlet φ : Θ → R be the function given by φ ( θ ) = 1 c · dπdπ ( θ ) , (23)Also, let k = min { n ′ ∈ N | n ′ ≥ γn } . Consider the following communicationstrategy. The sender always sends the same non-informative public message m = ∅ . She chooses a random target audience T ⊆ R consisting of exactly k receiversuniformly from { R ′ ⊆ R | k R ′ k = k } . She sends the (compound) message m ′ =( ∅ , m ) with probability 1 to all receivers in T regardless of the state. For eachstate θ ∈ Θ ∗ , each receiver r not in T receives message m ′ with probability µ ( m ′ | θ, r T ) = 11 − γ ( φ ( θ ) − γ ) , (24)and with the remaining probability he receives a different fixed compound message m ′′ = ( ∅ , m ). Condition ( iii ) implies that there exists E ∗ ∈ B Θ such that π ( E ∗ ) = 1 and Note that the sender only uses two private messages, and that public messages are notinformative. These facts play no role in this proof, but they are used in the proof of otherpropositions which rely on on this communication strategy. ( θ ) ∈ [ γ,
1] for every θ ∈ E ∗ . Hence, µ r ( m ′ | θ, r T ) ∈ [0 ,
1] for every θ ∈ E ∗ .Therefore, µ is a well defined communication strategy (up to a null event, in whichit can be redefined arbitrarily).A receiver r who receives message m ′ does not know whether he belongs to T or not. He thus updates based on the probability µ r ( m ′ | θ ) = (1 − γ ) µ r ( m ′ | θ, r T ) + γ = φ ( θ ) (25)Now, take any event E ⊆ E ∗ such that π ( E ) > µ r ( m ′ | E ) = Z E µ r ( m ′ | θ ) dπ ( θ | E ) = Z E φ ( θ ) · π ( E ) dπ ( θ )= 1 c π ( E ) Z E dπdπ ( θ ) dπ ( θ ) = 1 c π ( E ) · π ( E ) , (26)where the second equality follows because π ≪ π , and thus π ( E ) >
0; the thirdone follows from (23), and the last one follows from the Radon-Nikodim theorem.Therefore, using Bayes’ rule yields β r ( E | m ′ ) = π ( E ) µ r ( m ′ | E ) µ r ( m ′ ) = 1 c µ r ( m ′ ) · π ( E ) ∝ π ( E ) . (27)Since E was arbitrary and π ( E ∗ ) = 1, it follows that β r ( m ′ ) = π a.s.. Sincethere are always at least nγ receivers with m r = m ′ , we can conclude that π is γ -attained by β and µ . (cid:4) The following lemma is an intermediate step to prove Lemma 2.
Lemma 11
Given any γ ∈ (0 , , a γ -attainable belief π ∈ Π( γ ) is an extremepoint of Π( γ ) if and only if there exists c > such that dπ/dπ ∈ { γc , c } a.s..Proof. ( ⇐ ) Suppose that there exists c > dπ/dπ ∈ { γc , c } a.s..Lemma 1 implies that π is γ -attainable. Suppose towards a contradiction thatthere exist π ′ , π ′′ ∈ Π( γ ) \ { π } and λ ∈ (0 ,
1) such that π = λπ ′ + (1 − λ ) π ′′ .Since π = π ′ , it follows that π ( dπ/dπ ( θ ) = dπ ′ /dπ ( θ )) >
0. Without lossof generality, suppose that π ( dp ′ /dπ ( θ ) > c ) >
0. Lemma 1 then implies that π ( dπ ′ /dπ ( θ ) ≤ γc ) = 0. Since π ∈ co { π ′ , π ′′ } , it follows that dπ ′′ /dπ ( θ ) < γc a.s. on E − := { θ | dπ/dπ ( θ ) = γc } . Lemma 1 then implies that π ( dπ ′′ /dπ ( θ ) ≥ ) = 0. In particular, dπ ′′ /dπ ( θ ) < c a.s. on E + := { θ | dπ/dπ ( θ ) = c } . Hence, dπ ′′ /dπ < dπ/dπ a.s.. However, this implies the following contradiction: π ′′ (Θ) = Z Θ dπ ′′ dπ ( θ ) dπ ( θ ) < Z Θ dπdπ ( θ ) dπ ( θ ) = π (Θ) = 1 H . (28)Hence, we can conclude that π is an extreme point of Π( γ ).( ⇒ ) Fix a belief π ∈ Π( γ ). Let ¯ ν, ν ∈ R + ∪{∞} be the supremum and infimumof the support of dπ/dπ ( θ ), as defined in (18) and (19). The proof proceeds intwo steps. First, I will show that if π ( dπ/dπ ( θ ) ∈ { ν, ¯ ν } ) <
1, then π is not an extreme point of Π( γ ). Second, I will show that if ν = γ ¯ ν , then π is not anextreme point of Π( γ ). Step 1 —Suppose that π ( dπ/dπ ( θ ) ∈ { ν, ¯ ν } ) < . I will show that there exist π ′ , π ′′ ∈ Π( γ ) \ { π } such that π = 0 . π ′ + 0 . π ′′ . Consequently, π is not an extremepoint of Π( γ ) .Since π ( dπ/dπ ( θ ) ∈ [ ν, ¯ ν ]) = 1 and π ( dπ/dπ ( θ ) ∈ { ν, ¯ ν } ) < , there ex-ist numbers ν , ν ∈ ( ν, ¯ ν ) such that ν ≤ ν and π ( E ) > , where E := { θ ∈ Θ | dπ/dπ ( θ ) ∈ [ ν , ν ] } . Moreover, it follows from the definition of ¯ ν and ν that π ( E ) < . Since π ≪ π , it follows that π (Θ \ E ) > . Fix any number ǫ > such that ǫ < min (cid:26) , ν c , ¯ ν − ν ¯ ν + c , ν − νν + c (cid:27) , (29)where c = π (Θ \ E ) π ( E ) ∈ (0 , ∞ ) . (30)Let π ′ and π ′′ be the beliefs given by π ′ ( E ) = Z E dπ ′ dπ ( θ ) dπ ( θ ) and π ′′ ( E ) = Z E dπ ′′ dπ ( θ ) dπ ( θ ) , (31)where dπ ′ /dπ : Θ → R + and dπ ′′ /dπ : Θ → R + are given by dπ ′ dπ ( θ ) = dπdπ ( θ ) + ǫc if θ ∈ E (1 − ǫ ) dπdπ ( θ ) if θ E , (32)25nd dπ ′′ dπ ( θ ) = dπdπ ( θ ) − ǫc if θ ∈ E (1 + ǫ ) dπdπ ( θ ) if θ E . (33)It is straightforward to verify that dπ/dπ = 0 . dπ ′ /dπ + 0 . dπ ′′ /dπ and,consequently, π = 0 . π ′ + 0 . π ′′ . The first two bounds from (29) imply that dπ ′ /dπ ≥ and dπ ′′ /dπ ≥ a.s.. Also, note that π ′ (Θ) = Z E " dπdπ ( θ ) + ǫπ (Θ \ E ) π ( E ) dπ ( θ ) + Z Θ \ E (1 − ǫ ) dπdπ ( θ ) dπ ( θ )= π ( E ) + ǫπ (Θ \ E ) π ( E ) · π ( E ) + (1 − ǫ ) π (Θ \ E ) = π (Θ) = 1 , (34)and, by a similar argument π ′′ (Θ) = 1 . It follows that π ′ and π ′′ are well definedbeliefs and are absolutely continuous with respect to π .From the third bound in (29) it follows that ǫ < ¯ ν − ν ¯ ν + c ⇒ ¯ νǫ + c ǫ < ¯ ν − ν ⇒ dπdπ ( θ ) + ǫc ≤ ν + c ǫ < (1 − ǫ )¯ ν. (35)Therefore, dπ ′ /dπ ∈ [(1 − ǫ ) ν, (1 − ǫ )¯ ν ] a.s.. By a similar argument, the fourthbound in (29) implies that dπ ′′ /dπ ∈ [(1 + ǫ ) ν, (1 + ǫ )¯ ν ] a.s.. Lemma 1 impliesthat ν ≥ γ ¯ ν and thus π ′ and π ′′ are γ -attainable. Step 2 — Now suppose that dπ/dπ ∈ { ν, ¯ ν } a.s., but ν = γν . Again, I willshow that there exist π ′ , π ′′ ∈ Π( γ ) \ { π } such that π = 0 . π ′ + 0 . π ′′ . Since theargument is very similar to the one used in the previous step, I will omit somedetails.From Lemma 1, it follows that ν ≥ γ ¯ ν . Hence, it must be the case that ν > γ ¯ ν > . Since γ ≤ , this implies that ν = ¯ ν . Consider the events E + = { θ | dπ/dπ ( θ ) = ¯ ν } and E − = { θ | dπ/dπ ( θ ) = ν } . We know that π ( E + ) + π ( E − ) = 1 . Moreover, π ( E + ) = 1 or π ( E − ) = 1 would only be possible if26 ν = ν = 1 . Therefore, π ( E + ) π ( E − ) = 0 . Fix any ǫ > such that ǫ < min ( ν, ¯ νc , ¯ ν − ν c , ν − γ ¯ ν γc ) , (36)where c = π ( E − ) /π ( E + ) ∈ (0 , ∞ ) .Let π ′ and π ′′ be defined as in (31), but with dπ ′ /dπ and dπ ′′ /dπ given by dπ ′ dπ ( θ ) = ¯ ν + ǫc if θ ∈ E + ν − ǫ if θ ∈ E − , (37)and dπ ′′ dπ ( θ ) = ¯ ν − ǫc if θ ∈ E + ν + ǫ if θ ∈ E − . (38)Using an analogous argument to the one used in Step 1, it is possible to verify that π = 0 . π ′ + 0 . π ′′ and π ′ and π ′′ are well defined beliefs and π ′ , π ′′ ∈ Π( γ ) . (cid:4) Proof or Lemma 2. ( ⇒ ) Let π be an extreme point of Π( γ ) . From Lemma 11,there exists a constant c such that dπ/dπ ∈ { γc , c } a.s.. Let E + = { θ ∈ Θ | dπ/dπ ( θ ) = c } . We have dπ/dπ ( θ ) = γc a.s. on Θ \ E + . Hence, for everyevent E , π ( E ) = Z E ∩ E + dπdπ ( θ ) dπ ( θ ) + Z E \ E + dπdπ ( θ ) dπ ( θ )= c h π ( E ∩ E + ) + γπ ( E \ E + ) i . (39)In particular, for E = Θ , π (Θ) = c h π (Θ ∩ E + ) + γπ (Θ \ E + ) i . (40)Since π (Θ) = 1 , it follows that c = 1 / [ π (Θ ∩ Θ + ) + γπ (Θ \ Θ + )] .( ⇐ ) Suppose that there exists a measurable set E + ⊆ Θ such that π satisfies(3) for every event E ∈ B Θ . I claim that the Radon-Nikodym derivative of π with27espect to π is given by dπdπ ( θ ) = c h ( θ ∈ E + ) + γ · ( θ E + ) i , (41)where c = 1 / [ π ( E + ) + γπ (Θ \ E + )] . Indeed, note that for every event E , Z E dπdπ ( θ ) dπ ( θ ) = Z E ∩ E + c dπ ( θ ) + Z E \ E + c γ dπ ( θ )= π ( E ∩ E + ) + γπ ( E \ E + ) π ( E + ) + γπ (Θ \ E + ) . (42)Since γc ≤ dπ/dπ ( θ ) ≤ c for all θ ∈ Θ , the result follows from Lemma 1. (cid:4) A.2. Effective communication and persuasion
Proof of Proposition 3.
The threshold is given by ¯ γ = ν/ ¯ ν ∈ (0 , . If γ < ¯ γ ,then dπ ∗ /dπ ∈ [ γ ¯ ν, ¯ ν ] a.s.. Lemma 1 thus implies that π ∗ is γ -attainable bysome β and µ . Consider any strategy profile α such that all receivers choosebest responses and, in particular, α r ( m r , ω ) = a ∗ whenever β r ( m r ) = π ∗ . Thetupple ( α , β , µ ) constitutes an equilibrium. Let a be any action profile thatresults with positive probability in this equilibrium. By construction, at least γ of the receivers satisfy a r = a ∗ . Hence, it maximizes the sender’s utility. (cid:4) Proof of Proposition 4.
Let P ∗ = BR − ( a ∗ ) be the set of beliefs in ∆Θ for which a ∗ is a best response. Also, for each λ ∈ [0 , let π λ = λπ ′ + (1 − λ ) π ∗ . Let Λ ∗ = { λ ∈ [0 , | π λ ∈ P ∗ } . And let ¯ λ = sup Λ ∗ and ¯ π = π ¯ λ . ¯ λ ∗ is well defined an belongs to [0 , because ∈ Λ ∗ , and thus Λ ∗ = ∅ .Take any sequence ( λ k ) in Λ ∗ such that λ k −→ ¯ λ . For any event E , π λ k ( E ) =(1 − λ k ) π ′ ( E ) + λ k π ∗ ( E ) −→ (1 − ¯ λ ) π ′ ( E ) + ¯ λπ ∗ ( E ) = ¯ π . Hence π λ k convergeswith respect to the weak topology to ¯ π and, consequently, with also with respectto the weak* topology. Since u R is continuous, P ∗ is closed. Since π λ k ( E ) ∈ P ∗ for all k , ¯ π ∈ Λ ∗ . Moreover, since P ∗ is convex, so is Λ ∗ . Therefore, we can write Λ ∗ = [0 , ¯ π ] with ¯ π ∈ [0 , .Assumption 4 implies that π ′ ∼ π ∗ and, consequently, π λ ∼ π λ ′ for all λ, λ ′ ∈ , . Therefore, for every λ ∈ (¯ π, we have that d ¯ πdπ λ = d ¯ πdπ ∗ · dπ ∗ dπ λ = d ¯ π/dπ ∗ dπ λ /dπ ∗ = ¯ λdπ ′ /dπ ∗ + (1 − ¯ λ ) λdπ ′ /dπ ∗ + (1 − λ ) , (43)a.s., where the first and second equalities follow from the chain rule, and the thirdone from the linearity of Radon-Nikodym derivatives. Define f λ : R + → R by f λ ( x ) = ¯ λx + 1 − ¯ λλx + 1 − λ . (44)Notice that df λ dx ( x ) = ¯ λ − λ ( λx + 1 − λ ) < . (45)Hence f is decreasing, and using the fact that dπ ′ /dπ ∗ ∈ [ ν ′ , ¯ ν ′ ] a.s., it followsfrom (43) that d ¯ π/dπ λ ∈ [ f λ (¯ ν ′ ) , f λ ( ν ′ )] a.s..Since there is an excess audience, there exists ǫ > such that ǫ < (1 − γ ) / (1 + γ ) . Note that lim λ → ¯ λ + f λ ( ν ′ ) = lim λ → ¯ λ + f λ (¯ ν ′ ) = 1 . (46)Hence, there exists a number δ > such that if λ ∈ (¯ λ, ¯ λ + δ ) , then f λ ( ν ′ ) , f λ (¯ ν ′ ) ∈ (1 − ǫ, ǫ ) . This implies that f λ (¯ ν ′ ) /f λ ( ν ′ ) ≥ (1 − ǫ ) / (1 + ǫ ) > γ , and thus, d ¯ π/dπ λ ∈ [ γf λ ( ν ′ ) , f λ ( ν ′ )] a.s.. Lemma 1 thus implies that if π = π λ , then ¯ π is γ -attainable.Fix some λ ∈ (¯ λ, ¯ λ + δ ) , and suppose that π = π λ . Since π λ P ∗ , Assumption4 implies that v ( π ) < u S ( a ∗ , . . . , a ∗ ) . The sender payoff equals u S ( a ∗ , . . . , a ∗ ) inthe equilibrium constructed in the proof of Proposition 3. Hence, there is effectiveinformation transmission in this equilibrium. Therefore, π λ ∈ P = ∅ .It remains to show that if k Θ k < ∞ , then P has a nonempty interior. Forthat purpose, let Π( γ , π ) denote the set of beliefs that would be γ -attainable if pi = π . From Lemma 1, it follows that π ∈ Π( γ , ¯ π ) ⇔ π ≪ ¯ π and dπd ¯ π ∈ [ γ c , c ]a . s . ! ⇔ ¯ π ≪ π and d ¯ πdπ ∈ (cid:20) γ c , c (cid:21) a . s . ! ⇔ ¯ π ∈ Π( γ , π ) . (47)29hen γ < and k Θ k < + ∞ , ¯ π belongs to the interior of Π( γ , ¯ π ) . Therefore,there exists an open neighborhood P ′ ⊆ Π( γ , ¯ π ) of ¯ π such that ¯ π ∈ Π( γ , π ) forall π ∈ P ′ . Since ¯ π belongs to the boundary of P ∗ , it follows that P ′′ = P ′ \ P ∗ = ∅ .Since P ∗ is closed, P ′′ is open. And, from the same argument we used for π λ , itfollows that for every π ∈ P ′′ there exists an equilibrium with effective informationtransmission. Hence, P ′′ ⊆ P . (cid:4) A.3. Sender-optimal equilibriumLemma 12
Every equilibrium value can be attained by a symmetric equilibrium ( α , β , µ ) with α r = α r ′ , β r = β r ′ , and µ r = µ r ′ for all receivers r and r ′ .Proof. Suppose that an equilibrium value u ∗ S is generated by some equilibrium ( α , β , µ ) . Consider the alternative strategies ˜ µ and ˜ α obtained by shuffling identi-ties as follows. First, the sender draws a message profile m ∈ M × M n using theoriginal distribution µ , but does not deliver them. Then, she shuffles the identity ofthe receivers by drawing a permutation uniformly from { I : R → R | I is biyective } .She tells each receiver which function α I ( r ) ( m I ( r ) , · ) they would have used in theoriginal equilibrium with their swapped identity and message.Note that α I ( r ) ( m I ( r ) , · ) would be a best response for r had he been told theshuffling I and the message m I ( r ) . The sure thing principle then implies that it isalso a best response when this information is garbled. The new strategy profilethus induces an equilibrium that yields u ∗ S , and is symmetric. (cid:4) Lemma 13 ( Lipnowski and Ravid (2018) ) Under assumption 5, v ∗ ≥ env q v ( π ) .Proof. Let u ∗ S = env q v ( π ) . Consider the alternative environment with ˜ n = 1 and ˜ u S ( a ) = u s ( a, . . . , a ) , and Θ , A , π , and u R unchanged. Assumption 5 impliesthat the value function of the alternative environment coincides with the valuefunction of the original environment. Since there is only one receiver, and thesender’s utility does not depend on the state, this alternative environment satisfiesthe assumptions in Lipnowski and Ravid (2018). Hence, by their Theorem 2, thereexists an equilibrium ( ˜ α , ˜ β , ˜ µ ) which achieves u ∗ S .Consider the replica of this equilibrium given by α r ( m , m r , ω ) = ˜ α ( m , ω ) , β r = ˜ β , and µ ( m, ∅ , . . . , ∅| θ ) = ˜ µ ( m | θ ) , where ∅ denotes a fixed non-informative30rivate message. Note that this replica uses correlated strategies which guaranteethat all agents receive the same message and take the same action with probability . It is straightforward to verify that ( α , β , µ ) is an equilibrium of the originalenvironment and achieves u ∗ S . (cid:4) Proof of Theorem 5. Step 1 ( att env q v is well defined)—Since Θ and A are com-pact and u R is continuous, v is well defined and upper-semicontinuous. ByLemma 5 in Lipnowski and Ravid (2018), env q v is also well defined and upper-semicontinuous. From Lemma 1, Π is closed. Prokhorov’s theorem thus impliesthat Π is compact. Hence, att env q v is well defined by Weierstrass’ extreme-valuetheorem. Step 2 ( v ∗ ≥ att env q v ( π ) )—Let u ∗ S = att env q v ( π ) . There exists some ˆ π ∈ Π such that u ∗ S = env q v (ˆ π ) . Consider the alternative environment with ˜ n = 1 , ˜ u S ( a ) = U S ( a ) , ˜ π = ˆ π , and Θ , A , and u R unchanged. Assumption5 implies that the value function of the alternative environment coincides with v . Since there is only one receiver, and the sender’s utility does not depend onthe state, this alternative environment satisfies the assumptions in Lipnowski andRavid (2018). Hence by their Theorem 2, there exists an equilibrium ( ˜ α , ˜ β , ˜ µ ) which achieves env q v (ˆ π ) = u ∗ S . Moreover, we must have Z U S ( ˜ α ( m )) dω = u ∗ S , (48)for every (compound) message m such that ˜ µ ( m ) > . Let ( ˆ α , ˆ β , ˆ µ ) be thereplica of ( ˜ α , ˜ β , ˜ µ ) constructed as in the proof of Lemma 12. It would be anequilibrium of the original environment if π = ˆ π .Let ¯ x and ¯ β be the communication strategy and updating rule that γ -attain ˆ π from the proof of Lemma 1. Consider the tuple ( α , β , µ ) described as follows. Thesender first draws (but does not deliver) messages ˆ m using ˆ x , and ¯ m ∈ { m ′ , m ′′ } n using ¯ x . She only sends non-informative public messages m = ∅ . If ¯ m r = m ′ , then r receives the private message m pr = ( m ′ , ˆ m ) . Otherwise, he receives the privatemessage m pr = ( m ′′ ) . β is derived from µ using Bayes’ rule. Actions are given by α r ( ∅ , ( m ′ , ˆ m ) , ω ) = ˜ α ( ˆ m , ω ) , and α r ( ∅ , m ′′ , ω ) = a ′′ with a ′′ = min BR( β r ( m ′′ )) .By construction, we have that β ( ∅ , ( m ′ , ˆ m )) = ˜ β ( ˆ m ) . Therefore, the strat-egy α r ( ∅ , ( m ′ , ˆ m ) , ω ) is a best response for the senders. Since ( ˆ α , ˆ β , ˆ µ ) would bean equilibrium if π = ˆ π , the sender cannot benefit from manipulating ˆ m . Hence,if u ∗ S ≥ U S ( a ′′ ) , then ( α , β , µ ) is an equilibrium. Since there are always n players31ho receive message m ′ and their actions lead to u ∗ S (because of (48)), this wouldimply v ∗ ≥ att env q v ( π ) .Otherwise, u ∗ S < ˜ u S ( a ′′ ) . In this case, note that π is a convex combinationof ˆ π and β r ( ∅ , m ′′ ) . And, in turn ˆ p ∈ co { β ( ∅ , ( m ′ , ˆ m )) | ˜ µ ( ˆ m ) > } . Note that v ( β r ( ∅ , m ′′ ) ≥ U S ( a ′′ ) > u ∗ S , and v ( β r ( ∅ , ( m ′ , ˆ m )) ≥ u ∗ S (because of (48). Hence, env q v ( π ) ≥ u ∗ S = att env q v ( π ) , and the desired inequality v ∗ ≥ att env q v ( π ) follows from Lemma 13. Step 3 ( v ∗ ≤ att env q v ( π ) )—Let u ∗ S be an arbitrary equilibrium value. FromLemma 12, there exists a symmetric equilibrium ( α , β , µ ) that generates u ∗ S . Let m be a message profile such that µ ( m ) > . Under assumption 5, there must exista set R ( m ) with k R ( m ) k ≥ n and such that sup { U S ( α r ( m r , ω )) | ω ∈ [0 , } ≥ u ∗ S for every r ∈ R ( m ) . Let P be the set corresponding set of posterior beliefs, P = n β r ( m r ) (cid:12)(cid:12)(cid:12) µ ( m ) > and r ∈ R ( m ) o . (49)It follows that u ∗ S ≤ inf n v ( π ) (cid:12)(cid:12)(cid:12) π ∈ P o . (50)Consider the alternative communication strategy µ ′ with only two (compound)messages m ′ and m ′′ described as follows. The sender first draws a profile m according to µ (but does not deliver it). Receiver r receives message m ′ if andonly if r ∈ R ( m ) . Since ( α , β , µ ) is symmetric, ¯ π := β r ( m ′ ) does not depend on r .The martingale property of Bayes’ rule implies that ¯ π ∈ co( P ) . Since there arealways at least n receivers in R ( m ) , it follows that ¯ π is γ -attainable. Therefore u ∗ S ≤ inf π ∈ P v ( π ) ≤ inf π ∈ P env q v ( π ) ≤ inf π ∈ co( P ) env q v ( π ) ≤ env q v (¯ p ) ≤ att env q v ( π ) . (51)The first inequality is just (50). The second inequality follows because env q v majorizes v . The third one because env q v is quasiconcave. The fourth onebecause infimums are lower bounds. The last one from the fact that ¯ p is γ -attainable. Since, u ∗ S was an arbitrary equilibrium payoff, it follows that v ∗ ≤ att env q v ( π ) . (cid:4) Proof of Proposition 6. If v is quasi-concave then env q v = v . If v ∗ > env q v then Π = { π } . Lemma 1 thus implies γ < . (cid:4) roof of Proposition 7. Since Π is ⊆ -decreasing in γ , att env q v is weakly de-creasing. Hence ( i ) follows from Theorem 5. ( iii ) is a corollary of Proposition 3,and ( ii ) is a corollary of Proposition 4 and Assumption 5. (cid:4) A.4. Applications and extensions
Proof of Proposition 8.
We are interested in equilibria in which the event is backedfor sure. If the project gets backed, the expected utility for a receiver with beliefs π can be written as Z Θ u R ( a r , θ ) dπ ( θ ) = θ π log( w + ρ a r ) + (1 − θ π ) log( w − a r ) , (52)where θ π := R Θ θ dπ ( θ ) . The first order condition for an interior maximum is thus θ π ρw + ρa r − (1 − θ π ) 1 w − a r = 0 . (53)This condition yields the best response function BR( π ) = min ( , wρ ( θ π (1 + ρ ) − ) . (54)If ρ > , then R Θ θ dπ ( θ ) = 1 / > / (1 + ρ ) . This means that receivers arewilling to make pledge a positive amount BR( π ) = w ( ρ − / ρ > without anyinformation transmission. In this cases, it suffices to have n > ρη/w ( ρ − tohave a babbling equilibriun in which the project gets backed. The interesting caseis when ρ < , so that BR( π ) = 0 . In this case, receivers need to be persuadedto make a positive pledge.Let n ′ = min { k ∈ N k ≥ nρ / } and γ = n ′ /n ≥ ρ / . Using ρ < , it canbe shown that n ′ ≤ n − as long as n ≥ . From the analysis of Example 5 inSection 2, it follows that there is a γ -attainable belief π γ such that Z Θ θ dπ γ ( θ ) = 11 + √ γ . (55)Using the strategy that γ -attains this belief, the sender can guarantee a total33nvestment greater than n ′ BR( π γ ) = n ′ wρ ρ √ γ − ! ≥ nρ wρ ρ ρ/ − ! = n · wρ · ρ ρ > n · wρ ≥ ηwρ · wρ
12 = η, (56)where the first equality follows from (54) and (55), the first inequality from thedefinition of n ′ and γ , the second equality from simple algebra, the second inequal-ity from ρ < , and the last inequality from n ≥ η/wρ . Since the project getsbacked for sure, the sender has no incentive to deviate and the proposed strategyprofile constitutes a perfect Bayesian equilibrium. (cid:4) Proof of Theorem 9. ( ≥ ) Let u ∗ S = att env v ( π ) . There exists some ˆ p ∈ Π suchthat u ∗ S = env v (ˆ p ) . By Caratheodory’s theorem there exist beliefs p , . . . , p K and weights µ ∈ ∆ K with K ≤ n Θ + 1 such that ˆ p = P Kk =1 µ k p k , and u ∗ S = P Kk =1 µ k v ( p k ) . Under Assumption 5, there exist actions a , . . . , a K such that v ( p k ) = ˜ u S ( a k ) and a k ∈ BR( p k ) . The result then follows from Proposition 1 inKamenica and Gentzkow (2011).( ≤ ) Let u ∗ S be the expected payoff to the sender from a commitment pro-tocol ( α , β , µ ) . Note that the arguments from the proof of Lemma 12 can beapplied to commitment protocols. Hence, we can assume without loss of gen-erality that ( α , β , µ ) is symmetric. Also, since there is no incentive compati-bility constraint for the sender, we can assume without loss of generality that U S ( α r ( m r , ω )) = v ( β r ( m r )) for every receiver r , ω ∈ [0 , and every message m r such that µ r ( m r ) > .For every message profile m such that µ ( m ) > , let R ( m ) be a set consistingof exactly n receivers such that v ( β r ( m r )) ≥ v ( β r ′ ( m r ′ )) for every r ∈ R ( m ) and r R ( m ) . Note that u ∗ S = Z Θ Z M × M n X r ∈ R ( m ) v ( β r ( m r )) dx ( m | θ ) dπ ( θ ) ≤ Z Θ Z M × M env v (¯ π ( m )) dx ( m | θ ) dπ ( θ ) ≤ env v (¯ π ) , (57)where ¯ π ( m ) := P r ∈ R ( m ) β r ( m r ) /n , ¯ π := R Θ R M × M ¯ π ( m ) dx ( m | θ ) dπ ( θ ) . The firstinequality follows from the definition of env , and the second one from Jensen’s34nequality and the concavity of env v . Using a communication strategy analogousto the communication strategy µ ′ from the proof of Theorem 5, it can be shownthat ¯ π is γ -attainable. Hence, (57) implies that u ∗ S ≤ Π env v ( π ) . Since u ∗ S wasarbitrary, it follows that v ∗∗ ≤ Π env v ( π ) . (cid:4) Proof of Proposition 10.