Persuasion and receiver’s news

Economics Letters 141 (2016) 60–63

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Persuasion and receiver’s news Alessandro Ispano CREST, 15 Boulevard Gabriel Péri, 92245 Malakoff, France

highlights • In a persuasion game the receiver is endowed with independent access to information. • The receiver may be better off when she does not observe the content of the news. • The value of ignorance deteriorates when communication is two-sided.

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Article history: Received 1 October 2015 Received in revised form 18 January 2016 Accepted 29 January 2016 Available online 6 February 2016

abstract In a persuasion game with possibly missing evidence a receiver with access to news may make better decisions when she does not observe its content. The value of ignorance deteriorates when communication is two-sided. © 2016 Elsevier B.V. All rights reserved.

JEL classification: C72 D82 D83 Keywords: Disclosure Informed receiver Rational ignorance Two-sided communication

The model I study in this paper departs from the typical persuasion game with possibly missing evidence (Dye, 1985; Jung and Kwon, 1988; Shin, 1994) in that the receiver is endowed with some independent access to information. I compare the receiver’s equilibrium payoff when she may only know that evidence exists (indirect news) and when she may also discover its content (direct news). Due to strategic information retention considerations by the sender, the receiver is sometimes better off in the case of indirect news. Her equilibrium payoff always improves compared to when she has no access to information and it is maximal in one of these two polar cases even when her signal about the state can be more flexible than perfectly revealing or completely uninformative. The value of indirect news may however deteriorate when the receiver also has an opportunity to disclose. These results on the countervailing effect of direct news in a persuasion game echo findings from the recent literature on cheap-talk communication with an informed receiver (de Barreda,

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.econlet.2016.01.026 0165-1765/© 2016 Elsevier B.V. All rights reserved.

2010; Chen, 2012; Lai, 2014; Ishida and Shimizu, forthcoming). Besides, this effect is akin to that of repeated communication when information is soft but lies detectable (Dziuda, 2012). In hard information settings, instead, it is the quantity rather than the quality of news that has been shown to generate similar effects when biased experts compete to influence (Bhattacharya and Mukherjee, 2013).1 Also, exact grading may cause comparable effects in the case of voluntary costly certification (Harbaugh and Rasmusen, 2013). Finally, in the literature on principal–agent relations the use of ignorance as incentive device is a recurrent theme that dates back since Cremer (1995). 1. The model The game has two players: a sender (S) and a receiver (R). S aims at maximizing the action (a) that R takes, while R wants her action

1 Bhattacharya and Mukherjee (2013) construct examples in which the receiver’s payoff decreases with the experts’ likelihood of obtaining information. If we consider the receiver as an expert, in this paper the configuration of preferences of the two players is such that this effect never arises.

A. Ispano / Economics Letters 141 (2016) 60–63

Fig. 1. Disclosure(

) and nondisclosure(

) regions.

to match the true state (ω). Nature draws ω ∈ Ω = {0, v, 1} from a common prior distribution that assigns equal probability to each outcome, where v ∈ (0, 1). With probability p ∈ (0, 1) there exists hard evidence certifying the state. In this case, S automatically observes it, while R obtains information only with probability q ∈ (0, 1). I will consider two alternative scenarios about R’s information: direct news reveals to R the content of the evidence; indirect news reveals to R only that evidence exists. When evidence does not exist, an event that has probability 1 − p, both S and R receive no information. S’s payoff is a, R’s payoff is −(a − ω)2 and the timing of the game is the following: 0. nature draws ω and the information status of S and R; 1. S sends a message m to R, who chooses her action a; 2. ω becomes public and payoffs realize. Because evidence is hard, when S is uninformed she must remain silent (m = ∅). When she is informed, instead, she can either remain silent or disclose (m = ω). The relevant solution concept is sequential equilibrium (Kreps and Wilson, 1982). For ease of exposition, in the main body I neglect mixed strategy equilibria that exist for non-generic parameters by adopting the convention that in such cases S elects to disclose. The appendix contains all proofs. 2. Results Because type 0 will always conceal the evidence and type 1 will always reveal it,2 we can restrict our attention to the behavior of the middle type. Proposition 1 (Equilibrium Disclosure). There exist three cutoffs such that 0 < v i < v¯ i < vˆ d < 1/2 and: in the case of direct news the middle type discloses if and only if v ≥ vˆ d ; in the case of indirect news the middle type discloses when v ≥ v¯ i , does not disclose when v < v i and discloses, does not disclose depending on  or randomizes  the prevailing equilibrium when v ∈ v i , v¯ i . Fig. 1 represents the equilibrium strategy of the middle type in the case of direct and indirect news, where the dashed line indicates that disclosure is not the unique outcome.3 When v is relatively low the middle type has a strong incentive to attempt to pass for an uninformed high type and she conceals the evidence regardless of the nature of R’s news. Similarly, when v is relatively high the middle type has a strong incentive to separate from the low type and she always discloses. For intermediate values of v , instead, the nature of R’s news determines the extent of voluntary disclosure. In the case of indirect news R’s threat of choosing the lowest action upon discovering that S is informed induces the middle type to disclose, which in turn makes such a threat credible. In the case of direct news, instead, such a threat is not credible, so that the endogenous punishment for withholding information is lower and the middle type elects to remain silent.

2 Indeed, in any equilibrium 0 < E [a∗ (∅)] < 1, where E [a∗ (∅)] denotes the expected action R takes when m = ∅. 3 In that region, R’s beliefs are self-fulfilling in that both disclosing and withholding information can be rational for the middle type depending on the behavior R expects.

61

When indirect news induces the middle type to disclose, any additional information about ω is redundant. R’s payoff is therefore higher than in the case of direct news whenever the latter discourages disclosure. When even under indirect news the middle type remains silent, instead, direct news is more valuable to R because it allows separating the low and the intermediate state. Corollary 1 (Value of Direct and Indirect News). The receiver’s equilibrium payoff is higher in the case of indirect news when v ∈   v¯ i , vˆ d and in the case of direct news when v < v i . When selecting her equilibrium, indirect news is more valuable in the whole  favorite  v i , vˆ d region. I will derive the following comparative statics by focusing on R’s favorite equilibrium.4 The region in which indirect news is more valuable decreases with the likelihood that evidence exists and increases with R’s likelihood of obtaining it. In short, this occurs because if p is low or q is high, whenever R is uninformed S’s ignorance is plausible. Nondisclosure is then relatively attractive and the disincentive effect of direct news on voluntary revelation is strong.5 Corollary 2 (Comparative Statics). The region in which indirect news is more valuable decreases with p and increases with q. The region in which direct news is more valuable decreases with p and q. The two regions vanish as p converges to 1 or as q converges to 0. No matter the nature of R’s news, her equilibrium payoff increases with p and q and it is hence higher than when R obtains no information unless S discloses (q = 0). Corollary 3 (Value of Information). Both in the case of direct and indirect news the receiver’s equilibrium payoff is increasing in p and q and it is therefore higher than when she has no access to information. Arbitrary precision. Suppose R’s news consists of a realization σω ∈ {0, v, 1} of a signal σ such that, when the state is ω, σω = ω has probability s ∈ ( 31 , 1) and the two other realizations have s each. Direct and indirect news correspond to s = 1 probability 1− 2 1 and s = 3 respectively and R’s equilibrium payoff is necessarily maximal in one of these two cases.

Proposition 2 (Optimal Precision). When the precision of the receiver’s signal can be arbitrary, her equilibrium payoff cannot be higher than in the case of either direct or indirect news. Two-sided communication. Suppose now that in between time 0 and 1 R can send a verifiable message to S about her information status.6 In the case of direct news, the disclosure decisions of S and R are essentially independent and this additional communication stage has no impact. In the case of indirect news, instead, by interpreting R’s silence as ignorance S may induce R to reveal her information status and tailor her own disclosure decision accordingly. As a result, the equilibrium set enlarges and the region in which indirect news is more valuable no matter the selection vanishes.

4 As its proof shows, Corollary 2 equally applies to the region in which indirect news is more valuable no matter the selection. Corollary 3 also holds no matter the selection except that in the mixed strategy equilibrium R’s payoff is constant in q. 5 As p increases R takes a lower action in the absence of any information. While remaining silent becomes less attractive both in the case of direct and indirect news, the downward updating is stronger in the former. As q increases, instead, so does R’s action in the absence of any information. Due to this change, in the case of direct news the marginal type now elects to conceal information. On the contrary, in the case of indirect news she elects to disclose because this payoff increase is more than offset by a decrease in its likelihood. 6 There is no loss of generality in assuming that R’s message does not contain information about the state.

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A. Ispano / Economics Letters 141 (2016) 60–63

Proposition 3 (Two-Sided Communication). When communication is two-sided, there no longer exists a region in which indirect news is always more valuable than direct news.

Similarly, an equilibrium in which m∗i (v) = ∅ exists if and only if v ≤ (1 − q)h(v, 0) + q g (v, 0), that is, when



v ≤ v¯ i (p, q) ≡ Acknowledgments The author is grateful to Antonio Cabrales, Christian Hellwig, Doh-Shin Jeon, Perrin Lefebvre, In-Uck Park, Peter Schwardmann and an anonymous referee for useful comments. The author thanks the Laboratory of Excellence in Economics and Decision Sciences (ECODEC) for financial support.

2 1 − q − (1 − q) p



4 − q − (2 + q) p

∈ (0, 1/2).

Because g (v, x) < v , both v i and v¯ i are lower than vˆ d , where vˆ d is defined in Eq. (4). Also, because g (v, x) is always decreasing in x and h(v, x) is decreasing in x whenever v < vˆ d , v i < v¯ i . Finally, in an equilibrium in which type v discloses with probability β ∈ (0, 1) it must be that v = (1 − q)h(v, β) + q g (v, β), that is, 2 (1 − p − q + pq) − v 4 − q − p (2 + q)



β = ∗

Appendix

Let inf j and inf j represent the events that player j obtains and does not obtain information, respectively. Also, let m(v) represent the message of type v and a(m, k) represent the action R takes when her information status is k and S’s message is m. The subscripts d and i will indicate whether R’s news is direct or indirect, respectively. Direct news. Naturally, in any equilibrium a∗d (ω, k) = a∗d (∅, inf R ) = ω. If R believes that type v discloses with probability x, her optimal action when she is uninformed and m = ∅ is ad (∅, inf R ) = h(v, x), where

  

h(v, x) ≡ P inf S m = ∅, inf R , x E ωinf S





   + P inf S , ω = v m = ∅, inf R , x v =

(1 − v (2 − p) − p) (1 − q)

Proof of Corollary 1 Let U represent a terminal payoff of R and EU her expected utility. Consider first the region in which m∗ (v) = ∅ both in the case of direct and indirect news. In equilibrium histories in which S discloses, or in which S does not disclose and R is uninformed, Ud∗ = Ui∗ because R’s strategies a∗d and a∗i prescribe identical actions.7 In equilibrium histories in which S does not disclose and R is informed, instead, Ui∗ < Ud∗ = 0. Thus, EU ∗d > EU ∗i . Consider now a region in which m∗d (v) = ∅ but m∗i (v) = v and denote   by aˆ i the strategy that is equal to a∗i except that aˆ i ∅, inf R =





(1)

EU i aˆ i , m∗i − EU ∗d = p(1 − q)





Fix x. For md (v) = v to be sequentially rational it must be that

v ≥ E [ad (∅)] = (1 − q)h(v, x) + q v,

(2)

or, equivalently, that

v ≥ h(v, x).

(3)

Solving v = h(v, x) with respect to v yields 1−p 2 − p − qp

∈ (0, 1/2).

(4)

When v > vˆ d , v > h(v, x) and hence in the unique equilibrium m∗d (v) = v . Conversely, when v < vˆ d , v < h(v, x) and in the unique equilibrium m∗d (v) = ∅. A mixed equilibrium in which S discloses with arbitrary probability exists only for the non-generic parameter combination v = vˆ d . Indirect news. Naturally, in any equilibrium ai (ω, k) = ω. If R believes that type v discloses with probability x, her optimal strategy prescribes ai (∅, inf R ) = h(v, x) and ai (∅, inf R ) = g (v, x), where h(v, x) is defined in Eq. (1) and



1 3



g (v, x) = P ω = v m = ∅, inf R , x v =

1 3

(1 − x)

+ 13 (1 − x)

v ≥ v i (p, q) ≡

2 + q(1 − 2p) − p

∈ (0, 1/2).

  2  > 0. v − a∗d ∅, inf R

(8)

Proof of Corollary 2 An inspection of vˆ d (p, q), v i (p, q) and v¯ i (p, q), which are defined respectively in Eqs. (4), (6) and (7), reveals that ∂ vˆ d ∂p ∂ vˆ d ∂q

• •

<

∂ v¯ i ∂p

> 0,

< ∂v i ∂q

∂v i ∂p

< 0;

< 0 and

∂ v¯ i ∂q

< 0;

• limp→1 vˆ d = limp→1 v i = limp→1 v¯ i = 0; • limq→0 vˆ d = limq→0 v i = limq→0 v¯ i =

1 −p . 2 −p

7 Indeed, a∗ (ω, k) = ω and a∗ ∅, inf = h(v, 0), where h(·) is defined in R Eq. (1). 8 If we denote by a∗ the strategy of R in the mixed equilibrium, by a∗ the one β 1



(5)

An equilibrium in which m∗i (v) = v hence exists if and only if v ≥ (1 − q)h(v, 1) + q g (v, 1), that is, when 1 − q − (1 − q)p

3



v.

Fix x. For mi (v) = v to be sequentially rational it must be that

v ≥ E [ai (∅)] = (1 − q)h(v, x) + q g (v, x).

1

By the definition of conditional expectation R’s equilibrium strategy uniquely S’s strategy, so that EU ∗i =   maximizes  EU ∗given  ∗ maxa EU i a, mi ≥ EU i aˆ i , mi > EU ∗d . A similar argument demonstrates that, taking the nature of news as given, R’s expected utility is always higher in an equilibrium in which m∗ (v) = v than in one in which type v discloses with probability β ∈ [0, 1).8

∗



.

a∗d ∅, inf R . Then,

(1 − p) (1+v) + 1 (1 − x)p(1 − q)v 3  1 31  . (1 − p) + 3 + 3 (1 − x) (1 − q)p

v = vˆ d (p, q) ≡



This equilibrium exists only when v i < v < v¯ i , as only then β ∗ ∈ (0, 1).

Proof of Proposition 1



(7)

(6)



in the pure strategy equilibrium and by aˆ 1 the one that is equal to a∗1 except that









aˆ 1 ∅, inf R = a∗β ∅, inf R , we can write an analogous of Eq. (8) EU aˆ 1 , m∗1 − EU a∗β , m∗β ≥ p(1 − q)(1 − β)









1 3



   2 v − a∗β ∅, inf R > 0.

A. Ispano / Economics Letters 141 (2016) 60–63

Proof of Corollary 3 Let us consider the effect of a marginal increase from q to q′ . If this change does not alter m∗ (v), EU ∗q′ > EU ∗q , because R’s likelihood of detecting a type who conceals evidence increases.9 Similarly, if this change induces type v to disclose, EU ∗q′ > EU ∗q because ∂v

of the same argument used in the proof of Corollary 1.10 As ∂ qi < 0, it follows that in the case of indirect news ∗

dEU ∗ i

dq

> 0. Instead,

∂ vˆ d ∂q

>

0, so that in the case of direct news EU d can in principle be decreasing in q at v = vˆ d if it has a discontinuity. But when v = vˆ d , Eqs. (3) and (4) clarify that a∗d (ˆvd , k) = vˆ d = a∗d (∅, inf R ), so that EU ∗d is continuous.11 Analogous arguments apply to an increase from p to ∂v

∂ vˆ

p′ and, because ∂ pi < 0 and ∂ pd < 0,

dEU ∗ i dp

> 0 and

dEU ∗ d dp

> 0.

Proof of Proposition 2 Consider first an equilibrium in which m∗σ (v) = v , that is, in which σ induces type v to disclose.12 Because R’s posterior must satisfy P (ω = 0|m = ∅, σω ) = 1, the condition for the existence of this equilibrium is the same as in the case of indirect news, i.e. Eq. (5) must hold at x = 1. Thus, whenever v ≥ v i , the additional informational content of σ relative to indirect news is superfluous. When v < v i , instead, it must be that m∗σ (v) = ∅, as if a mixed equilibrium existed, there would also exist one in which m∗σ (v) = v ,13 yielding a contradiction. Direct news is then more valuable than σ because it allows separating type 0 and type v without noise. Proof of Proposition 3 Because R’s news is hard information, whenever she is uninformed she must remain silent (mR = ∅), while when she is informed she can either remain silent or reveal her status (mR = inf R ).

63

Direct news. Upon receiving mR = inf R , type v anticipates that a∗d = v and her payoff does not depend on her disclosure decision. Upon receiving mR = ∅, type v again finds it optimal to disclose if and only if Eq. (3) holds, which is independent from her belief about R’s disclosure strategy.14 As the payoff of an informed R does not depend on her own disclosure decision, the previous analysis applies and the equilibrium outcome is the same as when communication is one-sided.15 Indirect news. Consider a candidate equilibrium in which R discloses whenever informed and type v discloses if and only if R does so. The disclosure decision of an informed R is sequentially rational, as it allows separating state 0 and state v . As for type v , upon receiving mR = inf R , m∗ (v) = v is sequentially rational because otherwise a∗ (∅, inf R ) = 0; upon receiving mR = ∅, m∗ (v) = ∅ is sequentially rational if the reverse of inequality (3) holds at x = 0, namely, if v ≤ vˆ d . Thus, whenever v ≤ vˆ d we identified an equilibrium,16 which is outcome equivalent to the one prevailing in the case of direct news. References Bhattacharya, S., Mukherjee, A., 2013. Strategic information revelation when experts compete to influence. Rand J. Econ. 44 (3), 522–544. Chen, Y., 2012. Value of public information in sender–receiver games. Econom. Lett. 114 (3), 343–345. Cremer, J., 1995. Arm’s length relationships. Quart. J. Econ. 110 (2), 275–295. de Barreda, I.M., 2010. Cheap talk with two-sided private information, Working paper. Dye, R.A., 1985. Disclosure of nonproprietary information. J. Account. Res. 23 (1), 123–145. Dziuda, W., 2012. Communication with detectable deceit, Working paper. Harbaugh, R., Rasmusen, E., 2013. Coarse grades: Informing the Public by withholding information, Working paper. Ishida, J., Shimizu, T., 2015. Cheap talk with an informed receiver. Econ. Theory Bull. forthcoming. Jung, W.-O., Kwon, Y.K., 1988. Disclosure when the market is unsure of information Endowment of managers. J. Account. Res. 26 (1), 146–153. Kreps, D.M., Wilson, R., 1982. Sequential equilibria. Econometrica 50 (4), 863–894. Lai, E.K., 2014. Expert advice for amateurs. J. Econ. Behav. Organ. 103, 1–16. Shin, H.S., 1994. News management and the value of firms. Rand J. Econ. 25 (1), 58–71.

9 Formally, this argument follows from the envelop theorem. Because by the definition of conditional expectation R’s equilibrium strategy a∗ uniquely dEU ∗

∂ EU (a,m∗ ) 

∂ EU (a,m∗ ) 

q q  ∗ and naturally  ∗ > 0. maximizes EU q (a; m∗ ), dq q = ∂q ∂q a=a a=a 10 If we denote by a∗ the equilibrium strategy of R under q, by a∗ the one under ′ q q









q′ and by aˆ q′ the one that is equal to a∗q′ except that aˆ q′ ∅, inf R = a∗q ∅, inf R , we can write an analogous of Eq. (8) EU q′ (ˆaq′ , m∗q′ ) − EU ∗q > EU q (ˆaq′ , m∗q′ ) − EU ∗q ≥ p(1 − q)

1 3

v − a∗q (∅, inf R )

2

> 0.

11 EU ∗ is also differentiable and indeed d

 (1 − p )2 p  = > 0. dq v=ˆvd (2 − p − qp)2

dEU ∗d 

14 If type v believes that an informed R discloses with probability y, she finds it optimal to disclose if and only if

12 Should this equilibrium coexist with others, the proof of Corollary 1 clarifies it is R’s favorite. 13 In this region, type v ’s expected payoff from remaining silent is decreasing in

v≥

R’s belief x about her disclosure probability. Indeed, a∗σ (∅, inf R ) = h(v, x), which is decreasing in x whenever v < vˆ d , and a∗σ (∅, σω ) is decreasing in x for any signal realization σω . Thus, if v = E [aσ (∅)] for some x ∈ (0, 1), v > E [aσ (∅)] when x = 1.

1−q (1 − y)q h(v, x) + v, (1 − q ) + (1 − y )q (1 − q ) + (1 − y )q

which simplifies to Eq. (3). 15 For completeness, an informed R and a S who receives m = inf are indifferent R R and hence they disclose with arbitrary probability. 16 For completeness, when type 0 receives m = inf she is indifferent and hence R

she discloses with arbitrary probability.

R