Persuading Skeptics and Reaffirming Believers

Transcription

1 Persuading Skeptics and Reaffirming Believers May, 31 st, 2014 Becker-Friedman Institute Ricardo Alonso and Odilon Camara Marshall School of Business - USC

2 Introduction Sender wants to influence decisions of a Receiver. Influence through Expertise : Communicate/Disclose/Signal private information. Influence without Expertise : Indirectly affect Receiver s learning Signal jamming. Information control Design experiment/questions/production of information (as in Kamenica and Gentzkow 2011, Rayo and Segal 2010).

3 Introduction Examples: Clinical trials prior to FDA testing, Firm s New Product Testing What tests to run? Sample size? Questionnaires? Electoral debates, Media Slant. Moderator s questions shape learning about candidates. Media decides what aspects of a story to investigate. Under information control: Designer is no expert : does not have privileged information. Therefore, designer determines what can be learned rather than what will be learned. Influence comes from ability/authority to design experiment.

4 Regulation of New Products Example: Regulator s response to innovative activity. Adoption of new standards (e.g., electrical standard AC/DC), Unintended consequences dictates need for regulation (e.g. safety concerns over electricity, cancer risks of cell phones). Novelty makes it harder to agree on what to do. Receiver (regulator) chooses a policy (regulation). " Optimal policy depends on unknown state 0,1/ 2,1. { } ( ) " Policy maker optimal policy given beliefs is E R. " Sender (lobbyist) prefers higher policies: u S (a)=a. " Sender and Receiver hold different prior beliefs over θ.

5 Investigative Report Investigative report ( ) " Sender wants to maximize E R through an investigative report π.. " Report consists of set of signal realizations Z and likelihoods π(z θ), which are commonly understood by players. " After observing z, Policy Maker updates according to Bayes Rule. " With common priors, risk-neutral Sender does not gain from π.

6 Suppose priors are Motivation Persuading Skeptics θ=0 θ=1/2 θ=1 E[θ] p S = 1/3 1/3 1/3 E S [θ]=0.5 p R = E R [θ]=0.35 " Receiver is a skeptic : Absent info, Sender is better off facing a like-minded Receiver. " In particular, p S FOSD p R " With no signal, Sender s expected utility is " With a fully revealing signal, Sender s expected utility is 0.5. " Information control is valuable. However, is full disclosure optimal?

7 Motivation Persuading Skeptics Suppose priors are θ=0 θ=1/2 θ=1 E[θ] p S = 1/3 1/3 1/3 E S [θ]=0.5 p R = E R [θ]=0.35 " Consider signal { } z = z L if " 0, 1 2 z z L, z H # z = z H if { 1} $ % & " Now E S z # " L $ < E z # R " L$ " Sender s expected utility is E S " E R " z % 1 4 < 5 ( ' * & 18) { } ## $ $ = 1 3 * * 5 18 = > max E R " # $,E S " # $

8 Suppose priors are Motivation Reaffirming Believers θ=0 θ=1/2 θ=1 E[θ] p S = 1/3 1/3 1/3 E S [θ]=0.5 p R = E R [θ]=0.65 " Receiver is a believer : Absent info, Sender is worse off facing a likeminded Receiver. " In particular, p R FOSD p S. " With no signal, Sender s expected utility is " With a fully revealing signal, Sender s expected utility is 0.5. " Can Sender benefit from persuading a believer?

9 Motivation Reaffirming Believers Suppose priors are θ=0 θ=1/2 θ=1 E[θ] p S = 1/3 1/3 1/3 E S [θ]=0.5 p R = E R [θ]=0.65 " Consider signal z { z E, z } M z = z E if { 0,1} z = z M if { 1/ 2} " Now E S " z E # $ < E z R " E # $ 0.5 < 0.8 ( ) ( ) Pr S [z E ] > Pr R [z E ] 2 / 3 >1/ 2 " Sender expects receiver to implement policy E S { } E R " s# # " $ $ = 1 3 * * 8 10 = 0.7 > max E R " # $,E S " # $

10 Introduction Research Questions In a world where rational individuals i- Agree to disagree over the state of the world. ii- Sender designs a signal. iii- Agree on how signal realizations are generated. " When does the Sender benefit from designing the signal? " What is Sender s optimal choice of signal?

11 Related Literature Information Control: Brocas and Carrillo (2007), Kamenica and Gentzkow (2011), Rayo and Segal (2010), Ivanov (2010), Kolotilin (2012). Persuasion under heterogeneous priors: Che and Kartik (2009), Van den Steen (2009, 2010a).

12 Results Preview Value of information control: Set of (subjective) profiles of posterior beliefs induced by a signal. Necessary and sufficient conditions for Sender to profit Info Control. Pure Persuasion: If common priors, Sender can exploit convexities in R s actions. With open disagreement, Sender can resort to beneficial realizations whose likelihood Receiver underestimates. For rich state spaces, these realizations exist generically. If receiver s action is the expected state, then Sender generically benefits from designing the signal. Fully revealing signal often not optimal.

13 General Setup Finite state space Θ. Continuous utility functions u i (a,θ), i=s,r. Receiver selects a A. Heterogeneous priors: Sender and Receiver hold priors p R and p S. Priors are totally mixed (i.e. both have full support). Information Control: Sender can costlessly design a signal π,. = { Z, (. ") }, (. ") "( Z). { } " No cost differences between signals. Rich set of signals. Commonly understood signals: Sender and Receiver agree on likelihood functions of π.

14 Timing and Solution Concept Timing, Sender selects signal π. Signal realization z Z is observed by Receiver. Receiver updates beliefs to q R (z), selects a(q R (z)). Payoffs are realized and game ends. Solution concept: Language-Invariant Perfect Bayesian Equilibria. Language Invariance: Receiver s action depends ONLY on posterior beliefs Existence guaranteed: Sender-Preferred Perfect Bayesian Equilibrium is Language Invariant. Sender-Preferred PBE always exists.

15 Optimal Signal and Value of Information Control Sender s utility from signal π is then ( ) z E S E S u S a( q R (z) "# "# ), Sender needs to solve the following program ( ) s V ( p R, p S ) = sup E S E S u S a( q R (s) "# "# ), where the optimization is taking over all possible signals. Information control is valuable iff $ $ %& %& ( ) V ( p R, p S ) > E S u S a( p R $ "# ), %& $ $ %& %&

16 Plan of Talk Value of Information Control Pure Persuasion Conclusions

17 Preliminaries Bayes Plausible Distribution of Posteriors (I) Signal π induces a distribution over posterior beliefs. Language Invariance implies value of π determined by distribution over posteriors. If players disagree on BOTH the state and the signal likelihood functions, then knowledge of signal needed to recover full joint. If they agree on likelihood functions, however, joint distribution can be recovered from knowledge of marginal distribution of just one player.

18 Preliminaries Bayes Plausible Distribution of Posteriors (II) Lemma 1 Let r R be the likelihood ratio of prior beliefs R R pθ rθ = S pθ From Sender s perspective, a distribution τ over profiles of posterior beliefs is induced by a signal, if and only (i) If (q S,q R ) in Supp(τ) then (ii) E τ [q S ]=p S q R = q S r R q S R r ' ' ' Bayesian Updating + Commonly Understood Signal: knowledge of beliefs of one player suffices to infer the beliefs of another. Relation between posteriors independent of specific signal. Heterogenous priors does not lead to more ways to persuade.

19 Proof: Relation between Posteriors Consider any signal π and signal realization z. Then Define signal likelihood ratio Updated likelihood ratio: ratio of prior likelihood ratio to signal likelihood ratio. Adding over states we have q S (z) Pr S S [z] = (z ) = q R (z) Pr R R [z] p p q R (z) q S (z) = r R z = Pr [z] R Pr S [z] = # q S (z) r R = E S (r R z) " z R = Pr R [z] Pr S [z] R " z R

20 Preliminaries Equivalent Program The value to Sender from Information Control is V ( p R, p S ) = sup E S E S " u S (a(q R (z))) s# # " $ $ The restriction to common information then gives Lemma 2 The value of Information Control is V ( p R, p S ) = sup E " # q S u S (a(q R (q S ))) $ % s.t. q R (q S ) is given in Lemma 1 and E τ [q S ]=p S

21 Value of Information Control Suppose θ={θ L, θ H } and consider V S (q S ): V S S ( q ) H ( " " V S (q S S ) = * q u $ S a * $ $ ) # # q S r R % % + q S R r ', '- '- & &, p H S S q H

22 Value of Information Control Consider a signal that induces posteriors q S _ and q S + V S S ( q ) H q S p H S q + S

23 Value of Information Control Attainable payoffs: Intersection of convex hull of graph of V S (q S ) and the line q S =p S. V S S ( q ) H R S V( p, p ) p S

24 Value of Information Control Let ˆf (q) be the concave closure of the function f (q) Concave closure: smallest concave function that majorizes f. Proposition 1. Let V S (q S S r ) = q u S (a(q S ),) q S R r ' R and S V R (q R R p ) = q u S (a(q R,),) p R Then, the maximum expected utility of Sender is V ( p R, p S ) = ˆ V S ( p S ) = ˆ V R ( p R )

25 Value of Persuasion Geometric Conditions Computing concave closure often not immediate. However, simple condition equivalent to no value of Info Control. Corollary, There is no value of information control iff V S (q S ) admits a majorizing supporting hyperplane at q S =p S. In particular, if V S (q S ) is differentiable at p S, then there is no value of information control iff V S ( p S ), q S p S V S (q S ) V S ( p S ), q S "(#).

26 Value of Information Control No value of information control at p S. Clearly tangent hyperplane majorizes V S (.) at p S. V S (q S ) p S Immediately: If V S is globally concave then no value of info control, if V S is locally convex then info control valuable.

27 Value of Garbling Geometric Conditions Suppose that in the absence of Sender, Receiver perfectly learns the state. Sender garbles information by selecting a less informative signal. Corollary: Sender does not benefit from garbling if and only if # q S u S a(1 ), " ( ) V S (q S ), q S "(#). Intuition: Garbling not valuable iff full revelation weakly preferred for any posterior belief that Sender/Receiver may have.

28 Plan of Talk Value of Information Control Pure Persuasion Conclusions

29 Pure Persuasion under Monotone Preferences Pure Persuasion : Sender s utility independent of state u S (a,) = u S (a), with u' S > 0. Under pure persuasion V R (q R ) = u S (a(q R )) q R,r S Value of info control depends on: Sender s risk preferences, Curvature of receiver s action, Direction of belief disagreement r S. If common priors and u S (a(q R )) concave no value of info control.

30 Pure Persuasion under Monotone Preferences Proposition: If u S (a(q R )) is strictly concave then there exists δ>0 such that whenever p R -p S <δ there is no value of info control. For every bounded u S and totally mixed prior p R such that u S (a( p R )) < max a(q R ) q R "(#) there exists a totally mixed p S such that info control is valuable. So, small disagreement may not be enough, but there is always a level of disagreement that makes info control valuable.

31 Linear Actions Suppose that Receiver s action is the expectation of an r.v. x. a(q R ) = x q R This would be the case, for example, if u R (a,) = (a x ) 2 For simplicity, consider x θ =θ; Receiver selects E R [θ]. With this representation, two types of receivers. Receiver is a skeptic if Receiver is a believer if E R [ z] < E S [ z] E R [ z] E S [ z]

32 Belief Disagreement and Preferences Definition: Vectors v and w are negatively collinear w.r.t T iff there exist λ<0 such that their projections onto T satisfy v T = w T. Lemma: Let T={v:<v,1>=0} and r S be the likelihood ratio r S = p S / p R. Then, for any π and signal realization z, E R [ z] E R [] Pr R [z] Pr S [z] if, and only if, θ and r S are negatively collinear w.r.t. T.

33 Belief Disagreement and Preferences Recall that signal likelihood ratio is S z = Pr [z] S Pr R [z] = E [r S z] R Thus r S T is the direction of maximum disagreement Compare beliefs that increase actions to disagreement over those beliefs. E R r S =1 z S >1 r S p R E R = R p E R [ z] E R []

34 Value of Information Control under Pure Persuasion Proposition: Suppose that card(θ)>2, u S (a) is twice continuously differentiable, and p S p R. Then, Sender benefits from info control if θ and r S are not negatively collinear w.r.t T. If u S (a) is concave, Sender benefits from info control iff θ and r S are not negatively collinear w.r.t T. Implication: The set of priors beliefs that are either common or are negatively collinear w.r.t. T are two lines. If card(θ)>2, then the sender generically benefits from information control (in space of profiles of prior beliefs).

35 Sketch of proof (Sufficiency)) Skeptics and Believers q R Pr S q R + Pr R q R + > 0 z S >1 p R E R r S =1 E R = R p q R +

36 Optimality of Full Disclosure Generically, sender provides some info to receiver. Would the sender want to realize ALL information? Proposition: Suppose that u S is convex and p S strictly dominates p R in the likelihood ratio sense. Then full disclosure is optimal. Full disclosure optimal if both i. Sender not hurt by variability, and ii. Receiver remains a (weak) skeptic after every signal realization.

37 Optimality of Full Disclosure If any of those conditions fail, then full disclosure can be suboptimal Proposition: If there are two states θ< θ such that ( S r ) 2 ' u' S ( ') ( S r ) 2 u' S () < 0 then a perfectly informative signal is not optimal.

38 Plan of Talk Value of Information Control Pure Persuasion. Conclusions

39 Ongoing Projects Information Control exciting, underexplored perspective on information supply in organizations Ongoing Projects: Disagreement and Information Control. Common Information and Belief Disagreement Transparency and Political Disagreement Organizational responses to information control Persuading Voters Information Control in Organizations

40 Conclusions Persuasion under open disagreement With commonly understood signals open disagreement doesn t give Sender more ways of persuading Receiver. Tractable conditions for positive value of Information control. Main implications: New rationale for valuable persuasion: Sender exploits beneficial realizations over which she is more confident. If actions are the expected state, info control generically valuable. Typically, it is optimal To not fully inform an skeptic. To partially inform a believer.

41 Thank You