Persuasion Under Costly Lying

Transcription

1 Persuasion Under Costly Lying Teck Yong Tan Columbia University 1 / 43

2 Introduction Consider situations where agent designs learning environment (i.e. what additional information to generate) to persuade another agent : advertising and marketing communications in organizations medical testing and research media control/ lobbying Kamenica and Gentzkow (2011), hereafter KG: Model these situations as a strategic communication (or cheap talk) game with Sender commitment. Sender designs the information generating structure and can commit to truthfully reporting all his information to Receiver. 2 / 43

3 Leading Example of KG A prosecutor (S) wants to convince a judge (R) that a defendant is guilty: State ω {Guilty, I nnocent}, unknown to both. Judge chooses action a {Acquit, Convict}. Gets utility 1 when chosen the right action (acquit when ω = I, convict when ω = G); Gets utility 0 otherwise. = R chooses convict when believes Pr(ω = G) 1. 2 Prosecutor only wants to convict the defendant: Gets utility 0 if Judge chooses acquit. Gets utility U > 0 if Judge chooses convict. Common prior β 0 = 0.3. Judge chooses acquit. 3 / 43

4 Leading Example of KG (continued) Suppose Prosecutor can conduct an investigation and can design the informational content of his investigation costlessly. No bribery (i.e. no transfers or monetary incentives allowed). Full commitment assumption: Outcome of investigation is publicly observed. Formally, let signal space be S = {g, i} and Prosecutor chooses a family of distribution π( ω) on S. 4 / 43

5 Leading Example of KG (continued) No investigation π(s = i ω = I ) = 1 2 π(s = i ω = G) = 1 2 π(s = g ω = I ) = 1 2 π(s = g ω = G) = 1 2 Judge chooses acquit (since prior β = 0.3). 5 / 43

6 Leading Example of KG (continued) Full investigation π(s = i ω = I ) = 1 π(s = i ω = G) = 0 π(s = g ω = I ) = 0 π(s = g ω = G) = 1 Bayes rule: Pr(ω = G s = g) = 1, Pr(ω = I s = i) = 1. Judge chooses convict when observed s = g, acquit when observed s = i. = Judges chooses convict with probability Pr(s = g) = 0.3. Prosecutor's ex-ante expected utility is 0.3U. 6 / 43

7 Leading Example of KG (continued) Prosecutor can do better by being vague sometimes. Optimal investigation: π(s = i ω = I ) = 4 7 π(s = i ω = G) = 0 π(s = g ω = I ) = 3 7 π(s = g ω = G) = 1 Bayes rule: Pr(ω = G s = g) = 1, Pr(ω = I s = i) = 1. 2 Judge still chooses convict when observed s = g, acquit when observed s = i. But now, Pr(s = g) = 0.6. Prosecutor's expected utility is 0.6U. Judge convicts with ex-ante probability 0.6, even though Judge knows defendant is guilty with only 30% chance. This is the optimal signal structure (investigation). 7 / 43

8 The Full Commitment Assumption What if Judge cannot fully observe the investigation and relies on Prosecutor to report the ndings? What happens during the communication stage? If lying is costless: similar to cheap talk (Crawford and Sobel (1982)). Information will only be transmitted when preferences are suciently aligned. If Prosecutor must tell the truth but not necessarily the whole truth: similar to cheap talk under veriable information (Grossman (1981), Milgrom (1981)) There will be unravelling through skepticism. If Prosecutor can lie with a cost: similar to cheap talk under costly lying, or costly signaling (Kartik et al. (2007), Kartik (2009), Spence (1973)) Costly cheap-talk: There is language ination (exaggeration). 8 / 43

9 Example Allowing For Costly Lying Many reasons why Judge might need to rely on Prosecutor to report the ndings, and why Prosecutor can lie at a cost: Costly interpretation Tampering with ndings at a cost Suppose now that Prosecutor privately observes the signal realization s and then sends a message m {g, i} to the Judge. The structure of the investigation π( ω) is still public knowledge. Cost of lying: Incurs no cost for truthful reporting, but incurs a cost k > 0 for lying. k is common knowledge. 9 / 43

10 Large Lying Cost (k > U) k > U implies any possible gain from lying is outweighed by lying cost. Prosecutor will never lie. Lying cost has no bite. 10 / 43

11 Small Lying Cost (k < U) Suppose k < U. Consider the KG optimal investigation structure: π(s = i ω = I ) = 4 7 π(s = i ω = G) = 0 π(s = g ω = I ) = 3 7 π(s = g ω = G) = 1 Bayes rule: Pr(ω = G s = g) = 1, Pr(ω = I s = i) = 1. 2 Pr(s = g) = 0.6. If Judge always chooses convict when Prosecutor reports m = g, Prosecutor will always report m = g regardless of the signal. 11 / 43

12 Small Lying Cost (k < U) continued Can establish 3 three things: 1 Judge cannot disregard message and always choose convict (since prior µ 0 = 0.3). 2 Judge cannot disregard message and always choose acquit (since Prosecutor will want to tell the truth and avoid lying cost). 3 Judge cannot always choose convict after hearing a particular message (from previous argument). 12 / 43

13 Small Lying Cost (k < U) continued Suppose Judge becomes wary of the Prosecutor's message. Chooses convict k (0, 1) of the time when Prosecutor sends m = g. U This is still optimal for Judge since she is indierent at s = g (Pr(ω = G s = g) = 1 ). 2 Prosecutor is now indierent between lying and truth-telling for s = i. Willing to always report truthfully now. Ex-ante expected utility is Pr(s = g) k U = 0.6k < 0.6U. U Can the Prosecutor do better with a dierent investigation and possibly lie sometimes? No! The above investigation is still the optimal one (we will see why later). 13 / 43

14 Outline Consider strategic communication games where: Sender designs information acquisition structure (as in KG). Limited commitment in revealing his information truthfully in the form of costly lying. Main results for this talk: Limited commitment hurts the Sender. For general lying cost structure, Sender will design an information acquisition structure that gives him no incentive to lie. Social welfare is lower because learning must be vague. When Sender has state-independent preference, the optimal information acquisition structure resembles that of KG. 14 / 43

15 Setup Two players: Receiver (she) and Sender (he). True state of world ω Ω. Ω is nite. Both players uncertain about ω; shares common prior β0 (Ω). Stage one: Sender chooses signal structure π from a family of distribution Π = {π( ω)} ω Ω over a nite signal realization (or signal) space S. Choice of π is observed by Receiver. Stage two: Sender privately observes signal realization s S. Reports a message m S to Receiver. Can have richer form of message space. Key is every message carries a meaning of I observed signal s for a s S. Receiver then takes an action a A; A is compact. 15 / 43

16 Setup (continued) Receiver's expected utility is u(a, ω): can depend on both her action and true state. Sender's expected utility is v(a, ω) C(m, s): v(a, ω) is the payo; dependent on Receiver's action and true state C(m, s) is lying cost; Assume C(s, s) = 0 and C(s, s ) > 0 s s. Call it KG environment when C(s, s ) = s s Call it lying environment otherwise. 16 / 43

17 Setup (Strategies) Sender's strategy is σ = (π, µ): Choice of signal structure: π Π. Messaging strategy: µ : Π S (S). Let µ be the truthful reporting strategy (i.e. µ(s) = s s). Receiver's strategy is α : (Ω) (A): Dene a belief update operator ϕπ,µ : S (Ω): π is the chosen signal structure. µ is Receiver's conjecture of Sender's messaging strategy. 17 / 43

18 Posterior distribution A signal structure generates a distribution of posterior beliefs τ ( (Ω)) Supp(τ) = {βs} s S. Each posterior is derived via Bayes' rule: s β s(ω) = :µ(s )=s π(s ω)β 0(ω) ω Ω s :µ(s )=s π(s ω )β 0(ω ) τ(β) = π(s ω)β 0(ω) s:β s =β ω Ω A distribution of posteriors τ is Bayes plausible if: τ(β s ) β s = β 0 β s Supp(τ) Note: Not all Bayes plausible distribution can be generated when µ µ Eg. if µ(s) = s0 s. The only distribution of posterior generated is degenerate at the prior 18 / 43

19 Utility In Terms Of Posteriors Let a (β) := arg max a A E β u(a, ω). Assume that: A has at least two elements. a A, there exists a belief β (Ω) such that a a (β). Let â(β) := arg max a { Eβ v(a, ω) a a (β) }. â(β) is the action that generates the highest expected payo for Sender among Receiver's optimal actions under belief β. 19 / 43

20 Utility In Terms Of Posteriors Let Sender's expected payo when both players hold belief β and Receiver takes â(β) be ˆv(β) := E β v(â(β), ω). This allows us to dene Sender's utility in terms of posteriors generated: In KG, the Sender preferred eq restricts α( ) = â( ). This is not the case here. Let V be the concave closure of ˆv: V (β) := sup{z (β, z) co(ˆv)} where co(ˆv) is the convex hull of graph ˆv. 20 / 43

21 Persuasion In KG Environment In the KG environment (i.e. no possibility of lying), Sender's problem can be reduced to s.t. max τ E τ ˆv(β) Supp(τ) βτ(β) = β 0 21 / 43

22 Persuasion In KG Environment ˆv(β) ˆv(β) β 0 22 / 43

23 Persuasion In KG Environment ˆv(β) V (β) co(ˆv) ˆv(β) β 0 23 / 43

24 Persuasion In KG Environment ˆv(β h ) V (β) ˆv(β l ) co(ˆv) ˆv(β) β l β 0 β h 24 / 43

25 Persuasion In KG Environment Choose the set of posteriors {β s } to induce. Bayes plausibility is assured by: τ s = ω β s(ω) β 0 (ω). Signal structure is backed out by: π(s ω) = βs (ω) τs β 0(ω). 25 / 43

26 Persuasion In Lying Environment Suppose Receiver does not observe the signal and Sender reports the signal under costly lying now... Sender's expected utility is weakly lower in the lying environment than in the KG environment Proof: The distribution of posterior in the lying environment must be Bayes plausible and hence, can be generated in the KG environment as well. 26 / 43

27 Communication Subgame Restrict attention to principal-optimal equilibrium (PBE) An equilibrium here refers to equilibrium of the whole game, not just the communication subgame. Note that in an equilibrium, Receiver's conjecture of Sender's message strategy is correct. 27 / 43

28 Communication Subgame Lemma At equilibrium, there cannot be a full-separation with lying involved in the communication subgame. Full-separation with lying means Receiver knows exactly what Sender observed Proof: If not, Sender can just relabel the signals and save on lying costs. 28 / 43

29 Communication Subgame Lemma At equilibrium, there cannot be any pooling of messages in the communication subgame. Implies Sender will not play mixed reporting strategy as well. Proof: Suppose µ pools s 1 and s 2 and always report s 1. Hence, Sender incurs lying cost at realization s 2. Receiver's posterior will be β 1 Sender can choose a dierent π such that π(s 1 ω) = π(s 2 ω) ω, and β 1 = β 2 = β 1. This is Bayes plausible and expected payo is unchanged. 29 / 43

30 Communication Subgame Proposition At equilibrium, there is no lying at the communication subgame. Proof: Follows from the previous two lemmas Intuitively, any form of lying will be anticipated by Receiver (i.e. Receiver will not be manipulated by lying). Hence, lying will not improve Sender's payo. 30 / 43

31 Persuasion In Lying Environment In the lying environment, Sender's problem can be reduced to max τ E τ ˆv(β) s.t. (Bayes plausibility) Supp(τ) βτ(β) = β 0 (No incentive to lie)???????? What does the additional no incentive to lie constraint bring? Sender cannot choose too sharp an information structure. 31 / 43

32 What is the lying cost Ordering If lying from s1 to s 3 is higher than lying from s 1 to s 2, Sender can just relabel the signal (or not use it at all) Magnitude Should lying cost be proportional to the expected gain from lying? But expected gain is endogenous to the strategies. Shall assume C(s, s ) = k > 0 s s 32 / 43

33 State-independent preference for Sender Assumes v(a, ω) = v(a, ω ) for all ω, ω Ω If Sender's preference depends on states, the payo of lying towards s depends on the true realization s. Eg. lying from s to s yields payo E βs v(α(β s ), ω) and not ˆv(β s ) Instead, state-independent preference implies payo of reporting any message m is just ˆv(β m ) Relevant applications: Prosecution example Lobbying Advertising 33 / 43

34 Optimal Signal Structure Sender's problem is reduced to max τ E τ ˆv(β) s.t. (Bayes plausibility) Supp(τ) βτ(β) = β 0 (No incentive to lie) v(β s ) min β Supp(τ) ˆv(β) k β s Supp(τ) 34 / 43

35 Optimal Signal Structure (Some Notation) For E {KG, L}, (L for Lying) Let τ E be the optimal signal structure Supp(τ E ) = {β E,..., 1 βe n }, with ˆv(β E ) 1 ˆv(βE ) 2 ˆv(βE n ) State independence preference allows us to rank beliefs Say there is gain from persuasion in the E environment if E τ E ˆv(β E s ) > ˆv(β 0 ) Corollary If there is gain from persuasion in the KG environment and ˆv(β KG n then there is gain from from persuasion in the lying environment. ) ˆv(β KG 1 ) k, 35 / 43

36 Optimal Signal Structure In Lying Environment When is there gain from persuasion? What does the optimal signal structure look like? Proposition There is gain from persuasion in the lying environment if and only if (i) there is gain from persuasion in the KG and environment and (ii) ˆv(β 0 ) < ˆv(β KG 1 ) + k. Proposition Let β be such that ˆv(β) = ˆv(β KG 1 ) + k. If there is gain from persuasion in the lying environment, the optimal signal structure τ L with Supp(τ L ) = {β L 1,..., βl n} can be achieved by: β L s = { βs KG β if ˆv(β s KG ) ˆv(β 1 ) + k if ˆv(β s KG ) > ˆv(β 1 ) + k 36 / 43

37 Persuasion In KG Environment ˆv(β h ) V (β) ˆv(β l ) co(ˆv) ˆv(β) β l β 0 β h 37 / 43

38 Optimal Signal Structure In Lying Environment ˆv(β h ) > k ˆv(β l ) ˆv(β) β l β 0 β h 38 / 43

39 Optimal Signal Structure In Lying Environment ˆv(β h ) > k ˆv(β l ) = k ˆv(β) β l β 0 β β h 39 / 43

40 Prosecutor Example U ˆv(β) β 0 = / 43

41 Prosecutor Example U ˆv(β) V (β) β 0 = / 43

42 Prosecutor Example U ˆv(β) V (β) = k β 0 = / 43

43 Conclusion Inducing an agent to do something through persuasion is done through belief manipulation. This paper studies the extent of belief manipulation possible when Sender cannot commit to revealing all new information found. Showed that under costly lying, Sender will not lie in the communication stage. Would instead choose to induce a vague information acquisition process (any gain from sharper information is eroded away by Receiver's wariness of Sender lying). Under state-independent Sender preference and constant lying cost, optimal signal structure resembles that of KG. 43 / 43