joint with Jonathan Pogach (Penn) 2009 FESAMES at Tokyo Hong Kong University of Science and Technology Honesty vs. White Lies Kim & Pogach Overview

Transcription

1 vs. White vs. joint with Jonathan Pogach (Penn) Kyungmin (Teddy) Kim Hong Kong University of Science and Technology 2009 FESAMES at Tokyo

2 vs. White vs. Question Models Main Results Contributions 1 Honest Agent 2 White Liar

3 vs. White vs. Question Models Main Results Contributions 1 Honest Agent Always tell the truth. 2 White Liar

4 vs. White vs. Question Models Main Results Contributions 1 Honest Agent Always tell the truth. Behavioral 2 White Liar

5 vs. White vs. Question Models Main Results Contributions 1 Honest Agent Always tell the truth. Behavioral 2 White Liar May lie but only for the sake of the principal.

6 vs. White vs. Question Models Main Results Contributions 1 Honest Agent Always tell the truth. Behavioral 2 White Liar May lie but only for the sake of the principal. Rational with no bias

7 vs. White vs. Question Models Main Results Contributions 1 Honest Agent Always tell the truth. Behavioral 2 White Liar May lie but only for the sake of the principal. Rational with no bias Commitment vs. Flexibility

8 vs. White Models Question Models Main Results Contributions Two variants of the standard cheap-talk game

9 vs. White Models Question Models Main Results Contributions Two variants of the standard cheap-talk game 1 Model Agent is honest with prob. µ. Agent has bias b with prob. 1 µ.

10 vs. White Models Question Models Main Results Contributions Two variants of the standard cheap-talk game 1 Model Agent is honest with prob. µ. Agent has bias b with prob. 1 µ. 2 White Lie Model Agent has no bias with prob. µ (white liar). Agent has bias b with prob. 1 µ.

11 vs. White Main Results Question Models Main Results Contributions often wins! 1 For µ close to 1, honesty always wins. 2 For µ close to 0, honesty wins for at least 79.1% of bias values. White lie sometimes wins. 3 For intermediate values of µ, honesty often wins. * µ = probability that agent is either honest or a white liar.

12 vs. White Intuitions Question Models Main Results Contributions 1 Commitment simplifies communication protocol and thus reduces the loss due to strategic considerations. Suppose µ = 1 (honest or white liar for sure). : a unique equilibrium with perfect communication. White lie: Lots of equilibria (from babbling to perfect communication). The perfect communication equilibrium outcome in the white lie model is not lower hemi-continuous.

13 vs. White Intuitions Question Models Main Results Contributions 1 Commitment simplifies communication protocol and thus reduces the loss due to strategic considerations. Suppose µ = 1 (honest or white liar for sure). : a unique equilibrium with perfect communication. White lie: Lots of equilibria (from babbling to perfect communication). The perfect communication equilibrium outcome in the white lie model is not lower hemi-continuous. 2 Commitment enriches language used in communication. White lie: Essentially a finite number of messages are used as in the standard cheap-talk model. : All messages are fully used.

14 vs. White Question Models Main Results Contributions Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies.

15 vs. White Question Models Main Results Contributions Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007).

16 vs. White Question Models Main Results Contributions Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007). 3 Identify new channels through which commitment is valauble. 1 Simplify strategic interactions. 2 Enrich language.

17 vs. White Question Models Main Results Contributions Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007). 3 Identify new channels through which commitment is valauble. 1 Simplify strategic interactions. 2 Enrich language. 4 (Theoretically) More general equilibrium characterization in both models White lies: Morgan and Stocken (2003) : Chen, Kartik, and Sobel (2008)

18 vs. White Uniform-Quadratic Model (Crawford and Sobel) Model Equilibrium Characterization Two players: principal (receiver), agent (sender, expert) Agent observes θ U[0,1] and sends m M = [0,1]. Principal receives m and takes an action y Y = [0, 1]. P s utility: U P (y, θ) (y θ) 2 A s utility: U A (y, θ,b) (y (θ + b)) 2

19 vs. White Perfect Bayesian Equilibrium: Two Conditions Model Equilibrium Characterization An equilibrium is characterized by a partition {θ 0, θ 1,..., θ n } and a sequence {y 1,...,y n } such that 1 BR (Best Response): y k = θ k 1 + θ k,k = 1,...,n. 2 2 NA (No Arbitrage): (θ k + b) y k = y k+1 (θ k + b),k = 1,...,n 1.

20 vs. White Example Model Equilibrium Characterization Principal y 1 y 2 Agent 0 θ 1 θ 1 + b 1

21 vs. White Model Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium With probability µ, the agent is honest. Honest type s strategy: m(θ) = θ, θ [0, 1] With probability 1 µ, the agent is strategic and has bias b > 0.

22 vs. White No Communication Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Babbling is not an equilibrium.

23 vs. White No Communication Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Babbling is not an equilibrium. No communication equilibrium exists if (and only if) µ 1/2.

24 vs. White No Communication Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Babbling is not an equilibrium. No communication equilibrium exists if (and only if) µ 1/2. Consider the following strategy profile: { 1 µ θ, with probability 1 µ r b (θ) =, m U[0,1], with probability 1 2µ 1 µ, y(m) = 1 2, m.

25 vs. White No Communication Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Babbling is not an equilibrium. No communication equilibrium exists if (and only if) µ 1/2. Consider the following strategy profile: { 1 µ θ, with probability 1 µ r b (θ) =, m U[0,1], with probability 1 2µ 1 µ, y(m) = 1 2, m. Different posteriors, but constant conditional expectation.

26 vs. White CS Equilibrium Outcomes Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Proposition Any equilibrium outcome in CS can be supported as an equilibrium outcome in the honesty model if and only if µ 1/2.

27 vs. White Mass Balance Condition Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Lemma Suppose 0 m < m 1, 0 θ < θ 1, and y = B(µ,m,m, θ, θ ). There exists a collection of probability measures {r (θ), θ [θ, θ ]} ([m,m ]) such that E µ,r [θ M] = y for any Borel set M in [m,m ], if and only if µ(m m ) 2 (1 µ)(θ θ ) 2.

28 vs. White Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Least Communication Equilibrium The following strategy profile is a natural extension of no communication equilibrium: Principal y 1 Honest Agent m 0 Biased Agent 0 1

29 vs. White Least Communication Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium MB (Mass Balance) m 0 must be sufficiently high. IC (Incentive Compatibility) m 0 must be sufficiently low so that the biased agent does not want to deviate to below m 0.

30 vs. White Type I Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Two partitions, {0,m 0,m 1,...,m n = 1} and {θ 0 = 0, θ 1,..., θ n = 1} and a sequence, {y 1,...,y n }, constitute an equilibrium if 1 IC (Incentive Compatibility): y 1 b b m 0 and m 0 b if m 0 > 0. 2 NA (No Arbitrage): y k + y k+1 = 2(θ k + b) 3 BR (Best Response): y k = B (µ,m k 1,m k, θ k 1, θ k ) 4 MB (Mass Balance): µ(m k m k 1 ) 2 (1 µ)(θ k θ k 1 ) 2

31 vs. White Example of Type I Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Principal Honest Agent y 1 y 2 m 0 m 1 Biased Agent 0 θ 1 1

32 vs. White Type II Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium MB and IC cannot be simultaneously satisfied if µ is sufficiently large

33 vs. White Type II Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium MB and IC cannot be simultaneously satisfied if µ is sufficiently large The following is the ONLY possibility: r b (θ) = θ + b µ, if θ [0, θ 0] for some θ 0 > 0.

34 vs. White Type II Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Principal Honest Agent b b/µ θ 0 + b θ 0 + b/µ Biased Agent 0 θ 0 1 For example, suppose θ = 0. The biased agent sends message b/µ. The principal takes an action µ (b/µ) + (1 µ) 0 = b.

35 vs. White Type II Equilibrium Model CS Equilibrium Outcomes Mass Balance Type I Equilibrium Type II Equilibrium Two partitions, {0,m 0,...,m n } and {0, θ 0,..., θ n }, and a sequence, {y 1,...,y n }, constitute an equilibrium if 1 IC (Incentive Compatibility): b/µ θ 0 + b 2 EL (Equal Length) m 0 = θ 0 + b/µ 3 NA0 (No Arbitrage): y 1 = θ 0 + b 4 NA (No Arbitrage) 5 BR (Best Response) 6 MB (Mass Balance)

36 vs. White Model White Lie Model CS Equilibrium Outcomes Equilibrium Uncertainty over the agent s bias With probability µ, the agent has no bias. With probability 1 µ, the agent has bias b > 0.

37 vs. White CS Equilibrium Outcomes White Lie Model CS Equilibrium Outcomes Equilibrium Proposition For µ > 0, only the no communication outcome in CS is supported as an equilibrium outcome in the white lie model.

38 vs. White CS Equilibrium Outcomes White Lie Model CS Equilibrium Outcomes Equilibrium Proposition For µ > 0, only the no communication outcome in CS is supported as an equilibrium outcome in the white lie model. When the biased agent is indifferent between y 1 and y 2 at θ, the white liar prefers y 1 to y 2 at θ.

39 vs. White CS Equilibrium Outcomes White Lie Model CS Equilibrium Outcomes Equilibrium Proposition For µ > 0, only the no communication outcome in CS is supported as an equilibrium outcome in the white lie model. When the biased agent is indifferent between y 1 and y 2 at θ, the white liar prefers y 1 to y 2 at θ. New Equilibrium Condition: No Arbitrage condition for the White liar (NAW)

40 vs. White CS Equilibrium Outcomes White Lie Model CS Equilibrium Outcomes Equilibrium Proposition For µ > 0, only the no communication outcome in CS is supported as an equilibrium outcome in the white lie model. When the biased agent is indifferent between y 1 and y 2 at θ, the white liar prefers y 1 to y 2 at θ. New Equilibrium Condition: No Arbitrage condition for the White liar (NAW) If two actions, and no action in between, are induced by both types in equilibrium, then both the white liar and the biased agent must be indifferent between the two actions at their own boundary states.

41 vs. White Equilibrium White Lie Model CS Equilibrium Outcomes Equilibrium Principal White Liar y 1 y 2 θ 0 0 θ 0 1 Biased Agent 0 θ1 b 1

42 vs. White White Lie Equilibrium White Lie Model CS Equilibrium Outcomes Equilibrium Proposition Any equilibrium in the white lie model is characterized by two finite partitions, {0, θ0 0, θ0 1,..., θ0 n = 1} and {θ0 b = 0, θb 1,..., θb n = 1}, a finite sequence, {y 1,...,y n }, and n 0 N { } if θ0 0 > 0 such that y 1 + 2n 0 1 2n 0 θ0 0 2b if θ0 0 > 0, (ICB) y 1 = 2n 0+1 2n 0 θ0 0 if θ0 0 > 0, (ICW) y k + y k+1 = 2(θk b + b), (NAB) y k + y k+1 = 2θk 0, (NAW) y k = B(µ, θk 1 0, θ0 k, θb k 1, θb k ). (BR)

43 vs. White Almost vs. Almost Close to 1 Close to 0 Intermediate Cases Proposition For any b (0,1), if µ is sufficiently close to 1, honesty wins (that is, the principal is better off with a possibly honest agent than with a potential white liar).

44 vs. White Almost vs. Almost Close to 1 Close to 0 Intermediate Cases Proposition For any b (0,1), if µ is sufficiently close to 1, honesty wins (that is, the principal is better off with a possibly honest agent than with a potential white liar). equilibrium is close to perfect communication.

45 vs. White Almost vs. Almost Close to 1 Close to 0 Intermediate Cases Proposition For any b (0,1), if µ is sufficiently close to 1, honesty wins (that is, the principal is better off with a possibly honest agent than with a potential white liar). equilibrium is close to perfect communication. White lie equilibrium does not converge to the perfect communication equilibrium. The loss due to strategic considerations does not vanish even when the agent is the white liar almost for sure.

46 vs. White Almost Biased Agents 1 2n(n+1) 1 2n(n+0.5) 1 1 2n 2 2n(n 1) Close to 1 Close to 0 Intermediate Cases Mixed

47 vs. White Almost Biased Agents 1 2n(n+1) 1 2n(n+0.5) 1 1 2n 2 2n(n 1) Close to 1 Close to 0 Intermediate Cases Mixed For µ close to 0, the two models share equilibrium conditions other than MB for the former and NAW for the latter.

48 vs. White Almost Biased Agents 1 2n(n+1) 1 2n(n+0.5) 1 1 2n 2 2n(n 1) Close to 1 Close to 0 Intermediate Cases Mixed For µ close to 0, the two models share equilibrium conditions other than MB for the former and NAW for the latter. As µ tends to 0, MB becomes negligible, while NAW thus still binds.

49 vs. White Almost Biased Agents 1 2n(n+1) 1 2n(n+0.5) 1 1 2n 2 2n(n 1) Close to 1 Close to 0 Intermediate Cases Mixed For µ close to 0, the two models share equilibrium conditions other than MB for the former and NAW for the latter. As µ tends to 0, MB becomes negligible, while NAW thus still binds. White lie equilibrium outcome can be replicated in the honesty model.

50 vs. White Mixed Area: Example Welfare comparison for b (1/3,1/2) (one partition element) when µ is sufficiently small. Close to 1 Close to 0 Intermediate Cases 8 x U W (0,b)/ µ U H (0,b)/ µ b

51 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ.

52 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ. This is true in the white lie model.

53 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ. This is true in the white lie model. Not necessarily in the honesty model. MB is an inequality condition.

54 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ. This is true in the white lie model. Not necessarily in the honesty model. MB is an inequality condition. Two consequences

55 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ. This is true in the white lie model. Not necessarily in the honesty model. MB is an inequality condition. Two consequences 1 More uniform partition element sizes

56 vs. White Intermediate Cases Close to 1 Close to 0 Intermediate Cases In CS, partition element size increases for higher θ. This is true in the white lie model. Not necessarily in the honesty model. MB is an inequality condition. Two consequences 1 More uniform partition element sizes 2 No. of partition elements increases faster in the honesty model.

57 vs. White Related Literature Related Literature 1 Games with Uncertain Bias Morgan and Stocken (2003), Li and Madarasz (2007) 2 Honest Type Olszewski (2004), Chen, Kartik, and Sobel (2008) 3 with Uncertainty over Agent s Motive Sobel (1985), Benabou and Laroque (1992), Morris (2001)

58 vs. White Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. Related Literature

59 vs. White Related Literature Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007).

60 vs. White Related Literature Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007). 3 Identify new channels through which commitment is valauble. 1 Simplify strategic interactions. 2 Enrich language.

61 vs. White Related Literature Contributions 1 First formal model and results on a long-standing question on honesty vs. white lies. 2 Micro-foundation for the preference for honesty (more generally, character) 55% of those surveyed consider honesty, integrity and other values of character the most important qualities they look for in a presidential candidate (USA Today, 3/12/2007). 3 Identify new channels through which commitment is valauble. 1 Simplify strategic interactions. 2 Enrich language. 4 (Theoretically) More general equilibrium characterization in both models White lies: Morgan and Stocken (2003) : Chen, Kartik, and Sobel (2008)

62 vs. White IC Lemma Type I and II IC Lemma Lemma Let z(, ) : Θ T Y represent the outcome of the cheap talk game where T is the set of agent types and z(θ,b) is the policy chosen by the principal when the agent has bias b and observes state θ. Denote by V S (θ,b) the indirect utility function of the agent with bias b, that is, V S (θ,b) = U A (z(θ,b), θ,b). z(,b) is incentive compatible for the agent with bias b if and only if (i) z(,b) is nondecreasing, (ii) V S (,b) is absolutely continuous, and (iii) if z 1 (θ,b) = z(θ,b)/ θ exists, then U A (z(θ,b), θ,b) y z(θ,b) θ = 0.

63 vs. White Implication of IC Lemma IC Lemma Type I and II Θ = [0,1] can be decomposed into the following three subsets: { } Θ 1 = θ Θ : z(θ,b) = y A (θ,b) = θ + b = {θ Θ : most preferred policy for agent}, { Θ 2 = θ Θ : z 1 (θ,b) = z(θ,b) } = 0 θ = {θ Θ : the same action is implemented around θ}, Θ 3 = {θ Θ : z 1 (θ,b) does not exist}.

64 vs. White Two properties IC Lemma Type I and II 1 Convexity: The biased agent s reporting strategy is convex if there exists m 0 [0,1] such that the biased agent never sends messages below m 0 and sends all messages above m 0. 2 Monotonicity: The principal s strategy y is monotone if for all m > m such that m supp r b (θ) and m supp r b (θ ) for some θ and θ, y(m ) y(m). Proposition Any equilibrium in which the biased agent s strategy is convex and the principal s strategy is monotone is either Type I or II.

65 vs. White Two properties Principal y 1 y 1 y 2 IC Lemma Type I and II Honest Agent Biased Agent Non-convex equilibriumnon-monotone equilibrium