Clearly Biased Experts

Transcription

1 Clearly Biased Experts Archishman Chakraborty and Rick Harbaugh IIOC, May / 27

2 Cheap Talk Cheap Talk Sender has some private information of interest to Receiver I Stock analyst knows value of stock I Lobbyist knows value of a project I Salesperson knows quality of a product I Network knows about progress of a war But what if information on multiple dimensions? I Stock analyst - multiple stocks I Lobbyist - multiple projects I Salesperson - multiple goods I News network - multiple issues 2 / 27

3 Cheap Talk Cheap Talk Sender has some private information of interest to Receiver I Stock analyst knows value of stock I Lobbyist knows value of a project I Salesperson knows quality of a product I Network knows about progress of a war But what if information on multiple dimensions? I Stock analyst - multiple stocks I Lobbyist - multiple projects I Salesperson - multiple goods I News network - multiple issues 2 / 27

4 Cheap Talk Multidimensional cheap Talk When can make credible tradeo s across dimensions? When does sender bene t from such tradeo s? Do asymmetries across dimensions make tradeo s harder? Does transparency of sender s preferences make tradeo s easier? 3 / 27

5 Cheap Talk Chakraborty and Harbaugh (JET, 2007) existence result I Suppose Sender and Receiver payo s additively separable and supermodular I And environment (nearly) symmetric I Then comparative cheap talk is credible, informative even if strong incentive to exaggerate within each dimension I Trades o higher action on one dimension for lower action on another dimension I Supermodularity, separability, symmetry also used in multidimensional bargaining (Chakraborty and Harbaugh, Econ. Letters, 2003) and multi-object auctions (Chakraborty, Gupta, and Harbaugh, RAND, 2006) 4 / 27

6 Cheap Talk Chakraborty and Harbaugh (JET, 2007) existence result I Suppose Sender and Receiver payo s additively separable and supermodular I And environment (nearly) symmetric I Then comparative cheap talk is credible, informative even if strong incentive to exaggerate within each dimension I Trades o higher action on one dimension for lower action on another dimension I Supermodularity, separability, symmetry also used in multidimensional bargaining (Chakraborty and Harbaugh, Econ. Letters, 2003) and multi-object auctions (Chakraborty, Gupta, and Harbaugh, RAND, 2006) 4 / 27

7 Model assumptions I Sender knows θ = (θ 1,..., θ N ) 2 Θ where Θ is compact, convex subset of R N I Distribution F with density f has full support on Θ I Sender sends message m 2 M I Receiver action a = (a 1,..., a N ) = (E [θ 1 jm],..., E [θ N jm]) I Sender s preferences U(a) continuous in a i I Note that sender does not care directly about θ state independence or transparent motives 5 / 27

8 Model assumptions I Sender knows θ = (θ 1,..., θ N ) 2 Θ where Θ is compact, convex subset of R N I Distribution F with density f has full support on Θ I Sender sends message m 2 M I Receiver action a = (a 1,..., a N ) = (E [θ 1 jm],..., E [θ N jm]) I Sender s preferences U(a) continuous in a i I Note that sender does not care directly about θ state independence or transparent motives 5 / 27

9 Equilibrium Equilibrium I A hyperplane h divides Θ into halfspaces I Message m indicates in which halfspace of Θ is θ I Perfect Bayesian Equilibrium if h is such that sender has no incentive to misreport the halfspace I Note equilibrium is a convex partition of Θ 6 / 27

10 Media Bias Example: Media Bias I Seriousness of two scandals θ 1 and θ 2, uniform i.i.d. on [0, 1] I Network wants to push scandals to increase viewership I U = a 1 + a 2 so biased within but not across dimensions I Comparative cheap talk equilibrium 7 / 27

11 Media Bias Example: Media Bias I Seriousness of two scandals θ 1 and θ 2, uniform i.i.d. on [0, 1] I Network wants to push scandals to increase viewership I U = a 1 + a 2 so biased within but not across dimensions I Comparative cheap talk equilibrium 7 / 27

12 Media Bias Example: Media Bias I Seriousness of two scandals θ 1 and θ 2, uniform i.i.d. on [0, 1] I Network wants to push scandals to increase viewership I U = a 1 + a 2 so biased within but not across dimensions I Comparative cheap talk equilibrium 7 / 27

13 Media Bias Example: Media Bias I Seriousness of two scandals θ 1 and θ 2, uniform i.i.d. on [0, 1] I Network wants to push scandals to increase viewership I U = a 1 + a 2 so biased within but not across dimensions I Comparative cheap talk equilibrium 7 / 27

14 Media Bias Example: Media Bias I Partisan network: U = 4a 1 + a 2 I Simple ranking of θ 1 and θ 2 not credible I Can divide space with announcement line h to put actions on same indi erence curve? 8 / 27

15 Media Bias Example: Media Bias I Partisan network: U = 4a 1 + a 2 I Simple ranking of θ 1 and θ 2 not credible I Can divide space with announcement line h to put actions on same indi erence curve? 8 / 27

16 Media Bias Example: Media Bias I Partisan network: U = 4a 1 + a 2 I Simple ranking of θ 1 and θ 2 not credible I Can divide space with announcement line h to put actions on same indi erence curve? 8 / 27

17 Media Bias Example: Media Bias I Spin line around interior point c I Actions continuously move around point I Double back on each other when orientation of line is reversed I For some orientation must have actions on same indi erence curve 9 / 27

18 Media Bias Example: Media Bias I Spin line around interior point c I Actions continuously move around point I Double back on each other when orientation of line is reversed I For some orientation must have actions on same indi erence curve 9 / 27

19 Media Bias Example: Media Bias I Spin line around interior point c I Actions continuously move around point I Double back on each other when orientation of line is reversed I For some orientation must have actions on same indi erence curve 9 / 27

20 Media Bias Example: Media Bias I Spin line around interior point c I Actions continuously move around point I Double back on each other when orientation of line is reversed I For some orientation must have actions on same indi erence curve 9 / 27

21 Media Bias Example: Media Bias I Two actions not the same - equilibrium is in uential I Each action taken with positive probability - equilibrium is informative I If c = E [θ] then for any distribution each action taken with probability 1/3 or higher 10 / 27

22 Media Bias Example: Media Bias I Two actions not the same - equilibrium is in uential I Each action taken with positive probability - equilibrium is informative I If c = E [θ] then for any distribution each action taken with probability 1/3 or higher 10 / 27

23 Media Bias Example: Media Bias I Two actions not the same - equilibrium is in uential I Each action taken with positive probability - equilibrium is informative I If c = E [θ] then for any distribution each action taken with probability 1/3 or higher 10 / 27

24 Borsuk-Ulam Borsuk-Ulam Theorem I For every continuous, odd mapping G : S N 1! R N 1 there exists a point s 2 S N 1 satisfying G (s) = 0. I Odd mapping: G (s) = G ( s) I If temperature varies continuously around equator then must be two antipodal points on equator with same temperature I Let s be point on equator and G (s) be di erence in temperature between s and its antipode ( s) I Then G (s) = G ( s) I Stranger: if temperature and barometric pressure vary continuously around globe then there must be two antipodal points on globe with same temperature and same barometric pressure 11 / 27

25 Borsuk-Ulam Borsuk-Ulam Theorem I For every continuous, odd mapping G : S N 1! R N 1 there exists a point s 2 S N 1 satisfying G (s) = 0. I Odd mapping: G (s) = G ( s) I If temperature varies continuously around equator then must be two antipodal points on equator with same temperature I Let s be point on equator and G (s) be di erence in temperature between s and its antipode ( s) I Then G (s) = G ( s) I Stranger: if temperature and barometric pressure vary continuously around globe then there must be two antipodal points on globe with same temperature and same barometric pressure 11 / 27

26 Theorem 1 Existence of Informative Cheap Talk Theorem An in uential cheap talk equilibrium exists for all continuous U. I For instance, suppose N = 2 I s 2 S is orientation of line h dividing plane into two regions, R + and R I a + (s) = (E [θ 1 jθ 2 R + (s)], E [θ 2 jθ 2 R + (s)]) I a (s) = (E [θ 1 jθ 2 R (s)], E [θ 2 jθ 2 R (s)]) I G (s) = U(a + (s)) I G continuous and odd U(a (s)) I So there exists s such that G (s) = 0 12 / 27

27 Theorem 1 Existence of Informative Cheap Talk Theorem An in uential cheap talk equilibrium exists for all continuous U. I For instance, suppose N = 2 I s 2 S is orientation of line h dividing plane into two regions, R + and R I a + (s) = (E [θ 1 jθ 2 R + (s)], E [θ 2 jθ 2 R + (s)]) I a (s) = (E [θ 1 jθ 2 R (s)], E [θ 2 jθ 2 R (s)]) I G (s) = U(a + (s)) I G continuous and odd U(a (s)) I So there exists s such that G (s) = 0 12 / 27

28 Quasiconvex Sender and Receiver payo s Receiver always better o from communication Sender better o if utility function quasiconvex And worse o if utility function quasiconcave 13 / 27

29 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

30 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

31 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

32 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

33 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

34 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

35 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

36 Auction Example: Private Value Auction I Four buyers: I v 1 = a 1 I v 2 = (1/4)a 1 + (3/4)a 2 I v 3 = (3/4)a 1 + (1/4)a 2 I v 4 = a 2 I U=2nd maxfv j g I U quasiconvex so cheap talk helps seller I True generally for N 4 bidders 14 / 27

37 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

38 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

39 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

40 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

41 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

42 Voting Example: Voting I Defense lawyer I Two blocs of jurors I a i is probability of juror bloc i voting acquittal I U = p(a) is probability of acquittal I Need both blocs: p(a) = a 1 a 2 (quasiconcave) I Need either bloc: p(a) = a 1 + a 2 a 1 a 2 (quasiconvex) 15 / 27

43 Product Di erentiation Example: Horizontal and Vertical Di erentiation I Incumbent rm with known quality I Entrant rm with unknown quality θ 2 and unknown horizontal di erentiation θ 1 I Entrant makes cheap talk statement I (Not credible to claim both good and di erent) I Payo s for both rms (and consumers) increase 16 / 27

44 Product Di erentiation Example: Horizontal and Vertical Di erentiation I Incumbent rm with known quality I Entrant rm with unknown quality θ 2 and unknown horizontal di erentiation θ 1 I Entrant makes cheap talk statement I (Not credible to claim both good and di erent) I Payo s for both rms (and consumers) increase 16 / 27

45 Product Di erentiation Example: Horizontal and Vertical Di erentiation I Incumbent rm with known quality I Entrant rm with unknown quality θ 2 and unknown horizontal di erentiation θ 1 I Entrant makes cheap talk statement I (Not credible to claim both good and di erent) I Payo s for both rms (and consumers) increase 16 / 27

46 Product Di erentiation Example: Horizontal and Vertical Di erentiation I Incumbent rm with known quality I Entrant rm with unknown quality θ 2 and unknown horizontal di erentiation θ 1 I Entrant makes cheap talk statement I (Not credible to claim both good and di erent) I Payo s for both rms (and consumers) increase 16 / 27

47 Product Di erentiation Example: Horizontal and Vertical Di erentiation I Incumbent rm with known quality I Entrant rm with unknown quality θ 2 and unknown horizontal di erentiation θ 1 I Entrant makes cheap talk statement I (Not credible to claim both good and di erent) I Payo s for both rms (and consumers) increase 16 / 27

48 2-message equilibrium Informative? Sure that equilibrium is informative? I Maybe both messages have almost same meaning? I Or one message rarely sent? Not a problem I Messages are always distinct - convex regions I If c is centerpoint then each message sent with probability at least 1/(N + 1) I If density f logconcave then probability at least 1/e I If N > 2 can make each message probability 1/2 17 / 27

49 2-message equilibrium Informative? Sure that equilibrium is informative? I Maybe both messages have almost same meaning? I Or one message rarely sent? Not a problem I Messages are always distinct - convex regions I If c is centerpoint then each message sent with probability at least 1/(N + 1) I If density f logconcave then probability at least 1/e I If N > 2 can make each message probability 1/2 17 / 27

50 Media Bias example Consider again linear preferences I Example: negative advertising I U = 4a 1 a 2 I Tradeo higher action on both issues vs lower action on both issues I To make other side look bad have to make self look bad I Eq construction same as before 18 / 27

51 Media Bias example Consider again linear preferences I Example: negative advertising I U = 4a 1 a 2 I Tradeo higher action on both issues vs lower action on both issues I To make other side look bad have to make self look bad I Eq construction same as before 18 / 27

52 Media Bias example Consider again linear preferences I Example: negative advertising I U = 4a 1 a 2 I Tradeo higher action on both issues vs lower action on both issues I To make other side look bad have to make self look bad I Eq construction same as before 18 / 27

53 Media Bias example Consider again linear preferences I Example: negative advertising I U = 4a 1 a 2 I Tradeo higher action on both issues vs lower action on both issues I To make other side look bad have to make self look bad I Eq construction same as before 18 / 27

54 Media Bias example Consider again linear preferences I Example: negative advertising I U = 4a 1 a 2 I Tradeo higher action on both issues vs lower action on both issues I To make other side look bad have to make self look bad I Eq construction same as before 18 / 27

55 More Information? Can Reveal More Information? I Can do better for linear preferences? I If sender says in upper half-plane, can ask again I And again, and again... I So can reveal arbitrarily ne slice of plane 19 / 27

56 More Information? Can Reveal More Information? I Can do better for linear preferences? I If sender says in upper half-plane, can ask again I And again, and again... I So can reveal arbitrarily ne slice of plane 19 / 27

57 More Information? Can Reveal More Information? I Can do better for linear preferences? I If sender says in upper half-plane, can ask again I And again, and again... I So can reveal arbitrarily ne slice of plane 19 / 27

58 More Information? Can Reveal More Information? I Can do better for linear preferences? I If sender says in upper half-plane, can ask again I And again, and again... I So can reveal arbitrarily ne slice of plane 19 / 27

59 More Information? Can Reveal More Information? I Can do better for linear preferences? I If sender says in upper half-plane, can ask again I And again, and again... I So can reveal arbitrarily ne slice of plane 19 / 27

60 More Information? Can Reveal More Information? I Expectations for each pair of subregions must lie on line through expectation for whole region I So actions for every slice on same line I So truth-telling is subgame perfect I Either simultaneous or sequential cheap talk 20 / 27

61 More Information? Can Reveal More Information? I Expectations for each pair of subregions must lie on line through expectation for whole region I So actions for every slice on same line I So truth-telling is subgame perfect I Either simultaneous or sequential cheap talk 20 / 27

62 More Information? Can Reveal More Information? I Expectations for each pair of subregions must lie on line through expectation for whole region I So actions for every slice on same line I So truth-telling is subgame perfect I Either simultaneous or sequential cheap talk 20 / 27

63 More Information? Can Reveal More Information? I Expectations for each pair of subregions must lie on line through expectation for whole region I So actions for every slice on same line I So truth-telling is subgame perfect I Either simultaneous or sequential cheap talk 20 / 27

64 Theorem 2 Very Informative Cheap Talk with Linear Preferences U(a; t) = λ 1 (t)a λ N (t)a N Theorem A k-message cheap talk equilibrium exists for all linear U and all k. As k becomes large, the sender can reveal almost all information in N 1 dimensions. I Sender and receiver have no con ict on N 1 dimensions I With linear preferences can credibly reveal all this info I With nonlinear preferences can use mixed messages I Also k message equilibrium, also more informative as k increases, but noisy so not all info revealed 21 / 27

65 Theorem 2 Very Informative Cheap Talk with Linear Preferences U(a; t) = λ 1 (t)a λ N (t)a N Theorem A k-message cheap talk equilibrium exists for all linear U and all k. As k becomes large, the sender can reveal almost all information in N 1 dimensions. I Sender and receiver have no con ict on N 1 dimensions I With linear preferences can credibly reveal all this info I With nonlinear preferences can use mixed messages I Also k message equilibrium, also more informative as k increases, but noisy so not all info revealed 21 / 27

66 Theorem 2 Very Informative Cheap Talk with Linear Preferences U(a; t) = λ 1 (t)a λ N (t)a N Theorem A k-message cheap talk equilibrium exists for all linear U and all k. As k becomes large, the sender can reveal almost all information in N 1 dimensions. I Sender and receiver have no con ict on N 1 dimensions I With linear preferences can credibly reveal all this info I With nonlinear preferences can use mixed messages I Also k message equilibrium, also more informative as k increases, but noisy so not all info revealed 21 / 27

67 Robustness I Suppose Sender has T > 1 di erent types (preference orderings). Results still hold if N > T. I Suppose some small uncertainty about Sender s type - still holds with mild conditions I Suppose Sender has Euclidean preferences (not transparent!) - still holds for ε > 0 cost of lying if biases large enough 22 / 27

68 Euclidean Euclidean Preferences U(a; θ) = d(a, τ(θ)) = N i=1(a i (θ i + b i )) 2 1/2 I d(, ) is the Euclidean distance function I b = (b 1,..., b N ) 2 R N is bias in each dimension I τ(θ) = (θ 1 + b 1,..., θ N + b N ) is sender s ideal point I Interested in arbitrarily large biases: b = (ρ 1 B,..., ρ N B) for B 0 and ρ = (ρ 1,..., ρ N ) 6= 0 23 / 27

69 Euclidean Euclidean Preferences U(a; θ) = d(a, τ(θ)) = N i=1(a i (θ i + b i )) 2 1/2 I d(, ) is the Euclidean distance function I b = (b 1,..., b N ) 2 R N is bias in each dimension I τ(θ) = (θ 1 + b 1,..., θ N + b N ) is sender s ideal point I Interested in arbitrarily large biases: b = (ρ 1 B,..., ρ N B) for B 0 and ρ = (ρ 1,..., ρ N ) 6= 0 23 / 27

70 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

71 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

72 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

73 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

74 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

75 Euclidean Convergence to Linear Preferences I For small B indi erence curves very curved I As B increases become straighter I And become more similar for each θ I Converge uniformly to linear preferences as B! I So always cheap talk equilibrium in limit 24 / 27

76 Theorem 3 Epsilon Cheap Talk I Suppose there is some ε > 0 cost to lying I De ne ε cheap talk equilibrium for large biases as case where for any ε > 0 there is some B such that for all B > B the gain from deviating from candidate equilibrium is less than ε Theorem An announcement strategy is a k-message ε cheap talk equilibrium for Euclidean U with large B if and only if it is a cheap talk equilibrium for the limiting linear U with ρ = λ. 25 / 27

77 Theorem 3 Epsilon Cheap Talk I Suppose there is some ε > 0 cost to lying I De ne ε cheap talk equilibrium for large biases as case where for any ε > 0 there is some B such that for all B > B the gain from deviating from candidate equilibrium is less than ε Theorem An announcement strategy is a k-message ε cheap talk equilibrium for Euclidean U with large B if and only if it is a cheap talk equilibrium for the limiting linear U with ρ = λ. 25 / 27

78 Transparency Other transparency models I We model transparency as common knowledge of sender s preferences over actions I Not same as just knowing sender s biases I In canonical CS model still uncertainty over sender s preferences if receiver knows b because preferences are state-dependent I When transparency is modeled as receiver knowing b, transparency helps or hurts depending on size of b that is revealed (Morgan and Stocken, 2003; Dimitrakas and Sara das, 2004) I Li and Madarasz (2006) nd transparency can hurt when uncertainty over sign of large b 26 / 27

79 Conclusion I Su cient transparency implies existence of cheap talk equilibria I Asymmetries not a barrier when preferences are transparent I Simple conditions for when gain/lose from communication I Seems applicable to many environments 27 / 27