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Instability of Babbling Equilibria in Cheap Talk Games: Some Experimental Results

Instability of Babbling Equilibria in Cheap Talk Games: Some Experimental Results. Toshiji Kawagoe Future University – Hakodate and Hirokazu Takizawa Institute of Economy, Trade and Industry. Section 1. Cheap Talk Games, Sequential Equilibria, and its Refinements. 1. Cheap Talk Games (1).

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Instability of Babbling Equilibria in Cheap Talk Games: Some Experimental Results

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  1. Instability of Babbling Equilibria in Cheap Talk Games:Some Experimental Results Toshiji Kawagoe Future University – Hakodate and Hirokazu Takizawa Institute of Economy, Trade and Industry

  2. Section 1.Cheap Talk Games, Sequential Equilibria, and its Refinements

  3. 1. Cheap Talk Games (1) • Sender-Receiver Games • A sender, who has private information, sends a payoff-irrelevant message to a receiver, then the receiver chooses a payoff-relevant action. • Coordination via communication (persuasion) • Policy announcement by the Fed, Veto threats in congress, Sales talk, etc. • Research motivation • Comparing equilibrium selection/refinement theory in changing the degree of coordination between the sender and the receiver.

  4. 2. Cheap Talk Games (2) • Crawford & Sobel (1982)’s model • Sender’s type • sender’ message • receiver’s action • sender’s payoff • receiver’s payoff • coincidence of interests perfect partial

  5. X X Y a A b Y Z Z 0.5 N X X 0.5 Y Y a b Z B Z 3. Cheap Talk Games (3) Sender Receiver Receiver Sender

  6. 3. Cheap Talk Games (3) X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  7. 3. Cheap Talk Games (3) X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  8. 3. Cheap Talk Games (3) X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  9. 3. Cheap Talk Games (3) X X Sender Y a A b Y 1, 1 1, 1 Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  10. 4. Cheap Talk Games (4) Game1 [ b(A)=b(B)=0 ] Game2 [ b(A)=1/5, b(B)=－1/5 ] Game3 [ b(A)=0, b(B)=－1/3 ]

  11. b(t) b(t) b(t) t t t 5. Cheap Talk Games (5) Game2 Game1 0 0 Game3 0

  12. 6. Sequential Equilibria (1) • Separating equilibria • The sender reveals her type, then the receiver chooses an action according to the sender’s type. • Babbling equilibria • The receiver ignores the sender’s message, then chooses an action which maximizes expected payoff with the belief based on prior probability of the sender’s type. • There are pooling and mixed strategy babbling equilibria.

  13. X X Y a A b Y Z Z 0.5 N X X 0.5 Y Y a b Z B Z 7. Separating equilibria Sender Receiver Receiver Sender

  14. 7. Separating equilibria X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y a b Z B Z Sender

  15. 7. Separating equilibria X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  16. 7. Separating equilibria X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  17. X X Y a A b Y Z Z 0.5 N X X 0.5 Y Y a b Z B Z 8. Pooling babbling equilibria Sender Receiver Receiver Sender

  18. X X Y a A b Y Z Z 0.5 N X X 0.5 Y Y a b Z B Z 8. Pooling babbling equilibria Sender Receiver Receiver Sender

  19. X X Y a A b Y Z Z 0.5 N X X 0.5 Y Y a b Z B Z 8. Pooling babbling equilibria Sender Receiver Receiver Sender

  20. 8. Pooling babbling equilibria X X Sender Y a A b Y Z Z 0.5 Receiver N Receiver X X 0.5 Y Y a b Z B Z Sender

  21. 9. Refinements of Equilibria (1) • Farrell (1985)’s neologism-proofness • The sender never receives higher payoff than equilibrium payoff by deviating the equilibrium using off-the-equilibrium messages. • cf. Cho & Kreps (1987)’s intuitive criterion • Rabin and Sobel (1996)’s recurrent set • Consider further deviations from deviation from the equilibrium and find stable set of outcomes robust to such sequences of deviations.

  22. 10. Refinements of Equilibria (2) • Game1 • Deviation(aa,ZZ)⇒(ab,XY) ⇒(ab,XY) • Separating equilibria are only recurrent set.

  23. 11. Refinements of Equilibria (3) • Game2 • Deviation(ab,XY) ⇒(bb,ZZ) ⇒(bb,ZZ) • Pooling babbling equilibria are only recurrent set.

  24. 12. Refinements of Equilibria (4) • Game3 • (bb,ZZ) ⇒(ab,XY) ⇒(aa,ZZ) ⇒(aa,ZZ) • Though pooling babbling equilibria are onlyrecurrent set, deviation to separating equilibria may occur.

  25. Section 2.Experimentsand Bounded Rationality

  26. 13. Experimental Design • Each subject plays three sender-receiver games alternatively with different opponents each times (one shot game environment). • Subject receives monetary reward proportional to her payoff or draws lottery with winning probability proportional to her payoff. • Average reward is about 3,000 yen.

  27. 14. Hypotheses • Hypothesis 1 • Separating equilibria is played more frequently than babbling equilibria in Game 1 and 2. • Hypothesis 2 • Separating equilibria is played more frequently in Game 1 than in Game 2. • Hypothesis 3 • Babbling equilibria is played more frequently than any other outcomes in Game 3.

  28. 15. Predictions and initial results

  29. 16. Initial Results Session1, Lottery

  30. 17. New Design (1) Deviation from equilibrium or refinement prediction is severe in Game 2 and 3. Permuting labels Label on each strategy may induces separating equilibria in Game 2 and 3. Learning Repetition of same game may increase equilibrium plays.

  31. 18. New Design (2)

  32. 19. Bounded Rationality Deviations from equilibrium are still severe in Game 2 and 3 in new design. Subjects’ behavior are anomalous. Subjects’ behavior may be explained by bounded rationality or some noisy equilibrium model.

  33. 20. Quantal Response Equilibria • Consider best responses under stochastic error. • (cf. McFadden’s random utility model) • Prob.{i chooses strategy j} = • Expected payoff when i chooses j: • Fixed points of the equations below are QRE

  34. 21. Properties of QRE • λrepresents the degree of rationality • Whenλ＝0, random choice • λ→∞, Nash equilibria (sequential equilibria) • QRE exists. • QRE is a refinement of equilibrium.

  35. 22. QRE in Cheap Talk Games (1) • In Game1, 2, separating and a mixed strategy babbling equilibrium are QRE. • In Game3, a mixed strategy babbling equilibrium is AQRE. • Pooling babbling equilibria are not QRE. • Cf. neologism-proofness and recurrent set predicts pooling babbling equilibria.

  36. r1 r1 r2 r2 r3 r3 23. QRE in Cheap Talk Games (2) X s1 X p 1-p a A b Y s2 Y Z 0.5 Z s3 N X s1 X 0.5 Y Y s2 a b Z B Z s3 q 1-q

  37. 24. QRE in Cheap Talk Games (3)

  38. 25. Estimation procedures • Maximum likelihood method • Calculate a fixed point of QRE for givenλ, then evaluate log likelihood function (LL). Iterate this process and find aλthat maximizes LL using grid search method. • Bootstrap method • Confidence interval is calculated by bootstrap method using 1,000 resampling pseudo-data. • Model selection： • Goodness-of-fit：pseudo

  39. 26. AQRE for Sender (1)

  40. 27. AQRE for Sender (2)

  41. 28. AQRE for Sender (3)

  42. 29. AQRE for Receiver (1)

  43. 30. AQRE for Receiver (2)

  44. 31. AQRE for Receiver (3)

  45. 32. Other estimated models • Model based on equilibria • NNM-SE (noisy Nash model) • MIX-SE • POOL • POOL-SE

  46. 33. NNM-SE • NNM-SE • Convex combination of separating equilibria σwith probabilityγ and uniform distributionμwith probablity 1-γ • P＝γσ＋（１－γ）μ • Find aγthat maximizes log likelihood using grid search method. • Confidence intervals is calculated by bootstrap method. • Model selection: AIC, Goodness-of-fit:pseudo R2

  47. 34. MIX-SE • MIX-SE • Convex combination of separating equilibriaσwith probabilityγ and QRE correspondes tomixed strategy babbling equilibriumμwith probablity 1-γ • p＝γσ＋（１－γ）μ • Find aγthat maximizes log likelihood using grid search method. • Confidence intervals is calculated by bootstrap method. • Model selection: AIC, Goodness-of-fit:pseudo R2

  48. 35. POOL • POOL • Convex combination of pooling babbling equilibriaσwith probabilityγ and uniform distributionμwith probablity 1-γ • p＝γσ＋（１－γ）μ • Find aγthat maximizes log likelihood using grid search method. • Confidence intervals is calculated by bootstrap method. • Model selection: AIC, Goodness-of-fit:pseudo R2

  49. 36. POOL-SE • POOL-SE • Convex combination of pooling babbling equilibriaσwith probabilityγ (sender) orseparating equilibriaσwith probabilityγ (receiver) and uniform distributionμwith probablity 1-γ • p＝γσ＋（１－γ）μ • Find aγthat maximizes log likelihood using grid search method. • Confidence intervals is calculated by bootstrap method. • Model selection: AIC, Goodness-of-fit:pseudo R2

  50. 37. Estimation results

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