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) 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.

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

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

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

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

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

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

4. Cheap Talk Games (4) X Y Z A 4, 4 1, 1 3, 3 B X Y Z A 3, 4 2, 1 Game1 [ b(A)=b(B)=0 ] Game2 [ b(A)=1/5, b(B)=－1/5 ] X Y Z A 4, 4 1, 1 3, 3 B X Y Z A 3, 4 2, 1 4, 3 B Game3 [ b(A)=0, b(B)=－1/3 ] X Y Z A 4, 4 1, 1 2, 3 B 3, 1 2, 4 4, 3

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

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.

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

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

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

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

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

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

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

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

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.

10. Refinements of Equilibria (2) Game1 Deviation　(aa,ZZ)⇒(ab,XY) ⇒(ab,XY) Separating equilibria are only recurrent set. X Y Z A 4, 4 1, 1 3, 3 B

11. Refinements of Equilibria (3) Game2 Deviation　(ab,XY) ⇒(bb,ZZ) ⇒(bb,ZZ) Pooling babbling equilibria are only recurrent set. X Y Z A 3, 4 2, 1 4, 3 B

12. Refinements of Equilibria (4) Game3 (bb,ZZ) ⇒(ab,XY) ⇒(aa,ZZ) ⇒(aa,ZZ) Though pooling babbling equilibria are only recurrent set, deviation to separating equilibria may occur. X Y Z A 4, 4 1, 1 2, 3 B 3, 1 2, 4 4, 3

Section 2. Experiments and Bounded Rationality

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.

14. Hypotheses Hypothesis 1 Hypothesis 2 Hypothesis 3 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.

15. Predictions and initial results Sequential equilibria prediction Equilibrium refinements Experimental results Game1 Separating Babbling Game2 Game3 ???

16. Initial Results Game 1 2 3 Separating 25 (96%) 20 (77%) Babbling Session1, Lottery Game 1 2 3 Separating 25 (96%) 20 (77%) Babbling ( 4%) 10 (38%) Others ( 0%) 5 (19%) 16 (62%) Total 26

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.

18. New Design (2) 1-direct 13 1, 2, 3 1-lottery 2 3 26 1, 3 4 Session # of subjects Game Labelling Learning 1-direct 13 1, 2, 3 　one shot 1-lottery 　　2 Change 　　3 26 1, 3 repetition 　　4

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.

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

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.

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.

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

24. QRE in Cheap Talk Games (3)

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

26. AQRE for Sender (1)

27. AQRE for Sender (2)

28. AQRE for Sender (3)

29. AQRE for Receiver (1)

30. AQRE for Receiver (2)

31. AQRE for Receiver (3)

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

33. NNM-SE NNM-SE Convex combination of separating equilibria σwith probabilityγ and uniform distribution μwith 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

34. MIX-SE MIX-SE Convex combination of separating equilibria σwith probabilityγ and QRE correspondes to mixed strategy babbling equilibrium μwith 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

35. POOL POOL Convex combination of pooling babbling equilibria σwith probabilityγ and uniform distribution μwith 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

36. POOL-SE POOL-SE Convex combination of pooling babbling equilibria σwith probabilityγ (sender) or separating equilibria σwith probabilityγ (receiver) and uniform distribution μwith 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

37. Estimation results Session Game 1 Game 2 Game 3 1-direct MIX-SE (γ=0.92) (γ= 0.60) (γ= 0.62) 1-lottery AQRE-SE [λ=3.22] [λ= 2.67] POOL-SE (γ= 0.43) 2 [λ= 1.11] [λ= 1.76] (γ= 0.12) 3 (γ= 0.85) POOL (γ= 0.39) 4 (γ= 0.94) (γ= 0.33)

38. Fact 1 Separating equilibria were observed frequently in Game1 and 2. Coordination via communication works well. But equilibrium refinement theory predicts pooling babbling equilibria in Game 2.

39. Fact 2 Sender used pooling babbling equilibria, but receiver used pseudo separating equilibria in Game 3. Receiver tries to read meanings from sender’s message. But separating equilibrium is not equilibrium.

40. Conclusions There is no theory that can explain whole experimental results. Need for new theory… Why cannot the receiver ignore the sender’s message? Trust? Theory of Mind?

References Cho, I.-K. and D. Kreps (1987): “Signaling Games and Stable Equilibria,” Quarterly Journal of Economics, 102, 179-221 Crawford, V. and J Sobel (1982): “Strategic Information Transmission,” Econometrica, 50, 1431-1451 Farrell, J. (1993): “Meaning and Credibility in Cheap-Talk Games,” Games and Economic Behavior, 5, 514-531 McKelvey, R. D. and T. R. Palfrey (1995): “A Statistical Theory of Equilibrium in Games,” Japanese Economic Review, 47, 186-209 Rabin, M. and J. Sobel (1996): “Deviations, Dynamics, and Equilibrium Refinments,” Journal of Economic Theory, 68, 1-25