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

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

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

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

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) 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

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

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

14. 7. Separating equilibriaX 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 equilibriaX 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 equilibriaX 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. 8. Pooling babbling equilibriaX Y Z A B N
0.5 a b
Sender
Receiver
Receiver
Sender

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

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

20. 8. Pooling babbling equilibriaX 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. X Y Z A
4, 4
1, 1
3, 3 B

23. 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

24. 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

25. Section 2. Experiments and 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 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.

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

29. 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

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) 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

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. 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

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 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

47. 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

48. 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

49. 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

50. 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)

51. 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.

52. 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.

53. 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?

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