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 payoffirrelevant 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
t T
Sender’s type
sender’ message m  M
receiver’s action a  A
sender’s payoff uS  (a  (t  b(t ))2
receiver’s payoff uR  (a  t ) 2
 0
• coincidence of interests b(t )
 0
perfect
partial
3. Cheap Talk Games (3)
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
3. Cheap Talk Games (3)
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
3. Cheap Talk Games (3)
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
3. Cheap Talk Games (3)
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
3. Cheap Talk Games (3)
X
1, 1
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
1, 1
4. Cheap Talk Games (4)
Game1 [ b(A)=b(B)=0 ]
X
Y
Z
A
4, 4
1, 1
3, 3
B
1, 1
4, 4
3, 3
Game2 [ b(A)=1/5, b(B)=－1/5 ]
X
Y
Z
A
3, 4
2, 1
4, 3
B
2, 1
3, 4
4, 3
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)
b(t
)
b(t
)
Game1
0
Game2
0
t
uS  (a  (t  b(t ))2
uR  (a  t )2
t
b(t
)
Game3
0
t
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
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
7. Separating equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
b
Z
7. Separating equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
7. Separating equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
8. Pooling babbling equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
8. Pooling babbling equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
8. Pooling babbling equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
8. Pooling babbling equilibria
X
Y
a
Z
Sender
A
X
b
Z
0.5
Receiver
N
Receiver
X
X
0.5
Y
Z
Y
a
B
Sender
Y
b
Z
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
X
Y
Z
A
4, 4
1, 1
3, 3
B
1, 1
4, 4
3, 3
– Deviation (aa,ZZ)⇒(ab,XY) ⇒(ab,XY)
– Separating equilibria are only recurrent set.
11. Refinements of Equilibria (3)
• Game2
X
Y
Z
A
3, 4
2, 1
4, 3
B
2, 1
3, 4
4, 3
– Deviation (ab,XY) ⇒(bb,ZZ) ⇒(bb,ZZ)
– Pooling babbling equilibria are only recurrent
set.
12. Refinements of Equilibria (4)
• Game3
X
Y
Z
A
4, 4
1, 1
2, 3
B
3, 1
2, 4
4, 3
– (bb,ZZ) ⇒(ab,XY) ⇒(aa,ZZ) ⇒(aa,ZZ)
– Though pooling babbling equilibria are only
recurrent set, deviation to separating equilibria
may occur.
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
– 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
Game1
Game2
Game3
Sequential
equilibria
prediction
Separating
Babbling
Separating
Babbling
Babbling
Equilibrium
refinements
prediction
Separating
Experimental
results
Babbling
Separating
Babbling
???
Separating
16. Initial Results
Session1, Lottery
Game
Separating
Babbling
Others
Total
1
2
3
25
(96%)
1
( 4%)
0
( 0%)
26
20
(77%)
1
( 4%)
5
(19%)
26
10
(38%)
16
(62%)
26
17. New Design (1)
Deviation from equilibrium or refinement
prediction is severe in Game 2 and 3.
Permuting labels
Learning
Label on each strategy may
induces separating
equilibria in Game 2 and 3.
Repetition of same game
may increase equilibrium
plays.
18. New Design (2)
Session
# of
subjects
Game
Labelling
Learning
1-direct
13
1, 2, 3
one shot
1-lottery
13
1, 2, 3
one shot
2
13
1, 2, 3
Change
one shot
3
26
1, 3
Change
repetition
4
26
1, 3
Change
repetition
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)
j
• Prob.{i chooses strategy j} = pi
j
• Expected payoff when i chooses j:  i
• Fixed points of the equations
below are
j
  i
QRE
e
j
pi 

k
e
  ik
i, j
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
Y r2
p
a
Z r3
A
1-p
b
0.5
s1 X
s2 Y
s3 Z
N
X r1
Y r2
Z
r3
s1 X
0.5
a
q
B
b
1-q
s2 Y
Z
s3
24. QRE in Cheap Talk Games (3)
p
1
1 e
 ( s1  2 s 2  r1  2 r2 )
1
r1 
1 e
3 p  3 q

pq
e
 p2q

pq
1
s1 
1 e
3 p 3 q

2 p  q
e
1 p  2 q

2 p  q
q
1
1  e  ( 2 s1  s2  2 r1  r2 )
1
r2 
1 e

3 p 3 q
pq
e

2 p q
pq
1
s2 
1 e

3 p  3 q
2 p q
e
1 2 p  q
2 p q

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： AIC  2LL(ˆ)  2k
• Goodness-of-fit：pseudo R2  1  LL(ˆ) LL(0)
26. AQRE for Sender (1)
probability
Game1, separating
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
p
q
0
1
2
3
4
5
lambda
6
7
8
9
10
27. AQRE for Sender (2)
probability
Game2, separating
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
p
q
0
1
2
3
4
5
lambda
6
7
8
9
10
28. AQRE for Sender (3)
probability
Game3, babbling
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
p
q
0
1
2
3
4
5
lambda
6
7
8
9
10
29. AQRE for Receiver (1)
probablity
Game1, separating
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
r1
r3
0
1
2
3
4
5
lambda
6
7
8
9
10
30. AQRE for Receiver (2)
probability
Game2, separating
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
r1
r3
0
1
2
3
4
5
lambda
6
7
8
9
10
31. AQRE for Receiver (3)
probability
Game3, babbling
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
r1
r3
0
1
2
3
4
5
lambda
6
7
8
9
10
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)
AQRE-SE
[λ=3.22]
AQRE-SE
[λ= 1.11]
MIX-SE
(γ= 0.85)
MIX-SE
(γ= 0.94)
MIX-SE
(γ= 0.60)
AQRE-SE
[λ= 2.67]
AQRE-SE
[λ= 1.76]
MIX-SE
(γ= 0.62)
POOL-SE
(γ= 0.43)
POOL-SE
(γ= 0.12)
POOL
(γ= 0.39)
POOL
(γ= 0.33)
1-lottery
2
3
4
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

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