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

1
Instability of Babbling Equilibria in Cheap Talk
GamesSome 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
• Senders type
• sender message
• receivers action
• senders payoff
• receivers payoff
• coincidence of interests

perfect
partial
5
3. Cheap Talk Games (3)
Sender
Receiver
Receiver
Sender
6
3. Cheap Talk Games (3)
X
X
Sender
Y
A
Y
a
b
Z
Z
0.5
N
Receiver
Receiver
X
X
0.5
Y
Y
a
b
Z
B
Z
Sender
7
3. Cheap Talk Games (3)
X
X
Sender
Y
A
Y
a
b
Z
Z
0.5
N
Receiver
Receiver
X
X
0.5
Y
Y
a
b
Z
B
Z
Sender
8
3. Cheap Talk Games (3)
X
X
Sender
Y
A
Y
a
b
Z
Z
0.5
N
Receiver
Receiver
X
X
0.5
Y
Y
a
b
Z
B
Z
Sender
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3. Cheap Talk Games (3)
X
X
Sender
Y
A
Y
a
b
1, 1
1, 1
Z
Z
0.5
N
Receiver
Receiver
X
X
0.5
Y
Y
a
b
Z
B
Z
Sender
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4. Cheap Talk Games (4)
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 1, 1 4, 4 3, 3
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
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5. Cheap Talk Games (5)
Game2
Game1
0
0
Game3
0
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6. Sequential Equilibria (1)

• Separating equilibria
• The sender reveals her type, then the receiver
  chooses an action according to the senders type.
• Babbling equilibria
• The receiver ignores the senders message, then
  chooses an action which maximizes expected payoff
  with the belief based on prior probability of the
  senders type.
• There are pooling and mixed strategy babbling
  equilibria.

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

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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 1, 1 4, 4 3, 3
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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 2, 1 3, 4 4, 3
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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
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Section 2.Experimentsand Bounded Rationality
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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.

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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.

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15. Predictions and initial results
Sequential equilibria prediction Equilibrium refinements prediction Experimental results
Game1 Separating Babbling Separating Separating
Game2 Separating Babbling Babbling Separating
Game3 Babbling Babbling ???
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16. Initial Results
Session1, Lottery
Game 1 2 3
Separating 25 (96) 20 (77)
Babbling 1 ( 4) 1 ( 4) 10 (38)
Others 0 ( 0) 5 (19) 16 (62)
Total 26 26 26
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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.
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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
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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.
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20. Quantal Response Equilibria

• Consider best responses under stochastic error.
• (cf. McFaddens random utility model)
• Prob.i chooses strategy j
• Expected payoff when i chooses j
• Fixed points of the equations below are QRE

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21. Properties of QRE

• ?represents the degree of rationality
• When?0, random choice
• ??8, Nash equilibria (sequential equilibria)
• QRE exists.
• QRE is a refinement of equilibrium.

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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.

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23. QRE in Cheap Talk Games (2)
X
X
s1
p
1-p
A
a
b
Y
Y
s2
Z
Z
0.5
s3
N
X
X
s1
0.5
Y
Y
s2
a
b
Z
B
Z
s3
q
1-q
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24. QRE in Cheap Talk Games (3)
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25. Estimation procedures

• Maximum likelihood method
• Calculate a fixed point of QRE for given?, then
  evaluate log likelihood function (LL). Iterate
  this process and find a?that 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-fitpseudo

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26. AQRE for Sender (1)
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27. AQRE for Sender (2)
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28. AQRE for Sender (3)
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29. AQRE for Receiver (1)
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30. AQRE for Receiver (2)
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31. AQRE for Receiver (3)
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32. Other estimated models

• Model based on equilibria
• NNM-SE (noisy Nash model)
• MIX-SE
• POOL
• POOL-SE

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33. NNM-SE

• NNM-SE
• Convex combination of separating equilibria swith
  probability? and uniform distribution µwith
  probablity 1-?
• P?s(1-?)µ
• Find a?that maximizes log likelihood using grid
  search method.
• Confidence intervals is calculated by bootstrap
  method.
• Model selection AIC, Goodness-of-fitpseudo R2

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34. MIX-SE

• MIX-SE
• Convex combination of separating equilibria swith
  probability? and QRE correspondes to mixed
  strategy babbling equilibrium µwith probablity
  1-?
• p?s(1-?)µ
• Find a?that maximizes log likelihood using grid
  search method.
• Confidence intervals is calculated by bootstrap
  method.
• Model selection AIC, Goodness-of-fitpseudo R2

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35. POOL

• POOL
• Convex combination of pooling babbling equilibria
  swith probability? and uniform distribution µwith
  probablity 1-?
• p?s(1-?)µ
• Find a?that maximizes log likelihood using grid
  search method.
• Confidence intervals is calculated by bootstrap
  method.
• Model selection AIC, Goodness-of-fitpseudo R2

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36. POOL-SE

• POOL-SE
• Convex combination of pooling babbling equilibria
  swith probability? (sender) or separating
  equilibria swith probability? (receiver) and
  uniform distribution µwith probablity 1-?
• p?s(1-?)µ
• Find a?that maximizes log likelihood using grid
  search method.
• Confidence intervals is calculated by bootstrap
  method.
• Model selection AIC, Goodness-of-fitpseudo R2

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37. Estimation results
Session Game 1 Game 2 Game 3
1-direct MIX-SE (?0.92) MIX-SE (? 0.60) MIX-SE (? 0.62)
1-lottery AQRE-SE ?3.22 AQRE-SE ? 2.67 POOL-SE (? 0.43)
2 AQRE-SE ? 1.11 AQRE-SE ? 1.76 POOL-SE (? 0.12)
3 MIX-SE (? 0.85) POOL (? 0.39)
4 MIX-SE (? 0.94) POOL (? 0.33)
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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.
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39. Fact 2
Sender used pooling babbling equilibria, but
receiver used pseudo separating equilibria in
Game 3.
Receiver tries to read meanings from senders
message.
But separating equilibrium is not equilibrium.
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40. Conclusions

• There is no theory that can explain whole
  experimental results.
• Need for new theory
• Why cannot the receiver ignore the senders
  message?
• Trust?
• Theory of Mind?

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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