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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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**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**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**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**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**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**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)**Game1 [ b(A)=b(B)=0 ] Game2 [ b(A)=1/5, b(B)=－1/5 ] Game3 [ b(A)=0, b(B)=－1/3 ]**b(t)**b(t) b(t) t t t 5. Cheap Talk Games (5) Game2 Game1 0 0 Game3 0**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.**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**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**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**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**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**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.**11. Refinements of Equilibria (3)**• Game2 • Deviation(ab,XY) ⇒(bb,ZZ) ⇒(bb,ZZ) • Pooling babbling equilibria are only recurrent set.**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.**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.**16. Initial Results**Session1, Lottery**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.**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.**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**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**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 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**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**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**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

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