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Enrica Carbone (UniBA) Giovanni Ponti (UA-UniFE)

Positional Learning with Noise. Enrica Carbone (UniBA) Giovanni Ponti (UA-UniFE). ESA-Luiss–30/6/2007. Motivation. We deal with a standard model of positional learning Like in a standard signaling game, the public message reveals players’ private information on the true state of the world

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Enrica Carbone (UniBA) Giovanni Ponti (UA-UniFE)

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  1. Positional Learning with Noise Enrica Carbone (UniBA) Giovanni Ponti (UA-UniFE) ESA-Luiss–30/6/2007

  2. Motivation • We deal with a standard model of positional learning • Like in a standard signaling game, the public message reveals players’ private information on the true state of the world • Unlike a standard signaling game, players have no incentive to manipulate their public message, since they all win a fixed price if they are able to guess the true state of the world • We modify the basic protocol by targeting a player in the sequence. This player will win with some probability (known in advance to all players) if she guess right • To which extent this will affect her behavior? • To which extent this will affect her followers’ behavior?

  3. Related literature

  4. Feri et al. (2006): the “Chinos’ Game” • Each player hides in her hands a # of coins • In a pre-specified order players guess on the total # of coins in the hands of all the players • Information of a player Her own # of coins Predecessors’ guesses + • Our setup → simplified version: • 3 players • # of coins in the hands of a player: either 0 or 1 • Outcome of an exogenous iid random mechanism (p[s1=1]=.75) • Formally: multistage game with incomplete information

  5. Outcome function • All players who guess correctly win a prize: • Players’ incentives do not conflict • Unique Perfect Bayesian Equilibrium: Revelation • Perfect signal of the private information • After observing each player’s guess, any subsequent player can infer exactly the number of coins in the predecessors’ hands.

  6. WPBE for the Chinos Game • Players:i  N  {1, 2, 3} • Signal (coins): si S  {0, 1} • Random mechanism: P(si = 1) = ¾ (i.i.d.) • Guesses: gi G  {0, 1, 2, 3} • Information sets: I1=s1 I2=(s2, g1) I3=(s3, g1, g2)

  7. Player 2’s expectations • M(1)=1 • M(2)=2 • PBE: equilibrium guesses • g1 = 2 + s1 • g2 = (g1 - 1)+ s2 • g3 = (g2 - 1)+ s3 WPBE for the Chinos Game Player 1’s expectations • P(s2 + s3 ) = 0=(1-p)2=0.0625 • P(s2 + s3 ) = 1=2p(1-p)=0.375 • P(s2 + s3 ) = 2= p2=0.5625 • P(s3 = 0)=(1-p)=0.25 • P(s3 =1) = p=0.75

  8. C&P: Experimental design • Sessions: 2 held in March 2007 • Subjects: 48 students (UA), 24 per session (1 and 1/2 hour approx., € 19 average earning) • Software: z-Tree (Fischbacher, 2007) • Matching: Fixed group, fixed player positions • Independent observations: 2x(24/3=8)=16 • Information ex ante: identity of the “ELEGIDO” and associated a (probability of winning if guessing right) • Information ex post: after each round, agents where informed about everything (signal choices, outcome of the random shocks) • Random events: selection of the “ELEGIDO”, deterministic (and aggregate), everything else iid.

  9. Descriptive results: Outcomes • Feri et al. (2006): • Carbone and Ponti (2007):

  10. Descriptive results II: Behavior (Player 1) • Feri et al. (2006): • Carbone and Ponti (2007):

  11. Descriptive results II: Behavior (player 1)

  12. Descriptive results II: Behavior (Player 2) • Feri et al. (2006): • Carbone and Ponti (2007):

  13. Descriptive results II: Behavior (Player 2) • Carbone and Ponti (2007): Player 1

  14. (Logit) Quantal Response Equilibrium (QRE) • McKelvey & Palfrey (GEB) propose a notion of equilibrium with noise • In a QRE, each pure strategy is selected with some positive probability, with this probability increasing in expected payoff:

  15. QRE when N=2 • In the (modified) Chinos’ Game, Player 1’s expected payoff does not depend on Player 2’s mixed strategy: • As for h1=0, the corresponding QRE is as follows:

  16. Both alpha_h_10 and alpha_h_11 are significant Results 1: best-replies (for Player 1’s information set) • Let BR1 be =1 if player 1 is playing the best response and 0 otherwise. • H0: alpha_h_10=alpha_h_11: REJECTED (p=.0202) • Higher expected payoff when s1=0 (a.4 vs. a.36)

  17. When g1=3 we cannot expect dependency of br2 on alpha1 • What about the case when g1=2? Results: br2=f(alpha1,alpha2) (PRELIMINARY)

  18. Conclusions Preliminary results: • The introduction of α makes people’s choices less precise, both the first player and the other players play less the best strategy. • Error cascades persist in our noisy environment • Future research: the following players play less the best strategy • Introducing heterogeneity through a (using questionnaire answers)

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