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

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 1. To which extent this will affect her behavior? 2. To which extent this will affect her followers’ behavior?

Positional Learning with Noise Related literature ModelTheoryExperiment Info Cascades Mod. 1Bikhchandani et al, (1992)Anderson and Holt (1997) Info Cascades Mod. 2Banerjee (1992)Alsopp & Hey (2001) Guessing Sign SumÇelen and Kariv (2001)Çelen and Kariv (2003) Chinos’ GamePastor Abia et al. (2002)Feri et al. (2006)

Positional Learning with Noise 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[s 1 =1]=.75) Formally: multistage game with incomplete information

Positional Learning with Noise 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.

Positional Learning with Noise WPBE for the Chinos Game Players: i  N  {1, 2, 3} Signal (coins): s i  S  {0, 1} Random mechanism: P(s i = 1) = ¾ (i.i.d.) Guesses: g i  G  {0, 1, 2, 3} Information sets: I 1 =s 1 I 2 =(s 2, g 1 ) I 3 =(s 3, g 1, g 2 )

Positional Learning with Noise WPBE for the Chinos Game M(2)=2 P(s 2 + s 3 ) = 0=(1-p) 2 = P(s 2 + s 3 ) = 1=2p(1-p)=0.375 P(s 2 + s 3 ) = 2= p 2 = P(s 3 = 0)=(1-p)=0.25 P(s 3 =1) = p=0.75 Player 1’s expectationsPlayer 2’s expectations PBE: equilibrium guesses – g 1 = 2 + s 1 – g 2 = (g 1 - 1) + s 2 – g 3 = (g 2 - 1) + s 3 M(1)=1

Positional Learning with Noise 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  (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.

Positional Learning with Noise The Computer Interface

Positional Learning with Noise Descriptive results: Outcomes PlayerRight guesses 140.5% (56) 250.3% (75) 361.1% (100) Feri et al. (2006): Carbone and Ponti (2007): PlayerRight guesses 143.7% (56) 254.5% (75) 358.9% (100)

Positional Learning with Noise Descriptive results II: Behavior (Player 1) Info. set: Signal 1 Guess 1%EQ ,9326,8572, % 1 09,6237,9852,4 Feri et al. (2006): Carbone and Ponti (2007):

Positional Learning with Noise Descriptive results II: Behavior (player 1)

Positional Learning with Noise Descriptive results II: Behavior (Player 2) Info. Set pl2Guess 2% Eq. Play Guess1Signal ,2260, % 17,5557,5534, ,6975,863, Feri et al. (2006): Carbone and Ponti (2007):

Positional Learning with Noise Descriptive results II: Behavior (Player 2) Carbone and Ponti (2007): Player 1

Positional Learning with Noise (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:

Positional Learning with Noise 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 h 1 =0, the corresponding QRE is as follows:

Positional Learning with Noise Results 1: best-replies (for Player 1’s information set) Higher expected payoff when s1=0 (a.4 vs. a.36) 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) Both alpha_h_10 and alpha_h_11 are significant

Positional Learning with Noise Results: br2=f(alpha1,alpha2) (PRELIMINARY) When g1=3 we cannot expect dependency of br2 on alpha1 What about the case when g1=2?

Positional Learning with Noise 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  (using questionnaire answers)