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

ESA-Luiss–30/6/2007

Positional Learning with Noise

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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

Model Theory Experiment

Info Cascades Mod. 1 Bikhchandani et al, (1992) Anderson and Holt (1997)

Info Cascades Mod. 2 Banerjee (1992) Alsopp & Hey (2001)

Guessing Sign Sum Çelen and Kariv (2001) Çelen and Kariv (2003)

Chinos’ Game Pastor 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[s1=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): 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)

Positional Learning with Noise

WPBE for the Chinos Game

•M(2)=2

•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

Player 1’s expectations Player 2’s expectations

PBE: equilibrium guesses– g1 = 2 + s1

– g2 = (g1 - 1) + s2 – g3 = (g2 - 1) + s3

•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

Descriptive results: Outcomes

Player Right guesses

1 40.5% (56)

2 50.3% (75)

3 61.1% (100)

Feri et al. (2006):

Carbone and Ponti (2007):

Player Right guesses

1 43.7% (56)

2 54.5% (75)

3 58.9% (100)

Positional Learning with Noise

Descriptive results II: Behavior (Player 1)

Info. set:

Signal 1 Guess 1 %EQ

0 1 2 3

0 0,93 26,85 72,22 0

59 % 1 0 9,62 37,98 52,4

Feri et al. (2006):

Carbone and Ponti (2007):

Fichas1 0 1 2 30 4 19 57 8

% 4.55 21.59 64.77 9.091 8 26 110 152

% 2.70 8.78 37.16 51.35Total 12 45 167 160

3.13 11.72 43.49 41.67

guess1

54%

Positional Learning with Noise

Descriptive results II: Behavior (player 1)

Fichas1 0 1 2 30 4 19 57 8

% 4.55 21.59 64.77 9.091 8 26 110 152

% 2.70 8.78 37.16 51.35Total 12 45 167 160

3.13 11.72 43.49 41.67

guess1

54%

Fichas1 0 1 2 30 4 19 57 8

% 33.33 42.22 34.13 5.001 8 26 110 152

% 66.67 57.78 65.87 95.00

guess1

Positional Learning with Noise

Descriptive results II: Behavior (Player 2)

Info. Set pl2 Guess 2 % Eq.

PlayGuess1 Signal2 1 2 3

20 39,22 60,78 0

65 %1 7,55 57,55 34,91

30 20,69 75,86 3,45

1 0 10 90

Feri et al. (2006):

Carbone and Ponti (2007):

guess1 Fichas2 1 2 30 57 72 0

% 44.19 55.81 0.001 18 204 150

% 4.84 54.84 40.320 18 120 3

% 12.77 85.11 2.131 15 51 273

% 4.42 15.04 80.53

2% Eq. Play

66%3

guess2

Positional Learning with Noise

Descriptive results II: Behavior (Player 2)

Carbone and Ponti (2007): Player 1

Fichas1 0 1 2 30 4 19 57 8

% 33.33 42.22 34.13 5.001 8 26 110 152

% 66.67 57.78 65.87 95.00

guess1

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 h1=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)

br1 Coef. Std. Err. z P>z

alpha_h_10 1,425 0,329 4,330 0,000 0,781 2,069alpha_h_11 1,072 0,295 3,640 0,000 0,494 1,650

Round_2 -0,306 1,010 -0,300 0,762 -2,286 1,674Round_3 -0,967 1,027 -0,940 0,347 -2,980 1,047

Round dum._cons -2,278 0,881 -2,580 0,010 -4,006 -0,551

[95% Conf. Interval]

omitted

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)

Fichas1 0 1 2 30 4 19 57 8

% 33.33 42.22 34.13 5.001 8 26 110 152

% 66.67 57.78 65.87 95.00

Total 12 45 167 160100.00 100.00 100.00 100.00

guess1

br2 Coef. Std. Err. z P>z

alpha1 3,502 1,879 1,860 0,062 -1,813 7,185alpha2 2,079 2,731 0,760 0,447 -3,273 7,430

Round_2 -5,301 2,867 -1,850 0,064 -1,092 0,318Round_3 -0,967 1,027 -0,940 0,347 -2,980 1,047

Round dum._cons -2,278 0,881 -2,580 0,010 -4,006 -0,551

[95% Conf. Interval]

omitted

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)