Enrica Carbone UniBA

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Positional Learning with Noise

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

ESA-Luiss30/6/2007
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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?

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Related literature
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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
  (ps11.75)

• Formally multistage game with incomplete
  information

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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 players guess, any
  subsequent player can infer exactly the number of
  coins in the predecessors hands.

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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
• I1s1
• I2(s2, g1)
• I3(s3, g1, g2)

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WPBE for the Chinos Game
Player 1s expectations

• P(s2 s3 ) 0(1-p)20.0625
• P(s2 s3 ) 12p(1-p)0.375
• P(s2 s3 ) 2 p20.5625

• P(s3 0)(1-p)0.25
• P(s3 1) p0.75

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CP 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(24/38)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.

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The Computer Interface
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Descriptive results Outcomes

• Feri et al. (2006)

• Carbone and Ponti (2007)

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Descriptive results II Behavior (Player 1)

• Feri et al. (2006)

• Carbone and Ponti (2007)

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Descriptive results II Behavior (player 1)
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Descriptive results II Behavior (Player 2)

• Feri et al. (2006)

• Carbone and Ponti (2007)

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Descriptive results II Behavior (Player 2)

• Carbone and Ponti (2007) Player 1

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(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

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QRE when N2

• In the (modified) Chinos Game, Player 1s
  expected payoff does not depend on Player 2s
  mixed strategy

• As for h10, the corresponding QRE is as follows

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Results 1 best-replies (for Player 1s
information set)

• Let BR1 be 1 if player 1 is playing the best
  response and 0 otherwise.

• H0 alpha_h_10alpha_h_11 REJECTED (p.0202)

• Higher expected payoff when s10 (a.4 vs. a.36)

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Results br2f(alpha1,alpha2) (PRELIMINARY)
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Conclusions

• Preliminary results
• The introduction of a makes peoples 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)