Francesco Feri Innsbruck

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Error Cascades in Positional LearningAn
Experiment on the Chinos Game

• Francesco Feri (Innsbruck)
• MA Meléndez (Málaga)
• Giovanni Ponti (UA-UniFE)
• Fernando Vega (IUE)

2007ESA - LuissRM - 30/6/07
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Motivation

• Situations where agents have to take public
  decisions in sequence, along which
• Actions
• Identities

• Private valuable information is (may be) revealed
  through actions
• Financial markets
• Technological adoptions
• Firms business strategies (uncertain market
  conditions)

• Observational (Positional) Learning

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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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The Chinos Game Game-Form (2-players)
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Outcome function

• All players who guess correctly win a prize
• All Win Game (AWG)
• 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
• I1 ? S I1s1
• I2 ? S x G I2(s2, g1)
• I3 ? S x G2 I3(s3, g1, g2)

• PBE revelation
• g1 s1 2
• g2 g1 s2 - 1
• g3 g2 s3 - 1

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Reasonable beliefs

• (Out-of-equilibrium) beliefs are as such that
  later movers always belief that out-of
  equilibrium guesses are associated with the
  signal that would have yielded the highest
  expected payoff

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Experimental design

• Sessions 4 held in May 2005
• Subjects 48 students (UA), 12 per session (1 1/2
  hour approx., 19 average earning)
• Software z-Tree (Fischbacher, 2007)
• Matching Fixed group, fixed player positions
• Independent observations 4x(12/34)16
• Information ex ante private signal
• Information ex post everything about about
  everything (signals choices) about group
  members
• Random events everything (i.e. signals) iid.

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Descriptive results Outcomes

• Frequency of right guesses increases with player
  position
• Difference between theoretical and actual
  frequences also increases with player position

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Descriptive results Behavior (player 1)

• Behavioral strategies follow expected payoffs
• Better play when s10 (???)

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Descriptive results Behavior (Player 2)

• Adherence with equilibrium much higher when g13

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Descriptive results Behavior (Player 3)

• Adherence with equilibrium much higher when g13

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Towards a theory of error cascades

• ? is a measure how subjects do well from their
  own perspective
• ? is a measure how subjects do well from their
  followers perspective
• This interpretation (may) fall short out of the
  equilibrium path

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Towards a theory of error cascades
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Towards a theory of error cascades
Any other view risk relegating rational
players to the role of the unlucky bridge
expert who usually loses but explains that his
play is correct and would have led to his
winning if only the opponents had played
correctly Binmore (1987)

• Players are learning notionally if they play a
  best-response to the equilibrium strategy of
  their opponent

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Towards a theory of error cascades

• Players are learning optimally if they play a
  best response to their predecessors strategies
  (that they can infer by past experience)

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Thetas betas Player 2
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Thetas betas Player 3
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Error cascades in the Chinos Game
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Error cascades in the Chinos Game
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Error cascades in the Chinos Game
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(A)QRE A Theory of Error Cascades

• The basic question why error cascades?
• Assume that subjects' choices are also affected
  by other (unmodeled) external factors that make
  this process intrinsically noisy
• Why? Complexity of the game, limitation of
  subjects' computational ability, random
  preference shocks, etc
• A classic model of (endogenous) noise McKelvey
  and Palfreys 1995 Quantal Response Equilibrium
• The QRE approach is applied to the Agent Normal
  Form (McKelvey Palfrey, EE 1998)

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(Logit) Quantal Response Equilibrium (QRE)

• In a (A)QRE, (full support) behavioral strategies
  follow expected payoffs

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Estimating individual QRE noise parameters (I)

• Individual (static) estimates
• Common beliefs assumed
• All (24) observations considered

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Player 1s QRE
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Player 2s QRE
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Player 2s QRE
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Error cascades along the equibrium path (g12
s21)
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Error cascades along the equibrium path (g13
s21)
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Error cascades on the equibrium path Player 2
(s21)
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Error (QRE) cascades Player 3
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Further Research Conflicting interest

• Constant sum games
• One and only one player in the group wins the
  prize
• Agents incentives ? Pure conflict

• First win game (FWG)
• Winner ? the player who first guesses correctly
• If no one guess right ? the prize goes to player
  3
• Equilibrium ? revelation (but no repetition
  constraint)

• Last win game (LWG)
• Winner ? the last player who guesses correctly
• If no one guess right ? the prize goes to player
  1
• Equilibrium ? uninformative pooling

• Last, but not least ()
• Positional learning with noise (Carbone and
  Ponti, 2007)