francesco feri (innsbruck) ma mel é ndez (m á laga) giovanni ponti (ua-unife) fernando vega (iue)
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Error Cascades in Positional Learning An Experiment on the Chinos Game. Francesco Feri (Innsbruck) MA Mel é ndez (M á laga) Giovanni Ponti (UA-UniFE) Fernando Vega (IUE). 2007ESA - LuissRM - 30/6/07. Perfectly observed. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
Francesco Feri (Innsbruck)MA Meléndez (Málaga)Giovanni Ponti (UA-UniFE)Fernando Vega (IUE)
2007ESA - LuissRM - 30/6/07
Error Cascades in Positional Learning
An Experiment on the Chinos Game
Error Cascades in Observational Learning
Motivation
Situations where agents have to take public decisions in sequence, along which1. Actions2. Identities Perfectly observed
Private valuable information is (may be) revealed through actions
– Financial markets– Technological adoptions– Firms’ business strategies (uncertain market conditions)
Observational (“Positional”) Learning
Error Cascades in Observational Learning
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 Game Çelen and Kariv (2001) Çelen and Kariv (2003)
Chinos’ Game Pastor Abia et al. (2002) Feri et al. (2006)
Error Cascades in Observational Learning
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
Error Cascades in Observational Learning
The “Chinos’ Game”: Game-Form (2-players)
Error Cascades in Observational Learning
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 player’s guess, any subsequent player can
infer exactly the number of coins in the predecessors’ hands.
Error Cascades in Observational Learning
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 I1=s1
I2 S x GI2=(s2, g1)
I3 S x G2 I3=(s3, g1, g2)
PBE: revelation– g1 = s1 + 2– g2 = g1 + s2 - 1 – g3 = g2 + s3 - 1
Error Cascades in Observational Learning
“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
Error Cascades in Observational Learning
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(12/3=4)=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.
Error Cascades in Observational Learning
Descriptive results: Outcomes
Player Right guesses
1 40.5% (56)
2 50.3% (75)
3 61.1% (100)
Frequency of right guesses increases with player position
Difference between theoretical and actual frequences also increases with player position
Error Cascades in Observational Learning
Descriptive results: Behavior (player 1)
Behavioral strategies follow expected payoffs
Better play when s1=0 (???)
Error Cascades in Observational Learning
Descriptive results: Behavior (Player 2)
Adherence with equilibrium much higher when g1=3
Error Cascades in Observational Learning
Descriptive results: Behavior (Player 3)
Adherence with equilibrium much higher when g1=3
Error Cascades in Observational Learning
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
Error Cascades in Observational Learning
Towards a theory of “error cascades”
Error Cascades in Observational Learning
“… 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
Notional learning
Towards a theory of “error cascades”
Error Cascades in Observational Learning
Players are learning optimally if they play a best response to their predecessors’ strategies (that they can infer by past experience)
Optimal learning
Towards a theory of “error cascades”
Error Cascades in Observational Learning
Thetas & betas: Player 2
Error Cascades in Observational Learning
Thetas & betas: Player 3
Error Cascades in Observational Learning
Error cascades in the Chinos Game
Error Cascades in Observational Learning
Error cascades in the Chinos Game
Error Cascades in Observational Learning
Error cascades in the Chinos Game
Error Cascades in Observational Learning
(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 Palfrey’s [1995] Quantal Response Equilibrium
The QRE approach is applied to the “Agent Normal Form” (McKelvey & Palfrey, EE 1998)
Error Cascades in Observational Learning
(Logit) Quantal Response Equilibrium (QRE)
In a (A)QRE, (full support) behavioral strategies follow expected payoffs:
It is essentially a QRE IN BEHAVIORAL STRATEGIES
Error Cascades in Observational Learning
Estimating individual QRE noise parameters (I)
Individual (static) estimates Common beliefs assumed All (24) observations considered
Error Cascades in Observational Learning
Player 1’s QRE
Error Cascades in Observational Learning
Player 2’s QRE
Error Cascades in Observational Learning
Player 2’s QRE
Prop. 4.1
Prop. 4.2
Prop. 5
Error Cascades in Observational Learning
Error cascades along the equibrium path (g1=2 & s2=1)
2
1
2(3)
2(2)
Error Cascades in Observational Learning
Error cascades along the equibrium path (g1=3 & s2=1)
2
1
2(3)
2(2)
Error Cascades in Observational Learning
Error cascades on the equibrium path: Player 2 (s2=1)
Error Cascades in Observational Learning
Error (QRE) cascades: Player 3
Error Cascades in Observational Learning
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)