francesco feri (innsbruck) ma mel é ndez (m á laga) giovanni ponti (ua-unife) fernando vega (iue)

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)