OutlineIn-Class Experiment on Centipede Game

Test of Iterative Dominance Principle I: McKelvey and Palfrey (1992)

Test of Iterative Dominance Principle II: Ho, Camerer, and Weigelt (1988)

Four-move Centipede Game

Six-move Centipede Game

Variables and PredictionsProportion of Observations at each Terminal Node,

fj,(j=1-5 for four-move and j=1-7 for six-move games)

Implied Take Probability at Each Stage, pj (j=1-4 for four-move and j=1-6 for six move games)

Iterative Dominance Predictions fj = 1.0 for j=1 and 0 otherwisepj = 1.0 for all j.

Experimental Design

Basic Results: fj

Basic Results: pj

Basic Results: Cumulative Outcome Frequencies

Basic Results:Early versus Later Rounds

Summary of Basic Results All outcomes occur with strictly positive probability.

pj is higher at higher j.

Behaviors become “more rational” in later rounds.

pj is higher in 4-move game than in 6-move game for the same j.

For a given j, pn-j in a n-move game increases with n.

There are 9 players who chose PASS at every opportunity.

Basic Model“Gang of Four” (Kreps, Milgrom, Roberts, and

Wilson, JET, 1982) Story

Complete Incomplete information game where the prob. of a selfish individual equals q and the prob. of an altruist is 1-q. This is common knowledge.

Selfish individuals have an incentive to “mimic” the altruists by choosing to PASS in the earlier stages.

Properties of PredictionFor any q, Blue chooses TAKE with probability 1

on its last move.If 1-q > 1/7, both Red and Blue always choose

PASS, except on the last move, when Blue chooses TAKE.

If 0 < 1-q < 1/7, the equilibrium involves mixed strategies.

If q=1, then both Red and Blue always choose TAKE.

For 1-q> 1/49 in the 4-move game and 1-q > 1/243, the solution satisfies pi > pj whenever i > j.

Proportions of Outcomes as a Function of the Level of Altruism

Proportions of Outcomes as a Function of the Level of Altruism

Problems and SolutionsFor any 1-q, there is at least one outcome with 0

or close to 0 probability of occurrence.

Possibility of error in actionsTAKE with probability (1-t) p* and makes a random

move (50-50 chance of PASS and TAKE) with probability t.

Learning:

Heterogeneity in beliefs (errors in beliefs)Q (true) versus qi (drawn from beta distribution ())Each player plays the game as if it were common

knowledge that the opponent had the same belief.

)1( tt e

Equilibrium with Errors in Actions

Equilibrium with Errors in Actions

The Likelihood Function

2),,(

),,()1(2

),,(

tta

tstt

ts

vqP

vqpvqP

A player draws a belief qFor every t and every t, and for each of the player’s decision nodes, v, we have

the equilibrium prob. of TAKE given by:

Player i’s prob. of choosing TAKE given q:

t

sti

si

vts

sti

qq

vqPq

),,(),,(

),,(),,(

The Likelihood FunctionIf Q is the true proportion for the fraction of selfish players, then the likelihood

becomes:

The Likelihood function is:

)],,,,(log[),,,,(

),;(),,,(),,,,(

),,()1(),,(),,,(

1

1

0

QsQL

dqqBqQQs

qQqQqQ

ii

ii

ai

sii

Maximum Likelihood Estimates

Estimated Distribution of Beliefs

Tests of Nested Models

Differences in Noisy Actions Across Treatments

Predicted Versus Actual Choices

Predicted versus Actual Choices

Summary