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Outline. In-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 Predictions. - PowerPoint PPT Presentation
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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
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
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