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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.
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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)
Variables and Predictions • Proportion 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 otherwise • pj = 1.0 for all j.
Summary of Basic Results • All outcomes occur with strictly positive probability. • pjis 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 Prediction • For 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 Solutions • For any 1-q, there is at least one outcome with 0 or close to 0 probability of occurrence. • Possibility of error in actions • TAKE with probability (1-et) p* and makes a random move (50-50 chance of PASS and TAKE) with probability et. • Learning: • Heterogeneity in beliefs (errors in beliefs) • Q (true) versus qi(drawn from beta distribution (a, b)) • Each player plays the game as if it were common knowledge that the opponent had the same belief.
The Likelihood Function • A player draws a belief q • For every t and every et, 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:
The Likelihood Function • If Q is the true proportion for the fraction of selfish players, then the likelihood becomes: • The Likelihood function is:
