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  Proportion of Observations at each Terminal Node, f j, (j=1-5 for four-move and j=1-7 for six-move games)  Implied Take Probability at Each Stage, p j (j=1-4 for four-move and j=1-6 for six move games)  Iterative Dominance Predictions  f j = 1.0 for j=1 and 0 otherwise  p j = 1.0 for all j.

Experimental Design

Basic Results: f j

Basic Results: p j

Basic Results: Cumulative Outcome Frequencies

Basic Results: Early versus Later Rounds

Summary of Basic Results  All outcomes occur with strictly positive probability.  p j is higher at higher j.  Behaviors become “more rational” in later rounds.  p j is higher in 4-move game than in 6-move game for the same j.  For a given j, p n-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 p i > p j whenever i > j.

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-  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 q i (drawn from beta distribution (  ))  Each player plays the game as if it were common knowledge that the opponent had the same belief.

Equilibrium with Errors in Actions

The Likelihood Function  A player draws a belief q  For 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:

The Likelihood Function  If Q is the true proportion for the fraction of selfish players, then the likelihood becomes:  The Likelihood function is:

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