In a beauty contest game where players choose numbers from 0 to 100 and the winner is closest to 2/3 of the average, the Nash equilibrium is 0. In experiments, the average choice is typically around...
A0, confirming Nash equilibrium predictions
B33, consistent with one step of strategic reasoning (level-1 thinking: best-respond to uniform random play at 50)
C50, because players choose randomly
D67, because players multiply the maximum by 2/3
The typical experimental result clusters around 33, with a secondary cluster near 22 (level-2 thinking: best-respond to level-1 at 33). A level-0 player randomizes (mean 50), a level-1 player best-responds to level-0 by choosing 2/3 * 50 = 33, a level-2 player chooses 2/3 * 33 = 22, and so on converging to 0. The data suggest most players perform 1-2 steps of strategic reasoning, not the infinite recursion required for Nash equilibrium. With experience (repeated play), choices converge toward the Nash prediction, but initial play is well-described by level-k with low k.
Question 2 True / False
Quantal response equilibrium assumes that players always best-respond to their beliefs about other players' strategies.
TTrue
FFalse
Answer: False
QRE assumes players choose better strategies more often than worse ones, but with noise — they do not perfectly best-respond. The probability of choosing a strategy increases with its expected payoff according to a logistic (or similar) response function governed by a precision parameter lambda. When lambda is infinite, QRE collapses to Nash equilibrium (perfect best response). When lambda is zero, players randomize uniformly. QRE is an equilibrium concept because players' noisy strategies are mutually consistent — each player's choice probabilities are a quantal (noisy) best response to the other players' choice probabilities. This captures the intuition that people are roughly strategic but make mistakes, with costlier mistakes being rarer.
Question 3 Short Answer
What distinguishes level-k models from cognitive hierarchy models, and why does the distinction matter empirically?
Think about your answer, then reveal below.
Model answer: In level-k models, each player believes all other players are exactly one level below them — a level-2 player assumes everyone else is level-1. In cognitive hierarchy models, each level-k player best-responds to a mixture of all lower types (0 through k-1), weighted by a frequency distribution (typically Poisson). The distinction matters because level-k predicts sharp, level-specific behavior (a level-2 player ignores the existence of level-0 players), while cognitive hierarchy produces smoother predictions by averaging over the lower-level population. Cognitive hierarchy generally fits experimental data better because it accounts for the heterogeneity of opponents a player might face.
The Poisson distribution in the cognitive hierarchy model is governed by a single parameter tau (the average number of thinking steps), making it parsimonious. Camerer, Ho, and Chong found tau around 1.5 fits a wide range of experimental games — suggesting people average about 1.5 steps of strategic reasoning. This parameter is remarkably stable across different game forms, giving the model predictive power for new games rather than just post-hoc fit.