Outline

Motivation and Modeling Philosophy

Empirical Alternative I: Model of Cognitive Hierarchy (Camerer, Ho, and Chong, QJE, 2004)

Empirical Alternative II: Quantal Response Equilibrium (McKelvey and Palfrey, GEB, 1995)

Empirical Alternative III: Model of Noisy Introspection (Goeree and Holt, AER, 2001)

Modeling Principles

Principle Nash CH QRE NI

Strategic Thinking

Best Response

Mutual Consistency

Quantal Response Equilibrium

)( where1

ijij

J

k

x

x

ij

x

e

ei

ij

ij

Player i choosing strategy j with probability ij in equilibrium.

xij is i’s expected payoff of choosing j given that others choosetheir strategy according to equilibrium profile .

The values of depends on .

Example 1

q (1-q)

p(1-p)

Nash Equilibrium q (1-q)

p(1-p)

1.1141 q = 0.1238 (1- q) q = 0.101.1141 (1-p) = 1.1141 p p = 0.5

Example 1: =2

pp

p

qq

q

ee

eq

ee

ep

1141.12)1(1141.12

)1(1141.12'

)1(1238.021141.12

1141.12'

Iteration q p' q'1 0.500 0.729 0.2652 0.265 0.601 0.3903 0.390 0.672 0.3174 0.317 0.631 0.3585 0.358 0.654 0.3356 0.335 0.641 0.3487 0.348 0.649 0.3408 0.340 0.644 0.3449 0.344 0.647 0.342

10 0.342 0.645 0.34311 0.343 0.646 0.34312 0.343 0.646 0.343

Prediction as a Function of

Example 2

QRE Prediction: A = B = 5

QRE Prediction: A = B = 100

Lieberman’s 2-person Zero-sum Game

MLE Estimation Count Label

A1 78 N1A2 8 N2A3 216 N3B1 10 M1B2 90 M2B3 200 M3

321321 ))()(1()()())()(1()()()( 21212121MMMNNN qqqqppppLL

Best-fitted QRE Model

QRE Plots

O’Neill’s 2-person Zero-sum Game

QRE and its Special Cases

QRE Estimates Across Time

QRE Plots

Rapoport and Boebel’s 2-person Zero-sum Game

QRE Estimates; Experiment I

QRE Estimates: Experiment II

QRE Estimates Across Time

QRE Plots

Ochs’s 2-person Non-zero Sum Games

QRE Estimates

QRE Plots

QRE Estimates

Prediction

QRE Plots

QRE versus CHStahl & Cooper & Costa-Gomes

Data set Wilson (1995) Van Huyck et al. Mixed Entry Signal

Log-likelihood

Cognitive Hierarchy (Game-specific t ) 1 -360 -838 -264 -824 -150 -191Quantal Response (Game-specific ) -395 -871 -281 -897 -150 -199Cognitive Hierarchy (Common General Distribution f(k)) -388 -867 -271 -866 -150 -204Cognitive Hierarchy (Common t ) -458 -868 -274 -872 -150 -205Quantal Response (Common ) -421 -916 -292 -1005 -151 -207

Mean Squared DeviationCognitive Hierarchy (Game-specific t ) 0.0074 0.0090 0.0035 0.0097 0.0004 0.0480Quantal Response (Game-specific ) 0.0153 0.0157 0.0111 0.0203 0.0001 0.0489Cognitive Hierarchy (Common General Distribution f(k)) 0.0125 0.0142 0.0090 0.0175 0.0004 0.0481Cognitive Hierarchy (Common t ) 0.0327 0.0145 0.0097 0.0179 0.0005 0.0489Quantal Response (Common ) 0.0248 0.0276 0.0180 0.0456 0.0022 0.0506

Log-likelihoodCognitive Hierarchy (Common General Distribution f(k)) -422 -918 -338 -884 -154 -227Cognitive Hierarchy (Common t ) -469 -956 -293 -884 -154 -238Quantal Response (Common ) -431 -990 -374 -1402 -166 -216

Mean Squared DeviationCognitive Hierarchy (Common General Distribution f(k)) 0.0265 0.0270 0.0554 0.0194 0.0043 0.0759Cognitive Hierarchy (Common t ) 0.0416 0.0335 0.0237 0.0215 0.0046 0.0803Quantal Response (Common ) 0.0275 0.0506 0.0702 0.0487 0.0216 0.0632

Within-dataset Forecasting

Cross-dataset Forecasting

Table 2: Model Fit (Log Likelihood LL and Mean Squared Deviation MSD)

Summary