Preface ........................................................ ix
1 Introduction and Background .................................. 1
1.1 Errare Humanum Est ...................................... 1
1.2 Why a Statistical Version of Nash Equilibrium? .......... 5
1.3 Six Guiding Principles for Statistical Game Theory ...... 7
1.4 A Road Map .............................................. 8
2 Quantal Response Equilibrium in Normal-Form Games ........... 10
2.1 The Reduced-Form Approach: Regular QRE ................. 13
2.2 Two Simple Regular QRE Models .......................... 15
2.2 The Luce Model and a Power-Law Generalization .......... 16
2.2.2 QRE Based on Distribution Functions ............. 19
2.3 Properties of Logit QRE ................................ 22
2.3.1 Examples to Illustrate Logit-QRE-Based
Refinements ..................................... 24
2.4 Alternative Models Related to Regular QRE .............. 31
2.4.1 Rosenthal's Linear Response Model ............... 31
2.4.2 Van Damme's Control Cost Model .................. 32
2.5 Structural Approach to QRE ............................. 35
2.5.1 An Equilibrium Model of Action Trembles ......... 35
2.5.2 Structural QRE: An Equilibrium Model of Payoff
Trembles ........................................ 39
2.6 On the Empirical Content of QRE ........................ 42
2.6.1 Empirical Restrictions of Regular QRE ........... 43
2.6.2 Empirical Restrictions of Structural QRE ........ 46
2.6.3 Regularity with Structural QRE .................. 48
2.6.4 Structural versus Reduced-Form QRE .............. 49
2.7 QRE for Continuous Games ............................... 53
3 Quanta! Response Equilibrium in Extensive-Form Games ........ 63
3.1 Regular QRE for Extensive-Form Games ................... 67
3.2 Structural AQRE for Extensive-Form Games ............... 72
3.2.1 Notation ........................................ 12
3.2.2 Agent Quantal Response Equilibrium .............. 74
3.3 Logit AQRE ............................................. 76
3.4 AQRE Analysis of the Centipede Game .................... 85
4 Heterogeneity ............................................... 88
4.1 Skill and Role Heterogeneity ........................... 90
4.1.1 Systematic Heterogeneity Depending on Player
Roles ........................................... 90
4.1.2 Idiosyncratic Heterogeneity ..................... 90
4.2 Heterogeneous Quantal Response Equilibrium ............. 91
4.3 Skill Heterogeneity without Rational Expectations ...... 92
4.3.1 Egoistic Beliefs about Opponents ................ 92
4.3.2 Unconstrained Point Beliefs about Opponents'
Skill ........................................... 93
4.4 QRE without Rational Expectations ...................... 94
4.4.1 Subjective Heterogeneous QRE .................... 94
4.4.2 Truncated QRE and Cognitive Hierarchy ........... 96
4.5 Noisy Introspection ................................... 103
4.5.1 Noisy Rationalizability ........................ 107
4.5.2 Noisy Introspection in Extensive-Form Games .... 109
5 Dynamics and Learning ...................................... 112
5.1 QRE in Infinitely Repeated Games ...................... 113
5.1.1 Notation ....................................... 115
5.1.2 A Finite Recursive Form for Repeated Games ..... 116
5.2 QRE in Dynamic and Stochastic Games ................... 121
5.2.1 Markov Perfect Equilibrium ..................... 122
5.2.2 Logit Markov QRE ............................... 125
5.3 Evolutionary Dynamics and Logit QRE ................... 127
5.4 Stochastic Learning Equilibrium ....................... 133
5.4.1 Some Alternative Learning Rules ................ 134
5.4.2 Beliefs and Probabilistic Choice ............... 137
5.4.3 History Formation .............................. 138
5.4.4 Stochastic Learning Equilibrium ................ 138
6 QRE as a Structural Model for Estimation ................... 141
6.1 The QRE Estimation Approach ........................... 142
6.1.1 Estimation Program for Generalized Matching
Pennies ......................................... 151
6.2 Estimation and Methodological Issues .................. 151
6.2.1 Multiple Equilibria ............................ 152
6.2.2 Estimation Methods ............................. 154
6.2.3 Urwbserved Heterogeneity ....................... 158
7 Applications to Game Theory ................................ 161
7.1 The Traveler's Dilemma ................................ 164
7.1.1 Traveler's Dilemma Experimental Data ........... 165
7.1.2 Logit QRE Claim Distributions .................. 167
7.2 The Compromise Game ................................... 169
7.2.1 Simultaneous versus Sequential Compromise
Games .......................................... 170
7.2.2 The Logic of Equilibrium ....................... 171
7.2.3 QRE with Continuous Types and Binary Action
Spaces ......................................... 172
7.2.4 Estimating QRE, TQRE, and a-Cursed QRE ......... 177
7.3 Simplified Poker ...................................... 180
7.4 Signaling Games ....................................... 182
7.4.1 Banks, Camerer, and Porter (1994) .............. 184
7.4.2 Brandts and Holt (1992, 1993) .................. 188
7.5 Social Learning ....................................... 190
7.5.1 The Basic Model ................................ 191
7.5.2 Nash Equilibrium ............................... 193
7.5.3 AQRE in the Information Cascade Game ........... 194
7.5.4 Belief Dynamics Implied by QRE ................. 196
7.5.5 An Experiment on Self-Correcting Cascades ...... 197
7.5.6 Estimation ..................................... 202
8 Applications to Political Science .......................... 206
8.1 Participation Games ................................... 207
8.1.1 General Participation Games .................... 209
8.1.2 Participation Games with Negative
Externalities .................................. 210
8.1.3 Participation Games with Positive
Externalities .................................. 212
8.1.4 Positive arul Negative Externalities in
Voting Games ................................... 214
8.2 Incomplete Information and Voter Turnout .............. 220
8.2.1 QRE Model of Turnout with Private Information .. 221
8.2.2 Experimental Data .............................. 221
8.2.3 Asymptotic Voter Turnout Predicted by QRE ...... 224
8.3 Information Aggregation by Voting ..................... 225
8.3.1 Rational Strategic Voting in Juries ............ 227
8.3.2 Experimental Data .............................. 229
8.3.3 QRE Voting in Juries ........................... 231
8.4 Markov QRE and Dynamic Legislative Bargaining ......... 232
8.4.1 Approximating MPE by Logit MQRE ................ 234
8.4.2 Steady-State Equilibrium Dynamics .............. 234
8.4.3 Using MQRE for Estimation ...................... 239
8.5 Crisis Bargaining ..................................... 240
9 Applications to Economics .................................. 248
9.1 Imperfect Price Competition ........................... 249
9.1.1 Learning and Convergence to Equilibrium ........ 251
9.1.2 Logit QRE and Logit SLE Price Distributions .... 254
9.2 Minimum-Effort Coordination Games ..................... 257
9.2.1 Stochastic Potential and Logit QRE ............. 258
9.2.2 A Coordination-Game Experiment ................. 260
9.3 All-Pay Auctions ...................................... 261
9.3.1 Rent Dissipation ............................... 265
9.3.2 Own-Payoff Effects in Asymmetric All-Pay
Auctions ....................................... 267
9.4 Private-Value Auctions ................................ 269
9.4.1 Two Simple Auction Games ....................... 270
9.5 Common-Value Auctions ................................. 273
9.5.1 The Maximum-Value Auction Game ................. 275
9.5.2 Logit QRE for the Maximum-Value Auction Game ... 276
9.5.3 Estimating HQRE for the Maximum-Value Auction
Game ........................................... 276
10 Epilogue: Some Thoughts about Future Research .............. 281
10.1 Theory ................................................ 283
10.1.1 Repeated Games ................................. 283
10.1.2 Correlated Equilibrium, Preplay
Communication, and Mechanism Design ............ 284
10.1.3 Endogenizing the Quantal Response Parameter .... 285
10.2 Applications .......................................... 286
10.2.1 Bargaining Games: Legislative Bargaining,
Offer/Counteroffer ............................. 286
10.2.2 The Winner's Curse ............................. 286
10.3 Methodology ........................................... 287
10.4 Closing Remarks ....................................... 289
References .................................................... 291
Index ......................................................... 301
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