Theory of Learning in Games: Table of Contents

1. Introduction

2. Fictitious Play

3. The Replicator Dynamics and Related Deterministic Models of Evolution

4. Stochastic Fictitious Play and Mixed Strategy Equilibria

5. Adjustment Models with Persistent Randomness

6. Extensive form games and self-confirming equilibrium

7. Nash Equilibrium, Large Population Models, and Mutations in Extensive Form Games

8. Sophisticated Learning

Last updated: August 01, 2000

1. Introduction

1.1. Introduction

1.2. Large Populations and Matching Models

1.3. Three Common Models of Learning and /or Evolution

1.4. Cournot Adjustment

1.5. Analysis of Cournot Dynamics

1.6. Cournot Process with Lock-In

1.7. Review of Finite Simultaneous Move Games

1.7.1. Strategic-Form Games

1.7.2. Dominance and Iterated Dominance

1.7.3. Nash Equilibrium

1.7.4. Correlated Equilibrium

APPENDIX: Dynamical Systems and Local Stability

2. Fictitious Play

2.1. Introduction

2.2. Two Player Fictitious Play

2.3. The Asymptotic Behavior of Fictitious Play

2.4. The Interpretation of Cycles in Fictitious Play

2.5. Multi-Player Fictitious Play

2.6. Payoffs in Fictitious Play

2.7. Consistency and Correlated Equilibrium in 2 Strategy Games

2.8. Fictitious Play and the Best Response Dynamic

2.9. Generalizations of Fictitious Play

APPENDIX: Dirichlet Priors and Multinomial Sampling

3. The Replicator Dynamics and Related Deterministic Models of Evolution

3.1. Introduction

3.2. The Replicator Dynamics in a Homogeneous Population

3.3. Stability in the Homogeneous-Population Replicator Dynamic

3.4. Evolutionarily Stable Strategies

3.5. Asymmetric Replicator Models

3.6. Interpretation of the Replicator Equation

3.6.1. Overview

3.6.2. Social Learning

3.6.3. The Stimulus Response Model

3.7. Generalizations of the Replicator Dynamic and Iterated Strict Dominance

3.8. Myopic Adjustment Dynamics

3.8.1. Replicator versus Best-Response

3.8.2.Two by Two Symmetric Games

3.8.3. Stable Attractors and Strategic Stability

3.9. Set Valued Limit Points and Drift

3.10. Cheap Talk and the Secret Handshake

3.11. Discrete-Time Replicator Systems

APPENDIX: Liouville's Theorem

4. Stochastic Fictitious Play and Mixed Strategy Equilibria

4.1. Introduction

4.2. Notions of Convergence

4.3. Asymptotic Myopia and Asymptotic Empiricism

4.4. Randomly Perturbed Payoffs and Smoothed Best Responses

4.5. Smooth Fictitious Play and Stochastic Approximation

4.6. Partial Sampling

4.7. Universal Consistency and Smooth Fictitious Play

4.8. Stimulus-Response and Fictitious Play as Learning Models

4.8.1. Stimulus Response with Negative Reinforcement

4.8.2. Experimental Evidence

4.8.3. Learning Effectiveness

4.8.4. Fictitious Play as a Stimulus-Response Model

4.9. Learning About Strategy Spaces

APPENDIX: Stochastic Approximation Theory

5. Adjustment Models with Persistent Randomness

5.1. Introduction

5.2. Overview of Stochastic Adjustment Models

5.3. Kandori-Mailath-Rob Model

5.4. Discussion of Other Dynamics

5.5. Local Interaction

5.6. The Radius and Coradius of Basins of Attraction

5.7. The Modified Coradius

5.8. Uniform Random Matching with Heterogeneous Populations

5.9. Stochastic Replicator Dynamics

APPENDIX 1: Review of Finite Markov Chains

APPENDIX 2: Stochastic Stability Analysis

6. Extensive form games and self-confirming equilibrium

6.1. Introduction

6.2. An Example

6.3. Extensive Form Games

6.4. A Simple Learning Model

6.4.1. Beliefs

6.4.2. Behavior Given Beliefs

6.4.3 Equilibrium Notions

6.5. Stability of Self-Confirming Equilibrium

6.6. Heterogeneous Self-Confirming Equilibrium

6.7. Consistent Self-Confirming Equilibrium

6.8. Consistent Self Confirming Equilibria and Nash Equilibria

6.9. Rationalizable SCE and Prior Information on Opponents' Payoffs

6.9.1. Notation

6.9.2. Belief-Closed Sets and Extensive-Form Rationalizability

6.9.3. Robustness

6.9.4. Example 6.1 revisited

6.9.5. Experimental Evidence

7. Nash Equilibrium, Large Population Models, and Mutations in Extensive Form Games

7.1. Introduction

7.2. Relevant Information Sets and Nash equilibrium

7.3. Exogenous Experimentation

7.4. Learning in Games Compared to the Bandit Problem

7.5. Steady State Learning

7.6. Stochastic Adjustment and Backwards Induction in a Model of 'Fast Learning'

7.6.1. The Model

7.6.2. The Deterministic Dynamic

7.6.3. Dynamic with Mutations

7.7. Mutations and Fast Learning in Models of Cheap Talk

7.8. Experimentation and The Length of the Horizon

APPENDIX: Review of Bandit Problems

8. Sophisticated Learning

8.1. Introduction

8.2. Three Paradigms for Conditional Learning

8.3. The Bayesian Approach to Sophisticated Learning

8.4. Interpreting the Grain of Truth Assumption

8.5. Choosing Among Experts

8.6. Conditional Learning

8.7. Discounting

8.8. Does Sophisticated Learning Lead to Complex Dynamics?

8.9. Does Sophisticated Learning Lead to Stability?

8.10. Calibration and Correlated Equilibrium

8.11. Manipulating Learning Procedures