Handbook of Game Theory

Front Cover

Handbook of Game Theory

Copyright

Contents

Contributors

Preface

Acknowledgments

Introduction to the Series

Chapter 1: Rationality

1.1 Neoclassical Rationality

1.1.1 Substantive or procedural?

1.1.1.1 Evolution

1.1.2 Rationality as consistency

1.1.3 Positive or normative?

1.2 Revealed Preference

1.2.1 Independence of irrelevant alternatives

1.2.1.1 Aesop

1.2.1.2 Utility

1.2.1.3 Causal utility fallacy

1.2.2 Revealed preference in game theory

1.3 Decisions under Risk

1.3.1 VN&M utility functions

1.3.1.1 Attitudes to risk

1.3.1.2 Unbounded utility?

1.3.1.3 Utilitarianism

1.4 Bayesian Decision Theory

1.4.1 Savage's theory

1.4.1.1 Bayes' rule

1.4.2 Small worlds

1.4.2.1 Bayesianism?

1.4.2.2 Where do Savage's priors come from?

1.4.2.3 When are the worlds of game theory small?

1.4.2.4 Common priors?

1.5 Knowledge

1.5.1 Knowledge as commitment

1.5.1.1 Contradicting knowledge?

1.5.2 Common knowledge

1.5.3 Common knowledge of rationality?

1.5.3.1 Counterfactuals

1.6 Nash Equilibrium

1.6.1 Evolutionary game theory

1.6.2 Knowledge requirements

1.6.3 Equilibrium selection problem

1.6.3.1 Refinements of Nash equilibrium

1.7 Black Boxes

1.7.1 Nash program

1.7.2 Other preplay activity

1.8 Conclusion

Acknowledgments

References

Chapter 2: Advances in Zero-Sum Dynamic Games

2.1 Introduction

2.1.1 General model of repeated games (RG)

2.1.2 Compact evaluations

2.1.3 Asymptotic analysis

2.1.4 Uniform analysis

2.2 Recursive Structure

2.2.1 Discounted stochastic games

2.2.2 General discounted repeated games

2.2.2.1 Recursive structure

2.2.2.2 Specific classes of repeated games

2.2.3 Compact evaluations and continuous time extension

2.3 Asymptotic Analysis

2.3.1 Benchmark model

2.3.2 Basic results

2.3.2.1 Incomplete information

2.3.2.2 Stochastic games

2.3.3 Operator approach

2.3.3.1 Nonexpansive monotone maps

2.3.3.2 Applications to RG

2.3.4 Variational approach

2.3.4.1 Discounted values and variational inequalities

2.3.4.2 General RG and viscosity tools

2.3.4.3 Compact discounted games and comparison criteria

2.4 The Dual Game

2.4.1 Definition and basic results

2.4.2 Recursive structure and optimal strategies of the noninformed player

2.4.3 The dual differential game

2.4.4 Error term, control of martingales, and applications to price dynamics

2.5 Uniform Analysis

2.5.1 Basic results

2.5.1.1 Incomplete information

2.5.1.2 Stochastic games

2.5.1.3 Symmetric case

2.5.2 From asymptotic value to uniform value

2.5.3 Dynamic programming and MDP

2.5.4 Games with transition controlled by one player

2.5.5 Stochastic games with signals on actions

2.5.6 Further results

2.6 Differential Games

2.6.1 A short presentation of differential games (DG)

2.6.2 Quantitative differential games

2.6.3 Quantitative differential games with incomplete information

2.7 Approachability

2.7.1 Definition

2.7.2 Weak approachability and quantitative differential games

2.7.3 Approachability and B-sets

2.7.4 Approachability and qualitative differential games

2.7.5 Remarks and extensions

2.8 Alternative Tools and Topics

2.8.1 Alternative approaches

2.8.1.1 A different use of the recursive structure

2.8.1.2 No signals

2.8.1.3 State dependent signals

2.8.1.4 Incomplete information on the duration

2.8.1.5 Games with information lag

2.8.2 The ``Limit Game''

2.8.2.1 Presentation

2.8.2.2 Examples

2.8.2.3 Specific Properties

2.8.3 Repeated games and differential equations

2.8.3.1 RG and PDE

2.8.3.2 RG and evolution equations

2.8.4 Multimove games

2.8.4.1 Alternative evaluations

2.8.4.2 Evaluation on plays

2.8.4.3 Stopping games

2.9 Recent Advances

2.9.1 Dynamic programming and games with an informed controller

2.9.1.1 General evaluation and total variation

2.9.1.2 Dynamic programming and TV-asymptotic value

2.9.1.3 Dynamic programming and TV-uniform value

2.9.1.4 Games with a more informed controller

2.9.1.5 Comments

2.9.2 Markov games with incomplete information on both sides

2.9.3 Counter examples for the asymptotic approach

2.9.3.1 Counter example for finite state stochastic games with compact action spaces

2.9.3.2 Counter examples for games with finite parameter sets

2.9.3.3 Oscillations

2.9.3.4 Regularity and o-minimal structures

2.9.4 Control problem, martingales, and PDE

2.9.5 New links between discrete and continuous time games

2.9.5.1 Multistage approach

2.9.5.2 Discretization of a continuous time game

2.9.5.3 Stochastic games with short stage duration

2.9.5.4 Stochastic games in continuous time

2.9.5.5 Incomplete information games with short stage duration

2.9.6 Final comments

Acknowledgments

References

Chapter 3: Games on Networks

3.1 Introduction and Overview

3.2 Background Definitions

3.2.1 Players and networks

3.2.2 Games on networks

3.3 Strategic Complements and Strategic Substitutes

3.3.1 Defining strategic complements and substitutes

3.3.2 Existence of equilibrium

3.3.2.1 Games of strategic complements

3.3.2.2 Games of strategic substitutes and other games on networks

3.3.2.3 Games with strategic substitutes, continuous action spaces and linear best-replies

3.3.3 Two-action games on networks

3.3.3.1 Changes in behaviors as the network varies

3.3.3.2 Coordination games

3.3.3.3 Stochastically stable play in coordination games on networks

3.4 A Model with Continuous Actions, Quadratic Payoffs, and Strategic Complementarities

3.4.1 The benchmark quadratic model

3.4.1.1 Katz-Bonacich network centrality and strategic behavior

3.4.1.2 Nash equilibrium

3.4.1.3 Welfare

3.4.2 The model with global congestion

3.4.3 The model with ex ante heterogeneity

3.4.4 Some applications of the quadratic model

3.4.4.1 Crime

3.4.4.2 Education

3.4.4.3 Industrial organization

3.4.4.4 Cities

3.4.4.5 Conformity and conspicuous effects

3.5 Network Games with Incomplete Information

3.5.1 Incomplete information and contagion effects

3.5.1.1 A model of network games with incomplete information

3.5.1.2 Monotonicity of equilibria

3.5.1.3 A dynamic best reply process

3.5.2 Incomplete information about payoffs

3.5.3 Incomplete information with communication in networks

3.6 Choosing Both Actions and Links

3.6.1 Coordination games

3.6.2 Network formation in quadratic games

3.6.3 Games with strategic substitutes

3.7 Repeated Games and Network Structure

3.8 Concluding Remarks and Further Areas of Research

3.8.1 Bargaining and exchange on networks

3.8.2 Risk-sharing networks

3.8.3 Dynamic games and network structure

3.8.4 More empirical applications based on theory

3.8.5 Lab and field experiments

Acknowledgments

References

Chapter 4: Reputations in Repeated Games

4.1 Introduction

4.1.1 Reputations

4.1.2 The interpretive approach to reputations

4.1.3 The adverse selection approach to reputations

4.2 Reputations with Short-Lived Players

4.2.1 An example

4.2.2 The benchmark complete information game

4.2.3 The incomplete information game and commitment types

4.2.4 Reputation bounds

4.2.4.1 Relative entropy

4.2.4.2 Bounding the one-step ahead prediction errors

4.2.4.3 From prediction bounds to payoffs

4.2.4.4 The Stackelberg bound

4.2.5 More general monitoring structures

4.2.6 Temporary reputations under imperfect monitoring

4.2.6.1 The implications of reputations not disappearing

4.2.6.2 The contradiction and conclusion of the proof

4.2.7 Interpretation

4.2.8 Exogenously informative signals

4.3 Reputations with Two Long-Lived Players

4.3.1 Types vs. actions

4.3.2 An example: The failure of reputation effects

4.3.3 Minmax-action reputations

4.3.3.1 Minmax-action types

4.3.3.2 Conflicting interests

4.3.4 Discussion

4.3.4.1 Weaker payoff bounds for more general actions

4.3.4.2 Imperfect monitoring

4.3.4.3 Punishing commitment types

4.4 Persistent Reputations

4.4.1 Limited observability

4.4.2 Analogical reasoning

4.4.3 Changing types

4.4.3.1 Cyclic reputations

4.4.3.2 Permanent reputations

4.4.3.3 Reputation as separation

4.5 Discussion

4.5.1 Outside options and bad reputations

4.5.2 Investments in reputations

4.5.3 Continuous time

4.5.3.1 Characterizing behavior

4.5.3.2 Reputations without types

Acknowledgments

References

Chapter 5: Coalition Formation

5.1 Introduction

5.2 The Framework

5.2.1 Ingredients

5.2.2 Process of coalition formation

5.2.3 Equilibrium process of coalition formation

5.2.4 Some specific settings

5.2.4.1 State spaces

5.2.4.2 Characteristic functions and partition functions

5.2.4.3 Remarks on protocols and effectivity correspondences

5.2.5 Remarks on the response protocol

5.3 The Blocking Approach: Cooperative Games

5.3.1 The setting

5.3.2 Blocking

5.3.3 Consistency and farsightedness

5.3.4 The farsighted stable set

5.3.5 Internal blocking

5.3.5.1 A recursive definition

5.3.5.2 EPCF and the farsighted stable set with internal blocking

5.3.5.3 Characteristic functions

5.3.5.4 Effectivity without full support

5.3.5.5 Internal blocking in the presence of externalities

5.3.6 Beyond internal blocking

5.3.6.1 Farsighted stability for characteristic functions

5.3.6.2 Farsighted stability for games with externalities

5.4 The Bargaining Approach: Noncooperative Games

5.4.1 Ingredients of a coalitional bargaining model

5.4.1.1 The protocol

5.4.1.2 Possibilities for changing or renegotiating agreements

5.4.1.3 Payoffs in real time or not

5.4.1.4 Majority versus unanimity

5.4.2 Bargaining on partition functions

5.4.2.1 Equilibrium in a real-time bargaining model

5.4.2.2 Two elementary restrictions

5.4.2.3 EPCF and bargaining equilibrium

5.4.3 Some existing models of noncooperative coalition formation

5.4.3.1 The standard bargaining problem

5.4.3.2 Coalitional bargaining with irreversible agreements

5.4.3.3 Equilibrium coalition structure

5.4.4 Reversibility

5.5 The Welfare Economics of Coalition Formation

5.5.1 Two sources of inefficiency

5.5.2 Irreversible agreements and efficiency

5.5.3 Reversible agreements and efficiency

5.5.3.1 Temporary agreements

5.5.3.2 Renegotiation

5.6 Coalition Formation: The Road Ahead

Acknowledgments

References

Chapter 6: Stochastic Evolutionary Game Dynamics

6.1 Evolutionary Dynamics and Equilibrium Selection

6.1.1 Evolutionarily stable strategies

6.1.2 Stochastically stable sets

6.2 Equilibrium Selection in 2 2 Games

6.2.1 A simple model

6.2.2 The unperturbed process

6.2.3 The perturbed process

6.3 Stochastic Stability in Larger Games

6.3.1 A canonical model of adaptive learning

6.3.2 Markov processes and rooted trees

6.3.3 Equilibrium selection in larger games

6.4 Bargaining

6.4.1 An evolutionary model of bargaining

6.4.2 The case of heterogeneous agents

6.4.3 Extensions: Sophisticated agents and cooperative games

6.5 Public Goods

6.5.1 Teamwork

6.5.2 Bad apples

6.5.3 The volunteer's dilemma

6.5.4 General public-good games and potential

6.6 Network Games

6.7 Speed of Convergence

6.7.1 Autonomy

6.7.2 Close-knittedness

6.8 Concluding Remarks

References

Chapter 7: Advances in Auctions

7.1 Introduction

7.2 First-Price Auctions: Theoretical Advances

7.2.1 Mixed-strategy equilibria

7.2.2 Asymmetric buyers: Existence of mixed and pure-strategy equilibria

7.2.3 Relaxation of symmetry and independence

7.2.4 Monotonicity and the role of tie-breaking rules

7.2.5 Revenue comparisons

7.3 Multiunit Auctions

7.3.1 Efficient ascending-bid auctions

7.3.2 Multiple heterogeneous items

7.4 Dynamic Auctions

7.4.1 Dynamic population

7.4.2 Repeated ascending-price auctions

7.5 Externalities in Single-Object Auctions

7.5.1 A general social choice model

7.5.2 Complete information

7.5.3 Incomplete information

7.6 Auctions with Resale

7.6.1 First-price and second-price auctions

7.6.2 Seller's optimal mechanism

7.6.3 Further results

7.7 All-Pay Auctions

7.7.1 Complete information

7.7.2 Incomplete information

7.7.3 Multiple prizes

7.7.4 Bid-dependent rewards

7.7.5 Contests versus lotteries

7.7.6 All-pay auctions with spillovers

7.7.7 Bid caps

7.7.8 Research contests

7.7.9 Blotto games

7.7.10 Other topics

7.8 Incorporating Behavioral Economics

7.8.1 Regret

7.8.2 Impulse balance

7.8.3 Reference points

7.8.4 Buy-it-now options

7.8.5 Level-k bidding

7.8.6 Spite

7.8.7 Ambiguity

7.9 Position Auctions in Internet Search

7.9.1 First-price pay-per-click auctions

7.9.2 Second-price pay-per-click auctions

7.9.3 Other formats

7.10 Spectrum Auctions

7.10.1 3G auctions

7.10.2 4G auctions

7.11 Concluding Remarks

Acknowledgments

References

Chapter 8: Combinatorial Auctions

8.1 Introduction

8.2 Supporting Prices

8.2.1 The core

8.3 Incentives

8.3.1 When VCG is not in the core

8.3.2 Ascending implementations of VCG

8.3.3 What is an ascending auction?

8.3.4 The clock-proxy auction

8.4 Complexity Considerations

8.4.1 Exact methods

8.4.2 Approximation

References

Chapter 9: Algorithmic Mechanism Design: Through the lens of Multiunit auctions

9.1 Introduction

9.2 Algorithmic Mechanism Design and This Survey

9.2.1 The field of algorithmic mechanism design

9.2.2 Our example: multiunit auctions

9.2.3 Where are we going?

9.3 Representation

9.3.1 Bidding languages

9.3.2 Query access to the valuations

9.3.2.1 Value queries

9.3.2.2 General communication queries

9.4 Algorithms

9.4.1 Algorithmic efficiency

9.4.2 Downward sloping valuations

9.4.3 Intractability

9.4.3.1 NP-Completeness

9.4.3.2 Communication complexity

9.4.4 Approximation

9.5 Payments, Incentives, and Mechanisms

9.5.1 Vickrey-Clarke-Groves mechanisms

9.5.2 The clash between approximation and incentives

9.5.3 Maximum-in-range mechanisms

9.5.4 Single parameter mechanisms

9.5.5 Multiparameter mechanisms beyond VCG?

9.5.6 Randomization

9.6 Conclusion

Acknowledgments

References

Chapter 10: Behavioral Game Theory Experiments and Modeling

10.1 Introduction

10.2 Cognitive Hierarchy and Level-k Models

10.2.1 P-beauty contest

10.2.2 Market entry games

10.2.3 LUPI lottery

10.2.4 Summary

10.3 Quantal Response Equilibrium

10.3.1 Asymmetric hide-and-seek game

10.3.2 Maximum value auction

10.4 Learning

10.4.1 Parametric EWA learning: interpretation, uses, and limits

10.4.2 fEWA functions

10.4.2.1 The change-detector function 0=x"011Ei(t)

10.4.2.2 The attention function, 0=x"010Eij(t)

10.4.3 fEWA predictions

10.4.4 Example: mixed strategy games

10.4.5 Summary

10.5 Sophistication and Teaching

10.5.1 Sophistication

10.5.2 Strategic teaching

10.5.3 Summary

10.6 Sociality

10.6.1 Public goods

10.6.2 Public goods with punishment

10.6.3 Negative reciprocity: ultimatums

10.6.4 Impure altruism and social image: dictator games

10.6.5 Summary

10.7 Conclusion

References

Chapter 11: Evolutionary Game Theory in Biology

11.1 Strategic Analysis—What Matters to Biologists?

11.2 Sex Ratios—How the Spirit of Game Theory Emerged in Biology

11.2.1 There is a hitch with fitness

11.2.2 Düsing's solution—the first biological game

11.2.3 Fisher's treatment of sex-ratio theory

11.2.4 Does it suffice to count grandchildren—what is the utility?

11.2.5 Evolutionary dynamics

11.2.6 Reproductive value

11.2.7 Haplodiploid sex-ratio theory

11.3 The Empirical Success of Sex-Ratio Theory

11.3.1 Experimental evolution

11.3.2 Measuring the slices of the cake

11.3.3 Local mate competition

11.3.4 Environmental sex determination and the logic of randomization

11.4 Animal Fighting and the Official Birth of Evolutionary Game Theory

11.4.1 The basic idea of an evolutionary game

11.4.2 The concept of an evolutionarily stable strategy

11.4.3 What does the Hawk-Dove game tell us about animal fighting?

11.4.4 The current view on limited aggression

11.5 Evolutionary Dynamics

11.5.1 The replicator equation

11.5.2 Adaptive dynamics and invasion fitness

11.5.3 Gene-frequency dynamics and the multilocus mess

11.5.4 Finite populations and stochasticity

11.6 Intragenomic Conflict and Willful Passengers

11.6.1 Meiotic drive

11.6.2 Example of an ultraselfish chromosome

11.6.3 Endosymbionts in conflict with their hosts

11.6.4 Wolbachia—master manipulators of reproduction

11.7 Cooperation in Microbes and Higher Organisms

11.7.1 Reciprocal altruism

11.7.2 Indirect reciprocity

11.7.3 Tragedy of the commons

11.7.4 Common interest

11.7.5 Common interest through lifetime monogamy

11.8 Biological Trade and Markets

11.8.1 Pollination markets

11.8.2 Principal-agent models, sanctioning and partner choice

11.8.3 Supply and demand

11.9 Animal Signaling—Honesty or Deception?

11.9.1 Education and the Peacock's tail

11.9.2 Does the handicap principle work in practice?

11.9.3 The rush for ever more handicaps, has it come to an end?

11.9.4 Warning signals and mimicry

References

Chapter 12: Epistemic Game Theory

12.1 Introduction and Motivation

12.1.1 Philosophy/Methodology

12.2 Main Ingredients

12.2.1 Notation

12.2.2 Strategic-form games

12.2.3 Belief hierarchies

12.2.4 Type structures

12.2.5 Rationality and belief

12.2.6 Discussion

12.2.6.1 State dependence and nonexpected utility

12.2.6.2 Elicitation

12.2.6.3 Introspective beliefs and restrictions on strategies

12.2.6.4 Semantic/syntactic models

12.3 Strategic Games of Complete Information

12.3.1 Rationality and common belief in rationality

12.3.2 Discussion

12.3.3 -Rationalizability

12.4 Equilibrium Concepts

12.4.1 Introduction

12.4.2 Subjective correlated equilibrium

12.4.3 Objective correlated equilibrium

12.4.4 Nash equilibrium

12.4.5 The book-of-play assumption

12.4.6 Discussion

12.4.6.1 Condition AI

12.4.6.2 Comparison with aumann1987correlated

12.4.6.3 Nash equilibrium

12.5 Strategic-Form Refinements

12.6 Incomplete Information

12.6.1 Introduction

12.6.2 Interim correlated rationalizability

12.6.3 Interim independent rationalizability

12.6.4 Equilibrium concepts

12.6.5 -Rationalizability

12.6.6 Discussion

12.7 Extensive-Form Games

12.7.1 Introduction

12.7.2 Basic ingredients

12.7.3 Initial CBR

12.7.4 Forward induction

12.7.4.1 Strong belief

12.7.4.2 Examples

12.7.4.3 RCSBR and extensive-form rationalizability

12.7.4.4 Discussion

12.7.5 Backward induction

12.7.6 Equilibrium

12.7.7 Discussion

12.8 Admissibility

12.8.1 Basics

12.8.2 Assumption and mutual assumption of rationality

12.8.3 Characterization

12.8.4 Discussion

12.8.4.1 Issues in the characterization of IA

12.8.4.2 Extensive-form analysis and strategic-form refinements

Acknowledgement

References

Chapter 13: Population Games and Deterministic Evolutionary Dynamics

13.1 Introduction

13.2 Population Games

13.2.1 Definitions

13.2.2 Examples

13.2.3 The geometry of population games

13.3 Revision Protocols and Mean Dynamics

13.3.1 Revision protocols

13.3.2 Information requirements for revision protocols

13.3.3 The stochastic evolutionary process and mean dynamics

13.3.4 Finite horizon deterministic approximation

13.4 Deterministic Evolutionary Dynamics

13.4.1 Definition

13.4.2 Incentives and aggregate behavior

13.5 Families of Evolutionary Dynamics

13.5.1 Imitative dynamics

13.5.1.1 Definition

13.5.1.2 Examples

13.5.1.3 Basic properties

13.5.1.4 Inflow-outflow symmetry

13.5.2 The best response dynamic and related dynamics

13.5.2.1 Target protocols and target dynamics

13.5.2.2 The best response dynamic

13.5.2.3 Perturbed best response dynamics

13.5.3 Excess payoff and pairwise comparison dynamics

13.5.3.1 Excess payoff dynamics

13.5.3.2 Pairwise comparison dynamics

13.6 Potential Games

13.6.1 Population games and full population games

13.6.2 Definition, characterization, and interpretation

13.6.3 Examples

13.6.4 Characterization of equilibrium

13.6.5 Global convergence and local stability

13.6.6 Local stability of strict equilibria

13.7 ESS and Contractive Games

13.7.1 Evolutionarily stable states

13.7.2 Contractive games

13.7.3 Examples

13.7.4 Equilibrium in contractive games

13.7.5 Global convergence and local stability

13.7.5.1 Imitative dynamics

13.7.5.2 Target and pairwise comparison dynamics: global convergence in contractive games

13.7.5.3 Target and pairwise comparison dynamics: local stability of regular ESS

13.8 Iterative Solution Concepts, Supermodular Games, and Equilibrium Selection

13.8.1 Iterated strict dominance and never-a-best-response

13.8.2 Supermodular games and perturbed best response dynamics

13.8.3 Iterated p-dominance and equilibrium selection

13.9 Nonconvergence of Evolutionary Dynamics

13.9.1 Examples

13.9.2 Survival of strictly dominated strategies

13.10 Connections and Further Developments

13.10.1 Connections with stochastic stability theory

13.10.2 Connections with models of heuristic learning

13.10.3 Games with continuous strategy sets

13.10.4 Extensive form games and set-valued solution concepts

13.10.5 Applications

Acknowledgements

References

Chapter 14: The Complexity of Computing Equilibria

14.1 The Task

14.2 Problems and Algorithms

14.3 Good Algorithms

14.4 P and NP

14.5 Reductions and NP-complete Problems

14.6 The Complexity of Nash Equilibrium

14.7 Approximation, Succinctness, and Other Topics

Acknowledgments

References

Chapter 15: Theory of Combinatorial Games

15.1 Motivation and an Ancient Roman War-Game Strategy

15.2 The Classical Theory, Sum of Games, Complexity

15.2.1 Complexity, hardness, and completeness

15.3 Introducing Draws

15.4 Adding Interactions Between Tokens

15.5 Partizan Games

15.5.1 Two examples: Hackenbush and Domineering

15.5.2 Outcomes and sums

15.5.3 Values

15.5.4 Simplest forms

15.5.5 Numbers

15.5.6 Infinitesimals

15.5.7 Stops and the mean value

15.6 Misère Play

15.6.1 Misère Nim value

15.6.2 Genus theory

15.6.3 Misère canonical form

15.6.4 Misère quotients

15.7 Constraint Logic

15.7.1 The constraint-logic framework

15.7.2 One-player games

15.7.2.1 Bounded games

15.7.2.2 Unbounded games

15.7.3 Two-player games

15.7.4 Team games

15.8 Conclusion

Acknowledgment

References

Chapter 16: Game Theory and Distributed Control**

16.1 Introduction

16.2 Utility Design

16.2.1 Cost/Welfare-sharing games

16.2.2 Achieving potential game structures

16.2.3 Efficiency of equilibria

16.3 Learning Design

16.3.1 Preliminaries: repeated play of one-shot games

16.3.2 Learning Nash equilibria in potential games

16.3.2.1 Fictitious play and joint strategy fictitious play

16.3.2.2 Simple experimentation dynamics

16.3.2.3 Equilibrium selection: log-linear learning and its variants

16.3.2.4 Near potential games

16.3.3 Beyond potential games and equilibria: efficient action profiles

16.3.3.1 Learning efficient pure Nash equilibria

16.3.3.2 Learning pareto efficient action profiles

16.4 Exploiting the Engineering Agenda: State-Based Games

16.4.1 Limitations of strategic form games

16.4.1.1 Limitations of protocol design

16.4.1.2 Distributed optimization: consensus

16.4.2 State-based games

16.4.3 Illustrations

16.4.3.1 Protocol design

16.4.3.2 Distributed optimization

16.5 Concluding Remarks

References

Chapter 17: Ambiguity and Nonexpected Utility

17.1 Introduction

Part I Nonexpected Utility Theory Under Risk

17.2 Nonexpected Utility: Theories and Implications

17.2.1 Preliminaries

17.2.2 Three approaches

17.2.3 The existence of Nash equilibrium

17.2.4 Atemporal dynamic consistency

17.2.5 Implications for the theory of auctions

17.3 Rank-Dependent Utility Models

17.3.1 Introduction

17.3.2 Representations and interpretation

17.3.3 Risk attitudes and interpersonal comparisons ofrisk aversion

17.3.4 The dual theory

17.4 Cumulative Prospect Theory

17.4.1 Introduction

17.4.2 Trade-off consistency and representation

Part II Nonexpected Utility Theory Under Uncertainty

17.5 Decision Problems under Uncertainty

17.5.1 The decision problem

17.5.2 Mixed actions

17.5.3 Subjective expected utility

17.6 Uncertainty Aversion: Definition andRepresentation

17.6.1 The Ellsberg paradox

17.6.2 Uncertainty aversion

17.6.3 Uncertainty averse representations

17.7 Beyond Uncertainty Aversion

17.7.1 Incompleteness

17.7.2 Smooth ambiguity model

17.8 Alternative Approaches

17.8.1 Choquet expected utility

17.8.2 Uncertainty aversion revisited

17.8.3 Other models

17.9 Final Remarks

Acknowledgments

References

Chapter 18: Calibration and Expert Testing

18.1 Introduction

18.2 Terminology and Notation

18.3 Examples

18.3.1 Example 1

18.3.2 Example 2

18.3.3 Example 3

18.4 Calibration

18.4.1 Definition and result

18.4.2 Calibrated forecasting rule

18.4.3 Sketch of proof

18.4.4 Sketch of proof of Blackwell's theorem

18.5 Negative Results

18.5.1 Generalizations of Foster and Vohra's result

18.5.2 Prequential principle

18.5.3 Interpretations

18.6 Positive Results

18.6.1 Category tests

18.6.2 A simple example of a good test

18.6.3 Other good tests

18.6.4 Good “prequential” tests

18.7 Restricting the Class of Allowed Data-Generating Processes

18.8 Multiple Experts

18.9 Bayesian and Decision-Theoretic Approachesto Testing Experts

18.9.1 Bayesian approach

18.9.2 Decision-theoretic approach

18.10 Related Topics

18.10.1 Falsifiability and philosophy of science

18.10.2 Gaming performance fees by portfolio managers

Acknowledgment

References

Index