What is a game? Classically, a game is perceived as something played by human beings. Its mathematical analysis is human-centered, explores the structures of particular games, economic or social environments and tries to model supposedly "rational" human behavior in search of appropriate "winning strategies". This point of view places game theory into a very special scientific corner where mathematics, economics and psychology overlap and mingle. This book takes a novel approach to the subject. Its focus is on mathematical models that apply to game theory in particular but exhibit a universal character and thus extend the scope of game theory considerably. This textbook addresses anyone interested in a general game-theoretic view of the world. The reader should have mathematical knowledge at the level of a first course in real analysis and linear algebra. However, possibly more specialized aspects are further elaborated and pointers to relevant supplementary literature are given. Moreover, many examples invite the reader to participate "actively" when going through the material. The scope of the book can be covered in one course on Mathematical Game Theory at advanced undergraduate or graduate level.
Author(s): Ulrich Faigle
Edition: 1
Publisher: WSPC
Year: 2022
Language: English
Pages: 240
Tags: Mathematics; Game Theory
Contents
Preface
Part 1. Introduction
Chapter 1. Mathematical Models of the Real World
1. Mathematical modelling
2. Mathematical preliminaries
2.1. Functions and data representation
2.2. Algebra of functions and matrices
2.3. Numbers and algebra
2.4. Probabilities, information and entropy
3. Systems
3.1. Evolutions
4. Games
Part 2. 2-Person Games
Chapter 2. Combinatorial Games
1. Alternating players
2. Recursiveness
3. Combinatorial games
4. Winning strategies
5. Algebra of games
5.1. Congruent games
5.2. Strategic equivalence
6. Impartial games
6.1. Sums of GRUNDY numbers
Chapter 3. Zero-Sum Games
1. Matrix games
2. Equilibria
3. Convex zero-sum games
3.1. Randomized matrix games
3.2. Computational aspects
4. LAGRANGE games
4.1. Complementary slackness
4.2. The KKT-conditions
4.3. Shadow prices
4.4. Equilibria of convex LAGRANGE games
4.5. Linear programs
4.6. Linear programming games
Chapter 4. Investing and Betting
1. Proportional investing
1.1. Expected portfolio
1.2. Expected utility
1.3. The fortune formula
2. Fair odds
3. Betting on alternatives
3.1. Statistical frequencies
4. Betting and information
5. Common knowledge
5.1. Red and white hats
5.2. Information and knowledge functions
5.3. Common knowledge
5.4. Different opinions
Part 3. n-Person Games
Chapter 5. Potentials, Utilities and Equilibria
1. Potentials and utilities
1.1. Potentials
1.2. Utilities
2. Equilibria
2.1. Existence of equilibria
Chapter 6. n-Person Games
1. Dynamics of n-person games
2. Equilibria
3. Randomization of matrix games
4. Traffic flows
Chapter 7. Potentials and Temperature
1. Temperature
1.1. BOLTZMANN distributions
1.2. BOLTZMANN temperature
2. The METROPOLIS process
3. Temperature of matrix games
Chapter 8. Cooperative Games
1. Cooperative TU-games
2. Vector spaces of TU-games
2.1. Duality
2.2. MŌBIUS transform
2.3. Potentials and linear functionals
2.4. Marginal values
3. Examples of TU-games
3.1. Additive games
3.2. Production games
3.3. Network connection games
3.4. Voting games
4. Generalized coalitions and balanced games
5. The core
5.1. Stable sets
5.2. The core
6. Core relaxations
6.1. The open core and least cores
6.2. Nuclea
6.3. Nucleolus and nucleon
6.4. Excess minimization
7. MONGE vectors and supermodularity
7.1. The MONGE algorithm
7.2. The MONGE extension
7.3. Linear programming aspects
7.4. Concavity
7.5. Supermodularity
7.6. Submodularity
8. Values
8.1. Linear values
8.2. Random values
8.2.1. The value of Banzhaf
8.2.2. Marginal vectors and the Shapley value
9. Boltzmann values
10. Coalition formation
10.1. Individual greediness and public welfare
10.2. Equilibria in cooperative games
Chapter 9. Interaction Systems and Quantum Models
1. Algebraic preliminaries
1.1. Symmetry decomposition
2. Complex matrices
2.1. Spectral decomposition
2.2. Hermitian representation
3. Interaction systems
3.1. Interaction states
3.2. Interaction potentials
3.3. Interaction in cooperative games
3.4. Interaction in infinite sets
4. Quantum systems
4.1. The quantum model
4.2. Evolutions of quantum systems
4.3. The quantum perspective on interaction
4.4. The quantum perspective on cooperation
5. Quantum games
6. Final remarks
Appendix
1. Basic facts from real analysis
2. Convexity
3. Polyhedra and linear inequalities
4. BROUWER’s fixed-point theorem
5. The MONGE algorithm
6. Entropy and BOLTZMANN distributions
6.1. BOLTZMANN distributions
6.2. Entropy
7. MARKOV chains
Bibliography
Index
