Game Theory: A Classical Introduction, Mathematical Games, and the Tournament

Description

This book is a formalization of collected notes from an introductory game theory course taught at Queen's University. The course introduced traditional game theory and its formal analysis, but also moved to more modern approaches to game theory, providing a broad introduction to the current state of the discipline. Classical games, like the Prisoner's Dilemma and the Lady and the Tiger, are joined by a procedure for transforming mathematical games into card games. Included is an introduction and brief investigation into mathematical games, including combinatorial games such as Nim. The text examines techniques for creating tournaments, of the sort used in sports, and demonstrates how to obtain tournaments that are as fair as possible with regards to playing on courts. The tournaments are tested as in-class learning events, providing a novel curriculum item. Example tournaments are provided at the end of the book for instructors interested in running a tournament in their own classroom. The book is appropriate as a text or companion text for a one-semester course introducing the theory of games or for students who wish to get a sense of the scope and techniques of the field Content: 1. Introduction: the prisoner's dilemma and finite state automata -- 1.1 Introduction -- 1.1.1 Helpful sources of information -- 1.2 The prisoner's dilemma -- 1.3 Finite state automata -- 1.4 Exercises -- 2. Games in extensive form with complete information and backward induction -- 2.1 Introduction -- 2.2 The lady and the tiger game -- 2.3 Games in extensive form with complete information -- 2.4 Backward induction -- 2.5 The ultimatum game -- 2.5.1 The enhanced ultimatum game -- 2.5.2 What backward induction says -- 2.5.3 Backward induction is wrong about this one -- 2.6 The boat crash game -- 2.7 Continuous games -- 2.7.1 Two models of the Stackelberg duopology -- 2.8 The failings of backward induction -- 2.9 Exercises -- 3. Games in normal form and the Nash equilibrium -- 3.1 Introduction and definitions -- 3.2 The stag hunt -- 3.3 Dominated strategies -- 3.3.1 Iterated elimination of dominated strategies -- 3.4 The nacho game -- 3.4.1 The nacho game with K players -- 3.5 Nash equilibria -- 3.5.1 Finding the NE by IEDS -- 3.5.2 IEDS process -- 3.6 The vaccination game -- 3.6.1 The N-player vaccination game -- 3.7 Exercises -- 4. Mixed strategy Nash equilibria and two-player zero-sum games -- 4.0.1 The fundamental theorem of Nash equilibria -- 4.1 An example with a 3-by-3 payoff matrix -- 4.2 Two-player zero sum games -- 4.2.1 The game of odds and evens -- 4.3 Domination of two-player zero sum games -- 4.3.1 Saddle points -- 4.3.2 Solving two-by-two games -- 4.4 Goofspiel -- 4.5 Exercises -- 5. Mathematical games -- 5.1 Introduction -- 5.1.1 The subtraction game -- 5.2 Nim -- 5.2.1 Moore's Nim -- 5.3 Sprouts -- 5.4 The graph domination game -- 5.5 Deck-based games -- 5.5.1 Deck-based Prisoner's dilemma -- 5.5.2 Deck-based Rock-paper-scissors(-lizard-spock) -- 5.5.3 Deck-based Divide the dollar -- 5.5.4 FLUXX-like game mechanics -- 5.5.5 A note on adding new mechanics to mathematical games -- 5.6 Exercises -- 6. Tournaments and their design -- 6.1 Introduction -- 6.1.1 Some types of tournaments -- 6.2 Round robin scheduling -- 6.3 Round robin scheduling with courts -- 6.3.1 Balanced tournament designs -- 6.3.2 Court-balanced tournament designs -- 6.4 Cyclic permutation fractal tournament design -- 6.4.1 Case 1: n = 2k, k [epsilon] Z+ -- 6.4.2 A recursive generation of the minimum number of rounds -- 6.5 Exercises -- 7. Afterword -- 7.1 Conclusion and future directions -- A. Example tournaments -- A.1. Example tournaments -- The enhanced ultimatum game tournament -- The vaccination game tournament -- A different kind of IPD tournament -- A2. Some things to consider before running a tournament in a classroom -- Bibliography -- Author's biography.