Game Theory

A Comprehensive Introduction

Hans Keiding

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814623674
• Topic: Mathematik
• Subtopic: Spieltheorie

This book is intended as an introduction to game theory which goes beyond the field of application, economics, and which introduces the reader to as many different sides of game theory as possible within the limitations of an introduction. The main goal is to give an impression of the diversity of game theoretical models, while at the same time covering the standard topics. The book has an equal coverage of non-cooperative and cooperative games, and it covers several topics such as selecting Nash equilibria, non-transferable utility games, applications of game theory to logic, combinatorial and differential games.

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This book is intended as an introduction to game theory which goes beyond the field of application, economics, and which introduces the reader to as many different sides of game theory as possible within the limitations of an introduction. The main goal is to give an impression of the diversity of game theoretical models, while at the same time covering the standard topics. The book has an equal coverage of non-cooperative and cooperative games, and it covers several topics such as selecting Nash equilibria, non-transferable utility games, applications of game theory to logic, combinatorial and differential games.

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Contents:

• Non-Cooperative Game Theory:
  • Two Person Zero-Sum Games
  • Applications of Minimax Theory
  • Solutions for General Non-Cooperative Games
  • More About Nash Equilibria
  • Games with Incomplete Information
  • Choosing Among Nash Equilibria
  • Repeated Games
  • Selected Topics in Non-Cooperative Games
• Cooperative Game Theory:
  • Introduction to Cooperative Games
  • Bargaining
  • TU Games: Classical Solutions
  • TU Games: Other Solutions
  • Solutions of NTU Games: The Core
  • Values of NTU Games
  • The Theory of Game Forms

Key Features:

• Introduces the reader to as many different sides of game theory as possible, within the limitations of an introduction
• More detailed treatment of cooperative game theory
• More emphasis on parts of game theory which rarely show up in introductions, such as combinatorial games, applications to logic, differential games

Chapter 1

Introduction

1 What is game theory?

A convenient way of introducing a theory – and one which actually works reasonably well in our case – is to define a certain object, in this case a game, and then declare that the theory consists of the analysis of this object.

An obvious drawback of this approach is that it gives little or no impression of why the theory is worthwhile; one has to fight one’s way through a considerable body of theory before seeing what it can be used for.

Therefore, although we shall indeed use this method, defining first a game and then proceeding to analyze games over several chapters before we get to convincing applications, we begin with some more easy-going considerations as a motivation for what comes later.

Broadly speaking, game theory is the analysis of conflict situations. The type of conflict is irrelevant, indeed the attraction of game theory lies partly in the fact that superficially very different phenomena such as parlor games and war has common features, which can be studied with largely the same analytical tools, and probably it was this many-sidedness of game theory, which made it arise and survive as an independent scientific discipline.

On the other hand, not all conflict studies can be characterized as game theory, indeed game theory may constitute only a minor part of the general study of conflicts. It deals with the possibilities and the actual choices of the participants, when the conflict and its consequences are clearly delineated. General research in conflicts may also consider the emergence of conflicts as well as the way in which conflicts influence the minds of the parties.

Game theory is – at least in the version to be presented here – a mathematically formulated discipline, aiming at studying “rational” behavior in conflict situations. The very concept of rationality will occasionally be questioned, so we shall not overdo this program of finding rational behavior in conflicts. As in may other fields, the study of the theory will make us understand its limitations much better.

2 Definition of a game

In order to define a game we need some parties to the conflicts, to be called either agents or individuals or, most often, players. These players should have several possible choices, in the following referred to as strategies, and the conflict is described in such a way that the choices of a strategy by each player should result in an outcome for the players.

DEFINITION 1 A game form is an array G = (N, (S_i)_i∈N,π), where

(i) N is a nonempty set of players,

(ii) for each player i ∈ N, S_i is a set (of strategies of i),

(iii) π :

_i∈N S_i → X, where X is a nonempty set (of outcomes), and π is a map assigning to every strategy array s = (s_i)_i∈N an outcome π(s).

The set N of players can be arbitrarily large (in particular N may be an infinite set, something which turns out to be useful in some contexts although not in this book). In what follows we shall assume that N = {1,...,n}.

The outcome set X occurring in Definition 1 has been supressed in our notation, where it is implicit in the definition of the outcome map π. If we need to be explicit about the outcome space, then we say that G is a game form over X. If G = (N, (S_i)_i∈N,π) is a game form and f : X → X′ is a map, then we get another game form G = (N, (S_i)_i∈N, f

π) (over X′), where the outcome map is f

π, the composition of π with the map f. T...