Guillermo Owen - Game Theory
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GAME THEORY
FOURTH EDITION
FOURTH EDITION
BY
GUILLERMO OWEN
Department of Mathematics
Naval Postgraduate School
Monterey, California
United Kingdom North America Japan India Malaysia China
Emerald Group Publishing Limited
Howard House, Wagon Lane, Bingley BD16 1WA, UK
First edition 2013
Second edition 1982
Third edition 1995
Fourth edition 2013
Copyright 2013 Emerald Group Publishing Limited
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No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. Any opinions expressed in the chapters are those of the authors. Whilst Emerald makes every effort to ensure the quality and accuracy of its content, Emerald makes no representation implied or otherwise, as to the chapters suitability and application and disclaims any warranties, express or implied, to their use.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN: 978-1-78190-507-4
Game theory once again proves to be a dynamic field. In less than 100 years since the name was coined, it has reached the status of a major branch of both mathematics and economics. It has also proved quite useful in several of the social sciences. Even biology and finance have been touched (and, I hope, improved) by game-theoretic analysis. The field continues to expand, and so a book that I last published in 1995 requires some modification. I am happy that Emerald Publishers is giving me a chance to bring out a new edition.
It would be very difficult to include all the new developments in the field. I have therefore merely added a chapter explaining some of the most important advances in game theory. In particular, I have tried to describe the developments that have been central to several of the Nobel Memorial Prizes in Economics.
I also wish to thank people who have helped me, by vigorous and stimulating discussion, to improve this book. These include Francesc Carreras, Hlne Ferrer, Gianfranco Gambarelli, Maurice Koster, Ines Lindner, Conrado Manuel, Gordon McCormick, Martha Saboy, and Juan Tejada.
Finally, I am sad to mention the deaths of some members of the game theory community. Michael Maschler, my mentor and one of my best friends, was instrumental in the development of the bargaining set and related concepts. John Harsanyi, also a good friend, was one of the first to study games with incomplete information. I will always miss them both.
Guillermo Owen
The general idea of a game is that with which we are familiar in the context of parlor games. Starting from a given point, there is a sequence of personal moves, at each of which one of the players chooses from among several possibilities; interspersed among these there may also be chance, or random, moves such as throwing a die or shuffling a deck of cards.
Examples of this type of game are chess, in which there are no chance moves (except for the determination of who shall play first), bridge, in which chance plays a much greater part, but in which skill is still important, and roulette, which is entirely a game of chance in which skill plays no part.
The examples of bridge and chess help to point out another important element of a game. In fact, in a chess game each player knows every move that has been made so far, while in bridge a players knowledge is usually very imperfect. Thus, in some games, a player is unable to determine which of several possible moves has actually been made, either by an opposing player, or by chance. The practical result of this is that, when a player makes a move, he does not know the exact position of the game, and must make his move remembering that there are several possible actual positions.
Finally, at the end of a game, there is normally some payoff to the players (in the form of money, prestige, or satisfaction) which depends on the progress of the game. We may think of this as a function which assigns a payoff to each terminal position of the game.
In our general idea of a game, therefore, three elements enter: (1) alternation of moves, which can be either personal or random (chance) moves, (2) a possible lack of knowledge, and (3) a payoff function.
We define, first, a topological tree or game tree as a finite collection of nodes, called vertices, connected by lines, called arcs, so as to form a connected figure which includes no simple closed curves. Thus it follows that, given any two vertices A and B, there is a unique sequence of arcs and nodes joining A to B.
From this we obtain
1.2.1 Definition. Let be a topological tree with a distinguished vertex A. We say that a vertex C follows the vertex B if the sequence of arcs joining A to C passes through B. We say C follows B immediately if C follows B and, moreover, there is an arc joining B to C. A vertex X is said to be terminal if no vertex follows 
.
1.2.2 Definition. By an n-person game in extensive form is meant
a topological tree 
with a distinguished vertex 
called the starting point of 
;
a function, called the payoff function, which assigns an n-vector to each terminal vertex of 
;
a partition of the nonterminal vertices of 
into 
sets 
, called the player sets;
a probability distribution, defined at each vertex of 
, among the immediate followers of this vertex;
for each 
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