Binmore - Rational Decisions

Rational Decisions

The Gorman Lectures in Economics

Series Editor, Richard Blundell

Lawlessness and Economics: Alternative Modes of Governance,
Avinash K. Dixit

Rational Decisions,
Ken Binmore

A series statement appears at the back of the book

Ken Binmore

Princeton University Press
Princeton and Oxford

Copyright 2009 by Princeton University Press

Published by Princeton University Press,
41 William Street, Princeton, New Jersey 08540

In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW

All Rights Reserved

ISBN: 978-0-691-13074-3 (alk. paper)

Library of Congress Control Number: 2008938513

British Library Cataloging-in-Publication Data is available

This book has been composed in LucidaBright

Typeset by T&T Productions Ltd, London

Printed on acid-free paper

press.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

This book began with two lectures I gave in 2006 at
University College London in memory of

Terence Gorman

who was a great economist and a good friend. He would
have wished me to dedicate this book to the fondly
remembered teacher at his grammar school in Northern
Ireland who first inspired him with a love of mathematics.

What is rationality? What is the solution to the problem of scientific induction? I dont think it reasonable to expect sharp answers to such questions. One might as well ask for precise definitions of life or consciousness. But we can still try to push forward the frontier of rational decision theory beyond the Bayesian paradigm that represents the current orthodoxy.

Many people see no need for such an effort. They think that Bayesianism already provides the answers to all questions that might be asked. I believe that Bayesians of this stamp fail to understand that their theory applies only in what Jimmie Savage (1951) called a small world in his famous Foundations of Statistics. But the world of scientific inquiry is largeso much so that scientists of the future will look back with incredulity at a period in intellectual history when it was possible be taken seriously when claiming that Bayesian updating is the solution to the problem of scientific induction.

Jack Good once claimed to identify 46,656 different kinds of Bayesians. My first priority is therefore to clarify what I think should be regarded as the orthodoxy on Bayesian decision theorythe set of foundational assumptions that offer the fewest hostages to fortune. This takes up most of the book, since I take time out to review various aspects of probability theory along the way. My reason for spending so much time offering an ultra-orthodox review of standard decision theory is that I feel the need to deny numerous misapprehensions (both positive and negative) about what the theory really saysor what I think it ought to saybefore getting on to my own attempt to extend a version of Bayesian decision theory to worlds larger than those considered by Savage ().

I dont for one moment imagine that my extension of Bayesian decision theory comes anywhere near solving the problem of scientific induction, but I do think my approach will sometimes be found useful in applications. For example, my theory allows the mixed strategies of game theory to be extended to what I call muddled strategies (much as pure strategies were extended to mixed strategies by the creators of the theory).

What is the audience for this book? I hope that it will be read not just by the economics community from which I come myself, but also by statisticians and philosophers. If it only succeeds in bridging some of the gaps between these three communities, it will have been worthwhile. However, those seeking a survey of all recent research will need to look for a much bigger book than this. I have tried to include references to literatures that lie outside its scope, but I never stray very far from my own take on the issues. This streamlined approach means that the book may appeal to students who want to learn a little decision theory without being overwhelmed by masses of heavy mathematics or erudite philosophical reasoning, as well as to researchers in the foundations of decision theory.

Sections in which I havent succeeded in keeping the mathematics at a low level, or where the going gets tough for some other reason, are indicated with an arrow pointing downward in the margin. When such an arrow appears, you may wish to skip to the next full section.

Finally, I want to acknowledge the debt that everyone working on decision theory owes to Duncan Luce and Howard Raiffa, whose Games and Decisions remains a source of inspiration more than fifty years after it was written. I also want to acknowledge the personal debt I owe to Francesco Giovannoni, Larry Samuelson, Jack Stecher, Peter Wakker, and Zibo Xu for their many helpful comments on the first draft of this book.

Rational Decisions

A rational number is the ratio of two whole numbers. The ancients thought that all numbers were rational, but Pythagorass theorem shows that the length of the diagonal of a square of unit area is irrational. Tradition holds that the genius who actually made this discovery was drowned, lest he shake the Pythagorean faith in the ineffable nature of number. But nowadays everybody knows that there is nothing irrational about the square root of two, even though we still call it an irrational number.

There is similarly nothing irrational about a philosopher who isnt a rationalist. Rationalism in philosophy consists of arriving at substantive conclusions without appealing to any data. If you follow the scientific method, you are said to be an empiricist rather than a rationalist. But only creationists nowadays feel any urge to persecute scientists for being irrational.

What of rational decision theory? Here the controversy over what should count as rational is alive and kicking.

Bayesianism. Bayesianism is the doctrine that Bayesian decision theory is always rational. The doctrine entails, for example, that David Hume was wrong to argue that scientific induction cant be justified on rational grounds. Dennis Lindley (1988) is one of many scholars who are convinced that Bayesian inference has been shown to be the only coherent form of inference.

The orthodoxy promoted by Lindley and others has become increasingly claustrophobic in economics, but Gilboa and Schmeidler (2001) have shown that it is still possible to consider alternatives without suffering the metaphorical fate of the Pythagorean heretic who discovered the irrationality of . Encouraged by their success, I follow their example by asking three questions:

What is Bayesian decision theory?

When should we count Bayesian decision theory as rational?

What should we do when Bayesian decision theory isnt rational?

In answering the first question, I hope to distinguish Bayesian decision theory from Bayesianism. We can hold on to the virtues of the former without falling prey to the excesses of the latter.

In answering the second question, I shall note that Leonard (Jimmie) Savagenormally acknowledged as the creator of Bayesian decision theoryheld the view that it is only rational to apply Bayesian decision theory in small worlds. But what is a small world?