Rational Decisions
Ken Binmore

Contents
Preface ix
1    Revealed Preference 1
1.1     Rationality? 1
1.2     Modeling a Decision Problem 2
1.3     Reason Is the Slave of the Passions 3
1.4     Lessons from Aesop 5
1.5     Revealed Preference 7
1.6     Rationality and Evolution 12
1.7     Utility 14
1.8     Challenging Transitivity 17
1.9     Causal Utility Fallacy 19
1.10   Positive and Normative 22
2    Game Theory 25
2.1     Introduction 25
2.2     What Is a Game? 25
2.3     Paradox of Rationality? 26
2.4     Newcomb’s Problem 30
2.5     Extensive Form of a Game 31
3    Risk 35
3.1     Risk and Uncertainty 35
3.2     Von Neumann and Morgenstern 36
3.3     The St Petersburg Paradox 37
3.4     Expected Utility Theory 39
3.5     Paradoxes from A to Z 43
3.6     Utility Scales 46
3.7     Attitudes to Risk 50
3.8     Unbounded Utility? 55
3.9     Positive Applications? 58
4    Utilitarianism 60
4.1     Revealed Preference in Social Choice 60
4.2     Traditional Approaches to Utilitarianism 63
4.3     Intensity of Preference 66
4.4     Interpersonal Comparison of Utility 67viii Contents
5    Classical Probability 75
5.1     Origins 75
5.2     Measurable Sets 75
5.3     Kolmogorov’s Axioms 79
5.4     Probability on the Natural Numbers 82
5.5     Conditional Probability 83
5.6     Upper and Lower Probabilities 88
6    Frequency 94
6.1     Interpreting Classical Probability 94
6.2     Randomizing Devices 96
6.3     Richard von Mises 100
6.4     Refining von Mises’ Theory 104
6.5     Totally Muddling Boxes 113
7    Bayesian Decision Theory 116
7.1     Subjective Probability 116
7.2     Savage’s Theory 117
7.3     Dutch Books 123
7.4     Bayesian Updating 126
7.5     Constructing Priors 129
7.6     Bayesian Reasoning in Games 134
8    Epistemology 137
8.1     Knowledge 137
8.2     Bayesian Epistemology 137
8.3     Information Sets 139
8.4     Knowledge in a Large World 145
8.5     Revealed Knowledge? 149
9    Large Worlds 154
9.1     Complete Ignorance 154
9.2     Extending Bayesian Decision Theory 163
9.3     Muddled Strategies in Game Theory 169
9.4     Conclusion 174
10  Mathematical Notes 175
10.1   Compatible Preferences 175
10.2   Hausdorff’s Paradox of the Sphere 177
10.3   Conditioning on Zero-Probability Events 177
10.4   Applying the Hahn–Banach Theorem 179
10.5   Muddling Boxes 180
10.6   Solving a Functional Equation 181
10.7   Additivity 182
10.8   Muddled Equilibria in Game Theory 182
References 189
Index 197

Preface
What  is  rationality?  What  is  the  solution  to  the  problem  of  scien-
tific induction? I don’t 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 Bayesian-
ism 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
large—so  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 theory—the 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 prob-
ability theory along the way. My reason for spending so much time offer-
ing 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 says—or what I think it ought to say—before
getting on to my own attempt to extend a version of Bayesian decision
theory to worlds larger than those considered by Savage (chapter 9).
I don’t 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 applica-
tions. 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 somex Preface
of the gaps between these three communities, it will have been worth-
while. 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 haven’t 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 deci-
sion 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.