# Rational Decisions

Ken Binmore
Edition: STU - Student edition
Pages: 216
https://www.jstor.org/stable/j.ctt7szmq

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. 1 Revealed Preference
(pp. 1-24)

A rational number is the ratio of two whole numbers. The ancients thought that all numbers were rational, but Pythagoras’s 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 isn’t a rationalist. Rationalism in philosophy consists...

5. 2 Game Theory
(pp. 25-34)

Game theory is perhaps the most important arena for the application of rational decision theory. It is also a breeding ground for innumerable fallacies and paradoxes. However, this book isn’t the place to learn the subject, because I plan to say only enough to allow me to use a few examples here and there. My bookPlaying for Realis a fairly comprehensive introduction that isn’t mathematically demanding (Binmore 2007b).

A game arises when several players have to make decisions in a situation in which the outcome for each player is partly determined by the choices made by the other...

6. 3 Risk
(pp. 35-59)

Economists attach a precise meaning to the wordsriskanduncertainty. Pandora makes a decision under risk if unambiguous probabilities can be assigned to the states of the world in her belief spaceB. Otherwise, she decides under uncertainty. The importance of distinguishing decision problems in which unambiguous probabilities are available from those in which they aren’t was first brought to the attention of the world by Frank Knight (1921). For this reason, people often speak ofKnightianuncertainty to emphasize that they are using the word in its technical sense.

The archetypal case of risk is playing roulette in...

7. 4 Utilitarianism
(pp. 60-74)

Until relatively recently, it was an article of faith among economists that one can’t make meaningful comparisons of the utilities that different people may enjoy. This chapter is a good one to skip if you don’t care about this question.

Social choice.The theory of social choice is about how groups of people make decisions collectively. This chapter is an aside on the implications of applying the theory of revealed preference to such a social context. Even in the case of a utilitarian government, we shall therefore be restricting our attention to notions of utility that make it fallacious to...

8. 5 Classical Probability
(pp. 75-93)

Probability doesn’t come naturally to the human species. The ancients never came up with the idea at all, although they enjoyed gambling games just as much as we do. It was only in the seventeenth century that probability saw the light of day. The Chevalier de Méré will always be remembered for proposing a problem about gambling odds that succeeded in engaging the attention of two great mathematicians of the day, Pierre de Fermat and Blaise Pascal. Some of the letters they exchanged in 1654 still survive.¹ It is fascinating to learn how difficult they found ideas that we teach...

9. 6 Frequency
(pp. 94-115)

Kolmogorov’s classical theory of probability is entirely mathematical, and so says nothing whatever about the world. To apply the theory, we need to find an interpretation of the objects that appear in the theory that is consistent with whatever facts we are treating as given in whatever model of the world we choose to maintain.

The literature recognizes three major interpretations of probability:

Objective probability

Subjective probability

Logical probability

Donald Gillies’s (2000) excellent bookPhilosophical Theories of Probabilitysurveys the literature very thoroughly from a perspective close to mine, and so I need only sketch the different interpretations.

Objective probability....

10. 7 Bayesian Decision Theory
(pp. 116-136)

This chapter studies the theory of subjective probabilities invented independently by Frank Ramsey (1931) and Bruno de Finetti (1937). The version of the theory developed by Leonard Savage (1951) in his famousFoundations of Statisticsis now universally called Bayesian decision theory. My own simplification of his theory in section 7.2 differs from the usual textbook accounts, but the end product will be no less orthodox than my simplification of Von Neumann and Morgenstern’s theory of expected utility in section 3.4.

We last visited the theory of revealed preference when developing a version of Von Neumann and Morgenstern’s theory of...

11. 8 Epistemology
(pp. 137-153)

Philosophers traditionally treat knowledge as justified true belief, and then argue about what their definition means. This chapter contributes little to this debate, because it defends an entirely different way of thinking about knowledge. However, before describing the approach to knowledge that I think most useful in decision theory, it is necessary to review Bayesian epistemology—the study of how knowledge is treated in Bayesian decision theory.

When Pandora uses Bayes’ rule to update her prior probability prob(E) of an eventEto a posterior probability prob$(E\,|\,F)$on learning that the eventFhas occurred, everybody understands that we...

12. 9 Large Worlds
(pp. 154-174)

There was once a flourishing literature on rational decision theory in large worlds. Luce and Raiffa (1957, chapter 13) refer to this literature as decision making under complete ignorance. They classify what we now call Bayesian decision theory as decision making under partial ignorance (because Pandora can’t be completely ignorant if she is able to assign subjective probabilities to some events).

It says a lot about our academic culture that this literature should be all but forgotten. Presumably nobody reads Savage’s (1951)Foundations of Statisticsany more, since the latter half of the book is entirely devoted to his own...

13. 10 Mathematical Notes
(pp. 175-188)

This section justifies the representation of separable preferences of section 3.6.2.

Two preference relations${ \preccurlyeq _1}$and${ \preccurlyeq _2}$will be said to be compatible if it is always true that

$a{ \preccurlyeq _1}b$implies$a{ \preccurlyeq _2}b$.

(It then follows that$a{ \preccurlyeq _2}b$implies$a{ \preccurlyeq _1}b$.) If the two preference relations are defined on the set lott(C)of lotteries with prizes inCand satisfy the Von Neumann and Morgenstern postulates, then they may be described with Von Neumann and Morgenstern utility functions${v_1}:C \to \mathbb{R}$and${v_2}:C \to \mathbb{R}$. If they are also compatible on lott(C), then${v_1}\; = \;A{v_2}\; + \;B$or${v_2}\; = \;A{v_1} + B$, where$A \geqslant \;0$andBare constants. (If,$A > 0$, then${ \preccurlyeq _1}$and...

14. References
(pp. 189-196)
15. Index
(pp. 197-200)
16. Back Matter
(pp. 201-201)