It is widely held that Bayesian decision theory is the final word on how a rational person should make decisions. However, Leonard Savage--the inventor of Bayesian decision theory--argued that it would be ridiculous to use his theory outside the kind of small world in which it is always possible to "look before you leap." If taken seriously, this view makes Bayesian decision theory inappropriate for the large worlds of scientific discovery and macroeconomic enterprise. When is it correct to use Bayesian decision theory--and when does it need to be modified? Using a minimum of mathematics, Rational Decisions clearly explains the foundations of Bayesian decision theory and shows why Savage restricted the theory's application to small worlds.
The book is a wide-ranging exploration of standard theories of choice and belief under risk and uncertainty. Ken Binmore discusses the various philosophical attitudes related to the nature of probability and offers resolutions to paradoxes believed to hinder further progress. In arguing that the Bayesian approach to knowledge is inadequate in a large world, Binmore proposes an extension to Bayesian decision theory--allowing the idea of a mixed strategy in game theory to be expanded to a larger set of what Binmore refers to as "muddled" strategies.
Written by one of the world's leading game theorists, Rational Decisions is the touchstone for anyone needing a concise, accessible, and expert view on Bayesian decision making.
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By Ken Binmore
Princeton University PressCopyright © 2009 Princeton University Press
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Chapter OneRevealed Preference
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 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 can't 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 [square root of 2]. 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 isn't 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) Savage-normally acknowledged as the creator of Bayesian decision theory-held the view that it is only rational to apply Bayesian decision theory in small worlds. But what is a small world?
The worlds of macroeconomics and high finance most certainly don't fall into this category. What should we do when we have to make decisions in such large worlds? I am writing this book because I want to join the club of those who think they have the beginnings of an answer to this third question.
No formal definition of rationality will be offered. I don't believe in the kind of Platonic ideal that rationalist philosophers seem to have in mind when they appeal to Immanuel Kant's notion of Practical Reason. I think that rationality principles are invented rather than discovered. To insist on an a priori definition would be to make the Pythagorean mistake of prematurely closing our minds to possible future inventions. I therefore simply look for a minimal extension of orthodox decision theory to the case of large worlds without pretending that I have access to some metaphysical hotline to the nature of absolute truth.
1.2 Modeling a Decision Problem
When Pandora makes a decision, she chooses an action from those available. The result of her action will usually depend on the state of the world at the time she makes her decision. For example, if she chooses to step into the road, her future fate will depend on whether a car happens to be passing by.
We can capture such a scenario by modeling a decision problem as a function
D : A ? B [right arrow] C
in which A is the set of available actions, B is the set of possible states of the world, and ITLITL is the set of possible consequences. So if Pandora chooses action a when the world is in state b, the outcome will be c = D(a, b). Figure 1.1 illustrates a simple case.
An act in such a setting is any function [alpha] : B [right arrow] ITLITL. For example, if Pandora bets everything she owns on number 13 when playing roulette, then she chooses the act in which she will be wealthy if the little ball stops in the slot labeled 13, and ruined if it doesn't. Pandora's choice of an action always determines some kind of act, and so we can regard A as her set of feasible alternatives within the set of all possible acts.
If Pandora chooses action a from her feasible set A in a rational way, then we say that a is an optimal choice. The framework we have chosen therefore already excludes one of the most common forms of irrationality-that of choosing an optimal action without first considering what is feasible.
Knowledge. Ken Arrow (1971, p. 45) tells us that each state in B should be "a description of the world so complete that, if true and known, the consequences of every action would be known." But how does Pandora come to be so knowledgeable?
If we follow the philosophical tradition of treating knowledge as justified true belief, the answer is arguably never. I don't see that we are even entitled to assume that reality accords to some model that humans are able to envisage. However, we don't need a view on the metaphysical intelligibility of the universe to discuss decision theory intelligently. The models we use in trying to make sense of the world are merely human inventions. To say that Pandora knows what decision model she is facing can therefore be taken as meaning no more than that she is committed to proceeding as though her model were true (section 8.5).
1.3 Reason Is the Slave of the Passions
Thomas Hobbes characterized man in terms of his strength of body, his passions, his experience, and his reason. When Pandora is faced with a decision problem, we may identify her strength of body with the set A of all actions that she is physically able to choose. Her passions can be identified with her preferences over the set ITLITL of possible consequences, and her experience with her beliefs about the likelihood of the different possible states in the set B. In orthodox decision theory, her reason is identified with the manner in which she takes account of her preferences and beliefs in deciding what action to take.
The orthodox position therefore confines rationality to the determination of means rather than ends. To quote David Hume (1978): "Reason is, and ought only to be, the slave of the passions." As Hume extravagantly explained, he would be immune to accusations of irrationality even if he were to prefer the destruction of the entire universe to scratching his finger. Some philosophers hold to the contrary that rationality can tell you what you ought to like. Others maintain that rationality can tell you what you ought to choose without reference to your preferences. For example, Kant (1998) tells us that rationality demands that we honor his categorical imperative, whether or not we like the consequences.
My own view is that nothing is to be gained in the long run by inventing versions of rationality that allow their proponents to label brands of ethics or metaphysics other than their own as irrational. Instead of disputing whose ethical or metaphysical system should triumph, we are then reduced to disputing whose rationality principles should prevail. I prefer to emulate the logicians in seeking to take rationality out of the firing line by only adopting uncontroversial rationality principles.
Such a minimalist conception of rational decision theory isn't very glamorous, but then, neither is modern logic. However, as in logic, there is a reward for following the straight and narrow path of rectitude. By so doing, we will be able to avoid getting entangled in numerous thorny paradoxes that lie in wait on every side.
Consistency. The modern orthodoxy goes further than David Hume. It treats reason as the slave, not only of our passions, but also of our experience. Pandora's reason is assumed to be the slave of both her preferences and her beliefs.
It doesn't follow that rational decision theory imposes no constraints on our preferences or our beliefs. Everyone agrees that rational people won't fall prey to the equivalent of a logical contradiction. Their preferences and beliefs will therefore be consistent with each other in different situations. But what consistency criteria should we impose? We mustn't be casual about this question, because the words rationality and consistency are treated almost as synonyms in much modern work.
For example, Bayesianism focuses on how Pandora should respond to a new piece of data. It is said that Pandora must necessarily translate her prior beliefs into posterior beliefs using Bayes' rule if she is to act consistently. But there is seldom any serious discussion of why Pandora should be consistent in the sense required. However, this is a topic for a later chapter (section 7.5.2). We already have enough contentious issues in the current chapter to keep us busy for some time.
1.4 Lessons from Aesop
The fox in Aesop's fable was unable to reach some grapes and so decided that they must be sour. He thereby irrationally allowed his beliefs in domain B to be influenced by what actions are feasible in domain A.
If Aesop's fox were to decide that chickens must be available because they taste better than grapes, he would be guilty of the utopian mistake of allowing his assessment of what actions are available in domain A to be influenced by his preferences in domain ITLITL. The same kind of wishful thinking may lead him to judge that the grapes he can reach must be ripe because he likes ripe grapes better than sour grapes, or that he likes sour grapes better than ripe grapes because the only grapes that he can reach are probably sour. In both these cases, he fails to separate his beliefs in domain B from his preferences in domain ITLITL.
Aesop's principle. These observations motivate the following principle:
Pandora's preferences, her beliefs, and her assessments of what is feasible should all be independent of each other.
For example, the kind of pessimism that might make Pandora predict that it is bound to rain now that she has lost her umbrella is irrational. Equally irrational is the kind of optimism that Voltaire was mocking when he said that if God didn't exist, it would be necessary to invent Him.
1.4.1 Intrinsic Preferences?
It is easy to propose objections to Aesop's principle. For example, Pandora's preferences between an umbrella and an ice cream might well alter if it comes on to rain. Shouldn't we therefore accept that preferences will sometimes be state-dependent?
It is true that instrumental preferences are usually state-dependent. One can tell when one is dealing with an instrumental preference, because it advances matters to ask Pandora why she holds the preference. She might say, for example, that she prefers an umbrella to an ice cream because it looks like rain and she doesn't want to get wet. Or that she prefers driving nails home with a hammer rather than a screwdriver because it takes less time and trouble. More generally, any preference over actions is an instrumental preference.
One is dealing with intrinsic preferences when it no longer helps to ask Pandora why she likes one thing rather than another, because nothing relevant that might happen is capable of altering her position. For example, we could ask Pandora why she likes wearing a skirt that is two inches shorter than the skirts she liked last year. She might reply that she likes being in the fashion. Why does she like being in the fashion? Because she doesn't like being laughed at for being behind the times. Why doesn't she like being laughed at? Because girls who are ridiculed are less attractive to boys. Why does she like being attractive to boys? One could reply that evolution made most women this way, but such an answer doesn't take us anywhere, because we don't plan to consider alternative environments in which evolution did something else. The fact that Pandora likes boys can therefore usefully be treated as an intrinsic preference. Her liking for miniskirts is instrumental because it changes with her environment.
Economists are talking about intrinsic preferences when they quote the slogan: De gustibus, non est disputandum. In welfare economics, it is particularly important that the preferences that we seek to satisfy should be intrinsic. It wouldn't help very much, for example, to introduce a reform that everyone favors if it changes the environment in a way that reverses everyone's preferences.
1.4.2 Constructing a Decision Problem
Decision problems aren't somehow built into the structure of the universe. Pandora must decide how to formulate her decision problem for herself. There will often be many formulations available, some of which satisfy the basic assumptions of whatever decision theory she plans to apply-and others which don't. She might have to work hard, for example, to find a formulation in which her preferences on the set ITLITL of consequences are intrinsic. If she plans to apply Aesop's principle completely, she must work even harder and construct a model in which there are no linkages at all between any of the sets A, B, and ITLITL (other than those built into the function D) that are relevant to her decision.
For example, if Pandora's actions are predictions of the weather, she mustn't take the states in B to be correct or mistaken, because the true state of the world would then depend on her choice of action. In the case of an umbrella on a rainy day, it may be necessary to identify the set C of consequences with Pandora's states of mind rather than physical objects. One can then speak of the states of mind that accompany having an umbrella-on-a-sunny-day or having an umbrella-on-a-wet-day, rather than speaking just of an umbrella.
Critics complain that the use of such expedients makes the theory tautological, but what could be better when one's aim is to find an uncontroversial framework?
Every thing is what it is, and not something else. The price that has to be paid for an uncontroversial theory is that it can't be used to model everything that we might like to model. For example, we can't model the possibility that people might choose to change their intrinsic preferences by adopting behaviors that are likely to become habituated. Such restrictions are routinely ignored when appeals are made to rational decision theory, but to bowdlerize Bishop Butler: Every theory is good for what it is good for, and not for something else.
To ignore the wisdom of Bishop Butler is to indulge in precisely the kind of wishful thinking disbarred by Aesop's principle. We aren't even entitled to take for granted that Pandora can formulate her decision problem so that Aesop's principle applies. In fact, I shall be arguing later that it is a characteristic of decision making in large worlds that agents are unable to separate their preferences from their beliefs. However, this unwelcome consideration will be left on ice until chapter 9.
Finally, nothing says that one can't construct a rational decision theory that applies to decision problems that don't satisfy Aesop's principle. Richard Jeffrey (1965) offers a theory in which your own choice of action may provide evidence about the choices made by other people. Ed Karni (1985) offers a theory in which the consequences may be inescapably state-dependent. I am hesitant about such theories because I don't know how to evaluate their basic assumptions.
Excerpted from Rational Decisions by Ken Binmore Copyright © 2009 by Princeton University Press. Excerpted by permission.
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Table of ContentsPreface ix
Chapter 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
Chapter 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
Chapter 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
Chapter 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 67
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter 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
Chapter 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
What People are Saying About This
This book takes the reader from the very beginnings of decision theory through innovative new work, and it is written concisely and clearly enough to make the journey seem effortless. It will be essential reading for decision theorists and economists in general, whether new to the field or old hands.
Larry Samuelson, Yale University
Well-written, with many nice illustrative examples, this book contributes to the ongoing debate on whether the Bayesian method is the proper normative framework for making decisions and updating beliefs. The book is extremely impressive in its breadth of cited references: historical, mathematical, philosophical, and statistical.
Mark Machina, University of California, San Diego
Strong on ideas and opinions but low on jargon, this is one of the most lively discussions of the strengths and limitations of the Bayesian approach to decision making that I have ever come across. Clearly and strongly argued, controversial, and a pleasure to read.
Riccardo Rebonato, author of The Plight of the Fortune Tellers
Ken Binmore is one of our deepest thinkers on the foundations of economics and game theory. Here he gives us his personal take on standard decision theory and his own extension of the theory to the case in which decision makers cannot assign unambiguous probabilities to future events. This book will be of considerable interest to economists and philosophers alike.
Eric Maskin, Nobel Prize-winning economist
Ken Binmore's new book on rational decisions is a superb up-to-date introduction to the now large and complex literature of this subject. The touch is light, but the examples and theorems are precise. The references are more than ample to lead the reader to more details on those parts of the subject that are of most interest. Although the author is an economist, the content will appeal to philosophers, psychologists, and statisticians as well.
Patrick Suppes, Stanford University