The assumptions of the last chapter are idealizations, and they deal only with the special case of pure Newcomb situations. In this chapter, I will trade those assumptions for weaker ones relative to a more general setting with n states and m available acts. This more general setting may constitute a pure Newcomb situation, a mixed Newcomb situation or no Newcomb situation at all. After giving a probabilistic analysis of an agent's beliefs regarding the causal relations among the acts, the states and states of a special kind (like RФ(DM) of Chapter 6), I shall offer two theorems which indicate that PMCEU gives the right prescription wherever causal decision theory does. The main argument of this chapter will appeal to the plausibility of causal decision theory – in particular, PMKE – when its causal expectation is evaluated relative to appropriate partitions, in the sense of ‘appropriate’ given in Chapter 5. But before proceeding to the more general setting, I shall consider a kind of objection that might be made to the analysis of Chapter 6 to which the theory of this chapter provides an answer.

The “tickle defense”

Skyrms discusses a kind of defense of PMCEU which has come to be known as the “tickle defense” (1980a: 130–2). Consider again the cholesterol case. If the way in which the lesion caused high cholesterol intake was by producing a detectable tickle of a certain kind in the taste buds and if the agent knew this and noticed whether or not he had that tickle, then obviously PMCEU would give the correct prescription: the man would know whether or not he had the lesion and, either way, PMCEU would clearly recommend the eggs benedict. This is because having the tickle (and thus the lesion) – or not having the tickle (and thus not having the lesion) – screens off the causal, and thus the probabilistic, correlation between high cholesterol intake and hardening of the arteries: given the tickle – or given its absense – the probability of atherosclerosis is the same conditional on high cholesterol intake and conditional on low to medium cholesterol intake. And if the counterpart of the tickle exists for the other Newcomb situations, then PMCEU will give the right answers there as well.

Decision theory and other theories, maxims and principles about human behavior often have both normative and descriptive interpretations. Such is the case with SEU theory. As normative, it is a theory about how we should make decisions; as descriptive, it is a theory about how we do make decisions. Both as a normative and as a descriptive theory, it is a model of human rationality. As normative, the theory is an outline of part of a philosophical account of the concept of rationality: it is a theory about how one's actions, preferences, values and beliefs must be related to each other for them to be rationally so related. Thus, it is not surprising that the theory has been applied to philosophical problems in areas where the concept of rationality is central: epistemology and moral theory. In this chapter, I will explore some ways in which SEU theory can be applied in the theories of induction and value. As descriptive, SEU theory is an outline of a psychological account of human behavior. Thus, it is not surprising that the theory has been put to much empirical test. In this chapter, I will also describe some of this empirical research and discuss its descriptive and normative implications for SEU theory.

Finally, there is an aspect of SEU theory which, perhaps, is properly a part of the descriptive aspect but which, because of its special and detachable philosophical importance, I will distinguish as ‘the explanatory aspect’. According to this part of the theory, preference and choice phenomena are explainable in terms of theoretical degrees of belief and degrees of desire: one makes certain choices and has certain preferences because of the existence of certain theoretical entities: certain degrees of belief and desire. Thus, preference and choice are the observable manifestations of degrees of belief and desire. This suggests the possibility that these degrees can be measured from their observational manifestations. And, indeed, the representation theorems, described earlier, show that this can, in principle, be done. Of course it must be argued that the theoretical entities lying behind choice and preference are actually degrees of belief and degrees of desire. And a dispositional theory of belief and desire – what might be called the philosophical foundations of this explanatory aspect of SEU theory and this measurement technique – makes this plausible.

In this chapter, I will present two prima facie counterexamples to Jeffrey's PMCEU of the kind inspired by Newcomb's paradox, briefly describe a number of others of the same general kind and try to clarify their structure. Newcomb's paradox itself will be discussed in Chapter 8.

The prima facie counterexamples are best understood as establishing a prima facie conflict between PMCEU and another intuitively plausible and well-respected principle of choice which, though not very broadly applicable, is almost certainly true. The second principle is a revised version of Savage's principle of dominance that takes into account the possibility that the states are not act-independent. Henceforth, when referring to the principle of dominance (PDOM, for short), I shall be referring to the revised version, stated below.

PDOM will here be given a propositional formulation. That is, the acts, states and outcomes referred to in my statement of the principle will be propositions, à la Jeffrey, to the effect that the agent performs such and such an act, that such and such state of the world obtains and that such and such an outcome accrues to the agent, respectively. To state PDOM, I must first give the following definition: Where A and B are any available acts and {F1, … Fr} is any set of mutually exclusive and collectively exhaustive decision-relevant propositions, A dominates B in the agent's preferences relative to {F1, … Fr} if, and only if, D(A & Fi) > D(B & Fi), for i = 1, … r. PDOM states: If (i) some act, A, dominates every other act in your preferences relative to some partition and (ii) you believe that which act you perform does not causally affect which element of that partition is true, then do A.

Let the states and outcomes considered by the agent be as labeled in the section on Jeffrey in the previous chapter. Then a consequence of PDOM, a weaker version, relies on the following definition: For any acts A and B, A dominates B in the agent's preferences if, and only if, for any two (maximally specific) decision-relevant propositions of the forms A & Si & Oj and B & Si & Oj, D(A & Si, & Oj) > D(B & Si & Oj).

In this chapter, the basic elements of three developments of Bayesian decision theory will be presented: the theories of Ramsey's, Savage's and Jeffrey's. (Since the subsequent chapters will focus on Jeffrey's theory, some readers may wish to concentrate only on the third section of this chapter.) Not all the details will be given, and the discussion will be, for the most part, informal. However, sufficient detail will be given to make possible a discussion of the relative merits of the three theories. The two main problems to which the theories provide different answers are: (i) What are the entities to which the agent's subjective probabilities and desirabilities attach? And (ii) How are these subjective probabilities and desirabilities measured? The points of comparison among the three theories on which this chapter will focus involve the nature of the basic entities assumed by the theories and how they enter into the calculation of subjective expected utility.

Ramsey

In his essay, ‘Truth and Probability’ (1926), Ramsey is primarily concerned with the problem of defining ‘degree of belief’, and he suggests a method of measuring an agent's subjective probabilities and desirabilities from his preferences. The basic entities in Ramsey's theory are: outcomes, propositions, gambles and a preference relation on the set of outcomes and gambles. Ramsey calls the outcomes ‘possible worlds’; so the outcomes should be thought of as maximally specific relative to the set of eventualities which the agent considers to be relevant to his happiness. They are “the different totalities of events between which our subject chooses – the ultimate organic unities” (1926: 176–7, my italics). Gambles are constructed from outcomes and propositions. A gamble is an arrangement under which the agent gets some specified outcome if a given proposition is true and another specified outcome if the proposition is false. I will symbolize gambles according to the following pattern: ‘[Oi, p, Oj]’ denotes the gamble: The agent gets outcome Oi if proposition p is true and outcome Oj if p is false. Note that a gamble [Oi, p, Oj] is the same as [Oj, –p, Oi]. Also note that, as pointed out earlier, outcomes can be thought of as gambles of a kind: an outcome Oi can be thought of as the gamble [Oi, p, Oi], or [Oi, p ˅ –p, Oj].

In the Newcomb situations described in Chapter 4, the principle of dominance and our intuitions agreed that one act was correct while it seemed that PMCEU prescribed the other act. Newcomb's paradox, presented in the introduction, involves a decision situation which is, as we shall see, essentially identical in structure to the Newcomb situations of Chapter 4. However, people's intuitions seem not to be as committed to the correctness of the prescription of the principle of dominance in the decision situation of Newcomb's paradox (i.e., taking both boxes) as they are in the other, less fantastic, Newcomb situations. Indeed, there is much controversy as to which act is correct in Newcomb's paradox. As I indicated in the introduction, I believe that the correct act is to take the contents of both boxes. However, my goal in this chapter is not to convince the reader of this. I shall do three things in this chapter, the third of which is the most important. First, I shall summarize some of the main existing arguments for the correctness of one act or the other in Newcomb's paradox. My main reason for presenting these arguments – with minimal evaluation – is to help the reader gain a deeper understanding of the controversy and the kinds of considerations brought to bear on each side. Second, I shall suggest a way of making one of the arguments in favor of taking the contents of both boxes more persuasive. And third, I shall show how the analysis of Newcomb situations developed in the previous chapters can be applied to Newcomb's paradox. This will show that, contrary to what has generally been thought, one can consistently be both a CEU-maximizer and a “two-boxer”: the choice between being a one-boxer and being a two-boxer is not a choice between being and not being a CEU-maximizer.

I shall present three one-box arguments and three two-box arguments. The first one-box argument is the best known and was already presented in the introduction: Given that I take only the contents of the opaque box, the probability is almost 1 that the predictor predicted this, so that I would get the $1,000,000; given that I take the contents of both boxes, the probability is almost 1 that the predictor predicted this, so that I would get only $1,000; therefore I should take only the contents of the opaque box.

Bayesianism is usually characterized as the philosophical view that, for many philosophically important purposes, probability can usefully be interpreted subjectively, as an individual's “rational degree of belief,” and that the rational way to assimilate new information into one's structure of beliefs is by a process called ‘conditionalization’. The subjective interpretation of probability is connected, however, in very important ways with a mathematically precise and intuitively plausible theory of rational decision, called the ‘subjective expected utility maximization theory’. Because of this connection, and the nature of it, Bayesianism can alternatively be characterized as the view that (i) rational decision and rational preference go by subjective expected utility, (ii) subjective probabilities (and numerical subjective utilities) are more or less theoretical entities that “lie behind,” explain and are given partial empirical interpretation by, an individual's choices and preferences and (iii) learning goes by conditionalization. In this chapter and the next two, I will describe these three aspects of Bayesianism, discuss their plausibility and indicate various ways in which they are philosophically significant. The subsequent chapters will deal with a formidable challenge to this potentially very powerful philosophical theory.

Subjective expected utility

Deliberation is the process of envisaging the possible consequences of pursuing various possible courses of action and evaluating the merits of the possible courses of action in terms of their possible consequences. Roughly, the Bayesian model says that a course of action has merit to the extent that it makes good consequences probable and that a rational person pursues a course of action that makes the best consequences the most probable, where the goodnesses and probabilities of the consequences are the agent's subjective assessments thereof: how true, reasonable or otherwise objectively or morally sound these assessments are is regarded as a separate question.

This last point is quite important; it indicates an essential feature of Bayesianism and Bayesian decision theory which is worth fully noting at the outset. It is implicit in the theory that whether or not a given course of action in a given decision making situation is rational is not an absolute kind of thing: a course of action is rational only relative to a possessed body of information (beliefs and desires) in terms of which the merits of the available courses of action can be rationally evaluated.

Imagine yourself in the following situation. There are two boxes before you: one transparent and one opaque. You can see that there is $1,000 in the transparent box, and you know that there is either $1,000,000 or nothing in the opaque box. You must choose between the following two acts: take the contents only of the opaque box or take the contents of both boxes. Furthermore, there is a being in whose predictive powers you have enormous confidence, and you know that he has already determined the contents of the opaque box according to the following rules: If he predicted that you would take the contents only of the opaque box, he put the $1,000,000 in the opaque box, and if he predicted that you would take the contents of both boxes, he put nothing in the opaque box. What would you do?

A paradox, known as ‘Newcomb's paradox’, seems to arise. Robert Nozick (1969) discusses the paradox in detail. It appears that two principles of decision – both of which are well-respected and intuitively attractive – prescribe different courses of action in the decision situation described above. Consider this version of the principle of dominance: If (i) you must perform either act A or act B, (ii) which act you perform does not causally affect which of two states of affairs, S and –S, obtains, and (iii) no matter which of S and –S obtains, you are better off doing A than doing B, then do A. In the decision situation described above, (i) you must either take the contents only of the opaque box or take the contents of both boxes, (ii) which act you perform does not affect whether or not the $1,000,000 is in the opaque box, and (iii) whether the $1,000,000 is in the opaque box or not, you get $1,000 more by taking the contents of both boxes than you get by taking the contents only of the opaque box. So the principle of dominance recommends taking the contents of both boxes.

Now consider this rough statement of the principle of maximizing conditional expected utility (hereafter, PMCEU): perform the act that makes the most desirable outcomes the most probable.

This book deals mainly with the decision-theoretic paradox known as ‘Newcomb's paradox’, with related examples of decision problems and with the current philosophical controversy surrounding the principle of maximizing expected utility, “causal decision theory” and such decision problems. I have taken care, in two ways, to make the book self-contained and generally accessible. First, two appendixes review the elementary logic and probability theory occasionally used; and second, the first three chapters present, somewhat in the way of a review, the more general “Bayesian” philosophical ideas and theories in relation to which the issues dealt with in the bulk of the book are philosophically significant. Also, the bibliography, though incomplete in relation to all the published work in the area, should be helpful to those interested in further pursuing the questions dealt with here.

I first became interested in these issues as a graduate student at the University of California at Berkeley. I am especially grateful to Charles Chihara, who supervised my Ph.D. dissertation, for all of his help and advice in all aspects and stages of the development of that project, out of which this book has arisen, and for his continued interest and advice. I have also benefited from discussions with Ernest Adams. Discussions with Brian Skyrms inspired many and corrected many of my views on his theory of K-expectation. I have had the benefit of discussing the ideas of Chapter 6 at talks given at the University of Chicago, North Carolina State University at Raleigh, the University of Wisconsin at Milwaukee, the University of Illinois at Chicago Circle, State University of New York at Buffalo and the University of Wisconsin at Madison. Richard Jeffrey and David Lewis have offered helpful comments and supplied me with useful pre-publication manuscripts dealing with the relevant issues. I am also indebted to N. A. Blue, James H. Fetzer, Morry Lipson and especially D. H. Mellor, editor of the series, for many helpful and strategic suggestions in connection with the introductory chapters.