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THE ALLAIS PARADOX The Allais Paradox, discovered by Maurice Allais, provides an example in decision theory of preferences that violate the most widely accepted normative theory of decision making, expected utility theory. This entry briefly explains expected utility theory along with the paradox, and describes responses to the paradox. The Paradox Consider a decision maker choosing between lotteries, i.e., probability distributions over outcomes. According to expected utility theory, as long as the decision maker is rational, her preferences can be represented by a utility function of outcomes with the property that of any two lotteries she prefers the lottery with the higher expected utility value. The idea that decision makers maximize expected utility – the “expected utility hypothesis” – was put forth in part to account for the fact that many decision makers are risk averse in the sense that they would rather have, for example, a sure-thing amount of money than a lottery with the same average monetary value. While such behavior is not consistent with maximizing expected monetary value, it is consistent with maximizing expected utility, relative to a concave utility function of money. The link between a decision maker’s preferences and her utility is cemented by “representation theorems”: these theorems show that being (representable as) an expected utility maximizer is equivalent to having preferences that satisfy particular axioms. One of the earliest and most influential axiomatizations is that of John von Neumann and Oskar Morgenstern. The axioms of von Neumann and Morgenstern’s theorem – and those employed in representation theorems in general – seem to many to be requirements of rational preferences. The Allais Paradox is a counterexample to the expected utility hypothesis. Allais asks us to consider the following choice scenarios. First, we are asked whether we prefer situation A or situation B: Situation A: $100 million for certain. Situation B: a 10% chance of $500 million; an 89% chance of $100 million; a 1% chance of nothing. We are then asked whether we prefer situation C or situation D: Situation C: an 11% chance of $100 million; an 89% chance of nothing. Situation D: a 10% chance of $500 million; a 90% chance of nothing. Allais hypothesized that most people strictly prefer A to B and also strictly prefer C to D, on the grounds that in the first choice scenario the advantages of certain gain from A outweigh the perhaps higher but uncertain gain from B, but that in the choice between C and D, the much higher gain outweighs the slightly higher probability of a much lower gain. This pattern of preferences has been confirmed experimentally. As mentioned, this pattern of preferences violates the expected utility hypothesis: there is no possible assignment of utility values to $0, $100 million, and $500 million such that A has a higher expected utility than B and D has a higher expected utility than C. (Once can strictly prefer A to B and C to D, or one can strictly prefer B to A and D to C, but no other combination of strict preference satisfies the expected utility hypothesis). The particular axiom that these preferences violate is von Neumann and Morgenstern’s Independence Axiom. Responses Broadly speaking, there are two ways to take a decision theory: as an analysis of the canons of instrumental rationality (“normative” decision theory) or as a description of actual people’s preferences (“descriptive” decision theory). For normative decision theorists, the standard choices in Allais’s example are “paradoxical” in that seem rational to many people and yet they violate the dictates of the expected utility theory, which seems to correctly spell out the requirements of rationality. For descriptive decision theorists, the Allais choices aren’t so much paradoxical as they are a counterexample to the idea that expected utility theory is the correct descriptive theory. Descriptive theorists have responded to the paradox by formulating alternative theories that are compatible with the Allais choices. For normative decision theorists, there are three ways to respond to the paradox. The first is to claim that contrary to initial appearances, the Allais choices are simply irrational: although many people unreflectively have the standard Allais preferences, once an individual sees that her preferences violate the Independence Axiom, she ought to re-evaluate her preferences and bring them in line with expected utility theory. A second response to the paradox is to claim that contrary to initial appearances, the Allais choices do satisfy the expected utility hypothesis, and the apparent conflict is due to the fact that the choices have been under-described in the initial setup. Since decision makers prefer A to B on the grounds that A yields $100 million for certain, or on the grounds that an individual who takes B and ends up with nothing will feel regret, this response claims that the actual outcomes in the problem are not simply monetary amounts but also include the decision maker’s feelings about getting those outcomes in each situation. Since it is possible to assign utility values to, e.g., the outcomes $0, $100, $500, and $0 with regret such that the Allais choices maximize expected utility, they do not violate expected utility after all. Finally, normative decision theorists might respond to the paradox by denying the expected utility hypothesis and arguing that expected utility theory is inadequate as a theory of rationality. This response claims that the Allais choices genuinely violate the theory and that they are nonetheless rational. Theorists advocating this response may draw on the abovementioned descriptive theories and argue that the preferences of the decision makers they describe are in fact rational. Lara Buchak

DECISION THEORY by Lara Buchak Forthcoming in the Oxford Handbook of Probability and Philosophy, eds. Christopher Hitchcock and Alan Hájek. OUP. 0. Introduction Decision theory has at its core a set of mathematical theorems that connect rational preferences to functions with certain structural properties. The components of these theorems, as well as their bearing on questions surrounding rationality, can be interpreted in a variety of ways. Philosophy’s current interest in decision theory represents a convergence of two very different lines of thought, one concerned with the question of how one ought to act, and the other concerned with the question of what action consists in and what it reveals about the actor’s mental states. As a result, the theory has come to have two different uses in philosophy, which we might call the normative use and the interpretive use. It also has a related use that is largely within the domain of psychology, the descriptive use. The first two sections of this essay examine the historical development of normative decision theory and the range of current interpretations of the elements of the theory, while the third section explores how modern normative decision theory is supposed to capture the notion of rationality. The fourth section presents a history of interpretive decision theory, and the fifth section examines a problem that both uses of decision theory face. The sixth section explains the third use of decision theory, the descriptive use. Section seven considers the relationship between the three uses of decision theory. Finally, section eight examines some modifications to the standard theory and the conclusion makes some remarks about how we ought to think about the decision-theoretic project in light of a proliferation of theories. 1. Normative Decision Theory The first formal decision theory was developed by Blaise Pascal in correspondence with Pierre Fermat about “the problem of the points,” the problem of how to divide up the stakes of players involved in a game if the game ends prematurely.1 Pascal proposed that each gambler 1See Fermat and Pascal (1654). 2 should be given as his share of the pot the monetary expectation of his stake, and this proposal can be generalized to other contexts: the monetary value of a risky prospect is equal to the expected value of that prospect. Formally, if L = {$x1, p1; $x2, p2; … } represents a “lottery” which yields $xi with probability pi, then its value is: = This equivalence underlies a prescription: when faced with two lotteries, you ought to prefer the lottery with the higher expected value, and be indifferent if they have the same expected value. More generally, you ought to maximize expected value. This norm is attractive for a number of reasons. For one, it enjoins you to make the choice that would be better over the long run if repeated: over the long run, repeated trials of a gamble will average out to their expected value. For another, going back to the problem of the points, it ensures that players will be indifferent between continuing the game and leaving with their share. But there are several things to be said against the prescription. One is that it is easy to generate a lottery whose expected value is infinite, as shown by the St. Petersburg Paradox (first proposed by Nicolas Bernouilli). Under the norm in question, one ought to be willing to pay any finite amount of money for the lottery {$1, ½; $2, ¼; $4, 1/8; … }, but most people think that the value of this lottery should be considerably less. A second problem is that the prescription does not seem to account for the fact that whether one should take a gamble seems to depend on what one’s total fortune is: one ought not risk one’s last dollar for an even chance at $2, if losing the dollar means that one will be unable to eat. Finally, the prescription doesn’t seem to adequately account for the phenomenon of risk-aversion: most people would rather have a sure thing sum of $x than a gamble whose expectation is $x (for example, $100 rather than {$50, ½; $150, ½}) and don’t thereby seem irrational. In response to these problems, Daniel Bernouilli (1738) and Gabriel Cramer (see Bernouilli 1738: 33) each independently noted that the amount of satisfaction that money brings diminishes the more money one has, and proposed that the quantity whose expectation one ought to maximize is not money itself but rather the “utility” of one’s total wealth. (Note that for Bernouilli, the outcomes are total amounts of wealth rather than changes in wealth, as they were for Pascal.) Bernouilli proposed that an individual’s utility function of total wealth is u($x) = log($x). Therefore, the new prescription is to maximize: 3 = $ = $ This guarantees that the St. Petersburg lottery is worth a finite amount of money; that a gamble is worth a larger amount of one’s money the wealthier one is; and that the expected utility of any lottery is less than the utility of its monetary expectation. Notice that the norm associated with this proposal is objective in two ways: it takes the probabilities as given, and it assumes that everyone should maximize the same utility function. One might reasonably wonder, however, whether everyone does get the same amount of satisfaction from various amounts of money. Furthermore, non-monetary outcomes are plausibly of different value to different people, and the proposal tells us nothing about how we ought to value lotteries with non-monetary outcomes. A natural thought is to revise the norm to require that one maximize the expectation of one’s own, subjective utility function, and to allow that the utility function take any outcome as input. The problem with this thought is that it is not clear that individuals have access to their precise utility functions through introspection. Happily, it turns out that we can implement the proposal without such introspection: John von Neumann and Oskar Morgenstern (1944) discovered a representation theorem that allows us to determine whether an agent is maximizing expected utility merely from her pair-wise preferences, and, if she is, allows us to determine an agent’s entire utility function from these preferences. Von Neumann and Morgenstern identified a set of axioms on preferences over lotteries such that if an individual’s preferences conform to these axioms, then there exists a utility function of outcomes, unique up to positive affine transformation, that represents her as an expected utility maximizer.2 The utility function represents her in the following sense: for all lotteries L1 and L2, the agent weakly prefers L1 to L2 if and only if L1 has at least as high an expected utility as L2 according to the function. Thus, we can replace expected objective utility maximization with expected subjective utility maximization as an implementable norm, even if an agent’s utility function is opaque to her. 2I will often talk about an agent’s utility function when strictly speaking I mean the family of utility functions that represents her. However, facts about the utility function that are not preserved under affine transformation, such as the zero point, will not count as “real” facts about the agent’s utility values. 4 Leonard Savage’s (1954) representation theorem took the theory one step further. Like von Neumann and Morgenstern, Savage allowed that an individual’s values were up to her. But Savage was interested not primarily in how an agent should choose between lotteries when she is given the exact probabilities of outcomes, but rather in how an agent should choose between ordinary acts when she is uncertain about some feature of the world: for example, how she should choose between breaking a sixth egg into her omelet and refraining from doing so, when she does not know whether or not the egg is rotten. Savage noted that an act leads to different outcomes under different circumstances, and, taking an outcome to be specified so as to include everything an agent cares about, he defined the technical notion of an act as a function from possible states of the world to outcomes.3 For example, the act of breaking the egg is the function {egg is good -> I eat a 6-egg omelet; egg is rotten -> I throw away the omelet}. More generally, we can represent an act f as {x1, E1; … ; xn, En}, where Ei are mutually exclusive and exhaustive events (an event being a set of states), and each state in Ei results in outcome xi under act f.4 Savage’s representation theorem shows that an agent’s preferences over these acts suffice to determine both her subjective utility function of outcomes and her subjective probability function of events, provided her pair-wise preferences conform to the axioms of his theorem.5 Formally, u and p represent an agent’s preferences if and only if she prefers the act with the highest expected utility, relative to these two functions: = Savage’s theory therefore allows that both the probability function and the utility function are subjective. The accompanying prescription is to maximize expected utility, relative to these two functions. Since Savage, other representation theorems for subjective expected utility theory have been proposed, most of which are meant to respond to some supposed philosophical problem 3Savage used the terminology “consequence” where I am using “outcome.” 4Savage also treats the case in which the number of possible outcomes of an act is not finite (Savage (1954: 76-82), although his treatment requires bounding the utility function. Assuming each act has a finite number of outcomes will simplify the discussion. 5Again, the utility function is unique up to positive affine transformation. The probability function is fully unique. 5 with Savage’s theory.6 One set of issues surrounds what we should prefer when utility is unbounded and acts can have an infinite number of different outcomes, or when outcomes can have infinite utility value.7 Another set of issues concerns exactly what entities are the relevant ones to assign utility and probability to in decision-making. The developments in this area begin with Richard Jeffrey (1965), who objected to Savage’s separation between states, outcomes, and acts, and argued that the same objects ought to be the carriers of both probability and value. Jeffrey proposed a theory on which both the probability and utility function take propositions as inputs. Axiomatized by Ethan Bolker (see Jeffrey 1965: 142-3, 149), Jeffrey’s theory enjoins the agent to maximize:8 = | & where Si and A both stand for arbitrary propositions (they range over the same set), but Si is to play the role of a state and A of an act. Bolker’s (1965-67) representation theorem provides axioms on a preference relation over the set of propositions that allow us to extract p and u, although the uniqueness conditions are more relaxed than in the aforementioned theories. Jeffrey proposed that we ought to interpret the items that an agent has preferences over as “news items”; so, for example, one is asked whether one would prefer the news that one breaks the egg into one’s omelet or that one does not. The connection to action, of course, is that one has the ability to create the news when it comes to propositions about acts one is deciding between. Certain features of Jeffrey’s interpretation are inessential to the maximization equation. It is not necessary to follow Jeffrey in interpreting preferences as being about news items. Nor is there consensus that p and u ought to have as their domain the same set of objects.9 For 6See Fishburn (1981) for a helpful catalogue of some of these. 7See, for example, Vallentyne (1993), Nover and Hájek (2004), Bartha (2007), Colyvan (2008), and Easwaran (2008). 8Jeffrey used a slightly different, but equivalent, formulation. He also used functions named prob and des rather than p and u, but the difference is terminological. 9Of course, while this feature is inessential to Jeffrey’s maximization equation as written above, it is essential to Bolker’s representation theorem. 6 example, while it is clear that we can assign utility values to acts under our own control, Wolfgang Spohn (1977) and Isaac Levi (1991) each argue that we cannot assign these probability. Another issue with Jeffrey’s theory has been the source of a significant development in decision theory. Because the belief component of Jeffrey’s theory corresponds to conditional probabilities of states given acts, this component will have the same numerical value whether an act causes a particular outcome or is merely correlated with it. Therefore, agents will rank acts that are merely correlated with preferred outcomes the same as acts that tend to cause preferred outcomes. This is why Jeffrey’s theory has come to be known as evidential expected utility (EEU) theory: one might prefer an act in part because it gives one evidence that one’s preferred outcome obtains. Many have argued that this feature of the theory is problematic, and the problem can be brought out by a case know as Newcomb’s problem (first discussed by Robert Nozick (1969)). Here is the case. You are presented with two boxes, one closed and one open so that you can see its contents; and you may choose either to take only the closed box, or to take both boxes. The open box contains $1000. The contents of the closed box were determined as follows. A predictor predicted ahead of time whether you would choose to take the one box or both; if he predicted that you would take just the closed box, he’s put $1M in the closed box, but if he predicted that you would take both, he’s put nothing in the closed box. Furthermore, you know that many people have faced this choice and that he’s predicted correctly every time. Assuming you prefer more money to less, evidential EU theory recommends that you take only one box, since the relevant conditional probabilities are one and zero (or close thereto): p(there is $1M in the closed box | you take one box) ≈ 1, and p(there is $0 in the closed box | you take two boxes) ≈ 1. But many think that this is the wrong recommendation. After all, the closed box already contains what it contains, so your choice is between receiving whatever is in that box and receiving whatever is in that box plus an extra thousand dollars. Taking two boxes dominates taking one box, the argument goes: it is better in every possible world. We might diagnose the mis-recommendation of EEU theory as follows: p($1M | one box) is high because taking one box is correlated with getting $1M, but taking one box cannot cause $1M to be in the box because the contents of the box have been already determined; and so EEU gets the recommendation wrong because conditional probability does not distinguish between correlation 7 and causation. Not everyone accepts that two-boxing is the correct solution: those who advocate one-boxing point out that those who take only one box end up with more money, and since rationality ought to direct us to the action that will result in the outcome we prefer, it is rational to take one box. However, those who advocate two-boxing reply that even though those who take only one box end up with more money, this is a case in which they are essentially rewarded for behaving irrationally. For those who advocate two-boxing, one way to respond to this problem is to modify evidential EU theory by adding a condition like ratifiability (Jeffrey 1983: 19-20), which says that one can only pick an act if it still has the highest EU on the supposition that one has chosen it. However, this does not solve the general problem of distinguishing A’s being evidentially correlated with S from A’s causing S. To yield the two-boxing recommendation in the Newcomb case, as well as to address the more general problem, Allan Gibbard and William Harper (1978) proposed causal expected utility theory, drawing on a suggestion of Robert Stalnaker (1972). Causal expected utility theory enjoins an agent to maximize: = where □ ⟶ & □ ⟶ stands for the probability of the counterfactual “If I were to do A then Si would happen.”10 Armendt (1986) proved a representation theorem for the new theory, and Joyce (1999) provided a unified representation theorem for both evidential and causal expected utility theory. Causal expected utility theory recommends two-boxing if the pair of counterfactuals “If I were to take one box, there would be $0 in the closed box” and “If I were to take two boxes, there would be $0 in the opaque box” are assigned the same credence, and similarly for the corresponding pair involving $1M in the opaque box. This captures the idea that the contents of the closed box are independent of the agent’s choices, and vindicates the reasoning that taking two boxes will result in an extra thousand dollars: 1 2 = 1 □⟶ $0 $0+ 1 □⟶ $1 $1 = 2 □⟶ $0 $1 + 2 □⟶ $1 $1 +$1 10 Other formulations of causal decision theory include that of Lewis (1981) and Skyrms (1982). 8 To get this result, it is important that the counterfactuals in question are what Lewis (1981) calls “causal” counterfactuals rather than “back-tracking” counterfactuals. For there are two senses in which the counterfactuals “If I were to take one box, there would be $0 in the closed box” and “If I were to take two boxes, there would be $0 in the closed box” can be taken. In the backtracking sense, I would reason from the supposition that I take one box back to the conclusion that the predictor predicted I would take one box, and I would assign a very low credence to the former counterfactual; but I would by the same reasoning assign a very high credence to the latter. In the causal sense, I would hold fixed facts about the past, since I cannot now cause past events, and the supposition that I take one box would not touch facts about what the predictor did; and by this reasoning I would assign equal credence to both counterfactuals. It is worth considering how Savage’s original theory would treat the Newcomb problem. Savage’s theory uses unconditional credences, but correctly resolving the decision problem depends on specifying the states, outcomes, and acts in such a way that states are independent of acts. So, in effect, Savage’s theory is a kind of causal decision theory. Indeed, Lewis (1981: 13) thought of his version of causal decision theory as returning to Savage’s unconditional credences, but building the correct partition of states into the formalism itself rather than relying on an extra-theoretical principle about entity-specification. All of the modifications mentioned here leave the basic structure of the theory intact – probability and utility are multiplied and then summed – and treat both p and u as subjective, so we can put them all under the heading of subjective expected utility theory (hereafter EU theory). How should we understand the two functions, p and u, involved in EU theory? In the case of the probability function, although there is debate over whether p is defined by preferences (“betting behavior”) via a representation theorem or whether preferences are merely a way to discover p, it is widely acknowledged that p is supposed to represent an agent’s beliefs. In the case of the utility function, there are two philosophical disagreements. First, there is a disagreement about whether the utility function is defined by or merely discovered from preferences. If one thinks the utility function is defined by preferences, there is a further question about whether it is merely a convenient way to represent preferences or whether it refers to some pre-theoretical, psychologically real entity like strength of desire or perceived amount of satisfaction. Functionalists, for example, hold that utility is (at least partially) constituted by its role in preferences but also hold that utility is psychologically real. Since the 9 term “realism” is sometimes used to refer to the view that utility is independent of preferences, and sometimes used to refer to the view that utility is a psychologically real quantity, I will use the following terminology. I will call the view that utility is discovered from preferences nonconstructivist realism and the view that utility is defined from preferences constructivism. I will call the view that utility does correspond to something psychologically real psychological realism and the view that utility does not refer to any real entity formalism.11 Non-constructive realist views will be psychologically realist as well; however, functionalism counts as a constructivist, psychological realist view. Hereafter, when I am speaking of psychological realist theories, I will speak as if utility corresponds to desire, just as subjective probability corresponds to belief, though there may be other proposals about what utility corresponds to. 2. The Norm of Normative Decision Theory Representation theorems connect preferences conforming to a set of axioms on the one hand to utilities and probabilities such that preferences maximize expected utility on the other. Thus, representation theorems give us an equivalent way to state the prescription that one ought to maximize expected utility: one ought to have preferences that accord with the axioms. The upshot of this equivalence depends on which theory of utility one adopts. For psychological realists, both formulations of the norm may have some bite: the “maximization” norm is a norm about how preferences ought to be related to beliefs and desires, and the “axiom” norm is an internal norm on preferences. For formalists, since there is really no such thing as utility, the only sensible formulation of the norm is as the axiom norm. But for both interpretations, an important advantage of the representation theorems is that judgments about whether an agent did what she ought, as well as arguments about whether EU theory identifies a genuine prescription, can focus on the axioms. A point of clarification about the equivalent ways to state the norm of EU theory is needed. “Maximize expected utility” admits of two readings, one narrow-scope (“Given your utility function, maximize its expectation”) and one wide-scope (“Be such that there is a utility 11 The term constructivism comes from Dreier (1996), and the term formalism comes from Hanssen (1988). Bermúdez (2009) uses “operationalism” for what I call formalism. Zynda (2000) uses “strong realism” for what I call non-constructivist realism and “weak realism” for what I call psychological realism. 10 function whose expectation you maximize”). And the axiom norm is only equivalent to the wide-scope maximization norm. For the narrow-scope norm to apply in cases in which one fails to live up to it, one must be able to count as having a utility function even when one does not maximize its expectation. Clearly, this is possible according to the non-constructivist realist.12 I will also show that in many cases, it is possible according to all psychological realists. We can note the historical progression: in its original formulations, decision theory was narrow-scope, and the utility function (or its analogue) non-constructivist realist: money had an objective and fixed value. However, wide-scope, constructivist views are most popular nowadays. Relatedly, whereas originally a central justification of the norm was via how well someone who followed it did over the long run, such justifications have fallen out of favor and have been replaced by justification via arguments for the axioms. One final point of clarification. So far, we have been talking about the relationship of beliefs and desires to preferences. But one might have thought that the point of a theory about decision-making was to tell individuals what to choose. The final piece in the history of decision theory concerns the relationship between preference and choice. In the heyday of behaviorism, Samuelson’s (1938) idea of “revealed preference” was that preference can be cashed out in terms of what you would choose. However, nowadays philosophers mostly think the connection between preference and choice is not so tight. Throughout the rest of this article, I will use preference and choice interchangeably, while acknowledging that I take preference to be more basic and recognizing that the relationship between the two is not a settled question. There are two ways to take the norm of normative decision theory: to guide to one’s own actions or to assess from a third-person standpoint whether a decision-maker is doing what she 12 However, there is an additional problem with the wide-scope norm for the non-constructivist realist: maximizing the expectation of some utility function doesn’t guarantee that you’ve maximized the expectation of your own utility function. The connection between the utility function that is the output of a representation theorem and the decisionmaker’s actual utility function would need to be supplemented by some principle, such as a contingent version of Christensen’s (2001) “Representational Accuracy” or by his “Informed Preference.” 11 ought.13 Having explained the norm of normative decision theory, I now turn to the question of what sort of “ought” it is supposed to correspond to. 3. Rationality Decision theory is supposed to be a theory of rationality; but what concept of rationality does it analyze? Decision theory is sometimes said to be a theory of instrumental rationality – of taking the means to one’s ends – and sometimes said to be a theory of consistency. But it is far from obvious that instrumental rationality and consistency are equivalent. So it is worth spending time on what each is supposed to mean and how EU theory is supposed to analyze each; and in what sense instrumental rationality and consistency come to the same thing. Let us begin with instrumental rationality and with something else that is frequently said about decision theory: that it is “Humean.” Hume distinguished sharply between reason and the passions and said that reason is concerned with abstract reasoning and with cause and effect, and while a belief can be contrary to reason, a passion (or in our terminology, a desire) is an “original existence” and cannot itself be irrational. As his famous dictum goes, “’Tis not contrary to reason to prefer the destruction of the whole world to the scratching of my finger.”14 Hume thinks that although we cannot pass judgment on the ends an individual adopts, we can pass judgment if she chooses means insufficient for her ends. To see how decision theory might be thought to provide this kind of assessment, consider the psychological realist version of the theory in which an individual’s utility function corresponds to the strengths of her desires. This way of thinking about the theory gives rise to the natural suggestion that the utility function captures the strength of an agent’s desires for various ends, and the dictum to maximize expected utility formalizes the dictum to prefer (or choose) the means to one’s ends. 13 Bermúdez (2009) distinguishes these as two separate uses: what I call using normative decision theory to guide one’s own actions he calls the “action-guiding” use, and what I call using normative decision theory for third-person assessment he calls the “normative” use; however, he includes more in the normative use of decision theory than just assessing whether the agent has preferences that conform to the norm of EU theory, such as assessing how well she set up the decision problem and her substantive judgments of desirability. 14 Hume (1731: 416). 12 The equivalence of preferring the means to one’s ends and maximizing expected utility is not purely definitional. True, to prefer the means to one’s ends is to prefer the act with the highest utility: to prefer the act that leads to the outcome one desires most strongly. However, in the situations we are concerned with, it is not clear which act will lead to which outcome – one only knows that an act will lead to a particular outcome if a particular state obtains – so one cannot simply pick the act that will lead to the more preferred outcome. Therefore, there is a real philosophical question about what preferring the means to your ends requires in these situations. EU theory answers this substantive question by claiming that you ought to maximize the expectation of the utility function relative to your subjective probability function. So if we cash out EU theory in the means-ends idiom, it requires you not precisely to prefer the means to your ends, but to prefer the means that will, on average and by your own lights, lead to your ends. It also requires that you have a consistent subjective probability function and that the structure of desires is such that a number can be assigned to each outcome. So it makes demands on three kinds of entities: beliefs, desires, and preferences given these. This formulation of the maximization norm is compatible with both the narrow-scope and the wide-scope reading: if in concert with Hume’s position we think that desires cannot be changed by reason, there will be only one way to fulfill this requirement; but if we think that the agent might decide to alter her desires, there will be multiple ways to fulfill this requirement. A more modern formulation of the idea that decision theory precisifies what it is to take the means to one’s ends is that decision theory is consequentialist. This is to say that it is a principle of decision theory that acts must be valued only by their consequences. An important justification of the norm of EU theory as the unique consequentialist norm, and a justification that formalists and psychological realists can both avail themselves of, comes from Hammond (1988). Hammond considers sequential decision problems (decision problems in “extensive” form rather than “normal” form), where decision-makers are not choosing only once but instead can revise their plan of action as new information comes in. He argues that the assumption that decision-makers value acts only for their consequences, when cashed out in terms of some seemingly plausible principles about sequential choice, entails the substantive axioms of EU theory.15 15 For further discussion of this type of argument, see Seidenfeld (1988), McClennen (1990), and Levi (1991). 13 Even in the case of choice at a time, I think we can think of the axioms as trying to formalize necessary conditions to preferring the means to one’s ends. I don’t have space to pursue the suggestion here, but here is one example of what I have in mind. Consider the requirement of state-wise dominance, which says roughly that if act f is weakly preferred to act g in every state, and strictly preferred in some state that has positive probability, then you ought to strictly prefer f to g (this is a necessary condition of being representable as an EU maximizer). One plausible way to state what’s wrong with someone whose preferences don’t conform to this requirement is that they fail to prefer what they believe is superior in terms of satisfying their preferences, or they fail to connect their preferences about means to their preferences about ends. Not all of the axioms can be straightforwardly argued for in this way, but this can be a helpful way to think about the relationship of the axioms to instrumental rationality. Thus, normative EU theory may be supported by arguments to the effect that the maximization norm or the axiom norm spell out instrumental rationality (leaving aside whether these arguments are ultimately successful). The other notion of rationality that decision theory is often described as analyzing is consistency, and it seems that the axiom formulation of the norm coheres well with this. To understand why, it is helpful to consider the related idea that logic analyzes what it is to have consistent binary beliefs. There are two important standards at work in binary belief. First, an agent ought (roughly) to believe what is reasonable to believe, given her evidence. This is a requirement about the substance of her beliefs, or about the content of her beliefs vis-à-vis her evidence or what the world is like. Second, an agent’s beliefs ought to be consistent with one another in the sense elucidated by logic. This is a requirement about the structure of her beliefs, or about the content of her beliefs vis-à-vis the content of her other beliefs. This isn’t to say that everyone holds that agents must be logically perfect or omniscient, or that everyone holds that there is an external standard of adherence to the evidence, but the point is that these are two different kinds of norms and we can separately ask the questions of whether a believer conforms to each. Similarly, in evaluating preferences over acts, there are two questions we might ask: whether an agent’s preferences are reasonable, and whether they are consistent. Here, the axioms of decision theory are supposed to play a parallel role that the axioms of logic play in beliefs: without regard to the content of an agent’s preferences, we can tell whether they obey the axioms. So just as the axioms of logic are supposed to spell out what it is to have consistent 14 binary beliefs, so too are the axioms of decision theory supposed to spell out what it is to have consistent preferences. There are several ways in which it might be argued that the axioms correctly spell out what it is to have consistent preferences.16 One classic argument purports to show that violating one or more of them implies that you will be the subject of a “money pump,” a situation in which you will find a series or set of trades favorable but will disprefer the entire package, usually because taking all of them results in sure monetary loss for you.17 This amounts to valuing the same thing differently under different descriptions – as individual trades on the one hand and as a package on the other – and is thought to be an internal defect rather than a practical liability.18 A different argument, due to Peter Wakker (1988), purports to show that violating one of the axioms will entail that you will avoid certain cost-free information. What I want to propose is that consistency of preferences is an amalgamation of consistency in three different kinds of entities: consistency in preferences over outcomes, consistency in preferences about which event to bet on, and consistency in the relationship between these two kinds of preferences and preferences over acts.19 Or, psychological realists might say: consistency in desires, consistency in beliefs, and consistency in connecting these two things to preferences. Aside from the fact that adhering to the axioms does produce three separate functions (a utility function of outcomes, a probability function of states, and an expectational utility function of acts), which is not decisive, I offer two considerations in favor of this proposal. First, arguments for each of the axioms can focus more or less on each of these kinds of consistency. For example, an argument that transitivity is a rational requirement doesn’t need to say anything about beliefs or probability functions. Second, a weaker set of axioms than 16 Not all philosophers think that arguing for this conclusion is the right way to proceed. For example, Patrick Maher (1993: 62, 83) suggests that no knock-down intuitive argument can be given in favor of EU theory, but that we can justify it by the fruits it produces. 17 Original versions of this argument are due to Ramsey (1926) and de Finetti (1937). 18 See Christensen (1991), although he is mostly concerned with this type of argument as it relates to the subjective probability function. 19 By a preference to bet on E rather than F, I mean a preference to receive a favored outcome on E rather than to receive that outcome on F. 15 those of EU theory will produce a consistent probability function without a utility function relative to which the agent maximizes EU; and a weaker set of axioms than those of EU theory will produce a utility function of outcomes without a probability function relative to which the agent maximizes EU; and a weaker set of axioms than those of EU theory will produce a utility function and a probability function relative to which an agent maximizes something other than EU.20 Therefore, even if the justifications of each of the axioms are not separable into those based on each of the three kinds of consistency, the kinds of consistency are formally separable. And here is a difference, then, between logic and decision theory: logical consistency is an irreducible notion, whereas decision-theoretic consistency is a matter of being consistent in three different ways.21 Here, then, are the ways in which instrumental rationality and consistency are related. First, and most obviously, there are arguments that each is analyzed by EU theory; if these arguments are correct, then instrumental rationality and consistency come to the same thing. Second, given that consistency appears to involve consistency in the three kinds of entities instrumental rationality is concerned with, consistency in preferences can be seen as an internal 20 For an axiomatization of a theory that yields a probability function for a certain kind of non-EU maximizer, see Machina and Schmeidler (1992). For an axiomatization of a theory that yields a utility function for an agent who lacks a subjective (additive) probability function, see Gilboa (1987), or any non-expected utility theory that uses subjective decision weights that do not necessarily constitute a probability function. For an axiomatization of a theory that yields a utility and probability function relative to which an agent maximizes a different functional, see Buchak (forthcoming). 21 Note, however, that for non-constructive realists, there could be a case in which two of these things are inconsistent in the right way but preferences are still consistent. See Zynda (2000: 51-60), who provides an example of an agent whose beliefs are not governed by the probability calculus and whose norm is not expected utility maximization relative to his beliefs, but who has the same preferences as someone who maximizes EU relative to a probability function. 16 check on whether one really prefers the means to one’s ends relative to a set of consistent beliefs and desires.22 It was noted that even if binary beliefs are consistent, we might ask the further question of whether they are reasonable. Can a similar question be applied to preferences? Given that consistency applies to three entities, the question of reasonableness can also be separated into three questions: whether the subjective probability function is reasonable, whether the utility function is reasonable, and whether one’s norm is reasonable. The reasonableness question for subjective probability is an analogue of that for binary beliefs: are you in fact apportioning your beliefs to the evidence? For the formalist, the reasonableness question for utility, if it makes sense at all, will really be about preferences. But for the psychological realist, the reasonableness question for utility might be asked in different ways: whether the strength of your desires in fact tracks what would satisfy you, or whether they in fact track the good. In EU theory, there is only one norm consistent with taking the means to your ends – maximize expected utility – so the reasonableness question appears irrelevant; however, with the introduction of alternatives to EU theory, we might pose the question, and I will discuss this in section eight. 4. Interpretive Decision Theory The major historical developments in normative decision theory mostly came from considering the question of what we ought to do. By contrast, another strand of decision theory was moved forward by philosophical questions about mental states and their relationship to action. 22 Compare to Niko Kolodny’s proposal that wide-scope requirements of formal coherence as such may be reducible to narrow-scope requirements of reason. The “error theory” in Kolodny (2007) proposes that inconsistency in beliefs reveals that one is not adopting, on some proposition, the belief that reason requires; and the error theory in Kolodny (2008) proposes that inconsistency in intentions reveals that one is not adopting the intention that reason requires. Direct application of Kolodny’s proposal to the discussion here is complicated by the fact that some might see the maximization norm as wide-scope and some as narrow-scope. But those who see it as narrow-scope may take a Kolodny-inspired line and think that consistency of preferences is merely an epiphenomenon of preferring that which you have reason to prefer, given your beliefs and desires. 17 In 1926, Frank Ramsey was interested in a precise way to measure degrees of belief, since the prevailing view was that degrees of belief weren’t appropriate candidates to use in a philosophical theory unless there was a way to measure them in terms of behavior. Ramsey noted that since degrees of belief are the basis of action, we can measure the degree of a belief by the extent to which the individual would act on the belief in hypothetical circumstances. Ramsey created a method whereby a subject’s preferences in hypothetical choice situations are elicited and her degrees of belief (subjective probabilities) are inferred through these, without knowing her values ahead of time. For example, suppose a subject prefers getting a certain prize to not getting that prize, and suppose she is neutral about seeing the heads side of a coin or the tails side of a coin. Then if she is indifferent between the gamble on which she receives the prize if the coin lands heads and the gamble on which she receives the prize if the coin lands tails, it can be inferred that she believes to equal degree that the coin will land heads as that it will land tails, i.e., she believes each to degree 0.5. If she prefers getting the prize on the heads side, it can be inferred that she assigns a greater degree of belief to heads than to tails. Generalizing the insight that both beliefs and values can be elicited through preferences, Ramsey presented a representation theorem. Ramsey’s theorem was a precursor to Savage’s, and like Savage’s theorem, Ramsey’s connects preferences to a probability function and a value function, both subjective. Thus, like the normative decision theorists that came after him, Ramsey saw that maximizing expected utility with respect to one’s personal probability and utility functions is equivalent to having preferences that conform to certain structural requirements. However, Ramsey was not interested in using the equivalence to reformulate the maximization norm as a norm about preferences. Rather, he assumed that preferences do conform to the axioms, and used the equivalence to discover facts about the agent’s beliefs and desires. Related to Ramsey’s question of how to measure beliefs is the more general question of attributing mental states to individuals on the basis of their actions. Donald Davidson (1973) coined the term “radical interpretation” (a play on W.V.O. Quine’s “radical translation”) to refer to the process of interpreting a speaker’s beliefs, desires, and meanings from her behavior. For Davidson, this process is constrained by certain rules, among them a principle about the relationship between beliefs and desires on the one hand and actions on the other, which, as David Lewis (1974) made explicit, can be formalized using expected utility theory. Lewis’s 18 formulation of the “Rationalization Principle” is precisely that rational agents act so as to maximize their expectation given their beliefs and desires. Thus, Ramsey’s insight became a part of a more general theory about interpreting others. For theorists who make use of EU theory to interpret agents, maximizing EU is constitutive of (rational) action; indeed, Lewis (1974: 335) claims that the Rationalization Principle has a status akin to analyticity. An immediate tension arises between the following three facts. First, for interpretive theorists, anyone who cannot be interpreted via the Rationalization Principle will count as unintelligible. Second, actual human beings are supposed to be intelligible; after all, the point of the project is to formalize how we make sense of another person. Third, actual human beings appear to violate EU theory; otherwise, the normative theory wouldn’t identify an interesting norm. One line to take here is to retain the assumption that it is analytic that agents maximize EU, and to explain away the apparent violations. We will see a strategy for doing this in the next section, but I will argue there that adopting this strategy in such a way as to imply that EU maximization cannot be violated leads to uninformative ‘interpretations.’ A more promising line starts from the observation that when we try to make sense of another person’s preferences, we are trying to make sense of them as a whole, not of each considered in isolation. Consider an agent whose preferences mostly conform to the theory but fail to in a few particular instances, for example, an individual who generally gets up at 7 AM to go for a run but occasionally oversleeps her alarm. We would say that she prefers to exercise in the morning. Or consider an individual who generally brings an umbrella when the chance of rain is reported as at least 50%, but one time leaves it at home when she thinks it is almost certain to rain. We would say that she considers the burden of carrying around an umbrella only moderate in comparison to how much she does not like to get wet. In general, if a large set of an individual’s preferences cohere, the natural thing to say is that she has the beliefs and desires expressed by those preferences but that her preferences occasionally fail to match up with her beliefs and desires, perhaps because she is on occasion careless or confused or weak of will. This suggests what interpretive theorists ought to do in the case of non-ideal agents: take an agent’s actual preferences, consider the closest “ideal” set of preferences – the closest set of preferences that do conform to the axioms – and infer the agent’s beliefs and desires from these.