Decision theory is concerned with the reasoning underlying anagent’s choices, whether this is a mundane choice between takingthe bus or getting a taxi, or a more far-reaching choice made by apowerful agent about whether to pursue significant social change.(Note that “agent” here stands for an entity, usually anindividual person, that is capable of deliberation and action.)Standard thinking is that what an agent chooses to do on any givenoccasion is completely determined by her beliefs and desires orvalues, but this is not uncontroversial, as will be noted below. Inany case, decision theory is as much a theory of beliefs, desires andother relevant attitudes as it is a theory of choice; what matters ishow these various attitudes (call them “preferenceattitudes”) cohere together.

The focus of this entry is normative decision theory. That is, themain question of interest is what criteria an agent’s preferenceattitudesshould satisfy inany genericcircumstances. This amounts to a minimal account ofrationality, one that sets aside more substantial questionsabout appropriate desires and reasonable beliefs, given the situationat hand. The key issue for a minimal account is the treatment ofuncertainty. The orthodox normative decision theory,expectedutility (EU) theory, essentially says that, in situations ofuncertainty, one should prefer the option with greatestexpected desirability or value. (Note that in this context,“desirability” and “value” should beunderstood as desirability/valueaccording to the agent inquestion.) This simple maxim will be the focus of much of ourdiscussion.

The structure of this entry is as follows: Section 1 discusses thebasic notion of “preferences over prospects”, which liesat the heart of decision theory. Section 2 describes the developmentof normative decision theory in terms of ever more powerful andflexible measures of preferences. Section 3 discusses the twobest-known versions of EU theory. Section 4 considers the broadersignificance of EU theory for practical action, inference, andvaluing. Section 5 turns to prominent challenges to EU theory, whileSection 6 addresses sequential decisions, and how this richer settingbears on debates about rational preferences.

• 1. What arepreferences overprospects?
• 2. Utility measures of preference
  • 2.1 Ordinal utilities
  • 2.2 Cardinalizing utility
  • 2.3 The von Neumann and Morgenstern (vNM) representation theorem
• 3. Making real decisions
  • 3.1 Savage’s theory
  • 3.2 Jeffrey’s theory
• 4. Broader significance of Expected Utility (EU) theory
  • 4.1 On rational belief
  • 4.2 On rational desire
• 5. Challenges to EU theory
  • 5.1 On risk and regret attitudes
  • 5.2 On completeness: Vague beliefs and desires
  • 5.3 Catastrophic risk and the precautionary principle
  • 5.4 Unawareness
• 6. Sequential decisions
  • 6.1 Was Ulysses rational?
  • 6.2 The EU axioms revisited
• 7. Concluding remarks
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. What arepreferences overprospects?

The two central concepts in decision theory arepreferencesandprospects (or equivalently,options). Roughlyspeaking, when we (in this entry) say that an agent“prefers” the “option” \(A\) over \(B\) wemean that the agent takes \(A\) to be more desirable or choice-worthythan \(B\). This rough definition makes clear that preference is acomparative attitude. Beyond this, there is room for argument aboutwhat preferences over options actually amount to, or in other words,what it is about an agent that concerns us when we talk about theirpreferences over options. This section considers some elementaryissues of interpretation that set the stage for introducing (in thenext section) the decision tables and expected utility rule that formany is the familiar subject matter of decision theory. Furtherinterpretive questions regarding preferences and prospects will beaddressed later, as they arise.

We proceed by first introducing basic candidate properties of(rational) preference over options and only afterwards turning toquestions of interpretation. As noted above, preference concerns thecomparison of options; it is a relation between options. For a domainof options we speak of an agent’spreference ordering,this being the ordering of options that is generated by theagent’s preference between any two options in that domain.

In what follows, \(\preceq\) represents aweak preferencerelation. So \(A\preceq B\) means that the agent we are interested inconsiders option \(B\) to be at least as preferable as option \(A\).From the weak preference relation we can define thestrictpreference relation, \(\prec\), as follows: \(A\prec B\LeftrightarrowA\preceq B \ \& \ \neg (B\preceq A)\), where \(\neg X\) means“it is not the case that \(X\)”. The indifferencerelation, \(\sim\), is defined as: \(A\sim B \Leftrightarrow A\preceqB \ \& \ B\preceq A\). This represents that the agent we areinterested in considers \(A\) and \(B\) to be equally preferable.

We say that \(\preceq\)weakly orders a set \(S\) of optionswhenever it satisfies the following two conditions:

Axiom 1 (Completeness)
For any \(A, B\in S\): either \(A\preceq B\) or \(B\preceq A\).

Axiom 2 (Transitivity)
For any \(A, B, C\in S\): if \(A\preceq B\) and \(B\preceq C\) then\(A\preceq C\).

The above can be taken as a preliminary characterisation of rationalpreference over options. Even this limited characterisation iscontentious, however, and points to divergent interpretations of“preferences over prospects/options”.

Start with the Completeness axiom, which says that an agent cancompare, in terms of the weak preference relation, all pairs ofoptions in \(S\). Whether or not Completeness is a plausiblerationality constraint depends both on what sort of options are underconsideration, and how we interpret preferences over these options. Ifthe option set includes all kinds of states of affairs, thenCompleteness is not immediately compelling. For instance, it isquestionable whether an agent should be able to compare the optionwhereby two additional people in the world are made literate with theoption whereby two additional people reach the age of sixty. If, onthe other hand, all options in the set are quite similar to eachother, say, all options are investment portfolios, then Completenessis more compelling. But even if we do not restrict the kinds ofoptions under consideration, the question of whether or notCompleteness should be satisfied turns on the meaning of preference.For instance, if preferences merely represent choice behaviour orchoice dispositions, as they do according to the “revealedpreference theory” popular amongst economists (see Sen 1973),then Completeness is automatically satisfied, on the assumption that achoice must inevitably be made. By contrast, if preferences areunderstood rather as mental attitudes, typically considered judgmentsabout whether an option is better or more desirable than another, thenthe doubts about Completeness alluded to above are pertinent (forfurther discussion, see Mandler 2001).

Most philosophers and decision theorists subscribe to the latterinterpretation of preference as a kind of judgment that explains, asopposed to being identical with, choice dispositions and resultantchoice behaviour (see, e.g., Hausman 2011a, 2011b; Dietrich and List,2016a & 2016b; Bradley 2017; although see also Thoma 2021 andVredenburgh 2020 for recent defences of “revealed preferencetheory”, at least in the context of empirical economics).Moreover, many hold that Completeness is not rationally required,since they think that rationality makes demands only on the judgmentsan agent actually holds, but says nothing of whether a judgement mustbe held in the first place. Nevertheless, following Richard Jeffrey(1983), most decision theorists suggest that rationality requires thatpreferences becoherently extendible. This means that even ifyour preferences are not complete, it should be possible to completethem without violating any of the conditions that are rationallyrequired, in particular Transitivity.

This brings us to theTransitivity axiom, which says that if an option \(B\) is weakly preferred to \(A\), and\(C\) weakly preferred to \(B\), then \(C\) is weakly preferred to\(A\). A recent challenge to Transitivity turns on heterogeneous setsof options, as per the discussion of Completeness above. But here adifferent interpretation of preference is brought to bear on thecomparison of options. The idea is that preferences, or judgments ofdesirability, may be responsive to a salience condition. For example,suppose that the most salient feature when comparing cars \(A\) and\(B\) is how fast they can be driven, and \(B\) is no worse than \(A\)in this regard, yet the most salient feature when comparing cars \(B\)and \(C\) is how safe they are, and that \(C\) is no worse than \(B\)in this regard. Furthermore, when comparing \(A\) and \(C\), the mostsalient feature is their beauty. In such a case, some argue (e.g.,Temkin 2012) that there is no reason why Transitivity should besatisfied with respect to the preferences concerning \(A\), \(B\) and\(C\). Others (e.g., Broome 1991a) argue that Transitivity is part ofthe very meaning of the betterness relation (or objective comparativedesirability); if rational preference is a judgment of betterness ordesirability, then Transitivity is non-negotiable. With respect to thecar example, Broome would argue that the desirability of a fullyspecified option should not vary, simply in virtue of what otheroptions it is compared with. Either the choice context affects how theagent perceives the option at hand, in which case the description ofthe option should reflect this, or else the choice context does notaffect the option. Either way, Transitivity should be satisfied.

There is a more straightforward defence of Transitivity in preference;a defence that hinges on the sure losses that may befall anyone whoviolates the axiom. This is the so-calledmoney pump argument(see Davidson et. al. 1955 for an early argument of this sort, but forrecent discussion and revision of this argument, see Gustafsson 2010,2013 & 2022). It is based on the assumption that if you find \(X\)at least as desirable as \(Y\), then you should be happy to trade thelatter for the former. Suppose you violate Transitivity; for you:\(A\preceq B\), \(B\preceq C\) but \(C\prec A\). Moreover, suppose youpresently have \(A\). Then you should be willing to trade \(A\) for\(B\). The same goes for \(B\) and \(C\): you should be willing totrade \(B\) for \(C\). You strictly prefer \(A\) to \(C\), so youshould be willing to trade in \(C\) plus some sum \(\$x\) for \(A\).But now you are in the same situation as you started, having \(A\) andneither \(B\) nor \(C\), except that you have lost \(\$x\)! So in afew steps, each of which was consistent with your preferences, youfind yourself in a situation that is clearly worse, by your ownlights, than your original situation. The picture is made moredramatic if we imagine that the process could be repeated, turning youinto a “money pump”. Hence, the argument goes, there issomething (instrumentally) irrational about your intransitivepreferences. If your preferences were transitive, then you would notbe vulnerable to choosing a dominated option and serving as a moneypump. Therefore, your preferences should be transitive.

While the aforementioned controversies have not been settled, thefollowing assumptions will be made in the remainder of this entry: i)the objects of preference may be heterogeneous prospects,incorporating a rich and varied domain of properties, ii) preferencebetween options is a judgment of comparative desirability orchoice-worthiness, and iii) preferences satisfy both Completeness andTransitivity (although the former condition will be revisited inSection 5). The question that now arises is whether there are further generalconstraints on rational preference over options.

2. Utility measures of preference

In our continuing investigation of rational preferences overprospects, the numericalrepresentation (ormeasurement) of preference orderings will become important.The numerical measures in question are known asutilityfunctions. The two main types of utility function that will playa role are theordinal utility function and the moreinformation-richinterval-valued (orcardinal)utility function.

2.1 Ordinal utilities

It turns out that as long as the set of prospects/options, \(S\), isfinite, any weak order of the options in \(S\) can be represented byan ordinal utility function. Now let \(u\) be someutilityfunction with domain \(S\). We say that the function \(u\)represents the preference \(\preceq\) between the options in\(S\) just in case:

Another way to put this is that, when the above holds, the preferencerelation can be represented asmaximising utility, asmeasured byu, since it always favours an option with higherutility.

The only information contained in an ordinal utility representation ishow the agent whose preferences are being represented orders options,from least to most preferable. This means that if \(u\) is an ordinalutility function that represents the ordering \(\preceq\), then anyutility function \(u'\) that is an ordinal transformation of\(u\)—that is, any transformation of \(u\) that also satisfiesthe biconditional in (1)—represents \(\preceq\) just as well as\(u\) does. Hence, we say that an ordinal utility function isunique only up to ordinal transformations.

The result referred to above can be summarised as follows:

Theorem 1 (Ordinal representation). Let \(S\) be afinite set of prospects, and \(\preceq\) a weak preference relation on\(S\). Then there is an ordinal utility function that represents\(\preceq\) just in case \(\preceq\) is complete and transitive.

This theorem should not be too surprising. If \(\preceq\) is completeand transitive over \(S\), then the options in \(S\) can be put in anorder, from the most to the least preferred, where some options mayfall in the same position (if they are deemed equally desirable) butwhere there are no cycles, loops, or gaps.Theorem 1 just says that we can assign numbers to the options in \(S\) in a waythat represents this order. (For a simple proof of Theorem 1, exceptfor a strict rather than a weak preference relation, consult Peterson2009: 95.)

Note that ordinal utilities are not very mathematically“powerful”, so to speak. It does not make sense, forinstance, to compare the expected utilities of two prospects that aredescribed in terms of the probabilities they confer on differentordinal utilities. For example, consider the following two pairs ofprospects: the elements of the first pair are assigned ordinalutilities of 2 and 4, while those in the second pair are assignedordinal utilities of 0 and 5. Let us specify a “flat”probability distribution in each case, such that each element in thetwo pairs corresponds to a probability of 0.5. Relative to theseprobability and utility assignments, the expectation of the first pairof ordinal utilities is 3, which is larger than 2.5, the expectationof the second pair. Yet when we transform the ordinal utilities in apermissible way—for instance by increasing the highest utilityin the second pair from 5 to 10—the ordering of expectationsreverses; now the comparison is between 3 and 5. The significance ofthis point will become clearer in what follows, when we turn to thecomparative evaluation of lotteries and risky choices. Aninterval-valued or cardinal utility function is necessary forevaluating lotteries/risky prospects in a consistent way. By the sametoken, in order to construct or conceptualise a cardinal utilityfunction, one typically appeals to preferences over lotteries.(Although see Alt 1936 for a “risk-free” construction ofcardinal utility, that is, one that does not appeal to lotteries.)

2.2 Cardinalizing utility

In order to get a cardinal (interval-valued) utility representation ofa preference ordering—i.e., a measure that represents not onlyhow an agent orders the options but also says something about thedesirabilistic “distance” between options—we need aricher setting; the option set and the corresponding preferenceordering will need to have more structure than for an ordinal utilitymeasure. One such account, owing to John von Neumann and OskarMorgenstern (1944), will be cashed out in detail below. For now, it isuseful to focus on the kind of option that is key to understanding andconstructing a cardinal utility function: lotteries.^[1]

Consider first an ordering over three regular options, e.g., the threeholiday destinations Amsterdam, Bangkok and Cardiff, denoted \(A\),\(B\) and \(C\) respectively. Suppose your preference ordering is\(A\prec B \prec C\). This information suffices to ordinally representyour judgement; recall that any assignment of utilities is thenacceptable as long as \(C\) gets a higher value than \(B\) which getsa higher value than \(A\). But perhaps we want to know more than canbe inferred from such a utility function—we want to know howmuch \(C\) is preferred over \(B\), compared to how much \(B\) ispreferred over \(A\). For instance, it may be that Bangkok isconsidered almost as desirable as Cardiff, but Amsterdam is a long waybehind Bangkok, relatively speaking. Or else perhaps Bangkok is onlymarginally better than Amsterdam, compared to the extent to whichCardiff is better than Bangkok. This kind of information about therelative distance between options, in terms of strength of preferenceor desirability, is precisely what is given by an interval-valuedutility function. The problem is how to ascertain thisinformation.

To solve this problem, Ramsey (1926) and later von Neumann andMorgenstern (hereafter vNM) made the following suggestion: weconstruct a new option, alottery, \(L\), that has \(A\) and\(C\) as its possible “prizes”, and we figure out whatchance the lottery must confer on \(C\) for you to be indifferentbetween this lottery and a holiday in Bangkok. The basic idea is thatyour judgment about Bangkok, relative to Cardiff on the one hand andAmsterdam on the other, can be measured by the riskiness of thelottery \(L\) involving Cardiff and Amsterdam that you deem equallydesirable as Bangkok. For instance, if you are indifferent betweenBangkok and a lottery that provides a very low chance of winning atrip to Cardiff, then you evidently do not regard Bangkok to be muchbetter than Amsterdam, vis-à-vis Cardiff; for you, even a smallimprovement on Amsterdam, i.e., a lottery with a small chance ofCardiff but a large chance of Amsterdam, is enough to matchBangkok.

The above analysis presumes that lotteries are evaluated in terms oftheirexpected choice-worthiness or desirability. That is,the desirability of a lottery is the sum of the chances of each prizemultiplied by the desirability of that prize. Consider the followingexample: Suppose you are indifferent between the lottery and theholiday in Bangkok when the chance of the lottery resulting in aholiday in Cardiff is \(3/4\). Call this particular lottery \(L'\).The idea is that Bangkok is therefore three quarters of the way up adesirability scale that has Amsterdam at the bottom and Cardiff at thetop. If we stipulate that \(u(A)=0\) and \(u(C)=1\), then\(u(B)=u(L')=3/4\). This corresponds to theexpecteddesirability—or, as it is usually called, theexpectedutility—of the lottery, since \(1/4\cdot 0 + 3/4\cdot 1 =3/4 = u(L')\). That is, the desirability of the lottery is a weightedsum of the utilities of its prizes, where the weight on eachprize’s utility is determined by the probability that thelottery results in that prize.

We thus see that an interval-valued utility measure over options canbe constructed by introducing lottery options. As the name suggests,the interval-valued utility measure conveys information about therelative sizes of the intervals between the options according to somedesirability scale. That is, the utilities are unique after we havefixed the starting point of our measurement and the unit scale ofdesirability. In the above example, we could have, for instance,assigned a utility value of 1 to \(A\) and 5 to \(C\), in which casewe would have had to assign a utility value of 4 to \(B\), since 4 is3/4 of the way between 1 and 5. In other words, once we have assignedutility values to \(A\) and \(C\), the utility of \(L'\) and thus\(B\) has been determined. Let us call this second utility function\(u'\). It is related to our original function as follows: \(u'=4\cdotu +1\). This type of relationship always holds between two suchfunctions: If \(u\) is an interval-valued utility function thatrepresents a preference ordering, \(\preceq\), and \(u'\) is anotherutility function that also represents this same preference ordering,then there are constants \(a\) and \(b\), where \(a\) must bepositive, such that \(u'=a\cdot u + b\). This is to say thatinterval-valued utility functions areunique only up to positivelinear transformation.

Before concluding this discussion of measuring utility, two relatedlimitations regarding the information such measures convey should bementioned. First, since the utilities of options, whether ordinal orinterval-valued, can only be determinedrelative to theutilities of other options, there is no such thing as theabsolute utility of an option, at least not without further assumptions.^[2] Second, by the same reasoning, neither interval-valued nor ordinalutility measures, as discussed here, areinterpersonallycommensurable with respect to levels and units of utility. By wayof a quick illustration, suppose that both you and I have thepreference ordering described above over the holiday options: \(A\precB \prec C\). Suppose too that, as per the above, we are bothindifferent between \(B\) and the lottery \(L'\) that has a \(3/4\)chance of yielding \(C\) and a \(1/4\) chance of yielding \(A\). Canwe then say that granting me Cardiff and you Bangkok would amount tothe same amount of “total desirability” as granting youCardiff and me Bangkok? We are not entitled to say this. Our sharedpreference ordering is, for instance, consistent with me finding avacation in Cardiff a dream come true while you just find it the bestof a bad lot. Moreover, we are not even entitled to say that thedifference in desirability between Bangkok and Amsterdam is the samefor you as it is for me. According to me, the desirability of thethree options might range from living hell to a dream come true, whileaccording to you, from bad to quite bad; both evaluations areconsistent with the above preference ordering. In fact, the same mighthold for our preferences overall possible options, includinglotteries: even if we shared the same total preference ordering, itmight be the case that you are just of a negativedisposition—finding no option that great—while I am veryextreme—finding some options excellent but others a sheertorture. Hence, absent further assumptions, utility functions, whetherinterval-valued or ordinal, do not allow for meaningful interpersonalcomparisons. (Elster and Roemer 1993 contains a number of papersdiscussing these issues; see also the entry onsocial choice theory.)

2.3 The von Neumann and Morgenstern (vNM) representation theorem

The last section provided an interval-valued utility representation ofa person’s preferences over lotteries, on the assumption thatlotteries are evaluated in terms of expected utility. Some might findthis a bit quick. Why should we assume that people evaluate lotteriesin terms of their expected utilities? The vNM theorem effectivelyshores up the gaps in reasoning by shifting attention back to thepreference relation. In addition to Transitivity and Completeness, vNMintroduce further principles governing rational preferences overlotteries, and show that an agent’s preferences can berepresented as maximising expected utility whenever her preferencessatisfy these principles.

Let us first define, in formal terms, the expected utility of alottery: Let \(L_i\) be a lottery from the set \(\bL\) of lotteries,and \(O_{ik}\) the outcome, or prize, of lottery \(L_i\) that ariseswith probability \(p_{ik}\). The expected utility of \(L_i\) is thendefined as:

The assumption made earlier can now be formally stated:

When the above holds, we say that there is an expected utilityfunction that represents the agent’s preferences; in otherwords, the agent can be represented asmaximising expectedutility.

The question that vNM address is: What sort of preferences can be thusrepresented? To answer this question, we must return to the underlyingpreference relation \(\preceq\) over the set of options, in this caseinvolving lotteries. The vNM theorem requires the set \(\bL\) oflotteries to be rather extensive: it is closed under“probability mixture”, that is, if \(L_i, L_j\in \bL\),then compound lotteries that have \(L_i\) and \(L_j\) as possibleprizes are also in \(\bL\). (Another technical assumption, that willnot be discussed in detail, is that compound lotteries can always bereduced, in accordance with the laws of probability, to simplelotteries that only involve basic prizes.)

A basic rationality constraint on the preference relation has alreadybeen discussed—that it weakly orders the options (i.e.,satisfies Transitivity and Completeness). The following notation willbe used to introduce the two additional vNM axioms of preference:\(\{pA, (1-p)B\}\) denotes a lottery that results either in \(A\),with probability \(p\), or \(B\), with probability \(1-p\), where\(A\) and \(B\) can be final outcomes but can also be lotteries.

Continuity implies that no outcome \(A\) is so bad that you would notbe willing to take some gamble that might result in you ending up withthat outcome, but might otherwise result in you ending up with anoutcome (\(C\)) that you find to be a marginal improvement on yourstatus quo (\(B\)), provided that the chance of \(A\) is small enough.Intuitively, Continuity guarantees that an agent’s evaluationsof lotteries are appropriately sensitive to the probabilities of thelotteries’ prizes.

Independence implies that when two alternatives have the sameprobability for some particular outcome, our evaluation of the twoalternatives should be independent of our opinion of that outcome.Intuitively, this means that preferences between lotteries should begoverned only by the features of the lotteries that differ; thecommonalities between the lotteries should be effectively ignored.

Some people find theContinuity axiom an unreasonable constraint on rational preference. Is there anyprobability \(p\) such that you would be willing to accept a gamblethat has that probability of you losing your life and probability\((1-p)\) of you gaining $10? Many people think there is not. However,the very same people would presumably cross the street to pick up a$10 bill they had dropped. But that is just taking a gamble that has avery small probability of being killed by a car but a much higherprobability of gaining $10! More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to imminent death, and correspondingly very highchances of some modest reward. We shall return to Continuity inSection 5.4.

Independence seems a compelling requirement of rationality, whenconsidered in the abstract. Nevertheless, there are famous exampleswhere people typically violate Independence without seemingirrational. These examples involvecomplementarities betweenthe possible lottery outcomes. A particularly well-known such exampleis the so-calledAllais Paradox, which the French economistMaurice Allais (1953) first introduced in the early 1950s. The paradoxturns on comparing people’s preferences over two pairs oflotteries similar to those given in Table 1. The lotteries aredescribed in terms of the prizes that are associated with particularnumbered tickets, where one ticket will be drawn randomly (forinstance, \(L_1\) results in a prize of $2500 if one of the ticketsnumbered 2–34 is drawn).

In this situation, many people strictly prefer \(L_2\) over \(L_1\)but also \(L_3\) over \(L_4\) (as evidenced by their choice behaviour,as well as their testimony), a pair of preferences which will bereferred to asAllais’ preferences.^[3] A common way to rationalise Allais’ preferences is that in thefirst choice situation, the risk of ending up with nothing when onecould have had $2400 for sure does not justify the increased chance ofa higher prize. In the second choice situation, however, the minimumone stands to gain is $0 no matter which choice one makes. Therefore,in that case many people do think that the slight extra risk of $0 isworth the chance of a better prize.

While the above reasoning may seem compelling, Allais’preferences conflict with theIndependence axiom. The following is true of both choice situations: whatever choice youmake, you will get the same prize if one of the tickets in the lastcolumn is drawn. Moreover, the probability of getting a prize from,say, the first column is independent of what you choose (and similarlyfor the other two columns). Therefore, Independence implies that bothyour preference between \(L_1\) and \(L_2\) and your preferencebetween \(L_3\) and \(L_4\) should be independent of the prizes in thelast column. But when you ignore the last column, \(L_1\) becomesidentical to \(L_3\) and \(L_2\) to \(L_4\). Hence, if you prefer\(L_2\) over \(L_1\) but \(L_3\) over \(L_4\), there seems to be aninconsistency in your preference ordering. And there is definitely aviolation of Independence (given how the options have been described;an issue to which we return inSection 5.1). As a result, the pair of preferences under discussion cannot berepresented as maximising expected utility. (Thus the“paradox”: many people think that Independence is arequirement of rationality, but nevertheless also want to claim thatthere is nothing irrational about Allais’ preferences.)

Decision theorists have reacted in different ways to Allais’Paradox. This issue will be revisited inSection 5.1, when challenges to EU theory will be discussed. The present goal issimply to show that Continuity and Independence are compellingconstraints on rational preference, although not without theirdetractors. The result vNM proved can be summarised thus:

Theorem 2 (von Neumann-Morgenstern)
Let \(\bO\) be a finite set of outcomes, \(\bL\) a set ofcorresponding lotteries that is closed under probability mixture and\(\preceq\) a weak preference relation on \(\bL\). Then \(\preceq\)satisfies axioms 1–4 if and only if there exists a function\(u\), from \(\bO\) into the set of real numbers, that is unique up topositive linear transformation, and relative to which \(\preceq\) canbe represented as maximising expected utility.

David Kreps (1988) gives an accessible illustration of the proof ofthis theorem.

3. Making real decisions

The vNM theorem is a very important result for measuring the strengthof a rational agent’s preferences over sure options (thelotteries effectively facilitate a cardinal measure over sureoptions). But this does not get us all the way to making rationaldecisions in the real world; we do not yet really have a decisiontheory. The theorem is limited to evaluating options that come with aprobability distribution over outcomes—a situation decisiontheorists and economists often describe as “choice underrisk” (Knight 1921).

In most ordinary choice situations, the objects of choice, over whichwe must have or form preferences, are not like this. Rather,decision-makers must consulttheir own probabilistic beliefsabout whether one outcome or another will result from a specifiedoption. Decisions in such circumstances are often described as“choices under uncertainty” (Knight 1921). For example,consider the predicament of a mountaineer deciding whether or not toattempt a dangerous summit ascent, where the key factor for her is theweather. If she is lucky, she may have access to comprehensive weatherstatistics for the region. Nevertheless, the weather statistics differfrom the lottery set-up in that they do notdetermine theprobabilities of the possible outcomes of attempting versus notattempting the summit on a particular day. Not least, the mountaineermust consider how confident she is in the data-collection procedure,whether the statistics are applicable to the day in question, and soon, when assessing her options in light of the weather.

Some of the most celebrated results in decision theory address, tosome extent, these challenges. They consist in showing what conditionson preferences over “real world options” suffice for theexistence of a pair of utilityand probability functionsrelative to which the agent can be represented as maximising expectedutility. The standard interpretation is that, just as the utilityfunction represents the agent’s desires, so the probabilityfunction represents her beliefs. The theories are referred tocollectively assubjective expected utility (SEU) theory asthey concern an agent’s preferences over prospects that arecharacterised entirely in terms of her own beliefs and desires (but wewill continue to use the simpler labelEU theory). In thissection, two of these results will be briefly discussed: that ofLeonard Savage (1954) and Richard Jeffrey (1965).

Note that these EU theories apparently prescribe two things: (a) youshould have consistent preference attitudes, and (b) you should preferthe means to your ends, or at least you should prefer the means thatyou assess willon average lead to your ends (cf. Buchak2016). The question arises: What is the relationship between theseprescriptions? TheEU representation theorems that will beoutlined shortly seem to show that, despite appearances, the twoprescriptions are actually just one: anyone who has consistentattitudes prefers the means to her ends, and vice versa. But thepuzzle remains that there are many ways to have consistent preferenceattitudes, and perhaps not all of these correspond to preferring themeans to one’s ends. This puzzle is worth bearing in mind whenappraising EU theory in its various guises; it will come up againlater.

3.1 Savage’s theory

Leonard Savage’s decision theory, as presented in his (1954)The Foundations of Statistics, is without a doubt thebest-known normative theory of choice under uncertainty, in particularwithin economics and the decision sciences. In the book Savagepresents a set of axioms constraining preferences over a set ofoptions that guarantee the existence of a pair of probability andutility functions relative to which the preferences can be representedas maximising expected utility. Nearly three decades prior to thepublication of the book, Frank P. Ramsey (1926) had actually proposedthat a different set of axioms can generate more or less the sameresult. Nevertheless, Savage’s theory has been much moreinfluential than Ramsey’s, perhaps because Ramsey neither gave afull proof of his result nor provided much detail of how it would go(Bradley 2004). Savage’s result will not be described here infull detail. However, the ingredients and structure of his theoremwill be laid out, highlighting its strengths and weaknesses.

The options or prospects in Savage’s theory are similar tolotteries, except that the possible outcomes do not come withprobabilities but instead depend on whether a particular state of theworld is actual. Indeed, the primitives in Savage’s theory areoutcomes^[4] andstates (of the world). The former are the good or badhappenings that ultimately affect and matter to an agent, while thelatter are the features of the world that the agent has no controlover and which are the locus of her uncertainty about the world. Setsof states are calledevents. This distinction betweenoutcomes and states serves to neatly separate desire and belief: theformer are, according to Savage’s theory, the target of desire,while the latter are the target of belief.

The lottery-like options over which the agent has preferences are arich set ofacts that effectively amount to all the possibleassignments of outcomes to states of the world. That is, acts arefunctions from the state space to the outcome space, and theagent’s preference ordering is taken to be defined over all suchpossible functions. Some of these acts will look quite sensible:consider the act that assigns to the event “it rains” theoutcome “miserable wet stroll” and assigns to the event“it does not rain” the outcome “very comfortablestroll”. This is apparently the act of going for a strollwithout one’s umbrella. Other Savage acts will not look quite sosensible, such as theconstant act that assigns to both“it rains” and “it does not rain” the sameoutcome “miserable wet stroll”. (Note that the constantacts provide a way of including sure outcomes within the preferenceordering.) The problem with this act (and many others) is that it doesnot correspond to anything that an agent could even in principlechoose to do or perform.^[5]

Savage’s act/state(event)/outcome distinction can be naturallyrepresented in tabular form, with rows serving as acts that yield agiven outcome for each state/event column. Table 2 depicts the actsmentioned above: i) “go for stroll without umbrella”, ii)“go for stroll with umbrella”, and iii) the bizarreconstant act. Of course, the set of acts required for Savage’stheorem involve even more acts that account for all the possiblecombinations of states and outcomes.

Before discussing Savage’s axioms, let us state the result thatthey give rise to. The following notation will be used: \(f\), \(g\),etc, are various acts, i.e., functions from the set \(\bS\) of statesof the world to the set \(\bO\) of outcomes, with \(\bF\) the set ofthese functions. \(f(s_i)\) denotes the outcome of \(f\) when state\(s_i\in\bS\) is actual. The expected utility of \(f\), according toSavage’s theory, denoted \(U(f)\), is given by:

The result Savage proved can be stated as follows:^[6]

The above result may seem remarkable; in particular, the fact that aperson’s preferences can determine a unique probability functionthat represents her beliefs. On a closer look, however, it is evidentthat some of our beliefs can be determined by examining ourpreferences. Suppose you are offered a choice between two lotteries,one that results in you winning a nice prize if a coin comes up headsbut getting nothing if the coin comes up tails, another that resultsin you winning the same prize if the coin comes up tails but gettingnothing if the coin comes up heads. Then assuming that thedesirability of the prize (and similarly the desirability of no prize)is independent of how the coin lands, your preference between the twolotteries should be entirely determined by your comparative beliefsfor the two ways in which the coin can land. For instance, if youstrictly prefer the first lottery to the second, then that suggestsyou consider heads more likely than tails.

The above observation suggests that one can gauge an agent’scomparative beliefs, and perhaps more, from her preferences. Savagewent one step further than this, anddefined comparativebelief in terms of preferences. To state Savage’s definition,let \(\wcbrel\) be a weak comparative belief relation, defined on theset \(\bS\) of states of the world. (\(\cbrel\) and \(\wcbsim\) aredefined in terms of \(\wcbrel \) in the usual way.)

Definition 1 is based on the simple observation that one wouldgenerally prefer to stake a good outcome on a more rather than lessprobable event. But the idea that thisdefines comparativebelief might seem questionable. We could, for instance, imagine peoplewho are instrumentally irrational, and as a result fail to prefer\(g\) to \(f\), even when the above conditions all hold and they find\(F\) more likely than \(E\). Moreover, this definition raises thequestion of how to define the comparative belief for those who areindifferent betweenall outcomes (Eriksson and Hájek2007). Perhaps no such people exist (and Savage’saxiom P5 indeed makes clear that his result does not pertain to such people).Nevertheless, it seems a definition of comparative belief should notpreclude that such people, if existent, havestrictcomparative beliefs. Savage suggests that this definition ofcomparative belief is plausible in light of his axiom P4, which willbe stated below. In any case, it turns out that when a person’spreferences satisfy Savage’s axioms, we can read off herpreferences a comparative belief relation that can be represented by a(unique) probability function.

Without further ado, let us state Savage’s axioms in turn. Theseare intended as constraints on an agent’s preference relation,\(\preceq\), over a set of acts, \(\bF\), as described above. Thefirst of Savage’s axioms is the basic ordering axiom.

P1. (Ordering)
The relation \(\preceq\) is complete and transitive.

The next axiom is reminiscent of vNM’s Independence axiom. Wesay that alternative \(f\) “agrees with” \(g\) in event\(E\) if, for all states in event \(E\), \(f\) and \(g\) yield thesame outcome.

The idea behind the Sure Thing Principle (STP) is essentially the sameas that behind Independence: since we should be able to evaluate eachoutcome independently of other possible outcomes, we can safely ignorestates of the world where two acts that we are comparing result in thesame outcome. Putting the principle in tabular form may make this moreapparent. The setup involves four acts with the following form:

The intuition behind the STP is that if \(g\) is weakly preferred to\(f\), then that must be because the consequence \(Y\) is consideredat least as desirable as \(X\), which by the same reasoning impliesthat \(g'\) is weakly preferred to \(f'\).

Savage also requires that the desirability of an outcome beindependent of the state in which it occurs, as this is necessary forit to be possible to determine a comparative belief relation from anagent’s preferences. To formalise this requirement, Savageintroduces the notion of anull event, defined asfollows:

Definition 2 (Null)
EventE is null just in case for any alternatives\(f,g\in\bF\), \(f\sim g\) givenE.

The intuition is that null events are those events an agent is certainwill not occur. If and only if an agent is certain that \(E\) will notoccur, then she is indifferent betweenany acts given \(E\).The following axiom then stipulates that knowing what state is actualdoes not affect the preference ordering overoutcomes:

P3. (State Neutrality)
If \(f(s_i)=X\) and \(g(s_i)=Y\) whenever \(s_i\in E\) and \(E\) isnot null, then \(f\preceq g\) given \(E\) just in case \(X\preceqY\).

The next axiom is also necessary for it to be possible to determine acomparative belief relation from an agent’s preferences. Aboveit was suggested that by asking you to stake a prize on whether a coincomes up heads or tails, it can be determined which of these events,heads or tails, you find more likely. But that suggestion is onlyplausible if the size of the prize does not affect your judgement ofthe relative likelihood of these two events. That assumption iscaptured by the next axiom. Since the axiom is rather complicated itwill be stated in tabular form:

Less formally (and stated in terms of strict preference), the idea isthat if you prefer to stake the prize \(X\) on \(f\) rather than\(f'\), you must consider \(E\) more probable than \(F\). Therefore,you should prefer to stake the prize \(Y\) on \(g\) rather than \(g'\)since the prize itself does not affect the probability of theevents.

The next axiom is arguably not a rationality requirement, but one ofSavage’s “structural axioms” (Suppes 2002). An agentneeds to have some variation in preference for it to be possible toread off her comparative beliefs from her preferences; and, moregenerally, for it to be possible to represent her as maximisingexpected utility. To this end, the next axiom simply requires thatthere be some alternatives between which the agent is notindifferent:

P5.
There are some \(f,g\in\bF\) such that \(f\prec g\).

When these five axioms are satisfied, the agent’s preferencesgive rise to a comparative belief relation, \(\wcbrel \), which hasthe property of being aqualitative probability relation,which is necessary for it to be possible to represent \(\wcbrel \) bya probability function. In other words, \(\wcbrel \) satisfies thefollowing three conditions, for any events \(E\), \(F\) and \(G\):

1. \(\wcbrel \) is transitive and complete,

2. if \(E\cap G=\emptyset=F\cap G\), then \(E \wcbrel F\LeftrightarrowE\cup G \wcbrel F\cup G\),

3. \(\emptyset \wcbrel E,\) \(\emptyset \cbrel \bS\)

Being a qualitative probability relation is, however, not sufficientto ensure the possibility of probabilistic representation. To ensurethis possibility, Savage added the following structural axiom:

Like the Continuity axiom of vNM, Non-Atomicity implies that no matterhow bad an outcome \(X\) is, if \(g\) is already preferred to \(f\),then if we add \(X\) as one of the possible outcomes of\(g\)—thereby constructing a modified alternative\(g'\)—the modified alternative will still be preferred tof as long as the probability of \(X\) is sufficiently small.(Similarly, no matter howgood X is, we can add it as one ofthe possible outcomes off, thus creating \(f'\) which isstill dispreferred tog.) In effect, Non-Atomicity impliesthat \(\bS\) contains events of arbitrarily small probability. It isnot too difficult to imagine how that could be satisfied. Forinstance, any event \(F\) can be partitioned into two equiprobablesub-events according to whether some coin would come up heads or tailsif it were tossed. Each sub-event could be similarly partitionedaccording to the outcome of the second toss of the same coin, and soon.

Savage showed that whenever these six axioms are satisfied, thecomparative belief relation can be represented by auniqueprobability function. Having done so, he could rely on the vNMrepresentation theorem to show that an agent who satisfies all six axioms^[7] can be represented as maximising expected utility, relative to aunique probability function that plausibly represents theagent’s beliefs over the states and a cardinal utility functionthat plausibly represents the agent’s desires for ultimateoutcomes (recall the statement of Savage’s theorem above).^[8] Savage’s own proof is rather complicated, but Kreps (1988)provides a useful illustration of it.

There is no doubt that Savage’sexpected utilityrepresentation theorem is very powerful. There are, however, twoimportant questions to ask about whether Savage achieves his aims: 1)Does Savage characteriserational preferences, at least inthe generic sense? And 2) Does Savage’s theorem tell us how tomake rational decisions in the real world? Savage’s theory hasproblems meeting these two demands, taken together. Arguably the coreweakness of the theory is that its various constraints and assumptionspull in different directions when it comes to constructing realisticdecision models, and furthermore, at least one constraint (notably,the Sure Thing Principle) is only plausible under decision modellingassumptions that are supposed to be the output, not the input, of thetheory.

One well recognised decision-modelling requirement for Savage’stheory is that outcomes be maximally specific in every way thatmatters for their evaluation. If this were not the case, the axiom ofState Neutrality, for instance, would be a very implausiblerationality constraint. Suppose we are, for example, wondering whetherto buy cocoa or lemonade for the weekend, and assume that how good wefind each option depends on what the weather will be like. Then weneed to describe the outcomes such that they include the state of theweather. For if we do not, the desirability of the outcomes willdepend on what state is actual. Since lemonade is, let us suppose,better on hot days than cold, an outcome like “I drink lemonadethis weekend” would be more or less desirable depending onwhether it occurs in a state where it is hot or cold. This would becontrary to the axiom of State Neutrality. Therefore, the appropriateoutcomes in this case are those of the form “I drink lemonadethis weekend in hot weather”. (Of course, this outcome must besplit into even more fine-grained outcomes if there are furtherfeatures that would affect the choice at hand, such as sharing thedrink with a friend who loves lemonade versus sharing the drink with afriend who loves hot cocoa, and so on.)

The fact that the outcomes in the above case must be specific enoughto contain the state of the weather may seem rather innocuous.However, this requirement exacerbates the above-mentioned problem thatmany of the options/acts that Savage requires for his representationtheorem are nonsensical, in that the semantic content of state/outcomepairs is contradictory. Recall that the domain of the preferenceordering in Savage’s theory amounts toevery functionfrom the set of states to the set of outcomes (what Broome 1991arefers to as theRectangular Field Assumption). So if“I drink lemonade this weekend in hot weather” is one ofthe outcomes we are working with, and we have partitioned the set ofstates according to the weather, then there must, for instance, be anact that has this outcome in the state where it is cold! The moredetailed the outcomes (as required for the plausibility of StateNeutrality), the less plausible the Rectangular Field Assumption. Thisis an internal tension in Savage’s framework. Indeed, it isdifficult to see how/why a rational agent can/should form preferencesover nonsensical acts (although see Dreier 1996 for an argument thatthis is not such an important issue). Without this assumption,however, the agent’s preference ordering will not be adequatelyrich for Savage’s rationality constraints to yield the EUrepresentation result.^[9]

The axiom in Savage’s theory that has received most attention isthe Sure Thing Principle. It is not hard to see that this principleconflicts with Allais’ preferences for the same reason thesepreferences conflict with Independence (recallSection 2.3). Allais’ challenge will be discussed again later. For now, ourconcern is rather the Sure Thing Principle vis-à-vis theinternal logic of Savage’s theory. To begin with, the Sure ThingPrinciple, like State Neutrality, exacerbates concerns about theRectangular Field Assumption. This is because the Sure Thing Principleis only plausible if outcomes are specific enough to account for anysort of dependencies between outcomes in different states of theworld. For instance, if the fact that one could have chosen arisk-free alternative—and thereby guaranteed an acceptableoutcome—makes a difference to the desirability of receivingnothing after having taken a risk (as in Allais’ problem), thenthat has to be accounted for in the description of the outcomes. Butagain, if we account for such dependencies in the description of theoutcomes, we run into the problem that there will be acts in thepreference ordering that are nonsensical (see, e.g., Broome 1991a: ch.5).

There is a further internal problem with Savage’s theoryassociated with the Sure Thing Principle: the principle is onlyreasonable when the decision model is constructed such that there isprobabilistic independence between the acts an agent is consideringand the states of the world that determine the outcomes of these acts.Recall that the principle states that if we have four options with thefollowing form:

then if \(g\) is weakly preferred to \(f\), \(g'\) must be weaklypreferred to \(f'\). Suppose, however, that there is probabilisticdependency between the states of the world and the alternatives we areconsidering, and that we find \(Z\) to be better than both \(X\) and\(Y\), and we also find \(W\) to be better than both \(X\) and \(Y\).Moreover, suppose that \(g\) makes \(\neg E\) more likely than \(f\)does, and \(f'\) makes \(\neg E\) more likely than \(g'\) does. Thenit seems perfectly reasonable to prefer \(g\) over \(f\) but \(f'\)over \(g'\).

Why is the requirement of probabilistic independence problematic? Forone thing, in many real-world decision circumstances, it is hard toframe the decision model in such a way that states are intuitivelyprobabilistically independent of acts. For instance, suppose an agentenjoys smoking, and is trying to decide whether to quit or not. Howlong she lives is amongst the contingencies that affect thedesirability of smoking. It would be natural to partition the set ofstates according to how long the agent lives. But then it is obviousthat the options she is considering could, and arguably should, affecthow likely she finds each state of the world, since it is wellrecognised that life expectancy is reduced by smoking. Savage wouldthus require an alternative representation of the decisionproblem—the states do not reference life span directly, butrather the agent’s physiological propensity to react in acertain way to smoking.

Perhaps there is always a way to contrive decision models such thatacts are intuitively probabilistically independent of states. Buttherein lies the more serious problem. Recall that Savage was tryingto formulate a way of determining a rational agent’s beliefsfrom her preferences over acts, such that the beliefs can ultimatelybe represented by a probability function. If we are interested inreal-world decisions, then the acts in question ought to berecognisable options for the agent (which we have seen isquestionable). Moreover, now we see that one of Savage’srationality constraints on preference—the Sure ThingPrinciple—is plausible only if the modelled acts areprobabilistically independent of the states. In other words, thisindependence must be built into the decision model if it is tofacilitate appropriate measures of belief and desire. But this is toassume that we already have important information about the beliefs ofthe agent whose attitudes we are trying to represent; namely whatstate-partitions she considers probabilistically independent of heracts.

The above problems suggest there is a need for an alternative theoryof choice under uncertainty. Richard Jeffrey’s theory, whichwill be discussed next, avoids all of the problems that have beendiscussed so far. But as we will see, Jeffrey’s theory haswell-known problems of its own, albeit problems that are notinsurmountable.

3.2 Jeffrey’s theory

Richard Jeffrey’s expected utility theory differs fromSavage’s in terms of both theprospects (i.e., options)under consideration and therationality constraints onpreferences over these prospects. The distinct advantage ofJeffrey’s theory is that real-world decision problems can bemodelled just as the agent perceives them; the plausibility of therationality constraints on preference do not depend on decisionproblems being modelled in a particular way. We first describeJeffrey’s prospects or decision set-up and hisconditional expected utility formula, before turning to thepertinent rationality constraints on preferences and the correspondingrepresentation theorem.

Unlike Savage, Jeffrey does not make a distinction between the objectsof instrumental and non-instrumental desire (acts and outcomesrespectively) and the objects of belief (states of the world). Rather,Jeffrey assumes thatpropositions describing states ofaffairs are the objects of both desire and belief. On first sight,this seems unobjectionable: just as we can have views about whether itwill in fact rain, we can also have views about how desirable thatwould be. The uncomfortable part of this setup is that acts, too, arejust propositions—they are ordinary states of affairs aboutwhich an agent has both beliefs and desires. Just as the agent has apreference ordering over, say, possible weather scenarios for theweekend, she has a preference ordering over the possible acts that shemay perform, and in neither case is the most preferred state ofaffairs necessarily the most likely to be true. In other words, theonly thing that picks out acts as special is their substantivecontent—these are the propositions that the agent has the powerto choose/make true in the given situation. It is as if the agentassesses her own options for acting from, rather, a third-personperspective. If one holds that a decision model should convincinglyrepresent the subjective perspective of the agent in question, this isarguably a weakness of Jeffrey’s theory, although it may be onewithout consequence.^[10]

Before proceeding, a word about propositions may be helpful: they areabstract objects that can be either true or false, and are commonlyidentified with sets of possible worlds. A possible world can bethought of as an abstract representation of how things are or could be(Stalnaker 1987; see also entry onpossible worlds). The proposition that it rains at time \(t\), for example, is just theset of all worlds where it rains at time \(t\). And this particularproposition is true just in case the actual world happens to be amember of the set of all worlds where it rains at time \(t\).

The basic upshot of Jeffrey’s theory is that the desirability ofa proposition, including one representing acts, depends both on thedesirabilities of the different ways in which the proposition can betrue, and the relative probability that it is true in these respectiveways. To state this more precisely, \(p\), \(q\), etc., will denotepropositional variables. Let \(\{p_1, p_2, …, p_n\}\) be oneamongst many finite partitions of the proposition \(p\); that is, setsof mutually incompatible but jointly exhaustive ways in which theproposition \(p\) can be realised. For instance, if \(p\) is theproposition that it is raining, then we could partition thisproposition very coarsely according to whether we go to the beach ornot, but we could also partition \(p\) much more finely, for instanceaccording to the precise millimetres-per-hour amount of rain. Thedesirability of \(p\) according to Jeffrey, denoted \(Des(p)\), isgiven by:

Jeffrey’s equation.
\(Des(p)=\sum_i Des(p_i)\cdot P(p_i\mid p)\)

This is aconditional expected utility formula for evaluating\(p\). As noted, a special case is when the content of \(p\) is suchthat it is recognisably something the agent can choose to make true,i.e., an act.

One important difference between Jeffrey’s desirability formulaand Savage’s expected utility formula is that there is nodistinction made between desirability and “expected”desirability, unlike what has to be done in Savage’s theory,where there is a clear distinction between utility, measuring anagent’s fundamental desires for ultimate outcomes, and expectedutility, measuring an agent’s preferences over uncertainprospects or acts. This disanalogy is due to the fact that there is nosense in which the \(p_i\)s that \(p\) is evaluated in terms of needto be ultimate outcomes; they can themselves be thought of asuncertain prospects that are evaluated in terms of their differentpossible realisations.

Another important thing to notice about Jeffrey’s way ofcalculating desirability is that it does not assume probabilisticindependence between the alternative that is being evaluated, \(p\),and the possible ways, the \(p_i\)s, that the alternative may berealised. Indeed, the probability of each \(p_i\) is explicitlyconditional on the \(p\) in question. When it comes to evaluatingacts, this is to say (in Savage’s terminology) that theprobabilities for the possible state-outcome pairs for the act areconditional on the act in question. Thus we see why the agent candescribe her decision problem just as she sees it; there is norequirement that she identify a set of states (in Jeffrey’scase, this would be a partition of the proposition space that isorthogonal to the act partition) such that the states areappropriately fine-grained and probabilistically independent of theacts.

It should moreover be evident, given the discussion of the Sure ThingPrinciple (STP) inSection 3.1, that Jeffrey’s theory does not have this axiom. Since statesmay be probabilistically dependent on acts, an agent can berepresented as maximising the value of Jeffrey’s desirabilityfunction while violating the STP. Moreover, unlike Savage’s,Jeffrey’s representation theorem does not depend on anythinglike the Rectangular Field Assumption. The agent is not required tohave preferences over artificially constructed acts or propositionsthat turn out to be nonsensical, given the interpretation ofparticular states and outcomes. In fact, only those propositions theagent considers to be possible (in the sense that she assigns them aprobability greater than zero) are, according to Jeffrey’stheory, included in her preference ordering.

Of course, we still need certain structural assumptions in order toprove a representation theorem for Jeffrey’s theory. Inparticular, the set \(\Omega\), on which the preference ordering\(\preceq\) is defined, has to be anatomless Boolean algebraof propositions, from which the impossible propositions, denoted\(\bot\), have been removed. A Boolean algebra is just a set of e.g.propositions or sentences that is closed under the classical logicaloperators and negation. An algebra is atomless just in case all of itselements can be partitioned into finer elements. The assumption that\(\Omega\) is atomless is thus similar to Savage’sP6, and can be given a similar justification: any way \(p_i\) in which\(p\) can be true can be partitioned into two further propositionsaccording to how some coin would land if tossed.

So under what conditions can a preference relation \(\preceq\) on theset \(\Omega\) be represented as maximising desirability? Some of therequired conditions on preference should be familiar by now and willnot be discussed further. In particular, \(\preceq\) has to betransitive, complete and continuous (recall our discussion inSection 2.3 of vNM’s Continuity preference axiom).

The next two conditions are, however, not explicitly part of the tworepresentation theorems that have been considered so far:

Averaging is the distinguishing rationality condition inJeffrey’s theory. It can actually be seen as a weak version ofIndependence and the Sure Thing Principle, and it plays a similar rolein Jeffrey’s theory. But it is not directly inconsistent withAllais’ preferences, and its plausibility does not depend on thetype of probabilistic independence that the STP implies. The postulaterequires that no proposition be strictly better or worse than all ofits possible realisations, which seems to be a reasonable requirement.When \(p\) and \(q\) are mutually incompatible, \(p\cup q\) impliesthat either \(p\) or \(q\) is true, but not both. Hence, it seemsreasonable that \(p\cup q\) should be neither strictly more nor lessdesirable than both \(p\) and \(q\). Suppose one of \(p\) or \(q\) ismore desirable than the other. Then since \(p\cup q\) is compatiblewith the truth of either the more or the less desirable of the two,\(p\cup q\)’s desirability should fall strictly between that of\(p\) and that of \(q\). However, if \(p\) and \(q\) are equallydesirable, then \(p\cup q\) should be as desirable as each of thetwo.

The intuitive appeal of Impartiality, which plays a similar role inJeffrey’s theory as P4 does in Savage’s, is not as greatas that of Averaging. Jeffrey himself admitted as much in hiscomment:

The axiom is there because we need it, and it is justified by ourantecedent belief in the plausibility of the result we mean to deducefrom it. (1965: 147)

Nevertheless, it does seem that an argument can be made that anyreasonable person will satisfy this axiom. Suppose you are indifferentbetween two propositions, \(p\) and \(q\), that cannot besimultaneously true. And suppose now we find a proposition \(r\), thatis pairwise incompatible with both \(p\) and \(q\), and which you findmore desirable than both \(p\) and \(q\). Then if it turns out thatyou are indifferent between \(p\) joined with \(r\) and \(q\) joinedwith \(r\), that must be because you find \(p\) and \(q\) equallyprobable. Otherwise, you would prefer the union that contains the oneof \(p\) and \(q\) that you find less probable, since that gives you ahigher chance of the more desirable proposition \(r\). It then followsthat for any other proposition \(s\) that satisfies the aforementionedconditions that \(r\) satisfies, you should also be indifferentbetween \(p\cup s\) and \(q\cup s\), since, again, the two unions areequally likely to result in \(s\).

The first person to prove a theorem stating sufficient conditions fora preference relation to be representable as maximising the value of aJeffrey-desirability function was actually not Jeffrey himself, butthe mathematician Ethan Bolker (1966, 1967). He proved the following result:^[11]

Theorem 4 (Bolker)
Let \(\Omega\) be a complete and atomless Boolean algebra ofpropositions, and \(\preceq\) a continuous, transitive and completerelation on \(\Omega \setminus \bot \), that satisfies Averaging andImpartiality. Then there is a desirability function on \(\Omega\setminus \bot \)and a probability function on \(\Omega\) relative towhich \(\preceq\) can be represented as maximising desirability.

Unfortunately, Bolker’s representation theorem does not yield aresult anywhere near as unique as Savage’s. Even if aperson’s preferences satisfy all the conditions inBolker’s theorem, then it is neither guaranteed that there willbe just one probability function that represents her beliefs nor thatthe desirability function that represents her desires will be uniqueup to a positive linear transformation (unless her preferences areunbounded). Even worse, the same preference ordering satisfying allthese axioms could be represented as maximising desirability relativeto two probability functions that do not even agree on how to orderpropositions according to their probability.^[12]

For those who think that the only way to determine a person’scomparative beliefs is to look at her preferences, the lack ofuniqueness in Jeffrey’s theory is a big problem. Indeed, thismay be one of the main reasons why economists have largely ignoredJeffrey’s theory. Economists have traditionally been skepticalof any talk of a person’s desires and beliefs that goes beyondwhat can be established by examining the person’s preferences,which they take to be the only attitude that is directly revealed by aperson’s behaviour. For these economists, it is thereforeunwelcome news if we cannot even in principle determine thecomparative beliefs of a rational person by looking at herpreferences.

Those who are less inclined towards behaviourism might, however, notfind this lack of uniqueness in Bolker’s theorem to be aproblem. James Joyce (1999), for instance, thinks that Jeffrey’stheory gets things exactly right in this regard, since one should notexpect that reasonable conditions imposed on a person’spreferences would suffice to determine a unique probability functionrepresenting the person’s beliefs. It is only by imposing overlystrong conditions, as Savage does, that we can achieve this. However,if uniqueness is what we are after, then we can, as Joyce points out,supplement the Bolker-Jeffrey axioms with certain conditions on theagent’s comparative belief relation (e.g. those proposed byVillegas 1964) that, together with the Bolker-Jeffrey axioms, ensurethat the agent’s preferences can be represented by a uniqueprobability function and a desirability function that is unique up toa positive linear transformation.

Instead of adding specific belief-postulates to Jeffrey’stheory, as Joyce suggests, one can get the same uniqueness result byenriching the set of prospects. Richard Bradley (1998) has, forinstance, shown that if one extends the Boolean algebra inJeffrey’s theory to indicative conditionals, then a preferencerelation on the extended domain that satisfies the Bolker-Jeffreyaxioms (and some related axioms that specifically apply toconditionals) will be representable as maximising desirability, wherethe probability function is unique and the desirability function isunique up to a positive linear transformation.

4. Broader implications of Expected Utility (EU) theory

It was noted from the outset that EU theory is as much a theory ofrational choice, or overall preferences amongst acts, as it is atheory of rational belief and desire. This section expands, in turn,on the epistemological and evaluative commitments of EU theory.

4.1 On rational belief

Some refer to subjective expected utility theory asBayesiandecision theory. This label brings to the forefront thecommitment toprobabilism, i.e., that beliefs may come indegrees which, on pain of irrationality, can be representednumerically as probabilities. So there is a strong connection betweensubjective EU theory and probabilism, or more generally betweenrational preference and rational belief. (The finer details ofrational preference and associated rational belief are not the focushere; challenges to EU theory on this front are addressed inSection 5 below.)

Some take the connection between rational preference and rationalbelief to run very deep indeed. At the far end of the spectrum is theposition that the very meaning of belief involves preference. Indeed,recall this manoeuvre in Savage’s theory, discussed earlier inSection 3.1. Many question the plausibility, however, of equating comparativebelief with preferences over specially contrived prospects. A moremoderate position is to regard these preferences as entailed by, butnot identical with, the relevant comparative beliefs. Whether or notbeliefs merely ground or are defined in terms of preference, there isa further question as to whether the only justification for rationalbelief having a certain structure (say, conforming to the probabilitycalculus) is a pragmatic one, i.e., an argument resting on theagent’s preferences being otherwise inconsistent orself-defeating. A recent defender of this kind of pragmatism (albeitcast in more general terms) is Rinard (e.g., 2017). Others contendthat accounts of rational belief can and should be ultimatelyjustified on epistemic grounds; Joyce (1998), for instance, offers anon-pragmatic justification of probabilism that rests on the notion ofoverallaccuracy of one’s beliefs. (For furtherdevelopments of this position, see the entry onepistemic utility arguments for probabilism.)

Notwithstanding these finer disputes, Bayesians agree that pragmaticconsiderations play a significant role in managing beliefs. Oneimportant way in which an agent can interrogate her degrees of beliefis to reflect on their pragmatic implications. Furthermore, whether ornot to seek more evidence is a pragmatic issue; it depends on the“value of information” one expects to gain with respect tothe decision problem at hand. The idea is that seeking more evidenceis an action that is choice-worthy just in case the expected utilityof seeking further evidence before making one’s decision isgreater than the expected utility of making the decision on the basisof existing evidence. This reasoning was made prominent in a paper byGood (1967), where he proves that an expected utility maximiser willalways seek “free evidence” that may have a bearing on thedecision at hand. (Precursors of this theorem can be found in Ramsey1990, published posthumously, and Savage 1954.) Note that the theoremassumes the standard Bayesian learning rule known as“conditionalisation”, which requires that when one’slearning experience can be characterised as coming to be certain thatsome proposition (to which one had assigned positive probability) istrue, one’s new degrees of belief should be equal to one’sold degrees of belief conditional on the proposition that now hasprobability one. Indeed, the fact that conditionalisation plays acrucial role in Good’s result about the non-negative value offree evidence is taken by some as providing some justification forthis learning rule.

So EU theory or Bayesian decision theory underpins a powerful set ofepistemic norms. It has given rise to a whole school of statisticalinference and experimental design, including formal interpretations ofkey concepts like “evidence”, “evidentialsupport”, “induction” versus“abduction”, and the bearing of “coherence”and “explanatory power” on truth (see the relevantrelated entries). The major competitor to Bayesianism, as regards scientific inference,is arguably the collection of approaches known as Classical or Errorstatistics, which deny the cogency of “degrees of support”(probabilistic or otherwise) conferred on a hypothesis by evidence.These approaches focus instead on whether a hypothesis has survivedvarious “severe tests”, and inferences are made with aneye to the long-run properties of tests as opposed to how they performin any single case, which would require decision-theoretic reasoning(see the entry onphilosophy of statistics).

4.2 On rational desire

EU theory also has implications for the structure of rational desire.In this regard, the theory has been criticised on opposing fronts. Weconsider first the criticism that EU theory is too permissive withrespect to what may influence an agent’s desires. We then turnto the opposing criticism: that when it comes to desire, EU theory isnot permissive enough.

The worry that EU theory is too permissive with respect to desire isrelated to the worry that the theory isunfalsifiable. Theworry is that apparently irrational preferences by the lights of EUtheory can always be construed as rational, under a suitabledescription of the options under consideration. As discussed inSection 1 above, preferences that seem to violate Transitivity can be construedas consistent with this axiom so long as the options being comparedvary in their description depending on, amongst other things, theother options under consideration. The same goes for preferences thatseem to violate Independence and the separability condition discussedfurther inSection 5.1 below. One might argue that this is the right way to describe suchagents’ preferences. After all, an apt model of preference issupposedly one that captures, in the description of final outcomes andoptions, everything that matters to an agent. In that case, however,EU theory is effectively vacuous or impotent as a standard ofrationality to which agents can aspire. Moreover, it stretches thenotion of what are genuine properties of outcomes that can reasonablyconfer value or be desirable for an agent.

There are two ways one can react to the idea that an agent’spreferences are necessarily consistent with EU theory:

• One can resist the claim, asserting that there are additionalconstraints on thecontent of an agent’s preferences.On the one hand there may be empirical constraints whereby the contentof preferences is determined by some tradeoff between fit andsimplicity in representing the agent’s greater “web”of preference attitudes. On the other hand there may be normativeconstraints with respect to what sorts of outcomes an agent mayreasonably discriminate (for relevant discussion, see Tversky 1975;Broome 1991a & 1993; Pettit 1993; Dreier 1996; Guala 2006;Vredenburgh 2020).

• One can alternatively embrace the claim, interpreting EU theory not asa standard against which an agent may pass or fail, but rather as anorganising principle that enables the characterisation of anagent’s desires as well as her beliefs (see esp. Guala2008).

Either way, EU theory does not have the conceptual resources todescribe thereasons for an agent’s preferenceattitudes. Dietrich and List (2013 & 2016a) have proposed a moregeneral framework that fills this lacuna. In their framework,preferences satisfying some minimal constraints are representable asdependent on the bundle of properties in terms of which each option isperceived by the agent in a given context. Properties can, in turn, becategorised as eitheroption properties (which are intrinsicto the option),relational properties (which concern theoption in a particular context), orcontext properties (whichconcern the context of choice itself). Such a representation permitsmore detailed analysis of the reasons for an agent’s preferencesand captures different kinds of context-dependence in an agent’schoices. Furthermore, it permits explicit restrictions on what countsas a legitimate reason for preference, or in other words, whatproperties legitimately feature in an outcome description; suchrestrictions may help to clarify the normative commitments of EUtheory.

There are also less general models that offer templates forunderstanding the reasons underlying preferences. For instance, themultiple criteria decision framework (see, for instance,Keeney and Raiffa 1993) takes an agent’s overall preferenceordering over options to be an aggregate of the set of preferenceorderings corresponding to all the pertinent dimensions of value.Under certain assumptions, the overall or aggregate preferenceordering is compatible with EU theory. One might otherwise seek tounderstand the role of time, or the temporal position of goods, onpreferences. To this end, outcomes are described in terms oftemporally-indexed bundles of goods, orconsumption streams(for an early model of this kind see Ramsey 1928; a later influentialtreatment is Koopmans 1960). There may be systematic structure to anagent’s preferences over these consumption streams, over andabove the structure imposed by the EU axioms of preference. Forinstance, the aforementioned authors characterise preferences thatexhibitexponential time discounting.

Let’s turn now to the opposing kind of criticism: that thelimited constraints that EU theory imposes on rational preference anddesire are nonetheless overly restrictive. Here the focus will be onthe compatibility of EU theory with prominent ethical positionsregarding the choice-worthiness of acts, as well as meta-ethicalpositions regarding the nature of value and its relationship tobelief.

One may well wonder whether EU theory, indeed decision theory moregenerally, is neutral with respect to normative ethics, or whether itis compatible only withethical consequentialism, given thatthe ranking of an act is fully determined by the utilities andprobabilities of its possible outcomes or consequences. Such a modelseems at odds withnonconsequentialist ethical theories forwhich the choice-worthiness of acts purportedly depends on more thanthe moral value of their consequences. The model does not seem able toaccommodate basic deontological notions like agent relativity,absolute prohibitions or permissible and yet suboptimal acts.

An initial response is that one should not read too much into theformal concepts of decision theory. The utility measure over acts andoutcomes is simply a convenient way to represent an ordering, andleaves much scope for different ways of identifying and evaluatingoutcomes. Just as an agent’s utility function need not beinsensitive to ethical considerations in general (a commonmisconception due to the prevalence of selfish preferences in economicmodels; see, for instance, Sen 1977), nor need it be insensitive tospecifically nonconsequentialist or deontological ethicalconsiderations. It all depends on how acts and their outcomes aredistinguished and evaluated. For starters, the character of an act mayfeature as a property of all its possible outcomes. Moreover, whethersome event befalls or is perpetrated by the deciding agent or rathersomeone else may be relevant. That an act involves lying, say, can bereferenced in all possible outcomes of the act, and furthermore thislying on the part of the deciding agent can be distinguished from thelying of others. In general, acts and their outcomes can bedistinguished according to whatever matters morally, be it a complexrelational property to do with how and when the act is chosen, and/orin what way some state of affairs results from the act. For earlydiscussions on how a wide range of ethical considerations can beaccommodated in the description of acts and outcomes, see, forinstance, Sen (1982), Vallentyne (1988), Broome (1991b) and Dreier(1993). This idea has since been embraced by others associated withthe so-called “consequentializing” program, includingLouise (2004) and Portmore (2007). The idea is that the normativeadvice of putatively nonconsequentialist ethical theories can berepresented in terms of a ranking of acts/outcomes corresponding tosome value function, as per consequentialist ethical theories (see tooColyvan et al. 2010).

A sticking point for reconciling decision theory with all forms ofnonconsequentialism is the difficulty in accommodatingabsoluteprohibitions orside constraints (see Oddie and Milne1999; Jackson and Smith 2006). For instance, suppose there is a moralprohibition against killing an innocent person, whatever else is atstake. Perhaps such a constraint is best modelled in terms of alexical ranking and corresponding value function, whereby thekilling-innocents status of an act/outcome takes priority indetermining its relative rank/value. But this has counterintuitiveimplications in the face of risk since very many acts will have somechance, however small, of killing an innocent. The lesson here maysimply be that the theories in question require development; anymature ethical theory owes us an account of how acts are ranked underrisk or uncertainty. A further challenge for the reconciliation ofdecision theory and nonconsequentialism is the accommodation of“agent-centred options” and associated“supererogation”. Portmore (e.g., 2007) and Lazar (e.g.,2017) offer proposals to this effect, which appeal (in different ways)to the moral ranking of acts/outcomes as distinct from the personalcosts to the agent of pursuing these acts/outcomes.

To the extent that decision theory can be reconciled with the fullrange of ethical theories, should we say that there are no meaningfuldistinctions between these theories? Brown (2011) and Dietrich andList (2017) demonstrate that in fact the choice-theoreticrepresentation of ethical theories better facilitates distinctionsbetween them; terms like “(non)consequentialism” can beprecisely defined, albeit in debatable ways. More generally, we cancatalogue theories in terms of the kinds of properties (whetherintrinsic or in some sense relational) that distinguish acts/outcomesand also in terms of the nature of the ranking of acts/outcomes thatthey yield (whether it is fully consistent with EU theory or ratherviolates one or more of its axioms).

Indeed, some of the most compelling counterexamples to EU axioms ofpreference rest on ethical considerations. Recall our earlierdiscussion of the basic ordering axioms inSection 1. The Transitivity axiom has been challenged by appeal toethically-motivated examples of preference cycles (see Temkin 2012).The notion of a non-continuous lexical ordering was mentioned above inrelation to ethical side constraints. The dispensability of theCompleteness axiom, too, is often motivated by appeal to examplesinvolving competing ethical values, not least components of personalwelfare, that are difficult to tradeoff against each other (forvarious such accounts of value and its implications for preference,see, e.g., Levi 1986, Broome 1991a, Chang 2002, Rabinowicz 2008; formore recent discussions of the multi-dimensional nature of value, see,e.g., Elson 2017, the contributions in Andersson and Herlitz 2022,Dorr et al. 2023, Hedden and Muñoz 2024). Note that some ofthese challenges to EU theory are discussed in more depth inSection 5 below.

Finally, we turn to the potential meta-ethical commitments of EUtheory. David Lewis (1988, 1996) famously employed EU theory to argueagainstanti-Humeanism, the position that we are sometimesmoved entirely by our beliefs about what would be good, rather than byour desires as the Humean claims. He formulated the anti-Humean theoryas postulating a necessary connection between, on the one hand, anagent’s desire for any proposition \(A\), and, on the otherhand, her belief in a proposition about \(A\)’s goodness; andclaimed to prove that when such a connection is formulated in terms ofEU theory, the agent in question will be dynamically incoherent.Several people have criticised Lewis’s argument. For instance,Broome (1991c), Byrne and Hájek (1997) and Hájek andPettit (2004) suggest formulations of anti-Humeanism that are immuneto Lewis’ criticism, while Stefánsson (2014) and Bradleyand Stefánsson (2016) argue that Lewis’ proof relies on afalse assumption. Nevertheless, Lewis’ argument no doubtprovoked an interesting debate about the sorts of connections betweenbelief and desire that EU theory permits. There are, moreover, furtherquestions of meta-ethical relevance that one might investigateregarding the role and structure of desire in EU theory. For instance,Jeffrey (1974) and Sen (1977) offer some preliminary investigations asto whether the theory can accommodatehigher-orderdesires/preferences, and if so, how these relate to first-orderdesires/preferences.

5. Challenges to EU theory

Thus far the focus has been on prominent versions of the standardtheory of rational choice: EU theory. This section picks up on somekey criticisms of EU theory that have been developed into alternativeaccounts of rational choice. The proposed innovations to the standardtheory are distinct and so are discussed separately, but they are notnecessarily mutually exclusive. Note that we do not address allcriticisms of EU theory that have inspired alternative accounts ofrational choice. Two major omissions of this sort (for want of spaceand also because they have been thoroughly addressed in alternativeentries of this encyclopedia) are i) the problem of causal anomaliesand the development of causal decision theory (see the entry oncausal decision theory), and ii) the problem of infinite state spaces and the development ofalternatives like “relative expectation theory” (see theentries onnormative theories of rational choice: expected utility theory andthe St. Petersburg paradox).

5.1 On risk and regret attitudes

Expected utility theory has been criticised for not allowing for valueinteractions between outcomes in different, mutually incompatiblestates of the world. For instance, recall that when deciding betweentwo risky options you should, according to Savage’s version ofthe theory, ignore the states of the world where the two optionsresult in the same outcome. That seems very reasonable if we canassumeseparability between outcomes in different states ofthe world, i.e., if the contribution that an outcome in one state ofthe world makes towards the overall value of an option is independentof what other outcomes the option might result in. For then identicaloutcomes (with equal probabilities) should cancel each other out in acomparison of two options, which would entail that if two optionsshare an outcome in some state of the world, then when comparing theoptions, it does not matter what that shared outcome is.

The Allais paradox, discussed inSection 2.3 above, is a classic example where the aforementioned separabilityseems to fail. For ease of reference, the options that generate theparadox are reproduced as Table 3. Recall fromSection 2.3 that people tend to prefer \(L_2\) over \(L_1\) and \(L_3\) over\(L_4\)—referred to as theAllais’preferences—in violation of expected utility theory. Theviolation occurs precisely because the contribution that some of theseoutcomes make towards the overall value of an option is notindependent of the other outcomes that the option can have. Comparethe extra chance of outcome $0 that \(L_1\) has over \(L_2\) with thesame extra chance of $0 that \(L_3\) has over \(L_4\). Many peoplethink that this extra chance counts more heavily in the firstcomparison than the latter, i.e., that an extra 0.01 chance of $0contributes a greater negative value to \(L_1\) than to \(L_3\). Someexplain this by pointing out that theregret one wouldexperience by winning nothing when one could have had $2400 forsure—i.e., when choosing \(L_1\) over \(L_2\) and the firstticket is drawn—is much greater than the regret one wouldexperience by winning nothing when the option one turned down also hada high chance of resulting in $0—such as when choosing \(L_3\)over \(L_4\) (see, e.g., Loomes and Sugden 1982). But whether or notthe preference in question should be explained by the potential forregret, it would seem that the desirability of the $0-outcome dependson what could (or would) otherwise have been; in violation of theaforementioned assumption of separability. (See Thoma 2020 for arecent extensive discussion of this assumption.)

Various attempts have been made to make Allais’ preferencescompatible with some version of expected utility theory. A commonresponse is to suggest that the choice problem has been incorrectlydescribed. If it really is rational to evaluate $0 differentlydepending on which lottery it is part of, then perhaps this should beaccounted for in the description of the outcomes (Broome 1991a). Forinstance, we could add a variable to the $0 outcome that \(L_1\) mightresult in to represent the extra regret or risk associate with thatoutcome compared to the $0 outcomes from the other lotteries (as doneinTable 4). If we do that, Allais’ preferences are no longer inconsistentwith EU theory. The simplest way to see this is to note that when weignore the state of the world where the options that are beingcompared have the same outcome (i.e., when we ignore the last columnin Table 4), \(L_1\) is no longer identical to \(L_3\), which meansthat the Independence axiom of von Neumann and Morgenstern (andSavage’s Sure Thing Principle) no longer requires that oneprefer \(L_2\) over \(L_1\) only if one prefers \(L_4\) over\(L_3\).

The above “re-description strategy” could be employedwhenever the value and/or contribution of an outcome depends on otherpossible outcomes: just describe the outcomes in a way that accountsfor this dependency. But more worryingly, the strategy could beemployed whenever one comes acrossany violation of expectedutility theory or other theories of rationality (as discussed inSection 4.2).

There are various non-expected utility theories that can accommodateAllais’ preferences without re-describing the outcomes. LaraBuchak (2013) develops a decision theory on which the explanation forAllais’ preferences is not the differentvalue that theoutcome $0 has depending on what lottery it is part of. The outcomeitself has the same value. However, thecontribution that $0makes towards the overall value of an option partly depends on whatother outcomes are possible. Buchak introduces ariskfunction that represents people’s willingness to tradechances of something good for risks of something bad. And she showsthat if an agent satisfies a particular set of axioms, which isessentially Savage’s except that the Sure Thing Principle isreplaced with a strictly weaker one, then the agent’spreferences can be represented as maximisingrisk weightedexpected utility. More recently, Bottomley and Williamson (2024)have defended an alternative non-expected utility theory that theyargue has significant advantages over Buchak’s theory. On theirpreferredweighted linear utility theory, an agent’srisk attitude may vary with the utility that is at stake. Moreover,their theory, unlike Buchak’s, satisfies the requirement that ifan agent is indifferent between two options, then she should also beindifferent between either of those options and a lottery that amountsto randomising over them.

Bradley and Stefánsson (2017) also develop a new decisiontheory partly in response to the Allais paradox. But they suggest thatwhat explains Allais’ preferences is that the value of winingnothing from a chosen lottery partly depends on what would havehappened had one chosen differently. To accommodate this, they extendthe Boolean algebra in Jeffrey’s decision theory tocounterfactual propositions, and show that Jeffrey’sextended theory can represent the value-dependencies one often findsbetween counterfactual and actual outcomes. In particular, theirtheory can capture the intuition that the (un)desirability of winningnothing partly depends on whether or not one was guaranteed to winsomething had one chosen differently. Therefore, their theory canrepresent Allais’ preferences as maximising the value of anextended Jeffrey-desirability function.

Stefánsson and Bradley (2019) suggest yet another way ofaccounting for Allais’ preferences in an extension ofJeffrey’s decision theory; this time extended tochancepropositions, that is, propositions describing objectiveprobability distributions. The general idea is that the desirabilityof a particular increase or decrease in the chance of someoutcome—for instance, in the Allais case, a 0.01 increase in thechance of the $0-outcome—might depend on what the chances werebefore the increase or decrease. Stefánsson and Bradley’sextension of Jeffrey’s theory to chance propositions is alsomotivated by the fact that standard decision theories do notdistinguish between risk aversion with respect to some good andattitudes to quantities of that good (which is found problematic by,for instance, Hansson 1988, Rabin 2000, and Buchak 2013).

5.2 On completeness: Vague beliefs and desires

As noted inSection 4, criticisms of the EU requirement of a complete preference orderingare motivated by both epistemic and desire/value considerations. Onthe value side, many contend that a rational agent may simply find twooptionsincomparable or at least not have a determinatepreference between them. As in, the agent’s evaluations of thedesirability of sure options may not be representable by any preciseutility function. Likewise, on the belief side, some contend (notably,Joyce 2010 and Bradley 2017) that the evidence may be such that itdoes not commit a rational agent to precise degrees of beliefmeasurable by a unique probability function.

There are various alternative, “fuzzier” representationsof desire and belief that might be deemed more suitable. Halpern(2003), for instance, investigates different ways of conceptualisingand representing epistemic uncertainty, once we depart fromprobabilities. Presumably there are also various ways to representuncertain desire. Here the focus will be on just one proposal that ispopular amongst philosophers: the use ofsets of probabilityand utility functions to represent uncertainty in belief and desirerespectively. This is a minimal generalisation of the standard EUmodel, in the sense that probability and utility measures stillfeature. Roughly, the more severe the epistemic uncertainty, the moreprobability measures over the space of possibilities needed toconjointly represent the agent’s beliefs. This notion ofrational belief is referred to asimprecise probabilism (seethe entry onimprecise probabilities). Likewise, the more severe the evaluative uncertainty, the moreutility measures over the space of sure options needed to conjointlyrepresent the agent’s desires. Strictly speaking, we should nottreat belief and desire separately, but rather talk of theagent’s incomplete preferences being represented by a set ofprobability and utility pairs. Recall the requirement that incompletepreferences becoherently extendible (refer back toSection 1); on this representation, all the probability-utility pairs amount tocandidate extensions of the incomplete preferences.

The question then arises: Is there a conservative generalisation ofthe EU decision rule that can handle sets of probability and utilitypairs? Contender decision rules are standardly framed in terms ofchoice functions that take as input some set of available options andreturn as output a non-empty set of admissible options that is asubset of the available options. There is controversy even about basicconstraints on such a choice function. Many endorse an “EUnon-dominance” constraint, whereby an option is admissible onlyif it is not “EU-dominated” by some other availableoption, that is, there is not some other available option that has atleast as high EU according to all pairs of probability and utilityfunctions, and higher EU according to at least one pair of probabilityand utility functions. Levi (1986) proposes a slightly different basicconstraint: an option is admissible only if it has maximum EU for atleast one pair of probability and utility functions. In ordinarycircumstances where sets of probability and utility functions areclosed convex sets, Levi’s constraint is equivalent to the EUnon-dominance constraint (Schervish et al. 2003). More recently, someauthors have considered alternative basic constraints that mayconflict with EU non-dominance even in ordinary circumstances. Forinstance, Hare (2010) considers a state-wise non-dominance constraintand Lederman (forthcoming) considers a dimension-wise negativedominance constraint (the aim in each case being to reveal theparadoxical nature of violations of the completeness axiom ofpreference).

Even those who agree on a basic constraint like EU non-dominance maydisagree on further constraints when it comes to choice in the face ofincompleteness. See Bradley (2017) for extensive discussion of thevarious ways to proceed. A consideration that is often appealed to inorder to discriminate between candidate options is caution. TheMaxmin-EU rule, for instance, recommends picking the action withgreatest minimum expected utility (see Gilboa and Schmeidler 1989;Walley 1991). The rule is simple to use, but arguably much toocautious, paying no attention at all to the full spread of expectedutilities. The \(\alpha\)-Maxmin rule, by contrast, recommends takingthe action with the greatest \(\alpha\)-weighted sum of the minimumand maximum expected utilities associated with it. The relativeweights for the minimum and maximum expected utilities can be thoughtof as reflecting either the decision maker’s pessimism in theface of uncertainty or else her degree of caution (see Binmore2009).

There are more complicated choice rules that depend on a richerrepresentation of uncertainty involving a notion ofconfidence. For instance, Klibanoff et al. (2005) propose arule whereby choices are made between otherwise incomparable optionson the basis of confidence-weighted expected utility. It presupposesthat weights can be assigned to the various expected utilitiesassociated with an act, reflecting the agent’s confidence in thecorresponding probability and utility pairs. There are alternativerules that appeal to confidence even in the absence of precisecardinal weights. Gärdenfors and Sahlin (1982), for instance,suggest simply excluding from consideration any probability (andutility) functions that fall below a confidence threshold, and thenapplying the Maxmin-EU rule based on the remainder. Hill’s(2013) choice theory is somewhat similar, although confidencethresholds for probability and utility pairs are allowed to varydepending on the choice problem (and the term “confidence”is itself used differently). There are further proposals whereby actsare compared in terms of how much uncertainty they can tolerate (whichagain depends on levels of confidence) and yet still be a satisfactoryoption (see, e.g., Ben-Haim 2001). These rules are compelling, butthey do raise a host of difficult questions regarding how to interpretand measure the extra subjective attitudes that play a role, like“amount of confidence in a belief/desire” and“satisfactory level of desirability”.

5.3 Catastrophic risk and the precautionary principle

The standard approach in economics and policy analysis is to useexpected utility theory to decide in the face ofcatastrophicrisks, just like more mundane risks. This approach has recently beenquestioned in the context of climate policy, where so-calledintegrated assessment models have been criticised for theirreliance on EU theory (see, for instance Pindyck 2022 and Stern et. al2022). Instead, it has been suggested that something closer to theprecautionary principle should guide climate policyevaluation. Although there is no generally agreed upon definition ofthe principle—and some would resist calling itaprinciple (Hartzel-Nichols 2013)—the general idea is toencourage decision-makers to err on the side of caution, even if thatcan be expected to result in less utility, in particular in situationsof great uncertainty and when faced with potentially catastrophicoutcomes.

There is disagreement on what type of principle the precautionaryprinciple is. While the principle is often defended as adecision-rule (see, e.g., Steel 2014), some think that anysuch interpretation of the principle is implausible (Peterson 2006,Stefánsson 2019). Instead, some suggest we should understandthe principle as ameta-rule for how to structuredecision-problems, to which decision-rules are then applied(see, e.g., Steele 2006). An example of such a meta-rule is a ruleinstructing us to err on the side of caution when describing adecision-problem, for instance, by carefully identifying the full setof available options and by describing the possible outcomes of theseoptions in as much detail as is required to account for all therelevant evaluative differences between them. An example of adecision-rule is the EU rule, requiring that we maximise expectedutility in each decision-problem.

Some have suggested that when the precautionary principle isinterpreted as a decision-rule, the feature that most clearlydistinguishes it from the EU rule is that it does not satisfy the EUaxiom of Continuity (see, e.g., Bartha and DesRoches 2021). Recallfrom section 3.3 that this axiom implies that no outcome is so badthat we shouldn’t be willing to take a gamble that might resultin that outcome, provided that the gamble is sufficiently likely toinstead result in an outcome that provides an improvement on thestatus quo. In the context of climate policy, Continuity implies thatno climate catastrophe is so bad that we shouldn’t be willing toimplement a policy that might (with low probability) result in thatcatastrophe as long as the policy is sufficiently likely to insteadresult in some social gain.

A recently defended complete decision theory without Continuity is thelexicographic utility theory. Bartha and DesRoches (2021) andSteel and Bartha (2023) defend the theory as a normatively plausibleformalisation of the precautionary principle. Stefánsson (2024)however argues that the lexicographic theory has problematicimplications when applied in contexts where the disvalue to be avoidedcomes in degrees, such as when evaluating public health policies, towhich the principle is often applied. In particular, he criticises thetheory for beinghypersensitive to trivial valuedifference—a criticism that Steel, Bartha, and DesRoches (2025)respond to. But despite the lexicographic utility theory’spotential theoretical problems, the theory may be pragmatically usefulfor decision-makers with limited information or computationalabilities, in addition to being potentially a faithfulcharacterisation of the intention behind the precautionary principle.In any case, the quest for a catastrophic-risk averse alternative toEU theory continues.

5.4 Unawareness

There has been recent interest in yet a further challenge to expectedutility theory, namely, the challenge fromunawareness. Infact, unawareness presents a challenge for all extant normativetheories of choice. To keep things simple, we shall however focus onSavage’s expected utility theory to illustrate the challengeposed by unawareness.

As the reader will recall, Savage takes for granted a set of possibleoutcomes \(\bO\), and another set of possible states of the world\(\bS\), and defines the set of acts, \(\bF\), as the set of allfunctions from \(\bS\) to \(\bO\). Moreover, his representationtheorem has been interpreted as justifying the claim that a rationalperson always performs the act in \(\bF\) that maximises expectedutility, relative to a probability measure over \(\bS\) and a utilitymeasure over \(\bO\).

Now, Savage’s theory is neutral about how to interpret thestates in \(\bS\) and the outcomes in \(\bO\). For instance, thetheory is consistent with interpreting \(\bS\) and \(\bO\) asrespectively the sets ofall logically possible states andoutcomes, but it is also consistent with interpreting \(\bS\) and\(\bO\) as respectively the sets of states and outcomesthat somemodeller recognises, or the sets of states and outcomesthatthe decision-maker herself recognises.

If the theory is meant to describe the reasoning of a decision-maker,the first two interpretations would seem inferior to the third. Theproblem with the first two interpretations is that the decision-makermight beunaware of some of the logically possible states andoutcomes, as well as some of the states and outcomes that the modelleris aware of. (Having said that, one may identify the states andoutcomes that the agent is unaware of by reference to those of whichthe modeller is aware.)

When it comes to (partially) unaware decision-makers, an importantdistinction can be made between on the one hand what we might call“unawareness of unawareness”—that is, a situationwhere a decision-maker does not realise that there might be someoutcome or state that they are unaware of—and on the other hand“awareness of unawareness”—that is, a situationwhere a decision-maker at least suspects that there is some outcome orstate of which they are unaware.

From the perspective of decision-making, unawareness of unawareness isnot of much interest. After all, if one is not even aware of thepossibility that one is unaware of some state or outcome, then thatunawareness cannot play any role in one’s reasoning about whatto do. However, decision-theoretic models have been proposed for how arational person responds togrowth in awareness, that aremeant to apply even to people who previously were unaware of theirunawareness. In particular, economists Karni and Vierø (2013,2015) have extended standard Bayesian conditionalisation to suchlearning events. Their theory,Reverse Bayesianism,informally says that awareness growth should not affect the ratios ofprobabilities of the states/outcomes that the agent was aware ofbefore the growth. Richard Bradley (2017) defends a similar principlein the context of the more general Jeffrey-style framework, and sodoes Roussos (2020); but the view is criticised by Steele andStefánsson (2021a, 2021b) and by Mahtani (2020).

In contrast, awareness of unawareness would seem to be of greatinterest from the perspective of decision-making. If you suspect thatthere is some possible state, say, that you have not yet entertained,and some corresponding outcome, the content of which you are unaware,then you might want to at least come to some view about how likely youexpect this state to be, and how good or bad you expect thecorresponding outcome to be, before you make a decision.

A number of people have suggested models to represent agents who areaware of their unawareness (e.g., Walker & Dietz 2013, Piermont2017, Karni & Vierø 2017). Steele and Stefánsson(2021b) argue that there may not be anything especially distinctiveabout how a decision-maker reasons about states/outcomes of which sheis aware she is unaware, in terms of the confidence she has in herjudgments and how she manages risk. (Indeed, being aware ofunawareness may differ only in degree, and not kind, from theimperfect grasp that a decision maker has on the other states andoutcomes pertinent to her decision problem.) That said, the way shearrives at such judgments of probability and desirability is worthexploring further. Grant and Quiggin (2013a, 2013b), for instance,suggest that these judgments are made based on induction from pastsituations where one experienced awareness growth. de Canson (2024)argues that, in any case, an agent should assign at least somepositive probability to there being some states/outcomes pertinent toher decision problem of which she is currently unaware.

In general, the literature on unawareness has been rapidly growing.Bradley (2017) and Steele and Stefánsson (2021b) are newin-depth treatments of this topic within philosophy. Schippermaintains a bibliography on unawareness, mostly with papers ineconomics and computer science (see “The UnawarenessBibliography”, Other Internet Resource).

6. Sequential decisions

The decision theories of Savage and Jeffrey, as well as those of theircritics, apparently concern a single or “one shot only”decision; at issue is an agent’s preference ordering, andultimately her choice of act, at a particular point in time. One mayrefer to this as astatic decision problem. The questionarises as to whether this framework is adequate for handling morecomplex scenarios, in particular those involving a series or sequenceof decisions; these are referred to assequential decisionproblems.

On paper, at least, static and sequential decision models look verydifferent. The static model has familiar tabular ornormalform, with each row representing an available act/option, and columnsrepresenting the different possible states of the world that yield agiven outcome for each act. The sequential decision model, on theother hand, has tree orextensive form (such as inFigure 1). It depicts a series of anticipated choice points, where the branchesextending from a choice point represent the options at that choicepoint. Some of these branches lead to further choice points, oftenafter the resolution of some uncertainty due to new evidence.

These basic differences between static and sequential decision modelsraise questions about how, in fact, they relate to each other:

• Do static and sequential decision models depict the same kind ofdecision problem? If so, what is the static counterpart of asequential decision model?

• Does the sequential decision setting reveal any further(dis)advantages of EU theory? More generally does this setting shedlight on normative theories of choice?

These questions turn out to be rather controversial. They will beaddressed in turn, after the scene has been set with an old storyabout Ulysses.

6.1 Was Ulysses rational?

A well-known sequential decision problem is the one facing Ulysses onhis journey home to Ithaca in Homer’s great tale from antiquity.Ulysses must make a choice about the manner in which he will sail pastan island inhabited by sweet-singing sirens. He can choose to sailunrestrained or else tied to the mast. In the former case, Ulysseswill later have the choice, upon hearing the sirens, to eithercontinue sailing home to Ithaca or to stay on the island indefinitely.In the latter case, he will not be free to make further choices andthe ship will sail onwards to Ithaca past the sweet-singing sirens.The final outcome depends on what sequence of choices Ulysses makes.Ulysses’ decision problem is represented in tree (or extensive)form inFigure 1 (where the two boxes represent choice points for Ulysses).

Figure 1. Ulysses’ decisionproblem

We are told that, before embarking, Ulysses would most prefer tofreely hear the sirens and return home to Ithaca. The problem is thatUlysses predicts his future self will not comply: if he sailsunrestrained, he will later be seduced by the sirens and will not infact continue home to Ithaca but will rather remain on the islandindefinitely. Ulysses therefore reasons that it would be better to betied to the mast, because he would prefer the shame and discomfort ofbeing tied to the mast and making it home to remaining on thesirens’ island forever.

It is hard to deny that Ulysses makes a wise choice, given hispreferences, in being tied to the mast. Some hold, however, thatUlysses is nevertheless not an exemplary agent, since his present selfmust play against his future self who will be unwittingly seduced bythe sirens. While Ulysses is rationalat the time of thefirst choice node, his is not rationalover the time periodin question, since the sequence of choices that he inevitably pursuesis suboptimal. It would have been better were he able to sailunconstrained and continue on home to Ithaca.

While rationality-over-time may have import in assessing anagent’s preferences and her norms for changing those preferences(more on which below inSection 6.2), the preliminary issue is rationality-at-a-time. How should an agentrepresent her decision problem in static form and choose amongst herinitial options in light of her projected decision tree? This questionhas generated a surprising amount of controversy. Three majorapproaches to negotiating sequential decision trees have appeared inthe literature. These are thenaïve ormyopicapproach, thesophisticated approach and theresolute approach. These will be discussed in turn; it willbe suggested that the disputes may not be substantial but ratherindicate subtle differences in the interpretation of sequentialdecision models.

The so-called naïve approach to negotiating sequential decisionsserves as a useful contrast to the other two approaches. Thenaïve agent assumes that any path through the decision tree ispossible, and so sets off on whichever path is optimal, given theirpresent attitudes. For instance, a naïve Ulysses would simplypresume that he has three overall strategies to choose from: eitherordering the crew to tie him to the mast, or issuing no such order andlater stopping at the sirens’ island, or issuing no such orderand later sticking to his course. Ulysses prefers the outcomeassociated with the latter combination, and so he initiates thisstrategy by not ordering the crew to restrain him. Table 5 presentsthe static counterpart of naïve Ulysses’ decision problem.In effect, this decision model does not take into accountUlysses’ present knowledge of his future preferences, and henceadvises that he pursue an option that is predicted to beimpossible.

There is no need to labour the point that the naïve approach tosequential choice is aptly named. The hallmark of the sophisticatedapproach, by contrast, is its emphasis on backwards planning: thesophisticated chooser does not assume that all paths through thedecision tree, or in other words, all possible combinations of choicesat the various choice nodes, will be possible. The agent considers,rather, what they will be inclined to choose at later choice nodeswhen they get to the temporal position in question. SophisticatedUlysses would take note of the fact that, if he reaches the island ofthe sirens unrestrained, he will want to stop there indefinitely, dueto the transformative effect of the sirens’ song on hispreferences. This is then reflected in the static representation ofthe decision problem, as per Table 6. The states here concernUlysses’ future preferences, once he reaches the island. Sincethe second state has (by assumption) probability zero, the acts aredecided on the basis of the first state, so Ulysses wisely chooses tobe tied to the mast.

Resolute choice deviates from sophisticated choice only under certainconditions that are not fulfilled by Ulysses, given his change inattitudes. Defenders of resolute choice typically defend decisiontheories and associated preferences that violate the Independenceaxiom/Sure-Thing Principle (notably McClennen 1990 and Machina 1989;see also Rabinowicz 1995 and Buchak 2013 for discussion), and appealto resolute choice to make these preferences more palatable in thesequential-decision context (to be discussed further inSection 6.2 below). According to resolute choice, in appropriate contexts, theagent should at all choice points stick to the strategy that wasinitially deemed best. The question is whether this advice makessense, given the standard interpretation of a sequential decisionmodel. What would it mean for an agent to choose against herpreferences in order to fulfill a previously-selected plan? That wouldseem to defy the very notion of preference. Of course, an agent mayplace considerable importance on honouring previous commitments. Anysuch integrity concerns, however, should arguably be reflected in thespecification of outcomes and thus in the agent’s preferences atthe time in question. This is quite different from choosing out ofstep with one’s all-things-considered preferences at a time.

Defenders of resolute choice may have in mind a differentinterpretation of sequential decision models, whereby future“choice points” are not really points at which an agent isfree to choose according to her preferences at the time. If so, thiswould amount to a subtle shift in the question or problem of interest.In what follows, the standard interpretation of sequential decisionmodels will be assumed, and accordingly, it will be assumed thatrational agents pursue the sophisticated approach to choice (as perLevi 1991, Maher 1992, Seidenfeld 1994, amongst others).

6.2 The EU axioms revisited

We have seen that sequential decision trees can help an agent likeUlysses take stock of the consequences of his current choice, so thathe can better reflect on what to donow. The literature onsequential choice is primarily concerned, however, with more ambitiousquestions. The sequential-decision setting effectively offers new waysto “test” theories of rational preference and norms forpreference (or belief and desire) change. The question is whether ornot an agent’s decision theory in this broad sense is shown tobedynamically inconsistent or self-defeating.

Skyrms’ (1993) “diachronic Dutch book” argument forconditionalisation can be read in this way. The agent is assumed tohave EU preferences and to take a sophisticated (backwards reasoning)approach to sequential decision problems. Skyrms shows that any suchagent who plans to learn in a manner at odds with conditionalisationwill make self-defeating choices in some specially contrivedsequential decision situations. A conditionalising agent, by contrast,will never make choices that are self-defeating in this way. The kindof “self-defeating choices” at issue here are ones thatyield a sure loss. That is, the agent chooses a strategy that issurely worse, by her own lights, than another strategy that she mightotherwise have chosen, if only her learning rule was such that shewould choose differently at one or more future decision nodes.

A similar “dynamic consistency” argument can be used todefend EU preferences in addition to learning in accordance withconditionalisation (see Hammond 1976, 1977, 1988b,c). It is assumed,as before, that the agent takes a sophisticated approach to sequentialdecision problems. Hammond shows that only a fully Bayesian agent canplan to pursue any path in a sequential decision tree that is deemedoptimal at the initial choice node. This makes the Bayesian agentunique in that she will never make “self-defeatingchoices” on account of her preferences and norms for preferencechange. She will never choose a strategy that is worse by her ownlights than another strategy that she might otherwise have chosen, ifonly her preferences were such that she would choose differently atone or more future decision nodes.

Hammond’s argument for EU theory, and the notion of dynamicconsistency that it invokes, has been criticised from differentquarters, both by those who defend theories that violate theIndependence axiom but retain the Completeness and Transitivity (i.e.,Ordering) axioms of EU theory, and those who defend theories thatviolate the latter (for discussion, see Steele 2010). The approachtaken by some defenders of Independence-violating theories (notably,Machina 1989 and McClennen 1990) has already been alluded to: Theyreject the assumption of sophisticated choice underpinning the dynamicconsistency arguments. Seidenfeld (1988a,b, 1994, 2000a,b) ratherrejects Hammond’s notion of dynamic consistency in favour of amore subtle notion that discriminates between theories that violateOrdering and those that violate Independence alone; the former, unlikethe latter, pass Seidenfeld’s test that turns on future decisionnodes where the agent is indifferent between the best options. Thisargument too is not without its critics (see McClennen 1988, Hammond1988a, Rabinowicz 2000).

The sure losses that are borne in the sequential-decision context canbe interpreted as aversion to new information or else aversion toopportunities for greater choice in the future (see Al-Najjar andWeinstein 2009). Kadane et al. (2008) and Bradley and Steele (2016)focus on the former: the fact that “non-EU” agents may payto avoid free evidence. Buchak (2010, 2013) defends such payments byappeal to a distinction (that those erring towards pragmatism aboutbelief would contest) between epistemic and instrumentalrationality.

7. Concluding remarks

Let us conclude by summarising the main reasons why decision theory,as described above, is of philosophical interest. First, normativedecision theory is clearly a (minimal) theory ofpracticalrationality. The aim is to characterise the attitudes of agents whoare practically rational, and various (static and sequential)arguments are typically made to show that certain practical setbacksbefall agents who do not satisfy standard decision-theoreticconstraints. Under the assumption that an ethical choice must berational, the findings of decision theory have implications forethics, or more generally, for the ways in which we canvaluestates of affairs. Second, many of the decision-theoretic constraintsconcern the agents’beliefs. In particular, normativedecision theory requires that agents’ degrees of beliefs satisfythe probability axioms and that they respond to new information byconditionalisation. Therefore, decision theory has great implicationsfor debates in epistemology and philosophy of science; that is, fortheories ofepistemic rationality.

Finally, decision theory should be of great interest to philosophersof mind and psychology, and others who are interested in how peoplecan understand the behaviour and intentions of others; and, moregenerally, how we can interpret what goes on in other people’sminds. Decision theorists typically assume that a person’sbehaviour can be fully explained in terms of her beliefs and desires.But perhaps more interestingly, some of the most important results ofdecision theory—the various representation theorems, some ofwhich have discussed here—suggest that if a person satisfiescertain rationality requirements, then we can read her beliefs anddesires, and how strong these beliefs and desires are, from her choicedispositions (or preferences). How much these theorems really tell usis a matter of debate, as discussed above. But on an optimisticreading of these results, they assure us that we can meaningfully talkabout what goes on in other people’s minds without much evidencebeyond information about their dispositions to choose.

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