Expected utility theory, which holds that a decision-makerought to maximize expected utility, is the prevailing theory ofinstrumental rationality. Nonetheless, four major challenges havearisen to the claim that the theory characterizes all rationalpreferences. These challenges are the phenomena of infinite orunbounded value, incommensurable goods, imprecise probabilities, andrisk-aversion. The challenges have been accompanied by alternativetheories that attempt to do better.

Expected utility theory consists of three components. The first is autility function that assigns real numbers to consequences. The secondis a probability function that assigns real numbers between 0 and 1 toevery possible event. The final component is an “aggregationnorm” that holds that the value of an act is its expectedutility value relative to these two functions, and that rationalpreferences track expected utility value (see the entry onexpected utility theory). Each challenge to EU theorycan be thought of as a rejection of one or more of the threecomponents as normative, and alternatives generally replace or extendthe relevant component.

It has long been observed that typical individuals do not in factconform to expected utility theory, and in response, a number ofdescriptive alternatives have arisen, particularly in the field ofeconomics (see Starmer 2000, Sugden 2004, Schmidt 2004 for surveys;see also the entry ondescriptive decision theory). This article will primarily discuss views that have beenput forth as normative.

• 1. Expected Utility Theory
  • 1.1 The Theory
  • 1.2 Components
    • 1.2.1 Utility
    • 1.2.2 Probability
    • 1.2.3 Expectation
• 2. Infinite and Unbounded Utility
  • 2.1 The Challenge from Infinite and Unbounded Utility
  • 2.2 Proposals
• 3. Incommensurability
  • 3.1 The Challenge from Incommensurability
  • 3.2 Proposals: Definitions
  • 3.3 Proposals: Decision Rules
• 4. Imprecise Probabilities or Ambiguity
  • 4.1 The Challenge from Imprecise Probabilities or Ambiguity
  • 4.2 Proposals: Probability Representations
  • 4.3 Proposals: Decision Rules
    • 4.3.1 Aggregative Decision Rules using Sets of Probabilities
    • 4.3.2 Aggregative Decision Rules: Choquet Expected Utility
    • 4.3.3 Decision Rules that Select from a Menu
  • 4.4 Normative Questions
• 5. Risk Aversion
  • 5.1 The Challenge from Risk Aversion
  • 5.2 Proposals
  • 5.3 Normative Issues
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Expected Utility Theory

1.1 The Theory

Decision theory concerns individuals’ preferences among bothconsequences (sometimes called “outcomes”) andgambles. The theory as originally developed focused ondecisions with monetary consequences (e.g., receiving $10 in a game ofchance), but subsequent developments broadened their focus to includedecisions with non-monetary consequences as well (e.g., eating a largeomelet, eating a smaller omelet, being stuck in the rain without anumbrella, lugging an umbrella around in the sunshine). Mostcontemporary authors define consequences to include any facts about adecision-maker’s situation that matter to her—so monetaryconsequences must technically describe a decision-maker’s totalfortune, and non-monetary consequences must technically describe theentire world that the decision-maker finds herself in—thoughthese full descriptions are often omitted when a decision will notalter the surrounding facts. Let the consequence set be \(\cX.\) Autility function \(\uf: \cX \rightarrow \cR\) assigns valuesto consequences, with the constraint that the individual prefers (orshould prefer), of two consequences, the one with the higher utilityvalue, and is indifferent between any two consequences with the sameutility value. Thus the utility function in some sense represents howthe individual values consequences.

Gambles come in one of two forms, depending on whether we are dealingwith the “objective probability” or “subjectiveprobability” version of the theory. In the objective probabilityversion, a gamble is alottery that assigns probabilities toconsequences. Consider, for example, a lottery that yields $100 withprobability 0.5, $300 with probability 0.3, and $200 with probability0.2. We can represent this lottery as {$100, 0.5; $300, 0.3; $200,0.2}. More generally, lotteries have the form \(L = \{x_1, p_1;\ldots;x_n, p_n\},\) where \(x_i \in \cX\) and \(p_i\) is the probabilitythat consequence \(x_i\) obtains. Lotteries needn’t berestricted to a finite set of consequences; they could instead becontinuous.

In the subjective probability version, a gamble is anact(sometimes called as “Savage act”; Savage 1954) thatassigns consequences to possible states of the world. Consider, forexample, the act of cracking an extra egg into one’s omelet,when the egg may be rotten: if the egg is fresh, the consequence willbe a large omelet, but if the egg is rotten, the consequence will be aruined omelet. We can represent this act a as {extraegg is fresh, large omelet;extra egg isrotten, ruined omelet}, and we can represent the act of notcracking the extra egg as {extra egg is rotten orfresh, small omelet}. More generally, acts have the form \(g =\{E_1, x_1;\ldots; E_n, x_n\},\) where \(x_i \in\cX,\) \(E_i \subseteq\cS\) is anevent (a subset of the state space), and \(x_i\)obtains when the true state of the world is in \(E_i.\) Again, actsneedn’t be restricted to a finite set of consequences. In thesubjective probability version, the individual has aprobabilityfunction \(p\) that assigns to each \(E_i\) a number between 0and 1 (inclusive), which represents her subjective probabilities, alsocalled “degrees of belief” or “credences”. Theprobability function isadditive in the sense that if \(E\)and \(F\) are mutually exclusive, then \(p(E v F) = p(E) + p(F).\)(For some of the discussion, it does not matter if we are talkingabout lotteries or acts, so I will use the variablesA,B,C… to range over lotteries or acts.)

The core principle of expected utility theory concerns how the utilityvalues of gambles are related to the utility values of consequences.In particular, the slogan of expected utility theory is thatrational agents maximize expected utility. The expectedutility (EU) of a lottery, relative to an individual’s utilityfunction \(\uf,\) is:

The expected utility of an act, relative to an individual’sutility function \(\uf\) and probability function \(p,\) is:

Continuous versions of these are defined using an integral instead ofa sum.

Expected utility theory holds that an individual’s preferencesorder gambles according to their expected utility, or ought to do so:\(A \succcurlyeq B\) iff \(\EU(A) \ge \EU(B).\) Generally, weakpreference \((\succcurlyeq)\) is taken to be basic and strictpreference \((\succ)\) and indifference \((\sim)\) defined in theusual way (\(A \succ B\) iff \(A \succcurlyeq B\) and not-\((B\succcurlyeq A)\); \(A \sim B\) iff \(A \succcurlyeq B\) and \(B\succcurlyeq A\)).

We can take utility and probability to be basic, and the norm to tellus what to prefer; alternatively, we can take preferences to be basic,and the utility and probability functions to be derived from them. Saythat a utility (and probability) functionrepresents apreference relation \(\succcurlyeq\)under EU- maximizationjust in case:

• \(u\) represents \(\succcurlyeq\) under EU-maximization (objectiveprobabilities): for all lotteries \(L_1\) and \(L_2,\)

  \[L_1 \succcurlyeq L_2 \text{ iff } \EU(L_1) \ge \EU(L_2),\]

  where EU is calculated relative to \(u.\)

• \(u\) and \(p\) represent \(\succcurlyeq\) under EU-maximization(subjective probabilities): for all acts \(f\) and \(g,\)

  \[f \succcurlyeq g \text{ iff } \EU(f) \ge \EU(g), \]

  where EU is calculated relative to \(u\) and \(p.\)

Arepresentation theorem exhibits a set of axioms (call it[axioms]) such that the following relationshipholds:

“Unique up to positive affine transformation” means thatany utility function \(u'\) that also represents the preferencerelation can be transformed into \(u\) by multiplying by a constantand adding a constant (to borrow an example from a different domain:temperature scales are unique up to positive affine transformation,because although temperature can be represented by Celsius orFahrenheit, Celsius can be transformed into Fahrenheit by multiplyingby 9/5 and adding 32).

The first and most historically important axiomatization of theobjective probabilities version of expected utility theory is that ofvon Neumann and Morgenstern (1944). The axioms are as follows(slightly different versions appear in the original):

The most historically important axiomatization of the subjectiveprobabilities version of expected utility theory is that of Savage(1954), though other prominent versions include Ramsey (1926), Jeffrey(1965), Armendt (1986, 1988), and Joyce (1999). These generallyinclude axioms that are analogues of the vonNeumann-Morgenstern axioms, plus some additional axioms that will not be our focus.

The first two of the axioms that will be our focus are (as above)completeness andtransitivity:

The third is some version of continuity, sometimes called anArchimedean Axiom.

The final axiom is aseparability axiom. Savage’sversion of this axiom is known as the sure-thing principle. Where\(f_E h\) is an act that agrees withf on eventE andagrees withh elsewhere:

In other words, if two acts agree on what happens on not-E,then one’s preference between them should be determined only bywhat happens on E. Other separability axioms include Jeffrey’sAveraging (1965) and Köbberling and Wakker’s TradeoffConsistency (2003).

Because representation theorems link, on the one hand, preferencesthat accord with the axioms and, on the other hand, a utility (andprobability) function whose expectation the individual maximizes, achallenge to one of the three components of expected utility theorymust also be a challenge to one (or more) of the axioms.

1.2 Components

1.2.1 Utility

Since representation theorems show that a utility (and probability)function can be derived from preferences—that having aparticular expectational utility function is mathematically equivalentto having a particular preference ordering—they open up a numberof possibilities for understanding the utility function. There are twoquestions here: what the utility function corresponds to (themetaphysical question), and how we determine an individual’sutility function (the epistemic question).

The first question is whether the utility function corresponds to areal-world quantity, such as strength of desire or perceiveddesirability or perceived goodness, or whether it is merely aconvenient way to represent preferences. The former view is known aspsychological realism (Buchak 2013) orweak realism(Zynda 2000), and is heldby Allais (1953) and Weirich (2008, 2020), for example. The latterview is known asformalism (Hansson 1988),operationalism (Bérmudez 2009), or therepresentational viewpoint (Wakker 1994), and is particularlyassociated with decision theorists from the mid-twentieth century(Luce & Raiffa 1957, Arrow 1951, Harsanyi 1977), and with contemporary economists.

The second question is what facts are relevant to determiningsomeone’s utility function. Everyone in the debate accepts thatpreferences provide evidence for the utility function, but there isdisagreement about whether there may be other sources of evidence aswell.Constructivists hold that an individual’s utilityfunction is defined by her preferences—utility is“constructed” from preference—so there are can be noother relevant facts (discussed in Dreier 1996, Buchak 2013); thisview is also calledstrong realism (Zynda 2000).Non-constructiverealists, by contrast, hold that there are other sources ofevidence about the utility function: for example, an individual mighthave introspective access to her utility function. This latter viewonly makes sense if one is a psychological realist, though one canpair constructivism with either psychological realism orformalism.

A key fact to note about the utility function is that it isreal-valued: each consequence can be assigned a real number.This means that no consequence is ofinfinite value, and allconsequences arecomparable. As we will see, each of thesetwo properties invites a challenge.

1.2.2 Probability

Given that a probability function can also be derived frompreferences, a similar question arises about the nature anddetermination of the probability function. One could hold that theprobability function represents some real-world quantity, such aspartial belief; or one could hold that the probability function ismerely a way of representing some feature of betting behavior. Thereis also disagreement about what facts are relevant to determiningsomeone’s probability function: some hold that it is determinedfrom betting behavior or from the deliverances of a representationtheorem while others take it to be primitive (see Eriksson &Hájek 2007).

Probability isreal-valued andpointwise(“sharp”, “precise”), meaning that there is aunique number representing an individual’s belief in or evidencefor an event. Again, this property will invite a challenge.

1.2.3 Expectation

We can see thenorm of expected utility in one of two ways:maximize expected utility, or have preferences that obey the axioms.Because of this, normative arguments for expected utility can argueeither for the functional form itself or for the normativity of theaxioms. Examples of the former include the argument that expectedutility maximizers do better in the long run, though these argumentsfell out of favor somewhat as the popularity of the realistinterpretations of utility waned. Examples of the latter include theidea that each axiom is itself an obvious constraint and the idea thatthe axioms follow from consequentialist (or means-ends rationality)principles. Of particular note is a proof that non-EU maximizers willeither be inconsistent or non-consequentialist over time (Hammond1988); how alternative theories have fared under dynamic choice hasbeen a significant focus of arguments about their rationality.

The idea that EU maximization is the correct norm can be challenged onseveral different grounds, as we will see. Those who advocate fornon-EU theories respond to the arguments listed above by eitherarguing that the new norm doesn’t actually fall prey to theargument (e.g., provide a representation theorem with supposedly moreintuitive axioms) or that it is nonetheless okay if it does.

2. Infinite and Unbounded Utility

2.1 The Challenge from Infinite and Unbounded Utility

The first challenge to EU maximization stems from two ways thatinfinite utility can arise in decision situations.

First, some particular outcome might haveinfinite utility orinfinite disutility. For example,Pascal’s Wager ismotivated by the idea that eternal life with God has infinite value,so one should “wager for God” as long as one assigns somenon-zero probability to God’s existence (Pascal 1670). If aparticular outcome has infinite (dis)value, then the Continuity Axiomor the Archimedean Axiom will not hold. (See discussions in Hacking1972 and Hájek 2003, and a related issue for utilitarianism inVallentyne 1993, Vallentyne & Kagan 1997, and Bostrom 2011.)

Second, all outcomes might have finite utility value, but this valuemight beunbounded, which, in combination with allowing thatthere can be infinitely manystates, gives rise to variousparadoxes. The most famous of these is theSt. PetersburgParadox, first introduced by Nicolas Bernoulli in a 1713 letter(published in J. BernoulliDW). Imagine a gamble whoseoutcome is determined by flipping a fair coin until it comes upheads. If it lands heads for the first time on then^th flip, the recipient gets $\(2^n\); thus thegamble has infinite expected monetary value for the person who takesit (it has a \(\lfrac{1}{2}\) probability of yielding $2, a\(\lfrac{1}{4}\) probability of yielding $4, a \(\lfrac{1}{8}\)probability of yielding $8, and so forth, and

While this version can be resolved by allowing utility to diminishmarginally in money—so that the gamble has finite expectedutility—if the payoffs are in utility rather than money, thenthe gamble will have infinite expected utility.

Related paradoxes and problems abound. One centers around a pair ofgames, thePasadena game and theAltadena game(Nover & Hájek 2004). The Pasadena game is also played byflipping a fair coin until it lands heads; here, the player receives\(\$({-1})^{n-1}(2^n/n)\) if the first heads occurs on then^th flip. Thus, its payoffs alternate betweenpositive and negative values of increasing size, so that its terms canbe rearranged to yield any sum whatsoever, and its expectation doesnot exist. The Altadena game is identical to the Pasadena game, exceptthat every payoff is raised by a dollar. Again, its terms can berearranged to yield any value, and again its expectation does notexist. However, it seems (contra EU maximization) that the Altadenagame should be preferred to the Pasadena game, since the formerstate-wise dominates the latter—it is better in everypossible state of the world (see also Colyvan 2006, Fine 2008,Hájek & Nover 2008). Similarly, it seems that thePetrograd game, which increases each payoff of the St.Petersburg game by $1, should be preferred to the St. Petersburg game,even though EU maximization will say they have the same (infinite)expectation (Colyvan 2008).

(See also Broome’s (1995) discussion of the two-envelopeparadox; Arntzenius, Elga, and Hawthorne’s (2004) discussion ofdiachronic puzzles involving infinite utility; and McGee’s(1999) argument that the utility function ought to be bounded, whichwill dissolve the above paradoxes.)

2.2 Proposals

Several proposals retain the basic EU norm, but reject the idea thatthe utility function ranges only over the real numbers. Some hold thatthe utility function can takehyper-real values (Skala 1975,Sobel 1996). Others hold that the utility function can takesurreal values (Hájek 2003, Chen & Rubio 2020).These proposals allow for versions of the Continuity/ArchimedeanAxiom. Another alternative is to use a vector-valued (i.e.,lexicographic) utility function, which rejects these axioms(see discussion in Hájek 2003).

A different kind of response is to subsume EU maximization under amore general norm that also applies when utility is infinite. Bartha(2007, 2016) definesrelative utility, which is a three-placerelation that compares two outcomes or lotteries relative to a third“base point” that is worse than both. The relative utilityof \(A\) to \(B\) with base-point \(Z\) (written \((A, B; Z)\)) willbe:

• If \(A,\) \(B\) and \(Z\) are finitely valued gambles:

  \[\frac{u(A) - u(Z)}{u(B) - u(Z)},\]

  as in standard EU maximization

• If \(A\) is infinitely valued and \(B\) and \(Z\) are not:\(\infty\)

Relative utility ranges over the extended real numbers \(\{\cR \cup\infty\}.\) “Finite” and “infinite” values canbe determined from preferences. Furthermore, relative utility isexpectational

and has a representation theorem consisting of the standard EU axiomsminus Continuity. (See Bartha 2007 for application to infinite-utilityconsequences and Bartha 2016 for application to unbounded-utilityconsequences.)

When considering only the paradoxes of unbounded utility (not those ofinfinite utility), there are other ways to subsume EU maximizationunder a more general norm. Colyvan (2008) definesrelativeexpected utility (unrelated to Bartha’s relative utility)of act \(f = \{E_1, x_1;\ldots; E_n, x_n\}\) over \(g = \{E_1,y_1;\ldots; E_n, y_n\}\) as:

In other words, one takes the difference in utility between \(f\) and\(g\) in each state, and weights this value by the probability of eachstate. Colyvan similarly defines the infinite state-space case as

The new norm is that \(f \succcurlyeq g\) iff \(\REU(f, g) \ge 0.\)This rule agrees with EU maximization in cases of finite state spaces,but also agrees with state-wise dominance; so it can require that theAltadena game is preferred to the Pasadena game and the Petrograd gameis preferred to the St. Petersburg game. (See also Colyvan & Hájek 2016.)

A more modest extension of standard EU maximization is suggested byEaswaran (2008). He points out that although the Pasadena and Altadenagames lack a “strong” expectation, they do have a“weak” expectation. (The difference corresponds to thedifference between the strong and weak law of large numbers.). Thus,we can hold that a decision-maker is required to value a gamble at itsweak expectation, which is equivalent to its strong expectation if thelatter exists. (See also Gwiazda 2014, Easwaran 2014b; relatedly, Fine (2008) shows that these two games and the St. Petersberg paradox can be assigned finite values that are consistent with EU theory.).

Lauwers and Vallentyne (2016) combine an extension of Easwaran’sproposal toinfinite weak expectations with an extension ofColyvan’s proposal tocardinal relative expectationthat can beinterval-valued. Meacham (2019) extendsColyvan’s proposal to cover cases in which the acts to becompared have utilities that are to be compared in different states,and cases in which probabilities are act-dependent; hisdifferenceminimizing theory re-orders each gamble from worst consequence tobest consequence, before taking their relative expectation. A keydifference between these two extensions is that difference minimizingtheory adheres tostochastic dominance and a relatedprinciple calledstochastic equivalence. (See also discussionin Seidenfeld, Schervish, & Kadane 2009; Hájek 2014;Colyvan & Hájek 2016.)

In a more radical departure from standard EU maximization, Easwaran(2014a) develops an axiomatic decision theorybased onstate-wise dominance, that starts with utility and probability andderives a normative preference relation. In cases that fit thestandard parameters of EU maximization, this theory can be made toagree with EU maximization; but it also allows us to compare some actswith infinite value, and some acts that don’t fit the standardparameters (e.g., incommensurable acts, acts with probabilities thatare comparative but non-numerical).

Finally, one couldtruncate the norm of EU maximization. Somehave argued that that for a gamble involving very small probabilities,we should discount those probabilities down to zero, regardless of theutilities involved. When combined with a way of aggregating theremaining possibilities, this strategy will yield a finite value forthe unbounded-utility paradoxes, and also allow people who attribute avery small probability to God’s existence to avoid wagering forGod. (This idea traces back to Nicolaus Bernoulli, Daniel Bernoulli,d’Alambert, Buffon, and Borel [see Monton 2019 for a historicalsurvey]; contemporary proponents of this view include Jordan 1994,Smith 2014, Monton 2019.)

3. Incommensurability

3.1 The Challenge from Incommensurability

Another challenge to expected utility maximization is to the idea thatpreferences are totally ordered—to the idea that consequencescan be ranked according to a single, consistent utility function. Ineconomics, this idea goes back at least to Aumann (1962); inphilosophy, it has been taken up more recently by ethicists.Economists tend to frame the challenge as a challenge to the idea thatthepreference relation is complete, and ethicists to theidea that thebetterness relation is complete. I use\(\succcurlyeq\) to represent whichever relation is at issue,recognizing that some proposals may be more compelling in one casethan the other.

The key claim is that there are some pairs of options for which it isfalse that one is preferred to (or better than) the other, but it isalso false that they are equi-preferred (or equally good). Proposedexamples include both the mundane and the serious: a Mexicanrestaurant and a Chinese restaurant; a career in the military and acareer as a priest; and, in an example due to Sartre (1946), whetherto stay with one’s ailing mother or join the Free French. Takingup the second of these examples: it is not the case that a career inthe military is preferred to (or better than) a career as a priest,norvice versa; but it is also not the case that they areequi-preferred (or equally good). Call the relation that holds betweenoptions in these pairsincommensurability.

Incommensurability is most directly a challenge to Completeness, sinceon the most natural interpretation of \(\succcurlyeq,\) the fact that\(A\) and \(B\) are incommensurable means that neither \(A\succcurlyeq B\) nor \(B \succcurlyeq A.\) But incommensurability caninstead be framed as a challenge to Transitivity, if we assume thatincommensurability is indifference, or define \(A \succcurlyeq B\) asthe negation of \(B \succ A\) (thus assuming Completeness bydefinition). To see this, notice that if two options \(A\) and \(B\)are incommensurable, then “sweetening” \(A\) to a slightlybetter \(A^+\) will still leave \(A^+\) and \(B\) incommensurable. Forexample, if \(A\) is a career in the military and \(A^+\) is thiscareer but with a slightly higher salary, the latter is stillincommensurable with a career as a priest. This pattern suffices toshow that the relation \(\sim\) is intransitive, since \(A \sim B\)and \(B \sim {A^+},\) but \({A^+} \succ A\) (de Sousa 1974).

There are four options for understanding incommensurability.Epistemicists hold that there is always some fact of thematter about which of the three relations \((\succ,\) \(\prec,\)\(\sim)\) holds, but that it is sometimes difficult or impossible todetermine which one—thus incommensurability is merely apparent.They can model the decision problem in the standard way, but as aproblem of uncertainty about values: one does not know whether one isin a state in which \(A \succcurlyeq B,\) but one assigns someprobability to that state, and maximizes expected utility taking thesekinds of uncertainties into account.Indeterminists hold thatit is indeterminate which relation holds, because these relations arevague; thus incommensurability is a type of vagueness (Griffin 1986,Broome 1997, Sugden 2009, Constantinescu 2012).Incomparabilists hold that in cases of incommensurability,\(A\) and \(B\) simply cannot be compared (de Sousa 1974, Raz 1988,Sinnott-Armstrong 1988). Finally, those who hold thatincommensurability isparity hold that there is a fourthrelation than can obtain between \(A\) and \(B\): \(A\) and \(B\) are“on a par” if \(A\) and \(B\) can be compared but it isnot the case that one of the three relations holds (Chang 2002a,2002b, 2012, 2015, 2016). (Taxonomy from Chang 2002b; see also Chang1997.)

3.2 Proposals: Definitions

Aumann (1962) shows that if we have a partial but not total preferenceordering, then we can represent it by a utility function (notunique-up-to-positive-affine-transformation) such that \(A \succ B\)implies \(\EU(A) \gt \EU(B),\) but notvice versa. Aumannshows that there will be at least one utility function that representsthe preference ordering according to (objective) EU maximization.Thus, we can represent a preference ordering as theset of allutility functions that “one-way” represent thedecision-maker’s preferences. Letting \(\EU_u(A)\) be theexpected utility of \(A\) given utility function \(u\):

If there is no incommensurability, then there will be a single(expectational) utility function in \(\cU,\) as in standard EU theory.But when neither \(A \succ B\) nor \(B \succ A\) nor \(A \sim B,\)there will be some \(u \in\cU\) such that \(\EU_u (A) \gt \EU_u (B),\)and some \(u'\in\cU\) such that \(\EU_{u'}(B) \gt \EU_{u'}(A)\); andvice versa.

Chang (2002a,b) proposes a similar strategy, but she takes value facts tobe basic, and defines the betterness relation—plus a new“parity” relation we will denote“\(\parallel\)”—from them, instead of the reverse.In addition, she defines these relations in terms of theevaluative differences between \(A\) and \(B,\) i.e., \((A -B)\) is the set of all licensed differences in value between \(A\) and\(B.\) If \((A - B) = \varnothing,\) then \(A\) and \(B\) areincomparable; however, if \((A - B) \ne \varnothing,\) therelevant relations are:

• \(A \succ B\) iff \((A - B)\) contains only positive numbers
• \(B \succ A\) iff \((A - B)\) contains only negative numbers
• \(A \sim B\) iff \((A - B)\) contains only 0
• \(A \parallel B\) otherwise

\((A - B)\) might be generated by a set of utility functions, each ofwhich represents a possible substantive completion of the underlyingvalue that utility represents (discussed in Chang 2002b);alternatively, it might be that there is parity “all the waydown” (discussed in Chang 2016, where she also replaces thedefinition in terms of explicit numerical differences with one interms of bias).

Rabinowicz (2008)provides a model that allows for both parity and grades ofincomparability. On his proposal, the betterness relation isrepresented by a class \(K\) of “permissible preferenceorderings”, each of which may be complete or incomplete. Hedefines:

(He defines \(\succcurlyeq\) as the union of several“atomic” possibilities for \(K.\)) Letting \(x\relT y\)hold iff \(x \succ y\) or \(y \succ x\) or \(x \sim y,\) he thendefines:

• \(x\) and \(y\) are fully comparable iff \((\forall R \in K)(xT_Ry)\)
• \(x\) and \(y\) are fully on a par iff they are fully comparableand \(x\parallel y\)
• \(x\) and \(y\) are incomparable iff \((\forall R\inK)(\text{not-}(xT _Ry))\)
• \(x\) and \(y\) are weakly incomparable iff \((\exists R\inK)(\text{not-}(xT_Ry)\)

Class \(K\) won’t necessarily give rise to a utilityfunction.

3.3 Proposals: Decision Rules

If the decision-maker’s preferences are represented by a set ofutility functions \(\cU,\) then a number of possible decision rulessuggest themselves. All the proposed rules focus on selection ofpermissible options from a set of alternatives \(\cS,\)rather than an aggregation function or a complete ranking of options(we can always recover the former from the latter, but notviceversa). To understand the first three of these rules, we canimaginatively think of each possible utility function as a“committee member”, and posit a rule for choice based onfacts about the opinions of the committee.

First, we might choose any option whichsome committee memberendorses; that is, we might choose any option which maximizes expectedutility relative tosome utility function:

Levi (1974) terms this ruleE-admissibility, and Hare (2010)calls itprospectism. (Seesection 4.3.3 for Levi’s full proposal, and for extensions of this rule tothe case in which the decision-maker does not have a singleprobability function.)

Aumann suggests that we can choose anymaximal option: anyoption that isn’t worse, according to all committee members,than some other particular option; that is, an option to which noother option is (strictly) preferred (assigned a higher utility by allutility functions):

This is a more permissive rule than E-admissibility: everyE-admissible option will be maximal, but notvice versa. Tosee the difference between the two rules, notice that if thedecision-maker has two possible rankings, \(A \gt B \gt C\) and \(C\gt B \gt A,\) then all three options will be maximal but only \(A\)and \(C\) will be E-admissible (no particular option is definitivelypreferred to \(B,\) so \(B\) is maximal; but it is definitive thatsomething is preferred to \(B,\) so \(B\) is not E-admissible).

A final possibility is that we can choose any option that is notinterval dominated by another act (Schick 1979, Gert 2004),where an interval dominated option has a lower “best”value than some other option’s “worst” value:

This is a more permissive rule than both E-admissibility andmaximality.

A different type of rule first looks at the options’ possibleutility values in each state, before aggregating over states; this iswhat Hare’s (2010)deferentialism requires. To find outif an option O is permissible under deferentialism, we consider how itfares if we, in each state, make the assumptions most favorable to it.First, “regiment” the utility functions in \(\cU\) so thatthere’s some pair of consequences \(\{x, y\}\) such that\((\forall u \in\cU)(u(x) = 1 \amp u(y) = 0)\); this allows only onerepresentative utility function for each possible completion of thedecision-maker’s preference. Next, take all the possible“state-segments”—the possible utility assignments ineach state—and cross them together in every possible arrangementto get an “expanded” set of utility functions (forexample, this will contain every possible utility function in \(E\)coupled with every possible utility function not-\(E\)). Then \(A\) ispermissible iff \(A\) maximizes expected utility relative tosome utility function in this expanded set.

4. Imprecise Probabilities or Ambiguity

4.1 The Challenge from Imprecise Probabilities or Ambiguity

A third challenge to expected utility maximization holds thatsubjective probabilities need not be “sharp” or“precise”, i.e., need not be a single, point-wisefunction. (In economics, this phenomenon is typically calledambiguity.) There are three historically significantmotivations for imprecise probabilities.

The first is that decision makers treat subjective (or unknown)probabilities differently from objective probabilities in theirdecision-making behavior. The classic example of this is the EllsbergParadox (Ellsberg 1961, 2001). Imagine you face an urn filled with 90balls that are red, black, and yellow, from which a single ball willbe drawn. You know that 30 of the balls are red, but you know nothingabout the proportion of black and yellow balls. Do you prefer \(f_1\)or \(f_2\); and do you prefer \(f_3\) or \(f_4\)?

• \(f_1\): $100 if the ball is red; $0 if the ball is black oryellow.
• \(f_2\): $100 if the ball is black; $0 if the ball is red oryellow.
• \(f_3\): $100 if the ball is red or yellow; $0 if the ball isblack.
• \(f_4\): $100 if the ball is black or yellow; $0 if the ball isred.

Most people appear to (strictly) prefer \(f_1\) to \(f_2\) and(strictly) prefer \(f_4\) to \(f_3.\) They would rather bet on theknown or objective probability than the unknown or subjectiveone—in the first pair,red has anobjective probability of \(\lfrac{1}{3}\), whereasblackhas a possible objective probability rangingfrom 0 to \(\lfrac{2}{3}\); in the second pair,blackor yellow has an objective probability of \(\lfrac{2}{3}\)whereasred or yellow has a possible objectiveprobability ranging from \(\lfrac{1}{3}\) to 1. These preferencesviolate the Sure-thing Principle. (To see this, notice that the onlydifference between the two pairs of acts is that the first pair yields$0 onyellow and the second pair yields $100onyellow.)

The second motivation for imprecise probability is that even if allthe relevant probabilities are subjective, a decision-maker’sbetting behavior might depend on how reliable or well-supported byevidence those probabilities are. Consider a decision-maker who maybet on three different tennis matches: in the first, she knows a lotabout the players and knows they are very evenly matched; in thesecond, she knows nothing whatsoever about either player; and in thethird, she knows that one of the two players is much better than theother, but she does not know which one. In each of the matches, thedecision-maker should presumably assign equal probability to eachplayer winning, since her information in favor of each is symmetrical;nonetheless, it seems rational to bet only on the first match and noton the other two (Gärdenfors & Sahlin 1982; see also Ellsberg1961).

A final motivation for imprecise probability is that evidencedoesn’t always determine precise probabilities (Levi 1974, 1983;Walley 1991; Joyce 2005;Sturgeon 2008; White2009; Elga 2010). Assume a stranger approaches you and pulls threeitems out of a bag: a regular-sized tube of toothpaste, a livejellyfish, and a travel-sized tube of toothpaste; you are asked toassign probability to the proposition that the next item he pulls outwill be another tube of toothpaste—but it seems that you lackenough evidence to do so (Elga 2010).

4.2 Proposals: Probability Representations

To accommodate imprecise probabilities in decision-making, we needboth an alternative way to represent probabilities and an alternativedecision rule that operates on the newly-represented probabilities.There are two primary ways to represent imprecise probabilities.

The first is to assign aninterval, instead of a singlenumber, to each proposition. For example, in the Ellsberg case:

The second is to represent the individual’s beliefs as aset of probability functions. For example, in the Ellsbergcase:

This means,for example, that the probability distribution \(p(\rR, \rB, \rY) =\langle \lfrac{1}{3}, 0, \lfrac{2}{3}\rangle\) and the probabilitydistribution \(\langle \lfrac{1}{3}, \lfrac{1}{3},\lfrac{1}{3}\rangle\) are both compatible with the available evidenceor possible “completion” of the individual’sbeliefs.

Each set-probability representation gives rise to an interval representation (assuming the set ofprobability functions is convex); but the set-probabilityrepresentation provides more structure to the relationships betweenpropositions. (A different proposal retains a precise probabilityfunction but refines the objects over which utility and probabilityrange (Bradley 2015; Stefánsson & Bradley 2015, 2019); seediscussion insection 5.2.)

4.3 Proposals: Decision Rules

We will examine rules for decision-making with imprecise probabilitiesin terms of how they evaluate the Ellsberg choices; for ease ofexposition we will assume \(u(\$0) = 0\) and \(u(\$100) = 1.\) Allproposals here are equivalent to EU maximization when there is asingle probability distribution in the set, so all will assign utility\(\lfrac{1}{3}\) to \(f_1\) and \(\lfrac{2}{3}\) to \(f_4\) in theEllsberg gamble; they differ in how they value the other acts.

4.3.1 Aggregative Decision Rules using Sets of Probabilities

The first type of decision rule associates to each act a single value,and yields a complete ranking of acts; call theseaggregativerules. The rules in this section use sets of probabilities.

Before we discuss these rules, it will be helpful to keep in mindthree aggregative rules that operate under complete ignorance, i.e.,when we have no information whatsoever about the state of the world.The first is maximin, which says to pick the option with the highestminimum utility. The second is maximax, which says to pick the optionwith the highest maximum utility. The third, known as the Hurwiczcriterion, says to take a weighted average, for each option, of theminimum and maximum utility, where the weight \(\alpha \in [0, 1]\)represents a decision-maker’s level of optimism/pessimism(Hurwicz 1951a):

Using the set-probability representation, we can associate to eachprobability distribution an expected utility value, to yield a set ofexpected utility values. Let \(\EU_p(f)\) be the expected utility of\(f\) given probability distribution \(p.\)

One proposal is that the value of an act is its minimum expectedutility value; thus, a decision-maker shouldmaximize her minimumexpected utility (Wald 1950; Hurwicz 1951b; Good 1952; Gilboa& Schmeidler 1989):

This rule is also sometimes known as MMEU. For the Ellsberg choices,the minimum expected utilities are, for \(f_1\), \(f_2\), \(f_3\), and \(f_4\), respectively: \(\lfrac{1}{3},\) \(0,\) \(\lfrac{1}{3},\) and \(\lfrac{2}{3},\). These values rationalize the common preference for \(f_1 \gt f_2\) and \(f_4 \gtf_3.\) Conversely, an individual whomaximizes her maximumexpected utility—who uses Γ-maximax—would havethe reverse preferences.

Γ-maximin appears too pessimistic. We might instead use anEU-analogue of Hurwicz criterion: take aweighted average of theminimum expected utility and the maximum expected utility, withweight \(\alpha \in[0, 1]\) corresponding to a decision-maker’slevel of pessimism (Hurwicz 1951b; Shackle 1952; Luce & Raiffa1957; Ellsberg 2001; Ghirardato et al. 2003):

In the Ellsberg choice, this model will assign\(\alpha(\lfrac{2}{3})\) to \(f_2\) and \((1 - \alpha)(\lfrac{1}{3}) +\alpha(1)\) to \(f_3,\) making these acts disprefered to \(f_1\) and\(f_4,\) respectively, if \(\alpha \lt \lfrac{1}{2}\); preferred to\(f_1\) and \(f_4\) if \(\alpha \gt \lfrac{1}{2}\); and indifferent to\(f_1\) and \(f_4\) if \(\alpha = \lfrac{1}{2}.\)

Instead, we can assume the decision-maker considers two quantitieswhen evaluating an act: the EU of the act, according to her“best estimate” at the probabilities (\(\text{est}_p),\)and the minimum EU of the act as the probability ranges over \(\cQ\);she also assigns a degree of confidence \(\varrho \in[0, 1]\) to herestimated probability distribution. The value of an act will then be aweighted average of her best estimate EU and the minimum EU, with herbest estimate weighed by her degree of confidence (Hodges &Lehmann 1952; Ellsberg 1961):

In the Ellsberg pairs, assuming the “best estimate” isthatyellow andblackeach have probability \(\lfrac{1}{3}\), this will assign\(\varrho(\lfrac{1}{3})\) to \(f_2\) and \(\varrho(\lfrac{2}{3}) + (1- \varrho)(\lfrac{1}{3})\) to \(f_3,\) making these acts dispreferedto \(f_1\) and \(f_4,\) respectively, as long as \(\varrho \lt1.\)

We can also combine these two proposals (Ellsberg 2001):

This model will rationalize the common preferences for many choices of\(\varrho\) and \(\alpha\) (setting \(\varrho = 0\) or \(\alpha = 0\)yields previously-mentioned models).

We might add additional structure to the representation: to eachprobability function, the decision-maker assigns a degree of“reliability”, which tracks how much relevant informationthe decision-maker has about the states of nature (Good 1952;Gärdenfors & Sahlin 1982). A decision-maker selects a desiredthreshold level of epistemic reliability. She then considers allprobability functions above this threshold, and maximizes the minimumexpected utility (Γ-maximin) with respect to these probabilityfunctions. (In principle, a different decision rule could be used inthis step.) For the decision-maker deciding whether to bet on tennismatches, the above-threshold probability functions for the first matchmay include only \(p(P1 \text{WINS}) \approx 0.5,\) but for the secondand third match may also include \(p(P1 \text{WINS}) \approx 0\); thusbetting on P1 in the first match will have a higher value than bettingon P1 in the other matches.

4.3.2 Aggregative Decision Rules: Choquet Expected Utility

A different kind of rule isChoquet expected utility, alsoknown ascumulative utility (Schmeidler 1989; Gilboa 1987).This rule starts with a function \(v\) which, like a probabilityfunction, obeys \(v(E)\in[0, 1],\) \(v(0) = 0,\) \(v(1) = 1,\) and \(A\subseteq B\) implies \(v(A) \le v(A).\) Unlike a probabilityfunction, however, \(v\) is non-additive; and \(v\) is notstraightforwardly used to calculate an expectation. (Many economistsrefer to this function as a “non-additive subjective probabilityfunction”.). Choquet expected utility is a member of therank-dependent family (Quiggin 1982, Yaari 1987, Kahneman& Tversky 1979, Tversky & Kahneman 1992, Wakker 2010).Functions in this family let the weight of an event in an act’soverall value depend on both the probability-like element and theevent’s position in the ordering of an act, e.g., whether it isthe worst or best event for that act. Formally, let \(g' = \{E_1,x_1;\ldots; E_n, x_n\}\) be a re-ordering of act \(g\) from worstevent to best event, so that \(u(x_1) \le \ldots \le u(x_n).\) TheChoquet expected utility of \(g'\) (and therefore of \(g\))is:

If \(v\) is additive, then \(v\) is an (additive) probability functionand CEU reduces to EU. If \(v\) is convex \((v(E) + v(F) \le v(EF) +v(E v F)),\) then the individual isuncertainty-averse.

In the Ellsberg example, we are given \(p(\RED) = \lfrac{1}{3}\) and\(p(\BLACK \lor \YELLOW) = \lfrac{2}{3},\) and so we can assume\(v(\RED) = \lfrac{1}{3}\) and \(v(\BLACK \lor \YELLOW) =\lfrac{2}{3}.\) A person who is “ambiguity averse” willassign \(v(\BLACK) + v(\YELLOW) \le v(\BLACK \lor \YELLOW)\); let usassume \(v(\BLACK) = v(\YELLOW) = \lfrac{1}{9}.\) Similarly, she willassign \(v(\RED \lor \YELLOW) + v(\BLACK) \le 1\); let us assume\(v(\RED \lor \YELLOW) = \lfrac{4}{9}.\)

Then the values of the acts will be:

This assignment recovers the Ellsberg preferences.

Axiomatizations of CEU use a restricted version of the separabilitycondition (the “Comonotonic” Sure-thing Principle or“Comonotonic” Tradeoff Consistency): namely, the conditiononly holds when all of the acts in its domain arecomonotonic, i.e., when the worst-to-best ranking of theevents coincides for all the acts (Gilboa 1987, Schmeidler 1989,Wakker 1989, Chew & Wakker 1996, Köbberling & Wakker2003; see also Wakker 2010, who also notes the relationship betweenCEU and \(\alpha\)-maximin.)

4.3.3 Decision Rules that Select from a Menu

Another different type of proposal focuses on selection ofpermissible options from a set of alternatives \(\cS,\)rather than a complete ranking of options. As insection 3.3, we can imaginatively think of each possible probability distributionin a set \(\cQ\) as a “committee member”, and posit a rulefor choice based on facts about the opinions of the committee. (Thefirst three rules are versions of the rules for sets of utilityfunctions in 3.3, and can be combined to cover sets ofprobability/utility pairs.)

The first possibility is that a decision-maker is permitted to pick anact just in casesome committee member is permitted to pickit over all the alternatives: just in case it maximizes expectedutility relative tosome probability function in the set.This is known asE-admissibility (Levi 1974, 1983, 1986;Seidenfeld, Schervish, & Kadane 2010):

Levi in fact tells a more complicated story about what adecision-maker is permitted to choose, in terms of a procedure thatrules out successively more and more options. First, the procedureselects from all the options just the ones that are E-admissible.Next, the procedure selects from the E-admissible options just theones that areP-admissible: options that “dobest” at preserving the E-admissible options (the idea beingthat a rational agent should keep her options open). Finally, theprocedure selects from the P-admissible options just the ones that areS-admissible: options that maximize the minimum utility.(Note that this last step involves maximin, not Γ-maximin.)

A more permissive rule than E-admissibility permits a choice as longthere isno particular alternative that the committee membersunanimously (strictly) prefer. As in the case ofutility-sets, this rule is known asmaximality (Walley1991):

(Seesection 3.3 for an example of the difference between E-admissibility andmaximality.)

More permissive still is the rule that a choice is permissible as longas it is notinterval dominated (Schick 1979, Kyburg 1983):its maximum value isn’t lower than the minimum value of someother act.

For a proof showing that Γ-maximax implies E-admissibility;Γ-maximin implies maximality; E-admissibility impliesmaximality; and maximality implies interval dominance, see Troffaes(2007).

A final approach is to interpret ambiguity as indeterminacy: onecommittee member has the “true” probability function, butit is indeterminate which one. If all probability functions agree thatan option is permissible to choose, then it is determinatelypermissible to choose; if all agree that it is impermissible tochoose, it is determinately impermissible to choose; and if some holdthat it is permissible and others hold that it is impermissible, it isindeterminate whether it is permissible (Rinard 2015).

These rules in this section allow but do not obviouslyexplain the Ellsberg choices unless supplemented by anadditional rule (e.g., Levi’s more complicated story or one ofthe rules fromsection 4.3.1), since any choice between \(f_1\) and \(f_2\) and between \(f_3\) and\(f_4\) appears to be E-admissible, maximal, andnon-interval-dominated.

4.4 Normative Questions

For those who favor non-sharp probabilities, two sets of normativequestions arise. The first set is epistemic: whether it is epistemically rational notto have sharp probabilities (White 2009; Elga 2010; Joyce 2011;Hájek & Smithson 2012; Seidenfeld et al. 2012; Mayo-Wilson& Wheeler 2016; Schoenfield 2017; Vallinder 2018; Builes et al 2022; Konekms. – see Other Internet Resources).

The second set of questions question is practical. Some hold that ambiguity aversion is not well-motivated by practical reasons and so we have no reason to account for it (Schoenfield 2020). Others hold that some particular decision rule associated with non-sharp probabilities leads to bad consequences, e.g., in virtue of running afoul of the principles mentioned insection 1.2.3. Of particular interest is howthese decision rules can be extended to sequential choice (Seidenfeld1988a,b; Seidenfeld et al. 1990;Elga 2010; Bradley and Steele 2014; Chandler 2014; Moss 2015; Sud 2014; Rinard 2015).

5. Risk Aversion

5.1 The Challenge from Risk Aversion

A final challenge to expected utility maximization is to the normitself—to the idea that a gamble should be valued at itsexpectation. In particular, some claim that it is rationallypermissible for individuals to be risk-averse (or risk-seeking) in asense that conflicts with EU maximization.

Say that an individual isrisk-averse in money (or anynumerical good) if she prefers the consequences of a gamble to be less“spread out”; this concept is made precise by Rothschildand Stiglitz’s idea ofdispreferring mean-preservingspreads (1972). As a special case of this, a person who isrisk-averse in money will prefer \(\$x\) to any gamble whose meanmonetary value is \(\$x.\) If an EU maximizer is risk-averse in money,then her utility function will be concave (it diminishes marginally,i.e., each additional dollar adds less utility than the previousdollar); if an EU maximizer is risk-seeking in money, then her utilityfunction will be convex (Rothschild & Stiglitz 1972). Therefore,EU theory equates risk-aversion with having a diminishing marginalutility function.

However, there are intuitively at least two different reasons thatsomeone might have for being risk-averse. Consider a person who lovescoffee but cannot tolerate more than one cup. Consider another personwhose tolerance is very high, such that the first several cups areeach as pleasurable as the last, but who has a particular attitudetowards risk: it would take a very valuable upside in order for her togive up a guaranteed minimum number of cups. Both will prefer 1 cup ofcoffee to a coin-flip between 0 and 2, but intuitively they value cupsof coffee very differently, and have very different reasons for theirpreference. This example generalizes: we might consider a person whois easily saturated with respect to money (once she has a bit ofmoney, each additional bit matters less and less to her); and anotherperson who is a miser—he likes each dollar just as much as thelast—but nonetheless has the same attitude towards gambling asour coffee drinker. Both will disprefer mean-preserving spreads, butintuitively have different attitudes towards money and differentreasons for this preference. Call the attitude of the second person ineach pairglobal sensitivity (Allais 1953, Watkins 1977,Yaari 1987, Hansson 1988, Buchak 2013).

This kind of example gives rise to several problems. First, if EUmaximization is supposed toexplain why someone made aparticular choice, it ought to be able to distinguish these tworeasons for preference; but if global sensitivity can be captured atall, it will have to be captured by a diminishing marginal utilityfunction, identical to that of the first person in each pair. Second,if one adopts a view of the utility function according to which thedecision-maker has introspective access to it, a decision-maker mightreport that she has preferences like the tolerant coffee drinker orthe miser—her utility function is linear—but nonethelessif she maximizes EU her utility function will have to be concave; soEU maximization will get her utility function wrong. Finally, even ifone holds that a decision-maker does not have introspective access toher utility function, if a decision-maker displays global sensitivity,then she will have some preferences that cannot be captured by anexpectational utility function (Allais 1953, Hansson 1988, Buchak2013).

A related worry displays a troubling implication of EUmaximization’s equating risk-aversion with diminishing marginalutility. Rabin’s (2000)Calibration Theorem shows thatif an EU-maximizer is mildly risk-averse in modest-stakes gambles, shewill have to be absurdly risk-averse in high-stakes gambles. Forexample, if an individual would reject the gamble {−$100, 0.5;$110, 0.5} at any wealth level, then she must also reject the gamble{−$1000, 0.5; $n, 0.5} for any \(n\) whatsoever.

Finally, theAllais Paradox identifies a set of preferencesthat are intuitive but cannot be captured by any expectational utilityfunction (Allais 1953). Consider the choice between the following twolotteries:

• \(L_1 : \{\$5,000,000, 0.1; $0, 0.9\}\)
• \(L_2 : \{\$1,000,000, 0.11; \$0, 0.89\}\)

Separately, consider the choice between the following twolotteries:

• \(L_3 : \{\$1,000,000, 0.89; \$5,000,000, 0.1; \$0, 0.01\}\)
• \(L_4 : \{\$1,000,000, 1\}\)

Most people (strictly) prefer \(L_1\) to \(L_2,\) and (strictly)prefer \(L_4\) to \(L_3,\) but there are no values \(u(\$0),\)\(u(\$1\rM),\) and \(u(\$5\rM)\) such that \(\EU(L_1) \gt \EU(L_2)\)and \(\EU(L_4) \gt \EU(L_3).\) The Allais preferences violate theIndependence Axiom; when the lotteries are suitably reframed as acts(e.g., defined over events such as the drawing of a 100-ticketlottery), they violate the Sure-thing Principle or relatedseparability principles.

5.2 Proposals

There have been a number of descriptive attempts to explain globalsensitivity, by those who are either uninterested in normativequestions or assume that EU is the correct normative theory. The mostwell-known of these are prospect theory (Kahneman & Tversky 1979;Tversky & Kahneman 1992) and generalized utility theory (Machina1982, 1983, 1987); others are mentioned in the discussion below. SeeStarmer (2000) for an overview.

Some normative proposals seek to accommodate global sensitivitywithin expected utility theory, by making the inputs of theutility function more fine-grained. Proponents of this“refinement strategy” hold that decision-makers prefer\(L_4\) to \(L_3\) because the $0 outcome in \(L_3\) would induceregret that one forewent a sure $1M (alternatively, because \(L_4\)includes psychological certainty) and therefore that theconsequence-descriptions should include these facts. Thus, the correctdescription of \(L_3\) is:

Once these gambles are correctly described, there is no directconflict with EU maximization (Raiffa 1986, Weirich 1986, Schick 1991,Broome 1991, Bermúdez 2009, Pettigrew 2015, Buchak 2015). Theproblem of when two outcomes that appear the same should bedistinguished is taken up by Broome (1991), Pettit (1991), and Dreier(1996).

The thought that the value of a consequence depends on what might havebeen is systematized by Bradley and Stefánsson (2017). Theirproposal uses a version of expected utility theory developed byJeffrey (1965) and axiomatized by Bolker (1966). Jeffrey replaces theutility function by a more general “desirability” functionDes, which applies not just to consequences but also to prospects;indeed, it doesn’t distinguish between “ultimate”consequences and prospects, since its inputs are propositions. Bradleyand Stefánsson propose to widen the domain of Des to includecounterfactual propositions, thus allowing that preferences forpropositions can depend on counterfacts. For example, a decision-makercan prefer “I choose the risky options and get nothing, and Iwouldn”t have been guaranteed anything if I had chosendifferently’ to “I choose the risky option and getnothing, and I would have been guaranteed something if I had chosendifferently”, which will rationalize the Allais preferences.(Incidentally, their proposal can also rationalize preferences thatseemingly violate EU because of fairness considerations, as in anexample from Diamond (1967).)

In a different series of articles (Stefánsson & Bradley2015, 2019), these authors again employ Jeffrey’s framework, butthis time widen the domain of Des to includechancepropositions (in addition to factual prospects), propositionslike “the chance that I get $100 is 0.5”. They hold that arational decision-maker can have a preference between various chancesof \(X,\) even on the supposition that \(X\) obtains (she need notobey “Chance Neutrality”). They capture the idea ofdisliking risk as such by holding that even though a rational agentmust maximize expected desirability, she need not have a \(\Des\)function of \(X\) that is expectational with respect to the Desfunction of chance propositions about \(X\) (she need not obey“Linearity”). For example, \(\Des(\text{“I get\$100”})\) need not be equal to \(2(\Des(\text{“the chancethat I get \$100 is 0.5”})).\) (This does not conflict withmaximizing expected desirability, because it concerns only therelationship between particular inputs to the \(\Des\) function, anddoes not concern the decision-maker’s subjectiveprobabilities.). This proposal can also rationalize the Ellsbergpreferences (section 4.1), because it allows the decision-maker to assign differentprobabilities to the various chance propositions (see also Bradley2015).

Other proposals hold that we should reject the aggregation norm ofexpected utility. The earliest of these came from Allais himself, whoheld that decision-makers care not just about the mean utility of agamble, but also about the dispersion of values. He proposes thatindividuals maximize expected utility plus a measure of the riskinessof a gamble, which consists in a multiple of the standard deviation ofthe gamble and a multiple of its skewness. Formally, if \(s\) standsfor the standard deviation of \(L\) and \(m\) stands for the skewnessof \(L,\) then the utility value of \(L\) will be (Allais 1953, Hagen1979):

where \(\varepsilon\) is an error term. He thus proposes thatriskiness is an independently valuable property of a gamble, to becombined with (and traded off against) its expected utility. Thisproposal essentially treats the riskiness of a gamble as a propertythat is (dis)valuable in itself (see also Nozick 1993 on symbolicutility).

A final approach treats global sensitivity as a feature of thedecision-maker’s way of aggregating utility values. It might bethat a decision-maker’s utility and probability function are notyet enough to tell us what he should prefer; he must also decide howmuch weight to give to what happens in worse states versus whathappens in better states. Inrisk-weighted expected utility(Buchak 2013), a generalization of Quiggin’s (1982)anticipated utility and a member of the rank-dependent family(seesection 4.3.2), this decision is represented by hisrisk function.

Formally, let \(g' = \{E_1, x_1;\ldots; E_n, x_n\}\) be a re-orderingof act \(g\) from worst event to best event, so that \(u(x_1) \le\ldots \le u(x_n).\) Then therisk-weighted expected utilityof \(g\) is:

with \(0 \le r(p) \le 1,\) \(r(0) = 0\) and \(r(1) = 1,\) and \(r(p)\)non-decreasing.

The risk function measures the weight of the top \(p\)-portion ofconsequences in the evaluation of an act—how much thedecision-maker cares about benefits that obtain only in the top\(p\)-portion of states. (One could also think of the risk function asdescribing the solution to a distributive justice problem amongone’s future possible selves—it says how much weight adecision-maker gives to the interests of the top \(p\)-portion of hisfuture possible selves.) Arisk-avoidant person is someonewith a convex risk function: as benefits obtain in a smaller andsmaller portion of states, he gives proportionally less and lessweight to them. Arisk-inclined person is someone with aconcave risk function. And aglobally neutral person issomeone with a linear risk function, i.e., an EU maximizer.

Diminishing marginal value and global sensitivity are captured,respectively, by the utility function and the risk function.Furthermore, the Allais preferences can be accommodated by a convexrisk function (Segal 1987, Prelec 1998, Buchak 2013; but see Thoma& Weisberg 2017). Thus, REU maximization holds thatdecision-makers have the Allais preferences because they care moreabout what happens in worse scenarios than better scenarios, or aremore concerned with the minimum value than potential gains above theminimum.

The representation theorem for REU combines conditions from twoexisting theorems (Machina & Schmeidler 1992, Köbberling andWakker 2003), replacing the separability condition with two weakerconditions. One of these conditions fixes a unique probabilityfunction of events (Machina & Schmeidler’s “StrongComparative Probability”, 1992) and the other fixes a uniquerisk function of probabilities; the latter is a restricted version ofthe separability condition (Köbberling & Wakker’s[2003] “Comonotonic” Tradeoff Consistency; seesection 4.3.2). Since the representation theorem derives a unique probabilityfunction, a unique risk function, and a unique (up to positive affinetransformation) utility function, it separates the contribution ofdiminishing marginal value and global sensitivity to a givenpreference ordering. One can disprefer mean-preserving spreads as aresult of either type of risk-aversion, or a combination of both.

5.3 Normative Issues

Proposals that seek to retain EU but refine the outcome space face twoparticular worries. One of these is that the constraints of decisiontheory end up trivial (Dreier 1996); the other is that they saddle thedecision-maker with preferences over impossible objects (Broome1991).

For theories that reject the Sure-thing Principle or the IndependenceAxiom, several potential worries arise, including the worry that theseaxioms are intuitively correct (Harsanyi 1977, Savage 1954, Samuelson1952; see discussion in McClennen 1983, 1990); that decision-makerswill evaluate consequences inconsistently (Samuelson 1952, Broome1991); and that decision-makers will reject cost-free information(Good 1967, Wakker 1988, Buchak 2013, Ahmed & Salow 2019,Campbell-Moore & Salow 2020). The most widely discussed worry isthat these theories will leave decision-makers open to diachronicinconsistency (Raiffa 1968; Machina 1989; Hammond 1988; McClennen1988, 1990; Seidenfeld 1988a,b; Maher 1993; Rabinowicz 1995, 1997;Buchak 2013, 2015, 2017; Briggs 2015; Joyce 2017; Thoma 2019).

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• Konek, Jason, manuscript, “Epistemic Conservativity and Imprecise Credence”

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