## 1. Expected Utility Theory


### 1.1 The Theory



Decision theory concerns individuals' preferences among both
*consequences* (sometimes called "outcomes") and
*gambles*. The theory as originally developed focused on
decisions with monetary consequences (e.g., receiving $10 in a game of
chance), but subsequent developments broadened their focus to include
decisions with non-monetary consequences as well (e.g., eating a large
omelet, eating a smaller omelet, being stuck in the rain without an
umbrella, lugging an umbrella around in the sunshine). Most
contemporary authors define consequences to include any facts about a
decision-maker's situation that matter to her--so monetary
consequences must technically describe a decision-maker's total
fortune, and non-monetary consequences must technically describe the
entire world that the decision-maker finds herself in--though
these full descriptions are often omitted when a decision will not
alter the surrounding facts. Let the consequence set be \(\cX.\) A
*utility function* \(\uf: \cX \rightarrow \cR\) assigns values
to consequences, with the constraint that the individual prefers (or
should prefer), of two consequences, the one with the higher utility
value, and is indifferent between any two consequences with the same
utility value. Thus the utility function in some sense represents how
the individual values consequences.



Gambles come in one of two forms, depending on whether we are dealing
with the "objective probability" or "subjective
probability" version of the theory. In the objective probability
version, a gamble is a *lottery* that assigns probabilities to
consequences. Consider, for example, a lottery that yields $100 with
probability 0.5, $300 with probability 0.3, and $200 with probability
0.2. 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 probability
that consequence \(x\_i\) obtains. Lotteries needn't be
restricted to a finite set of consequences; they could instead be
continuous.



In the subjective probability version, a gamble is an *act*
(sometimes called as "Savage act"; Savage 1954) that
assigns consequences to possible states of the world. Consider, for
example, the act of cracking an extra egg into one's omelet,
when the egg may be rotten: if the egg is fresh, the consequence will
be a large omelet, but if the egg is rotten, the consequence will be a
ruined omelet. We can represent this act a as {extra
egg is fresh, large omelet; extra egg is
rotten, ruined omelet}, and we can represent the act of not
cracking the extra egg as {extra egg is rotten or
fresh, 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 an *event* (a subset of the state space), and \(x\_i\)
obtains when the true state of the world is in \(E\_i.\) Again, acts
needn't be restricted to a finite set of consequences. In the
subjective probability version, the individual has a *probability
function* \(p\) that assigns to each \(E\_i\) a number between 0
and 1 (inclusive), which represents her subjective probabilities, also
called "degrees of belief" or "credences". The
probability function is *additive* 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 talking
about lotteries or acts, so I will use the variables *A*,
*B*, *C*... to range over lotteries or acts.)



The core principle of expected utility theory concerns how the utility
values of gambles are related to the utility values of consequences.
In particular, the slogan of expected utility theory is that
*rational agents maximize expected utility*. The expected
utility (EU) of a lottery, relative to an individual's utility
function \(\uf,\) is:

 

\[
\EU(L) = \sum\_{i = 1}^{n} p\_{i} \uf(x\_{i})
\]


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



\[
\EU(g) = \sum\_{i = 1}^{n} p(E\_{i}) \uf(x)\_{i}
\]


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



Expected utility theory holds that an individual's preferences
order gambles according to their expected utility, or ought to do so:
\(A \succcurlyeq B\) iff \(\EU(A) \ge \EU(B).\) Generally, weak
preference \((\succcurlyeq)\) is taken to be basic and strict
preference \((\succ)\) and indifference \((\sim)\) defined in the
usual 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 tell
us what to prefer; alternatively, we can take preferences to be basic,
and the utility and probability functions to be derived from them. Say
that a utility (and probability) function *represents* a
preference relation \(\succcurlyeq\) *under EU- maximization*
just in case:


* \(u\) represents \(\succcurlyeq\) under EU-maximization (objective
probabilities): 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 \(p.\)



A *representation theorem* exhibits a set of axioms (call it
[axioms]) such that the following relationship
holds:



* Representation Theorem (objective probabilities): If a preference
relation over lotteries satisfies [axioms],
then there is a utility function \(u,\) unique up to positive affine
transformation, that represents the preference relation under
EU-maximization.
* Representation Theorem (subjective probabilities): If a preference
relation over lotteries satisfies [axioms],
then there is a utility function \(u,\) unique up to positive affine
transformation, and a unique probability function \(p\) that represent
the preference relation under EU-maximization.




"Unique up to positive affine transformation" means that
any utility function \(u'\) that also represents the preference
relation can be transformed into \(u\) by multiplying by a constant
and 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 or
Fahrenheit, Celsius can be transformed into Fahrenheit by multiplying
by 9/5 and adding 32).



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



* Completeness: For all lotteries \(L\_1\) and \(L\_2 : L\_1
\succcurlyeq L\_2\) or \(L\_2 \succcurlyeq L\_1.\)
* Transitivity: For all lotteries \(L\_1, L\_2,\) and \(L\_3\): If
\(L\_1 \succcurlyeq L\_2\) and \(L\_2 \succcurlyeq L\_3,\) then \(L\_1
\succcurlyeq L\_3.\)
* Continuity: For all lotteries \(L\_1, L\_2,\) and \(L\_3\): If \(L\_1
\succcurlyeq L\_2\) and \(L\_2 \succcurlyeq L\_3,\) then there is some
real number \(p\in [0, 1]\) such that \(L\_2 \sim\{L\_1, p\); \(L\_3, 1 -
p\}\)
* Independence: For all lotteries \(L\_1,\) \(L\_2,\) \(L\_3\) and all \(0
\lt p \le 1\):

\[
L\_1 \succcurlyeq L\_2 \Leftrightarrow \{L\_1, p; L\_3, 1 - p\} \succcurlyeq \{L\_2, p; L\_3, 1 - p\}
\]




The most historically important axiomatization of the subjective
probabilities 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 generally
include axioms that are analogues of the von
Neumann-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* and
*transitivity*:



* Completeness: For all acts \(f\) and \(g\): \(f \succcurlyeq g\)
or \(f \succcurlyeq g.\)
* Transitivity: For all acts \(f,\) \(g,\) and \(h\): If \(f
\succcurlyeq g\) and \(g \succcurlyeq h,\) then \(f \succcurlyeq
h.\)




The third is some version of continuity, sometimes called an
*Archimedean Axiom*.



The final axiom is a *separability* axiom. Savage's
version of this axiom is known as the sure-thing principle. Where
\(f\_E h\) is an act that agrees with *f* on event *E* and
agrees with *h* elsewhere:



* Sure-thing Principle: For all acts \(f\_E h,\) \(g\_E h,\) \(f\_E j,\) and \(g\_E j\):

 \[f\_E h \succcurlyeq g\_E h \Leftrightarrow f\_E j \succcurlyeq g\_E j\]




In other words, if two acts agree on what happens on not-*E*,
then one's preference between them should be determined only by
what happens on E. Other separability axioms include Jeffrey's
Averaging (1965) and Kobberling and Wakker's Tradeoff
Consistency (2003).



Because representation theorems link, on the one hand, preferences
that accord with the axioms and, on the other hand, a utility (and
probability) function whose expectation the individual maximizes, a
challenge to one of the three components of expected utility theory
must 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 a
particular expectational utility function is mathematically equivalent
to having a particular preference ordering--they open up a number
of possibilities for understanding the utility function. There are two
questions here: what the utility function corresponds to (the
metaphysical question), and how we determine an individual's
utility function (the epistemic question).



The first question is whether the utility function corresponds to a
real-world quantity, such as strength of desire or perceived
desirability or perceived goodness, or whether it is merely a
convenient way to represent preferences. The former view is known as
*psychological realism* (Buchak 2013) or *weak realism*
(Zynda 2000), and is held
by Allais (1953) and Weirich (2008, 2020), for example. The latter
view is known as *formalism* (Hansson 1988),
*operationalism* (Bermudez 2009), or the
*representational viewpoint* (Wakker 1994), and is particularly
associated 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 determining
someone's utility function. Everyone in the debate accepts that
preferences provide evidence for the utility function, but there is
disagreement about whether there may be other sources of evidence as
well. *Constructivists* hold that an individual's utility
function is defined by her preferences--utility is
"constructed" from preference--so there are can be no
other relevant facts (discussed in Dreier 1996, Buchak 2013); this
view is also called *strong realism* (Zynda 2000). *Non-constructive
realists*, by contrast, hold that there are other sources of
evidence about the utility function: for example, an individual might
have introspective access to her utility function. This latter view
only makes sense if one is a psychological realist, though one can
pair constructivism with either psychological realism or
formalism.



A key fact to note about the utility function is that it is
*real-valued*: each consequence can be assigned a real number.
This means that no consequence is of *infinite value*, and all
consequences are *comparable*. As we will see, each of these
two properties invites a challenge.


#### 1.2.2 Probability



Given that a probability function can also be derived from
preferences, a similar question arises about the nature and
determination of the probability function. One could hold that the
probability function represents some real-world quantity, such as
partial belief; or one could hold that the probability function is
merely a way of representing some feature of betting behavior. There
is also disagreement about what facts are relevant to determining
someone's probability function: some hold that it is determined
from betting behavior or from the deliverances of a representation
theorem while others take it to be primitive (see Eriksson &
Hajek 2007).



Probability is *real-valued* and *pointwise*
("sharp", "precise"), meaning that there is a
unique number representing an individual's belief in or evidence
for an event. Again, this property will invite a challenge.


#### 1.2.3 Expectation



We can see the *norm* 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 argue
either for the functional form itself or for the normativity of the
axioms. Examples of the former include the argument that expected
utility maximizers do better in the long run, though these arguments
fell out of favor somewhat as the popularity of the realist
interpretations of utility waned. Examples of the latter include the
idea that each axiom is itself an obvious constraint and the idea that
the axioms follow from consequentialist (or means-ends rationality)
principles. Of particular note is a proof that non-EU maximizers will
either be inconsistent or non-consequentialist over time (Hammond
1988); how alternative theories have fared under dynamic choice has
been a significant focus of arguments about their rationality.



The idea that EU maximization is the correct norm can be challenged on
several different grounds, as we will see. Those who advocate for
non-EU theories respond to the arguments listed above by either
arguing that the new norm doesn't actually fall prey to the
argument (e.g., provide a representation theorem with supposedly more
intuitive 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 that
infinite utility can arise in decision situations.



First, some particular outcome might have *infinite utility* or
infinite disutility. For example, *Pascal's Wager* is
motivated by the idea that eternal life with God has infinite value,
so one should "wager for God" as long as one assigns some
non-zero probability to God's existence (Pascal 1670). If a
particular outcome has infinite (dis)value, then the Continuity Axiom
or the Archimedean Axiom will not hold. (See discussions in Hacking
1972 and Hajek 2003, and a related issue for utilitarianism in
Vallentyne 1993, Vallentyne & Kagan 1997, and Bostrom 2011.)



Second, all outcomes might have finite utility value, but this value
might be *unbounded*, which, in combination with allowing that
there can be infinitely many *states*, gives rise to various
paradoxes. The most famous of these is the *St. Petersburg
Paradox*, first introduced by Nicolas Bernoulli in a 1713 letter
(published in J. Bernoulli *DW*). Imagine a gamble whose
outcome is determined by flipping a fair coin until it comes up
heads. If it lands heads for the first time on the
*n*th flip, the recipient gets $\(2^n\); thus the
gamble has infinite expected monetary value for the person who takes
it (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

 

\[\left(\frac{1}{2}\right)(2) + \left(\frac{1}{4}\right)(4) + \left(\frac{1}{8}\right)(8) + \ldots \rightarrow \infty.\]


While this version can be resolved by allowing utility to diminish
marginally in money--so that the gamble has finite expected
utility--if the payoffs are in utility rather than money, then
the gamble will have infinite expected utility.



Related paradoxes and problems abound. One centers around a pair of
games, the *Pasadena game* and the *Altadena game*
(Nover & Hajek 2004). The Pasadena game is also played by
flipping a fair coin until it lands heads; here, the player receives
\(\$({-1})^{n-1}(2^n/n)\) if the first heads occurs on the
*n*th flip. Thus, its payoffs alternate between
positive and negative values of increasing size, so that its terms can
be rearranged to yield any sum whatsoever, and its expectation does
not exist. The Altadena game is identical to the Pasadena game, except
that every payoff is raised by a dollar. Again, its terms can be
rearranged to yield any value, and again its expectation does not
exist. However, it seems (contra EU maximization) that the Altadena
game should be preferred to the Pasadena game, since the former
*state-wise dominates* the latter--it is better in every
possible state of the world (see also Colyvan 2006, Fine 2008,
Hajek & Nover 2008). Similarly, it seems that the
*Petrograd 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-envelope
paradox; Arntzenius, Elga, and Hawthorne's (2004) discussion of
diachronic puzzles involving infinite utility; and McGee's
(1999) argument that the utility function ought to be bounded, which
will dissolve the above paradoxes.)


### 2.2 Proposals



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



A different kind of response is to subsume EU maximization under a
more general norm that also applies when utility is infinite. Bartha
(2007, 2016) defines *relative utility*, which is a three-place
relation that compares two outcomes or lotteries relative to a third
"base point" that is worse than both. The relative utility
of \(A\) to \(B\) with base-point \(Z\) (written \((A, B; Z)\)) will
be:


* 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 can
be determined from preferences. Furthermore, relative utility is
expectational

 

\[U(\{A, p; A', 1-p\}, B; Z) = pU(A, B; Z) + (1-p)U(A', B; Z)\]


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



When considering only the paradoxes of unbounded utility (not those of
infinite utility), there are other ways to subsume EU maximization
under a more general norm. Colyvan (2008) defines *relative
expected 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:

 

\[\REU(f,g) = \sum\_{i = 1}^{n} p(E\_{i}) \left(u(x\_{i}) - u(y\_{i})\right)\]


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



\[\REU(f,g) = \sum\_{i = 1}^{\infty} p(E\_{i}) \left(u(x\_{i}) - u(y\_{i})\right).\]


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 the
Altadena game is preferred to the Pasadena game and the Petrograd game
is preferred to the St. Petersburg game. (See also Colyvan & Hajek 2016.)



A more modest extension of standard EU maximization is suggested by
Easwaran (2008). He points out that although the Pasadena and Altadena
games lack a "strong" expectation, they do have a
"weak" expectation. (The difference corresponds to the
difference between the strong and weak law of large numbers.). Thus,
we can hold that a decision-maker is required to value a gamble at its
weak expectation, which is equivalent to its strong expectation if the
latter 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's
proposal to *infinite* weak expectations with an extension of
Colyvan's proposal to *cardinal* relative expectation
that can be *interval-valued*. Meacham (2019) extends
Colyvan's proposal to cover cases in which the acts to be
compared have utilities that are to be compared in different states,
and cases in which probabilities are act-dependent; his *difference
minimizing theory* re-orders each gamble from worst consequence to
best consequence, before taking their relative expectation. A key
difference between these two extensions is that difference minimizing
theory adheres to *stochastic dominance* and a related
principle called *stochastic equivalence*. (See also discussion
in Seidenfeld, Schervish, & Kadane 2009; Hajek 2014;
Colyvan & Hajek 2016.)



In a more radical departure from standard EU maximization, Easwaran
(2014a) develops an axiomatic decision theory *based* on
state-wise dominance, that starts with utility and probability and
derives a normative preference relation. In cases that fit the
standard parameters of EU maximization, this theory can be made to
agree with EU maximization; but it also allows us to compare some acts
with infinite value, and some acts that don't fit the standard
parameters (e.g., incommensurable acts, acts with probabilities that
are comparative but non-numerical).



Finally, one could *truncate* the norm of EU maximization. Some
have argued that that for a gamble involving very small probabilities,
we should discount those probabilities down to zero, regardless of the
utilities involved. When combined with a way of aggregating the
remaining possibilities, this strategy will yield a finite value for
the unbounded-utility paradoxes, and also allow people who attribute a
very small probability to God's existence to avoid wagering for
God. (This idea traces back to Nicolaus Bernoulli, Daniel Bernoulli,
d'Alambert, Buffon, and Borel [see Monton 2019 for a historical
survey]; 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 that
preferences are totally ordered--to the idea that consequences
can be ranked according to a single, consistent utility function. In
economics, this idea goes back at least to Aumann (1962); in
philosophy, it has been taken up more recently by ethicists.
Economists tend to frame the challenge as a challenge to the idea that
the *preference* relation is complete, and ethicists to the
idea that the *betterness* relation is complete. I use
\(\succcurlyeq\) to represent whichever relation is at issue,
recognizing that some proposals may be more compelling in one case
than the other.



The key claim is that there are some pairs of options for which it is
false that one is preferred to (or better than) the other, but it is
also false that they are equi-preferred (or equally good). Proposed
examples include both the mundane and the serious: a Mexican
restaurant and a Chinese restaurant; a career in the military and a
career as a priest; and, in an example due to Sartre (1946), whether
to stay with one's ailing mother or join the Free French. Taking
up the second of these examples: it is not the case that a career in
the military is preferred to (or better than) a career as a priest,
nor *vice versa*; but it is also not the case that they are
equi-preferred (or equally good). Call the relation that holds between
options in these pairs *incommensurability*.



Incommensurability is most directly a challenge to Completeness, since
on 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 can
instead be framed as a challenge to Transitivity, if we assume that
incommensurability is indifference, or define \(A \succcurlyeq B\) as
the negation of \(B \succ A\) (thus assuming Completeness by
definition). To see this, notice that if two options \(A\) and \(B\)
are incommensurable, then "sweetening" \(A\) to a slightly
better \(A^+\) will still leave \(A^+\) and \(B\) incommensurable. For
example, if \(A\) is a career in the military and \(A^+\) is this
career but with a slightly higher salary, the latter is still
incommensurable with a career as a priest. This pattern suffices to
show 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 the
matter about which of the three relations \((\succ,\) \(\prec,\)
\(\sim)\) holds, but that it is sometimes difficult or impossible to
determine which one--thus incommensurability is merely apparent.
They can model the decision problem in the standard way, but as a
problem of uncertainty about values: one does not know whether one is
in a state in which \(A \succcurlyeq B,\) but one assigns some
probability to that state, and maximizes expected utility taking these
kinds of uncertainties into account. *Indeterminists* hold that
it is indeterminate which relation holds, because these relations are
vague; 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 that
incommensurability is *parity* hold that there is a fourth
relation than can obtain between \(A\) and \(B\): \(A\) and \(B\) are
"on a par" if \(A\) and \(B\) can be compared but it is
not the case that one of the three relations holds (Chang 2002a,
2002b, 2012, 2015, 2016). (Taxonomy from Chang 2002b; see also Chang
1997.)


### 3.2 Proposals: Definitions



Aumann (1962) shows that if we have a partial but not total preference
ordering, then we can represent it by a utility function (not
unique-up-to-positive-affine-transformation) such that \(A \succ B\)
implies \(\EU(A) \gt \EU(B),\) but not *vice versa*. Aumann
shows that there will be at least one utility function that represents
the preference ordering according to (objective) EU maximization.
Thus, we can represent a preference ordering as the *set of all
utility functions* that "one-way" represent the
decision-maker's preferences. Letting \(\EU\_u(A)\) be the
expected utility of \(A\) given utility function \(u\):



\[\begin{align}
\cU = \big\{u \mid {}&(A \succ B \Rightarrow \EU\_u (A) \succ \EU\_u (B)) \\
 &{}\amp (A \sim B \Rightarrow \EU\_u (A) = \EU\_u (B))\big\}.
\end{align}\]


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 \(\EU\_{u'}(B) \gt \EU\_{u'}(A)\); and
*vice versa*.



Chang (2002a,b) proposes a similar strategy, but she takes value facts to
be 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 the
*evaluative 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 \(B\) are
*incomparable*; however, if \((A - B) \ne \varnothing,\) the
relevant 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 of
which represents a possible substantive completion of the underlying
value that utility represents (discussed in Chang 2002b);
alternatively, it might be that there is parity "all the way
down" (discussed in Chang 2016, where she also replaces the
definition in terms of explicit numerical differences with one in
terms of bias).



Rabinowicz (2008)
provides a model that allows for both parity and grades of
incomparability. On his proposal, the betterness relation is
represented by a class \(K\) of "permissible preference
orderings", each of which may be complete or incomplete. He
defines:

 

\[
\begin{align}
x \succ y & \text{ iff } (\forall R\in K)(x \succ\_R y)\\
x \sim y & \text{ iff } (\forall R\in K)(x \sim \_R y)\\
x \parallel y & \text{ iff } (\exists R\in K)(x \succ\_R y) \amp (\exists R \in K)(x \succ\_R x)\\
\end{align}
\]


(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 then
defines:


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



Class \(K\) won't necessarily give rise to a utility
function.


### 3.3 Proposals: Decision Rules



If the decision-maker's preferences are represented by a set of
utility functions \(\cU,\) then a number of possible decision rules
suggest themselves. All the proposed rules focus on selection of
*permissible* 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 not *vice
versa*). To understand the first three of these rules, we can
imaginatively think of each possible utility function as a
"committee member", and posit a rule for choice based on
facts about the opinions of the committee.



First, we might choose any option which *some* committee member
endorses; that is, we might choose any option which maximizes expected
utility relative to *some* utility function:

 

\[
\text{Permissible choice set } = \{ A \mid (\exists u \in \cU) (\forall B \in \cS)(\EU\_u (A) \ge \EU\_u (B))\}\]


Levi (1974) terms this rule *E-admissibility*, and Hare (2010)
calls it *prospectism*. (See
 section 4.3.3
 for Levi's full proposal, and for extensions of this rule to
the case in which the decision-maker does not have a single
probability function.)



Aumann suggests that we can choose any *maximal* option: any
option that isn't worse, according to all committee members,
than some other particular option; that is, an option to which no
other option is (strictly) preferred (assigned a higher utility by all
utility functions):

 

\[
\text{Permissible choice set } = \{A \mid (\forall B \in \cS ) (\exists u \in \cU)(\EU\_u (A) \ge \EU\_u (B)) \}\]


This is a more permissive rule than E-admissibility: every
E-admissible option will be maximal, but not *vice versa*. To
see the difference between the two rules, notice that if the
decision-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 definitively
preferred to \(B,\) so \(B\) is maximal; but it is definitive that
something is preferred to \(B,\) so \(B\) is not E-admissible).



A final possibility is that we can choose any option that is not
*interval 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:



\[
\text{Permissible choice set } = \{ A \mid (\forall B \in \cS )(\text{max}\_u \EU\_u (A) \ge \text{min}\_u \EU\_u (B))\} \]


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



A different type of rule first looks at the options' possible
utility values in each state, before aggregating over states; this is
what Hare's (2010) *deferentialism* requires. To find out
if an option O is permissible under deferentialism, we consider how it
fares if we, in each state, make the assumptions most favorable to it.
First, "regiment" the utility functions in \(\cU\) so that
there's some pair of consequences \(\{x, y\}\) such that
\((\forall u \in\cU)(u(x) = 1 \amp u(y) = 0)\); this allows only one
representative utility function for each possible completion of the
decision-maker's preference. Next, take all the possible
"state-segments"--the possible utility assignments in
each state--and cross them together in every possible arrangement
to get an "expanded" set of utility functions (for
example, this will contain every possible utility function in \(E\)
coupled with every possible utility function not-\(E\)). Then \(A\) is
permissible iff \(A\) maximizes expected utility relative to
*some* 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 that
subjective probabilities need not be "sharp" or
"precise", i.e., need not be a single, point-wise
function. (In economics, this phenomenon is typically called
*ambiguity*.) There are three historically significant
motivations for imprecise probabilities.



The first is that decision makers treat subjective (or unknown)
probabilities differently from objective probabilities in their
decision-making behavior. The classic example of this is the Ellsberg
Paradox (Ellsberg 1961, 2001). Imagine you face an urn filled with 90
balls that are red, black, and yellow, from which a single ball will
be drawn. You know that 30 of the balls are red, but you know nothing
about 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 or
yellow.
* \(f\_2\): $100 if the ball is black; $0 if the ball is red or
yellow.
* \(f\_3\): $100 if the ball is red or yellow; $0 if the ball is
black.
* \(f\_4\): $100 if the ball is black or yellow; $0 if the ball is
red.



Most people appear to (strictly) prefer \(f\_1\) to \(f\_2\) and
(strictly) prefer \(f\_4\) to \(f\_3.\) They would rather bet on the
known or objective probability than the unknown or subjective
one--in the first pair, red has an
objective probability of \(\lfrac{1}{3}\), whereas black
has a possible objective probability ranging
from 0 to \(\lfrac{2}{3}\); in the second pair, black
or yellow has an objective probability of \(\lfrac{2}{3}\)
whereas red or yellow has a possible objective
probability ranging from \(\lfrac{1}{3}\) to 1. These preferences
violate the Sure-thing Principle. (To see this, notice that the only
difference between the two pairs of acts is that the first pair yields
$0 on yellow and the second pair yields $100
on yellow.)



The second motivation for imprecise probability is that even if all
the relevant probabilities are subjective, a decision-maker's
betting behavior might depend on how reliable or well-supported by
evidence those probabilities are. Consider a decision-maker who may
bet on three different tennis matches: in the first, she knows a lot
about the players and knows they are very evenly matched; in the
second, she knows nothing whatsoever about either player; and in the
third, she knows that one of the two players is much better than the
other, but she does not know which one. In each of the matches, the
decision-maker should presumably assign equal probability to each
player winning, since her information in favor of each is symmetrical;
nonetheless, it seems rational to bet only on the first match and not
on the other two (Gardenfors & Sahlin 1982; see also Ellsberg
1961).



A final motivation for imprecise probability is that evidence
doesn't always determine precise probabilities (Levi 1974, 1983;
Walley 1991; Joyce 2005;
Sturgeon 2008; White
2009; Elga 2010). Assume a stranger approaches you and pulls three
items out of a bag: a regular-sized tube of toothpaste, a live
jellyfish, and a travel-sized tube of toothpaste; you are asked to
assign probability to the proposition that the next item he pulls out
will be another tube of toothpaste--but it seems that you lack
enough evidence to do so (Elga 2010).


### 4.2 Proposals: Probability Representations



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



The first is to assign an *interval*, instead of a single
number, to each proposition. For example, in the Ellsberg case: 



\[
\begin{align}
p(\RED) & = [\lfrac{1}{3}], \\
p(\YELLOW) & = [0, \lfrac{2}{3}]; \\
p(\BLACK) & = [0, \lfrac{2}{3}].\\
\end{align}
\]


The second is to represent the individual's beliefs as a
*set* of probability functions. For example, in the Ellsberg
case:



\[ 
\cQ = \{p\in\cP \mid p(\RED) = \lfrac{1}{3}\} 
\]


This means,
for example, that the probability distribution \(p(\rR, \rB, \rY) =
\langle \lfrac{1}{3}, 0, \lfrac{2}{3}\rangle\) and the probability
distribution \(\langle \lfrac{1}{3}, \lfrac{1}{3},
\lfrac{1}{3}\rangle\) are both compatible with the available evidence
or possible "completion" of the individual's
beliefs.



Each set-probability representation gives rise to an interval representation (assuming the set of
probability functions is convex); but the set-probability
representation provides more structure to the relationships between
propositions. (A different proposal retains a precise probability
function but refines the objects over which utility and probability
range (Bradley 2015; Stefansson & Bradley 2015, 2019); see
discussion in
 section 5.2.)


### 4.3 Proposals: Decision Rules



We will examine rules for decision-making with imprecise probabilities
in terms of how they evaluate the Ellsberg choices; for ease of
exposition we will assume \(u(\$0) = 0\) and \(u(\$100) = 1.\) All
proposals here are equivalent to EU maximization when there is a
single probability distribution in the set, so all will assign utility
\(\lfrac{1}{3}\) to \(f\_1\) and \(\lfrac{2}{3}\) to \(f\_4\) in the
Ellsberg 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 these *aggregative
rules*. The rules in this section use sets of probabilities.



Before we discuss these rules, it will be helpful to keep in mind
three 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 highest
minimum utility. The second is maximax, which says to pick the option
with the highest maximum utility. The third, known as the Hurwicz
criterion, says to take a weighted average, for each option, of the
minimum and maximum utility, where the weight \(\alpha \in [0, 1]\)
represents a decision-maker's level of optimism/pessimism
(Hurwicz 1951a):

 

\[
H(f) = (1 - \alpha)(\text{min}\_u(f)) + \alpha(\text{max}\_u(f))\]


Using the set-probability representation, we can associate to each
probability distribution an expected utility value, to yield a set of
expected 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 expected
utility value; thus, a decision-maker should *maximize her minimum
expected utility* (Wald 1950; Hurwicz 1951b; Good 1952; Gilboa
& Schmeidler 1989):

 

\[\text{G-maximin}(f) = \text{min}\_p (\EU\_p(f))\]


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 \(f\_4 \gt
f\_3.\) Conversely, an individual who *maximizes her maximum
expected utility*--who uses G-maximax--would have
the reverse preferences.



G-maximin appears too pessimistic. We might instead use an
EU-analogue of Hurwicz criterion: take a *weighted average of the
minimum expected utility and the maximum expected utility*, with
weight \(\alpha \in[0, 1]\) corresponding to a decision-maker's
level of pessimism (Hurwicz 1951b; Shackle 1952; Luce & Raiffa
1957; Ellsberg 2001; Ghirardato et al. 2003):

 

\[\alpha\text{-maximin}(f) = (1 - \alpha)(\text{min}\_p (\EU\_p (f))) + \alpha(\text{max}\_p (\EU\_p(f)))\]


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 quantities
when 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 her
estimated probability distribution. The value of an act will then be a
weighted average of her best estimate EU and the minimum EU, with her
best estimate weighed by her degree of confidence (Hodges &
Lehmann 1952; Ellsberg 1961):

 

\[E(f) = \varrho(\text{est}\_p (\EU\_p(f))) + (1 - \varrho)(\text{min}\_p (\EU\_p(f)))\]


In the Ellsberg pairs, assuming the "best estimate" is
that yellow and black
each 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 disprefered
to \(f\_1\) and \(f\_4,\) respectively, as long as \(\varrho \lt
1.\)



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

 

\[
E(f) = \varrho(\text{est}\_p \EU\_p(f)) + (1 - \varrho)
[(1 - \alpha)(\text{min}\_p \EU\_p(f)) + \alpha(\text{max}\_p \EU\_p(f))]\]


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 each
probability function, the decision-maker assigns a degree of
"reliability", which tracks how much relevant information
the decision-maker has about the states of nature (Good 1952;
Gardenfors & Sahlin 1982). A decision-maker selects a desired
threshold level of epistemic reliability. She then considers all
probability functions above this threshold, and maximizes the minimum
expected utility (G-maximin) with respect to these probability
functions. (In principle, a different decision rule could be used in
this step.) For the decision-maker deciding whether to bet on tennis
matches, the above-threshold probability functions for the first match
may include only \(p(P1 \text{WINS}) \approx 0.5,\) but for the second
and third match may also include \(p(P1 \text{WINS}) \approx 0\); thus
betting on P1 in the first match will have a higher value than betting
on P1 in the other matches.


#### 4.3.2 Aggregative Decision Rules: Choquet Expected Utility



A different kind of rule is *Choquet expected utility*, also
known as *cumulative* utility (Schmeidler 1989; Gilboa 1987).
This rule starts with a function \(v\) which, like a probability
function, obeys \(v(E)\in[0, 1],\) \(v(0) = 0,\) \(v(1) = 1,\) and \(A
\subseteq B\) implies \(v(A) \le v(A).\) Unlike a probability
function, however, \(v\) is non-additive; and \(v\) is not
straightforwardly used to calculate an expectation. (Many economists
refer to this function as a "non-additive subjective probability
function".). Choquet expected utility is a member of the
*rank-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's
overall value depend on both the probability-like element and the
event's position in the ordering of an act, e.g., whether it is
the 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 worst
event to best event, so that \(u(x\_1) \le \ldots \le u(x\_n).\) The
*Choquet expected utility* of \(g'\) (and therefore of \(g\))
is:

 

\[
\CEU(g') = u(x\_{1}) + \sum\_{i = 2}^{n} v\left(\bigcup\_{j = i}^{n} E\_{j}\right) \left(u(x\_{i}) - u(x\_{i - 1})\right)\]


If \(v\) is additive, then \(v\) is an (additive) probability function
and CEU reduces to EU. If \(v\) is convex \((v(E) + v(F) \le v(EF) +
v(E v F)),\) then the individual is *uncertainty-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" will
assign \(v(\BLACK) + v(\YELLOW) \le v(\BLACK \lor \YELLOW)\); let us
assume \(v(\BLACK) = v(\YELLOW) = \lfrac{1}{9}.\) Similarly, she will
assign \(v(\RED \lor \YELLOW) + v(\BLACK) \le 1\); let us assume
\(v(\RED \lor \YELLOW) = \lfrac{4}{9}.\)



Then the values of the acts will be:

 

\[
\begin{align}
\CEU(f\_1) & = 0 + v(\RED)(1 - 0) & = \lfrac{1}{3}\\
\CEU(f\_2) & = 0 + v(\BLACK)(1 - 0) & = \lfrac{1}{9}\\
\CEU(f\_3) & = 0 + v(\RED \lor \YELLOW)(1 - 0) &= \lfrac{4}{9}\\
\CEU(f\_4) & = 0 + v(\BLACK \lor \YELLOW)(1 - 0) &= \lfrac{2}{3}\\
\end{align}
\]


This assignment recovers the Ellsberg preferences.



Axiomatizations of CEU use a restricted version of the separability
condition (the "Comonotonic" Sure-thing Principle or
"Comonotonic" Tradeoff Consistency): namely, the condition
only holds when all of the acts in its domain are
*comonotonic*, i.e., when the worst-to-best ranking of the
events coincides for all the acts (Gilboa 1987, Schmeidler 1989,
Wakker 1989, Chew & Wakker 1996, Kobberling & Wakker
2003; see also Wakker 2010, who also notes the relationship between
CEU and \(\alpha\)-maximin.)


#### 4.3.3 Decision Rules that Select from a Menu



Another different type of proposal focuses on selection of
*permissible* options from a set of alternatives \(\cS,\)
rather than a complete ranking of options. As in
 section 3.3,
 we can imaginatively think of each possible probability distribution
in a set \(\cQ\) as a "committee member", and posit a rule
for choice based on facts about the opinions of the committee. (The
first three rules are versions of the rules for sets of utility
functions in 3.3, and can be combined to cover sets of
probability/utility pairs.)



The first possibility is that a decision-maker is permitted to pick an
act just in case *some* committee member is permitted to pick
it over all the alternatives: just in case it maximizes expected
utility relative to *some* probability function in the set.
This is known as *E-admissibility* (Levi 1974, 1983, 1986;
Seidenfeld, Schervish, & Kadane 2010):

 

\[
\text{Permissible choice set } = \{A \mid (\exists p\in\cQ )(\forall B \in \cS)(\EU\_p(A) \ge \EU\_p(B))\}\]


Levi in fact tells a more complicated story about what a
decision-maker is permitted to choose, in terms of a procedure that
rules out successively more and more options. First, the procedure
selects from all the options just the ones that are E-admissible.
Next, the procedure selects from the E-admissible options just the
ones that are *P-admissible*: options that "do
best" at preserving the E-admissible options (the idea being
that a rational agent should keep her options open). Finally, the
procedure selects from the P-admissible options just the ones that are
*S-admissible*: options that maximize the minimum utility.
(Note that this last step involves maximin, not G-maximin.)



A more permissive rule than E-admissibility permits a choice as long
there is *no* particular alternative that the committee members
*unanimously* (strictly) prefer. As in the case of
utility-sets, this rule is known as *maximality* (Walley
1991):

 

\[
\text{Permissible choice set } = \{A \mid (\forall B\in\cS)(\exists p \in\cQ )(\EU\_p(A) \ge \EU\_p (B))\}\]


(See
 section 3.3
 for an example of the difference between E-admissibility and
maximality.)



More permissive still is the rule that a choice is permissible as long
as it is not *interval dominated* (Schick 1979, Kyburg 1983):
its maximum value isn't lower than the minimum value of some
other act.

 

\[\text{Permissible choice set } = \{A \mid (\forall B\in \cS)(\text{max}\_p \EU\_p(A) \ge \text{min}\_p \EU\_p(B))\}\]


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



A final approach is to interpret ambiguity as indeterminacy: one
committee member has the "true" probability function, but
it is indeterminate which one. If all probability functions agree that
an option is permissible to choose, then it is determinately
permissible to choose; if all agree that it is impermissible to
choose, it is determinately impermissible to choose; and if some hold
that it is permissible and others hold that it is impermissible, it is
indeterminate whether it is permissible (Rinard 2015).



These rules in this section allow but do not obviously
*explain* the Ellsberg choices unless supplemented by an
additional rule (e.g., Levi's more complicated story or one of
the rules from
 section 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, and
non-interval-dominated.


### 4.4 Normative Questions



For those who favor non-sharp probabilities, two sets of normative
questions arise. The first set is epistemic: whether it is epistemically rational not
to have sharp probabilities (White 2009; Elga 2010; Joyce 2011;
Hajek & Smithson 2012; Seidenfeld et al. 2012; Mayo-Wilson
& Wheeler 2016; Schoenfield 2017; Vallinder 2018; Builes et al 2022; Konek
ms. - 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 in
 section 1.2.3. Of particular interest is how
these decision rules can be extended to sequential choice (Seidenfeld
1988a,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 norm
itself--to the idea that a gamble should be valued at its
*expectation*. In particular, some claim that it is rationally
permissible for individuals to be risk-averse (or risk-seeking) in a
sense that conflicts with EU maximization.



Say that an individual is *risk-averse* in money (or any
numerical good) if she prefers the consequences of a gamble to be less
"spread out"; this concept is made precise by Rothschild
and Stiglitz's idea of *dispreferring mean-preserving
spreads* (1972). As a special case of this, a person who is
risk-averse in money will prefer \(\$x\) to any gamble whose mean
monetary 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 previous
dollar); if an EU maximizer is risk-seeking in money, then her utility
function will be convex (Rothschild & Stiglitz 1972). Therefore,
EU theory equates risk-aversion with having a diminishing marginal
utility function.



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



This kind of example gives rise to several problems. First, if EU
maximization is supposed to *explain why* someone made a
particular choice, it ought to be able to distinguish these two
reasons for preference; but if global sensitivity can be captured at
all, it will have to be captured by a diminishing marginal utility
function, identical to that of the first person in each pair. Second,
if one adopts a view of the utility function according to which the
decision-maker has introspective access to it, a decision-maker might
report that she has preferences like the tolerant coffee drinker or
the miser--her utility function is linear--but nonetheless
if she maximizes EU her utility function will have to be concave; so
EU maximization will get her utility function wrong. Finally, even if
one holds that a decision-maker does not have introspective access to
her utility function, if a decision-maker displays global sensitivity,
then she will have some preferences that cannot be captured by an
expectational utility function (Allais 1953, Hansson 1988, Buchak
2013).



A related worry displays a troubling implication of EU
maximization's equating risk-aversion with diminishing marginal
utility. Rabin's (2000) *Calibration Theorem* shows that
if an EU-maximizer is mildly risk-averse in modest-stakes gambles, she
will have to be absurdly risk-averse in high-stakes gambles. For
example, 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, the *Allais Paradox* identifies a set of preferences
that are intuitive but cannot be captured by any expectational utility
function (Allais 1953). Consider the choice between the following two
lotteries:


* \(L\_1 : \{\$5,000,000, 0.1; $0, 0.9\}\)
* \(L\_2 : \{\$1,000,000, 0.11; \$0, 0.89\}\)



Separately, consider the choice between the following two
lotteries:


* \(L\_3 : \{\$1,000,000, 0.89; \$5,000,000, 0.1; \$0, 0.01\}\)
* \(L\_4 : \{\$1,000,000, 1\}\)



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 the
Independence Axiom; when the lotteries are suitably reframed as acts
(e.g., defined over events such as the drawing of a 100-ticket
lottery), they violate the Sure-thing Principle or related
separability principles.


### 5.2 Proposals



There have been a number of descriptive attempts to explain global
sensitivity, by those who are either uninterested in normative
questions or assume that EU is the correct normative theory. The most
well-known of these are prospect theory (Kahneman & Tversky 1979;
Tversky & Kahneman 1992) and generalized utility theory (Machina
1982, 1983, 1987); others are mentioned in the discussion below. See
Starmer (2000) for an overview.



Some normative proposals seek to accommodate global sensitivity
*within* expected utility theory, by making the inputs of the
utility 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 induce
regret that one forewent a sure $1M (alternatively, because \(L\_4\)
includes psychological certainty) and therefore that the
consequence-descriptions should include these facts. Thus, the correct
description of \(L\_3\) is:

 

\[L\_3 : \{\$1,000,000, 0.89; \$5,000,000, 0.1; \$0 \textit{ and regret, } 0.01\}\]


Once these gambles are correctly described, there is no direct
conflict with EU maximization (Raiffa 1986, Weirich 1986, Schick 1991,
Broome 1991, Bermudez 2009, Pettigrew 2015, Buchak 2015). The
problem of when two outcomes that appear the same should be
distinguished is taken up by Broome (1991), Pettit (1991), and Dreier
(1996).



The thought that the value of a consequence depends on what might have
been is systematized by Bradley and Stefansson (2017). Their
proposal uses a version of expected utility theory developed by
Jeffrey (1965) and axiomatized by Bolker (1966). Jeffrey replaces the
utility function by a more general "desirability" function
Des, 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. Bradley
and Stefansson propose to widen the domain of Des to include
counterfactual propositions, thus allowing that preferences for
propositions can depend on counterfacts. For example, a decision-maker
can prefer "I choose the risky options and get nothing, and I
wouldn"t have been guaranteed anything if I had chosen
differently' to "I choose the risky option and get
nothing, and I would have been guaranteed something if I had chosen
differently", which will rationalize the Allais preferences.
(Incidentally, their proposal can also rationalize preferences that
seemingly violate EU because of fairness considerations, as in an
example from Diamond (1967).)



In a different series of articles (Stefansson & Bradley
2015, 2019), these authors again employ Jeffrey's framework, but
this time widen the domain of Des to include *chance
propositions* (in addition to factual prospects), propositions
like "the chance that I get $100 is 0.5". They hold that a
rational decision-maker can have a preference between various chances
of \(X,\) even on the supposition that \(X\) obtains (she need not
obey "Chance Neutrality"). They capture the idea of
disliking risk as such by holding that even though a rational agent
must maximize expected desirability, she need not have a \(\Des\)
function of \(X\) that is expectational with respect to the Des
function of chance propositions about \(X\) (she need not obey
"Linearity"). For example, \(\Des(\text{"I get
\$100"})\) need not be equal to \(2(\Des(\text{"the chance
that I get \$100 is 0.5"})).\) (This does not conflict with
maximizing expected desirability, because it concerns only the
relationship between particular inputs to the \(\Des\) function, and
does not concern the decision-maker's subjective
probabilities.). This proposal can also rationalize the Ellsberg
preferences
 (section 4.1),
 because it allows the decision-maker to assign different
probabilities to the various chance propositions (see also Bradley
2015).



Other proposals hold that we should reject the aggregation norm of
expected utility. The earliest of these came from Allais himself, who
held that decision-makers care not just about the mean utility of a
gamble, but also about the dispersion of values. He proposes that
individuals maximize expected utility plus a measure of the riskiness
of a gamble, which consists in a multiple of the standard deviation of
the gamble and a multiple of its skewness. Formally, if \(s\) stands
for the standard deviation of \(L\) and \(m\) stands for the skewness
of \(L,\) then the utility value of \(L\) will be (Allais 1953, Hagen
1979):

 

\[\text{AH}(L) = \EU(L) + F(s, m/s^2) + \varepsilon\]


where \(\varepsilon\) is an error term. He thus proposes that
riskiness is an independently valuable property of a gamble, to be
combined with (and traded off against) its expected utility. This
proposal essentially treats the riskiness of a gamble as a property
that is (dis)valuable in itself (see also Nozick 1993 on symbolic
utility).



A final approach treats global sensitivity as a feature of the
decision-maker's way of aggregating utility values. It might be
that a decision-maker's utility and probability function are not
yet enough to tell us what he should prefer; he must also decide how
much weight to give to what happens in worse states versus what
happens in better states. In *risk-weighted expected utility*
(Buchak 2013), a generalization of Quiggin's (1982)
*anticipated utility* and a member of the rank-dependent family
(see
 section 4.3.2),
 this decision is represented by his *risk function*.



Formally, let \(g' = \{E\_1, x\_1;\ldots; E\_n, x\_n\}\) be a re-ordering
of act \(g\) from worst event to best event, so that \(u(x\_1) \le
\ldots \le u(x\_n).\) Then the *risk-weighted expected utility*
of \(g\) is:

 

\[
\REU (g') = u(x\_{1}) + \sum\_{i = 2}^{n} r
\left(\sum\_{j = i}^{n} p(E\_{j})\right) \left(u(x\_{i}) - u( x\_{i - 1})\right)\]


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 of
consequences in the evaluation of an act--how much the
decision-maker cares about benefits that obtain only in the top
\(p\)-portion of states. (One could also think of the risk function as
describing the solution to a distributive justice problem among
one's future possible selves--it says how much weight a
decision-maker gives to the interests of the top \(p\)-portion of his
future possible selves.) A *risk-avoidant* person is someone
with a convex risk function: as benefits obtain in a smaller and
smaller portion of states, he gives proportionally less and less
weight to them. A *risk-inclined* person is someone with a
concave risk function. And a *globally neutral person* is
someone 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 convex
risk function (Segal 1987, Prelec 1998, Buchak 2013; but see Thoma
& Weisberg 2017). Thus, REU maximization holds that
decision-makers have the Allais preferences because they care more
about what happens in worse scenarios than better scenarios, or are
more concerned with the minimum value than potential gains above the
minimum.



The representation theorem for REU combines conditions from two
existing theorems (Machina & Schmeidler 1992, Kobberling and
Wakker 2003), replacing the separability condition with two weaker
conditions. One of these conditions fixes a unique probability
function of events (Machina & Schmeidler's "Strong
Comparative Probability", 1992) and the other fixes a unique
risk function of probabilities; the latter is a restricted version of
the separability condition (Kobberling & Wakker's
[2003] "Comonotonic" Tradeoff Consistency; see
 section 4.3.2).
 Since the representation theorem derives a unique probability
function, a unique risk function, and a unique (up to positive affine
transformation) utility function, it separates the contribution of
diminishing marginal value and global sensitivity to a given
preference ordering. One can disprefer mean-preserving spreads as a
result 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 two
particular worries. One of these is that the constraints of decision
theory end up trivial (Dreier 1996); the other is that they saddle the
decision-maker with preferences over impossible objects (Broome
1991).



For theories that reject the Sure-thing Principle or the Independence
Axiom, several potential worries arise, including the worry that these
axioms are intuitively correct (Harsanyi 1977, Savage 1954, Samuelson
1952; see discussion in McClennen 1983, 1990); that decision-makers
will evaluate consequences inconsistently (Samuelson 1952, Broome
1991); 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 is
that these theories will leave decision-makers open to diachronic
inconsistency (Raiffa 1968; Machina 1989; Hammond 1988; McClennen
1988, 1990; Seidenfeld 1988a,b; Maher 1993; Rabinowicz 1995, 1997;
Buchak 2013, 2015, 2017; Briggs 2015; Joyce 2017; Thoma 2019).