## 1. Defining Expected Utility




The concept of expected utility is best illustrated by
example. Suppose I am planning a long walk, and need to decide whether
to bring my umbrella. I would rather not tote the umbrella on a sunny
day, but I would rather face rain with the umbrella than without
it. There are two acts available to me: taking my umbrella, and
leaving it at home. Which of these acts should I choose?




This informal problem description can be recast, slightly more
formally, in terms of three sorts of entities. First, there are
*outcomes*--objects of non-instrumental preferences. In
the example, we might distinguish three outcomes: either I end up dry
and unencumbered; I end up dry and encumbered by an unwieldy umbrella;
or I end up wet. Second, there are *states*--things
outside the decision-maker's control which influence the outcome of
the decision. In the example, there are two states: either it is
raining, or it is not. Finally, there are *acts*--objects
of the decision-maker's instrumental preferences, and in some sense,
things that she can do. In the example, there are two acts: I may
either bring the umbrella; or leave it at home. Expected utility
theory provides a way of ranking the acts according to how
*choiceworthy* they are: the higher the expected utility, the
better it is to choose the act. (It is therefore best to choose the
act with the highest expected utility--or one of them, in the
event that several acts are tied.)




Following general convention, I will make the following assumptions
about the relationships between acts, states, and outcomes.


* States, acts, and outcomes are propositions, i.e., sets of
possibilities. There is a maximal set of possibilities, \(\Omega\), of
which each state, act, or outcome is a subset.
* The set of acts, the set of states, and the set of outcomes are
all partitions on \(\Omega\). In other words, acts and states are
individuated so that every possibility in \(\Omega\) is one where exactly
one state obtains, the agent performs exactly one act, and exactly one
outcome ensues.
* Acts and states are logically independent, so that no state rules
out the performance of any act.
* I will assume for the moment that, given a state of the world,
each act has exactly one possible outcome. (Section 1.1 briefly
discusses how one might weaken this assumption.)




So the example of the umbrella can be depicted in the following
matrix, where each column corresponds to a state of the world; each row
corresponds to an act; and each entry corresponds to the outcome that
results when the act is performed in the state of the world.




|  |  |
| --- | --- |
|  | **states** |
| *it rains* | *it does not rain* |
| **acts** | *take umbrella* | encumbered, dry | encumbered, dry |
| *leave umbrella* | wet | free, dry |




Having set up the basic framework, I can now rigorously define
expected utility. The expected utility of an act \(A\) (for
instance, taking my umbrella) depends on two features of the
problem:


* The value of each outcome, measured by a real number called a
*utility*.
* The probability of each outcome conditional on \(A\).




Given these three pieces of information, \(A\)'s expected
utility is defined as:



\[
EU(A) = \sum\_{o \in O} P\_{A}(o) U(o)
\]



where \(O\) is is the set of outcomes,
\(P\_{A}(o)\) is the probability of outcome \(o\) conditional on \(A\), and 
\(U(o)\) is the utility of \(o\).




The next two subsections will unpack the conditional probability
function \(P\_A\) and the utility function \(U\).


### 1.1 Conditional Probabilities




The term \(P\_{A}(o)\) represents the probability
of \(o\) given \(A\)--roughly, how likely it is that
outcome \(o\) will occur, on the supposition that the agent
chooses act \(A\). (For the axioms of probability, see the entry
on 
 interpretations of probability.)
 To understand what this means, we
must answer two questions. First, which interpretation of probability
is appropriate? And second, what does it mean to assign a probability
on the supposition that the agent chooses act \(A\)?




Expected utility theorists often interpret probability as measuring
individual degree of belief, so that a proposition \(E\) is likely
(for an agent) to the extent that that agent is confident of \(E\)
(see, for instance, Ramsey 1926, Savage 1972, Jeffrey 1983). But
nothing in the formalism of expected utility theory forces this
interpretation on us. We could instead interpret probabilities as
objective chances (as in von Neumann and Morgenstern 1944), or as the
degrees of belief that are warranted by the evidence, if we thought
these were a better guide to rational action. (See the entry on
 interpretations of probability
 for discussion of these and other options.)




What is it to have a probability on the supposition that the agent
chooses \(A\)? Here, there are two basic types of answer,
corresponding to evidential decision theory and causal decision
theory.




According to evidential decision theory, endorsed by Jeffrey (1983),
the relevant suppositional probability \(P\_{A}(o)\)
is the conditional probability 
\(P(o \mid A)\), 
defined as the ratio of two unconditional probabilities:
\(P(A \amp o) / P(A)\).




Against Jeffrey's definition of expected utility, Spohn (1977)
and Levi (1991) object that a decision-maker should not assign
probabilities to the very acts under deliberation: when freely deciding
whether to perform an act \(A\), you shouldn't take into
account your beliefs about whether you will perform \(A\). 
If Spohn and Levi are right, then Jeffrey's ratio is undefined
(since its denominator is undefined).




Nozick (1969) raises another objection: Jeffrey's definition gives
strange results in the *Newcomb Problem*. A predictor
hands you a closed box, containing either $0 or $1 million, and offers
you an open box, containing an additional $1,000. You can either
refuse the open box ("one-box") or take the open box
("two-box"). But there's a catch: the predictor
has predicted your choice beforehand, and all her predictions are 90%
accurate. In other words, the probability that you one-box, given
that she predicts you one-box, is 90%, and the probability that you
two-box, given that she predicts you two-box, is 90%. Finally,
the contents of the closed box depend on the prediction: if the
predictor thought you would two-box, she put nothing in the closed box,
while if she thought you would one-box, she put $1 million in the
closed box. The matrix for your decision looks like this:




|  |  |
| --- | --- |
|  | **states** |
| *$1 million in closed box* | *$0 in closed box* |
| **acts** | *one-box* | $1,000,000 | $0 |
| *two-box* | $1,001,000 | $1,000 |




Two-boxing *dominates* one-boxing: in every state, two-boxing
yields a better outcome. Yet on Jeffrey's definition of conditional
probability, one-boxing has a higher expected utility than
two-boxing. There is a high conditional probability of finding $1
million is in the closed box, given that you one-box, so one-boxing
has a high expected utility. Likewise, there is a high conditional
probability of finding nothing in the closed box, given that you
two-box, so two-boxing has a low expected utility.



Causal decision theory is an
alternative proposal that gets around these problems. It does not
require (but still permits) acts to have probabilities, and it
recommends two-boxing in the Newcomb problem.




Causal decision theory comes in many varieties, but I'll consider a
representative version proposed by Savage (1972), which calculates
\(P\_{A}(o)\) by summing the probabilities of states that, when
combined with the act \(A\), lead to the outcome \(o\). Let
\(f\_{A,s}(o)\) be a of outcomes, which maps \(o\) to 1 if \(o\)
results from performing \(A\) in state *s*, maps \(o\) to 0
otherwise. Then



\[
P\_{A}(o) = \sum\_{s\in S}P(s)f\_{A,s}(o)
\]



On Savage's proposal, two-boxing comes out with a higher
expected utility than one-boxing. This result holds no matter
which probabilities you assign to the states prior to your
decision. Let \(x\) be the probability you assign to the
state that the closed box contains $1 million. According to
Savage, the expected utilities of one-boxing and two-boxing,
respectively, are:



\[
x {\cdot} U({$1,000,000}) + (1 - x) {\cdot} U($0)
\]



and



\[
x {\cdot} U({$1,001,000}) + (1 - x) {\cdot} U({$1,000})
\]



As long as the larger monetary amounts are assigned strictly larger
utilities, the second sum (the utility of two-boxing) is guaranteed to
be larger than the first (the utility of one-boxing).





Savage assumes that each act and state are enough to uniquely
determine an outcome. But there are cases where this assumption
breaks down. Suppose you offer to sell me the following gamble:
you will toss a coin; if the coin lands heads, I win $100; and if the
coin lands tails, I lose $100. But I refuse the gamble, and the
coin is never tossed. There is no outcome that *would*
have resulted, had the coin been tossed--I might have won $100,
and I might have lost $100.




We can generalze Savage's proposal by letting \(f\_{A,s}\) be a
probability function that maps outcomes to real numbers in the \([0,
1]\) interval. Lewis (1981), Skyrms (1980), and Sobel (1994) equate
\(f\_{A,s}\) with the objective chance that \(o\) would be the outcome
if state \(s\) obtained and the agent chose action \(A\).




In some cases--most famously the Newcomb problem--the
Jeffrey definition and the Savage definition of expected utility come
apart. But whenever the following two conditions are satisfied,
they agree.


* Acts are
probabilistically independent of states. In formal terms, for all
acts \(A\) and states \(s\), 
\[
P(s) = P(s \mid A) = \frac{P(s \amp A)}{P(A)}.
\]
(This is the condition that is violated in the Newcomb problem.)
* For all outcomes
\(o\), acts \(A\), and states \(s\),
\(f\_{A,s}(o)\) is equal to the conditional
probability of \(o\) given \(A\) and \(s\); in formal
terms, 
\[
f\_{A,s}(o) = P(o \mid A \amp s) = \frac{P(o \amp A \amp s)}{P(A \amp s)}.\]

(The need for this condition arises when acts and states fail to
uniquely determine an outcome; see Lewis 1981.)


### 1.2 Outcome Utilities




The term \(U(o)\) represents the utility of the outcome
\(o\)--roughly, how valuable \(o\) is. Formally, \(U\) is a
function that assigns a real number to each of the outcomes. (The
units associated with \(U\) are typically called *utiles*,
so that if \(U(o) = 2\), we say that \(o\) is worth 2 utiles.) The greater
the utility, the more valuable the outcome.




What kind of value is measured in utiles? Utiles are typically
*not* taken to be units of currency, like dollars, pounds, or
yen. Bernoulli (1738) argued that money and other goods have
diminishing marginal utility: as an agent gets richer, every successive
dollar (or gold watch, or apple) is less valuable to her than the
last. He gives the following example: It makes rational sense for
a rich man, but not for a pauper, to pay 9,000 ducats in exchange for a
lottery ticket that yields a 50% chance at 20,000 ducats and a 50%
chance at nothing. Since the lottery gives the two men the same
chance at each monetary prize, the prizes must have different values
depending on whether the player is poor or rich.




Classic utilitarians such as Bentham (1789), Mill (1861), and Sidgwick
(1907) interpreted utility as a measure of pleasure or happiness. For
these authors, to say \(A\) has greater utility than \(B\) (for an
agent or a group of agents) is to say that \(A\) results in more
pleasure or happiness than \(B\) (for that agent or group of
agents).




One objection to this interpretation of utility is that there may
not be a single good (or indeed any good) which rationality requires us
to seek. But if we understand "utility" broadly
enough to include all potentially desirable ends--pleasure,
knowledge, friendship, health and so on--it's not clear that
there is a unique correct way to make the tradeoffs between different
goods so that each outcome receives a utility. There may be no
good answer to the question of whether the life of an ascetic monk
contains more or less good than the life of a happy libertine--but
assigning utilities to these options forces us to compare them.




Contemporary decision theorists typically interpret utility as a
measure of preference, so that to say that \(A\) has greater
utility than \(B\) (for an agent) is simply to say that the agent
prefers \(A\) to \(B\). It is crucial to this approach
that preferences hold not just between outcomes (such as amounts of
pleasure, or combinations of pleasure and knowledge), but also between
uncertain prospects (such as a lottery that pays $1 million dollars if
a particular coin lands heads, and results in an hour of painful
electric shocks if the coin lands tails). Section 2 of this
article addresses the formal relationship between preference and choice
in detail.




Expected utility theory does *not* require that preferences
be selfish or self-interested. Someone can prefer giving money to
charity over spending the money on lavish dinners, or prefer
sacrificing his own life over allowing his child to die. Sen
(1977) suggests that each person's psychology is best represented
using three rankings: one representing the person's narrow
self-interest, a second representing the person's self-interest
construed more broadly to account for feelings of sympathy (e.g.,
suffering when watching another person suffer), and a third
representing the person's commitments, which may require her to
act against her self-interest broadly construed.




Broome (1991, Ch. 6) interprets utilities as measuring comparisons of
objective betterness and worseness, rather than personal preferences:
to say that \(A\) has a greater utility than \(B\) is to say that
\(A\) is objectively better than \(B\), or that a rational person
would prefer \(A\) to \(B\). Just as there is nothing in the formalism
of probability theory that requires us to use subjective rather than
objective probabilities, so there is nothing in the formalism of
expected utility theory that requires us to use subjective rather than
objective values.




Those who interpret utilities in terms of personal preference face a
special challenge: the so-called *problem of interpersonal utility
comparisons*. When making decisions about how to distribute
shared resources, we often want to know if our acts would make Alice
better off than Bob--and if so, how much better off. But if
utility is a measure of individual preference, there is no clear,
meaningful way of making these comparisons. Alice's
utilities are constituted by Alice's preferences, Bob's
utilities are constituted by Bob's preferences, and there are no
preferences spanning Alice and Bob. We can't assume that
Alice's utility 10 is equivalent to Bob's utility 10, any
more than we can assume that getting an A grade in differential
equations is equivalent to getting an A grade in basket weaving.




Now is a good time to consider which features of the utility
function carry meaningful information. Comparisons are
informative: if \(U(o\_1) \gt U(o\_2)\) (for a person), then
\(o\_1\) is better than (or preferred to)
\(o\_2\). But it is not *only* comparisons
that are informative--the utility function must carry other
information, if expected utility theory is to give meaningful
results.




To see why, consider the umbrella example again. This time,
I've filled in a probability for each state, and a utility for each
outcome.




|  |  |
| --- | --- |
|  | **states** |
| *it rains* \((P = 0.6)\) | *it does not rain* \((P = 0.4)\) |
| **acts** | *take umbrella* | encumbered, dry \((U = 5)\) | encumbered, dry \((U = 5)\) |
| *leave umbrella* | wet \((U = 0)\) | free, dry \((U =10)\) |




The expected utility of taking the umbrella is



\[
\begin{align}
EU(\take)&= P\_{\take}(\encumbered, \dry) \cdot 5 \\
 & \quad + P\_{\take}(\wet) \cdot 0 \\
 & \quad + P\_{\take}(\free, dry) \cdot 10 \\
 &=5
\end{align}
\]



while the expected utility of leaving the umbrella is



\[
\begin{align}
EU(\leave)&= P\_{\leave}(\encumbered, \dry) \cdot 5 \\
 & \quad + P\_{\leave}(\wet) \cdot 0 \\
 & \quad + P\_{\leave}(\free, dry) \cdot 10 \\
 &=4
\end{align}
\]



Since \(EU(\take) \gt EU(\leave)\), expected utility theory tells me
that taking the umbrella is better than leaving it.




But now, suppose we change the utilities of the outcomes: instead of
using \(U\), we use \(U'\).




|  |  |
| --- | --- |
|  | **states** |
| *it rains* \((P=0.6)\) | *it does not rain* \((P=0.4)\) |
| **acts** | *take umbrella* | encumbered, dry \((U'=4)\) | encumbered, dry \((U'=4)\) |
| *leave umbrella* | wet \((U'=2)\) | free, dry \((U'=8)\) |




The new expected utility of taking the umbrella is



\[
\begin{align}
EU'(\take)&= P\_{\take}(\encumbered, \dry) \cdot 4 \\
 & \quad + P\_{\take}(\wet) \cdot 2 \\
 & \quad + P\_{\take}(\free, dry) \cdot 8 \\
 &= 4
\end{align}
\]



while the new expected utility of leaving the umbrella is



\[
\begin{align}
EU'(\leave)&= P\_{\leave}(\encumbered, \dry) \cdot 4 \\
 & \quad + P\_{\leave}(\wet) \cdot 2 \\
 & \quad + P\_{\leave}(\free, dry) \cdot 8 \\
 &= 4.4
\end{align}
\]



Since \(EU'(\take) \lt EU'(\leave)\), expected utility theory tells me
that leaving the umbrella is better than taking it.




The utility functions \(U\) and \(U'\) rank the outcomes
in exactly the same way: *free, dry* is best;
*encumbered, dry* ranks in the middle; and *wet*
is worst. Yet expected utility theory gives different advice in
the two versions of the problem. So there must be some
substantive difference between preferences appropriately described by
\(U\), and preferences appropriately described by
\(U'\). Otherwise, expected utility theory is fickle, and
liable to change its advice when fed different descriptions of the same
problem.




When do two utility functions represent the same basic state of
affairs? Measurement theory answers the question by characterizing the
*allowable transformations* of a utility function--ways of
changing it that leave all of its meaningful features intact. If we
characterize the allowable transformations of a utility function, we
have thereby specified which of its features are meaningful.




Defenders of expected utility theory typically require that utility
be measured by a *linear scale*, where the
allowable transformations are all and only the *positive linear*
transformations, i.e., functions \(f\) of the form



\[
f(U(o))=x {\cdot} U(o)+y
\]



for real numbers \(x \gt 0\) and \(y\).




Positive linear transformations of outcome utilities will never
affect the verdicts of expected utility theory: if \(A\) has
greater expected utility than \(B\) where utility is measured by
function \(U\), then \(A\) will also have greater expected
utility than \(B\) where utility is measured by any positive
linear transformation of \(U\). 


## 2. Arguments for Expected Utility Theory




Why choose acts that maximize expected utility? One possible
answer is that expected utility theory is rational bedrock--that
means-end rationality essentially involves maximizing expected
utility. For those who find this answer unsatisfying, however,
there are two further sources of justification. First, there are
long-run arguments, which rely on evidence that expected-utility
maximization is a profitable policy in the long term. 
Second, there are arguments based on representation theorems, which
suggest that certain rational constraints on preference entail that all
rational agents maximize expected utility.


### 2.1 Long-Run Arguments




One reason for maximizing expected utility is that it makes for good
policy in the long run. Feller (1968) gives a version of this
argument. He relies on two mathematical facts about
probabilities: the *strong* and *weak* *laws of large
numbers*. Both these facts concern sequences of independent,
identically distributed trials--the sort of setup that results
from repeatedly betting the same way on a sequence of roulette spins or
craps games. Both the weak and strong laws of large numbers
say, roughly, that over the long run, the average amount of utility
gained per trial is overwhelmingly likely to be close to the expected
value of an individual trial.




The weak law of large numbers states that where each trial has an
expected value of \(\mu\), for any arbitrarily small real numbers
\(\epsilon \gt 0\) and \(\delta \gt 0\), there is some finite number of
trials \(n\), such that for all \(m\) greater than or equal
to \(n\), with probability at least \(1-\delta\), the gambler's
average gains for the first \(m\) trials will fall within
\(\epsilon\) of \(\mu\). In other words, in a long run of similar
gamble, the average gain per trial is highly likely to become
arbitrarily close to the gamble's expected value within a finite
amount of time. So in the finite long run, the average value
associated with a gamble is overwhelmingly likely to be close to its
expected value.




The strong law of large numbers states that where each trial has an
expected value of \(\mu\), with probability 1, for any arbitrarily small real number \(\epsilon \gt 0\),as the number of trials increases, the gambler's average winnings per trial will fall within \(\epsilon\) of
\(\mu\). In other words, as the number of repetitions of a
gamble approaches infinity, the average gain per trial will become
arbitrarily close to the gamble's expected value with probability
1. So in the long run, the average value associated with a gamble is
virtually certain to equal its expected value.




There are several objections to these long run arguments. 
First, many decisions *cannot* be repeated over indefinitely
many similar trials. Decisions about which career to pursue, whom
to marry, and where to live, for instance, are made at best a small
finite number of times. Furthermore, where these decisions are
made more than once, different trials involve different possible
outcomes, with different probabilities. It is not clear why
long-run considerations about repeated gambles should bear on these
single-case choices.




Second, the argument relies on two independence assumptions, one or
both of which may fail. One assumption holds that the
probabilities of the different trials are independent. This is
true of casino gambles, but not true of other choices where we wish to
use decision theory--e.g., choices about medical treatment. 
My remaining sick after one course of antibiotics makes it more likely
I will remain sick after the next course, since it increases the chance
that antibiotic-resistant bacteria will spread through my body. The
argument also requires that the utilities of different trials be
independent, so that winning a prize on one trial makes the same
contribution to the decision-maker's overall utility no matter
what she wins on other trials. But this assumption is violated in
many real-world cases. Due to the diminishing marginal utility of
money, winning $10 million on ten games of roulette is not worth ten times as
much as winning $1 million on one game of roulette.




A third problem is that the strong and weak laws of large numbers are
modally weak. Neither law entails that if a gamble were repeated
indefinitely (under the appropriate assumptions), the average utility
gain per trial *would* be close to the game's expected
utility. They establish only that the average utility gain per trial
would *with high probability* be close to the game's expected
utility. But high probability--even probability 1--is not
certainty. (Standard probability theory rejects *Cournot's
Principle*, which says events with low or zero probability will
not happen. But see Shafer (2005) for a defense of Cournot's
Principle.) For any sequence of independent, identically distributed
trials, it is possible for the average utility payoff per trial to
diverge arbitrarily far from the expected utility of an individual
trial.


### 2.2 Representation Theorems


 

A second type of argument for expected utility theory relies on
so-called representation theorems. We follow Zynda's (2000) formulation
of this argument--slightly modified to reflect the role of
utilities as well as probabilities. The argument has three
premises:



*The Rationality Condition.*  
 The axioms of expected
utility theory are the axioms of rational preference.



*Representability.*  
 If a person's preferences obey the
axioms of expected utility theory, then she can be represented as
having degrees of belief that obey the laws of the probability
calculus [and a utility function such that she prefers acts with
higher expected utility].



*The Reality Condition.*  
 If a person can be
represented as having degrees of belief that obey the probability
calculus [and a utility function such that she prefers acts with higher
expected utility], then the person really has degrees of belief that
obey the laws of the probability calculus [and really does prefer acts
with higher expected utility].




These premises entail the following conclusion.



> 
> 
> If a person [fails to prefer acts with higher expected utility], then
> that person violates at least one of the axioms of rational
> preference.




If the premises are true, the argument shows that there is something
wrong with people whose preferences are at odds with expected utility
theory--they violate the axioms of rational preference. Let
us consider each of the premises in greater detail, beginning with the
key premise, Representability.




A probability function and a utility function together
*represent* a set of preferences just in case the following
formula holds for all values of \(A\) and \(B\) in the domain
of the preference relation



\[
EU(A) \gt EU(B) \text{ if and only if } A \text{ is preferred to } B.
\]



Mathematical proofs of Representability are called
*representation theorems*. Section 2.1 surveys three of
the most influential representation theorems, each of which relies on a
different set of axioms.




No matter which set of axioms we use, the Rationality Condition is
controversial. In some cases, preferences that seem rationally
permissible--perhaps even rationally required--violate the
axioms of expected utility theory. Section 3 discusses such cases
in detail.




The Reality Condition is also controversial. Hampton (1994), Zynda
(2000), and Meacham and Weisberg (2011) all point out that to be
*representable* using a probability and utility function is not
to *have* a probability and utility function. After all,
an agent who can be represented as an expected utility maximizer with
degrees of belief that obey the probability calculus, can also be
represented as someone who fails to maximize expected utility with
degrees of belief that *violate* the probability calculus. 
Why think the expected utility representation is the right one?




There are several options. Perhaps the defender of
representation theorems can stipulate that what it is to have
particular degrees of belief and utilities is just to have the
corresponding preferences. The main challenge for defenders of
this response is to explain why representations in terms of expected
utility are explanatorily useful, and why they are better than
alternative representations. Or perhaps probabilities and utilities are
a good cleaned-up theoretical substitutes for our folk notions of
belief and desire--precise scientific substitutes for our folk
concepts. Meacham and Weisberg challenge this response, arguing
that probabilities and utilities are poor stand-ins for our folk
notions. A third possibility, suggested by Zynda, is that facts
about degrees of belief are made true independently of the
agent's preferences, and provide a principled way to restrict the
range of acceptable representations. The challenge for defenders
of this type of response is to specify what these additional facts
are.




I now turn to consider three influential representation
theorems. These representation theorems differ from each other in
three of philosophically significant ways. 




First, different representation theorems disagree about the objects
of preference and utility. Are they repeatable? Must they be wholly within the agent's control




Second, representation theorems differ in their treatment of
probability. They disagree about which entities have
probabilities, and about whether the same objects can have both
probabilities and utilities.




Third, while every representation theorem proves that for a suitable
preference ordering, there *exist* a probability and utility
function representing the preference ordering, they differ how
*unique* this probability and utility function are. In
other words, they differ as to which transformations of the probability
and utility functions are allowable.


#### 2.2.1 Ramsey




The idea of a representation theorem for expected utility dates back
to Ramsey (1926). (His sketch of a representation theorem is
subsequently filled in by Bradley (2004) and Elliott (2017).) Ramsey
assumes that preferences are defined over a domain of gambles, which
yield one prize on the condition that a proposition \(P\) is true, and
a different prize on the condition that \(P\) is false. (Examples of
gambles: you receive a onesie if you're having a baby and a bottle of
scotch otherwise; you receive twenty dollars if Bojack wins the
Kentucky Derby and lose a dollar otherwise.) 




Ramsey calls a proposition ethically neutral when "two possible
worlds differing only in regard to [its truth] are always of equal
value". For an ethically neutral proposition, probability 1/2
can be defined in terms of preference: such a proposition has
probability 1/2 just in case you are indifferent as to which side of
it you bet on. (So if *Bojack wins the Kentucky Derby* is an
ethically neutral proposition, it has probability 1/2 just in case you
are indifferent between winning twenty dollars if it's true and losing
a dollar otherwise, and winning twenty dollars if it's false and
losing a dollar otherwise.) 




By positing an ethically neutral proposition with probability 1/2,
together with a rich space of prizes, Ramsey defines numerical
utilities for prizes. (The rough idea is that if you are indifferent
between receiving a middling prize \(m\) for certain, and a gamble
that yields a better prize \(b\) if the ethically neutral proposition
is true and a worse prize \(w\) if it is falls, then the utility of
\(m\) is halfway between the utilities of \(b\) and \(w\).) Using
these numerical utilities, he then exploits the definition of expected
utility to define probabilities for all other propositions.



The rough idea is to exploit the richness of the space of prizes,
which ensures that for any gamble \(g\) that yields better prize \(b\)
if \(E\) is true and worse prize \(w\) if \(E\) is false, the agent is
indifferent between \(g\) and some middling prize \(m\). This means
that \(EU(g) = EU(m)\). Using some algebra, plus the fact that
\(EU(g) = P(E)U(b) + (1-P(E))U(w)\), Ramsey shows that



\[
P(E) = \frac{(1 - U(m)}{(U(b) - U(w))}
\]

#### 2.2.2 Von Neumann and Morgenstern




Von Neumann and Morgenstern (1944) claim that preferences are defined
over a domain of *lotteries*. Some of these lotteries are
*constant*, and yield a single prize with certainty. (Prizes
might include a banana, a million dollars, a million dollars' worth of
debt, death, or a new car.) Lotteries can also have other lotteries
as prizes, so that one can have a lottery with a 40% chance of
yielding a banana, and a 60% chance of yielding a 50-50 gamble between
a million dollars and death.) The domain of lotteries is closed under
a mixing operation, so that if \(L\) and \(L'\) are lotteries and \(x\) is a
real number in the \([0, 1]\) interval, then there is a lottery \(x L +
(1-x) L'\) that yields \(L\) with probability \(x\) and \(L'\) with
probability \(1-x\). They show that every preference relation obeying
certain axioms can be represented by the probabilities used to define
the lotteries, together with a utility function which is unique up to
positive linear transformation.


#### 2.2.3 Savage




Instead of taking probabilities for granted, as von Neumann and
Morgenstern do, Savage (1972) defines them in terms of preferences
over acts. Savage posits three separate domains. Probability attaches
to *events*, which we can think of as disjunctions of states,
while utility and intrinsic preference attach to *outcomes*.
Expected utility and non-intrinsic preference attach
to *acts*. 




For Savage, acts, states, and outcomes must satisfy certain
constraints. Acts must be wholly under the agent's control (so
publishing my paper in *Mind* is not an act, since it depends
partly on the editor's decision, which I do not control). Outcomes
must have the same utility regardless of which state obtains (so "I
win a fancy car" is not an outcome, since the utility of the fancy car
will be greater in states where the person I most want to impress
wishes I had a fancy car, and less in states where I lose my driver's
license). No state can rule out the performance of any act, and an act
and a state together must determine an outcome with certainty. For
each outcome \(o\), there is a constant act which yields \(o\) in
every state. (Thus, if world peace is an outcome, there is an act that
results in world peace, no matter what the state of the world.)
Finally, he assumes for any two acts \(A\) and \(B\) and any event
\(E\), there is a mixed act \(A\_E \amp B\_{\sim E}\) that yields the
same outcome as \(A\) if \(E\) is true, and the same outcome as \(B\)
otherwise. (Thus, if world peace and the end of the world are both
outcomes, then there is a mixed act that results in world peace if a
certain coin lands heads, and the end of the world otherwise.)




Savage postulates a preference relation over acts, and gives axioms
governing that preference relation. He then defines subjective
probabilities, or degrees of belief, in terms of preferences. The key
move is to define an "at least as likely as" relation
between events; I paraphrase here. 



> 
> 
> Suppose \(A\) and \(B\) are constant acts such that \(A\) is preferred
> to \(B\). Then \(E\) is at least as likely as \(F\) just in case the
> agent either prefers \(A\_E \amp B\_{\sim E}\) (the act that yields
> \(A\) if \(E\) obtains, and \(B\) otherwise) to \(A\_F \amp B\_{\sim
> F}\) (the act that yields \(A\) if \(F\) obtains, and \(B\)
> otherwise), or else is indifferent between \(A\_E \amp B\_{\sim E}\) and
> \(A\_F \amp B\_{\sim F}\).
> 




The thought behind the definition is that the agent considers \(E\) at
least as likely as \(F\) just in case she would *not* rather
bet on \(F\) than on \(E\)).




Savage then gives axioms constraining rational preference, and shows
that any set of preferences satisfying those axioms yields an
"at least as likely" relation that can be uniquely
represented by a probability function. In other words, there is one
and only one probability function \(P\) such that for all \(E\) and \(F\),
\(P(E) \ge P(F)\) if and only if \(E\) is at least as likely as \(F\). Every
preference relation obeying Savage's axioms is represented by this
probability function \(P\), together with a utility function which is
unique up to positive linear transformation.




Savage's representation theorem gives strong results: starting
with a preference ordering alone, we can find a single probability
function, and a narrow class of utility functions, which represent that
preference ordering. The downside, however, is that Savage has to
build implausibly strong assumptions about the domain of acts.




Luce and Suppes (1965) point out that Savage's constant acts
are implausible. (Recall that constant acts yield the same
outcome and the same amount of value in every state.) Take some
very good outcome--total bliss for everyone. Is there really
a constant act that has this outcome in every possible state, including
states where the human race is wiped out by a meteor? 
Savage's reliance on a rich space of mixed acts is also
problematic. Savage has had to assume that any two outcomes and
any event, there is a mixed act that yields the first outcome if the
event occurs, and the second outcome otherwise? Is there really an act
that yields total bliss if everyone is killed by an
antibiotic-resistant plague, and total misery otherwise? Luce and
Krantz (1971) suggest ways of reformulating Savage's
representation theorem that weaken these assumptions, but Joyce (1999)
argues that even on the weakened assumptions, the domain of acts
remains implausibly rich.


#### 2.2.4 Bolker and Jeffrey




Bolker (1966) proves a general representation theorem about
mathematical expectations, which Jeffrey (1983) uses as the basis for a
philosophical account of expected utility theory. Bolker's
theorem assumes a single domain of propositions, which are objects of
preference, utility, and probability alike. Thus, the proposition that
it will rain today has a utility, as well as a probability. 
Jeffrey interprets this utility as the proposition's *news
value*--a measure of how happy or disappointed I would be to
learn that the proposition was true. By convention, he sets the
value of the necessary proposition at 0--the necessary proposition
is no news at all! Likewise, the proposition that I take my
umbrella to work, which is an act, has a probability as well as a
utility. Jeffrey interprets this to mean that I have degrees of
belief about what I will do.




Bolker gives axioms constraining preference, and shows that any
preferences satisfying his axioms can be represented by a probability
measure \(P\) and a utility measure \(U\). However,
Bolker's axioms do not ensure that \(P\) is unique, or that
\(U\) is unique up to positive linear transformation. Nor do
they allow us to define comparative probability in terms of
preference. Instead, where \(P\) and \(U\) jointly
represent a preference ordering, Bolker shows that the pair
\(\langle P, U \rangle\) is unique up to a fractional linear
transformation. 




In technical terms, where \(U\) is a utility function
normalized so that \(U(\Omega) = 0\), \(inf\) is the
greatest lower bound of the values assigned by \(U\), \(sup\)
is the least upper bound of the values assigned by by \(U\), and
\(\lambda\) is a parameter falling between \(-1/inf\) and
\(-1/sup\), the fractional linear transformation
\(\langle P\_{\lambda}, U\_{\lambda} \rangle\) of
\(\langle P, U \rangle\) corresponding to \(\lambda\) is given
by:



\[
\begin{align}
 P\_{\lambda} &= P(x)(1 + \lambda U(x)) \\
 U\_{\lambda} &= U(x)((1+\lambda)/(1 + \lambda U(x))
\end{align}
\]



Notice that fractional linear transformations of a
probability-utility pair can disagree with the original pair about
which propositions are likelier than which others.




Joyce (1999) shows that with additional resources, Bolker's
theorem can be modified to pin down a unique \(P\), and a
\(U\) that is unique up to positive linear transformation. 
We need only supplement the preference ordering with a primitive
"more likely than" relation, governed by its own set of
axioms, and linked to belief by several additional axioms. Joyce
modifies Bolker's result to show that given these additional
axioms, the "more likely than" relation is represented by a
unique \(P\), and the preference ordering is represented by
\(P\) together with a utility function that is unique up to
positive linear transformation.


#### 2.2.5 Summary




Together, these four representation theorems above can be summed up
in the following table.




|  |  |  |  |  |
| --- | --- | --- | --- | --- |
| **Theorem** | **Objects ofpreference** | **Order ofconstruction** | **Allowabletransformations:probability** | **Allowabletransformations: utility** |
| Ramsey | gambles | preference - utility - probability | identity | positive linear |
| von Neumann/Morgenstern | lotteries | (preference & probability) - utility | N/A | positive linear |
| Savage | acts | preference - probability - utility | identity | positive linear |
| Jeffrey/Bolker | propositions | preference - (probability & utility) | -- fractional linear -- |




Notice that the order of construction differs between theorems: Ramsey
constructs a representation of probability using utility, while von
Neumann and Morgenstern begin with probabilities and construct a
representation of utility. Thus, although the arrows represent a
mathematical relationship of representation, they cannot represent a
metaphysical relationship of grounding. The Reality Condition needs to
be justified independently of any representation theorem.




Suitably structured ordinal probabilities (the relations picked out
by "at least as likely as", "more likely than",
and "equally likely") stand in one-to-one correspondence
with the cardinal probability functions. Finally, the grey line
from preferences to ordinal probabilities indicates that every
probability function satisfying Savage's axioms is represented by
a unique cardinal probability--but this result does not hold for
Jeffrey's axioms.




Notice that it is often possible to follow the arrows in
circles--from preference to ordinal probability, from ordinal
probability to cardinal probability, from cardinal probability and
preference to expected utility, and from expected utility back to
preference. Thus, although the arrows represent a mathematical
relationship of representation, they do not represent a metaphysical
relationship of grounding. This fact drives home the importance

of independently justifying the Reality Condition--representation
theorems cannot justify expected utility theory without additional
assumptions.


## 3. Objections to Expected Utility Theory


### 3.1 Maximizing Expected Utility is Impossible




Ought implies can, but is it humanly possible to maximize expected
utility? March and Simon (1958) point out that in order to compute
expected utilities, an agent needs a dauntingly complex understanding
of the available acts, the possible outcomes, and the values of those
outcomes, and that choosing the best act is much more demanding than
choosing an act that is merely good enough. Similar points appear in
Lindblom (1959), Feldman (2006), and Smith (2010).




McGee (1991) argues that maximizing expected utility is not
mathematically possible even for an ideal computer with limitless
memory. In order to maximize expected utility, we would have to
accept any bet we were offered on the truths of arithmetic, and reject
any bet we were offered on false sentences in the language of
arithmetic. But arithmetic is undecidable, so no Turing machine can
determine whether a given arithmetical sentence is true or false.




One response to these difficulties is the
 bounded rationality approach, which
aims to replace expected utility theory with some more tractable
rules. Another is to argue that the demands of expected utility
theory are more tractable than they appear (Burch-Brown 2014; see also
Greaves 2016), or that the relevant "ought implies can"
principle is false (Srinivasan 2015).


### 3.2 Maximizing Expected Utility is Irrational




A variety of authors have given examples in which expected utility
theory seems to give the wrong prescriptions. Sections 3.2.1 and 3.2.2 discuss
examples where rationality seems to permit preferences inconsistent
with expected utility theory. These examples suggest that
maximizing expected utility is not *necessary* for
rationality. Section 3.2.3 discusses examples where expected
utility theory permits preferences that seem irrational. These
examples suggest that maximizing expected utility is not
*sufficient* for rationality. Section 3.2.4 discusses an
example where expected utility theory requires preferences that seem
rationally forbidden--a challenge to both the necessity and the
sufficiency of expected utility for rationality.


#### 3.2.1 Counterexamples Involving Transitivity and Completeness




Expected utility theory implies that the structure of preferences
mirrors the structure of the greater-than relation between real
numbers. Thus, according to expected utility theory, preferences
must be *transitive*: If \(A\) is preferred to \(B\)
(so that \(U(A) \gt U(B)\)), and
\(B\) is preferred to \(C\) (so that \(U(B)
\gt U(C)\)), then \(A\) must be preferred to
\(C\) (since it must be that \(U(A) \gt U(C)\)).
Likewise, preferences must be
*complete*: for any two options, either one must be preferred to
the other, or the agent must be indifferent between them (since of
their two utilities, either one must be greater or the two must be
equal). But there are cases where rationality seems to permit (or
perhaps even require) failures of transitivity and failures of
completeness.




An example of preferences that are not transitive, but nonetheless
seem rationally permissible, is Quinn's puzzle of the
self-torturer (1990). The self-torturer is hooked up to a machine
with a dial with settings labeled 0 to 1,000, where setting 0 does
nothing, and each successive setting delivers a slightly more powerful
electric shock. Setting 0 is painless, while setting 1,000 causes
excruciating agony, but the difference between any two adjacent
settings is so small as to be imperceptible. The dial is fitted
with a ratchet, so that it can be turned up but never down. 
Suppose that at each setting, the self-torturer is offered $10,000 to
move up to the next, so that for tolerating setting \(n\), he
receives a payoff of \(n {\cdot} {$10,000}\). It is permissible for the
self-torturer to prefer setting \(n+1\) to setting \(n\) for
each \(n\) between 0 and 999 (since the difference in pain is
imperceptible, while the difference in monetary payoffs is
significant), but *not* to prefer setting 1,000 to setting 0
(since the pain of setting 1,000 may be so unbearable that no amount of
money will make up for it.




It also seems rationally permissible to have incomplete
preferences. For some pairs of actions, an agent may have no
considered view about which she prefers. Consider Jane, an
electrician who has never given much thought to becoming a professional
singer or a professional astronaut. (Perhaps both of these
options are infeasible, or perhaps she considers both of them much
worse than her steady job as an electrician). It is false that
Jane prefers becoming a singer to becoming an astronaut, and it is
false that she prefers becoming an astronaut to becoming a
singer. But it is also false that she is indifferent between
becoming a singer and becoming an astronaut. She prefers becoming
a singer and receiving a $100 bonus to becoming a singer, and if she
were indifferent between becoming a singer and becoming an astronaut,
she would be rationally compelled to prefer being a singer and
receiving a $100 bonus to becoming an astronaut.




There is one key difference between the two examples considered
above. Jane's preferences can be *extended*, by
adding new preferences without removing any of the ones she has, in a
way that lets us represent her as an expected utility maximizer. 
On the other hand, there is no way of extended the
self-torturer's preferences so that he can be represented as an
expected utility maximizer. Some of his preferences would have to
be altered. One popular response to incomplete preferences is to
claim that, while rational preferences need not satisfy the axioms of a
given representation theorem (see section 2.2), it must be possible to
extend them so that they satisfy the axioms. From this weaker
requirement on preferences--that they be extendible to a
preference ordering that satisfies the relevant axioms--one can
prove the *existence* halves of the relevant representation
theorems. However, one can no longer establish that each
preference ordering has a representation which is *unique* up to
allowable transformations.




No such response is available in the case of the self-torturer,
whose preferences cannot be extended to satisfy the axioms of expected
utility theory. See the entry on
 preferences
 for a more extended discussion of the self-torturer case.


#### 3.2.2 Counterexamples Involving Independence




Allais (1953) and Ellsberg (1961) propose examples of preferences that
cannot be represented by an expected utility function, but that
nonetheless seem rational. Both examples involve violations of
Savage's Independence axiom:



> 
> 
> *Independence*. Suppose that \(A\) and \(A^\*\) are two
> acts that produce the same outcomes in the event that \(E\) is
> false. Then, for any act \(B\), one must have
> 
> 
> * \(A\) is preferred
> to \(A^\*\) if and only if \(A\_E \amp B\_{\sim E}\)
> is preferred to \(A^\*\_E \amp B\_{\sim E}\)
> * The agent is
> indifferent between \(A\) and \(A^\*\) if and only if she is
> indifferent between 
> \(A\_E \amp B\_{\sim E}\) and \(A^\*\_E \amp B\_{\sim E}\)
> 
> 
> 




In other words, if two acts have the same consequences whenever
\(E\) is false, then the agent's preferences between those
two acts should depend only on their consequences when \(E\) is
true. On Savage's definition of expected utility, expected
utility theory entails Independence. And on Jeffrey's
definition, expected utility theory entails Independence in the
presence of the assumption that the states are probabilistically
independent of the acts.




The first counterexample, the Allais Paradox, involves two separate
decision problems in which a ticket with a number between 1 and 100 is
drawn at random. In the first problem, the agent must choose
between these two lotteries:


* *Lottery* \(A\)
* *
$100 million with certainty


* *Lottery* \(B\)
* *
$500 million if one of tickets 1-10 is drawn
* *
$100 million if one of tickets 12-100 is drawn
* *
Nothing if ticket 11 is drawn



In the second decision problem, the agent must choose between these
two lotteries:


* *Lottery* \(C\)
* *
$100 million if one of tickets 1-11 is drawn
* *
Nothing otherwise


* *Lottery* \(D\)
* *
$500 million if one of tickets 1-10 is drawn
* *
Nothing otherwise




It seems reasonable to prefer \(A\) (which offers a sure $100
million) to \(B\) (where the added 10% chance at $500 million is
more than offset by the risk of getting nothing). It also seems
reasonable to prefer \(D\) (an 10% chance at a $500 million prize)
to \(C\) (a slightly larger 11% chance at a much smaller $100
million prize). But together, these preferences (call them the
*Allais preferences*) violate Independence. Lotteries
\(A\) and \(C\) yield the same $100 million prize for
tickets 12-100. They can be converted into lotteries \(B\)
and \(D\) by replacing this $100 million prize with $0.




Because they violate Independence, the Allais preferences are
incompatible with expected utility theory. This incompatibility does
not require any assumptions about the relative utilities of the $0,
the $100 million, and the $500 million. Where $500 million has
utility \(x\), $100 million has utility \(y\), and $0 has utility \(z\),
the expected utilities of the lotteries are as follows.



\[
\begin{align}
 EU(A) &= 0.11y + 0.89y \\
 EU(B) &= 0.10x + 0.01z + 0.89y \\
 EU(C) &= 0.11y + 0.89z \\
 EU(D) &= 0.10x + 0.01z + 0.89z
\end{align}
\]



It is easy to see that the condition under which \(EU(A) \gt EU(B)\) is
exactly the same as the condition under which \(EU(C) \gt EU(D)\): both
inequalities obtain just in case \(0.11y \gt 0.10x +
0.01z\) 




The Ellsberg Paradox also involves two decision problems that generate
a violation of the sure-thing principle. In each of them, a ball is
drawn from an urn containing 30 red balls, and 60 balls that are
either white or yellow in unknown proportions. In the first decision
problem, the agent must choose between the following lotteries:


* *Lottery* \(R\)
* *
Win $100 if a red ball is drawn
* *
Lose $100 otherwise


* *Lottery* \(W\)
* *
Win $100 if a white ball is drawn
* *
Lose $100 otherwise



In the second decision problem, the agent must choose between the
following lotteries:


* *Lottery* \(RY\)
* *
Win $100 if a red or yellow ball is drawn
* *
Lose $100 otherwise


* *Lottery* \(WY\)
* *
Win $100 if a white or yellow ball is drawn
* *
Lose $100 otherwise




It seems reasonable to prefer \(R\) to \(W\), but at the same time prefer
\(WY\) to \(RY\). (Call this combination of preferences the *Ellsberg
preferences*.) Like the Allais preferences, the Ellsberg
preferences violate Independence. Lotteries
\(W\) and \(R\) yield a $100 loss if a yellow ball is
drawn; they can be converted to lotteries \(RY\) and \(WY\)
simply by replacing this $100 loss with a sure $100 gain.




Because they violate independence, the Ellsberg preferences are
incompatible with expected utility theory. Again, this incompatibility
does not require any assumptions about the relative utilities of
winning $100 and losing $100. Nor do we need any assumptions about where
between 0 and 1/3 the probability of drawing a yellow ball
falls. Where winning $100 has utility \(w\) and losing $100 has
utility \(l\),



\[
\begin{align}
 EU(R) &= \tfrac{1}{3} w + P(W)l + P(Y)l \\
 EU(W) &= \tfrac{1}{3} l + P(W)w + P(Y)l \\
 EU(RY)&= \tfrac{1}{3} w + P(W)l + P(Y)w \\
 EU(WY)&= \tfrac{1}{3} l + P(W)w + P(Y)w
\end{align}
\]



It is easy to see that the condition in which 
\(EU(R) \gt EU(W)\) is exactly the same as the condition
under which 
\(EU(RY) \gt EU(WY)\):
both inequalities obtain just in case 
\(1/3\,w + P(W)l \gt 1/3\,l + P(W)w\).



 

There are three notable responses to the Allais and Ellsberg
paradoxes. First, one might follow Savage (101 ff) and Raiffa (1968,
80-86), and defend expected utility theory on the grounds that
the Allais and Ellsberg preferences are irrational. 


 

Second, one might follow Buchak (2013) and claim that that the
Allais and Ellsberg preferences are rationally permissible, so that
expected utility theory fails as a normative theory of
rationality. Buchak develops an a more permissive theory of
rationality, with an extra parameter representing the decision-maker's
attitude toward risk. This risk parameter interacts with the utilities
of outcomes and their conditional probabilities on acts to determine
the values of acts. One setting of the risk parameter yields expected
utility theory as a special case, but other, "risk-averse"
settings rationalise the Allais preferences.




Third, one might follow Loomes and Sugden (1986), Weirich (1986),
and Pope (1995) and argue that the outcomes in the Allais and Ellsberg
paradoxes can be re-described to accommodate the Allais and Ellsberg
preferences. The alleged conflict between the Allais and Ellsberg
preferences on the one hand, and expected utility theory on the other,
was based on the assumption that a given sum of money has the same
utility no matter how it is obtained. Some authors challenge this
assumption. Loomes and Sugden suggest that in addition to monetary
amounts, the outcomes of the gambles include feelings of disappointment
(or elation) at getting less (or more) than expected. Pope
distinguishes "post-outcome" feelings of elation or
disappointment from "pre-outcome" feelings of excitement,
fear, boredom, or safety, and points out that both may affect outcome
utilities. Weirich suggests that the value of a monetary sum
depends partly on the risks that went into obtaining it, irrespective
of the gambler's feelings, so that (for instance) $100 million as
the result of a sure bet is more than $100 million from a gamble that
might have paid nothing. 




Broome (1991, Ch. 5) raises a worry about this re-description
solution. *Any* preferences can be justified by re-describing
the space of outcomes, thus rendering the axioms of expected utility
theory devoid of content. Broome rebuts this objection by suggesting
an additional constraint on preference: if \(A\) is preferred to
\(B\), then \(A\) and \(B\) must differ in some way that justifies
preferring one to the other. An expected utility theorist can then
count the Allais and Ellsberg preferences as rational if, and only if,
there is a non-monetary difference that justifies placing outcomes of
equal monetary value at different spots in one's preference
ordering.


#### 3.2.3 Counterexamples Involving Probability 0 Events




Above, we've seen purported examples of rational preferences
that violate expected utility theory. There are also
purported examples of irrational preferences that satisfy expected
utility theory.




On a typical understanding of expected utility theory, when two acts
are tied for having the highest expected utility, agents are required
to be indifferent between them. Skyrms (1980, p. 74) points out
that this view lets us derive strange conclusions about events with
probability 0. For instance, suppose you are about to throw a
point-sized dart at a round dartboard. Classical probability
theory countenances situations in which the dart has probability 0 of
hitting any particular point. You offer me the following lousy
deal: if the dart hits the board at its exact center, then you will
charge me $100; otherwise, no money will change hands. My
decision problem can be captured with the following matrix:




|  |  |
| --- | --- |
|  | **states** |
| *hit center* (\(P=0\)) | *miss center* (\(P=1\)) |
| **acts** | *accept deal* | \(-100\) | \(0\) |
| *refuse deal* | \(0\) | \(0\) |




Expected utility theory says that it is permissible for me to accept
the deal--accepting has expected utility of 0. (This is so on
both the Jeffrey definition and the Savage definition, if we assume
that how the dart lands is probabilistically independent of how you
bet.) But common sense says it is not permissible for me to accept the
deal. Refusing *weakly dominates* accepting: it yields a better
outcome in some states, and a worse outcome in no state. 




Skyrms suggests augmenting the laws of classical probability with an
extra requirement that only impossibilities are assigned probability 0.
Easwaran (2014) argues that we should instead reject the view that
expected utility theory commands indifference between acts with equal
expected utility. Instead, expected utility theory is not a
complete theory of rationality: when two acts have the same expected
utility, it does not tell us which to prefer. We can use
non-expected-utility considerations like weak dominance as
tiebreakers.


#### 3.2.4 Counterexamples Involving Unbounded Utility




A utility function \(U\) is bounded above if there is a limit to how
good things can be according to \(U\), or more formally, if there is
some least natural number \(sup\) such that for every \(A\) in \(U\)'s
domain, \(U(A) \le sup\). Likewise, \(U\) is bounded below if there is a
limit to how bad things can be according to \(U\), or more formally, if
there is some greatest natural number \(inf\) such that for every
\(A\) in \(U\)'s domain, \(U(A) \ge inf\). Expected utility
theory can run into trouble when utility functions are unbounded
above, below, or both.




One problematic example is the St. Petersburg game, originally
published by Bernoulli. Suppose that a coin is tossed until it lands
tails for the first time. If it lands tails on the first toss, you win
$2; if it lands tails on the second toss, you win $4; if it lands
tails on the third toss, you win $8, and if it lands tails on the
\(n\)th toss, you win $\(2^n\). Assuming each dollar is worth one
utile, the expected value of the St Petersburg game is



\[
(\tfrac{1}{2} \cdot 2) + (\tfrac{1}{4} \cdot 4) + (\tfrac{1}{8} \cdot 8) + \cdots 
 + (\tfrac{1}{2^n} \cdot 2^n) + \cdots
\]

or

\[
 1 + 1 + 1 + \cdots = \infty
\]



It turns out that this sum diverges; the St Petersburg game has
infinite expected utility. Thus, according to expected utility
theory, you should prefer the opportunity to play the St Petersburg
game to any finite sum of money, no matter how large. 
Furthermore, since an infinite expected utility multiplied by any
nonzero chance is still infinite, anything that has a positive
probability of yielding the St Petersburg game has infinite expected
utility. Thus, according to expected utility theory, you should
prefer any *chance* at playing the St Petersburg game, however
slim, to any finite sum of money, however large.




Nover and Hajek (2004) argue that in addition to the
St. Petersburg game, which has infinite expected utility, there are
other infinitary games whose expected utilities are undefined, even
though rationality mandates certain preferences among them.




One response to these problematic infinitary games is to argue that
the decision problems themselves are ill-posed (Jeffrey (1983, 154);
another is to adopt a modified version of expected utility theory that
agrees with its verdicts in the ordinary case, but yields intuitively
reasonable verdicts about the infinitary games (Thalos and Richardson
2013) (Fine 2008) (Colyvan 2006, 2008) (Easwaran 2008).


## 4. Applications


### 4.1 Economics and Public Policy




In the 1940s and 50s, expected utility theory gained currency in the
US for its potential to provide a mechanism that would explain the
behavior of macro-economic variables. As it became apparent that
expected utility theory did not accurately predict the behaviors of
real people, its proponents instead advanced the view that it might
serve instead as a theory of how rational people should respond to
uncertainty (see Herfeld 2017).




Expected utility theory has a variety of applications in public
policy. In welfare economics, Harsanyi (1953) reasons from expected
utility theory to the claim that the most socially just arrangement is
the one that maximizes total welfare distributed across a society
society. The theory of expected utility also has more direct
applications. Howard (1980) introduces the concept of
a *micromort*, or a one-in-a-million chance of death, and uses
expected utility calculations to gauge which mortality risks are
acceptable. In health policy, quality-adjusted life years, or QALYs,
are measures of the expected utilities of different health
interventions used to guide health policy (see Weinstein et al
2009). McAskill (2015) uses expected utility theory to address the
central question of *effective altruism*: "How can I do
the most good?" (Utilties in these applications are most
naturally interpreted as measuring something like happiness or
wellbeing, rather than subjective preference satisfaction for an
individual agent.)




Another area where expected utility theory finds applications is in
insurance sales. Like casinos, insurance companies take on
calculated risks with the aim of long-term financial gain, and must
take into account the chance of going broke in the short run.


### 4.2 Ethics




Utilitarians, along with their descendants contemporary
consequentialists, hold that the rightness or wrongness of an act is
determined by the moral goodness or badness of its consequences. Some
consequentialists, such as (Railton 1984), interpret this to mean that
we ought to do whatever will in fact have the best consequences. But
it is difficult--perhaps impossible--to know the long-term
consequences of our acts (Lenman 2000, Howard-Snyder 2007). In light
of this observation, Jackson (1991) argues that the right act is the
one with the greatest expected moral value, not the one that will in
fact yield the best consequences.




As Jackson notes, the expected moral value of an act depends on
which probability function we work with. Jackson argues that,
while every probability function is associated with an
"ought", the "ought" that matters most to
action is the one associated with the decision-maker's degrees of
belief at the time of action. Other authors claim priority for
other "oughts": Mason (2013) favors the probability
function that is most reasonable for the agent to adopt in response to
her evidence, given her epistemic limitations, while Oddie and Menzies
(1992) favor the objective chance function as a measure of objective
rightness. (They appeal to a more complicated probability
function to define a notion of "subjective rightness" for
decisionmakers who are ignorant of the objective chances.)




Still others (Smart 1973, Timmons 2002) argue that even if that we
ought to do whatever will have the best consequences, expected utility
theory can play the role of a decision procedure when we are uncertain
what consequences our acts will have. Feldman (2006) objects that
expected utility calculations are horribly impractical. In most
real life decisions, the steps required to compute expected utilities
are beyond our ken: listing the possible outcomes of our acts,
assigning each outcome a utility and a conditional probability given
each act, and performing the arithmetic necessary to expected utility
calculations.



The expected-utility-maximizing version of consequentialism is not
strictly speaking a theory of *rational* choice. It is a theory
of moral choice, but whether rationality requires us to do what is
morally best is up for debate.



### 4.3 Epistemology




Expected utility theory can be used to address practical questions
in epistemology. One such question is when to accept a
hypothesis. In typical cases, the evidence is logically
compatible with multiple hypotheses, including hypotheses to which it
lends little inductive support. Furthermore, scientists do not
typically accept only those hypotheses that are most probable given
their data. When is a hypothesis likely enough to deserve
acceptance?




Bayesians, such as Maher (1993), suggest that this decision be made
on expected utility grounds. Whether to accept a hypothesis is a
decision problem, with acceptance and rejection as acts. It can
be captured by the following decision matrix:




|  |  |
| --- | --- |
|  | **states** |
| *hypothesis is true* | *hypothesis is false* |
| **acts** | *accept* | correctly accept | erroneously accept |
| *reject* | erroneously reject | correctly reject |




On Savage's definition, the expected utility of accepting the
hypothesis is determined by the probability of the hypothesis, together
with the utilities of each of the four outcomes. (We can expect
Jeffrey's definition to agree with Savage's on the
plausible assumption that, given the evidence in our possession, the
hypothesis is probabilistically independent of whether we accept or
reject it.) Here, the utilities can be understood as purely
epistemic values, since it is epistemically valuable to believe
interesting truths, and to reject falsehoods.




Critics of the Bayesian approach, such as Mayo (1996), object that
scientific hypotheses cannot sensibly be given probabilities. 
Mayo argues that in order to assign a useful probability to an event,
we need statistical evidence about the frequencies of similar
events. But scientific hypotheses are either true once and for
all, or false once and for all--there is no population of worlds
like ours from which we can meaningfully draw statistics. Nor can
we use subjective probabilities for scientific purposes, since this
would be unacceptably arbitrary. Therefore, the expected
utilities of acceptance and rejection are undefined, and we ought to
use the methods of traditional statistics, which rely on comparing the
probabilities of our evidence conditional on each of the
hypotheses.




Expected utility theory also provides guidance about when to gather
evidence. Good (1967) argues on expected utility grounds that it is
always rational to gather evidence before acting, provided that
evidence is free of cost. The act with the highest expected utility
after the extra evidence is in will always be always at least as good
as the act with the highest expected utility beforehand. 




In *epistemic decision theory*, expected utilities are used to
assess belief states as rational or irrational. If we think of belief
formation as a mental act, facts about the contents of the agent's
beliefs as events, and closeness to truth as a desirable feature of
outcomes, then we can use expected utility theory to evaluate degrees
of belief in terms of their expected closeness to truth. The entry
on epistemic utility arguments for
probabilism includes an overview of expected utility arguments for
a variety of epistemic norms, including conditionalization and the
Principal Principle.



### 4.4 Law




Kaplan (1968), argues that expected utility considerations can be used
to fix a standard of proof in legal trials. A jury deciding whether
to acquit or convict faces the following decision problem:




|  |  |
| --- | --- |
|  | **states** |
| *guilty* | *innocent* |
| **acts** | *convict* | true conviction | false conviction |
| *acquit* | false acquittal | true acquittal |



Kaplan shows that \(EU(convict) > EU(acquit)\) whenever 



\[ P(guilty) > \frac{1}{1+
\frac{U(\mathrm{true~conviction})-U(\mathrm{false~acquittal})}{U(\mathrm{true~acquittal})-U(\mathrm{false~conviction})}}
\]



Qualitatively, this means that the standard of proof increases as the
disutility of convicting an innocent person
\((U(\mathrm{true~conviction})-U(\mathrm{false~acquittal}))\)
increases, or as the disutility of acquitting a guilty person
\((U(\mathrm{true~acquittal})-U(\mathrm{false~conviction}))\)
decreases.




Critics of this decision-theoretic approach, such as Laudan (2006),
argue that it's difficult or impossible to bridge the gap between the
evidence admissible in court, and the real probability of the
defendant's guilt. The probability guilt depends on three factors: the
distribution of apparent guilt among the genuinely guilty, the
distribution of apparent guilt among the genuinely innocent, and the
ratio of genuinely guilty to genuinely innocent defendants who go to
trial (see Bell 1987). Obstacles to calculating any of these factors
will block the inference from a judge or jury's perception of apparent
guilt to a true probability of guilt.