## 1. Expected Utility



Suppose that a student is considering whether to study for an exam. He
reasons that if he will pass the exam, then studying is wasted effort.
Also, if he will not pass the exam, then studying is wasted effort. He
concludes that because whatever will happen, studying is wasted
effort, it is better not to study. This reasoning errs because
studying raises the probability of passing the exam. Deliberations
should take account of an act's influence on the probability of
its possible outcomes.


An act's expected utility is a probability-weighted average
of its possible outcomes' utilities. Possible states of the
world that are mutually exclusive and jointly exhaustive, and so form
a partition, generate an act's possible outcomes. An act-state
pair specifies an outcome. In the example, the act of studying and the
state of passing form an outcome comprising the effort of studying and
the benefit of passing. The expected utility of studying is the
probability of passing if one studies times the utility of studying
and passing plus the probability of not passing if one studies times
the utility of studying and not passing. In compact notation,



\[
\textit{EU} (S) = P(P \mbox{ if } S) \util (S \amp P) + P({\sim}P \mbox{ if } S) \util (S \amp{\sim}P).
\] 


Each product specifies the probability and utility of a possible
outcome. The sum is a probability-weighted average of the possible
outcomes' utilities.



How should decision theory interpret the probability of a state \(S\)
if one performs an act \(A\), that is, \(P(S \mbox{ if }A)\)?
Probability theory offers a handy suggestion. It has an account of
conditional probabilities that decision theory may adopt. Decision
theory may take \(P(S \mbox{ if }A)\) as the probability of the state
conditional on the act. Then \(P(S \mbox{ if }A)\) equals \(P(S\mid A)\),
which probability theory defines as \(P(S \amp A)/P(A)\) when \(P(A)
\ne 0\). Some theorists call expected utility computed using
conditional probabilities conditional expected utility. I call it
expected utility *tout court* because the formula using
conditional probabilities generalizes a simpler formula for expected
utility that uses nonconditional probabilities of states. Also, some
theorists call an act's expected utility its utility *tout
court* because an act's expected utility appraises the act
and yields the act's utility in ideal cases. I call it expected
utility because a person by mistake may attach more or less utility to
a bet than its expected utility warrants. The equality of an
act's utility and its expected utility is normative rather than
definitional.


Expected utilities obtained from conditional probabilities steer
the student's deliberations in the right
direction.



\[\textit{EU} (S) = P(P\mid S)\util (S \amp P) + P({\sim}P\mid S)\util
(S \amp{\sim}P), \]

and

 

\[\textit{EU} ({\sim}S) = P(P\mid {\sim}S)\util ({\sim}S \amp P) +
P({\sim}P\mid {\sim}S)\util ({\sim}S \amp{\sim}P).
\] 


Because of studying's effect on the probability of passing,
\(P(P\mid S) \gt P(P\mid {\sim}S)\) and \(P({\sim}P\mid S) \lt
P({\sim}P\mid {\sim}S)\). So \(\textit{EU} (S) \gt \textit{EU}
({\sim}S)\), assuming that studying's increase in the
probability of passing compensates for the effort of
studying. Maximization of expected utility recommends studying.



The handy interpretation of the probability of a state if one performs
an act, however, is not completely satisfactory. Suppose that one
tosses a coin with an unknown bias and obtains heads. This result is
evidence that the next toss will yield heads, although it does not
causally influence the next toss's result. An event's
probability conditional on another event indicates the evidence that
the second event provides for the first. If the two events are
correlated, the second may provide evidence for the first without
causally influencing it. Causation entails correlation, but
correlation does not entail causation. Deliberations should attend to
an act's causal influence on a state rather than an act's
evidence for a state. A good decision aims to produce a good outcome
rather than evidence of a good outcome. It aims for the good and not
just signs of the good. Often efficacy and auspiciousness go hand in
hand. When they come apart, an agent should perform an efficacious act
rather than an auspicious act.



Consider the Prisoner's Dilemma, a stock example of game theory.
Two people isolated from each other may each act either cooperatively
or uncooperatively. They each do better if they each act cooperatively
than if they each act uncooperatively. However, each does better if he
acts uncooperatively, no matter what the other does. Acting
uncooperatively dominates acting cooperatively. Suppose, in addition,
that the two players are psychological twins. Each thinks as the other
thinks. Moreover, they know this fact about themselves. Then if one
player acts cooperatively, he concludes that his counterpart also acts
cooperatively. His acting cooperatively is good evidence that his
counterpart does the same. Nonetheless, his acting cooperatively does
not cause his counterpart to act cooperatively. He has no contact with
his counterpart. Because he is better off not acting cooperatively
whatever his counterpart does, not acting cooperatively is the better
course. Acting cooperatively is auspicious but not efficacious.



To make expected utility track efficacy rather than auspiciousness,
causal decision theory interprets the probability of a state if one
performs an act as a type of causal probability rather than as a
standard conditional probability. In the Prisoner's Dilemma with
twins, consider the probability of one player's acting
cooperatively given that the other player does. This conditional
probability is high. Next, consider the causal probability of one
player's acting cooperatively if the other player does. Because
the players are isolated, this probability equals the probability of
the first player's acting cooperatively. It is low if that
player follows dominance. Using conditional probabilities, the
expected utility of acting cooperatively exceeds the expected utility
of acting uncooperatively. However, using causal probabilities, the
expected utility of acting uncooperatively exceeds the expected
utility of acting cooperatively. Switching from conditional to causal
probabilities makes expected-utility maximization yield acting
uncooperatively.


## 2. History



This section tours causal decision theory's history and along
the way presents various formulations of the theory.


### 2.1 Newcomb's Problem



Robert Nozick (1969) presented a dilemma for decision theory. He
constructed an example in which the standard principle of dominance
conflicts with the standard principle of expected-utility
maximization. Nozick called the example Newcomb's Problem after
the physicist, William Newcomb, who first formulated the problem.



In Newcomb's Problem an agent may choose either to take an
opaque box or to take both the opaque box and a transparent box. The
transparent box contains one thousand dollars that the agent plainly
sees. The opaque box contains either nothing or one million dollars,
depending on a prediction already made. The prediction was about the
agent's choice. If the prediction was that the agent will take
both boxes, then the opaque box is empty. On the other hand, if the
prediction was that the agent will take just the opaque box, then the
opaque box contains a million dollars. The prediction is reliable. The
agent knows all these features of his decision problem.



Figure 1 displays the agent's options and their outcomes. A row
represents an option, a column a state of the world, and a cell an
option's outcome in a state of the world.





|  |  |  |
| --- | --- | --- |
|  | Prediction

one-boxing | Prediction

two-boxing |
| Take one box | \(\$M\) | \(\$0\) |
| Take two boxes | \(\$M + \$T\) | \(\$T\) |



Figure 1. Newcomb's Problem




Because the outcome of two-boxing is better by \(\$T\) than the
outcome of one-boxing given each prediction, two-boxing dominates
one-boxing. Two-boxing is the rational choice according to the
principle of dominance. Because the prediction is reliable, a
prediction of one-boxing has a high probability given one-boxing.
Similarly, a prediction of two-boxing has a high probability given
two-boxing. Hence, using conditional probabilities to compute expected
utilities, one-boxing's expected utility exceeds
two-boxing's expected utility. One-boxing is the rational choice
according to the principle of expected-utility maximization.



Decision theory should address all possible decision problems and not
just realistic decision problems. However, if Newcomb's problem
seems untroubling because unrealistic, realistic versions of the
problem are plentiful. The essential feature of Newcomb's
problem is an inferior act's correlation with a good state that
it does not causally promote. In realistic, medical Newcomb problems,
a medical condition and a behavioral symptom have a common cause and
are correlated although neither causes the other. If the behavior is
attractive, dominance recommends it although expected utility
maximization prohibits it. Also, Allan Gibbard and William Harper
(1978: Sec. 12) and David Lewis (1979) observe that a Prisoner's
Dilemma with psychological twins poses a Newcomb problem for each
player. For each player, the other player's act is a state
affecting the outcome. Acting cooperatively is a sign, but not a
cause, of the other player's acting cooperatively. Dominance
recommends acting uncooperatively, whereas expected utility computed
with conditional probabilities recommends acting cooperatively. In
some realistic instances of the Prisoner's Dilemma, the
players' anticipated similarity of thought creates a conflict
between the principle of dominance and the principle of
expected-utility maximization.



Arif Ahmed (2018) offers a collection of essays on Newcomb's
problem. Kenny Easwaran (forthcoming) distinguishes Newcomb-like problems
according to opportunities for causal intervention.


### 2.2 Stalnaker's Solution



Robert Stalnaker (1968) presented truth conditions for subjunctive
conditionals. A subjunctive conditional is true if and only if in the
nearest antecedent-world, its consequent is true. (This analysis is
understood so that a subjunctive conditional is true if its antecedent
is true in no world.) Stalnaker used analysis of subjunctive
conditionals to ground their role in decision theory and in a
resolution of Newcomb's problem.


In a letter to Lewis, Stalnaker (1972) proposed a way of
reconciling decision principles in Newcomb's problem. He
suggested calculating an act's expected utility using
probabilities of conditionals in place of conditional
probabilities. Accordingly,



\[
\textit{EU} (A) = \sum\_i P(A \gt S\_i)\util (A \amp S\_i),
\] 


where \(A \gt S\_i\) stands for the conditional that if \(A\) were
performed then \(S\_i\) would obtain. Thus, instead of using the
probability of a prediction of one-boxing given one-boxing, one should
use the probability of the conditional that if the agent were to pick
just one box, then the prediction would have been one-boxing. Because
the agent's act does not cause the prediction, the probability
of the conditional equals the probability that the prediction is
one-boxing. Also, consider the conditional that if the agent were to
pick both boxes, then the prediction would have been one-boxing. Its
probability similarly equals the probability that the prediction is
one-boxing. The act the agent performs does not affect any
prediction's probability because the prediction occurs prior to
the act. Consequently, using probabilities of conditionals to compute
expected utility, two-boxing's expected utility exceeds
one-boxing's expected utility. Therefore, the principle of
expected-utility maximization makes the same recommendation as does
the principle of dominance.



Gibbard and Harper (1978) elaborated and made public Stalnaker's
resolution of Newcomb's problem. They distinguished causal
decision theory, which uses probabilities of subjunctive conditionals,
from evidential decision theory, which uses conditional probabilities.
Because in decision problems probabilities of subjunctive conditionals
track causal relations, using them to calculate an option's
expected utility makes decision theory causal.



Gibbard and Harper distinguished two types of expected utility. One
type they called value and represented with \(V\). It indicates
news-value or auspiciousness. The other type they called utility and
represented with \(U\). It indicates efficacy in attainment of goals.
A calculation of an act's expected value uses conditional
probabilities, and a calculation of its expected utility uses
probabilities of conditionals. They argued that expected utility,
calculated with probabilities of conditionals, yields genuine expected
utility.



As Gibbard and Harper introduce \(V\) and \(U\), both rest on an
assessment \(D\) (for desirability) of maximally specific outcomes.
Instead of adopting a formula for expected utility that uses an
assessment of outcomes neutral with respect to evidential and causal
decision theory, this essay follows Stalnaker (1972) in adopting a
formula that uses utility to evaluate outcomes.


### 2.3 Variants



Consider a conditional asserting that if an option were adopted, then
a certain state would obtain. Gibbard and Harper assume, to illustrate
the main ideas of causal decision theory, that the conditional has a
truth-value, and that, given its falsity, if the option were adopted,
then the state would not obtain. This assumption may be unwarranted if
the option is flipping a coin, and the relevant state is obtaining
heads. It may be false (or indeterminate) that if the agent were to
flip the coin, he would obtain heads. Similarly, the corresponding
conditional about obtaining tails may be false (or indeterminate).
Then probabilities of conditionals are not suitable for calculating
the option's expected utility. The relevant probabilities do not
sum to one (or do not even exist). To circumvent such impasses, some
theorists calculate causally-sensitive expected utilities without
probabilities of subjunctive conditionals. Causal decision theory has
many formulations.


Brian Skyrms (1980: Sec IIC; 1982) presented a version of causal
decision theory that dispenses with probabilities of subjunctive
conditionals. His theory separates factors that the agent's act
may influence from factors that the agent's act may not
influence. It lets \(K\_i\) stand for a possible full specification of
factors that an agent may not influence and lets \(C\_j\) stand for a
possible (but not necessarily full) specification of factors that the
agent may influence. The set of \(K\_i\) forms a partition, and the set
of \(C\_j\) forms a partition. The formula for an act's expected
utility first calculates its expected utility using factors the agent
may influence, with respect to each possible combination of factors
outside the agent's influence. Then it computes a
probability-weighted average of those conditional expected
utilities. An act's expected utility calculated this way is the
act's \(K\)-expectation, \(\textit{EU}\_k(A)\). According to
Skyrms's definition,



\[\textit{EU}\_k(A) = \sum\_i P(K\_i)\sum\_j P(C\_j \mid K\_i \amp A)\util (C\_j
\amp K\_i \amp A).\]


Skyrms holds that an agent should select an act that maximizes
\(K\)-expectation.


Lewis (1981) presented a version of causal decision theory that
calculates expected utility using probabilities of dependency
hypotheses instead of probabilities of subjunctive conditionals. A
dependency hypothesis for an agent at a time is a maximally specific
proposition about how the things the agent cares about do and do not
depend causally on his present acts. An option's expected
utility is its probability-weighted average utility with respect to a
partition of dependency hypotheses \(K\_i\). Lewis defines the expected
utility of an option \(A\) as



\[
\textit{EU} (A) = \sum\_i P(K\_i)\util (K\_i \amp A)
\] 


and holds that to act rationally is to realize an option that
maximizes expected utility. His formula for an option's expected
utility is the same as Skyrms's assuming that \(U(K\_i \amp A)\)
may be expanded with respect to a partition of factors the agent may
influence, using the formula



\[
U(K\_i \amp A) = \sum\_j P(C\_j\mid K\_i \amp A)\util (C\_j \amp K\_i \amp A).
\]


Skyrms's and Lewis's calculations of expected utility
dispense with causal probabilities. They build causality into states
of the world so that causal probabilities are unnecessary. In cases
such as Newcomb's problem, their calculations yield the same
recommendations as calculations of expected utility employing
probabilities of subjunctive conditionals. The various versions of
causal decision theory make equivalent recommendations when cases meet
their background assumptions. Adam Bales (2016) compares versions in
special cases that do not meet the background assumptions.


### 2.4 Representation Theorems



Decision theory often introduces probability and utility with
representation theorems. These theorems show that if preferences among
acts meet certain constraints, such as transitivity, then there exist
a probability function and a utility function (given a choice of
scale) that generate expected utilities agreeing with preferences.
David Krantz, R. Duncan Luce, Patrick Suppes, and Amos Tversky (1971)
offer a good, general introduction to the purposes and methods of
constructing representations theorems. In
 Section 3.1,
 I discuss the theorems' function in decision theory.



Richard Jeffrey ([1965] 1983) presented a representation theorem for
evidential decision theory, using its formula for expected utility.
Brad Armendt (1986, 1988a) presented a representation theorem for
causal decision theory, using its formula for expected utility. James
Joyce (1999) constructed a very general representation theorem that
yields either causal or evidential decision theory depending on the
interpretation of probability that the formula for expected utility
adopts.


### 2.5 Objections



The most common objection to causal decision theory is that it yields
the wrong choice in Newcomb's problem. It yields two-boxing,
whereas one-boxing is correct. Terry Horgan (1981 [1985]), Paul
Horwich (1987: Chap. 11), and Caspar Hare and Brian Hedden (2016) for
example, promote one-boxing. The main rationale for one-boxing is that
one-boxers fare better than do two-boxers. Causal decision theorists
respond that Newcomb's problem is an unusual case that rewards
irrationality. One-boxing is irrational even if one-boxers
prosper. Bales (2018) rejects the argument that two-boxing is
irrational



Some theorists hold that one-boxing is plainly rational if the
prediction is completely reliable. They maintain that if the
prediction is certainly accurate, then choice reduces to taking
\(\$M\) or taking \(\$T\). This view oversimplifies. If an agent
one-boxes, then that act is certain to yield \(\$M\). However, the
agent still would have done better by taking both boxes. Dominance
still recommends two-boxing. Making the prediction certain to be
accurate does not change the character of the problem. Efficacy still
trumps auspiciousness, as Howard Sobel (1994: Chap. 5) argues.



A way of reconciling the two sides of the debate about Newcomb's
problem acknowledges that a rational person should prepare for the
problem by cultivating a disposition to one-box. Then whenever the
problem arises, the disposition will prompt a prediction of one-boxing
and afterwards the act of one-boxing (still freely chosen). Causal
decision theory may acknowledge the value of this preparation. It may
conclude that cultivating a disposition to one-box is rational
although one-boxing itself is irrational. Hence, if in Newcomb's
problem an agent two-boxes, causal decision theory may concede that
the agent did not rationally prepare for the problem. It nonetheless
maintains that two-boxing itself is rational. Although two-boxing is
not the act of a maximally rational agent, it is rational given the
circumstances of Newcomb's problem.



Causal decision theory may also explain that it advances a claim about
the evaluation of an act given the agent's circumstances in
Newcomb's problem. It asserts two-boxing's conditional
rationality. Conditional and nonconditional rationality treat mistakes
differently. In contrast with conditional rationality, nonconditional
rationality does not grant past mistakes. It evaluates an act taking
account of the influence of past mistakes. However, conditional
rationality accepts present circumstances as they are and does not
discredit an act because it stems from past mistakes. Causal decision
theory maintains that two-boxing is rational, granting the
agent's circumstances and so ignoring any mistakes leading to
those circumstances, such as irrational preparation for
Newcomb's problem.



Another objection to causal decision theory concedes that two-boxing
is the rational choice in Newcomb's problem but rejects causal
principles of choice that yield two-boxing. It seeks noncausal
principles that yield two-boxing. Positivism is a source of aversion
to decision principles incorporating causation. Some decision
theorists shun causation because no positivist account specifies its
nature. Without a definition of causation in terms of observable
phenomena, they prefer that decision theory avoid causation. Causal
decision theory's response to this objection is both to
discredit positivism and also to clarify causation so that puzzles
concerning it no longer give decision theory any reason to avoid
it.



Evidential decision theory has weaker metaphysical assumptions than
has causal decision theory, even if causation has impeccable
metaphysical credentials. Some decision theorists do not omit
causation because of metaphysical scruples but for conceptual economy.
Jeffrey ([1965] 1983, 2004), for the sake of parsimony, formulates
decision principles that do not rely on causal relations.



Ellery Eells (1981, 1982) contends that evidential decision theory
yields causal decision theory's recommendations but, more
economically, without reliance on causal apparatus. In particular,
evidential decision theory yields two-boxing in Newcomb's
problem. An agent's reflection on his evidence makes conditional
probabilities support two-boxing.



A noncontentious elaboration of Newcomb's problem posits that
the agent's choice and its prediction have a common cause. The
agent's choice is evidence of the common cause and evidence of
the choice's prediction. Once an agent acquires the probability
of the common cause, he may put aside the evidence his choice provides
about the prediction. That evidence is superfluous. Given the
probability of the common cause, the probability of a prediction of
one-boxing is constant with respect to his options. Similarly, the
probability of a prediction of two-boxing is constant with respect to
his options. Because the probability of a prediction is the same
conditional on either option, the expected utility of two-boxing
exceeds the expected utility of one-boxing according to evidential
decision theory. Horgan (1981 [1985]) and Huw Price (1986) make
similar points.



Suppose that an event \(S\) is a sign of a cause \(C\) that produces
an effect \(E\). For the probability of \(E\), knowing whether \(C\)
holds makes superfluous knowing whether \(S\) holds. Observation of
\(C\) *screens off* the evidence that \(S\) provides for \(E\).
That is, \(P(E\mid C \amp S) = P(E\mid C)\). In Newcomb's problem,
assuming that the agent is rational, his beliefs and desires are a
common cause of his choice and the prediction. So his choice is a sign
of the prediction's content. For the probability of a prediction
of one-boxing, knowing one's beliefs and desires makes
superfluous knowing the choice that they yield. Knowledge of the
common cause screens off evidence that the choice provides about the
prediction. Hence, the probability of a prediction of one-boxing is
constant with respect to one's choice, and maximization of
evidential expected-utility agrees with the principle of dominance.
This defense of evidential decision theory is called the tickle
defense because it assumes that an introspected condition screens off
the correlation between choice and prediction.



Eells's defense of evidential decision theory assumes that an
agent chooses according to beliefs and desires and knows his beliefs
and desires. Some agents may not choose this way and may not have this
knowledge. Decision theory should prescribe a rational choice for such
agents, and evidential decision theory may not do that correctly, as
Lewis (1981: 10-11) and John Pollock (2010) argue. Armendt
(1988b: 326-329) and David Papineau (2001: 252-255) concur
that the phenomenon of screening off does not in all cases make
evidential decision theory yield the results of causal decision
theory.



Horwich (1987: Chap. 11) rejects Eells's argument because, even
if an agent knows that her choice springs from her beliefs and
desires, she may be unaware of the mechanism by which her beliefs and
desires produce her choice. The agent may doubt that she chooses by
maximizing expected utility. Then in Newcomb's problem her
choice may offer relevant evidence about the prediction. Eells (1984a)
constructs a dynamic version of the tickle defense to meet this
objection. Sobel (1994: Chap. 2) discusses that version of the
defense. He argues that it does not yield evidential decision
theory's agreement with causal decision theory in all decision
problems in which an act furnishes evidence concerning the state of
the world. Moreover, it does not establish that an evidential theory
of rational desire agrees with a causal theory of rational desire. He
concludes that even in cases where evidential decision theory yields
the right recommendation, it does not yield it for the right
reasons.



Price (2012) proposes a blend of evidential and causal decision theory
and motivates it with an analysis of cases in which an agent has
foreknowledge of an event occurring by chance. Causal decision theory
on its own accommodates such cases, argues Bales (2016). 
Ahmed (2014a) champions evidential decision theory and advances several
objections to causal decision theory. His objections assume some
controversial points about rational choice, including a controversial
principle for sequences of choices.



A common view distinguishes principles for evaluating choices from
principles for evaluating sequences of choices. The principle of
utility maximization evaluates an agent's choice as a resolution
of a decision problem only if the agent has direct control of each
option in the decision problem, that is, only if the agent can at will
immediately adopt any option in the decision problem. The principle
does not evaluate an agent's sequence of multiple choices
because the agent does not have direct control of such a sequence. She
realizes a sequence of multiple choices only by making each choice in
the sequence at the time for it; she cannot at will immediately
realize the entire sequence. Rationality evaluates an option in an
agent's direct control by comparing it with alternatives but
evaluates a sequence in an agent's indirect control by
evaluating the directly controlled options in the sequence; a sequence
of choices is rational if the choices in the sequence are rational.
Adopting this common method of evaluating sequences of choices fends
off objections to causal decision theory that assume rival
methods.


## 3. Current Issues



Decision theory is an active area of research. Current work addresses
a number of problems. Causal decision theory's approach to those
problems arises from its nonpositivistic methodology and its attention
to causation. This section mentions some topics on causal decision
theory's agenda.


### 3.1 Probability and Utility



Principles of causal decision theory use probabilities and utilities.
The interpretation of probabilities and utilities is a matter of
debate. One tradition defines them in terms of functions that
representation theorems introduce to depict preferences. The
representation theorems show that if preferences meet certain
structural axioms, then if they also meet certain normative axioms,
they are as if they follow expected utility. That is, preferences
follow expected utility calculated using probability and utility
functions constructed so that preferences follow expected utility.
Expected utility calculated this way differs from expected utility
calculated using probability and utility assignments grounded in
attitudes toward possible outcomes. For example, a person confused
about bets concerning a coin toss may have preferences among those
bets that are as if he assigns probability 60% to heads, when, in
fact, the evidence of past tosses leads him to assign probability 40%
to heads. Consequently, when preferences meet a representation
theorem's structural axioms, the theorem's normative
axioms justify only conformity with expected utility fabricated to
agree with preferences and do not justify conformity with expected
utility in the traditional sense. Defining probability and utility
using the representation theorems thus weakens the traditional
principle of expected utility. It becomes merely a principle of
coherence among preferences.



Instead of using the representation theorems to define probabilities
and utilities, decision theory may use them to establish
probabilities' and utilities' measurability when
preferences meet structural and normative axioms. This employment of
the representation theorems allows decision theory to advance the
traditional principle of expected utility and thereby enrich its
treatment of rational decisions. Decision theory may justify that
traditional principle by deriving it from general principles of
evaluation, as in Weirich (2001).



A broad account of probabilities and utilities takes them to indicate
attitudes toward propositions. They are rational degrees of belief and
rational degrees of desire, respectively. This account of
probabilities and utilities recognizes their existence in cases where
they are not inferable from preferences or their other effects but
instead are inferable from their causes, such as an agent's
information about objective probabilities, or are not inferable at all
(except perhaps by introspection). The account relies on arguments
that degrees of belief and degrees of desire, if rational, conform to
standard principles of probability and utility. Bolstering these
arguments is work for causal decision theory.



Besides clarifying its general interpretation of probability and
utility, causal decision theory searches for the particular
probabilities and utilities that yield the best version of its
principle to maximize expected utility. The causal probabilities in
its formula for expected utility may be probabilities of subjunctive
conditionals or various substitutes. Versions that use probabilities
of subjunctive conditionals must settle on an analysis of those
conditionals. Lewis (1973: Chap. 1) modifies Stalnaker's
analysis to count a subjunctive conditional true if and only if as
antecedent worlds come closer and closer to the actual world, there is
a point beyond which the consequent is true in all the worlds at least
that close. Joyce (1999: 161-180) advances probability images,
as Lewis (1976) introduces them, as substitutes for probabilities of
subjunctive conditionals. The probability image of a state \(S\) under
subjunctive supposition of an act \(A\) is the probability of \(S\)
according to an assignment that shifts the probability of
\({\sim}A\)-worlds to nearby \(A\)-worlds. Causal relations among an
act and possible states guide probability's reassignment.


A common formula for an act's expected utility takes the
utility for an act-state pair, the utility of the act's outcome
in the state, to be the utility of the act's and the
state's conjunction:



\[
\textit{EU} (A) = \sum\_i P(A \gt S\_i)\util (A \amp S\_i).
\] 


Does causal decision theory need an alternative, more
causally-sensitive utility for an act-state pair? Weirich (1980)
argues that it does. A person contemplating a wager that the capital
of Missouri is Jefferson City entertains the consequences if
he *were* to make the wager given that St. Louis *is*
Missouri's capital. A rational deliberator subjunctively
supposes an act attending to causal relations and indicatively
supposes a state attending to evidential relations, but can suppose an
act's and a state's conjunction only one way. Furthermore,
using the utility of an act's and a state's conjunction
prevents an act's expected utility from being
partition-invariant. The next subsection elaborates this point.


### 3.2 Partition Invariance



An act's expected utility is partition invariant if and only if
it is the same under all partitions of states. Partition invariance is
a vital property of an act's expected utility. If acts'
expected utilities lack this property, then decision theory may use
only expected utilities computed from selected partitions. Expected
utility's partition invariance makes an act's expected
utility independent of selection of a partition of states and thereby
increases expected utility's explanatory power.



Partition invariance ensures that various representations of the same
decision problem yield solutions that agree. Take Newcomb's
problem with Figure 2's representation.





|  |  |  |
| --- | --- | --- |
|  | Right prediction | Wrong prediction |
| Take only one box | \(\$M\) | $0 |
| Take two boxes | \(\$T\) |  \(\$M + \$T\) |



Figure 2. New States for Newcomb's
Problem




Dominance does not apply to this representation. It nonetheless
settles the problem's solution because it applies to a decision
problem if it applies to any accurate representation of the problem,
such as Figure 1's representation of the problem. If expected
utilities are partition-sensitive, then acts that maximize expected
utility may be partition-sensitive. The principle of expected utility
does not yield a decision problem's solution, however, if acts
of maximum expected-utility change from one partition to another. In
that case an act is not a solution to a decision problem simply
because it maximizes expected utility under some accurate
representation of the problem. Too many acts have the same
credential.



The expected utility principle, using probabilities of conditionals,
applies to Figure 2's representation of Newcomb's problem.
Letting \(P1\) stand for a prediction of one-boxing and \(P2\) stand
for a prediction of two-boxing, the acts' expected utilities
are:

 

\[
\begin{align}
\textit{EU} (1) & = P(1 \gt R)\util (\$M) + P(1 \gt W)0\\
 & = P(P1)\util (\$M)\\
 \textit{EU} (2) & = P(2 \gt R)\util (\$T) + P(2 \gt W)\util (\$M + \$T)\\
 & = P(P2)\util (\$T) + P(P1)\util (\$M + \$T)\\
\end{align}
\]


Hence \(\textit{EU}(1) \lt EU(2)\). This result agrees with the
verdict of causal decision theory given other accurate representations
of the problem. Provided that causal decision theory uses a
partition-invariant formula for expected utility, its recommendations
are independent of a decision problem's representation.


Lewis (1981: 12-13) observes that the formula



\[
EU(A) = \sum\_i P(S\_i)\util (A \amp S\_i)
\] 


is not partition invariant. Its results depend on the partition of
states. If a state is a set of worlds with equal utilities, then with
respect to a partition of such states every act has the same expected
utility. An element \(S\_i\) of the partition obscures the effects of
\(A\) that the utility of an outcome should evaluate. Lewis overcomes
this problem by using only partitions of dependency
hypotheses. However, causal decision theory may craft a
partition-invariant formula for expected utility by adopting a
substitute for \(U(A \amp S\_i)\).


Sobel (1994: Chap. 9) investigates partition invariance. Putting
his work in this essay's notation, he proceeds as
follows. First, he takes a canonical computation of an option's
expected utility to use worlds as states. His basic formula is



\[
\textit{EU} (A) = \sum\_i P(A \gt W\_i)\util (W\_i).
\] 


A world \(W\_i\) absorbs an act performed in it. Only the worlds in
which \(A\) holds contribute positive probabilities and so affect the
sum. Next, Sobel searches for other computations, using coarse-grained
states, that are equivalent to the canonical computation. A suitable
specification of utilities achieves partition invariance given his
assumptions. According to a theorem he proves (1994: 185),



\[
U(A) = \sum\_i P(S\_i)\util (A \mbox{ given } S\_i)
\] 


for any partition of states.


Joyce (2000: S11) also articulates for causal decision theory a
partition-invariant formula for an act's expected utility. He
achieves partition invariance, assuming that



\[
\textit{EU} (A) = \sum\_i P(A \gt S\_i)\util (A \amp S\_i),
\] 

by stipulating that \(U(A \amp S\_i)\) equals



\[
\sum\_{ij} P^A(W\_j\mid S\_i)\util (W\_j),
\] 


where \(W\_j\) is a world and \(P^A\) stands for the probability image
of \(A\). Weirich (2001: Secs. 3.2, 4.2.2), as Sobel does, substitutes
\(U(A \mbox{ given }S\_i)\) for \(U(A \amp S\_i)\) in the formula for
expected utility and interprets \(U(A \mbox{ given }S\_i)\) as the
utility of the outcome that \(A\)'s realization would produce if
\(S\) obtains. Accordingly, \(U(A \mbox{ given }S\_i)\) responds to
\(A\)'s causal consequences in worlds where \(S\_i\) holds. Then
the formula



\[
\textit{EU} (A) = \sum\_i P(S\_i) \util (A \mbox{ given }S\_i)
\] 


is invariant with respect to partitions in which states are
probabilistically independent of the act. A more complex formula,



\[
\textit{EU} (A) = \sum\_i P(S\_i \mbox{ if }A)\util (A \mbox{ given }
(S\_i \mbox{ if } A)),
\] 


assuming a causal interpretation of its probabilities, relaxes all
restriction on partitions. \(U(A \mbox{ given }(S\_i \mbox{ if }A))\)
is the utility of the outcome if \(A\) were realized, given that it is
the case that \(S\_i\) would obtain if \(A\) were realized.


### 3.3 Outcomes



One issue concerning outcomes is their comprehensiveness. Are an
act's outcomes possible worlds, temporal aftermaths, or causal
consequences? Gibbard and Harper ([1978] 1981: 166-168) mention
the possibility of narrowing outcomes to causal consequences, as
practical applicability advocates. The narrowing must be judicious,
however, because the expected-utility principle requires that outcomes
include every relevant consideration. For example, if an agent is
averse to risk, then each of a risky act's possible outcomes
must include the risk the act generates. Its inclusion tends to lower
each possible outcome's utility.


In Sobel's canonical formula for expected utility,



\[
\textit{EU} (A) = \sum\_i P(A \gt W\_i)\util (W\_i).
\] 


The formula, from one perspective, omits states of the world because
the outcomes themselves form a partition. The distinction between
states and outcomes dissolves because worlds play the role of both
states and outcomes. States are dispensable means of generating
outcomes that are exclusive and exhaustive. According to a basic
principle, an act's expected utility is a probability-weighted
average of possible outcomes that are exclusive and exhaustive, such
as the worlds to which the act may lead.


Suppose that a world's utility comes from realization of
basic intrinsic desires and aversions. Granting that the utilities of
their realizations are additive, the utility of a world is a sum of
the utilities of their realizations. Then besides being a
probability-weighted average of the utilities of worlds to which it
may lead, an option's expected utility is also a
probability-weighted average of the realizations of basic intrinsic
desires and aversions. In this formula for its expected utility,
states play no explicit role:



\[
\textit{EU} (A) = \sum\_i P(A \gt B\_i)\util (B\_i),
\] 


where \(B\_i\) ranges over possible realizations of basic intrinsic
desires and aversions. The formula considers for each basic desire and
aversion the prospect of its realization if the act were performed. It
takes the act's expected utility as the sum of the
prospects' utilities. The formula provides an economical
representation of an act's expected utility. It eliminates
states and obtains expected utility directly from outcomes taken as
realizations of basic desires and aversions.



To illustrate calculation of an act's expected utility using
basic intrinsic desires and aversions, suppose that an agent has no
basic intrinsic aversions and just two basic intrinsic desires, one
for health and the other for wisdom. The utility of health is 4, and
the utility of wisdom is 8. In the formula for expected utility, a
world covers only matters about which the agent cares. In the example,
a world is a proposition specifying whether the agent has health and
whether he has wisdom. Accordingly, there are four worlds:

\[
\begin{align}
H \amp W, \\
H \amp {\sim}W, \\
{\sim}H \amp W, \\
{\sim}H \amp {\sim}W.\\
\end{align}
\] 

 Suppose that \(A\) is equally likely to generate any
world. Using worlds, 

\[
\begin{align}
\textit{EU} (A) & = P(A \gt(H \amp W))\util (H \amp W) \\
&\qquad + P(A \gt(H \amp{\sim}W))\util (H \amp{\sim}W) \\
&\qquad + P(A \gt({\sim}H \amp W))\util ({\sim}H \amp W) \\
&\qquad + P(A \gt({\sim}H \amp{\sim}W))\util ({\sim}H \amp{\sim}W) \\
& = (0.25)(12) + (0.25)(4) + (0.25)(8) + (0.25)(0) \\
& = 6.\\
\end{align}

\] 

 Using basic intrinsic attitudes,

\[
\begin{align}
\textit{EU} (A) &= P(A \gt H)\util (H) + P(A \gt W)\util (W) \\
& = (0.5)(4) + (0.5)(8) \\
& = 6.
\end{align}
\]

 The two methods of computing an option's utility
are equivalent given that, under supposition of an act's
realization, the probability of a basic intrinsic desire's or
aversion's realization is the sum of the probabilities of the
worlds that realize it.


### 3.4 Acts



In deliberations, a first-person action proposition represents an act.
The proposition has a subject-predicate structure and refers directly
to the agent, its subject, without the intermediary of a concept of
the agent. A centered world represents the proposition. Such a world
not only specifies individuals and their properties and relations, but
also specifies which individual is the agent and where and when his
decision problem arises. Realization of the act is realization of a
world with, at its center, the agent at the time and place of his
decision problem.



Isaac Levi (2000) objects to any decision theory that attaches
probabilities to acts. He holds that deliberation crowds out
prediction. While deliberating, an agent does not have beliefs or
degrees of belief about the act that she will perform. Levi holds that
Newcomb's problem, and evidential and causal decision theories
that address it, involve mistaken assignments of probabilities to an
agent's acts. He rejects both Jeffrey's ([1965] 1983)
evidential decision theory and Joyce's (1999) causal decision
theory because they allow an agent to assign probabilities to her acts
during deliberation.



In opposition to Levi's views, Joyce (2002) argues that (1)
causal decision theory need not accommodate an agent's assigning
probabilities to her acts, but (2) a deliberating agent may
legitimately assign probabilities to her acts. Evidential decision
theory computes an act's expected utility using the probability
of a state given the act, \(P(S\mid A)\), defined as \(P(S \amp A)/P(A)\).
The fraction's denominator assigns a probability to an act.
Causal decision theory replaces \(P(S\mid A)\) with \(P(A \gt S)\) or a
similar causal probability. It need not assign a probability to an
act.



May an agent deliberating assign probabilities to her possible acts?
Yes, a deliberator may sensibly assign probabilities to any events,
including her acts. Causal decision theory may accommodate such
probabilities by forgoing their measurement with betting quotients.
According to that method of measurement, willingness to make bets
indicates probabilities. Suppose that a person is willing to take
either side of a bet in which the stake for the event is \(x\) and the
stake against the event is \(y\). Then the probability the person
assigns to the event is the betting quotient \(x/(x + y)\). This
method of measurement may fail when the event is an agent's own
future act. A bet on an act's realization may influence the
act's probability, as a thermometer's temperature may
influence the temperature of a liquid it measures.



Joyce (2007: 552-561) considers whether Newcomb problems are
genuine decision problems despite strong correlations between states
and acts. He concludes that, yes, despite those correlations, an agent
may view her decision as causing her act. An agent's decision
supports a belief about her act independently of prior correlations
between states and her act. According to a principle of evidential
autonomy (2007: 557), 



> 
> 
> A deliberating agent who regards herself as free need not proportion
> her beliefs about her own acts to the antecedent evidence that she has
> for thinking that she will perform them. 
> 
> 
> 



She should proportion her beliefs to her total evidence, including her
self-supporting beliefs about her own acts. Those beliefs provide new
relevant evidence about her acts.



How should an agent deliberating about an act understand the
background for her act? She should not adopt a backtracking
supposition of her act. Standing on the edge of a cliff, she should
not suppose that if she were to jump, she would have a parachute to
break her fall. Also, she should not imagine gratuitous changes in her
basic desires. She should not imagine that if she were to choose
chocolate instead of vanilla, despite currently preferring vanilla,
that she would then prefer chocolate. She should imagine that her
basic desires are constant as she imagines the various acts she may
perform, and, moreover, should adopt during deliberations the pretense
that her will generates her act independently of her basic desires and
aversions.



Christopher Hitchcock (1996) holds that an agent should pretend that
her act is free of causal influence. Doing this makes partitions of
states yielding probabilities for decision agree with partitions of
states yielding probabilities defining causal relevance. As a result,
probabilities in causal decision theory may form a foundation for
probabilities in the probabilistic theory of causation. Causal
decision theory, in particular, the version using dependency
hypotheses, grounds theories of probabilistic causation.


### 3.5 Generalizing Expected Utility



Problems such as Pascal's Wager and the St. Petersburg paradox
suggest that decision theory needs a means of handling infinite
utilities and expected utilities. Suppose that an option's
possible outcomes all have finite utilities. Nonetheless, if those
utilities are infinitely many and unbounded, then the option's
expected utility may be infinite. Alan Hajek and Harris Nover
(2006) also show that the option may have no expected utility. The
order of possible outcomes, which is arbitrary, may affect convergence
of their utilities' probability-weighted average and the value
to which the average converges if it does converge. Causal decision
theory should generalize its principle of expected-utility
maximization to handle such cases.



Also, common principles of causal decision theory advance standards of
rationality that are too demanding to apply to humans. They are
standards for ideal agents in ideal circumstances (a precise
formulation of the idealizations may vary from theorist to theorist).
Making causal decision theory realistic requires relaxing
idealizations that its principles assume. A generalization of the
principle of expected-utility maximization, for example, may relax
idealizations to accommodate limited cognitive abilities. Weirich
(2004) and Pollock (2006) take steps in this direction. Appropriate
generalizations distinguish taking maximization of expected utility as
a procedure for making a decision and taking it as a standard for
evaluating a decision even after the decision has been made.


### 3.6 Ratification



Gibbard and Harper (1978: Sec. 11) present a problem for causal
decision theory using an example drawn from literature. A man in
Damascus knows that he has an appointment with Death at midnight. He
will escape Death if he manages at midnight not to be at the place of
his appointment. He can be in either Damascus or Aleppo at midnight.
As the man knows, Death is a good predictor of his whereabouts. If he
stays in Damascus, he thereby has evidence that Death will look for
him in Damascus. However, if he goes to Aleppo he thereby has evidence
that Death will look for him in Aleppo. Wherever he decides to be at
midnight, he has evidence that he would be better off at the other
place. No decision is stable. Decision instability arises in cases in
which a choice provides evidence for its outcome, and each choice
provides evidence that another choice would have been better. Reed
Richter (1984, 1986) uses cases of decision instability to argue
against causal decision theory. The theory needs a resolution of the
problem of decision instability.



A common analysis of the problem classifies options as either
self-ratifying or not self-ratifying. Jeffrey ([1965] 1983) introduced
ratification as a component of evidential decision theory. His version
of the theory evaluates a decision according to the expected utility
of the act it selects. The distinction between an act and a decision
to perform the act grounds his definition of an option's
self-ratification and his principle to make self-ratifying, or
ratifiable, decisions. According to his definition ([1965] 1983: 16),




> 
> 
> A ratifiable decision is a decision to perform an act of maximum
> estimated desirability relative to the probability matrix the agent
> thinks he would have if he finally decided to perform that act. 
> 
> 
> 



Estimated desirability is expected utility. An agent's
probability matrix is an array of rows and columns for acts and
states, respectively, with each cell formed by the intersection of an
act's row and a state's column containing the probability
of the state given that the agent is about to perform the act. Before
performing an act, an agent may assess the act in light of a decision
to perform it. Information the decision carries may affect the
act's expected utility and its ranking with respect to other
acts.



Jeffrey used ratification as a means of making evidential decision
theory yield the same recommendations as causal decision theory. In
Newcomb's problem, for instance, two-boxing is the only
self-ratifying option. However, Jeffrey (2004: 113n) concedes that
evidential decision theory's reliance on ratification does not
make it agree with causal decision theory in all cases. Moreover,
Joyce (2007) argues that the motivation for ratification appeals to
causal relations, so that even if it yields correct recommendations
using Jeffrey's formula for expected-utility, it still does not
yield a purely evidential decision theory.



Causal decision theory's account of self-ratification may put
aside Jeffrey's method of evaluating a decision by evaluating
the act it selects. Because the decision and the act differ, they may
have different consequences. For example, a decision may fail to
generate the act it selects. Hence, the decision's expected
utility may differ from the act's expected utility. Driving
through a flooded section of highway may have high expected utility
because it minimizes travel time to one's destination. However,
the decision to drive through the flooded section may have low
expected utility because for all one knows the water may be deep
enough to swamp the car. Using an act's expected utility to
assess a decision to perform the act leads to faulty evaluations of
decisions. It is better to evaluate a decision by comparing its
expected utility to the expected utilities of rival decisions. A
decision's expected utility depends on the probability of its
execution as well as the expected consequences of the act it
selects.



Weirich (1985) and Harper (1986) define ratification in terms of an
option's expected utility given its realization rather than
given a decision to realize it. An option is self-ratifying if and
only if it maximizes expected utility given its realization. This
account of ratification accommodates cases in which an option and a
decision to realize it have different expected utilities. Weirich and
Harper also assume causal decision theory's formula for expected
utility. In the case of Death in Damascus, causal decision theory
concludes that the threatened man lacks a self-ratifying option. A
self-ratifying option emerges, however, if the man may flip a coin to
make his decision. Adopting the probability distribution for locations
is called a mixed strategy, whereas choices of location are called
pure strategies. Assuming that Death cannot predict the coin
flip's outcome, the mixed strategy is self-ratifying.



During deliberations to resolve a decision problem, an agent may
revise the probabilities she assigns to pure strategies in light of
computations of their expected utilities using earlier probability
assignments. The process of revision may culminate in a stable
probability assignment that represents a mixed strategy. Skyrms (1982,
1990) and Eells (1984b) investigate these dynamics of deliberation.
Some open issues are whether adoption of a mixed strategy resolves a
decision problem and whether a pure strategy arising from a mixed
strategy that constitutes an equilibrium of deliberations is rational
if the pure strategy itself is not self-ratifying.



Andy Egan (2007) argues that causal decision theory yields the wrong
recommendation in decision problems with an option that provides
evidence concerning its outcome. He entertains the case of an assassin
who deliberates about pulling the trigger, knowing that the
option's realization provides evidence of a brain lesion that
ruins his aim. Egan maintains that causal decision theory mistakenly
ignores the evidence that the option provides. However, versions of
causal decision theory that incorporate ratification are innocent of
the charges. Ratification takes account of evidence an option provides
concerning its outcome.



Any version of the expected utility principle, whether it uses
conditional probabilities or probabilities of conditionals, must
specify the information that guides assignments of probabilities and
utilities. Principles of nonconditional expected-utility maximization
use the same information for all options, and hence exclude
information about an option's realization. The principle of
ratification uses for each option information that includes the
option's realization. It is a principle of conditional
expected-utility maximization. Egan's cases count against
nonconditional expected-utility maximization, and not against causal
decision theory. Conditional expected-utility maximization using
causal decision theory's formula for expected utility addresses
the cases he presents.



Egan's examples do not refute causal decision theory but present
a challenge for it. Suppose that in a decision problem no
self-ratifying option exists, or multiple self-ratifying options
exist. How should a rational agent proceed, granting that a decision
principle should take account of information that an option provides?
This is an open problem in causal decision theory (and in any decision
theory acknowledging that an option's realization may constitute
evidence concerning its outcome). Ratification analyzes decision
instability but is not a complete response to it.



In response to Egan, Frank Arntzenius (2008) and Joyce (2012) argue
that in some decision problems an agent's rational deliberations
using freely available information do not settle on a single option
but instead settle on a probability distribution over options. They
acknowledge that the agent may regret the option issuing from these
deliberations but differ about the regret's significance.
Arntzenius holds that the regret counts against the option's
rationality, whereas Joyce denies this. Ahmed (2012) and Ralph
Wedgwood (2013) reject Arntzenius's and Joyce's responses
to Egan because they hold that deliberations should settle on an
option. Wedgwood introduces a novel decision principle to accommodate
Egan's decision problems. Ahmed contends that Egan's
analysis of these decision problems has a flaw because when it is
extended to some other decision problems, it declares every option
irrational.



Ahmed (2014b) criticizes causal decision theory in cases of
decision instability. Also, in such cases, Jack Spencer and Ian Wells
(2019, preprint 2017) criticize a principle of causal dominance
attributed to causal decision theory. This decision principle
prohibits adopting an option if another causally dominates it. Joyce
(2018) defends causal decision theory against Ahmed's and Spencer and
Wells's charges using points about deliberational dynamics. Armendt
(2019) and Bales (2020) also defend causal decision theory's handling
of decision instability.



Points about ratification in decision problems clarify points about
equilibrium in game theory because in games of strategy a
player's choice often furnishes evidence about other
players' choices. Decision theory underlies game theory because
a game's solution identifies rational choices in the decision
problems the game creates for the players. Solutions to games
distinguish correlation and causation, as do decision principles.
Because in simultaneous-move games two agent's strategies may be
correlated but not related as cause and effect, solutions to such
games do not have the same properties as solutions to sequential
games. Causal decision theory attends to distinctions on which
solutions to games depend. It supports game theory's account of
interactive decisions. Joyce and Gibbard (2016) describe the role of
ratification in game theory, and Stalnaker (2018) describes causal
decision theory's place in game theory.



The existence of self-ratifying mixed strategies in decision problems
such as Death in Damascus suggests that ratification, as causal
decision theory explains it, supports participation in a Nash
equilibrium of a game. Such an equilibrium assigns a strategy to each
player so that each strategy in the assignment is a best response to
the others. Suppose that two people are playing Matching Pennies.
Simultaneously, each displays a penny. One player tries to make the
sides match, and the other player tries to prevent a match. If the
first player succeeds, he gets both pennies. Otherwise, the second
player gets both pennies. Suppose that each player is good at
predicting the other player, and each player knows this. Then if the
first player displays heads, he has reason to think that the second
player displays tails. Also, if the first player displays tails, he
has reason to think that the second player displays heads. Because
Matching Pennies is a simultaneous-move game, neither player's
strategy influences the other player's strategy, but each
player's strategy is evidence of the other player's
strategy. Mixed strategies help resolve decision instability in this
case. If the first player flips his penny to settle the side to
display, then his mixed strategy is self-ratifying. The second
player's situation is similar, and she also reaches a
self-ratifying strategy by flipping her penny. The combination of
self-ratifying strategies is a Nash equilibrium of the game.



Weirich (2004: Chap. 9) presents a method of selecting among multiple
self-ratifying strategies, and hence a method by which a group of
players may coordinate to realize a particular Nash equilibrium when
several exist. Although decision instability is an open problem,
causal decision theory has resources for addressing it. The
theory's eventual resolution of the problem will offer game
theory a justification for participation in a Nash equilibrium of a
game.


## 4. Related Topics and Concluding Remarks



Causal decision theory has foundations in various areas of philosophy.
For example, it relies on metaphysics for an account of causation. It
also relies on inductive logic for an account of inferences concerning
causation. A comprehensive causal decision theory treats not only
causal probabilities' generation of options' expected
utilities, but also evidence's generation of causal
probabilities.



Research concerning causation contributes to the metaphysical
foundations of causal decision theory. Nancy Cartwright (1979), for
example, draws on ideas about causation to flesh out details of causal
decision theory. Also, some accounts of causation distinguish types of
causes. Both oxygen and a flame are metaphysical causes of
tinder's combustion. However, only the flame is causally
responsible for, and so a normative cause of, the combustion. Causal
responsibility for an event accrues to just the salient metaphysical
causes of the event. Causal decision theory is interested not only in
events for which an act is causally responsible, but also in other
events for which an act is a metaphysical cause. Expected utilities
that guide decisions are comprehensive.



Judea Pearl (2000) and also Peter Spirtes, Clark Glymour, and Richard
Scheines (2000) present methods of inferring causal relations from
statistical data. They use directed acyclic graphs and associated
probability distributions to construct causal models. In a decision
problem, a causal model yields a way of calculating an act's
effect. A causal graph and its probability distribution express a
dependency hypothesis and yield each act's causal influence
given that hypothesis. They specify the causal probability of a state
under supposition of an act. An act's expected utility is a
probability-weighted average of its expected utility according to the
dependency hypotheses that candidate causal models represent, as
Weirich (2015: 225-236) explains.



A causal model's directed graph and probability distribution
indicate causal relations among event types. As Pearl (2000: 30) and
Sprites et al. (2000: 11) explain, a causal model meets the causal
Markov condition if and only if with respect to its probability
distribution each event type in its directed graph is independent of
all the event type's nondescendants, given its parents. Given a
model meeting the condition, knowledge of all an event's direct
causes makes other information statistically irrelevant to the
event's occurrence, except for information about the event and
its effects. Knowledge of an event's direct causes screens off
evidence from indirect causes and independent effects of its causes.
Given a typical causal model for Newcomb's problem, knowledge of
the common cause of a decision and a prediction screens off the
correlation between the decision and the prediction.



Directed acyclic graphs present causal structure clearly, and so
clarify in decision theory points that depend on causal structure. For
example, Eells (2000) observes that choice is not genuine unless a
decision screens off an act's correlation with states. Joyce
(2007: 546) uses a causal graph to depict how this may happen in a
Newcomb problem that arises in a Prisoner's Dilemma with a
psychological twin. He shows that the Newcomb problem is a genuine
choice despite correlation of acts and states because a decision
screens off that correlation. Wolfgang Spohn (2012) constructs for
Newcomb's problem a causal model that distinguishes a decision
and its execution and argues that given the model causal decision
theory recommends one-boxing. An act in a decision problem may
constitute an intervention in the causal model for the decision
problem, as Meek and Glamour (1994) explain. Hitchcock (2016) and Joyce and Gibbard (2016)
maintain that treating an act as an intervention enriches causal
decision theory.



Timothy Williamson (2007: Chap. 5) studies the epistemology of
counterfactual, or subjunctive, conditionals. He points out their role
in contingency planning and decision making. According to his account,
one learns a subjunctive conditional if one robustly obtains its
consequent when imagining its antecedent. Experience disciplines
imagination. The experience leading to a judgment that a subjunctive
conditional holds may be neither strictly enabling nor strictly
evidential so that knowledge of the conditional is neither purely
*a priori* nor purely *a posteriori*. Williamson claims
that knowledge of subjunctive conditionals is foundational so that
decision theory appropriately grounds knowledge of an act's
choiceworthiness in knowledge of such conditionals.



Most texts on decision theory are consistent with causal decision
theory. Many do not treat the special cases, such as Newcomb's
problem, that motivate a distinction between causal and evidential
decision theory. For example, Leonard Savage (1954) analyzes only
decision problems in which options do not affect probabilities of
states, as his account of utility makes clear (1954: 73). Causal and
evidential decision theories reach the same recommendations in these
problems. Causal decision theory is the prevailing form of decision
theory among those who distinguish causal and evidential decision
theory.