## 1. The Basic Concept of Preference



In common parlance, the term "preference" assumes
different meanings, including that of comparative evaluation,
prioritisation or favouring, and choice ranking (See for instance
the *Oxford English Dictionary*). In this entry, we discuss the
notion of preference as *subjective comparative evaluations*,
of the form "Agent \(A\) prefers \(X\) to \(Y\)". This
characterisation distinguishes preference from other evaluative
concepts. 



Preferences are *evaluations*: they concern matters of value,
typically in relation to practical reasoning, i.e. questions about
what should be done, or should have been done. This distinguishes
preferences from concepts that concern matters of fact. 



Furthermore, preferences are *subjective* in that the
evaluation is typically attributed to an agent - where this
agent might be either an individual or a collective. This
distinguishes them from statements to the effect that "\(X\) is
better than \(Y\)" in an objective or otherwise impersonal
sense. The logic of preference has often also been used to represent
such objective evaluations (e.g. Broome 1991b), but the substantial
notion of preference includes this subjective element. 



Finally, preferences are *comparative* in that they express the
evaluation of an item \(X\)  *relative to* another item
\(Y\). This distinguishes them from monadic concepts like
"good", "is desired", etc. which only evaluate
one item. 



Most philosophers take the evaluated items to be propositions. In
contrast to this, economists commonly conceive of items as bundles of
 goods, represented as 
 vectors.[1] 
However, this approach has a difficult ambiguity. If
preferences are subjective evaluations of the alternatives, then what
matters are the results that can be obtained with the help of these
goods, not the goods themselves. Whether an agent has a preference
e.g. for a batch of wood over a crate of bricks will depend on whether
she intends to use it to generate warmth, build a shelter or create a
sculpture. Economists have tried to solve this ambiguity by coupling
preferences over goods with household production functions (Lancaster
1966, Becker and Michael 1973); but as these functions are very
difficult to determine, it is often thought more parsimonious to stick
with the sentential or propositional representations of states of the
world. 



Serious engagement with preferences began in the 20th
century. In the social sciences, the preference concept became
important for explanatory and predictive purposes with Irving
Fisher's (1892) and Vilfredo Pareto's (1909)
methodological criticisms of hedonistic cardinal utility. Previously,
economists largely agreed that decisions were motivated by the
individual's quest for pleasure, and that the difference in
quantity of pleasure derived from different alternatives was an
important influence on decisions. In that framework, the notion of
preference, to the extent that it was used at all, was merely derived
from hedonistic utility: \(X\) is preferred to \(Y\) iff
\(X\) yields more utility than \(Y\). Pareto argued that
because an accurate measurement procedure for cardinal hedonic utility
was not available, social scientists should constrain themselves to
merely ordinal comparisons (Bruni and Guala 2001). This argument
turned preference into a fundamental notion of the social sciences,
replacing (hedonic) utility. 



Economists in the 1930s (Hicks and Allen 1934) radicalised
Pareto's idea and argued that cardinal utility should be
excluded in order to expunge economics from psychological hedonism.
However, their concept of preference retained psychological content:
people are assumed to act purposefully and therefore to have
preferences that really constitute mental evaluations, rather than
being ex-post rationalisations of behaviour (Lewin 1996). Furthermore,
Ramsey (1926) and later von Neumann and Morgenstern (1944) devised
formal tools allowing the representation of preference magnitudes as
utility functions on an interval scale. This new utility, however, was very different from
the older hedonic concept: the preference concept was basic. The cardinal utility function was constructed to reflect the preference ordering, but was not completely determined by it. 



Psychologists also sought to move away from the old psychophysical
assumptions and began seeing mental concepts like preferences with
increased suspicion. Instead, they sought not only to connect and
measure psychological events, but indeed replace them by the
behavioural criteria with which they were hitherto connected. (See the
entry on
 behaviorism).
 Again, it was an economist, Paul Samuelson, who formulated this
principle most explicitly for the concept of preference. In 1938 he
suggested to "start anew ... dropping off the last vestiges
of the utility analysis" (1938, pp. 61-62). Preferences
were supposed to be defined in terms of choice, thus eliminating
reference to mental states altogether. Although this approach was highly influential at
the time, economists have largely not followed Samuelson in this
radical proposal (Hausman 2012), and it might indeed be the case that
Samuelson himself later changed his mind (Hands 2014). With the
increasing convergence of (parts of) economics and psychology, the
ordinal psychological interpretation of preferences appears to
currently dominate in these disciplines. However, there is an ongoing
discussion amongst philosophers whether the current concept of
preference used by economists is this mental,
"folk-theoretic" notion or a separate theoretical concept
(Maki 2000, Ross 2014, Thoma 2021). 



In philosophy, the concept of preference gained increased attention in
the wake of the conceptual developments in the social sciences.
Because the hedonic utility notion was increasingly questioned,
utilitarian philosophers sought alternative foundations for their
ethical theories. Today, *preferentialism* defends satisfaction
of individual preferences as the only intrinsic value bearer, and thus
is a subcategory of the broad welfarist family of value theories,
which identify intrinsic value with well-being. Few people defend the
view that well-being is constituted by the satisfaction of
*any* preference, but a number of authors defend refined
versions of preferentialism (e.g. Rawls 1971, Scanlon 1998).
Philosophers have also discussed the formal properties of preferences
in *preference logic*. To this we turn in the next section.



## 2. Preference logic



Although not all philosophical references to preference make use of
formal tools, preferences are almost always assumed to have structural
properties of a type that is best described in a formalized language.
The study of the structural properties of preferences can be traced
back to Book III of Aristotle's *Topics*. Since the early
twentieth century several philosophers have studied the structure of
preferences with logical tools. In 1957 and in 1963, respectively,
Soren Hallden and Georg Henrik von Wright proposed the
first complete systems of preference logic (Hallden 1957, von
Wright 1963). The subject also has important roots in utility theory
and in the theory of games and decisions. The preferences studied in
preference logic are usually the preferences of rational individuals,
but preference logic is also used in psychology and behavioural
economics, where the emphasis is on actual preferences as revealed in
behaviour.


### 2.1 Concepts and notation



There are two fundamental comparative value concepts, namely
"better" (strict preference) and "equal in value
to" (indifference) (Hallden 1957, 10). These terms are
used to express the wishes of persons, but they are also used for
other purposes, for instance to express objective or intersubjectively
valid betterness that does not coincide with the pattern of wishes of
any individual person. However, the structural (logical) properties of
betterness and value equality do not seem to differ between the cases
when they correspond to what we usually call "preferences"
and the cases when they do not. The term "preference
logic" is standardly used to cover the logic of these concepts
even in cases when we would typically not use the term
"preference" in a non-formalized context.



The relations of preference and indifference between alternatives are
usually denoted by the symbols \(\succ\) and \(\sim\) or alternatively by
\(P\) and \(I\). In accordance with a long-standing
philosophical tradition, \(A\succ B\) is taken to
represent "\(B\) is worse than \(A\)", as well
as "\(A\) is better than \(B\)".

 The objects
of preference are represented by the relata of the preference relation
\((A\) and \(B\) in \(A\succ B)\). In order
to make the formal structure determinate enough, every preference
relation is assumed to range over a specified set of relata. In most
applications, the relata are assumed to be mutually exclusive, i.e.
none of them is compatible with, or included in, any of the others.
Preferences over a set of mutually exclusive relata are referred to as
*exclusionary* preferences. When the relata are mutually
exclusive, it is customary to call the set of relata
an *alternative set*, or set of alternatives, \(\mathcal{A}\).


The following four properties of the two exclusionary comparative
relations are usually taken to be part of the meaning of the concepts
of (strict) preference and indifference:



\[\begin{align}
\tag{1}
A\succ B \rightarrow \neg(B\succ A) &\quad\text{(asymmetry of preference)}\\

\tag{2}
A\sim B \rightarrow B\sim A &\quad\text{(symmetry of indifference)}\\

\tag{3}
A\sim A &\quad\text{(reflexivity of indifference)}\\

\tag{4}
A\succ B \rightarrow \neg(A\sim B) &\quad\text{(incompatibility of preference} \\
&\quad\text{ and indifference)}
\end{align}\]


It follows from (1) that strict preference is irreflexive, i.e. that
\(\neg(A\succ A)\).



The relation \(\succcurlyeq\), "at least as good as" (or more
precisely: "better than or equal in value to"), can be
defined as follows:



\[
A\succcurlyeq B \leftrightarrow A\succ B \vee A\sim B \quad\text{(weak preference)}
\]


The alternative notation \(R\) is sometimes used instead of
\(\succcurlyeq\).



For reasons of convenience, weak preference is usually taken to be the
primitive relation of preference logic. Then both (strict) preference
and indifference are introduced as derived relations, as follows:



\[\begin{align}
A\succ B &\text{ if and only if } A\succcurlyeq B \text{ and } \neg(B\succcurlyeq A) \\

A\sim B &\text{ if and only if } A\succcurlyeq B \text{ and } B\succcurlyeq A
\end{align}\]


\(\succ\) is the *strict part* of \(\succcurlyeq\) and \(\sim\) its
*symmetric part*.



Two common notational conventions should be mentioned. First, chains
of relations can be contracted. Hence, \(A\succcurlyeq B\succcurlyeq
C\) abbreviates \(A\succcurlyeq B \wedge B\succcurlyeq C\), and
\(A\succ B\succ C\sim D\) abbreviates \(A\succ B \wedge B\succ C
\wedge C\sim D\). Second, the ancestral symbol \* is used to contract
repeated uses of the same relation; hence \(\succ\)\* stands for
\(\succ\) repeated any finite non-zero number of times (and similarly
for the other relations). Thus \(A\succ \*C\) denotes that either
\(A\succ C\) or there are \(B\_1 \ldots B\_n\) such that



\[
A\succ B\_1 \wedge B\_1 \succ B\_2 \wedge \ldots B\_{n-1}\succ B\_n \wedge B\_n\succ C.
\]

### 2.2 Completeness



In most applications of preference logic, it is taken for granted that
the following property, called *completeness* or
*connectedness*, should be satisfied:



\[
A\succcurlyeq B \vee B\succcurlyeq A
\]


or equivalently:



\[
A\succ B \vee A\sim B \vee B\succ A
\]


The following weaker version of the property is sometimes useful: 



\[
\text{If } A\ne B, \text{ then } 
 A\succcurlyeq B \vee B\succcurlyeq A \quad\text{(weak connectivity)}
\]

(*Completeness* holds if and only if both *weak
connectivity* and *reflexivity of indifference* hold.)




Completeness (connectedness) is commonly assumed in many applications,
not least in economics. Bayesian decision theory is a case in point.
The Bayesian decision maker is assumed to make her choices in
accordance with a complete preference ordering over the available
options. However, in many everyday cases, we do not have, and do not
need, complete preferences. Consider a person who has to choose
between five available objects \(A, B, C, D\), and \(E\). If she knows
that she prefers \(A\) to the others, then she does not have to make
up her mind about the relative ranking among \(B, C, D\), and
\(E\).



In terms of resolvability, there are three major types of preference
incompleteness. First, incompleteness may be *uniquely
resolvable*, i.e. resolvable in exactly one way. The most natural
reason for this type of incompleteness is lack of knowledge or
reflection. Behind what we perceive as an incomplete preference
relation there may be a complete preference relation that we can
arrive at through observation, introspection, logical inference, or
some other means of discovery. 



Secondly, incompleteness may be *multiply resolvable*, i.e.
possible to resolve in several different ways. In this case it is
genuinely undetermined what will be the outcome of extending the
relation to cover the previously uncovered cases.



Thirdly, incompleteness may be *irresolvable*. The most natural
reason for this is that the alternatives differ in terms of advantages
or disadvantages that we are unable to put on the same footing. A
person may be unable to say which she prefers--the death of two
specified acquaintances or the death of a specified friend. She may
also be unable to say whether she prefers the destruction of the
pyramids in Giza or the extinction of the giant panda. In
environmental economics, as a third example, it is a controversial
issue whether and to what extent environmental damage is comparable to
monetary loss.



Two alternatives are called "incommensurable" whenever it
is impossible to measure them with the same unit of measurement. Cases
of irresolvable incompleteness are often also cases of
incommensurability (Chang 1997). In moral philosophy, irresolvable
incompleteness is usually discussed in terms of the related notion of
a moral dilemma.


### 2.3 Transitivity



By far the most discussed logical property of preferences is the
following:



\[
A\succcurlyeq B \wedge B\succcurlyeq C \rightarrow A\succcurlyeq C
\quad\text{(transitivity of weak preference)}
\]


The corresponding properties of the other two relations are defined
analogously:



\[
A\sim B \wedge B\sim C \rightarrow A\sim C
\quad\text{(transitivity of indifference)}
\]

\[
A\succ B \wedge B\succ C \rightarrow A\succ C
\quad\text{(transitivity of strict preference)}
\]


A weak preference relation \(\succcurlyeq\) is
called *quasi-transitive* if its strict part \(\succ\) is
transitive.



Many other properties have been defined that are related to
transitivity. The following three are among the most important of
these:


* \(A\sim B \wedge B\succ C \rightarrow A\succ C
\quad\text{(IP-transitivity)}\)
* \(A\succ B \wedge B\sim C \rightarrow A\succ C
\quad\text{(PI-transitivity)}\)
* There is no series \(A\_1 ,\ldots ,A\_n\) of alternatives such that
\(A\_1 \succ \ldots \succ A\_n\succ A\_1\) 
\(\text{(acyclicity)}\)



All of these are ways of weakening the transitivity of \(\succcurlyeq\). Thus, if \(\succcurlyeq\) satisfies transitivity then \(\succ\) and \(\sim\)
are also transitive, and furthermore, IP-transitivity, PI-transitivity
and acyclicity hold.



Furthermore, if \(\succcurlyeq\) is transitive, then no cycles
containing \(\succ\) are possible, i.e. there are no \(A\) and \(B\)
such that \(A\succcurlyeq \*B\succ A\). Preferences with such a
\(\succ\)-containing cycle are called *cyclic preferences*.



Transitivity is a controversial property, and many examples have been
offered to show that it does not hold in general. A classic type of
counterexample to transitivity is the so-called Sorites Paradox. It
employs a series of objects that are so arranged that we cannot
distinguish between two adjacent members of the series, whereas we can
distinguish between members at greater distance (Armstrong 1939,
Armstrong 1948, Luce 1956). Consider 1000 cups of coffee, numbered
\(C\_0, C\_1, C\_2,\ldots\) up to \(C\_{999}\). Cup \(C\_0\) contains no
sugar, cup \(C\_1\) one grain of sugar, cup \(C\_2\) two grains
etc. Since one cannot taste the difference between \(C\_{999}\) and
\(C\_{998}\), one might consider them to be equally good (of equal
value), \(C\_{999}\sim C\_{998}\). For the same reason, we have
\(C\_{998}\sim C\_{997}\), etc. all the way to \(C\_1\sim C\_0\), but
clearly \(C\_0 \succ C\_{999}\). This contradicts transitivity of
indifference, and therefore also transitivity of weak preference.



In a related example proposed by Warren S. Quinn, a device has been
implanted into the body of a person (the self-torturer). The device
has 1001 settings, from 0 (off) to 1000. Each increase leads to a
negligible increase in pain. Each week, the self-torturer "has
only two options--to stay put or to advance the dial one setting.
But he may advance only one step each week, and he may *never*
retreat. *At each advance he gets $10,000.*" In this way
he may "eventually reach settings that will be so painful that
he would then gladly relinquish his fortune and return to 0"
(Quinn 1990, 79).



In an important type of counterexamples to transitivity of strict
preference, different properties of the alternatives dominate in
different pairwise comparisons. Consider an agent choosing between
three boxes of Christmas ornaments (Schumm 1987). Each box contains
three balls, coloured green, red and blue, respectively; they are
represented by the vectors \(\langle R\_1,G\_1,B\_1\rangle,\) \(\langle
R\_2,G\_2,B\_2\rangle\), and \(\langle R\_3,G\_3,B\_3\rangle\). The agent
strictly prefers box 1 to box 2, since they contain (to her) equally
attractive blue and green balls, but the red ball of box 1 is more
attractive than that of box 2. She prefers box 2 to box 3, since they
are equal but for the green ball of box 2, which is more attractive
than that of box 3. And finally, she prefers box 3 to box 1, since
they are equal but for the blue ball of box 3, which is more
attractive than that of box 1. Thus,



\[R\_1\succ R\_2\sim R\_3\sim R\_1,\]

\[G\_1\sim G\_2\succ G\_3\sim G\_1,\]

\[B\_1\sim B\_2\sim B\_3\succ B\_1;\]


and from this follows



\[
\langle R\_1,G\_1,B\_1\rangle \succ \langle R\_2,G\_2,B\_2\rangle \succ \langle R\_3,G\_3,B\_3\rangle \succ \langle R\_1,G\_1,B\_1\rangle.
\]


The described situation yields a preference cycle, which contradicts
transitivity of strict preference. (Notice the structural similarity
to Condorcet's Paradox, see the entry on
 voting methods.)



These and similar examples can be used to show that actual human
beings may have cyclic preferences. It does not necessarily follow,
however, that the same applies to the idealized *rational*
agents of preference logic. Perhaps such patterns are due to
irrationality or to factors, such as lack of knowledge or
discrimination, that prevent actual humans from being rational. There
is a strong tradition, not least in economic applications, to regard
full \(\succcurlyeq\)-transitivity as a necessary prerequisite of
rationality.



The most famous argument in favour of preference transitivity is the
money pump argument. The basic idea was developed by F.P. Ramsey
(1928a, 182), who pointed out that if a subject's behaviour
violated certain axioms of probability and preference, then 
he would be willing to buy a bet that yields a gain to the seller, and a loss to the buyer, no matter what happens. The argument was developed in more detail
in Davidson et al. (1955). 



The following example can be used to show how the argument works in a
non-probabilistic context: A certain stamp-collector has cyclic
preferences with respect to three stamps, denoted \(A, B\), and
\(C\). She prefers \(A\) to \(B, B\) to \(C\), and \(C\) to \(A\).
Following Ramsey, we may assume that there is an amount of money, say
10 cents, that she is prepared to pay for exchanging \(B\) for \(A,
C\) for \(B\), or \(A\) for \(C\). She comes into a stamp shop with
stamp \(A\). The stamp-dealer offers her to trade in \(A\) for \(C\),
if she pays 10 cents. She accepts the deal.



For a precise notation, let \(\langle X,V\rangle\) denote
that the collector owns stamp \(X\) and has paid \(V\) cents
to the dealer. She has now moved from the state
\(\langle A,0\rangle\) to the state \(\langle C,10\rangle\).



Next, the stamp-dealer takes out stamp \(B\) from a drawer, and
offers her to swap \(C\) for \(B\), against another payment
of 10 cents. She accepts, thus moving from the state
\(\langle C,10\rangle\) to \(\langle B,20\rangle\). The shop-owner
can go on like this forever. What causes the trouble is the following
sequence of preferences:



\[\begin{align}
\langle C,10\rangle &\succ \langle A,0\rangle \\
\langle B,20\rangle &\succ \langle C,10\rangle \\
\langle A,30\rangle &\succ \langle B,20\rangle \\
\langle C,40\rangle &\succ \langle A,30\rangle \\
\langle B,50\rangle &\succ \langle C,40\rangle \\
\langle A,60\rangle &\succ \langle B,50\rangle \\
&\vdots
\end{align}\]


The money-pump argument relies on a particular, far from
uncontroversial, way to combine preferences in two dimensions, which
is only possible if two crucial assumptions are satisfied: (1) The
primary alternatives (the stamps) can be combined with some other
commodity (money) to form composite alternatives. (2) Irrespectively
of the previous transactions there is always, for each preferred
change of primary alternatives, some non-zero loss of the auxiliary
commodity (money) that is worth that change. The money-pump can be
used to extract money from a subject with cyclic preferences only if
these two conditions are satisfied.



Another argument for the normative appropriateness of preference
transitivity suggests that transitivity is constitutive of the meaning
of preference, in addition to the minimal properties mentioned in
section 2.1. Drawing an analogy to length measurement, Davidson (1976,
273) asks: "If length is not transitive, what does it mean to
use a number to measure length at all? We could find or invent an
answer, but unless or until we do, we must strive to interpret
'longer than' so that it comes out transitive. Similarly
for 'preferred to'". Violating transitivity,
Davidson claims, thus undermines the very meaning of preferring an
option over others. 



Yet another argument rests on the importance of preferences for
choice. When agents choose from an 
alternative set, then preferences should be *choice guiding*.
They should have such a structure that they can be used to guide our
choice among the elements of that set. But when choosing e.g. from
\(\{A,B,C\}\), a preference relation \(\succ\) such that
\(A\succ B\succ C\succ A\) does not
guide choice at all: any or none of the alternatives should be chosen
according to \(\succ\). The transitivity of preference, it is therefore
suggested, is a necessary condition for a meaningful connection
between preferences and choice. A critic, however, can point out that
preferences are important even when they cannot guide choices. Take
e.g. preferences over lottery outcomes: these are real preferences,
regardless of the fact that one cannot choose between lottery
outcomes. Further, the necessary criteria for choice guidance are much
weaker than weak transitivity (Hansson 2001, 23-25; compare also
versions of decision theory in which transitivity fails, e.g. Fishburn
1991). Last, the indifference relation does not satisfy choice
guidance either. That does not make it irrational to be indifferent
between alternatives. Thus, choice guidance can be an argument for the
normative appropriateness of transitivity only under certain
restrictions, if at all (For further discussion, see Anand 1993). 


### 2.4 Order typology



One more property of preference relations needs to be specified. A
relation is *antisymmetric* if



\[
A\succcurlyeq B \wedge B\succcurlyeq A \rightarrow A=B
\text{ (antisymmetry of preference)}
\]


The categories summarized in the table below (based on Sen 1970a) are
standardly used to denominate preference relations that satisfy
certain logical properties.




|  |  |  |
| --- | --- | --- |
|  | *Properties* | *Name(s)* |
| 1. | reflexive, transitive | Preorder, Quasi-order |
| 2. | reflexive, transitive,
anti-symmetric | Partial order |
| 3. | irreflexive, transitive | Strict partial order |
| 4. | reflexive, transitive, complete | Total preorder, Complete quasi-ordering, Weak ordering |
| 5. | reflexive, transitive, complete,
anti-symmetric | Chain, Linear ordering, Complete
ordering |
| 6. | asymmetric, transitive, weakly
connected | Strict total order, Strong ordering |


### 2.5 Combinative preferences



Sections 2.1-2.4 were devoted to exclusionary preferences, i.e.
preferences that refer to a set of mutually exclusive alternatives. In
practice, people also have preferences between relata that are not
mutually exclusive. These are called *combinative preferences*.




Relata of combinative preferences are not specified enough
to be mutually exclusive. To say that one prefers having a dog over
having a cat neglects the possibility that one may have both at
the same time. Depending on how one interprets it, this preference
expression may say very different things. It may mean that one prefers
a dog (and no cat) to a cat (and no dog). Or, if one already has a
cat, it may mean that one prefers a dog and a cat to just having a
cat. Or, if one already has a dog, it may mean that one prefers just a
dog to both a cat and a dog. Combinative preferences are usually taken
to have states of affairs as their relata. These are represented by
sentences in sentential logic. It is usually assumed that logically
equivalent expressions can be substituted for each other.



Properties such as completeness, transitivity and acyclicity can be
transferred from exclusionary to combinative preferences. In addition,
there are interesting logical properties that can be expressed with
combinative preferences but not with exclusionary preferences. The
following are some examples of these (some of which are
controversial):



\[\begin{align}
p\succcurlyeq q \rightarrow p\succcurlyeq(p\vee q)\succcurlyeq q 
 &\quad\text{(disjunctive interpolation)} \\

p\succcurlyeq q \rightarrow \neg q\succcurlyeq \neg p
 &\quad\text{(contraposition of weak preference)} \\

p\sim q \rightarrow \neg q\sim \neg p
 &\quad\text{(contraposition of indifference)} \\

p\succ q \rightarrow \neg q\succ \neg p
 &\quad\text{(contraposition of strict preference)} \\

p\succcurlyeq q \leftrightarrow (p\wedge \neg q)\succcurlyeq(q\wedge \neg p)
 &\quad\text{(conjunctive expansion of} \\
 &\quad\text{ weak preference)} \\

p\succ q \leftrightarrow (p\wedge \neg q)\succ(q\wedge \neg p)
 &\quad\text{(conjunctive expansion of} \\
 &\quad\text{ strict preference)} \\

p\sim q \leftrightarrow (p\wedge \neg q)\sim(q\wedge \neg p)
 &\quad\text{(conjunctive expansion of indifference)} \\

(p\vee q)\succcurlyeq r \leftrightarrow p\succcurlyeq r \wedge q\succcurlyeq r
 &\quad\text{(left disjunctive distribution of } \succcurlyeq) \\

p\succcurlyeq(q\vee r) \leftrightarrow p\succcurlyeq q \wedge p\succcurlyeq r
 &\quad\text{(right disjunctive distribution of } \succcurlyeq)
\end{align}\]


Combinative preferences can be derived from exclusionary preferences,
which are then taken to be more basic. In most variants of this
approach, the underlying alternatives (to which the exclusionary
preferences refer) are possible worlds, represented by maximal
consistent subsets of the language (Rescher 1967, von Wright 1972). 
However, it has been argued that a more realistic
approach should be based on smaller alternatives that cover all the
aspects under consideration - but not all the aspects that might
have been considered. This approach may be seen as an application of
Simon's "bounded rationality view" (Simon 1957,
196-200).



The derivation of combinative preferences from exclusionary
preferences can be obtained with a representation function. By this is
meant a function \(f\) that takes us from a pair
\(\langle p,q\rangle\) of sentences to a set
\(f(\langle p,q\rangle)\) of pairs of alternatives
(perhaps possible worlds). Then
\(p\succcurlyeq\)*\(\_f\)q* holds if and only if
\(A\succcurlyeq B\) for all \(\langle A,B\rangle \in f(\langle p,q\rangle)\) (Hansson 2001,
70-73).


### 2.6 Preference-based monadic value predicates



In addition to the comparative notions, "better" and
"of equal value", informal discourse on values contains
monadic (one-place) value predicates, such as "good",
"best", "very bad", "fairly good",
etc. Predicates representing these notions can be inserted into a
formal structure that contains a preference relation.



Two major attempts have been made to define the principal monadic
predicates "good" and "bad" in terms of the
preference relation. One of these defines "good" as
"better than its negation" and "bad" as
"worse than its negation" (Brogan 1919).



\[\begin{align}
G\_N p \leftrightarrow p\succ \neg p &\quad\text{(negation-related good)} \\
B\_N p \leftrightarrow \neg p\succ p &\quad\text{(negation-related bad)}
\end{align}\]


The other definition requires that we introduce, prior to
"good" and "bad", a set of neutral
propositions. Goodness is predicated of everything that is better than
some neutral proposition, and badness of everything that is worse than
some neutral proposition. The best-known variant of this approach was
proposed by Chisholm and Sosa (1966). According to these authors, a
state of affairs is indifferent if and only if it is neither better
nor worse than its negation. Furthermore, a state of affairs is good
if and only if it is better than some indifferent state of affairs,
and bad if and only if some indifferent state of affairs is better
than it.



\[\begin{align}
G\_I p \leftrightarrow (\exists q)(p\succ q\sim \neg q)
 &\quad\text{(indifference-related good)} \\
B\_I p \leftrightarrow (\exists q)(\neg q\sim q\succ p)
 &\quad\text{(indifference-related bad)}
\end{align}\]


The negation-related and the indifference-related "good"
respectively "bad" do not necessarily coincide. Both
definitions have been developed with complete preference relations in
mind, but formal models are available that do not require completeness.
(Hansson 2001)



A proposal for defining preferences in terms of the monadic predicates
for "good" and "bad" was put forward by van
Benthem (1982, p. 195). It assumes that goodness and badness are
defined in relation to an alternative set, so that for instance
\(G\_{\{x,y\}}x\) means that \(x\) is good among the alternatives in
\(\{x,y\}\) and \(B\_{\{x,z,w\}}x\) that \(x\) is bad among the
alternatives in \(\{x,z,w\}\). This gives rise to the following
definitions: 



\(x \succ y\) if and only if
\(G\_{\{x,y\}}x \amp \neg G\_{\{x,y\}}y\)  (goodness-based preference)




\(x \succ y\) if and only if
\(B\_{\{x,y\}}y \amp \neg B\_{\{x,y\}}x\)  (badness-based preference)




However, these two definitions are not equivalent, and neither of them
is plausible in all cases. For instance, let \(x\) be good and not bad
in the context \(\{x,y\}\), and let \(y\) be neither good
nor bad in the same context. Then \(x \succ y\) holds according to
first definition but \(\neg(x \succ y)\) according to the second. To
avoid such problems, Hansson and Liu (2014) proposed the following
definition: 



\(x \succ y\) if and only if either
\(G\_{\{x,y\}}x \amp \neg G\_{\{x,y\}}y\) or
\(B\_{\{x,y\}}y \amp \neg B\_{\{x,y\}}x\) (bivalently based preference)



## 3. Numerical Representation of Preference



Preferences can be represented numerically.
\(A\succ B\) is then expressed by a set if numerical utility
functions \(\{u\_i\}\), each of which assigns a higher value to \(A\) than to
\(B\), while \(A\sim B\) is represented by
assigning the same value to the two. Such numerical representations
might serve different purposes, one being that utility functions can
be analysed with the tools of maximisation under constraints, as done
in economics. It is important, however, to stress the limitations of
such representations. First, not all preferences can be represented
numerically. Second, there are different scales by which preferences
can be represented, which require premises of different strengths.
Third, the resulting utility representation must be clearly
distinguished from the older hedonistic concept of utility. 


### 3.1 Representing preferences ordinally



The simplest form of numerical representation stipulates the following
equivalence: 



\[
A\succ B \text{ iff } u(A)\gt u(B) \quad\text{(Ordinal representation)}
\]


Any function \(u\) that assigns a larger number to \(A\)
than to \(B\) will work as such a representation. Consequently,
the function \(u\) can be replaced with any function
*u'* as long as *u'* is a *positive
monotone transformation* of \(u\). As this transformation
property is the defining characteristic of *ordinal scales*, we
call this an ordinal preference representation (See the entry on
 measurement in science.)



A preference relation has an ordinal representation only if it
satisfies both completeness and transitivity. However, even if
\(\mathcal{A}\) is finite, there can be
complete and transitive preference relations on \(\mathcal{A}\)
that cannot be represented by a utility
function (for a counter-example based on a lexicographic preference
relation, see Debreu
 1954).[2]




An *incomplete* preference ordering also has a value
representation of the following type:



\[
\text{If } A\succ B, \text{ then } u(A)\gt u(B)
\]


The inverse is obviously not true. However, under fairly wide
circumstances, given the set of *all* utility functions thus
defined, one can find the preference relation (Aumann 1962). 


### 3.2 Representing preferences cardinally



Ordinal numerical representations of preference are just a convenient
tool - they do not represent any information that cannot be
represented by the relation \(\succ\) itself. However, there is
relevant information about preferences that is not represented by the
relation \(\succ\) itself. For example, when an agent expresses two
preferences, say \(A\succ B\) and \(C\succ B\), one might ask *how
much* the agent prefers \(A\) to \(B\), in particular in
comparison to how much she prefers \(C\) to \(B\). To answer this
question, one needs to determine both a measurement procedure for
measuring preference intensities and a measurement scale for
representing these measurements. 



Measurement scales that represent magnitudes of intervals between
properties, or even magnitudes of ratios between properties, are
called *cardinal* scales. Although the discussion in the social
sciences often merely distinguishes between ordinal and cardinal
preference measures, it is important to further distinguish between
interval and ratio scales among the latter, as these require different
assumptions to hold. An interval scale allows for meaningful
comparisons of differences (e.g. "43degC is as much hotter than 41
degC as 29degC is hotter than 27degC"). In addition, a ratio
scale also allows for meaningful comparisons of ratios (e.g. "12m is twice
as long as 6m"). Although there have been some attempts to measure
preferences on a ratio scale (in particular, see Kahneman and
Tversky's (1979) *Prospect Theory*, which requires a
natural zero point and thus a ratio scale), most efforts have focussed
on measuring preferences on an interval scale. 



The basic idea of interval preference measurement is to assume that
acts have uncertain consequences, and that each act is equivalent to a
lottery between these outcomes. An agent who expresses a preference
for an act over others by choosing it thus expresses a preference for
the equivalent lottery over the lotteries equivalent to other acts.
The utilities of these acts are then determined as the expected
utilities of the equivalent lotteries, calculated as the
probability-weighted average of the lottery's consequences. This
approach was pioneered by Ramsey (1928) and refined by von Neumann and
Morgenstern (1944); other approaches have been presented by Savage
(1954/72) and Jeffrey (1965/90). There are substantial differences
between these approaches and their respective assumptions. For more
detail, see
 decision theory.


### 3.3 Alternative utility-based representations of preference



As mentioned in section 3.1, all transitive and complete preference
relations can be represented by a utility function according to the
following simple relationship



\[
A\succ B \text{ iff } u(A) \gt u(B)
\]


However, as can be seen from the Sorites paradox discussed in section
2.3, this recipe for the representation of preferences is too
demanding for some purposes. If \(u(A)\gt u(B)\), but \(u(A)-u(B)\) is
so small that it cannot be discerned, then \(A\succ B\) cannot be
expected to hold. One way to represent this feature is to employ a
cardinal utility function and to introduce a fixed limit of
indiscernibility, such that \(A\succ B\) holds if and only if
\(u(A)-u(B)\) is larger than that limit. Such a limit is commonly
called a *just noticeable difference* (JND).



\[
A\succ B \text{ iff } u(A)-u(B) \gt \delta,
 \quad(\text{JND representation, } \delta \gt 0)
\]


If the set of alternatives is finite, then \(\succcurlyeq\) has a JND
representation if and only if \(\succcurlyeq\) is complete, and
satisfies the two properties that for all (A, B, C, D):



\[\begin{align}
&A\succ B \wedge B\succ C \rightarrow A\succ D \vee D\succ C \text{ and } \\

&A\succ B \wedge C\succ D \rightarrow A\succ D \vee C\succ B.
\end{align}\]



Another interesting construction is to assign to each alternative an
interval instead of a single number. This requires two real-valued
functions, \(u\_{max}\) and \(u\_{min}\), such that for all \(A,
u\_{max}(A)\ge u\_{min}(A)\). Here, \(u\_{max}(A)\) represents the upper
limit of the interval assigned to \(A\), and \(u\_{min}(A)\) its lower
limit. \(A \succ B\) holds if and only if all elements of the interval
assigned to \(A\) have higher value than all elements of the \(B\)
interval:



\[
A\succ B \text{ iff } u\_{min}(A)\gt u\_{max}(B) 
 \quad\text{(Interval representation)}
\]


It has been shown that a preference relation \(\succcurlyeq\) has an
interval representation if and only if it satisfies completeness and
the property that for all \(A, B, C, D\):



\[
A\succ B \wedge C\succ D \rightarrow A\succ D \vee C\succ B
\]

(Fishburn 1970).



A final generalization is to let the threshold of discrimination
depend on both relata.



\[\begin{align}
A\succ B \text{ iff } u(A)-u(B)\gt \sigma(A, B) &\quad(\text{Doubly variable threshold}\\ 
 &\quad\text{ representation } \sigma(A,B)\gt 0) 
\end{align}\]



If the set of alternatives is finite, then \(\succcurlyeq\) has a doubly
variable threshold representation if and only if it satisfies
acyclicity.



For more details on these representations, see Scott and Suppes (1958)
and Abbas (1995).


## 4. Preferences combination



In practical decision making, there are often several preference
relations that have to be taken into account. The different preference
relations represent different aspects of the subject matter concerned
by the decision. For instance, when choosing among alternative
architectural designs for a new building, we will have a whole set of
aspects, each of which can be expressed with a preference relation:
costs, sustainability, aesthetics, fire safety etc. In other cases, the
various preference relations represent the wishes or interests of
different persons. This applies for instance when a group of people
with different preferences plan a joint vacation trip. 


### 4.1 Managing preference conflicts



The most convenient way to represent problems with multiple preference
aspects is to introduce a vector \(\langle \succcurlyeq\_1 ,\ldots
,\succcurlyeq\_n \rangle\) whose elements are the preference relations
we have to take into account. For simplicity, we can assume that all
these preferences are complete (or we can treat incompleteness as
indifference). We can call such a vector *conflict free* if and
only if it has no elements \(\succcurlyeq\_k\) and \(\succcurlyeq\_m\)
such that \(X \succ\_k Y\) and \(Y \succ\_m X\) for any alternatives
\(X\) and \(Y\). If \(\langle \succcurlyeq\_1 ,\ldots ,\succcurlyeq\_n
\rangle\) is conflict free, then we can define a combined preference
relation \(\succcurlyeq\) such that (1) \(X\succ Y\) if there is some
\(\succcurlyeq\_k\) such that \(X\succ\_k Y\), (2) \(Y\succ X\) if there
is some \(\succcurlyeq\_k\) such that \(Y\succ\_k X\), and (3) otherwise
\(X\sim Y\). This is a plausible construction for conflict free
preferences, since the combined preference relation does not
contradict any of the strict preferences expressed in the component
vectors. 



For *conflictual* preference vectors, i.e. vectors that are not
conflict free, there is no such simple solution that is plausible in
all applications. There are five common ways to deal with conflicts
among preferences.



*1. Reduction to a single dimension.* Such reductions are
usually performed by first translating all preference relations into
some numerical value, and then, for each alternative, adding up the
values assigned to it for all aspects. In utilitarian moral
philosophy, a fictional value unit, "utile", is used for
this purpose. In economics, cost-benefit analysis (CBA) instead uses monetary units. However in many cases
there is uncertainty or disagreement on how the reductions should be
performed. 



*2. Assuming that all conflicts cancel each other out.* This
amounts to extending the above definition of a combined preference
relation so that \(X\sim Y\) will hold in all cases of conflict,
i.e. whenever there are \(\succcurlyeq\_k\) and \(\succcurlyeq\_m\) such
that \(X \succ\_k Y\) and \(Y \succ\_m X\). Although usually not
expressed in this way, this is the effect of applying efficiency as
the sole criterion (e.g. Pareto efficiency as the sole criterion in a
multi-person case). This method has the obvious disadvantage that it
sometimes lets a small disadvantage in one dimension outweigh a large
advantage in another dimension. 



*3. Majoritarian solutions.* Another way to deal with conflicts
is to look for the alternatives that are favoured by most (but not necessarily all) of the preference relations. This requires that the aspects
covered by the different preference relations are valued equally.
Therefore, this solution is commonly used when the elements of the
vector correspond to the wishes or interests of different persons, but
not when they correspond to more general aspects of a decision (such
as sustainability and aesthetics in the example of of choosing an
architectural design). 



*4. Intuitive weighing.* In practice, decision makers often
weigh different preference dimensions against each other intuitively,
without any prior attempt to reduce the multi-dimensionality of the
decision. This way of dealing with multiple preferences has practical
advantages, but it also has the disadvantage of lacking efficient
mechanisms for ensuring consistency in decision-making. 



*5. Modifying at least one of the conflicting preference
relations.* This is what happens when people involved in
negotiations or discussions approach each other's views in ways that
make their preference relations less conflicting. The Delphi method is
a systematized procedure that can be used to reduce interindividual
differences in preferences. On an intraindividual level, strivings for
a reflective equilibrium can take the form of adjusting preference
relations that concern different aspects of an issue to each
other. From a psychological point of view, such changes can be
described as reductions of cognitive dissonance in value issues.



Voting procedures are often described as methods for aggregating or
combining preferences. Such aggregation can also be performed by a
benevolent planner striving to take the wishes and/or interests of all
concerned persons into account. The aggregation of preferences is a
major topic of social choice theory. See the entry on 
 social choice theory.


### 4.2 Total and partial preferences



Consider again the choice among alternative architectural designs for
a new building. As indicated above, our preferences can be expressed
with a vector \(\langle \succcurlyeq\_1 ,\ldots ,\succcurlyeq\_n
\rangle\), each of whose elements represents our *partial*
preferences with respect to some particular aspect such as
sustainability or aesthetics. If we manage to aggregate the vector to
a single preference relation \(\succcurlyeq\), then \(\succcurlyeq\)
represents our *total preferences*, or, as they are also
called, our "preferences *tout court*" or
"preferences all things considered". 



Some authors have argued that the preference notion in economics
always refers to total preferences (Hausman 2012). However, there are
also economists who recognize partial preferences, often identifying
them with preferences over properties or characteristics of economic
goods (Lancaster 1966). In contrast, philosophers often treat partial
preferences as referring to different reasons that one may have to
prefer one of the options to another (Pettit 1991, Osherson and
Weinstein 2012). 



Authors who recognize partial preferences usually give them priority,
and consider total preferences to be completely determined by the
partial preferences. In other words, they assume that a total
preference relation is uniquely determined by the partial preference
relations through a process of aggregation. There are different views
on the nature of this process. According to a *quantitative*
approach, each partial preference is connected with a cardinal partial
utility function for the aspect in question, and the total preference
relation can be obtained by aggregating these partial utility
functions using an appropriate set of weights. This requires strong
assumptions of preference independence in order to justify additivity
of utility (Keeney and Raiffa 1993). 



An alternative strategy employs *tools from social choice
theory* to map a vector of partial preferences into a total
preference relation. This approach only makes use of ordinal
information, and disregards any utility information that has no impact
on the partial preference relations. Unsurprisingly, the impossibility
results of social choice theory affect this method. Steedman and
Krause (1986) have shown that there is no rule for deriving total
preferences from a preference vector that satisfies four seemingly
plausible conditions and also yields a transitive and complete total
preference ordering. When applied to an intrapersonal conflict this
means that an agent may be rational in the sense of having a complete
and transitive (partial) relation for each of the aspects, but may
still be irrational either in the sense of not satisfying plausible
conditions on the relations between partial and total preferences, or
in the sense of not having a complete and transitive total preference
ordering for her overall appraisal of the options in question. This
argument connects with a long philosophical tradition, including Plato
and Bishop Butler, that draws an analogy between intrapersonal
conflicts and citizens' conflicting preferences within a state.




There are also authors who reject the idea that total preferences are
uniquely derivable from partial preferences. Instead they claim that
total preferences are *constructed* at the moment of
elicitation, and thus influenced by contexts and framings of the
elicitation procedure that are not encoded in pre-existing partial
preferences (Payne, Bettman and Johnson 1993). Total preferences seem
to be influenced by direct affective responses that are independent of
cognitive processes (Zajone 1980). For instance, food preferences seem
to be partly determined by habituation and are therefore difficult to
explain as the outcome of a process exclusively based on well-behaved
partial preferences. According to this view, partial preferences are
in many cases *ex post* rationalisations of total preferences,
rather than the basis from which total preferences are derived. 



A closely related standpoint was expressed by Nozick (1981, 244):
"Reasons do not come with previously assigned weights; the
decision process is not one of discovering such precise weights but of
assigning them. The process not only weighs reasons, it (also) weights
them." According to Kranz (1991, 34), "[p]eople do *and
should* act as problem solvers, not maximizers, because they have
many different and incommensurable ... goals to achieve".
Much in the same vein, Levi (1986, 246) maintained that "an
agent may terminate deliberation and take decisions without having
resolved the moral, political, economic and aesthetic conflicts
relevant to their predicaments". 


## 5. Preferences and choice



There is a strong tradition, particularly in economics, to relate
preference to choice. Preference is linked to hypothetical choice, and
choice to revealed preference. We begin this section by presenting
choice functions and some of their main properties. We then proceed to
discuss how choice functions and their properties can be derived from
preferences. Finally, we view the relationship from the other end, and
introduce some approaches to inferring preferences from observed
choices.


### 5.1 Choice functions and their properties



Given an alternative set \(\mathcal{A}\), we can represent
(hypothetical) choice as a function \(\mathbf{C}\) that, for any given
subset \(\mathcal{B}\) of \(\mathcal{A}\), delivers those elements of
\(\mathcal{B}\) that a deliberating agent has not ruled out for
choice. For brevity's sake we will call them 'chosen
elements'. The formal definition of a choice function is as
follows:




\(\mathbf{C}\) is a choice function for \(\mathcal{A}\) if and only if
it is a function such that for all \(\mathcal{B} \subseteq
\mathcal{A}\):


1. \(\mathbf{C}(\mathcal{B}) \subseteq
\mathcal{B}\), and
2. if \(\mathcal{B} \ne \varnothing\), then
\(\mathbf{C}(\mathcal{B}) \ne \varnothing\).




A large number of rationality properties have been proposed for choice
functions. The two most important of these are described here.



\[
\text{If } \mathcal{B} \subseteq \mathcal{A} \text{ then }
\mathcal{B} \cap \mathbf{C}(\mathcal{A})
\subseteq \mathbf{C}(\mathcal{B})
\quad\text{(Property } \alpha, \text{ "Chernoff")}
\]


This property states that if some element of a subset \(\mathcal{B}\)
of \(\mathcal{A}\) is chosen from \(\mathcal{A}\), then it is also
chosen from \(\mathcal{B}\). According to property \(\alpha\),
removing some of the alternatives that are not chosen does not
influence choice. This property has often been assumed to hold, but
counterarguments have been raised against it. Consider the following
example. Erna is invited to an acquaintance's house for
dinner. Her choice for dessert is between an apple (which is the last
piece of fruit in the fruit basket) \((X)\) and nothing instead
\((Y)\). Because Erna is polite, she chooses \(Y\). Had she faced a
choice between an apple \((X)\), nothing \((Y)\) and an orange
\((Z)\), she would have taken the apple. Thus her choices are:



\[
\mathbf{C}(\{X,Y,Z\}) =\{X\}
\]


and



\[\mathbf{C}(\{X,Y\}) =\{Y\},
\]


which violate property \(\alpha\). More generally, property \(\alpha\) has
been contested with reference to cases when alternatives are preferred
for their position in an alternative set, when the set of alternatives
itself constitutes important information about the alternative chosen,
or when certain alternatives provide the chooser with the freedom to
reject them (Sen 1993,
 501-503).[3]



The second property states that if \(X\) and \(Y\) are both
chosen from the subset \(\mathcal{B}\) of \(\mathcal{A}\),
then one of them cannot be chosen in \(\mathcal{A}\)
without the other also being chosen.



\[\begin{align}
\text{If } &\mathcal{B} \subseteq \mathcal{A}
\text{ and } X, Y \in \mathbf{C}(\mathcal{B}), \text{ then }
X \in \mathbf{C}(\mathcal{A}) \\
&\text{ iff } Y \in \mathbf{C}(\mathcal{A}) 
 \quad(\text{Property } \beta)
\end{align}\]


To exemplify property \(\beta\), suppose that we have three
restaurants, \(X, Y\) and \(Z\), within walking distance, and two
additional restaurants, \(V\) and \(W\), that we can reach by
car. Furthermore, suppose that we begin by choosing among the
restaurants within walking distance. We agree that the choice is
between \(X\) and \(Y\), but we find no reasons to choose one of them
rather than the other, i.e. \(\mathbf{C}(\{X, Y, Z\})=\{X,Y\}\). 
Then we find out that we do in fact have access to a car.
Property \(\beta\) says that if \(X\) is one of the chosen restaurants
in this situation, then so is \(Y\), and vice versa. In other words:
\(X\in \mathbf{C}(\{X, Y, Z, V, W\})\) holds if and only if \(Y \in
\mathbf{C}(\{X, Y, Z, V, W\})\). 



A third property, \(\gamma\), is described in
 footnote.[4]




The above properties are mainly found in the social choice literature.
A related property in the economic literature is the so-called
*Weak Axiom of Revealed Preferences* (WARP). It says that if
\(X\) is chosen when \(Y\) is available, then there must not
be an alternative set \(\mathcal{B}\) containing both
alternatives for which \(Y\) is chosen and \(X\) is not.




If \(X,Y\in \mathcal{A}\) and \(X\in \mathbf{C}(\mathcal{A})\), then
for all \(\mathcal{B}\), if \(X\in \mathcal{B}\), and \(Y\in
\mathbf{C}(\mathcal{B})\), then \(X\in \mathbf{C}(\mathcal{B})\)
  (WARP)




WARP is equivalent to the combination of properties \(\alpha\) and \(\beta\)
(Sen 1971, 50). A stronger version, SARP, is discussed in the first
part of the supplementary document.



The Strong Axiom of Revealed Preference




Counterexamples have been offered to show that these properties are
not plausible in all situations. Consider an agent who chooses to stay
at a friend's house for a cup of tea \((T)\) rather than to go
home \((H)\), but who leaves in a hurry when the friend offers a
choice between tea and cocaine \((C)\) at his next visit. Then
\(\mathbf{C}(\{T,H\}) =\{T\}\) and \(\mathbf{C}(\{T,H,C\}) =\{H\}\),
and hence the visitor violates both properties \(\alpha\) and
\(\beta\). 


### 5.2 Determining choice from preference



A choice function that is defined on the basis of a preference
relation is called *relational* (also *binary*). The
most obvious way to construct a choice function from a preference
relation \(\succcurlyeq\) is to have the function always choose the elements
that are best according to \(\succcurlyeq\):



*The best choice connection*  

\(\mathbf{C}^B(\mathcal{B}) = \{X \in \mathcal{B} \mid
\forall Y \in \mathcal{B}: (X\succcurlyeq Y)\}\) 




\(\mathbf{C}^B\) is a choice function (i.e. satisfies the defining
criteria for a choice function given in section 5.1) if and only if
\(\succcurlyeq\) is complete and acyclical. It will then also satisfy
properties \(\alpha\) and \(\gamma\). Furthermore, \(\mathbf{C}^B\)
satisfies property \(\beta\) if and only if \(\succcurlyeq\) is
transitive and complete (Sen 1970a, 19).



When the underlying preference relation is incomplete, there may not
be an element that is preferred to all other elements. A function
\(\mathbf{C}\) constructed according to the best choice connection
will then be empty, and hence not a choice function. To avoid this, an
alternative connection constructs the choice function as choosing
those elements that are not dispreferred to any other elements of the
set:



*The non-dominance choice connection*  

\(\mathbf{C}^L(\mathcal{B}) = \{X \in \mathcal{B} \mid
\forall Y \in \mathcal{B}: \neg(Y\succ X)\}\)




\(\mathbf{C}^L\) is a choice function if and only if \(\succcurlyeq\)
is acyclical. It will then satisfy properties \(\alpha\) and
\(\gamma\). Furthermore, \(\mathbf{C}^L\) satisfies property \(\beta\)
if and only if \(\succcurlyeq\) is transitive and complete (Herzberger
1973).



When the preference relation over \(\mathcal{A}\) is cyclical, neither
\(\mathbf{C}^B\) nor \(\mathbf{C}^L\) may be a relational choice
function for \(\mathcal{A}\). In the simplest case, with a cyclical
preference \(A\succ B\succ C\succ A\), \(\mathbf{C}^B(A,B,C) =
\mathbf{C}^L(A,B,C) = \varnothing\). Schwarz (1972) therefore proposes
a third relational choice function, which operates even on the basis
of cyclical preferences. Its basic idea is to select elements that
are not dominated by non-cyclical preference.


### 5.3 Inferring preference from choice



The close connections between preference axioms and choice axioms can
also be employed to construct a preference ordering from a choice
function that satisfies certain axioms. In economics, the *revealed
preference* approach has been used to define preference in terms
of choice. Historically, this approach developed out of the pursuit of
behaviouristic foundations for economic theories--i.e. the
attempt to eliminate the preference framework altogether. Today, it
serves to derive preference orderings from an agent's observed
choices, and to test the empirical validity of the preference axioms
by testing for the violation of choice axioms (Grune-Yanoff
2004).



There are many ways to construct preference relations from observed
choices. The simplest method defines an alternative \(X\) as "at
least as good as" an alternative \(Y\) if and only if
\(X\) is chosen from some set of alternatives that also contains
\(Y\).



\[\begin{align}
\tag{1}
X\succcurlyeq^S Y &\text{ iff for some }
 \mathcal{B}, X\in \mathbf{C}(\mathcal{B}) \text{ and } Y\in \mathcal{B} \\

X\succ^S Y &\text{ iff } X\succcurlyeq^S Y \text{ and not } 
 Y\succcurlyeq^S X \\

X\sim^S Y &\text{ iff } X\succcurlyeq^S Y \text{ and }
 Y\succcurlyeq^S X
\end{align}\]


If the choice function is defined over all subsets of \(\mathcal{B},
\succcurlyeq^S\) is complete. \(\succ^S\) does not necessarily
satisfy transitivity of strict preference, transitivity of
indifference, IP- or PI-transitivity. Two further methods are
described in the supplementary document:



The Strong Axiom of Revealed Preference




The formal relation to choice raises the question of the ontological status of preferences. Are preferences prior to choices, and function potentially as their cause? Or are preferences merely representations of actual or potential choice patterns? Debates about how to interpret preferences in economics have a long history, and in recent years the topic has received renewed attention. This discussion tended to focus on an opposition between behaviourist vs. mentalist interpretations. Many (but certainly not all) economists have interpreted the dominant revealed preference theory in behaviourist terms, claiming for example that "Standard economics does not address mental processes, and as a result, economic abstractions are typically not appropriate for describing them" (Gul and Pesendorfer 2008, 21). Consequently, these authors take the formal relations presented in this section as the determinants of the ontological question what preferences are. Furthermore, they also reject the interpretation of preferences as causes of choice, instead insisting that they only capture choice patterns (Binmore 2008, 19-22).



Mentalists, in contrast, insist that choices and preferences are
entities of quite different categories: Preferences are *states of
mind* whereas choices are *actions*. Consequently, they
reject the strong behaviourist program (Hausman 2012). An important
mentalist argument against behaviourist interpretations concerns the
role of beliefs in decision making. According to this argument, some
choices are not based on stable preferences over actions, but are
constructed from more basic cognitive and evaluative elements. A
simple choice - like choosing between two pieces of
candy - might be based on a preference for a world in which
one eats candy \(X\) over a world in which one eats candy \(Y\). But
more complex choices - e.g. choosing one's higher
education - depend on what one *believes* these choices to
bring about, and how one evaluates these consequences. In those cases,
a more complex framework specifies beliefs about the probability or
plausibility of possible states of the world, preferences over the
consequences of choices in those worlds, and an aggregation mechanism
of these preferences under those beliefs. The need for such a
cognitive structure to account for preferences has been proposed as an
argument against the ontological reduction of preferences to choices
(Hausman (2012).



Another argument against behaviourist interpretations points to the
apparent existence of preferences over alternatives that one cannot
choose between - for example preferences for winning a
certain prize of a lottery, or for particular configurations of
Paradise. This contradicts the claim that preferences exclusively
transpire from choices. One way to substantiate preferences over
alternatives that one cannot choose between is to ask people what they
prefer. Their answers can be interpreted as further choice
evidence - as verbal or writing *behaviour*. This
interpretation treats their answers on a par with all other forms of
behaviour. Alternatively, their answers can be interpreted
as *introspective* reports. This interpretation treats answers
as agents' privileged access to their own minds. Furthermore,
mentalists also distinguish between those agents who indeed have
preferences as states of minds - e.g. humans, and maybe
higher animals - and those agents who do
not - e.g. machines, plants or institutions. The former
category may choose on the basis of their preferences, and hence the
above-discussed effort can aim at eliciting the preferences on which
their choices are based. The latter category, despite their lack of
states of mind, may nevertheless exhibit behaviour that can be
interpreted as relational choice.



Functionalists' accounts of preferences deny such an
intrinsically mental interpretation of preferences, without reducing
them to behaviour. In functionalism, the nature of preferences, like
that of other mental states, is determined by what roles they play
- e.g. in evaluating choice alternatives, and in motivating
actions. Yet who assigns these roles, and what justifies such
assignments? Most authors agree that it is "the ontological
commitments of our best theory or theories in the relevant area"
that perform this assignment (List and Dietrich 2016, 257). How
preferences are individuated, then, is dependent on the purposes for
ascribing them. For example, revealed preferences are considered real
if they are non-redundantly useful for describing regularities in an
individual's behaviour:



> 
> To show that RPT [revealed preference theory] is useful we must find
> some real structures that are usefully measured - where
> "usefully" means nonredundantly relevant to explanation
> and prediction-using coefficients and relations defined by its
> axioms. (Ross 2005, 143)
> 



Revealed preference theory is nonredundantly relevant, Ross claims,
because it makes sense of patterns like intentionality and agency that
don't reduce to physical patterns. To the extent that this is
not merely a predictive but an explanatory task, one needs to admit
the existence of these patterns, thus endorsing "intentional
stance functionalism without sliding into instrumentalism" (Ross
2005, 143-4). According to such a view, preferences are
identified by the causal roles they play in generating human, animal
and machine behaviour; and they are real to the extent that the best
theories of such behaviour require their attribution.



Recently, some authors have sought to dissolve the simple dialectic
between mentalism and behaviourism. Guala (2019, 389) argues that the
debate is mainly motivated by a methodological, rather than an
ontological concern: whether or not economics should accept a healthy
injection of psychological theory and methods depends on the
explanatory tasks at hand. Clark (2020) reasons that whether and how
to treat preferences as genuine cognitive variables is a question that
depends on the epistemic purposes of the investigation (or as he
phrased it, "the ultimate aims of economics"). Similarly,
Thoma (2021) and Vredenburgh (2020) argue that for their specific
purposes, economists often have good reasons to largely "black
box" the causes of choice in their modelling of economic
behaviour.


## 6. Preference and welfare



Preference relates to welfare in rather intricate ways. Welfare is a
fundamental concept in moral philosophy and economics. It refers to
the fundamental good for individual human beings, and it is therefore
an anthropocentric and individualist concept. In order to clarify the
relationship between preference and welfare we need to distinguish
between three variants of the concept of welfare. 



According to the *material view*, a person's welfare is a
matter of her material conditions, such as access to food, shelter,
healthcare and, generally speaking, the necessities and perhaps
luxuries of life. This view of welfare has been criticized for being
materialistic in the sense of pursuing material possessions at the
expense of higher values. It also has to face the difficulties
inherent in weighing different material goods against each other. 



The other two principal views both treat welfare as a mental rather
than a material issue. The *wish-based mental view* uses each
person's wishes (i.e. preferences in the informal sense of that
word) as the criterion of welfare (Sen 1979). A person is considered
to have more welfare, the more her wishes are satisfied. If this view
is applied within a utilitarian framework, then it gives rise to
preference utilitarianism. This view of welfare has difficulties in
dealing with misinformed and self-defeating wishes. It can also have
difficulties with certain types of other-regarding wishes, e.g.
malevolent ones. The usual way to deal with this is to require that
preferences are filtered ("laundered") and/or refined
before they are used to judge a person's welfare. 



The filtering ("laundering") of preferences can be
justified by the everyday experience that some preferences are much
more important for a person's well-being than others. It can be
argued that a plausible preference-based account of welfare cannot be
based on total preferences, but would have to be based on a subset of
"core" preferences that are important for the individual.
The determination of that subset is expectedly contentious. If it is
to be determined by others than the individual whose welfare is
concerned, then problems of paternalism will be difficult to avoid.




Refinement of preferences is usually assumed to result in the
person's (hypothetical) "rational" or "ideally
considered" preferences. "My ideally considered
preferences are those I would have if I were to engage in
thoroughgoing deliberation about my preferences with full pertinent
information, in a calm mood, while thinking clearly and making no
reasoning errors." (Arneson 1989, 83; cf. Brandt 1979 and Rosati 2009) A problem
for this view is that there does not seem to be any easy way to
determine, based on a person's preferences and circumstances,
what her ideally considered preferences would be. 



Finally, the *state-dependent mental view* identifies a
person's welfare with some mental state such as happiness or
satisfaction. It is assumed that the relevant states of mind can at
least in principle be judged by external assessors. If such a view is
applied within a utilitarian framework, then it can give rise to
hedonistic utilitarianism. This view can be criticized for paternalism
and for being uncritical towards arrangements in which individuals are
happy in spite of being oppressed or deprived. Furthermore, the
difficulties involved in comparing mental properties such as happiness
in different persons creates a problem for views that depend on such
comparisons for the determination of welfare. However, recent work on
the measurement of happiness and life satisfaction has challenged that
view, and may have opened up new avenues for research in both
economics and moral philosophy. (Ng 1997) 



Only one of these three views, namely the wish-based one, refers to
what we usually call preferences. However, the other two can also be
expressed in terms of preference (betterness) relations. We can use
such relations to order material conditions respectively mental states
in terms of how they satisfy the criteria of welfare that we have
chosen to apply. It is important to distinguish between on the one
hand preferences in the common sense of comparative likings, and on
the other hand the use of a preference relation to express grades of
any property whose structure satisfies the common formal requirement
of such a relation. 



A common problem for attempts to account for welfare in terms of
preference is that we expect a person's view of her own welfare
to be essentially self-regarding, but her preferences can refer to
concerns that are not self-regarding, such as her concerns for other
people, her views on social justice, and her commitments for instance
to traditions, conventions, and moral ideas (Sen 1977). Our choices
are influenced by this wide range of preferences. Therefore, it does
not seem possible to link preferences strongly to welfare and at the
same time link them strongly to choice. 


## 7. Preference change



Preferences relate to time in several ways. Preference at one point in
time can refer to what happens or happened at other points in time.
Furthermore, preferences can change over time, due to changes in
beliefs, values, tastes, or a combination of these. Section 7.1
explains why preference change requires explanatory and theoretical
 treatment. In the following sections, three types of explanatory models are
discussed. (For more detail, see Grune-Yanoff and Hansson 2009.) 
*Time preference* models (section 7.2) only refer to
the temporal relationship between the occurrence of a preference and
the objects it refers to. *Doxastic* change models (section
7.3) investigate how a change of an agent's beliefs leads to a
change in her preferences. *Valuational* change models (section
7.4) investigate how a change in an agent's basic evaluations
leads to a change in her preferences.


### 7.1 Evidence for preference change



Some authors have argued that preference change is only a superficial
perception, and that the underlying preferences remain stable over
time. But there are at least four arguments to the effect that
people's preferences really do change over time. First, many
successful explanations of behavioural change have interpreted the
empirical behavioural evidence as preference change. These
explanations can be differentiated into models of *external
influences* and models of *internal coherence*. External
influence models attempt to establish general links between external
events and agents' preference formations. They include, for
example, social imitation (Leibenstein 1950), parental influence
(Cavalli-Sforza 1973), habit formation (Pollack 1976), and the effect
of production patterns on consumption (Duesenberry 1949). Internal
coherence models take certain external influences as given, and model
preference change as an accommodation of these external influences.
They include, for example, religious convictions (Iannaccone 1990) and
the effect of cognitive dissonance on preferences (Elster 1982).



A second argument for preference change is based on the correlations
between physiological changes and changes in behaviour. Changes in
blood sugar levels, for example, are correlated to feeding behaviour,
sexual behaviour varies with hormonal changes, and many behavioural
patterns change with increasing age (for references and discussion,
see Loewenstein 1996). These correlations are not deterministic; such
behavioural changes can be resisted in many cases. It is plausible to
incorporate these potential physiological effects as *visceral
preferences* in the general preference framework, and to treat the
relevant physiological changes as closely connected with preference
changes.



Third, most humans have introspective evidence for their own
preferences changing over time. The favourite activities of a child
are replaced by new pleasures as we grow up. Thus was the experience
of Shakespeare's Benedick: "...but doth not the
appetite alter? A man loves the meat in his youth that he cannot
endure in his age" (*Much ado about nothing* Act II,
Scene III). Even in adult life, we are literally overcome by sudden
and very radical ("transformative") changes of preference. Paul (2014) has investigated the types of decision-making that this involves. It would be strange to
claim in such cases that it is only our beliefs about the different types of
activities that change. Explanations in terms of preference change are
much more in line with how we spontaneously interpret our experiences
of such changes.



Last, certain concepts like taste refinement or self-restraint cannot
easily be understood without a notion of real preference change. In
particular, self-restraint presupposes that the motivational
components of one's self can change, for example, through
maturation or social influence; and that one can and should plan
one's future self by curbing certain appetites or by designing
the environment in ways that affect one's preferences.


### 7.2 Time preferences



The value that we assign to obtaining an advantage or disadvantage
usually varies with the point in time when we obtain it. In typical
cases, values decrease with time. For instance, most of us would
prefer receiving a large sum of money now to receiving it five years
later. Analytically, this temporal factor of evaluations is often
separated from time-independent factors of evaluations.



The standard approach to this issue in economic analysis treats
preference as based on value. Value is dealt with in a bifactorial
model, in which the value of a future good is assumed to be equal to
the product of two factors. One of these factors is a time-independent
evaluation of the good in question, i.e. the value of obtaining it
immediately. The other factor represents the subject's pure time
preferences. It is a function of the length of the delay, and is the
same for all types of goods. The most common type of time preference
function can be written



\[
v(A,t) = \frac{v(A,t\_0)}{(1 + r)^{t-t\_0}}
\]


where \(r\) is a discount rate and \(t-t\_0\) the duration of the
delay. This is the *exponetentially discounted utility model* (DU), proposed by
Samuelson (1937), which still dominates in economic analysis.



The choice of a discount rate can have a large impact on the
calculated values. It is therefore often politically controversial. As
one example of this, the discount rate used in assessing the economic
effects of climate change can have significant consequences for the
policy recommendations that are based on these assessments.



There is some evidence that the standard discounted utility model
does not adequately represent human behaviour. For a simple example,
consider a person who prefers one apple today to two apples tomorrow,
but yet (today) prefers two apples in 51 days to one apple in 50 days.
Although this is a plausible preference pattern, it is incompatible
with the *exponentially* discounted utility model. It can
however be accounted for in a bifactorial model with a declining
discount rate. Pioneered by Ainslie (1992), psychologists and
behavioural economists have therefore proposed to replace
Samuelson's exponential discounting model with a model of
hyperbolic discounting. The hyperbolic model discounts the future
consumption with a parameter inversely proportional to the delay of
the consumption, and can therefore cover examples like the above. 



Other deviations from the discounted utility model have also been
demonstrated. Experimental evidence indicates that we tend to discount
gains more than losses, and small amounts more than large amounts.
Discount rates also differ between different goods (such as money and
health). For some--but only some--types of goods, improving
sequences of outcomes are preferred to declining sequences. For instance, many would prefer an increasing rather than a decreasing living standard across their work life. These are
all patterns that cannot be handled in the bifactorial model with its
object-independent time preferences (Loewenstein et al 2002). Given
the empirical evidence, it is an open question whether the concept of
"time preferences" is at all descriptively adequate.



It is a separate question whether pure time preferences are rational.
Critics argue that one should want one's life, *as a
whole*, to go as well as possible, and that counting some parts of
life more than others interferes with this goal (Pigou, 1920; Ramsey,
1928b; Rawls 1971). According to this view, it is irrational to prefer
a smaller immediate good to a greater future good, because now and
later are equally parts of one life. Choosing the smaller good or the
greater bad makes one's life, as a whole, turn out worse:
"Rationality requires an impartial concern for all parts of our
life. The mere difference of location in time, of something's
being earlier or later, is not a rational ground for having more or
less regard for it" (Rawls 1971, 293). Critics of pure temporal
preferences often attribute apparent departures from temporal
neutrality to a *cognitive illusion* (which causes people to
see future pleasures or pains in some diminished form) or to a
*weakness of will* (which causes people to choose options
against their better judgment).



Against the temporal neutrality of preferences, some have argued that
there is no enduring, irreducible entity over time to whom all future
utility can be ascribed; they deny that all parts of one's
future are *equally* parts of oneself (Parfit 1984). They
argue, instead, that a person is a succession of overlapping selves
related to varying degrees by memories, physical continuities, and
similarities of character and interests, etc. By this view, it may be
just as rational to discount one's "own" future
preferences, as to discount the preferences of another distinct
individual, because the divisions between the stages of one's
life may be as "deep" as the distinctions between
individuals.


 
If pure time preferences are rational, the question arises whether
they are rationally required to adopt a certain form. Economists
widely consider the EDU model to be the rational standard, to the
extent that hyperbolic discounting is considered an indicator of
irrationality worthy of policy intervention (Strotz 1956,
O'Donoghue and Rabin 2015). Samuelson (1937, 160), the inventor
of the EDU, was more cautious:



> 
> any connection between utility as discussed here and any welfare
> concept is disavowed. The idea that such a [mathematical]
> investigation could have any influence upon ethical judgments of
> policy is one which deserves the impatience of modern
> economists.



Recently, some authors developed critiques of the normative validity
of EDU. Hedden (2015) argues that defending EDU would force one to
make untestable distinctions between actual and ultimate
preferences. Callender (2021, see also replies in this special issue)
argues that EDU is not a universally valid standard, but instead is
dependent on contextual factors as yet unspecified; and that many of
the current policy recommendations derived from this purported
normative standard are questionable. 



A quite different critical approach to discounting is connected with
the idea of sustainability. If sustainability is interpreted as
meaning that future generations should have access to the same
resources as those that the present generation has at its disposal,
then sustainability is sure to be in conflict with economic policies
based on exponential discounting. However, there are also views on
sustainability that allow us to use up natural resources if we replace
them by non-natural resources such as new technologies that will
compensate for the loss. Such a "weak" notion of
sustainability appears to be compatible with policies based on
discounting of future effects, but it has been criticized for putting future generations at disadvantage. (Ng 2005)


### 7.3 Doxastic preference change



Two kinds of beliefs are especially important for doxastic models. The
first is the belief that the presence of state \(X\) makes a
desired state \(Y\) more likely. Take for example the belief that
fluoride prevents dental cavities. This can lead a person to prefer
fluoride toothpaste to others. If she comes to disbelieve this
connection, she may well abandon this preference. More generally, if
\(X\wedge Y\) is preferred to
\(X\wedge \neg Y\), then a rise of the probability that
\(Y\) given \(X\) will result in a rise in the desirability
of \(X\), and vice versa.



The second kind of belief relevant for doxastic preference change
concerns prospects that influence the preference for other prospects
without being probabilistically related. For example, one's
preference for winning a trip to Florida in the lottery will crucially
depend on one's belief about the weather there during the
specified travel time, even though these two prospects are
probabilistically unrelated. More generally, if
\(X\wedge Y\) is preferred to
\(X\wedge \neg Y\), with \(X\) and \(Y\)
probabilistically not correlated, then a rise of the probability that
\(X\) will result in a rise in the desirability of \(Y\)
(even if it does not affect the probability of \(Y)\), and vice
versa.



Jeffrey (1977) provides a simple model of preference change as the
consequence of an agent coming to believe a proposition \(A\) to
be true. His model incorporates both kinds of belief relevant for
doxastic preference change. It is based on the notion of conditional
preferences. Jeffrey treats preferences as a relation over
propositions, viz. sets of possible worlds. They are represented by a
utility function \(\mathbf{U}\) (see section 2), such that:



\[
X\succcurlyeq Y \text{ iff } \mathbf{U}(X) \ge \mathbf{U}(Y).
\]


\(\mathbf{U}(X)\) in turn is defined as the probability-weighted
average of the utility \(u\) of all the possible worlds \(w\) in which
\(X\) is true. For unconditional preferences the weighing is built on
the probability function \(P\) that represents the agent's
actual information. The conditional preference ordering
\(\succcurlyeq\_A\), in contrast, is based on \(P\_A\), the probability
distribution representing the counterfactual scenario that the agent
accepts proposition \(A\) as true. (For more discussion on the
existence and uniqueness conditions of conditional preferences, see
Luce and Krantz 1971, Joyce 1999, chapter 4, Bradley 1999). Jeffrey
shows that the posterior utility function \(\mathbf{U}\_A\) is related
to the prior utility function \(\mathbf{U}\) as follows:



\[
\mathbf{U}\_A (X) = \mathbf{U}(A\cap X)
\]


Conditional preferences allow modelling doxastic preference change.
What matters for an agent's evaluation and behaviour are his
unconditional preferences, \(\succcurlyeq\_t\), which are
unconditional only in the sense that they rely on the agent's
actual information at time \(t\). When the agent accepts a new
proposition \(A\) at time \(t+1\), his
conditional-on-\(A\) preferences become his unconditional
preferences at \(t+1\): \(\succcurlyeq\_{t+1} = \succcurlyeq\_{A}.\)



Jeffrey's model is restricted in two ways. First, it requires an
unchanging evaluative function \(u\) defined over the atoms of the
propositional space, viz. possible worlds. Thus for all doxastically
changed preference orderings, the preferences over worlds remain
identical. Second, the model only considers the effects of a belief
change to certainty. But it is plausible that one's preference
- say, for a vacation in Florida - changes just because
one believes that it is more likely that there will be a hurricane
next week. Jeffrey's model can be generalised by introducing a
more general probability updating rule (e.g., Jeffrey
conditionalisation). An alternative solution was proposed by Bradley
(2005). It is based on relatively strong assumptions on the relation
between prior and posterior unconditional preferences.



An important discussion from economics needs mentioning, namely the
question whether models of doxastic preference change are capable in
principle to represent *all* preference changes. This question
originates with an important paper by Stigler and Becker (1977), who
argued that a wide range of phenomena which are commonly thought of as
preference changes--like addiction, habitual behaviour, fashions
and the effects of marketing--can be explained by stable,
well-behaved preferences. In a rather informal fashion, they argue
that such explanations involve only changes in information (more
precisely: prices and income), while leaving preferences intact. As a
result of this, economists largely abandoned the discussion of
preference change, believing that all preference change phenomena can
be explained in this way. Prima facie, their proposed explanations
exhibit important similarities to the discussed accounts of doxastic
preference change. The results from that discussion, which show that
models of doxastic preference change are subject to relatively strong
constraints, may therefore put doubt on the orthodox position in
economics that models of doxastic preference change are capable in
principle to represent *all* preference changes.


### 7.4 Valuational preference change



If an agent forms a specific preference as a result of some
experience, further changes in her overall preference state are often
necessary to regain consistency. A model of preference change can
therefore be constructed as an input-output model in the same style as
standard models of belief change. (Hansson 1995, Liu 2011) Changes in
preference are triggered by inputs that are represented by sentences
expressing new preference patterns. Hence, if the subject grows tired
of her previous favourite brand of mustard, \(A\), and starts to like
brand \(C\) better, then this will be represented by a change with the
sentence "\(C\) is better than \(A\)", in formal language
\(C\succ A\), as an input. However, a change in which the previous
preference \(A\succ C\) is replaced by the new preference \(C\succ A\)
can take place in different ways. For instance, there may be a third
brand \(B\) that was previously placed between \(A\) and \(C\) in the
preference ordering. The instruction to make the new preference
relation satisfy \(C\succ A\) does not tell us where \(B\) should be
placed in the new ordering. The new ordering may for instance be
either \(C\succ A\succ B\) or \(C\succ B\succ A\). One way to deal
with this is to include additional information in the input, for
instance specifying which element(s) of the alternative set should be
moved while the others keep their previous positions. In our example,
if only \(C\) is going to be moved, then the outcome should satisfy
\(C\succ A\succ B\). These and other considerations make it necessary
to modify the standard model of belief change in order to accommodate
the subject matter of preferences.


## 8. Preference criticism



In scholarly discussions, preferences are usually taken to be open to
rational criticism only insofar as (i) they have been inconsistent,
violating some of the rationally justifiable preference axioms, or
(ii) they (in combination with beliefs) commit the agent to
inconsistent inferences. 



Can there be rationally justifiable claims that certain
*intrinsic* preferences - i.e. preferences that are not
dependent on other preferences - are wrong, or should be
changed? The Humean position answers no. Hume distinguished reason
from the passion, and argued "that reason alone can never be a
motive to any action of the will; that it can never oppose passion in
the direction of the will" (*Treatise*, Book II, Part
III, Section III). Humeans often took this distinction between beliefs
and desires to imply not only that beliefs alone cannot motivate
action, but also that desires are not open to similar rational
criticism as beliefs. Therefore, Humeans conclude, preferences can
only be criticised if they are *extrinsic* - i.e.
instrumentally derived from other preferences on the basis of beliefs
- or inconsistent. Such criticism of extrinsic preferences would
seem ultimately to be a criticism of false beliefs, and it could be
argued that it is therefore not really criticism of preferences
(Broome 1993). 



Several authors have argued for a more substantial criticism of
preferences, including that of intrinsic ones. Some critics argue that
some or all preferences are in fact a kind of belief, and hence open
to the same rational criticism as beliefs. Two defences have been
presented to counter this challenge. First, it has been claimed that
that desires (standing for motivation in general) are fundamentally
distinct from epistemic states in their *direction of fit*.
Beliefs are directed to fit the world; hence their insufficient fit
provides the basis for their criticism. Desires are directed to fit
the world to them; hence they lack this basis for criticism (Smith
1987). Second, Humeans have argued that treating desires as beliefs is
incompatible with Bayesian decision theory and also with other,
non-quantitative, decision theories (Lewis 1988, Collins 1991,
Byrne/Hajek 1997). 



Some proponents of the criticizability of preferences have referred to
second-order preferences. An addict may prefer not to prefer smoking;
a malevolent person may prefer not to prefer evil actions; an indolent
may prefer not to prefer to shun work; a daydreamer may prefer not to
prefer what cannot be realised, etc. First-order preferences are
criticisable if they do not comply with second-order preferences. (For
accounts of second-order preferences, see Frankfurt 1971, Sen 1977.)
Second-order preferences may trigger attempts to change one's
preferences. Methods of self-restraint, self-command and
self-improvement have been extensively described (Schelling 1984,
Elster 1989, 2000). Already Hume described the possibility of
rationally choosing such expedience (Grune-Yanoff & McClennen
2006). 



Critics have argued against the possibility of rationally choosing
such indirect preference-modifying strategies. Millgram (1998) argues
that knowledge of the way such desires-at-will were brought about
makes it impossible for them to actually function as the desires they
are intended to be. He gives the example of a car salesman, who, in
order to be successful in his work, makes himself prefer the various
useless knick-knacks that the brand he represents offers for its cars.
When the salesman is laid off, the car-dealer offers him a car with
all the useless extras that he made himself prefer. Because he
remembers how he acquired these preferences, he chooses not to act on
them. So, Millgram argues, the desire-at-will was not genuine. What is
missing, he points out, are the backward-directed inferential
commitments that genuine preferences bring with them. Only if one
forgets that one acquired a specific preference at will, or if one
also acquires the inferential commitments of such a preference, can
preferring-at-will be successful.