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Preferences (Stanford Encyclopedia of Philosophy)

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. 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
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.[1]
They are represented as vectors, where each position in the
vector represents a specific good, and the scalar at that position
denotes the number of units of that good. For convenience, economists
often operate in two-goods worlds, where one good—the
numeraire—stands for all the other goods (money is the
most common numeraire). 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 components 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. This new utility, however, was very different from
the older hedonic concept: here the preference concept is basic, and
the cardinal utility function merely derived.

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 seperate theoretical concept
(Mäki 2000, Ross 2014).

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.

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,
Sören Halldén and Georg Henrik von Wright proposed the
first complete systems of preference logic (Halldén 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) (Halldén 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 ≻ and ∼ or alternatively by
P and I. In accordance with a long-standing
philosophical tradition, AB 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 AB). 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 set of relata are mutually
exclusive, it is customary to call them an alternative set,
or set of alternatives, 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:

(1) AB
¬(BA)    (asymmetry of
preference
)

(2) ABBA
   (symmetry of indifference)

(3) AA    (reflexivity of
indifference
)

(4) AB
¬(AB)
   (incompatibility of preference and
indifference
)

It follows from (1) that strict preference is irreflexive, i.e. that
¬(AA).

The relation ≽, “at least as good as” (or more
precisely: “better than or equal in value to”), can be
defined as follows:

ABAB
AB    (weak preference)

The alternative notation R is sometimes used instead of
≽.

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:

AB if and only if AB
and ¬(BA)

AB if and only if AB
and BA

≻ is the strict part of ≽ and ∼ its
symmetric part.

Two common notational conventions should be mentioned. First, chains
of relations can be contracted. Hence,
ABC abbreviates
ABBC, and
ABCD
abbreviates AB
BCCD. Second,
the ancestral symbol * is used to contract repeated uses of the same
relation; hence ≻* stands for ≻ repeated any finite
non-zero number of times (and similarly for the other relations). Thus
A≻*C denotes that either
AC or there are B1
Bn such that
AB1
B1B2 ∧ …
Bn−1Bn
BnC.

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:

ABBA

or equivalently:

ABAB
BA

The following weaker version of the property is sometimes useful:

If AB, then AB
BA (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 objects A, B, C, D,
and E. If she knows that she prefers A to the
others, 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:

ABBC
AC    (transitivity of
weak preference
)

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

ABBC
AC    (transitivity of
indifference
)

ABBC
AC    (transitivity of
strict preference
)

A weak preference relation ≽ is called quasi-transitive
if its strict part ≻ is transitive.

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

ABBC
AC
   (IP-transitivity)

ABBC
AC
   (PI-transitivity)

There is no series
A1,…,An of
alternatives such that A1
≻…≻AnA1
   (acyclicity)

All of these are weakenings of the transitivity of ≽. In other
words, if ≽ satisfies transitivity then ≻  and ∼
are also transitive, and furthermore, IP-transitivity, PI-transitivity
and acyclicity hold.

Furthermore, if ≽ is transitive, then no cycles containing
≻ are possible, i.e. there are no A and B such
that A≽*BA. Preferences with
such a ≻-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
C0, C1,
C2, … up to C999. Cup
C0 contains no sugar, cup C1
one grain of sugar, cup C2 two grains etc. Since
one cannot taste the difference between C999 and
C998, one might consider them to be equally good
(of equal value),
C999C998. For the same
reason, we have C998C997,
etc. all the way up to
C1C0, but clearly
C0C999. This
contradicts transitivity of indifference, and therefore also
transitivity of weak preference.

In a famous 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 counterexample 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 red, blue and green, respectively; they are
represented by the vectors
R1,G1,B1〉,
R2,G2,B2〉,
and
R3,G3,B3〉.
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,

R1R2R3R1,

G1G2G3G1,

B1B2B3B1;
and

R1,G1,B1〉≻〈R2,G2,B2〉≻〈R3,G3,B3〉≻〈R1,G1,B1〉.

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 ≽-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 axioms of probability and preference, then “[h]e could
have a book made against him by a cunning better and would then stand
to lose in any event”. The argument is 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 〈X,V〉 denote
that the collector owns stamp X and has paid V cents
to the dealer. She has now moved from the state
A,0〉 to the state 〈C,10〉.

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
C,10〉 to 〈B,20〉. The shop-owner
can go on like this forever. What causes the trouble is the following
sequence of preferences:

C,10〉 ≻ 〈A,0〉

B,20〉 ≻ 〈C,10〉

A,30〉 ≻ 〈B,20〉

C,40〉 ≻ 〈A,30〉

B,50〉 ≻ 〈C,40〉

A,60〉 ≻ 〈B,50〉

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 on
option over others.

Yet another argument rests on the importance of preferences for
choice. When agents choose at once from all the elements of 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 ≻ such that
ABCA does not
guide choice at all: any or none of the alternatives should be chosen
according to ≻. 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

ABBA
A=B    (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 typically are not specified enough
to be mutually exclusive. To say that one prefers having a dog over
having a cat does neglect 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):

pq
p≽(pq)≽q
   (disjunctive interpolation)

pq
¬q≽¬p
   (contraposition of weak preference)

pq → ¬q∼¬p
   (contraposition of indifference)

pq → ¬q
¬p    (contraposition of strict
preference
)

pq
(p∧¬q)≽(q∧¬p)
   (conjunctive expansion of weak preference)

pq
(p∧¬q)≻(q∧¬p)
   (conjunctive expansion of strict
preference
)

pq
(p∧¬q)∼(q∧¬p)
   (conjunctive expansion of indifference)

(pq)≽r
prqr
   (left disjunctive distribution of ≽)

p≽(qr) ↔
pqpr
   (right disjunctive distribution of ≽)

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) have been 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 produced with a representation function. By this is
meant a function f that takes us from a pair
p,q〉 of sentences to a set
f(〈p,q〉) of pairs of alternatives
(perhaps possible worlds). Then
pfq holds if and only if
AB for all 〈A,B
f(〈p,q〉) (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).

GN pp≻¬p
   (negation-related good)

BN p ↔ ¬pp
   (negation-related bad)

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.

GI p
(∃q)(pq∼¬q)
   (indifference-related good)

BI p
(∃q)(¬qqp)
   (indifference-related bad)

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 extensions are available that cover the more general cases.
(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:

xy if and only if
G{x,y}x &
¬G{x,y}y (goodness-based
preference)

xy if and only if
B{x,y}y &
¬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 xy
holds according to first definition but ¬(x
y) according to the second. To avoid such problems, Hansson
and Liu (2014) proposed the following definition:

xy if and only if either
G{x,y}x &
¬G{x,y}y or
B{x,y}y &
¬B{x,y}x (bivalently based
preference)

Preferences can be represented numerically.
AB is then expressed by a numerical utility
function u that assigns a higher value to A than to
B, while AB 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:

AB iff
u(A)>u(B)
   (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
A is finite, there can be
complete and transitive preference relations on 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:

If AB, then
u(A)>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 itself. However, there is
relevant information about preferences that is not represented by the
relation itself. For example, when an agent expresses
two preferences, say AB and
CB, one might ask how much the agent
prefers A to B, in particular in comparison 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. (43 °C is as much hotter than 41
°C as 29 °C is hotter than 27 °C.) In addition, a ratio
scale is allows for meaningful comparisons of ratios. (12 m is twice
as long as 6 m.) 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

AB iff
u(A)
>
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)>
u(B), but
u(A)-u(B) is so small that it
cannot be discerned, then AB 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 AB 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).

AB iff
u(A)−u(B)>δ,
   (JND representation, δ>0)

If the set of alternatives is finite, then ≽ has a JND
representation if and only if ≽ is complete, and satisfies the
two properties that for all
A,B,C,D:
ABBC
ADDC and
ABCD
ADCB.

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

AB iff
umin(A)>
umax(B)    (Interval
representation
)

It has been shown that a preference relation ≽ has an interval
representation if and only if it satisfies completeness and the
property that for all A,B,C,D:
ABCD
ADCB.

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

AB iff
u(A)−u(B)>σ(A,
B
)    (Doubly variable threshold
representation
, σ(A,B)>0)

If the set of alternatives is finite, then ≽ 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)

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 some 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 〈
1,…,≽n 〉 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
k and ≽m such
that XkY and Y
mX for any alternatives X
and Y. If 〈
1,…,≽n 〉 is
conflict free, then we can define a combined preference relation
≽ such that (1) XY if there is some
k such that
XkY, (2)
YX if there is some
k such that
YkX, and (3) otherwise
XY. 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 four 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, monetary units are used. Such reductions
are standardly used in cost-benefit analysis. 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 XY will hold in all cases of
conflict, i.e. whenever there are ≽k and
m such that X
kY and Y
mX. 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 (although
not 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 〈
1,…,≽n 〉, 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 ≽, then ≽ 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”.

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 A, we can
represent (hypothetical) choice as a function C that,
for any given subset B of A,
delivers those elements of 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:

C is a choice function for A
if and only if it is a function such that
for all BA:
(1) C(B)
B,
and (2) if B
≠ ∅, then
C(B)

∅.

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

If BA
then B
C(A)
C(B)
   (Property
α, “Chernoff”)

This property states that if some element of subset B
of A is chosen
from A, then it is also chosen from
B. According to property α,
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:

C({X,Y,Z}) ={X}

and

C({X,Y}) ={Y},

which violate property α. More generally, property α 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 B, a subset of A,
then one of them cannot be chosen in A
without the other also being chosen.

If BA
and X, Y
C(B), then
XC(A)
iff YC(A)
   (Property
β)

To exemplify property β, 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. C({X, Y,
Z})={X, Y}. Then we find out that we do in fact have access to a car.
Property β says that if X is one of the chosen
restaurants in this situation, then so is Y, and vice versa.
In other words: XC({X,
Y, Z, V, W}) holds if and only if
YC({X, Y, Z,
V, W}).

A third property, γ, 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 B containing both
alternatives for which Y is chosen and X is not.

If X,YA
and XC(A),
then for all B,
if XB,
and
YC(B),
then XC(B)
   (WARP)

WARP is equivalent to the combination of properties α and β
(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
C({T,H}) ={T} and
C({T,H,C}) ={H}, and hence the
visitor violates both properties α and β.

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 ≽ is to have the function always choose the elements
that are best according to ≽:

The best choice connection
CB(B) =
{XB |
YB:
(XY)}

CB is a choice function (i.e. satisfies
the defining criteria for a choice function given in section 5.1) if
and only if ≽ is complete and acyclical. It will then
satisfy properties α and γ. Furthermore,
CB satisfies property β if and only
if ≽ 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
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
CL(B) =
{XB |
YB:
¬(YX)}

CL is a choice function if and only if
≽ is acyclical. It will then satisfy properties
α and γ. Furthermore, CL
satisfies property β if and only if ≽ is transitive and
complete (Herzberger 1973).

When the preference relation over A is
cyclical, neither CB nor
CL may be a relational choice function for
A. In the simplest case, with a cyclical
preference
ABCA,
CB(A,B,C) =
CL(A,B,C) =
∅. 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 (Grüne-Yanoff
2004).

There are many ways to construct preference relations from observed
choices. The simplest 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.

XSY iff for some
B,
XC(B)
and YB
(1)
XSY iff
XSY and not
YSX
XSY iff
XSY and
YSX

If the choice function is defined over all subsets of B,
S is complete.
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 above-discussed theoretical effort sees preferences as completely
tied to choices. Various caveats concerning this connection are in
order. First, it must be pointed out that choices and preferences are
in fact entities of quite different categories. Preferences are
states of mind whereas choices are actions. The
strong behaviourist program, which sought to eliminate the notion of
preference by reducing it to choices, is therefore rightly regarded
with suspicion today (Hausman 2012).

Secondly, it is necessary to 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. In those cases, one can only speak of preferences
reconstructed from choice, without claiming that these
preferences describe mental states at all (Gul and Pesendorfer 2008,
Hands 2013, Ross 2014).

Thirdly, it seems obvious that there are 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. It thus acknowledges that their answers are possibly
influenced by other preferences, e.g. for privacy, status or pretend
play, and explains why answers may not be true. Alternatively, their
answers can be interpreted as introspective reports. This
interpretation treats answers as agents’ privileged access to
their own minds; and it becomes difficult to explain
“false” answers.

Fourthly, some choices are not based on stable preferences over
actions, but are constructed from more basic cognitive and evaluative
elements. A simple choice—like e.g. 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 the consequences of these choices.
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. Often,
this framework yields a preference relation over actions much like the
simple case. However, alternative frameworks have been developed in
which this is not the case (e.g. Loomes and Sugden 1982).

Last, the introspective concept of preference is closely connected to
the notion of welfare. An agent who prefers X to
Y is expected to judge herself to be better off with
X than with Y. But if preferences are tightly linked
to choice, the welfare interpretation is jeopardized. As Sen argues,
people choose not only on the basis of their concern for their own
welfare, but also on the basis of commitments—e.g.
traditions, habits, moral maxims, etc. (Sen 1977). So it seems that
preferences can either be interpreted as welfare judgements,
or as the basis of choices, but not as both at the same time.
We discuss the relation of preference and welfare in the next
section.

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.

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.[5]
In the following sections, three types of explanatory models are
discussed. (For more detail, see Grüne-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. Laurie A. 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,ti) =
v(A,ti−1) / (1 +
r)t

where r is a discount rate and t =
titi−1 the duration of the
delay. This is the discounted utility model, 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 a 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 hence covers 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. 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.

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 their compatibility with
sustainability has been challenged. (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
XY is preferred to
X∧¬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
XY is preferred to
X∧¬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 U (see section 2), such that:

XY iff U(X) ≥
U(Y)

U(X) in turn is defined as the weighted
average of the utility u of all the possible worlds
w in which X is true:

U(X) = [1/P(X)] ∑wX [u(w)
× P(w|X)]

where P is the probability weight (Jeffrey’s original
notation here is adapted to the discrete case).
u,P〉 represents the unconditional
preference ≽ if P is the probability distribution based
on the agent’s actual information. The conditional preference
ordering ≽A, in contrast, is represented by
the tuple 〈u,PA〉, where
PA is the probability distribution based on the
counterfactual scenario that the agent accepts proposition A
as true. That is, the agent imagines that if he changed his whole
belief system from P to PA (and
hence specifically P(A)<1 to PA(A)=1), then he would have the
preference relation ≽A as represented by
u,PA〉 (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
UA is related to the prior
utility function U as follows:

UA(X) = [1/PA(X)] ∑wX [u(w)
× PA(w|X)]
= [P(A)/P(AX)] ∑wXA
[u(w) ×
P(w|AX)/P(A)]
= U(AX)

Conditional preferences allow modelling doxastic preference change.
What matters for an agent’s evaluation and behaviour are his
unconditional preferences, ≽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: ≽t+1 =
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 CA, as
an input. However, a change in which the previous preference
AC is replaced by the new preference
CA 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
CA does not tell us where B should
be placed in the new ordering. The new ordering may for instance be
either CAB or
CBA. 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
CAB. These and other
considerations make it necessary to modify the standard model of
belief change in order to accommodate the subject matter of
preferences.

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 (Grüne-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.

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