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New trends in preference, utility, and choice:
from a mono-approach to a multi-approach
Alfio Giarlotta∗
Abstract
We give an overview of some new trends in preference modeling, utility representation, and
choice rationalization. Several recent contributions on these topics point in the same direc-
tion: the use of multiple tools –may they be binary relations, utility functions, or rationales
explaining a choice behavior– in place of a single one, in order to more faithfully model
economic phenomena. In this stream of research, the two traditional tenets of economic
rationality, completeness and transitivity, are partially (and naturally) given up. Here we
describe some recent approaches of this kind, namely: (1) utility representations having
multiple orderings as a codomain, (2) multi-utility and modal utility representations, (3) a
finer classifications of preference structures and forms of choice rationalizability by means of
generalized Ferrers properties, (4) a descriptive characterization of all semiorders in terms
of shifted types of lexicographic products, (5) bi-preference structures, and, in particular,
necessary and possible preferences, (6) simultaneous and sequential multi-rationalizations of
choices, and (7) multiple, iterated, and hierarchical resolutions of choice spaces. As multi-
ple criteria decision analysis provides broader models to better fit reality, so does a multi-
approach to preference, utility, and choice. The overall goal of this survey is to suggest the
naturalness of this general setting, as well as its advantages over the classical mono-approach.
Key words: Preference modeling; utility representation; choice rationalization; complete-
ness; transitivity; lexicographic order; semiorder; Z-product; (m, n)-Ferrers property; bi-
preference; necessary and possible preference; robust ordinal regression; multi-utility repre-
sentation; modal utility representation; multi-rationalization; choice resolution.
1 Introduction
In the field of mathematical economics, the modelization of an agent’s preference structure is
traditionally done by means of a mono-approach, which uses a single binary relation satisfying
the two basic tenets of economic rationality: (1) completeness, and (2) transitivity. Under
topological conditions of separability, these two properties guarantee the existence of a utility
representation of preferences by a continuous real-valued function [6, 40, 69]. Similarly, the
traditional approach of revealed preference theory [17, 224] often employs complete and transitive
binary relations to justify an agent’s choice behavior. In some cases, the satisfaction of the two
properties of completeness and transitivity has even guided the design of new economic theories:
a striking instance of kind is given by the classical book “Games and Economic Behavior” of
von Neumann and Morgenstern [198].
∗Department of Economics and Business, University of Catania, Italy. Email: giarlott@unict.it
1
2A. Giarlotta
By partially giving up these two properties, here we depart from traditional approaches, and
examine: (a) alternative types of utility representations, (b) more refined kinds of preference
structures, and (c) new forms of bounded rationality for choices. In fact, the general question
that motivates this survey is the following:
(Q0) Can we design sound theories of preference modeling, utility representation, and choice
rationalization, which give up, partially or totally, the basic tenets of economic rationality?
This paper illustrates some possible answers to question (Q0).
Specifically, first we deal with preference representations in a lexicographically ordered
codomain [60, 94], thus extending the classical real-valued representation. This approach pro-
vides a description of preferences that fail to have a real-valued representation [27, 28]. Succes-
sively, we describe some novel types of preference structures, which are formed by nested and
intertwined pairs of binary relations [125]. In this bi-preference approach, the two properties
of transitivity and completeness are coherently spread over the two components. This feature
makes these structures well suited to applications in operations research and economics. In
particular, special types of bi-preferences, called necessary and possible [116], have already been
successfully employed as a modeling tool in multiple criteria analysis [133]. Under suitable con-
ditions, bi-preferences can be represented by a doubly indexed family of utility functions: this
is the so-called modal utility representation [116], which adapts to bi-preferences the recently
introduced multi-utility representation of a preorder [83, 200].
In parallel to a multi-approach to preference and utility, we also develop a theory of choice
multi-rationalization. Samuelson’s theory of revealed preferences [17, 144, 224] postulates that
choices are observed, and preferences can be derived from them. The class of rationalizable
choices is especially significative in this respect, since it codifies all types of choice behavior that
can be explained by means of the maximization of a single binary relation. However, the theory
of revealed preferences yields a sharp rational/irrational dichotomy, since any non-rationalizable
choice behavior is bluntly classified as “irrational”. With the goal of smoothening this dichotomy,
several new theories of bounded rationality [233, 234] have naturally emerged over the last few
years [57, 150, 172, 175, 222]. Here we describe a general setting for the multi-rationalizability of
a choice [50], which may employ more than one binary preference to explain the behavior of an
economic agent, thus broadening the classical notion of mono-rationalizability. We also sketch
the main features of a recently introduced methodology in choice theory, called “resolution”.
This methodology, which is an adaptation of an analogous technique in general topology [89, 246],
studies the inner structure of a complex choice process [48] on the basis of a notion of delegations
of tasks. This yields a decomposition (and explanation) of a complex selection process into
independent and simpler decisional units, typically distributed in a hierarchical way.
Multiple criteria decision analysis [131, 132] provides powerful analytical tools to handle
complex real life problems, offering more flexible modelizations than mono-criterion techniques.
Similarly, mutatis mutandis, a multi-approach to the theories of preference, utility representa-
tion, and choice rationalization yields a more realistic representation of economic phenomena
rather than the classical mono-approach. The purpose of this work is to give an overview of a
multi-approach to these theories, also suggesting its naturalness, feasibility, and potential.
Organization of the paper
The remainder of this survey is organized into three main sections, a conclusive section, and an
appendix.
New trends in preference, utility, and choice: from mono to multi 3
Section 2 (The mono-approach). We start in (2.1) with a historical discussion about the
two properties of transitivity and completeness. Successively, we provide an overview of
basic notions and classical results in (2.2) preference modeling, (2.3) utility representation,
and (2.4) choice rationalization. These theories use a single tool for the description of an
agent’s behavior/attitude. In summarizing their main achievements, we shall also detect
some shortcomings, and indicate possible ways of coping with the arising issues.
Section 3 (The transition). Here we sketch a few recent approaches to the theories described
in (2.2)–(2.4). These techniques, which suggest the use of multiple tools to represent eco-
nomic behavior, address some shortcomings of classical theories and pave the way for more
general approaches to these topics. Specifically, we describe: (3.1) utility representations
using lexicographic orderings as a codomain, (3.2) universal characterizations of semiorders
based on shifted lexicographic products, (3.3) Ferrers properties describing a discrete evolu-
tion of transitivity, (3.4) choice correspondences rationalizable by well-structured revealed
preferences, and (3.5) a process detecting the inner structure of a choice in terms of delega-
tions of tasks. The goal of this section is to provide the reader with a natural justification
and a smooth transition toward a multi-approach.
Section 4 (The multi-approach). Here we finally describe some very recent developments in
the theories described in (2.2)–(2.4), which employ multiple tools rather than a single one.
Specifically, in (4.1) we introduce bi-preference structures, and describe their advantages
over mono-preferences. In (4.2), we deal with particular types of bi-preferences, called
necessary and possible, which have been already used in multiple criteria decision analysis.
In (4.3), we recall the notion of a multi-utility representation, and show how bi-preferences
are representable by a suitably indexed type of multi-utility representation, called modal.
Within the theory of choice rationalization, we provide in (4.4) an overview of the recent
bounded rationality approaches, which use multiple binary rationales to explain a choice
behavior. Finally, in (4.5) we describe a natural extension of the notion of choice resolution
to a multiple and iterated setting.
Section 5 concludes this contribution.
The Appendix contains two figures, which graphically describe some results.
Neither original results nor proofs appear in this survey.
2 The mono-approach
To keep the presentation as much self-contained as possible, this section recalls the classical
setting of the theories of preference modeling, utility representation, and choice rationalization.
2.1 The two classical tenets of rationality
A preference structure on a set Xof alternatives is usually modeled by a binary relation Ron
X. Traditionally, Ris assumed to “behave well”, in the sense that it satisfies suitable ordering
properties. The two classical properties that are assumed to hold for Rare:
(Completeness) for any distinct x, y ∈X, either xRy or yRx (or both);1
1Notice that, since xand yare distinct, this formulation of completeness does not imply reflexivity.
4A. Giarlotta
(Transitivity) for any x, y, z ∈X, if xRy and yRz, then xRz.
The reasons for which Ris often supposed to be both complete and transitive are several, some
being related to their economic significance, some others to their mathematical tractability.
However, both properties have been questioned by eminent scholars over time.
In their monumental work Theory of Games and Economic Behavior [198], von Neumann
and Morgenstern already acknowledged, albeit rather elusively, that preferences may naturally
be incomplete (p. 19–20):
“We have conceded that one may doubt whether a person can always decide which of two
alternatives ... he prefers. If the general comparability assumption is not made, a mathe-
matical theory ... is still possible. It leads to what may be described as a many dimensional
vector concept of utility. This is a more complicated and less satisfactory set-up, but we do
not propose to treat it systematically at this time.”
In fact, von Neumann and Morgenstern limited their analysis to complete (and transitive) pref-
erences, due to the mathematical amenability of this simplified setting, and never published
details about the mentioned “many dimensional vector concept of utility”.
In his seminal paper on incomplete preferences, Aumann [23] suggested (p. 449) an interpre-
tation of von Neumann and Morgenstern’s statement:
“What they probably had in mind was some kind of mapping from the space of lotteries to a
canonical partially ordered euclidian space, rather than the real-valued mappings we use here;
but it is not clear to me how this approach can be worked out.”
Aumann’s criticism of the completeness property was quite direct (p. 446):
“Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable.
Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we
find it hard to accept even from the normative viewpoint.”
Since Aumann’s work, many other authors started abandoning the axiom of completeness as
a basic feature of rational behavior. On the topic, Bewley [32] and Ok [200] attentively elaborate
on the links between the notion of rationality and the incompleteness of preferences.
In their systematic analysis of the multi-utility representation of preferences, Evren and
Ok [83] mention several behavioral phenomena which naturally yield incompleteness, e.g., status-
quo bias [14, 177], intransitive choice [172], choice deferral [154], and indecisiveness in revealed
preferences [80]. Similarly, incompleteness has been a main focus in various decision models used
in operations research and management science [66, 133, 178], financial economics [216], political
economics [161, 217], and game theory [24]. Further, several recent studies on (in)decisions under
risk and uncertainty use incomplete preorders to model preferences [77, 107, 108, 128, 167, 196,
201]. Last but not least, following the seminal work of Bernard Roy [218, 219, 220], there is
a large number of multiple criteria decision methodologies which explicitly take into account
incompleteness of preferences as a natural feature of the decision maker’s attitude [132].
The axiom of transitivity was possibly harder to abandon, even if probably questioned be-
fore completeness. In his well-known paper, Tversky [240] was still advocating the importance
of transitivity in the modelization of preferences, since its violation could cause unpleasant
phenomena of “money pump” [67].2This attitude was however contrasted by other authors,
2See Section 3.3 of this survey for a discussion on this point in relation to the so-called (m, n)-Ferrers properties.
New trends in preference, utility, and choice: from mono to multi 5
who had already been designing economic models in which transitivity was partially or totally
abandoned. The probabilistic choice model proposed by Luce [165] in 1959 can be regarded
as a pioneering example of intransitive preferences in economic theory. The obstinate insis-
tence of some economists to employ transitive models even brought Sen [230] to declare that
revealed preference theory is “obsessed with transitivity”. In their recent paper, Bleichrodt and
Wakker [35] argue that the year 1982 was a sort of “breaking point” in the economic literature,
since transitivity was given up in three seminal papers related to regret theory: the axiomatic
approach of Fishburn [96], a decision analysis oriented paper by Bell [30], and the fundamental
contribution of Loomes and Sudgen [163]. From an experimental point of view, there are many
papers in mathematical psychology explaining intransitivity of preferences by random models,
insofar as the subject’s preferences vary over time from one type of ordering to another: see,
e.g., [211, 212] for some models of this kind, and [68] for a recent method to test these models.
In the same stream of research that opposes the blunt assumption of fully transitive pref-
erences, we ought to mention the extraordinary amount of literature on semiorders, interval
orders, and similar preference structures, which describe forms of rational behavior character-
ized by weaker forms of transitivity. Anticipated by the intuitions of Fechner [88], Poincar´e [207],
Georgescu-Roegen [105], Armstrong [16], and Halphen [136], research on intransitive preference
structures had its definitive consecration by the seminal papers of Luce [164] and Fishburn [91],
who formally introduced the notions of semiorder and interval order, respectively. Their ap-
proaches are based on the idea of weakening the axiom of transitivity, rather than abandoning
it all together. Indeed, Luce’s famous coffee/sugar example suggests that the transitivity of the
associated indifference should be somehow weakened and regulated, whereas the transitivity of
the strict preference may be retained as a natural assumption of rational behavior.
The recently introduced weak (m, n)-Ferrers properties go exactly in the direction of con-
sidering binary structures with a transitive strict preference but a possibly intransitive indiffer-
ence [119]. Originally designed to provide a combinatorial extension of the Ferrers condition and
semitransitivity –which coincide, respectively, with weak (2,2)-Ferrers and weak (3,1)-Ferrers–
these properties display a finite taxonomy of enhanced forms of the transitivity of the strict
preference. In fact, roughly speaking, weak (m, n)-Ferrers properties classify transitive strict
preferences by means of the types of forbidden mixed cycles of preference/indifference (see Sec-
tion 4.2 in [46]). It follows that such an approach may be relevant for economic applications
insofar as weak (m, n)-Ferrers properties prompt a possible recognition of money-pump effects
due to the presence of mixed cycles of a certain length and type.
Strict (m, n)-Ferrers properties [119, 124, 202] go even further in weakening the assumption
of transitivity, since they do not even postulate the transitivity of the strict preference. These
properties yield an infinite taxonomy of intransitive preference structures, which are connected
to other types of money-pump phenomena.
In this paper, we shall also mention some new approaches to preference modeling in which
both basic tenets of economic rationality are only partially retained, being “spread over” two
binary relations (see Sections 4.1 and 4.2 on bi-preferences and NaP-preferences, respectively).
2.2 Preference modeling
Here we summarize the basic terminology in preference theory. Two good sources of information
on this topic –as well as on utility representations, which is the topic of the next section– are
the textbooks by Bridges and Mehta [40] and Aleskerov, Bouyssou and Monjardet [6].
6A. Giarlotta
Henceforth, Xis a nonempty (possibly infinite) set of alternatives (courses of action, etc.),
and ∆(X) = {(x, x) : x∈X}is the diagonal of X.
Definition 2.1 A reflexive binary relation on Xis referred to as a weak preference on X, and
is henceforth denoted by %; the pair (X, %) is generically called an ordered set. The following
relations are derived from a weak preference %on X: its strict preference ≻(the asymmetric part
of %), its indifference ∼(the symmetric part of %), and its incomparability ⊥(the symmetric
part of the complement of %). These relations are formally defined as follows for each x, y ∈X:
x≻ydef
⇐⇒ (x%y)∧ ¬(y%x)
x∼ydef
⇐⇒ (x%y)∧(y%x)
x⊥ydef
⇐⇒ ¬(x%y)∧ ¬(y%x).
Given an ordered set (X, %), the set of maximal elements of A⊆Xis defined by
max(A, %) := {x∈A: (∄y∈A)y≻x}.
The composition of two weak preferences %1and %2on Xis the binary relation %1◦%2on X
defined as follows for all x, y ∈X:
x(%1◦%2)ydef
⇐⇒ (∃z∈X)x%1z%2y .
Notice that a weak preference %is (i) complete if and only if its incomparability ⊥is empty,
and (ii) transitive if and only the inclusion %◦%⊆%holds. Whenever %is complete, the set
of maximal elements of A⊆Xcan be also written as max(A, %) := {x∈A: (∀y∈A)x%y}.
Finally, observe that, even when Xis finite, the set max(A, %) may be empty, due to the possible
presence of strict cycles (see Definition 2.2).
Definition 2.2 A weak preference %on Xis called (x, y, z , w are arbitrary elements of X):
-complete (or total or connected) if x%yor y%xalways holds (x6=y);
-antisymmetric if x%yand y%ximplies x=y(equivalently, ∼is the diagonal of X);
-acyclic if there are no x1, x2,...,xn∈X, with n≥3, such that x1≻x2≻...≻xn≻x1;
-quasi-transitive if ≻is transitive, i.e., (x≻yand y≻z) implies x≻z;3
-Ferrers if (x%yand z%w) implies (x%wor z%y);
-semitransitive if (x%yand y%z) implies (x%wor w%z);
- an interval order if it is Ferrers;
- a semiorder if it is Ferrers and semitransitive;
- a (partial) preorder if it is transitive;
- a partial order if it is an antisymmetric preorder;
- a total preorder if it is a complete preorder;
- a linear order if it is an antisymmetric total preorder.
Accordingly, the pair (X, %) is called, e.g., a semiordered set, a preordered set, a partially ordered
set (also called a poset), a linearly ordered set (also called a linear ordering or a chain), etc.
3In case %is complete, then the following statements are equivalent: (i) %is quasi-transitive; (ii) for each
x, y, z ∈X,x≻y%zimplies x%z; (iii) for each x, y, z ∈X,x%y≻zimplies x%z.
New trends in preference, utility, and choice: from mono to multi 7
Notice that (i) any total preorder is trivially a semiorder, (ii) any semiorder is trivially an
interval order, (iii) an interval order is both complete and quasi-transitive, and (iv) any quasi-
transitive weak preference is acyclic. Moreover, the indifference derived from a preorder is an
equivalence relation, but the same does not hold for the indifference associated to a semiorder
(hence, a fortiori, for that of an interval order). Observe also that if Xis finite, then an acyclic
relation on Xalways has maximal elements for each nonempty subset of X.
Next, we recall some notions due to Fishburn [91], which play an important role in the theory
of preferences, especially for defining notions of (semi)continuity as well as for preferences that
are interval orders and semiorders (but also for bi-preference structures, see Sections 4.1 and 4.2):
the “traces” of a weak preference.
Definition 2.3 Let %be a weak preference on X. For each x∈X, let
(weak lower section of x)x↓,%:= {w∈X:x%w},
(weak upper section of x)x↑,%:= {w∈X:w%x},
(strict lower section of x)x↓,≻:= {w∈X:x≻w},
(strict upper section of x)x↑,≻:= {w∈X:w≻x}.
Define three binary relations4on Xas follows for each x, y ∈X:
(left trace of %)x%∗ydef
⇐⇒ y↓,%⊆x↓,%,
(right trace of %)x%∗∗ ydef
⇐⇒ x↑,%⊆y↑,%,
(global trace of %)x%0ydef
⇐⇒ x%∗y∧x%∗∗ y.
The next lemma collects some enlightening results about traces: see, e.g., [97, 190, 205].
Lemma 2.4 Let %be a weak preference on X.
•%∗,%∗∗,%0are preorders contained in %.
•%∗◦%⊆%and %◦%∗∗ ⊆%.
•%0◦%⊆%and %◦%0⊆%.
•%is an interval order ⇐⇒ %∗is a total preorder ⇐⇒ %∗∗ is a total preorder.
•%is a semiorder ⇐⇒ %0is a total preorder.
•%is a preorder ⇐⇒ %=%0.
•%is a total preorder ⇐⇒ %=%0is complete.
Many classical results on preferences are related to the possibility of (continuously) repre-
senting them by a utility function, a topic that is analyzed in the next section. There are also
other issues arising from the traditional mono-approach to preference modeling, mostly due to
the limited expressive power of a single binary relation. In this respect, a general question is:
(Q1) Can we use binary relations to represent preferences in a more flexible and realistic way?
4We follow the approach described in Bouyssou and Pirlot [39], defining all traces in terms of weak sections,
instead of defining strict traces first and then deriving weak traces. The difference is immaterial whenever dealing
with complete and quasi-transitive preferences, in particular for interval orders and semiorders. Notice also that
the notion of global trace has been recently revised from a different perspective, and renamed transitive core [199].
8A. Giarlotta
We shall address question (Q1) in Sections 4.1 and 4.2, where we suggest how a bi-preference
approach may enhance the modeling power of a binary representation of agents’ preference
structures by taking into account two different kinds of “attitudes”.
2.3 Utility representations
In this section we deal with the classical setting of real-valued utility representations of binary
preferences. Two are the basic issues, the first purely order-theoretic and the second topological:
(Q2) Can we can represent a total preference relation by a real-valued utility function?
(Q3) Can we make this utility function continuous?
To start, we give the basic elements to properly formulate and then address question (Q2).5
Definition 2.5 A binary relation %on Xis representable in Rif there is a function u:X→R
such that, for all x, y ∈X, we have
x%y⇐⇒ u(x)≥u(y).
In this case, the function uis a utility representation of (X, %) in R. (We also say that (X, %)
is order-embeddable or embeddable in R.) The chain (R,≥) is the base of the representation.
An obvious necessary condition for the representability of a weak preference %in Ris that
%must be a total preorder, i.e., it satisfies the two classical properties of transitivity and
completeness. This condition is also sufficient for the cases in which the ground set Xis finite or
countably infinite (see, e.g., Chapter 1 of Bridges and Mehta [40]). In the general case, however,
we need an additional property of “separability” to ensure representability.
The first characterization of representability in Ris most likely the following [51, 188]:
Theorem 2.6 (Cantor, 1895, Milgram, 1939) A linear ordering (X, %)is order-embeddable
in Rif and only if it includes a countable subset that is weakly order-dense in X.6
Similar characterizations were given by Birkhoff [33]. Nevertheless, due to an imperfect commu-
nication in the scientific community, until the early 1950’s economists considered all preference
relations as representable in R. In other words, the concepts of “preference” and “utility” were
(wrongly) considered equivalent. For a salient instance of this kind, let us cite Hicks [143] (p. 19):
“If a set of items is strongly ordered, it is such that each item has a place of its own in the
order; it could, in principle, be given a number.”
If the above statement were to hold, then every total preorder would be representable in R, and
the concepts of preference and utility would coincide, which is false.
In his celebrated paper on the Open Gap Lemma, Debreu [69] finally exhibited an example
of a natural preference that is non-representable in R: the lexicographic plane R2
lex = (R2,%lex).
Several characterizations of representability followed, for instance [100]:
5The literature also examines weaker forms of representability of a single binary relation, e.g., the existence
of (continuous, semicontinuous) Richter-Peleg utility functions [4, 203, 214]. We shall deal with this topic in
Section 4.3, where we also discuss some shortcomings of this notion, and introduce multi-utility representations.
6A set Y⊆Xis weakly order-dense in Xif, for each x1, x2∈Xsuch that x1≻x2, there is y∈Ywith the
property that x1%y%x2. Such a set is often called Debreu order-dense, and the existence of a countable Debreu
order-dense is referred to as Debreu-separability [40].
New trends in preference, utility, and choice: from mono to multi 9
Theorem 2.7 (Fleischer, 1961) A chain (X, %)is representable in Rif and only if it has at
most countably many jumps and the topological space (X, τ%)is separable.7
For an extensive overview of the topic, the reader is referred to [40, 187].
In 2002, Beardon et al. [27, 28] systematically analyzed the structure of total and transitive
preferences that fail to be representable in R, and obtain a striking subordering classification of
them. Their characterization [27] can be suggestively rephrased as follows:
Theorem 2.8 (Beardon et al., 2002) A chain is non-representable in Rif and only if it is
(i) long or (ii) large or (iii) wild.8
(Here by “long” we mean that it contains a copy of the first uncountable ordinal9ω1or its
reverse ordering ω1∗; by “large” we mean that it contains a copy of a non-representable sub-
ordering of the lexicographic plane R2
lex; and by “wild” we mean that it contains a copy of an
Aronszajn line, which is defined as an uncountable chain such that neither ω1nor ω1∗nor an
uncountable subordering of Rembeds into it.) Some more recent results in this direction, which
use lexicographic orders as modeling tools, are mentioned in Section 3.1.
Next, we deal with question (Q3), that is, the existence of a continuous real-valued rep-
resentation. To describe the topological setting, we recall the notions of (i) the continuity of
a preorder, and (ii) the order topology induced by a preorder. (For all undefined topological
notions, the reader may consult the classical textbook by Munkres [193].)
Definition 2.9 Given a topological space (X, τ ), a preorder %on Xis continuous10 if %is a
closed subset of the topological product X×X.
It can be shown that a complete preorder %on (X, τ ) is continuous if and only if (i) all weak
upper sections x↑,%and lower sections x↓,%are closed subsets of (X, τ ) if and only if (ii) all strict
upper sections x↑,≻and lower sections x↓,≻are open subsets of (X, τ ). Conditions (i) and (ii)
are sometimes called, respectively, closed semicontinuity and open semicontinuity, whereas their
joint satisfaction is called bi-semicontinuity: see Section 4.1. Notice that bi-semicontinuity does
not imply continuity for incomplete preorders.11
7Ajump in an ordered space (X, %) is a pair (a, b)∈X2such that a≻band there is no point c∈Xsuch
that a≻c≻b. The topology τ%is the order topology induced by %. The topological space (X , τ%) is separable if
it contains a countable set Dthat intersects each nonempty open set. See Munkres [193] for topological notions.
8This is not the terminology originally used by the authors.
9An ordinal is a well-ordered set (X, <) such that each x∈Xis equal to its initial segment {y∈X:y < x}.
The finite ordinals are the natural numbers. The first infinite ordinal is the set ω0of all natural numbers, endowed
with the usual order. The first uncountable ordinal is the set ω1of all countable ordinals, endowed with the natural
order. The famous continuum hypothesis, formulated by George Cantor in 1878, says that the cardinality of R
is equal to ω1(as a cardinal). In 1963, Paul Cohen proved that the continuum hypothesis is independent from
the axioms of ZFC (Zermelo-Fraenkel axiomatic set theory, plus the Axiom of Choice), in sense that there are
models in which it is true, and models in which it is false (because |R|> ω1holds). See the classical textbook by
Kunen [157] for ZFC axiomatic set theory.
10Here we use the notion of continuity employed in some standard textbooks in microeconomic theory, such as
Mas-Colell et al. [174] p. 46. Other authors sometimes employ a weaker notion of continuity: see, e.g., Section 1.6
of Bridges and Mehta [40]. However, from the point of view of applications, the distinction between the various
notions of continuity is often immaterial. See also Evren and Ok [83] p. 555, and Geras´ımou [106] p. 2–3.
11Herden and Pallack [141] provide a very simple counterexample to the equivalence between continuity
and bi-semicontinuity for incomplete preferences: in fact, they show that the relation of equality is a bi-
semicontinuous non-continuous preorder in any topological space that is T1but not Hausdorff. On the topic,
see also Geras´ımou [106], who characterizes continuity in terms of closed semicontinuity and a property of “local
expansion” of transitivity (Theorem 1 in [106]).
10 A. Giarlotta
Definition 2.10 Given a preordered set (X, %), the order topology τ%on Xinduced by %is
the topology having as a subbasis the family of all strict upper and lower sections (equivalently,
the topology having as a basis the family of all open intervals).
An immediate consequence of Definitions 2.9 and 2.10 is that for any totally preordered set
(X, %), the order topology τ%is the coarsest topology on Xsuch that %is continuous.
There are many results dealing with continuous real-valued utility representations of a total
preorder. The most classical theorems in this field are due to Eilenberg [79] and Debreu [69, 70]:
Theorem 2.11 (Eilenberg, 1941) In a connected separable topological space, any continuous
total preorder is continuously representable in R.
Theorem 2.12 (Debreu, 1954, 1964) In a second countable topological space, any continu-
ous total preorder is continuously representable in R.
A miscellany of representation results followed (in the 1970’s): let us recall, among others,
the approaches due to Jaffray [145], Neuefeind [197], Peleg [203], Richter [215], and Sonder-
mann [235]. A common denominator of many approaches to the topic is the Open Gap Lemma,
which was (incorrectly) proved by Debreu [69] in 1954, and then corrected by the same author
ten years later [70]. For our purpose, the most relevant consequence of this result is the following:
Corollary 2.13 If a total preorder on a topological space is representable in R, then it is con-
tinuously representable in R.
The above result brings back the problem of the continuous representability of a total pre-
order to that of its mere representability, on which Theorem 2.8 by Beardon et al. [27] certainly
sheds some light. However, Theorem 2.8 mostly provides negative information, since several
total preorders typically fail to be representable. Thus, it appears natural to seek more refined
classifications of non-representable preferences. More precisely, the (new) questions are:
(Q2′)Can we detect weaker forms of representability for non-representable preferences?
(Q3′)Can we make these weaker forms of representability continuous?
A possible approach to questions (Q2′) and (Q3′) is to establish a “degree of representability”
of total preferences by using more descriptive codomains rather than the set of real numbers. In
this respect, codomains (different from R) ensuring that the content of Corollary 2.13 is preserved
–in the sense that the representability of a total preorder implies its continuous representability–
look quite appealing. This brought Herden and Mehta [140] to formulate the notion of a Debreu
chain, which is a linear ordering such that the representability in it also ensures the existence
of a continuous representation. (Thus, by Corollary 2.13 the linear ordering of the reals is the
prototype of a Debreu chain; however, it is not the only one.)
In the same direction of research, some other authors extended the notion of a Debreu
chain to that of a pointwise Debreu and locally Debreu chain [53], also considering lexicographic
products satisfying these properties [117]. We shall deal with these recent approaches that
aim at enlarging the representability of preference relations in Section 3.1, where we consider
representations with lexicographic codomains. Further, in Section 3.2 we will present a universal
description of semiorders by means of embeddings into modified forms of lexicographic products.
New trends in preference, utility, and choice: from mono to multi 11
Nevertheless, the issues mentioned in the last two paragraphs are not the only ones. In fact,
further problems on representability arise for the lack of representations of preferences that fail
to fully possess the classical tenets of economic rationality. More precisely, the issue –which is
obviously related to the question (Q1) formulated in Section 2.2– is the following:
(Q4) How can we represent more refined preference structures by means of utility functions?
We shall present possible ways to address question (Q4) in Section 4.3, where we deal with
multiple and modal utility representations of both a single preference and a pair of preferences.
2.4 Choice rationalization
Here we recall some elementary definitions on choices. We also summarize the basics of the theory
of revealed preferences, pioneered by Samuelson [224] and successively developed by several
eminent scholars: see, among many others, [17, 18, 58, 137, 142, 144, 206, 214, 230, 231, 232]. For
further details, the reader is referred to some textbooks on the topic, such as Aleskerov et al. [6]
and Suzumura [237], as well as the very recent monograph by Chambers and Echenique [55].
Definition 2.14 Let Ωbe a family of nonempty subsets of X, which contains all singletons and
is closed under the operation of taking finite unions (hence Ωcontains all nonempty finite subsets
of X).12 Achoice correspondence on Xis a map c:Ω→Ωsuch that the inclusion c(A)⊆A
holds for any A∈Ω. In particular, a choice function is a single-valued choice correspondence,
that is, |c(A)|= 1 for all A∈Ω. The set Ωis the domain of c, elements of Ωare menus, and
elements of a menu are items. A choice space is a pair (Ω, c), where cis a choice correspondence
on Xhaving Ωas domain. A choice space (Ω, c) is complete if Ωis the family 2Xof all nonempty
subsets of X, and is finite if Ωis the family of all finite nonempty subsets of X.
The nonempty set c(A) collects all items of Adeemed “selectable” by the economic agent; in
case the problem requires that a single item is to be chosen, this is usually done at a later time
and with a different procedure. However, in the special case of a choice function, a single item
is immediately selected from each menu: this is the original setting under which Samuelson was
working in his seminal paper [224], later extended to the general case of choice correspondences.
Next, we recall the classical notion of the preference revealed by a choice, which is typically
employed in order to identify all cases of rational behavior.
Definition 2.15 Let (Ω, c) be a choice space. The preference revealed by c, denoted by %c, is
the binary relation on Xdefined as follows for each x, y ∈X:
x%cydef
⇐⇒ there is a menu A∈Ωsuch that x, y ∈Aand x∈c(A).
Then cis called rationalizable if it can be retrieved from %cby maximization, that is, for all
menus A∈Ω, the equality c(A) = max(A, %c) holds. Equivalently, cis rationalizable if there is
a (not necessarily complete) binary relation %on Xsuch that c(A) = max(A, %) for all A∈Ω.
The next example illustrates the notions introduced so far.
12The literature on choice theory also consider other types of domains, e.g., for the case of choices arising from
consumer demand theory. For the sake of simplicity, here we limit our analysis to the case in which Ωsatisfies
some rather mild closure properties (see [46, 80] for a justification of this assumption).
12 A. Giarlotta
Example 2.16 Consider the following choice correspondences on X={x, y, z}:13
(c1)xy z , x y , x z , y z ,
(c2)xy z , x y , x z , y z ,
(c3)xy z , x y , x z , y z .
The three relations of revealed preferences %c1,%c2, and %c3are respectively defined by
(c1)x≻c1y , x ≻c1z , y ≻c1z ,
(c2)x∼c2y , x ≻c2z , y ∼c2z ,
(c3)x∼c3y , x ∼c3z , y ∼c3z .
Notice that %c1is a linear order, %c2is quasi-transitive but not transitive, and %3is an equiv-
alence relation. Further, c1and c2are rationalizable, whereas c3is not.
(Mono-)rationalizability coincides with the existence of an underlying preference relation
that fully describes the observed choice behavior. It is clear that a tiny percentage of choices
are rational according to this notion, since the size of the family of choices on a set Xis much
larger, in general, than the family of acyclic binary relations on X. In other words, Definition 2.15
implies that the large majority of choices are labeled as “irrational”. This situation naturally
calls for new, more refined notions of rationalizability, which should aim at smoothening the
sharp dichotomy between rational and irrational choices, possibly identifying weaker notions of
rationality. We shall deal with some recent approaches of this kind in Section 4.4.
Most of the existing results on the rationalizability of a choice are stated in terms of the
satisfaction of axioms of choice consistency. These are properties codifying rules of coherent
behavior, which ought to be respected in order to qualify a selection process as consistent. Here
are a few of the plethora of axioms introduced in the literature during the last 80 years:
✸Property (α)(Standard Contraction Consistency):
If x∈A⊆Band x∈c(B), then x∈c(A).
✸Property (β)(Symmetric Expansion Consistency):
If A⊆B,x, y ∈c(A), and y∈c(B), then x∈c(B).
✸Property (γ)(Standard Expansion Consistency):
If x∈c(Ai) for all i∈I, then x∈cSi∈IAi.
✸Property (ρ)(Standard Replacement Consistency):
If y∈c(A) and y /∈c(A∪ {x}), then x∈c(A∪ {x}).
✸WARP (Weak Axiom of Revealed Preference):
If x∈Aand there are y∈c(A) and B∈Ωsuch that y∈Band x∈c(B), then x∈c(A).
✸PI (Path Independence):
c(A∪B) = c(c(A)∪c(B)).
13Selected items are underlined: thus, xy z means c({x, y , z}) = {x},y z means c({y, z }) = {y, z}, etc. Notice
that, by the very definition of a choice correspondence, we always have c({a}) = {a}for each a∈X: thus, it
suffices to indicate how choices are defined for menus of size at least two.
New trends in preference, utility, and choice: from mono to multi 13
(A universal quantification over menus and items is implicit.)
The first three properties are classical, respectively introduced by Chernoff [58] for (α), and
by Sen [230] for (β) and (γ); on the contrary, property (ρ) is very recent [46]. WARP, due to
Samuelson [224], is the most well known axiom in choice theory. PI is a very elegant axiom due
to Plott [206].
The semantics of these axioms of choice consistency is simple. Property (α) says that if an
item xis selected from a menu B, then xis also selected from any submenu A⊆Bcontaining
it. Property (β) states that any two items x, y selected from a menu Aare simultaneously either
selected or rejected in any larger menu B. Property (γ) says that if an item xis selected from
all menus in a family A, then xis also selected from the menu obtained as the union of the
elements of A. Property (ρ) states that if an item yis selected from a menu Abut is rejected
as soon as a new item xis adjoined to A, then the new item xis selected from the larger menu
A∪ {x}.WARP says that an item xis always selected from a menu Awhenever there is an
item yselected from Asuch that xis revealed to be preferred to y. Finally, PI states that if
the dynamic process of selection proceeds in a “divide and conquer” manner,14 then the final
outcome is independent of the way the menu is initially divided for consideration.
Example 2.17 For the choices defined in Example 2.16, the following holds:
(1) c1satisfies all listed axioms of choice consistency;
(2) c2satisfies (α), (γ), (ρ), and PI, but (β) and WARP fail;
(3) c3only satisfies (α), but none of the other properties hold for it.
We conclude this overview by listing some relationships between forms of rationalizability
of a choice and the axioms of choice consistency introduced above, which hold under very mild
conditions on the choice domain: see, among several references on the topic, the classical papers
by Arrow [17] ad Sen [230], as well as the recent results in [46].
Theorem 2.18 The following equivalences hold for a choice space (Ω, c):
(i) cis rationalizable ⇐⇒ (α) & (γ)hold.
(ii) cis rationalizable by a total preorder ⇐⇒ WARP holds ⇐⇒ (α) & (β)hold.
(iii) cis rationalizable by a preorder ⇐⇒ (α) & (γ) & (ρ)hold.
The following questions naturally arise:
(Q5) Can we refine the classification of rationalizable choices given by Theorem 2.18?
(Q6) Can we smoothen the classical rational/irrational dichotomy, providing a classification of
non-rationalizable choices by means of “degrees of rationality”?
Questions (Q5) and (Q6) will be addressed in Sections 3.4 and 4.4, respectively.
14By a “divide and conquer” manner, we mean: the menu is split up into smaller sets, a choice is made over
each of these sets, the selected items are collected, and finally a choice is made from them.
14 A. Giarlotta
3 The transition
In this section we start a process of transition toward a multi-approach. Specifically, we describe
some alternative tools in preference modeling, utility representations, and choice rationalization,
all of which suggest the opportunity to pursue a multi-approach to a full extent. These techniques
do solve a few of the issues arising from the classical mono-approach. However, they are not
completely satisfactory, inasmuch as they fail to address some other important problems.
3.1 Utilities with lexicographic codomains
As already recalled in the previous sections, several well-behaved preferences that naturally
appear in applied fields fail to be representable by a real-valued utility function. In fact, even
in the desirable scenario in which an agent’s preferences are transitive and complete, their
representability by real-valued embeddings is not guaranteed in general. This consideration
brought Herden and Mehta [140] to formulate the following question:
(Q7) Why do we only consider R-valued utility functions as representations of preferences?
As extensively discussed in Mehta [187], the literature on utility representations mostly
deals with utility functions with values in the linear ordering (R,≥). Regrettably, the very same
literature lacks a systematic and convincing discussion explaining why Ris the only considered
codomain. The rationale of such a choice is possibly connected to the fact that economists
naturally identify the utility of a bundle of goods by a real number. In addition, the mathematical
amenability of the linearly ordered topological space (R,≥, τ≥) –which is metrizable, complete,
separable, etc.– provides further reasons of opportunity to universally implement this choice.
However, Herden and Mehta [140] argue that these arguments do not suffice. In fact, the two
authors identify several types of problems connected to the inveterate use of Ras the codomain
of utility functions. Following the presentation given in [53], we collect these issues in two groups:
(a) mathematical, which in turn can be ordinal or cardinal;15 and (b) theoretical.
(a) Historically, the most significant example of ordinal obstruction to the representability in
Ris the lexicographic plane R2
lex [69]: this linear ordering is not representable in Rbecause
it does not satisfy the countable chain condition (i.e., there are uncountably many pairwise
disjoint nonempty open intervals). Another example of non-representability in Rdue to an
ordinal obstruction is the long line, that is, the lexicographic product ω1×lex [0,1) with its
minimum (0,0) removed. The importance of the latter linear ordering in economic theory
is widely acknowledged [81, 191]. The structural reason for which the long line cannot be
embedded into Ris that it contains a copy of ω1, the first uncountable ordinal. (For a
throughout discussion of ordinal obstructions to representability, see [27, 28].)
Cardinal obstructions to the representability in Rare quite frequent as well. Herden and
Mehta [140] give some examples of commodity spaces studied in economic theory, which
fail to be representable in Rbecause their cardinality is greater than the continuum. A
first example of this kind is the infinite-dimensional commodity space L∞(µ) of essentially
bounded measurable functions on a measure space; in most models used in general equilib-
rium theory [31], this linear ordering is too large to be embedded in R. Another example
15We should also distinguish between purely ordinal codomains, and those which also have an algebraic struc-
ture. Among the latter, let us mention (without getting into details) representations that employ non-Archimedean
ordered fields, introduced by Louis Narens [195].
New trends in preference, utility, and choice: from mono to multi 15
of a linear preference that is not embeddable in Rfor cardinal reasons is the space (Rn)R
of all functions from Rto the commodity space Rn, used in capital theory [71].
(b) From the theoretical point of view, the use of Rto represent preferences may even clash
with the very concept of utility. In his paper on the foundations of utility, Chipman [59]
argues that utility is not a real number, but a vector that is inherently lexicographic in
nature. Accordingly, he proposes to employ the lexicographic power 2α
lex as a base of
utility representations. (Here 2 = {0,1}is the linear ordering with two elements, and α
is a suitable ordinal number.) Chipman points out the convenience to use of a transfinite
sequence of length αin place of a real number to represent preferences: mathematically,
every linear ordering becomes representable; economically, the concept of utility becomes
easier to understand. Last but not least, representability of a preference space (X, %) in
Rrequires the topological space (X, τ%) to have a countable base, which has no intuitive
meaning from the economic point of view [60]. For an extensive analysis of a notion of
lexicographic utility and alternative types of utility representations, the reader is referred
to the (dated but always valuable) survey by Fishburn [94].
In the light of the above discussion, it seems natural to consider alternative utility represen-
tations, which use a base chain different from R. The most frequent base chains employed in the
literature are lexicographic products, e.g., 2α
lex (as in Chipman [60]), R×lex 2 (as in Wakker [245]),
Rn
lex (as in Knoblauch [153]), and the long line (as in Campi´on et al. [41]). Thus, it appears
natural to develop a theory of utility representations in which the base chain is a lexicographic
product of linear orderings. To start, we recall the basic definition of lexicographic product.
Definition 3.1 Let X={(Xj,%j) : j∈J}be a nonempty family of chains indexed over a
well-ordered set (J, ≤). The lexicographic product of Xis the chain Qlex
j∈JXj=Qj∈JXj,%lex,
where the strict linear order ≻lex is defined as follows for all x= (xj)j∈J, y = (yj)j∈J∈Qj∈JXj:
x≻lex ydef
⇐⇒ there is δ∈Jsuch that xδ≻δyδand xj=yjfor all j < δ.
In particular, X×lex Ydenotes the lexicographic product of the two chains Xand Y. In case
the well-ordered index set Jis a nonzero ordinal α, we denote the corresponding lexicographic
product by Qlex
ξ<α Xξ. Further, Xα
lex is the lexicographic power of α-many copies of X.
The use of lexicographic products as a codomain of utility representations can be naturally
motivated when modeling multidimensional preferences. In fact, in order to endow a Cartesian
product of some given chains with a linear order, lexicographic utility structures come very
handy, since they are linked to the existence of some factors which are “overwhelmingly more
important” than others.
For instance, assume that there are nfactors X1,...,Xnof concern to the decision maker.
An element xj∈Xjis a “level of the factor Xj” (e.g., in an allocation problem, xjrepresents
the resources allocated to the j-th activity). Then X=X1× · · · × Xnis the set on which a
preference %has to be established by the decision maker. A lexicographic modeling of utilities
requires finding whether there exist nindividual utility functions uj:X→R,j= 1,...,n,
such that, for each x= (x1,...,xn), y= (y1,...,yn)∈X, we have x%yif and only if
(u1(x),...,un(x)) %lex (u1(y),...,un(y)), where %lex is the lexicographic ordering on Rn. In
this way, preferences are classified according to a measure of their “lexicographic complexity”.
16 A. Giarlotta
For instance, if a chain (X1,%1) can be order-embedded into the lexicographic power R2
lex but
not in R, and another chain (X2,%2) can be order-embedded into R4
lex but not in R3
lex, then the
lexicographic complexity of the latter is greater than the lexicographic complexity of the former.
Formally, we can define the notion of the representability number of a chain as follows [113]:
Definition 3.2 A chain (X, %) is α-representable in Rif it can be embedded into the lex-
icographic power Rα
lex, where αis an ordinal number. The least ordinal αsuch that Xis
α-representable in Ris the representability number of Xin R, denoted by reprR(X). More gen-
erally, given a base chain B, the representability number of Xin B, denoted by reprB(X), is the
least ordinal αsuch that Xcan be embedded into the lexicographic power Bα
lex.
The α-representability of a chain (X, %) in Rcorresponds to having a representation of the
preference ordering %in Xby a well-ordered family of utility functions uξ:X→Rindexed by
the ordinal numbers ξ < α. Then, for any x, y ∈X, we have x≻yif and only if uδ(x)> uδ(y)
holds, where δis the least ordinal number below αat which uδ(x) and uδ(y) differ. One can
think of the ordinal indices as determining the relative importance of the utility functions uξ.
In connection with the findings of Theorem 2.8, it is well-known that long chains are not α-
representable in Rfor any countable ordinal α(see Fleischer [101]): thus, their representability
number in Ris ω1. It follows that the family of all chains can be partitioned in three classes:
(i) long chains;
(ii) short (i.e., not long) chains with uncountable representability number in R;
(iii) chains with countable representability number in R.
The two classes (ii) and (iii) are very rich in variety. For instance, it is not surprising that
Aronszajn lines belong to class (ii). On the other hand, rather unexpectedly, in (ii) we can
also find several hierarchies of small chains, i.e., in the terminology of Theorem 2.8, chains that
are neither long nor wild: see [111], Chapter 5. Even more surprisingly, class (iii) contains
many types of linear orderings. For instance, Giarlotta and Watson [118] exhibit a hierarchy
of chains having representability number in Requal to ω(the first infinite ordinal). Finally,
in [112] lexicographic products that are representable in R(i.e., such that reprR(X) = 1) are
characterized in terms of suitable features of their factors.
Concerning the case of base chains different from R, in [113] the author determines the value
of reprB(X) for several base chains Band represented chains X, again in relation to Theorem 2.8.
Specifically, the following results hold:16
Theorem 3.3 (i) If κis a regular cardinal that is not embeddable into B, then reprB(κ) = κ.
(ii) If Bis an uncountable chain such that A×lex 2is not embeddable in Bfor any uncountable
A⊆B, then reprB(Bα
lex) = αfor any ordinal α.
(iii) If Xis an Aronszajn line or a Souslin line, then reprR(X) = ω1.
In particular, Theorem 3.3(ii) yields the following known fact [156]:
Corollary 3.4 reprR(Rα
lex) = αfor any ordinal α.
Some additional instances of theoretical results concerning the representations of lexico-
graphic preferences are given in [43, 120, 156].
16See Kunen [157] for the undefined notions of regular cardinal and Souslin line.
New trends in preference, utility, and choice: from mono to multi 17
3.2 Universal semiorders
Semiorders are among the most studied categories of binary relations in preference modeling.
This is due to their capability to model many phenomena in economics and psychology, whenever
the agent exhibits preferences/choices with a “threshold of perception or discrimination” (also
called just noticeable difference, see [169]). The reader may consult Chapter 2 of the monograph
by Pirlot and Vincke [205] for an extensive list of possible applications.
The notion of a semiorder originally appeared (under a different name) in 1914, in the work
of Wiener [247] (see [99]). Nevertheless, the introduction of semiorders in economics is usually
attributed to Luce [164], who was the first to use this model to study choices in settings where the
agent’s indifference is naturally intransitive. Luce’s original definition is based on the reciprocal
behavior of the associated relations of strict preference and indifference. Nowadays, a semiorder
is defined as either a reflexive relation that is Ferrers and semitransitive, or, equivalently, an
asymmetric relation that is Ferrers and semitransitive (sometimes called a strict semiorder).
Since Luce’s seminal paper, research on semiorders has been abundant, due the universally
acknowledged importance of this type of ordered structure. Several contributions on the topic are
concerned with real-valued representations of semiorders [29, 42, 44, 104, 155, 159, 169, 190, 194],
whereas many others deal with the more general notion of an interval order (see, e.g., [29, 36]
and references therein), a preference structure introduced by Fishburn in the 1970’s [91, 93, 97].
Semiorders have been also studied in connection to the assessment of knowledge and learning:
on the topic, the interested reader may consult the monographs by Doignon and Falmagne on
Knowledge Spaces [73] and Learning Spaces [87], as well as some papers describing stochastic
theories for the evolution of preference structures [72, 84, 85, 86].
Concerning the utility representation of semiorders, a main contribution on the topic is the
classical paper by Scott and Suppes [229], in which semiorders are described by the existence
of a “shifted” type of utility function (see also Rabinovitch [208]). Formally, a Scott-Suppes
representation of a semiordered set (X, %) is a function u:X→Rsuch that the equivalence
“x%y⇔u(x) + 1 ≥u(y)” holds for all x, y ∈X. (Here 1 is the threshold of perception or
discrimination.) It is well known that not all semiorders admit a Scott-Suppes representation: in
fact, its existence imposes strong structural restrictions, as pointed out by `
Swistak [238]. In this
respect, a recent result by Candeal and Indur´ain [44] characterizes Scott-Suppes representable
semiorders in terms of the properties of regularity and s-separability. Despite these restrictions,
Scott-Suppes representations have been given a lot of attention, due to their relevance in several
fields of research, e.g., modelizations of choice with errors [1], choice theory under risk [90],
extensive measurement in mathematical psychology [155, 159], decision making under risk [221].
Very recently, the structure of an arbitrary semiorder has been fully described by Giarlotta
and Watson [121]. This description has the flavour of a Scott-Suppes representation, insofar as
it uses “shifted” forms of lexicographic products. In fact, any semiorder can be order-embedded
into a modified form of lexicographic product of three total preorders: here the modification
is given by a shift operator, which typically creates intransitive indifferences. Since the middle
factor of this modified product is the usual ordering (Z,≥) of the integers, and the shift operator
is applied to it, these structures are called Z-products. In particular, a Z-line is a Z-product in
which the first and the third factors are linear orderings. The formal notions are as follows:
Definition 3.5 The Z-product of two totally preordered sets (A, %A) and (B, %B) is the triple
P, ⊕1,%⊕1
lex, where:
18 A. Giarlotta
•Pis the Cartesian product A×Z×B;
• ⊕1 is the unary operation on Pdefined by (a, n, b)⊕1 := (a, n + 1, b) for each (a, n, b)∈P;
•%⊕1
lex is the canonical completion17 of the asymmetric relation ≻⊕1
lex on Pdefined by
(a, n, b)≻⊕1
lex (a′, n′, b′)def
⇐⇒ (a, n, b)≻lex (a′, n′, b′)⊕1
for each (a, n, b),(a′, n′, b′)∈P, with ≻lex being the standard lexicographic order on P.
A⊙ZBdenotes the Z-product of the total preorders (A, %A) and (B, %B). The Z-product of
two linear orderings is a Z-line.
It turns out that Z-products (and Z-lines) are universal semiorders, in the sense that any
semiorder order-embeds into a Z-product. The process to construct such an embedding is rather
technical, but it can be summarized in the following three main steps:
(1) first consider a “macro-ordering”, given by the transitive closure18 of the semiorder;
(2) then partition each equivalence class of the macro-ordering into “vertical slices” indexed
by the integers, allowing only certain relationships between pairs of slices;
(3) finally establish a “micro-ordering” to further refine the distinction among elements of the
semiorder, and obtain an order-embedding into a Z-product.
The binary relations used at each stage are total preorders. This fact is clear for the macro-
ordering at stage (1). At stage (2), the partition of each indifference class of the transitive
closure uses a locally monotonic integer slicer (LMIS), which is an integer-valued map having
some ordering properties. The micro-ordering employed at stage (3) is a modified form of trace,
called sliced trace, which allows “backward paths” with respect to an LMIS. The reader is referred
to [121] for several examples of LMIS and the associated sliced traces. Then, we have [121]:
Theorem 3.6 The following statements are equivalent for a reflexive and complete (X, %):
(i) (X, %)is a semiordered space;
(ii) (X, %)order-embeds into a Z-product;
(iii) (X, %)order-embeds into a Z-line;
(iv) (X, %)order-embeds into (X, %tc )⊙Z(X, %ζ).
(In (iv), %tc is the transitive closure of the semiorder %, and %ζis the sliced trace associated
to some LMIS ζ:R→Z.) The following three consequences of Theorem 4.7 are noteworthy:
Corollary 3.7 Z-lines are universal semiorders.
Corollary 3.8 The Z-line Q⊙ZQis a universal countable semiorder.
17The canonical completion of an asymmetric relation transforms incomparability into indifference.
18The transitive closure of a binary relation %is the smallest transitive relation %tc containing %.
New trends in preference, utility, and choice: from mono to multi 19
Corollary 3.9 (Rabinovitch, 1978) The dimension19 of a strict semiorder is at most 3.
In addition to the above consequences, the descriptive characterization of all semiorders
established in Theorem 4.7 may provide a unifying view of several results that are currently
scattered throughout the literature. For instance, many notions of separability –Cantor, Debreu,
Jaffray, strong, weak, topological, interval order, semiorder, etc.– that have been extensively
studied in the past (see, e.g., [29, 45] and references therein) can be characterized by suitable
properties of embedding into Z-lines. Similarly, the geometric representations of semiorders given
by Beja and Gilboa (see Theorems 3.7, 3.8, 4.4, and 4.5 in [29]) as well as the characterization
of Scott-Suppes representability given by Candeal and Indur´ain [44] can be described in terms
of properties of embeddability into special Z-lines.20
3.3 (m, n)-Ferrers preferences
As recalled in Section 2.2, an interval order can be equivalently defined as (1) a reflexive rela-
tion satisfying the Ferrers property, or (2) an asymmetric relation satisfying the strict Ferrers
property:21 to distinguish the two cases, we shall speak of a strict interval order in case (2).
The two settings are equivalent because the canonical completion of a strict interval order is an
interval order, and, conversely, the asymmetric part of an interval order is a strict interval order.
Similarly, a semiorder can be equivalently defined as (1) a reflexive relation satisfying both
the Ferrers and the semitransitive properties, or (2) an asymmetric relation satisfying both the
strict Ferrers and the strict semitransitive properties: for clarity, we speak of a strict semiorder in
case (2). Again, the difference between (1) and (2) is immaterial, since the canonical completion
of a strict semiorder is a semiorder, and the asymmetric part of a semiorder is a strict semiorder.
Interval orders and semiorders have been employed in the literature on preference modeling
as a sound alternative to total preorders, due to their ability to realistically describe situations
in which the agent displays intransitive preferences. In fact, interval orders (hence semiorders)
always have a transitive strict part, but the associated indifference fails, in general, to be tran-
sitive. The main difference between modelizations based on interval orders and those based on
semiorders is that in the former case the threshold of discrimination need not be constant.
Quite recently, in the process of defining broader types of preferences for which the associated
indifference may be intransitive –and, specifically, to generalize some variations of semiorders
proposed by Fishburn [98]–, ¨
Ozt¨urk introduced the notion of (m, n)-Ferrers properties. These
properties require that the first and the last elements of two sequences of preferences having
length mand nmust be suitably related to each other. In particular, the classical Ferrers
condition is the (2,2)-Ferrers property, whereas semitransitivity is the (3,1)-Ferrers property.
However, ¨
Ozt¨urk’s definition is limited to an asymmetric (and transitive) relation, and so
it does not allow one to systematically deal with “degrees of transitivity” of preferences. This
motivated a further extension of her approach by Giarlotta and Watson [119], who distinguish
two types of (m, n)-Ferrers properties: weak and strict, respectively related to sequences of
preferences that are either reflexive or asymmetric.
19The dimension of a strict semiorder ≻is the least number of strict linear orders whose intersection gives ≻.
20This is a work in progress [126].
21The strict Ferrers property and the strict semitransitive property are respectively defined exactly as the Ferrers
property and the semitransitive property in Definition 2.2, with ≻in place of %.
20 A. Giarlotta
Definition 3.10 Let %be a weak preference on X, and ≻its asymmetric part. For fixed
integers m≥n≥1, we say that %satisfies the weak (m, n)-Ferrers property (or it is weakly
(m, n)-Ferrers) if the implication
(x1%...%xm)∧(y1%... %yn) =⇒x1%yn∨y1%xm(1)
holds for all x1,...,xm, y1,...,yn∈X. The notion of strict (m, n)-Ferrers property is defined
similarly, substituting %by ≻in (1).
Notice that the (strict or weak) (2,2)-Ferrers property is the classical Ferrers condition,
whereas the (strict or weak) (3,1)-Ferrers property is semitransitivity. Said differently, weak and
strict (m, n)-Ferrers properties coincide for m+n= 4, i.e., for interval orders and semiorders.
However, they behave quite oppositely as mand ngrow:
Lemma 3.11 Let %be a total weak preference on X. For all integers m, n, p, q such that
m≥n≥1,p≥q≥1,m≥p,n≥q, and m+n≥3, we have:
•if %is weakly (m, n)-Ferrers, then %is weakly (p, q)-Ferrers;
•if %is strictly (p, q)-Ferrers and ≻is transitive, then %is strictly (m, n)-Ferrers.
In other words, weak (m, n)-Ferrers properties display an increasing strength as mand n
grow, whereas strict (m, n)-Ferrers properties becomes weaker and weaker (under the hypothesis
of quasi-transitivity) as mand ngrow.
Weak (m, n)-Ferrers properties are simpler to study, since they display a finite taxonomy.
In fact, the family of weak (m, n)-Ferrers properties forms a finite lattice under implication,
having as maximum the (3,3)-Ferrers property, which corresponds to transitivity. Figure 6 in
the Appendix (taken from [46]) describes all implications among combinations of weak (m, n)-
Ferrers properties (in the gray boxes): see Theorem 3.1 in [119]. All reverse implications do
not hold: see Examples 3.3–3.10 in [119]. Roughly speaking, weak (m, n)-Ferrers properties
are linked to the transitivity of the associated relation of indifference. In this respect, Figure 6
describes a sort of discrete evolution of the transitive property: from possibly no shade of
transitivity (at (1,1)-Ferrers), to quasi-transitivity (at (2,1)-Ferrers), to the classical Ferrers
condition (at (2,2)-Ferrers) and semitransitivity (at (3,1)-Ferrers), until its full satisfaction (at
(3,3)-Ferrers), after having described several forms of transitivity on the path to full transitivity.
To give an idea of the possible “shape” of some weak (m, n)-Ferrers preferences, Figure 1
(taken from [114]) describes the geometric form of a few of them, whenever these preferences
happen to be extensions of a linear continuum.22 For all eight pictures in Figure 1, the dark
gray area represents the strict preference, whereas the light gray area is the indifference: for
instance, in picture (2) we have 7 ≻5 and 7 ∼8, in picture (8) we have 3 ≻2.5 and 7 ∼2.5,
etc. Further, by strong semiorder we mean weakly (3,2)- and (4,1)-Ferrers, whereas by strong
interval order we mean weakly (3,2)-Ferrers.
22Alinear continuum is a linear ordering with the properties that (i) every nonempty subset with an upper
bound has a least upper bound, and (ii) for every pair of distinct elements, we can always find another element
strictly in between them.
New trends in preference, utility, and choice: from mono to multi 21
(1) The total order ≥
0 10
10
0 101 2 3 5 6.5 9
1
2
3
5
6.5
9
10
(2) A total preorder
0 1 2 3 4 5 6.5 8 10
1
2
3
4
5
6.5
8
10
(3) A strong semiorder
0 1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
8
9
10
(4) A strong interval order
0 1 10
1
10
(5) A semiorder
0 1 3 4 5 7 8 10
1
3
4
5
7
8
10
(6) An interval order
0 5 6 7 8 10
5
6
7
8
10
(7) A total quasi-preorder
0 2 3 7 8 10
2
3
7
8
10
(8) A non-transitive relation
Figure 1: A geometric representation of some extensions of the linear ordering ([0,10],≥)
Contrary to weak (m, n)-Ferrers properties, strict (m, n)-Ferrers properties are much more
complicated to classify. Roughly speaking, these properties are linked to the transitivity of the
associated relation of strict preference, hence they refine the graph given in Figure 6 (in the
Appendix) in its lowest part (especially for the so-called “extended preferences”).
It turns out that even the case of strict (m, 1)-Ferrers properties is difficult to analyze, since
it gives rise to an infinite taxonomy of preferences. Furthermore, even if strict (m, 1)-Ferrers
properties somehow become less and less strong as mincreases, they do not display a monotonic
behavior. Specifically, the strongest strict (m, 1)-Ferrers property is (2,1), which implies all the
other strict (m, 1)-Ferrers properties for m≥3: in fact, a strictly (2,1)-Ferrers preference is a
total preorder. The second strongest property is (3,1), since it implies all strict (m, 1)-Ferrers
properties for m≥4: in fact, a strictly (3,1)-Ferrers preference is always quasi-transitive.
However, starting from the strict (4,1)-Ferrers property, this apparent regularity of behavior
vanishes, since (4,1) implies neither (5,1) nor quasi-transitivity.
This erratic behavior of strict (m, 1)-Ferrers properties induced Giarlotta and Watson [124] to
perform a combinatorial analysis of them, which yielded the following nontrivial characterization:
Theorem 3.12 The following statements are equivalent for all distinct integers m, n ≥2:
(i) the strict (n, 1)-Ferrers property implies the strict (m, 1)-Ferrers property;
(ii) n < m and exactly one of the following conditions holds:
(ii.1) m < 2n−3and (2n−3−m)divides (n−3);
(ii.2) m= 2n−3and nis odd;
(ii.3) m > 2n−3.
An interesting consequence of Theorem 3.12 is that the implications among strict (m, 1)-
Ferrers “eventually stabilize”, in the sense that a strict (m, 1)-Ferrers property implies all strict
(p, 1)-Ferrers properties for plarge enough. To formally state this result we need a notion:
22 A. Giarlotta
Definition 3.13 Given an integer m≥2, the Ferrers stabilizer of m, denoted by st(m), is the
least integer p≥mwith the property that the strict (m, 1)-Ferrers property implies the strict
(q, 1)-Ferrers property for all q≥p.
Roughly speaking, the Ferrers stabilizer of an integer is an index of its “limit strength” for
what concerns the satisfaction of the transitive property: the higher this number, the less strong
the property. For instance, st(2) = 2, st(3) = 3, st(4) = 6, st(11) = 17, st(23) = 41, st(63) = 117,
etc. The formula to compute the Ferrers stabilizer of an integer is surprisingly simple [124]:
Corollary 3.14 st(m) = 2m−3−run(m)for each m≥2.
The notation run(m) in Corollary 3.14 stands for the running index of m, defined by
run(m) :=
−1 for even m
0 for m= 3
max p < m −3 : {1,...,p} ⊆ Div(m−3)otherwise,
with Div(m−3) being the set of divisors of m−3, including 1. Thus, in particular, the running
index of an odd number m≥5 is the largest integer less than or equal to m−3
2, which leads a
running sequence of divisors of m−3. For instance, run(5) = run(7) = run(11) = run(13) =
run(17) = 2, run(9) = run(21) = run(33) = 3, run(15) = run(27) = 4, run(63) = 6, etc.
The preceding discussion might suggest that weak and strict (m, n)-Ferrers properties are a
mere numerical/combinatorial curiosity, being totally unsuited for potential applications to real
life problems. However, such an impression would be incorrect. In fact, (m, n)-Ferrers properties
turn out to be linked to money-pump phenomena, which have been carefully analyzed in several
fields of research, such as economics, psychology, and philosophy [67, 135, 138, 186, 204, 210,
213, 225, 227, 240]. Originally observed by Davidson et al. [67], these phenomena are described
by Tversky [240] in relation to the failure of the (strict) transitive property:
“Transitivity, however, is one of the basic and the most compelling principles of rational
behaviour. For if one violates transitivity, it is a well known conclusion that he is acting, in
effect, as a “money-pump”. Suppose an individual prefers yto x,zto y, and xto z. It is
reasonable to assume that he is willing to pay a sum of money to replace xby y. Similarly,
he should be willing to pay some amount of money to replace yby z, and still a third amount
to replace zby x. Thus, he ends up with the alternative he started with but with less money.”
It is apparent that the presence of a strict cycle of preferences puts the economic agent at the
risk of losing all her money, since she may get involved in another cycle of money-pump, and
continue in this fashion until her financial resources are exhausted.
Admittedly, the above money-pump effect requires strict cycles of preferences, which are
forbidden starting from the satisfaction of the weak (2,1)-Ferrers property (which is equivalent
to quasi-transitivity). However, many contributions to the economic literature show that a
money-pump effect may also arise in the presence of mixed cycles of strict preferences and
indifferences: see, e.g., Restle [213], who argues that a strict cycle can be easily induced by a
mixed cycle using a “small bonus” approach.23 Moreover, several other ways to induce a money-
pump from mixed cycles of strict preferences/indifferences have been proposed in the literature,
23For some recent examples of this approach, see Hansson [138] and Rabinowicz [210].
New trends in preference, utility, and choice: from mono to multi 23
e.g., by Schumm [227] in a multiple-criteria set up, as well as by Gustafsson [135] using the
notion of dominance in cases of preferences under uncertainty.24
In Section 4.3 of their paper on choices that are rationalizable by (m, n)-Ferrers preferences,
Cantone et al. [46] introduce a simple model of transactions of goods, which is well suited to
describe the semantics of weak (m, n)-Ferrers properties. Specifically, they show that, in this
model, whenever the binary relation modeling the agent’s preference structure satisfies a fixed
(m, n)-Ferrers property, there exists a strategy that prevents the agent from getting involved
in mixed indifference/strict preference cycles of a certain type. In fact, the authors exhibit a
numeric relationship between the level of transitivity of an economic agent’s preference structure
(i.e., the satisfaction of a certain weak (m, n)-Ferrers property) on one hand, and the caution
that she has to exercise whenever indulging in certain types of transactions (i.e., the avoidance
of money-pump phenomena) on the other hand.
A similar type of argument applies to strict (m, 1)-Ferrers properties. To that end, Giarlotta
and Watson [124] introduce a simple notion of “cash-value” preference as follows:
Definition 3.15 Given goods Xand Y, if there is a (perfectly divisible and fungible) good
Gsuch that Xis weakly preferred to G, and Gis weakly preferred to Y, then we say Xis
cash-value preferred to Y.
Typically, Gwill be money. Essentially cash-value preference is the strengthened weak preference
an agent arrives at when required to assign cash-value to goods: Xis cash-value preferred to Y
if Xis weakly preferred to an amount of cash which is weakly preferred to Y. Then, we have:
Proposition 3.16 A preference %satisfies the strict (m, 1)-Ferrers property if and only if we
never have a sequence of the type x1≻x2≻... ≻xmwhere xmis cash-value preferred to x1.
3.4 (m, n)-rationalizable choices
Here we answer question (Q5) in Section 2.4, refining the classification of rationalizable choices
provided by Theorem 2.18. This topic is based on a recent paper by Cantone et al. [46].
The basic idea of this approach to revealed preference theory is to systematically separate
two issues: (1) the rationalizability of a choice, and (2) the internal structure of its revealed
preference. This goal is achieved by designing a class of axioms of replacement consistency, all
having the same flavor: in fact, these properties examine how the addition of an item to a menu
causes a substitution in the subset of selected elements. We have already examined a property of
this kind in Section 2.4: the standard axiom (δ) of replacement consistency, which characterizes
rationalizable choices with a quasi-transitive revealed preference (see Theorem 2.18(iii)). The
natural extension of this approach to additional properties of the same kind aims at character-
izing rationalizable choices whose revealed preference satisfies different levels of transitivity.
Specifically, first we examine those cases in which the revealed preference is an interval
order, a semiorder, or a total preorder: this yields an axiomatization that is alternative to those
given by Jamison and Lau [147, 148], Fishburn [95], Schwartz [228], and Bandyopadhyay and
Sengupta [25, 26]. Successively, in order to complete a taxonomic classification of rationalizable
24On the other hand, Schick [225] and McClennen [186] argue against the possibility of a money-pump phe-
nomenon, observing that, after transactions between indifferent alternatives, an economic agent may well refuse a
transaction between strictly preferred alternatives. However, as Piper [204] notes, the above solutions are based
on the (unlikely) circumstance that the economic agent remembers the past and accordingly plans the future.
24 A. Giarlotta
choices, we also characterize choices with a weakly (m, n)-Ferrers revealed preference by means
of additional axioms of replacement consistency. In this way, we provide a uniform treatment
of the topic by introducing properties of choice consistency that belong to a single category.
To start, we state three new axioms of replacement consistency:
✸Property (ρF)(Ferrers Replacement Consistency):
If x∈c(A), y∈A,z∈c(B), and z /∈c(B∪ {y}), then x∈c(B∪ {x}).
✸Property (ρst)(Semitransitive Replacement Consistency):
If y∈c(A), z∈A,z∈c(B), and y /∈c(A∪ {x}), then x∈c(B∪ {x}).
✸Property (ρt)(Transitive Replacement Consistency):
If y∈c(A) and y /∈c(A∪ {x}), then c(A∪ {x}) = {x}.
(As usual, a universal quantification over menus and items is implicit.)
The rationale of the above properties is similar to that of the standard axiom (ρ) of replace-
ment consistency, in the sense that, under suitable conditions, a new item “replaces” an old
item in the selection taste of the economic agent. The statement of (ρt) only involves two items
and a single menu, hence its semantics is quite simple to understand. In fact, the antecedent of
(ρt) is exactly the same as that of (ρ), but its consequent is drastically stronger: if yis selected
from Abut is rejected from it as soon as xis adjoined to A, then x“fully replaces” yin the
selection taste of the agent, being the unique item selected from the larger menu A∪ {x}. On
the other hand, the rationale of axioms (ρF) and (ρst ), despite being of the same nature, is
more subtle, since their statements simultaneously involve three items and two menus. To give
a better insight into their semantics, in what follows we reformulate all axioms of replacement
consistency (ρ), (ρF), (ρst), and (ρt) using a model-theoretic notation.
First, we associate to any choice correspondence c:Ω→Ωtwo new preference relations %+
c
and Dc, both inspired by the replacement paradigm:
x%+
cydef
⇐⇒ (∃A∈Ω)y∈A∧c(A∪ {x}) = {x}
xDcydef
⇐⇒ (∃A∈Ω)y∈c(A)∧y /∈c(A∪ {x}).
Second, we employ the following model-theoretic notation:
A|=x%cystands for y∈A∧x∈c(A),
A|=x%+
cystands for y∈A∧c(A∪ {x}) = {x},
A|=xDcystands for y∈c(A)∧y /∈c(A∪ {x}),
where A∈Ωand x, y ∈X. According to the standard model theory semantics of the employed
notation, A|=x%cymeans that menu A“witnesses” a revealed preference of xover y; the
meaning of A|=x%+
cyand A|=xDcyis similar. Finally, we reformulate the four axioms of
replacement consistency using the above notation:
(ρ)A|=xDcy=⇒A∪ {x} |=x%cy
(ρF)A|=x%cy∧B|=yDcz∧z∈c(B) =⇒B∪ {x} |=x%cz
(ρst)A|=xDcy∧A|=y%cz∧z∈c(B) =⇒B∪ {x} |=x%cz
(ρt)A|=xDcy=⇒A∪ {x} |=x%+
cy.
New trends in preference, utility, and choice: from mono to multi 25
Note that this alternative formulation of the four axioms of replacement consistency reveals a
complementarity of (ρF) and (ρst ), since they both state a type of “transitive coherence” of the
two binary relations %cand Dc.25
One of the main results in [46] connects these properties of replacement consistency to levels
of transitivity of the rationalizing preference, thus partially answering question (Q5):
Theorem 3.17 Let c:Ω→Ωbe a rationalizable choice correspondence, and %cits revealed
preference. The following equivalences hold:
(i) %cis quasi-transitive ⇐⇒ csatisfies axiom (ρ);
(ii) %cis Ferrers ⇐⇒ csatisfies axiom (ρF);
(iii) %cis semitransitive ⇐⇒ csatisfies axiom (ρst);
(iv) %cis transitive ⇐⇒ csatisfies axiom (ρt).
Theorem 3.17 readily yields
Corollary 3.18 The following equivalences hold for an arbitrary choice correspondence c:
(i) cis rationalizable by a preorder ⇐⇒ properties (α),(γ), and (ρ)hold;
(ii) cis rationalizable by an interval order ⇐⇒ properties (α),(γ), and (ρF)hold;
(iii) cis rationalizable by a semiorder ⇐⇒ properties (α),(γ),(ρF), and (ρst )hold;
(iv) cis rationalizable by a total preorder ⇐⇒ properties (α),(γ), and (ρt)hold.
The analysis conducted in [46] goes further in the direction of classifying rationalizable
preferences in terms of the transitive structure of their revealed preferences. In fact, the authors
design, for each relevant pair (m, n) of positive integers, a property (ρm,n) of (m, n)-replacement
consistency, finally proving the following result:
Theorem 3.19 A choice correspondence is rationalizable by an (m, n)-Ferrers preference if and
only if properties (α),(γ), and (ρm,n)hold for it.
We refer the reader to the paper [46] for further details about the described approach, as
well as for future directions of research on the topic.
3.5 Resolutions of choices
In this section, which is entirely based on a very recent research by Cantone, Giarlotta, and
Watson [48], we introduce a novel notion for choices, called “resolution”. This notion is designed
to better understand the inner structure of an observed choice behavior, because it provides a
constructive way to possibly decompose the overall selection process in terms of smaller choices.
The general concept of resolution originated, however, from a different field of research.
In fact, resolutions were first introduced by Fedor˘cuk [89] in 1968 for the class of topological
spaces. The successive development of this notion by Watson [246] has proven to be very useful
25For the formal notion of the transitive coherence of two binary relations, see Section 4.1 on bi-preferences.
26 A. Giarlotta
in providing a common point of view of many seemingly different topological spaces (as well as
for linearly ordered spaces [52]): see the large amount of references in [246]. The idea underlying
the notion of topological resolution is natural: given a base topological space, a family of fibre
topological spaces indexed by the base space, and a family of continuous maps also indexed by
the base space, the output is a larger topological space, the resolution, in which every point
is substituted by the associated fibre space. For instance, the double arrow space –i.e., the
lexicographic product R×lex 2 endowed with the order topology, examined by Wakker [245] in
his study on lexicographic preferences– can be seen as a resolution of Rat all points xinto the
discrete space 2 = {0,1}by the functions fx:R\ {x} → 2, defined by fx(x′) := 0 if x′< x, and
fx(x′) := 1 if x′> x.
Cantone et al. [48] adapt the notion of topological resolution to choice theory: in this new
setting, a resolution describes how to build up a complete choice from independent choices on
smaller ground sets. In a nutshell, the process of resolving a choice space into a larger choice
space metaphorically consists of taking a magnifying glass, observing one special item of the
primitive space as being a menu with its own choice structure, and obtaining a larger choice
according to this new piece of information.
Before getting into technicalities, it may be useful to suggest possible interpretations of this
notion in some familiar settings. For instance, a resolution of a choice can be seen as follows:
(1) in a corporation, as the delegation of tasks from the top management to a department;
(2) in a restaurant menu, as the opening of a submenu at a specific (type of) item;
(3) in a portfolio, as the choice of investments including stocks recommended by a broker;
(4) in a budgeted hiring, as the hiring of employees including a particular class of workers.
On a more formal basis, assume that we are given a complete26 choice space (X, cX), called
“base” choice space. We identify a distinguished alternative xin X: the peculiarity of xis that
it can be viewed as a menu itself, which may open up at a “fibre” choice space (Y, cY). To make
a selection in the resolved space (Z, cZ), the process goes as follows. First, we choose from X,
where one of the selections is the (currently closed) menu x. If we do not pick x, then we do
not bother opening the fibre menu Y. On the other hand, if we choose x, then we open Y, and
use the inner structure of the fibre space (Y, cY) to make choices there as well. The following
definition makes the above process formal:
Definition 3.20 Let (X, cX) and (Y , cY) be two complete choice spaces defined on disjoint
ground sets Xand Y. Select x∈X, and let Z:= (X\ {x})∪Y. Define a map π:Z→Xby
π(z) := (zif z∈X\ {x}
xif z∈Y.
The resolution of (X, cX)at xinto (Y, cY), denoted by
(Z, cZ) = (X, cX)⊗x(Y, cY),
26According to the notation employed in Definition 2.14, we should denote these choice space by (2X, cX).
However, for the sake of simplicity, here we prefer to use the more direct notation (X, cX).
New trends in preference, utility, and choice: from mono to multi 27
is the complete choice space on Zwhose choice correspondence cZ: 2Z→2Zis given by
cZ(A) := (cX(π(A)) \ {x}∪cY(A∩Y) if x∈cX(π(A))
cX(π(A)) otherwise. (2)
The two factors (X, cX) and (Y, cY) are, respectively, the base choice space and the fibre choice
space (at x), whereas the distinguished item x∈Xis the base point of the resolution. The
surjective map πis the projection of the resolution. If both Xand Ycontain at least two items,
then the resolution (X, cX)⊗x(Y, cY) is nontrivial ; otherwise, it is trivial.
Figure 2 intuitively describes the semantics of a resolution (Z, cZ) = (X, cX)⊗x(Y , cY).
According to Definition 3.20, if A⊆Zis any menu in (Z, cZ), then the choice set cZ(A) is
obtained as follows. First, look at the trace of Ain X, computing π(A): thus, we have π(A) = A
if Adoes not intersect Y, and π(A) = (A∩X)∪ {x}otherwise. Second, we distinguish two
cases: (i) xis selected from π(A); (ii) xis not selected from π(A). In case (i), cZ(A) is obtained
as the union of what is selected from the trace of Ain X(minus x) and what is selected from
the trace of Ain Y: this is the first line of (2). In case (ii), we do not open Yat all, and cZ(A)
is exclusively computed on the basis of the trace of Ain X: this is the second line of (2).
(X, cX)
x
(Y, cY)
Figure 2: A resolution (Z, cZ) = (X, cX)⊗x(Y, cY)
Definition 3.21 A choice space (Z, c) is resolvable if it is isomorphic27 to a nontrivial resolution;
otherwise, it is irresolvable. If (Z, c) is a resolvable space, then we also say that cis resolvable.
The next example clarifies the above notions.
Example 3.22 Let X={x, x′}and Y={y, z}. Below we resolve some choices cXon Xat the
base point x∈Xinto some choices cYon Y. We list the corresponding results as resolutions
(Z, cZ), where Z:= (X\ {x})∪Y={x′, y, z}(the notation is suggestive of the meaning):
(i) x x′⊗xy z =x′y z , x′y , x′z , y z ,
(ii) x x′⊗xy z =x′y z , x′y , x′z , y z ,
(iii) x x′⊗xy z =x′y z , x′y , x′z , y z ,
(iv) xx′⊗xy z =x′y z , x′y , x′z , y z ,
(v) x x′⊗xy z =x′y z , x′y , x′z , y z .
27Two choice spaces (X, cX) and (W, cW) are isomorphic if there exists a bijection σ:X→Wthat preserves
the choice structure, i.e., the equality σ(cX(A)) = cW(σ(A)) holds for each menu A∈2X.
28 A. Giarlotta
For instance, (i) means that the choice cXis defined by cX({x, x′}) = {x′}, the choice cYis
defined by cY({y, z}) = {y}, and the choice cZis defined by cZ({x′, y, z}) = {x′},cZ({x′, y}) =
{x′},cZ({x′, z}) = {x′}, and cZ({y, z}) = {y}.
Notice that the five resolutions (i)-(v) produce rationalizable choices. In particular, (i)-(iv)
are rationalizable by a total preorder, in fact they are the unique (up to isomorphisms) choices on
a 3-element set satisfying WARP. The resolution (v) is slightly different from the others, because
WARP does not hold for it: in fact, it is the unique (up to isomorphisms) rationalizable choice
on a 3-element ground set having a quasi-transitive but intransitive revealed preference. Finally,
observe that there is only one (up to isomorphisms) choice on a 3-element set Z={x′, y, z}that
is rationalizable but irresolvable, namely,
(vi) x′y z , x′y , x′z , y z .
Among the six rationalizable choices on a 3-element set, the irresolvable choice (vi) is the unique
having a revealed preference that fails to be quasi-transitive (cf. Theorem 2.18(iii)): indeed,
properties (α) and (γ) hold, whereas (ρ) does not (since zis chosen in A={x′, z}and not in
A∪ {y}=Z, but yfails to be selected from Z).
The main problem that arises in this context is how to characterize the process that reverses
a resolution, expressing a resolved choice in terms of its factors. Formally, the question is:
(Q8) Can we determine whether a choice is resolvable (i.e., it is a resolution of simpler choices)?
A constructive answer to question (Q8) is provided in [48], where resolvable choices are char-
acterized by three properties, each of which has an immediate economic interpretation. These
properties yield the notion of “contractible” menu, defined as follows:28
Definition 3.23 Let (X, c) be a complete choice. A menu E∈2Xis contractible if the following
three conditions hold for each A∈2X:
(R1) A∩E6=∅=⇒c(A)\E=c(A∪E)\E,
(R2) c(A)∩E6=∅=⇒c(A)∩E=c(A∩E) , and
(R3) A∩E6=∅=⇒c(A∪E)∩E6=∅ ⇐⇒ c(A)∩E6=∅.
In the four settings described at the beginning of this section, a contractible menu reveals (1)
an autonomous department within a corporation, or (2) an implicit catch-of-the-day submenu
within a menu, or (3) an implicit trusted stock broker on whom an investor relies, or (4) an
implicit hiring budget for, e.g., engineers. The semantics of the three conditions (R1)–(R3) is
natural. For instance, in a corporation with a CEO and a VP of marketing:
(R1) the non-marketing tasks selected by the CEO are independent of which marketing tasks
are available, as long as there is at least one;
(R2) the tasks selected by the VP are independent of which non-marketing tasks are available;
28Contractibility is a form of “outer indiscernibility”, in the sense that a contractible menu cannot be dis-
tinguished from outside, but it typically has an internally distinguishable structure. In fact, contractibility is
a weaker version of revealed indiscernibility, introduced by Cantone et al. [49] in the process of dealing with
congruence relations (i.e., structure-preserving equivalence relations) on a choice space.
New trends in preference, utility, and choice: from mono to multi 29
(R3) whether or not a marketing task is selected is unaffected by which marketing tasks are
available, as long as there is at least one.
The following characterization of the resolvability of a choice is a consequence of the con-
structive approach undertaken in [48]:
Corollary 3.24 A choice is resolvable if and only if there is a contractible proper29 menu.
We conclude the first part of this section with an illustrative example, which provides an
instance of how to constructively answer question (Q8).
Example 3.25 Assume that a diner Dgoes every weekend to a restaurant, which offers a
wide variety of dishes. Over time, we observe the following selections of Dover all menus on
Z={p, c, s, t}, where pis pizza, cis chips, sis salmon, and tis tuna:
p c s t , pc s , p ct , p s t, c s t , pc , p s , p t , c s , c t , s t .
Thus, p c s t says that salmon and chips are selected by Dwhen pizza and tuna are available
in addition to them, ps t states that Dchooses pizza over salmon and tuna, etc. At a first
look, this choice cannot be readily justified. However, a less superficial analysis may reveal
some underlying principles of selection. Thus, the natural question is: Can we explain this
choice better as a resolution of simpler choices? The technique developed in [48] allows us to
constructively answer this question in a positive way.
Specifically, first we argue that the proper menu {s, t}is contractible, since it satisfies prop-
erties (R1)-(R3); thus, by Corollary 3.24, the diner’s choice is resolvable. Second, denoted by
fthe abstract item “fish”, we compute the so-called base choice induced by the menu {s, t}on
the base set X={p, c, f }, which is p c f , p c , pf , c f . Finally, the main result in [48] yields
p c s t , p c s , p c t , p s t , c s t , p c , p s , p t , cs , c t , st =p c f , p c , p f , c f ⊗fs t .
In more descriptive terms, as soon as we establish the contractibility of the menu {s, t}=
{salmon, tuna}, the selection made by Dcan be justified in terms of more elementary choices
as follows: (1) the two fish items salmon and tuna become a fish submenu, in which Dselects
salmon over tuna; (2) the induced base choice on {p, c, f}={pizza, chips, fish}shows that the
diner selects fish and chips, if both are available, and selects pizza alone over either fish and
chips if only one of those is available.
The second part of this section is devoted to exploring the relationship between resolutions
and revealed preference theory. In this context, the following natural question can be formulated:
(Q9) If an axiom of consistency holds for both the base choice and the fibre choice, does it also
hold for the resolved choice?
The answer to question (Q9) is positive for the majority of the mentioned properties of choice
consistency. In fact, we have [48]:
Theorem 3.26 A resolution satisfies an axiom in (α),(γ),(ρ)if and only if so do its factors.
A resolution is path independent if and only so are its factors.
29A menu is proper if it contains more than one item and is different from the ground set.
30 A. Giarlotta
Theorems 3.26 and 2.18 readily yield the following interesting consequence:
Corollary 3.27 A resolution is rationalizable if and only if so are both factors. A resolution is
rationalizable by a preorder if and only if so are both factors.
Thus, for instance, in a corporate structure setting, if the CEO makes a rational selection
and delegates all marketing choices to a rational VP, then the overall selection is still rational.
However, it turns out that WARP (and (β) as well) is not preserved by resolutions, unless
in very special cases. This last fact raises further doubts –in addition to those already raised in
the literature (see, e.g., [80])– on considering this property as an undisputed feature of rational
choice behavior. To clarify the tight boundaries of the preservation of WARP, we need a notion.
Definition 3.28 An alternative xin a choice space (X, c) is a repellent point if either c(A) = {x}
or x /∈c(A) holds for any menu A∈2X.
Thus, whether a repellent point is selected or not from a menu depends on the other available
items; however, if it is selected, then it is unique. Then, we have [48]:
Theorem 3.29 A resolution satisfies WARP if and only if both factors satisfy WARP and either
the base point is repellent or the fibre choice correspondence is the identity.
Theorem 3.29 says that WARP (equivalently, rationalizability by a total preorder) is only
preserved by resolutions in either extreme or trivial cases.
4 The multi-approach
In this section we briefly describe some possible answers to the general question (Q0) formulated
at the beginning of this survey. All these answers will rely on a multi-approach.
4.1 Bi-preferences
Here we specifically address question (Q1) posed in Section 2.2. To that end, we depart from the
traditional mono-relation approach to preference theory, and sketch a theory of bi-preferences.
This section is entirely based on a very recent paper by Giarlotta and Watson [125].
4.1.1 Definition, examples, and motivation
Informally, a bi-preference is a pair of nested binary relations on the same set of alternatives,
which provides two different yet connected types of information about the preference structure of
an agent (or a set of agents). In the general setting, we have: (1) a “rigid” preference %R, which
codifies the very core of the agent’s preference attitude, and is assumed to be fully rational; and
(2) a “soft” preference %S, which summarizes the agent’s tolerance, her willingness/capability
to compromise, and is assumed to be partially rational. In other words, %Rand %Sdescribe,
respectively, what “must” and “may” happen. As a consequence of their semantics, the rigid
component %Rof a bi-preference is transitive, whereas the soft component %Sis a coherent
extension of %R. The formal definition is as follows:
New trends in preference, utility, and choice: from mono to multi 31
Definition 4.1 Abi-preference on Xis a pair %R,%Sof binary relations on Xsuch that30
(Core Transitivity) %Ris a preorder,
(Soft Extension) %Scontains %R, and
(Transitive Coherence) %R◦%S⊆%Sand %S◦%R⊆%S.
The relation %Ris the rigid preference, and %Sis the soft preference. A bi-preference %R,%S
is complete if so is %S. The irreflexive relation %G:= %S\%Ris the gap of %R,%S.
Transitivity is the fundamental property that shapes the structure of a bi-preference. In fact,
Core Transitivity ensures that the rigid part %Rof an economic agent’s preference structure is
rational, whereas Soft Extension and Transitive Coherence require that the soft preference %S
expands %Rin a way that rationality is locally preserved with respect to %R. However, Transitive
Coherence does not guarantee the global transitivity of %S, since the soft preference may even
fail to be quasi-transitive. Notice that no assumption of completeness is made in the general
setting; nevertheless, some types of bi-preferences turn out to be complete, e.g., NaP-preferences
(see Section 4.2). The difference between what may and must happen, codified by the gap %G,
provides information about the agent’s indecisiveness, and prompts a partial ordering describing
the stability of the given information: see Section 4.2 for a formal description of the related poset.
The next example exhibits several natural instances of bi-preferences (see also [125]).
Example 4.2 Let %be a weak preference on X, and %0its trace. Further, let ≡be a generic
equivalence relation on X, and ∆(X) = {(x, x) : x∈X}the diagonal of X. Finally, let
c:Ω→Ωbe a choice correspondence on X, and %cthe preference revealed by c.
(i) (%,%) is a bi-preference if and only if %is a preorder.
(iii) (%, X2) is a complete bi-preference if and only if %is a preorder.
(iii) (∆(X),%) is a bi-preference.
(iv) (∆(X), X2) is the bi-preference on Xwith the largest possible gap.
(v) (%0,%) is a bi-preference.
(vi) If csatisfies properties (α) and (ρ), then the pair (%′
c,%c) is a complete bi-preference,
where %′
cis defined by x%′
cyif y∈c(A) implies x∈c(A) for all A∈Ωwith x, y ∈A.
(vii) (≡,≎) is a bi-preference, where ≎is any symmetric extension of ≡satisfying transitive
coherence (i.e, the two compositions of ≡and ≎are contained into ≎).
(viii) (≡, X2) is a complete bi-preference.
30This notion was originally introduced by Giarlotta and Greco [116] under the name of partial NaP-preference.
Here we switch to a simpler and more agile terminology, which allows us to qualify special types of bi-preferences,
such as “uniform”, “monotonic”, “comonotonic”, etc. (see later in this section).
32 A. Giarlotta
The interest in bi-preferences stems from economic motivations, since, in many applications
to decision making, these structures provide a more accurate modelization rather than mono-
preferences. A relevant advantage of a bi-preference approach is that the two layers of informa-
tion –rigid and soft– allow one to explicitly identify the four binary relations of strict preference,
indifference, incomparability, and indecisiveness. This yields an enrichment of the descriptive
power of the model, since the mono-preference approach is able, at its best, to only distinguish
three relations, namely, strict preference, indifference, and incomparability. As a matter of
fact, in most classical modelizations, indecisiveness is assumed to coincide with incomparability,
which in turn implies that an economic agent with a complete preference structure displays by
definition no indecisiveness whatsoever. On the contrary, in a bi-preference approach, indeci-
siveness is naturally definable at an aggregate level in the form of a transition of states, going
from an incomparability in the (usually incomplete) rigid component to an indifference in the
(possibly complete) soft one. The latter point is better clarified in dealing with special types of
bi-preferences, called “uniform”, which are the subject of the next section.
4.1.2 Uniform bi-preferences
How do bi-preference actually arise in a decision process? In [125], the authors provide two
possible answers to this question, which are designed according to the formative process of
the two components of a bi-preference. In fact, they distinguish between (I) simultaneous and
(II) sequential bi-preferences, depending on the timing of their formation.31 In case (I), the
two preferences are constructed at the same time. Typical examples of this kind are necessary
and possible preferences stemming from applications of the robust ordinal regression approach
in multiple criteria decision analysis (see next section). Another instance of type (I) is the
bi-preference revealed by a “replaceable” choice (see Example 4.2(vi)).32
On the other hand, there are many bi-preferences which happen to be sequential, especially
in a collective decision making setting. Whenever the components of a bi-preference are formed
in temporally distinct stages, we identify a primitive preference (which can be either the rigid
or the soft component) and a derived preference. Typical examples of type (II) are “tracing”
bi-preferences associated to primitive soft components (see Example 4.2(v)). Other examples
of sequential bi-preferences are those in which the primitive component describes the (rational)
inner attitude of an agent, whereas the soft component is a derived extension of the former,
obtained by consistently enriching the core evaluation in view of a specific goal (“sharpening”
or “smoothening” the primitive judgement). The next definition provides instances of this kind.
Definition 4.3 A bi-preference (%R,%S) is uniform if the strict preferences associated to the
two components are nested one inside the other. In particular, it is monotonic if the inclusion
≻R⊆ ≻Sholds, and comonotonic if the reverse inclusion ≻S⊆ ≻Rholds.
The logics underlying the two types of uniform bi-preferences are similar (in their objective)
but dual (in their philosophy). Specifically, their similarity consists of the fact that their com-
mon goal is to enrich the primitive rigid judgement by providing additional soft information;
furthermore, the criterion used in this enrichment is “uniformly” applied to all alternatives.
On the other hand, their duality lies in the philosophy used to construct the bi-preference: in
31The philosophy underlying this distinction will be used again in dealing with (1) simultaneous and sequential
multi-rationalizations (Section 4.4), and (2) multiple and iterated resolutions (Section 4.5).
32A choice is replaceable if the consistency properties (α) and (ρ) hold for it.
New trends in preference, utility, and choice: from mono to multi 33
fact, in a monotonic approach the soft component “sharpens” the rigid judgement, whereas in
a comonotonic one the soft component “smoothens” it. Let us sketch how the two philosophies
of sharpening and smoothening the primitive judgement operate.
Assume that a preorder %Ris given on a set Xof alternatives. There are three possible
configurations between two generic alternatives x, y ∈X, namely:
(1) a rigid indifference x∼Ry;
(2) a rigid preference x≻Ry(or, dually, y≻Rx);
(3) a rigid incomparability x⊥Ry.
Uniform bi-preferences create two additional cases, according to the specific logic used in each
extension. In a monotonic bi-preference %R,%S, we obtain the following five configurations:
(M1) a pure indifference x∼Ry;
(M2) a pure strict preference x≻Ry(or, dually, y≻Rx);
(M3) a pure gap preference x≻Gy(or, dually, y≻Gx), i.e., x≻Syand x⊥Ry;
(M4) a pure indecisiveness x∼Gy, i.e., x∼Syand x⊥Ry;
(M5) a pure incomparability x⊥Sy.
Notice that the configurations (M3), (M4), and (M5) all stem from the rigid configuration (3),
and are obtained by sharpening the judgment of incomparability given at the primitive level.
Figure 3 –taken from [125]– describes the three rigid configurations (1)-(3) and their successive
enrichment by the five monotonic configurations (M1)-(M5).33 Any thick black arrow in Figure 3
represents a rigid preference (from the source over the tail), whereas a thin gray arrow stands
for a gap preference.
As extensively explained in [125], monotonic bi-preferences may naturally arise, for instance,
in the process of selecting the set of best alternatives –possibly a single one– in each feasible
menu. In this case, the economic agent employs the rigid preference %Rto pre-select some
items according to a logic of maximality: xis pre-selected in a menu Awhenever there is no
other alternative y∈Asuch that y≻Rx. Then, in order to make a more accurate selection,
the economic agent may ask external sources to rank the remaining non-dominated options,
introducing new strict preferences via %Sin a transitively coherent way: in Figure 3 this is
obtained by passing from a configuration of type (3) to one of type (M3) or (M3′).34
Also in a comonotonic bi-preference %R,%Sthere are five possible configurations, even if
they are created in a different way:
(C1) a pure indifference x∼Ry;
(C2) a pure strict preference x≻Sy(or, dually, y≻Sx);
(C3) a balanced preference x≻Ryand x∼Sy(or, dually, y≻Rxand x∼Sy);
(C4) a pure indecisiveness x∼Gy, i.e., x∼Syand x⊥Ry;
(C5) a pure incomparability x⊥Sy.
Notice that the configurations (C2) and (C3) stem from the rigid configuration (2), whereas the
configurations (C4) and (C5) stem from the rigid configuration (3). All comonotonic configura-
tions are obtained by smoothening the judgment of either strict preference or incomparability
given at the primitive level. This fact is especially clear for the configuration (C3), where a rigid
preference of xover yis smoothened by a reverse soft preference of yover x(e.g., because there
33Configurations (2)′, (M2)′, and (M3)′are dual to, respectively, (2), (M2), and (M3).
34This procedure is reminiscent of the rational shortlist method, a bounded rationality approach to individual
choice recently introduced by Manzini and Mariotti [172]. However, in the latter case, the two sequential rationales
need not be nested one inside the other, and they fail in general to be transitively coherent. On the point, see
Section 4.4 of this survey.
34 A. Giarlotta
(1)
x∼Ry
(2)
x≻Ry
(2)′
y≻Rx
(3)
x⊥Ry
xyxyxyxy
(M1)
x∼Ry
(M2)
x≻Ry
(M2)′
y≻Rx
(M3)
x≻Gy
(M3)′
y≻Gx
(M4)
x∼Gy
(M5)
x⊥Sy
xyxyxyxyxyxyxy
Figure 3: The five types of admissible configurations in a monotonic bi-preference
are scenarios in which ymay be preferred to x). Figure 4 describes the three rigid configurations
(1)-(3) and their successive enrichment by the five comonotonic configurations (C1)-(C5).35
Comonotonic bi-preferences arise in cases when there is a necessity to consider different types
of arguments to evaluate a preference of an alternative over another one. Imagine, for instance,
that in the process of establishing a rigid relationship between two alternatives xand y, the
economic agent decides that the most likely judgement is that xis strictly preferred to y, i.e.,
x≻Ryholds. However, she is not perfectly convinced of this judgement, because there are (less
likely yet possible) scenarios in which she feels that yis either indifferent to xor even preferred
over x. In this circumstance, the role of %Sis to weaken the primitive judgement modelled by
%R, thus transforming the strict rigid preference x≻Ryinto a soft indifference x∼Sy: this is
exactly what happens in Figure 4 in passing from a configuration (2) of strict rigid preference
to a configuration (C3) of balanced preference.
(1)
x∼Ry
(2)
x≻Ry
(2)′
y≻Rx
(3)
x⊥Ry
xyxyxyxy
(C1)
x∼Ry
(C2)
x≻Sy
(C2)′
y≻Sx
(C3)
x≻Ry∧x∼Sy
(C3)′
y≻Rx∧y∼Sx
(C4)
x∼Gy
(C5)
x⊥Sy
xyxyxyxyxyxyxy
Figure 4: The five types of admissible configurations in a comonotonic bi-preference
4.1.3 An extension of Schmeidler’s theorem
We conclude this section on bi-preferences by providing an interesting theoretical application.
In 1971, Schmeidler [226] proved the following elegant –and maybe surprising– result connecting
the continuity of a preorder to its completeness:
35Configurations (2)′, (C2)′, and (C3)′are dual to, respectively, (2), (C2), and (C3).
New trends in preference, utility, and choice: from mono to multi 35
Theorem 4.4 (Schmeidler, 1971) A nontrivial bi-semicontinuous preorder on a connected
topological space is complete.
Here “nontrivial” means that the asymmetric part of the preorder is nonempty, whereas “bi-
semicontinuous” means that it is both closed semicontinuous (i.e., all lower and upper weak
sections are closed) and open semicontinuous (i.e., all lower and upper strict sections are open).36
The proof of Theorem 4.4 given by Schmeidler is neat and compact. However, two different
arguments –one order-theoretic and one topological– are quite intertwined, and this fact prevents
one from fully understanding to what extent the two hypotheses of connectedness (of the space)
and bi-continuity (of the preorder) are needed in the proof. In an attempt to refine Schmeidler’s
argument, Giarlotta and Watson [125] have very recently generalized Theorem 4.4 to a bi-
preference setting. Here we illustrate one of these extensions, which yields Schmeidler’s theorem
as a corollary (but Schmeidler’s theorem does not allow one to derive it). To begin with, we
need a few new notions.
Definition 4.5 A bi-preference %R,%Sis quasi-monotonic if ≻R∩(%S◦ ≻S)⊆ ≻Sand
≻R∩(≻S◦%S)⊆ ≻S.
Notice that quasi-monotonicity is a weakening of the property of monotonicity introduced in
Definition 4.3, which requires the inclusion ≻R⊆ ≻Sto hold regardless of any condition pointing
in that direction. Instead, quasi-monotonicity states that a strict rigid preference x≻Ryimplies
a strict soft preference x≻Syonly if there are elements in Xalready suggesting that possibility,
in the sense that either x%Sz≻Syor x≻Sz%Syholds for some z∈X. It follows that
quasi-monotonicity is a rather mild assumption in several economic scenarios.
Definition 4.6 A bi-preference %R,%Sis strongly comonotonic if it is comonotonic and quasi-
monotonic.
If %Sis a preorder, then any comonotonic bi-preference %R,%Sis obviously strongly
comonotonic. However, transitivity of the soft component is not needed to ensure strong comono-
tonicity of a bi-preference [125].37 Finally, we can state what we were after:
Theorem 4.7 Let %R,%Sbe a strongly comonotonic bi-preference on a connected topological
space. If %Ris closed semicontinuous, and ≻Sis nontrivial and open semicontinuous, then
(%R,%S)is a NaP-preference and %Sis complete.
Schmeidler’s theorem follows from Theorem 4.7 by taking a bi-preference %R,%Rsuch
that %Ris a bi-semicontinuous preorder on a connected topological space.
4.2 Necessary and possible preferences
In this section, we give a more refined answer to question (Q1) posed in Section 2.2, describing
special types of (comonotonic) bi-preferences: necessary and possible preferences. The specialty
of these bi-preference structures lies in the fact that they have already proven to be useful in
several applications within multiple criteria decision analysis.
36Cf. with the notion of continuity given in Section 2.3.
37In fact, Giarlotta and Watson [125] constructively characterize strongly comonotonic bi-preferences as those
that can be obtained from simpler types of bi-preferences by an operation of resolution. (Resolutions of preference
structures do have the same flavor as the operation of choice resolution described in Section 3.5.)
36 A. Giarlotta
4.2.1 NaP-preferences, robust ordinal regression, and decisions theory
We start with the main notion:
Definition 4.8 Anecessary and possible preference (NaP-preference) on Xis a comonotonic
bi-preference %N,%Pon Xsuch that the following additional property holds:
(Mixed Completeness) for each x, y ∈X, either x%Nyor y%Pxholds.
In this case, %Nand %Pare, respectively, the necessary preference and the possible preference.
The complement of %Pin X2is the impossible preference ≻I. A NaP-preference is normalized
if its necessary component is a partial order. We denote by NaP(X) the family of all NaP-
preferences on X, and by NaPnor(X) the subfamily of all normalized NaP-preferences on X.
Notice that Mixed Completeness implies that the possible component %Pof a NaP-preference
%N,%Pis complete; in particular, configuration (C5) in Figure 4 is ruled out.
NaP-preferences arise quite naturally in applications that require the considerations of several
points of view, for instance within the framework of Multiple Criteria Decision Analysis (MCDA).
In fact, the first appearance of NaP-preferences dates to 2008, in the seminal paper on Robust
Ordinal Regression (ROR)by Greco, Mousseau, and S lowi´nski [133]. (See also Angilella et al. [13]
for a non-additive ROR model based on the Choquet integral, as well as Greco et al. [134] for
an overview of the ROR methodology.)
The ROR approach was originally designed to provide a consistent extension of the UTA
method of Jacquet-Lagreze and Siskos [149], which only considered special types of utility func-
tions fitting the information provided by the decision maker. Instead, in a ROR approach, all
compatible utility functions are taken into account, which in turns yields the creation of a more
refined preference structure: a necessary and possible preference.
Nowadays, the ROR is among the most used methodologies employed in MCDA, as witnessed
by the very large amount of applications in several fields: see, among many others, Angilella
et al. [7] for an application to urban and territorial planning, Angilella et al. [9] for a customer
satisfaction analysis based on a multiplicity of interacting criteria, and Corrente et al. [62] for
applications of ROR to decisions under uncertainty and risk. We refer the reader to the recent
paper by Corrente et al. [61] for a survey on ROR in preference leaning and ranking.
In a ROR approach, the pieces of information provided by an economic agent on a set Xof
n-dimensional alternatives (i.e., in the presence of a set of n≥2 evaluation criteria gi:X→R,
1≤i≤n) are used to build a set Uof global value functions u:Rn→R, which do not contradict
data. In this multi-dimensional setting, two binary relations %Nand %Pon Xnaturally arise
by using, respectively, universal and existential quantification over U:
x%Nydef
⇐⇒ (∀u∈ U)u(x)≥u(y),
x%Pydef
⇐⇒ (∃u∈ U)u(x)≥u(y),(3)
where x, y ∈Xare arbitrary. Then the pair %N,%Pis a NaP-preference on A.
MCDA is not the unique setting in which NaP-preferences (and bi-preferences in general) nat-
urally arise. In fact, the realm of Decision Theory under uncertainty offers another environment
that is well suited to be described by these types of bi-preference structures. For instance, in an
Anscombe-Aumann setting [19], prototypes of a necessary preference and a possible preference
New trends in preference, utility, and choice: from mono to multi 37
are well known, being respectively described by Bewley’s Knightian preferences [32] and Lehrer-
Teper’s justifiable preferences [158]. Specifically, given a set of priors, a Knightian approach
states that an act fis preferred to another act gif this preference holds for all priors; on the
other hand, a model of justifiable preferences considers fbetter than gif this preference holds
for at least one prior. In a von Neumann-Morgenstern’s setting [198], a further example of a
necessary preference extendable by means of a possible preference is given by the incomplete
preorder modeled as in Dubra et al. [77]: here the authors consider a set of utility functions U
such that a lottery pis preferred to another lottery qif the expected utility of pis not smaller
than the expected utility of qfor all functions in U.
Indeed, earlier approaches to Decision Theory use bi-preferences. For instance, Gilboa et
al. [128] define two relations in an Anscombe-Aumann setting: an objective preference %∗and
asubjective preference %∧. The objective relation %∗is a Knightian preference, which models
cases such that the decision maker can convince everybody that he is right. The subjective
relation %∧codifies the maxmin expected utility of Gilboa-Schmeidler [129], and represents
preferences such that the decision maker cannot be convinced by anybody that he is wrong.
The objective preference %∗is a preorder, whereas the subjective preference %∧is a complete
preorder that extends %∗. However, although this model explicitly uses a bi-preference approach,
its underlying philosophy is quite different from that of NaP-preferences: in particular, no
transitive coherence is assumed to hold between the objective and the subjective preferences.
On the other hand, some very recent contributions in Decision Theory employ bi-preferences
more in the spirit of a NaP-preferences approach. For instance, in the model proposed by Cerreia-
Vioglio et al. [54] within an Anscombe-Aumann setting, two types of consistent preferences are
used: the first reflects the decision maker’s judgments about well-being (her mental preferences),
whereas the second represents the decision maker’s choice behavior (her behavioral preferences).
The authors propose axioms that describe the relationship between these preferences, that is,
between mind and behavior. Under standard expected utility assumptions, two representations
are obtained: the first uniquely infers choice behavior from mental preferences; the second uses
mental preferences to direct choice behavior, however leaving room for biases and framing effects.
Some of the results proved in [54] concern NaP-preferences.
Finally, we mention a necessary and possible extension of a very recent approach to sequential
decision making, introduced by Chambers and Miller [56]. The two authors develop a normative
theory of incomplete preferences, called “benchmarking”. Their theory aims at simplifying a
decision making process by modeling its preliminary stage: for instance, in the hiring process for
an academic job, a committee may employ some objective criteria to make a first screening among
candidates. Chambers and Miller characterize benchmarking rules, which are binary relations
satisfying four natural properties: transitivity, monotonicity with respect to set-containment,
incomparability of marginal gains, and incomparability of marginal losses. In Giarlotta and
Watson [127], it is shown that these rules are indifference-induced, that is, the combination of
their symmetric part with set-containment fully describes them. In the same spirit, additional
judgements of pure similarity provided by (groups of) decision makers allow one to enhance
these structures, and give rise to NaP-benchmarking rules [127].
4.2.2 Characterizations, properties, and semantics
In what follows, we give an overview of the main features and the semantics of NaP-preferences.
The following order-theoretic characterization holds (Giarlotta and Greco [116]):
38 A. Giarlotta
Theorem 4.9 (AC)38 A pair %N,%Pof binary relations on Xis a NaP-preference on Xif
and only if there is a family Tof total preorders on Xsuch that %N=TTand %P=ST.
In 1998, Donaldson and Weymark [76] proved a famous result, which says that any preorder
can be written as an intersection of total preorders. This result, later proved again by Bossert [38]
using a simpler technique, strengthens Lemma 15.4 in Fishburn [92] as well as Theorem A(4) in
Suzumura [237]. Donaldson and Weymark’s result is an immediate consequence of Theorem 4.9:
Corollary 4.10 (Donaldson and Weymark, 1998, Bossert, 1999) Every preorder is the
intersection of a collection of total preorders.
The proof of Corollary 4.10 given by Donaldson and Weymark [76] is direct, and makes no
use of related results for partial orders. Instead, the proof given by Bossert [38] is elementary,
since it makes use of a notorious result by Dushnik and Miller [78], which says that every partial
order is the intersection of a collection of linear orders.
Theorem 4.9 is the natural abstraction of the representation (3) stemming from a family
Uof real-valued utility functions (cf. Section 4.3). In fact, Theorem 4.9 can be equivalently
formulated by saying that %N,%Pis a NaP-preference on Xif and only if there exists a
family Tof total preorders on Xsuch that, for all x, y ∈X, the following two equivalences hold:
x%Ny⇐⇒ (∀%∈ T )x%y ,
x%Py⇐⇒ (∃%∈ T )x%y . (4)
Thus, if the total preorders in Tare Debreu-separable,39 then the two representations (3) and
(4) essentially coincide (since every total preorder in Tis representable in Rby Theorem 2.6).
Many other examples of NaP-preferences naturally arise in both theoretical settings and
practical applications. We have already discussed real decision problems in MCDA which may
benefit from a necessary and possible approach. For some instances of theoretical applications,
notice that the four examples of preorders (written as intersection of total preorders) presented
by Donaldson and Weymark [76] immediately generalize to NaP-preferences:
(1) the strong Pareto preorder, which is the first line of representation (3);
(2) the weak Pareto preorder, which is a suitable manipulation of utility representation (3);
(3) the dominance preorder [34], which extends the strong Pareto preorder by regarding per-
mutations of utility vectors as being indifferent to each other;
(4) the hull of dominance preorder [34], which extends the dominance quasi-ordering by using
bi-stochastic matrices.40
An alternative characterization of NaP-preferences, which emphasizes different aspects of
these structures, is given by Giarlotta and Watson [123] (Lemma 2.4):
38The Axiom of Choice (AC) is needed in the proof of Theorem 4.9 to apply Zorn’s Lemma in the case of an
uncountable ground set X.
39See Footnote 6.
40A square matrix is bi-stochastic if (i) all of its entries are non-negative, and (ii) all the row and column sums
are equal to one.
New trends in preference, utility, and choice: from mono to multi 39
Theorem 4.11 A pair %N,%Pof binary relations on Xis a NaP-preference if and only if
Core Transitivity, Soft Extension and the following three additional properties hold:
(Rigid Strict Extension) ≻Nincludes ≻P,
(Soft Completeness) %Pis complete, and
(Mixed Transitivity) %N◦ ≻P⊆ ≻Pand ≻P◦%N⊆ ≻P.
In particular, a bi-preference is a NaP-preference if and only if it is comonotonic and complete.
Rigid Strict Extension and Mixed Transitivity describe the characteristic features of the
strict possible preference ≻P: this relation is stronger than the strict necessary preference ≻N
insofar as it models a situation of preference in which no compensation in the opposite sense is
allowed (see configurations (C2) and (C2)′vs. configurations (C3) and (C3)′in Figure 4). In
fact, not only ≻Pis a strict partial order included in ≻N, but also it enjoys a property of mixed
transitivity whenever combined with the weak necessary preference.
The semantics of a NaP-preference is related to the type of information provided by the
economic agent on the set Xof alternatives. Indeed, Definition 4.8 yields a partition of X2
into three (possibly empty) classes, namely, %N,%G, and ≻I. The union %N∪ ≻Icodifies
the “total information” provided by the economic agent, and the gap %Grepresents a gray
area of “indecisiveness”. The total information provided by the agent can be, in turn, split
into two subtypes: the necessary preference %Nmodels its positive part (what must happen),
and the impossible preference ≻Iits negative part (what cannot happen). According to this
interpretation of NaP-preferences, we can make the family NaP(X) into a poset, using a binary
relation that ranks bi-preferences according to a measure of their “informative content/stability”.
Definition 4.12 Let ⊑be the binary relation on NaP(X) defined by
%N
1,%P
1⊑%N
2,%P
2def
⇐⇒ %N
1⊆%N
2and %P
1⊇%P
2.(5)
If (5) holds, then %N
2,%P
2is an informative refinement of %N
1,%P
1.
Clearly, the pair NaP(X),⊑is a poset. Upon observing that the condition %P
1⊇%P
2can
be equivalently rewritten as ≻I
1⊆ ≻I
2, the semantics of the binary relation ⊑becomes apparent.
Indeed, an informative refinement of a NaP-preference is characterized by an enlargement of both
types –positive and negative– of information, allowing no compensation whatsoever between the
two types. The maximal elements of this poset represent situations in which the gap is empty:
in these cases, the economic agent’s preference structure has no gray area of indecisiveness, and
so it is perfectly stable. These maximal NaP-preferences “are” the total preorders on X. On
the other hand, the unique minimum element of this poset represents a situation of complete
absence of either positive or negative information: in this case, everything may happen, and so
the agent’s preference structure is totally unstable. The next result describes the features of the
poset NaP(X),⊑(see Lemma 5.4 in [116]).
Theorem 4.13 (NaP(X),⊑)is a meet-semilattice,41 having ∆(X), X2as its unique mini-
mum element, and all pairs (%,%)as its maximal elements, with %any total preorder on X.
41Meet-semilattice means that (NaP(X),⊑) is a poset, and for each %N
1,%P
1,%N
2,%P
2∈NaP(X), there is a
greatest element %N
3,%P
3in NaP(X) such that %N
3,%P
3⊑%N
1,%P
1and %N
3,%P
3⊑%N
2,%P
2.
40 A. Giarlotta
An extended discussion on the topic can be found in [116]. For a graphical representation of
the meet-semilattice NaP(X),⊑, the reader may glimpse at either Figure 3 in [116] (for the
simplest case |X|= 2), or Figure 6 in [114] (for the already complicated case |X|= 3). For the
sake of completeness, this last figure is reported in the Appendix as Figure 7.
In view of Theorem 4.13 and its interpretation in terms of informative content, it becomes
very interesting to determine whether, in the case of a finite ground set X, this poset is “well-
graded” in the sense of Doignon and Falmagne [72]. Let us recall their notion of well-gradedness:
Definition 4.14 Let Xa finite set, and d: 2X×2X→Ra metric on the collection of nonempty
subsets of X. A family F ⊆ 2Xis well-graded if, for each R, S ∈ F at distance n, there is a
sequence of sets R=F0, F1,...,Fn=Sin Fsuch that d(Fi−1, Fi) = 1 for each i= 1,...,n.
Upon regarding families of preference relations on Xas sets of pairs, and taking as metric
on 2Xthe classical distance between sets, that is, the size of the symmetric difference, we have:
Theorem 4.15 (Doignon and Falmagne, 1997) The classes of partial orders, semiorders
and interval orders on a finite set are well-graded.
In other words, a uniform family Fof preferences (i.e., preferences of the same type, e.g.,
semiorders) on a set is well-graded if for any two distinct relations R, S ∈ F whose symmetric
difference has size n,Rand Scan be connected by a “path” of length n, that is, a sequence of
nelementary steps within Fwhich smoothly transforms Rinto Sby changing (i.e., eliminating
or adding) one edge a time. The well-gradedness of a uniform family of binary relations is a
fundamental property, which is needed in order to develop a stochastic theory describing the
evolution of preferences through the random occurrence of quantum tokens of information [84,
85, 86]. In the very same direction, Giarlotta and Watson [122] prove the following fact, which
paves the way toward the development of a stochastic theory of NaP-preferences:
Theorem 4.16 The class of normalized NaP-preferences on a finite set is well-graded.
Apart from those aspects already mentioned, NaP-preferences have also been studied from
several other perspectives:
- properties of transitive coherence linking the two components, and their relationship with
the genesis of interval orders and semiorders [114];
- asymmetric and normalized forms of NaP-preferences [115];
- symmetric counterparts of NaP-preferences, called NaP-indifferences [123], which also cod-
ify forms of revealed similarity in individual choice theory (see Section 3.2 of [123], which
is related to the notion of a congruence relation on a choice space [49]).
4.2.3 Some related approaches in fuzzy set theory
We conclude this section by summarizing the main features of two very recent approaches in
fuzzy set theory [249], which have employed NaP-preferences (and bi-preferences, in general) as
a source of inspiration:
(1) fuzzy politics;
(2) NaP-hesitant fuzzy sets.
New trends in preference, utility, and choice: from mono to multi 41
Concerning (1), Alcantud, Biondo, and Giarlotta [3] design a model for the genesis of parties,
which is based of a fuzzy elaboration of a necessary and possible approach. In this model, for
each topic of interest for the political campaign, a candidate is described by means of a PaP-
profile (private and public profile): the private one is known only among politicians, whereas
the public one is available to every citizen and accordingly displayed on the media. Both
components of a PaP-profile are trapezoidal or quasi-trapezoidal42 fuzzy sets on the interval
[−1,1], where −1 represents extreme left, 0 perfect centre, and 1 extreme right. The private
profile is always contained –in the sense of fuzzy set theory– in the public one. A candidate’s
private profile describes the very core of his political ideas on a topic, which he naturally shares
with other politicians in an attempt to form aggregations of powers to better pursue his goals.
On the other hand, the public profile extends the private one by summarizing the politician’s
tolerance/willingness to compromise on the topics of the campaign. All candidates are then
paired up according to the similarity of their PaP-profiles on all topics: this procedure creates
the so-called matching graph of politicians. Finally, an algorithm, which is based on the size and
the cohesion of the cliques of the matching graph, determines the family of newborn parties.
Concerning (2), Zadeh’s fuzzy set theory [249] deals with impreciseness/vagueness of data and
evaluations by imputing degrees to which objects belong to a set. The appearance of fuzzy sets
induced the rise of several related theories, which codify subjectivity, uncertainty, imprecision,
or roughness of evaluations. The rationale of these theories is to create new and more flexible
methodologies, which allow one to realistically model a variety of concrete decision problems.
In this direction, Torra [239] recently extended the notion of fuzzy sets by that of hesitant fuzzy
sets: these are maps assigning to any element of Xa subset of [0,1] (instead of a single element
of [0,1] as for fuzzy sets). Hesitant fuzzy sets permit the modelization of phenomena that cannot
be handled by classical fuzzy set theory: for instance, collective decision making is a natural
outlet for hesitant fuzzy models [2]. Alcantud and Giarlotta [5] propose an extension of Torra’s
notion of hesitant fuzzy set, which fits quite well group decision making. In fact, indecisiveness in
judgements is described by two nested hesitant fuzzy sets, which form a NaP-hesitant fuzzy set:
the smaller (necessary) component collects membership values determined according to a rigid
evaluation, whereas the larger (possible) component comprises socially acceptable membership
values. This novel approach displays structural similarities with Atanassov’s intuitionistic fuzzy
set theory [20, 21], but has rather different features and goals.
4.3 Multiple and modal utility representations
Classical utility representations of preferences fall short under many points of views, as pointed
out in Sections 2.3 and 3.1. Here we examine two ways of dealing with some shortcomings of
traditional (and less traditional) approaches, which are similar yet they address different issues:
(1) multi-utility representations, and (2) modal utility representations.
4.3.1 Multi-utility representations
A traditional stream of research concentrates on the analysis of the continuous and semicon-
tinuous representability of all preorders. Since an incomplete preorder obviously admits no
representation by means of a single utility function, two alternative approaches to the topic
42For the notions of trapezoidal and quasi-trapezoidal profiles, as well as for their canonical representations by
means of quadruples of real numbers in [−1,1], see Section 2.3 in [3].
42 A. Giarlotta
have been proposed over time, using either (i) a single utility function in a weaker form, or (ii) a
family of utility functions. Approach (i), due to Richter [214] and Peleg [203], is quite classical:
Definition 4.17 A preorder %on a set Xis Richter-Peleg representable if there exists a map
u:X→Rsuch that the following two implications hold for each x, y ∈X:
x%y=⇒u(x)≥u(y) and x≻y=⇒u(x)> u(y).
In this case, the function uis a Richter-Peleg representation of %.
A lot is known about this notion, due to the work on analytic order theory by Herden [139],
Jaffray [146], Levin [160], and Sonderman [235]. However, the use of a Richter-Peleg representa-
tion of a preorder is limited by the fact that it determines a loss of information, since one cannot
recover the primitive preference from its representation [168]. This has recently brought sev-
eral authors to consider hybrid approaches to the topic, as that of a Richter-Peleg multi-utility
representation : see Minguzzi [189], as well as Alcantud et al. [4].
These hybrid solutions to the mentioned problem bring us to discuss the second approach,
namely, (ii) multi-utility representations of a preorder. Originally introduced by Ok [200], the
topic of multi-utility representation has benefited from many important contributions by Kamin-
ski [152], Evren and Ok [83], Bosi and Herden [37], and Evren [82]. Here is the formal notion:
Definition 4.18 Amulti-utility representation of a preorder %on Xis a family Uof functions
u:X→Rsuch that the following equivalence holds for each x, y ∈X:
x%y⇐⇒ (∀u∈ U)u(x)≥u(y).(6)
(Cf. (6) with the first line of (3) in Section 4.2.)
Proposition 1 of Evren and Ok [83] easily establishes that every preorder admits a (semicon-
tinuous) multi-utility representation.43 On the other hand, the problem of finding continuous
multi-utility representations of a preorder poses some difficulties. Upon extending Herden’s [139]
approach, Evren and Ok [83] derive a theoretical characterization for the existence of a continu-
ous multi-utility representation of a preorder, which is linked to the solution of a Urysohn-type
separation problem. Regrettably, this characterization offers no insight in practical cases. Thus,
the two authors establish two sufficient conditions, which are useful in applications. The first
result imposes severe restrictions of the topological space and mild conditions on the preorder:
Theorem 4.19 (Evren and Ok, 2011) Every continuous preorder on a σ-compact and lo-
cally compact Hausdorff space has a continuous multi-utility representation.
The following consequence of Theorem 4.19 make it a useful tool in applications when the
ground topological space has nice features:
Corollary 4.20 (Evren and Ok, 2011) Every continuous preorder on a topological space that
is either compact or a nonempty closed subset of a Euclidean space has a continuous multi-utility
representation.
43See also Ok [200] (Theorem 3) and Mandler [170] (Theorem 1) for the existence of special multi-utility
representations under suitable order-separability conditions.
New trends in preference, utility, and choice: from mono to multi 43
The second result of Evren and Ok is complementary to the first, insofar as it requires less
from the topological space and more from the preorder:
Theorem 4.21 (Evren and Ok, 2011) Every “nice” semicontinuous preorder satisfying strong
local non-satiation has a continuous multi-utility representation.
The two properties of “strong local non-satiation” and “niceness” are rather undemanding con-
ditions, which are often satisfied by preferences encountered in dynamic consumer theory and
decision making under uncertainty.
We conclude the discussion on the multi-utility representation of a preorder by emphasizing
that this approach has been used in many recent preference models under uncertainty, which deal
with either a single potentially incomplete preference, or a suitable pair of linked preferences:
see, e.g., [54, 77, 108, 128, 201].
4.3.2 Modal utility representations
The notion analyzed here is a simple variation of a multi-utility representation. Specifically, the
motivating question is the following:
(Q10) Given a preorder %and a suitable extension of %, can we obtain a multi-utility represen-
tation of %that simultaneously represents (in a different way) its extension?
Such a representation would describe the preorder “globally” and its extension “locally”. The
formal notion is the following [116]:
Definition 4.22 Let %R,%Sbe a pair of binary relations on X. A modal utility representation
of %R,%Sis a nonempty family
U=uk
h:h∈H∧k∈Kh
of utility functions uk
h:X→Rindexed over the set Sh∈H(h, k) : k∈Khsuch that the
following properties hold for each x, y ∈X:
(M1) x%Ry⇐⇒ (∀h∈H) (∀k∈Kh)uk
h(x)≥uk
h(y);
(M2) x%Sy⇐⇒ (∃h∈H) (∀k∈Kh)uk
h(x)≥uk
h(y).
A pair %R,%Sthat admits a modal utility representation Uis modally representable: in this
case, His the set of modes of U, whereas Khis the extent of mode h∈H. In particular, Uis
unimodal if |H|= 1, and simple if |Kh|= 1 for each h∈H.
Notice that unimodal and simple representations only need one parameter to be described:
indeed, a unimodal representation Ucan be written as U={uk
0:k∈K0}, whereas a simple
modal representation can be written as U={u0
h:h∈H}. The modal representability of a pair
of weak preferences can be characterized as follows [116]:
Theorem 4.23 A pair %R,%Sof binary relations is modally representable if and only if it is
a bi-preference.
44 A. Giarlotta
The next definition describes a special case of modal representability:44
Definition 4.24 A bi-preference %R,%Son Xis quantifier-representable if there is a family
Uof utility functions on Xsuch that the following properties hold for each x, y ∈X:
(U1) x%Ry⇐⇒ (∀u∈U)u(x)≥u(y);
(U2) x%Sy⇐⇒ (∃u∈ U)u(x)≥u(y).
Quantifier-representability implies modal representability, but not vice versa [116]:
Proposition 4.25 Let %R,%Sbe a bi-preference on X.
•%R,%Shas a simple modal representation if and only if it is quantifier-representable.
•%R,%Shas a unimodal representation if and only if it has a multi-utility representation
if and only if %R=%Sis a (possibly incomplete) preorder.
•%R,%Shas a simple unimodal representation if and only if %R=%Sis a representable
total preorder.
4.4 Multi-rationalizable choices
In this section we provide a possible answer to the question (Q6) posed in Section 2.4.
In 1955, Herbert Simon described a behavioral choice model of bounded rationality [233, 234].
In this pioneering work, an economic agent makes her choices according to a list of elements in
the ground set X, a binary preference over X, and a satisfactory threshold x∗∈X. Then, she
selects a unique element, which is either the first element in the list that is not inferior to x∗,
or, if there is none, the last element in the list.
Quite recently, Rubinstein and Salant [222] create a very rich framework in which an agent
makes choices from a (finite) list rather than from a set. This choice model encompasses both
the classical setting of revealed preference theory and Simon’s approach, as well as several
other models of rational behavior (place-dependent rationality,reference point dictatorship [243],
successive choice [223], contrast effect, etc.).
Many additional models of bounded rationality have recently been proposed in the framework
of choice theory. These models pursue either a “simultaneous approach” (i.e., all justifying
preferences are applied at the same time to explain the selection from the feasible menus) or a
“sequential approach” (i.e., the justifying preferences are applied in some order, possibly with
different procedures, to explain the selection process): see, among the most recent contributions
of both kinds, Apestegu´ıa and Ballester [15], Au and Kawai [22], Cherepanov, Feddersen, and
Sandroni [57], Garc´ıa-Sanz and Alcantud [103], Kalai, Rubinstein, and Spiegler [150], Manzini
and Mariotti [172, 173], Masatliouglu and Nakajima [175], and Tyson [244].
Here, following the most recent trends in the literature, we lay down a comprehensive frame-
work for a theory of choice multi-rationalization, which aims at refining the classical theory of
revealed preferences. This refinement is pursued by associating a degree of binary rationality to
each choice: this is defined as the least number (degree) of binary relations (binary) that are
needed to explain (rationality) the observed choice behavior of an economic agent or a group of
44This type of representation has already been mentioned: see (3) in Section 4.2.
New trends in preference, utility, and choice: from mono to multi 45
economic agents. This in turn yields a natural classification of choices that are non-rationalizable
according to Definition 2.15. In this way, the amount of choices possessing features of rational-
ity is enlarged, and the rational/irrational dichotomy arising from revealed preference theory is
smoothened.
To start, we classify the approaches that use binary relations (henceforth called rationales)
to justify a choice behavior, listing the main variables under consideration.
(1) Ground set X:
(a) finite,
(b) infinite.
(2) Choice domain Ω:
(a) total (the powerset of the ground set minus the empty set),
(b) partial (a nonempty subset of a total domain, usually subject to closure properties).
(3) Selection mode c:
(a) choice function (single valued),
(b) choice correspondence (multi valued),
(c) quasi-choice function (zero/one valued),
(d) quasi-choice correspondence (zero/multi valued).45
(4) Internal structure of rationales:
(a) none (no property),
(b) partially transitive (acyclic, quasi-transitive, (m, n)-Ferrers),
(c) fully transitive (total preorder, linear order).
(5) Retrieval modality:
(a) by binary maximization (classical approach),
(b) by psychological maximization (behavioral approach),
(c) by type (general approach, model theoretic).
(6) Number of rationales:
(a) mono-rationalization (one preference),
(b) multi-rationalization (a nonempty set of preferences).
(7) Interactions among rationales:
(a) free (no interaction),
(b) monotonic (order-preserving with respect to reverse set-containment),
(c) strongly coherent (monotonic & transitively coherent),
(d) listable (guided by an underlying linear order), etc.
(8) Philosophy/Timing:
(a) simultaneous (selection made in one step, considering all rationales at the same time),
(b) sequential (selection made in sequential steps, each step with its own modality).
45Aquasi-choice function on Xis a map c:Ω→Ω∪ {∅} such that c(A)⊆Aand 0 ≤ |c(A)| ≤ 1 for all
A∈Ω. Thus, the agent selects either a single item or no item at all from each menu. Similarly, a quasi-choice
correspondence on Xis a map c:Ω→Ω∪ {∅} such that c(A)⊆Aand 0 ≤ |c(A)| ≤ |A|for all A∈Ω.
46 A. Giarlotta
(Of course, features (7)–(8) apply only in the case that several rationales can be used, that is, in
(6b).) Many approaches to choice rationalizability can be identified by the above features. For
instance, most traditional models using a single binary rationale typically fall in the category
identified by the features (1a)-(1b), (2a)-(2b), (3a)-(3b), (4b)-(4c), and (5a). Several recent ap-
proaches of bounded rationality can be classified by the same parameters, often using a retrieval
modality of (5b) “psychological maximization” [57, 222, 223, 233]. In the remainder of this sec-
tion, we provide the reader with an overview of some selected approaches of this kind, separately
dealing with (8a) simultaneous multi-rationalization, and (8b) sequential multi-rationalization.
4.4.1 Simultaneous multi-rationalization
A relatively recent multi-approach to the theory of choice rationalization is the rationalization
by multiple rationales (RMR), due to Kalai, Rubinstein, and Spiegler [150]. Their approach falls
into the category identified by the features (1a), (2a), (3a), (4c), (5a), (6b), (7a), and (8a). The
goal of RMR is to provide a discrete measure of the rationalizability of any total choice function
on a finite set: this is accomplished by determining the minimum number of linear orders such
that the unique item selected in each menu is maximal for some rationale. Kalai et al. [150] show
that any single valued choice on an n-element set can always be rationalized by (n−1) linear
orders, and the likelihood that this maximum value is attained tends to one as the size nof the
ground set goes to infinity. The RMR approach is neat and direct, but has some shortcomings.
In fact, it can only be employed if (1a) the ground set Xis finite, (2a) the choice domain Ωis
total, (3a) the selection mode is a single valued choice function, and (4c) the binary rationales
are linear orders; moreover, (7a) no interaction/relation among the rationales needs to exist.
Some of the above issues can be addressed by designing a general theory of simultaneous
multi-rationalization, which also takes into account the following cases: (2b) partial choice do-
mains, (3b) multi-valued choice correspondences, (4b) acyclic, quasi-transitive or (m, n)-Ferrers
rationales, and (7b)-(7c)-(7d) various types of interactions among rationales. The next few
definitions, given by Cantone, Giarlotta, and Watson [50], are motivated by this goal.46
Definition 4.26 For any finite nonempty set X, let Pref(X) be the family of all reflexive and
complete binary relations on X. In what follows, we denote by Pref ac(X), Pref qt (X), Pref tra(X),
and Pref lin(X), the subfamilies of Pref(X) composed of relations that are, respectively, acyclic,
quasi-transitive, transitive, and linear.
The notion of (simultaneous) multi-rationalization described by the next definition assigns
to each menu a binary rationale belonging to a pre-selected family of preferences: for instance,
in RMR, all admissible rationalizing preferences are in Pref lin(X). For less demanding theories
of choice multi-rationalizability, one may assume that either Pref tra(X) or Pref qt(X) are em-
ployed instead. Below we impose a minimal condition of internal consistency on the rationales:
acyclicity. This assumption agrees with common practice (see, e.g., [222]).
Definition 4.27 A choice correspondence c:Ω→Ωon a finite ground set Xis freely multi-
rationalizable (FMR) if there exists a map f:Ω→Pref ac(X) with the following property:
(Local Rationalization) c(A) = max(A, f (A)) for all A∈Ω.
46In order to avoid dealing with the theory of infinite cardinals, we limit our analysis to case (1a), that is, we
assume that the ground set Xis finite.
New trends in preference, utility, and choice: from mono to multi 47
A function fwith the above properties is a free rationalizer for c, and the cardinality of its
image is its rank. An FMR choice correspondence cis freely p-rationalizable if there exists a free
rationalizer for chaving rank p. Further, the free rationalizability number rat →
free(c) of cis the
least positive integer psuch that cis freely p-rationalizable, that is,
rat →
free(c) := min p∈N:chas a free rationalizer with rank p.
Next, we define an order-preserving map rat →
free :N\ {0} → N\ {0}as follows for each n≥1:
rat →
free(n) := min p∈N: (∀c∈Choice(n)) rat →
free(c)≤p
where Choice(n) denotes the set of all choice correspondences on a set of size n. Finally, denoting
by Choice−(n) the set of all choice functions (i.e., single valued) on a set of size n, we define a
map rat−
free :N\ {0} → N\ {0}as follows for each n≥1:
rat−
free(n) := min p∈N: (∀c∈Choice−(n)) rat →
free(c)≤p.
(Obviously, the inequality rat−
free(n)≤rat →
free(n) holds for all integers n≥1.)
The definition of a freely multi-rationalizable choice correspondence calls for the existence of
a family of minimally consistent binary rationales, which globally justifies the selection process
by locally using the classical maximization paradigm. Notice that a free rationalizer provides
each feasible menu with its own acyclic justification, which is in general independent of the
rationales associated to the other menus: in this sense we use the adjective “free”.
Intuitively, rat →
free(c) says how many “rational states of mind” are needed to explain the
observed choice behavior cof an economic agent. Consequently, rat →
free(n) can be thought as
a discrete measure of how irrational an arbitrary choice on an n-element set may be, and the
lower bounds exhibit the “least rational” choice behaviors. It is apparent that each choice
correspondence c:Ω→Ωis always FMR, and the upper bound rat →
free(c)≤ |Ω|holds.
Example 4.28 Consider the following total choice correspondences on X={x, y , z}:
(c1)xy , x z , y z , x y z
(c2)xy , x z , y z , x y z
(c3)x y , x z , y z , x y z .
Then, we have rat →
free(ci) = ifor i= 1,2,3.
The estimation of the free rationalizability number is not difficult for total choice functions,
i.e., in the case examined by Kalai et al. [150]. A first interesting fact is that in this case it is
immaterial whether we take Pref lin(X) or Pref ac(X) as family of rationales. Indeed, denoted by
rat RMR(c) the free rationalizability number by means of linear orders on X, we have [50]:
Lemma 4.29 rat →
free(c) = rat RMR (c)for each total choice function c.
In view of Lemma 4.29, the main results of [150] can be restated as follows:
Theorem 4.30 (Kalai et al., 2002) The equality rat−
free(n) = n−1holds for each n≥1.
Further, the fraction of total choice functions on Xhaving the maximum free rationalizability
number tends to 1as the size nof Xgoes to infinity.
48 A. Giarlotta
The findings of Theorem 4.30 are appealing, albeit somehow expected: the larger the size of
the ground set, the higher the percentage of chaotic choice functions (where “chaotic” means that
they require the maximum number of rationales). Regrettably, the situation becomes far more
complicated for choice correspondences. A first, simple result on rat →
free(n) is the following [50]:
Proposition 4.31 rat →
free(n)≤2n−1for each n≥1.
The upper bound given by Proposition 4.31 already fails to be tight for n= 3, because
rat →
free(3) = 3. In fact, a (rather complicated) combinatorial analysis of rat →
free(n) suggests that
finding better bounds is highly nontrivial [50]. An even more difficult problem is the following:
Problem 4.32 Determine rat →
free(n)for each n≥1.
Going back to RMR, Kalai et al. [150] (p. 2487) conclude their contribution by explicitly
recognizing that a serious issue of their approach is given by feature (7a), i.e., a total absence
of interactions among rationales:47
“We fully acknowledge the crudeness of this approach. The appeal of the RMR proposed
for “Luce and Raiffa’s dinner” does not emanate only from its small number of orderings,
but also from the simplicity of describing in which cases each of them is applied. ... More
research is needed to define and investigate “structured” forms of rationalization.”
Along the path suggested by them, we now sketch a “structured” type of simultaneous multi-
rationalization, called monotonic: the reader is referred to [50] for other types of structured
multi-rationalizations (strongly coherent,listable, etc.).
Definition 4.33 A choice correspondence c:Ω→Ωis monotonically multi-rationalizable (MMR)
if there exists a free rationalizer f:Ω→Pref ac(X) satisfying the following additional property:
(Monotonic Coherence) for all A, B ∈Ω, if A⊆B, then f(A)⊇f(B).
Such a function fis a monotonic rationalizer for c. A choice correspondence is monotonically
p-rationalizable if there exists a monotonic rationalizer for chaving rank p. If a choice corre-
spondence cis MMR, then its monotonic rationalizability number rat →
mon(c) is defined as the
least positive integer psuch that cis monotonically p-rationalizable, that is,
rat →
mon(c) := min{p∈N:chas a monotonic rationalizer with rank p}.
On the other hand, if chas no monotonic rationalizer, then we set by definition rat →
mon(c) := ∞.
Monotonic Coherence requires that an expansion of a menu induces a contraction of the
associated preference; said differently, whenever going to larger menus, the local rationale might
be less informative. This possible loss of information is due to the natural difficulty of a human
mind to make comparisons among a large number of items.48 For instance, in the process of
47“Luce and Raiffa’s dinner” mentioned in the quotation below will be described in Example 4.35.
48It is worth noticing that this is exactly the same underlying philosophy which has inspired those MCDA
methodologies that limit comparisons to few points of view a time: the prototype of such an approach is the
Pairwise Criterion Comparison Approach (PCCA), originally developed by Matarazzo [179, 180, 181, 182, 183,
184, 185] and later on by his followers [11, 12, 109, 110, 130, 131].
New trends in preference, utility, and choice: from mono to multi 49
selecting within larger and larger menus, the economic agent may start dropping some relation-
ships not so well established in her mind, transforming an indifference between two items into an
incomparability. This approach finds its theoretical justification in well established theories in
psychology, which advocate the use of bounded rationality heuristics and simplified strategies in
making judgements, especially in complicated settings: see the classical work by Tversky [241],
Tversky and Kahneman [242], and Kahneman, Slovic and Tversky [151].
Notice that, in the simplest case, the notion of monotonic multi-rationalizability generalizes
the classical notion of rationalizability. In fact, for any choice correspondence c:Ω→Ω,
crationalizable ⇐⇒ cmonotonically 1-rationalizable ⇐⇒ rat →
free(c) = rat →
mon(c) = 1.
The next example provides some simple instances of choices on a 3-element ground set, which
are monotonically rationalizable or fail to be so.
Example 4.34 Consider the following total choice correspondences on X={x, y , z}:
(c1)xy , x z , y z , x y z
(c2)x y , x z , y z , x y z
(c3)x y , x z , y z , x y z
(c∞)x y , x z , y z , x y z
(c′
∞)xy , x z , y z , x y z .
Then, we have rat →
mon(ci) = ifor i= 1,2,3, and rat →
mon(c∞) = rat →
mon(c′
∞) = ∞. It is worth
examining the (abstraction of the) pathologies displayed by the choices c∞and c′
∞, which prevent
them from being MMR: see [50].
To provide a motivation for a monotonic approach, next we examine a well known instance
of choice reversal, described by Luce and Raiffa [166]. In this example, the consumer exhibits
a switch in her (single valued) selection process whenever going from a menu to a larger one.
This choice reversal phenomenon –which might look somehow unexpected, but is motivated by
the so-called epistemic value of the menu described by Sen [232]– causes the choice function to
be non-rationalizable by a single binary relation. Kalai et al. [150] provide a a “non-structured”
RMR justification of this choice behavior; below we also exhibit a “structured” justification,
which may give further insight in the selection process.
Example 4.35 (Luce and Raiffa’s dinner.) A diner chooses a main course from a restaurant’s
menu. If the menu consists of chicken (x) and steak (y) only, then she chooses chicken. On the
other hand, if the menu consists of chicken, steak, and frog’s legs (z), then she selects steak.
Thus, the (partial) selection process is modeled by any choice con X={x, y, z}such that
xyand xyz. The classical theory of revealed preferences cannot formally explain the diner’s
behavior, since the (partial) choice function cfails to be rationalizable. On the other hand, this
choice behavior can be formally justified by employing a multi-rationalization model instead.
For instance, after suitably extending the definition of cto a total choice function bc: 2X→2X,
Kalai et al. [150] exhibit a free 2-rationalization of bcby linear orders. However, this multi-
rationalization does not provide any links between the two rationales, and so the diner’s choice
reversal phenomenon remains somehow unexplained by it.
A monotonic multi-rationalization may provide a sound explanation of this behavior. Let %1
and %2be the preferences on Xdefined by %1:= X2\ {(y , x)}and %2:= {(y, z),(z, x)} ∪∆(X).
50 A. Giarlotta
Then, the map f:Ω→Pref ac(X), defined by f({x, y}) := %1and f(X) := %2, is a monotonic
rationalizer of chaving rank two (hence rat →
mon(c) = 2). The two rationales %1and %2suggest
an interesting interpretation of the choice reversal phenomenon. In fact, %1can be seen as a
direct rationale, in the sense that the strict preference x≻1ylocally motivates the selection of
chicken over steak whenever they are the only available main courses. On the other hand, %2is
an indirect rationale, in the sense that the added item zinduces a sequence of strict preferences
y≻2z≻2x, which in turn motivates the selection of steak from the expanded menu. Said
differently, the selection of steak from the full menu “factors through” the new item (frog’s
legs), since the latter provides an indirect rationale for the observed switch: for example, the
presence of frog’s legs in the menu gives information about the (good) quality of the chef, or it
simply grosses the diner out and induces her to avoid chicken as well.
We conclude this example by observing that there are completions of cthat are MMR, and
others that fail to be so. For instance, the choice correspondence ec, which extends cby defining
x z and y z, is MMR with rat →
mon(ec) = 2. On the contrary, any extension of csuch that y z (i.e.,
frog legs are always selected versus steak) fails to be MMR.49
The next definition, which refines the notion of monotonic multi-rationalizability, provides
a surprising link with the classical theory of revealed preferences.
Definition 4.36 A choice correspondence c:Ω→Ωis elementarily MMR (eMMR ) if it has a
monotonic rationalizer f:Ω→Pref ac(X) such that all restrictions f(A)↾Aare antisymmetric.
The following characterization of eMMR choices is noteworthy [50]:
Theorem 4.37 A total choice correspondence is eMMR if and only if it satisfies (γ).
The above result has some interesting consequences, which shed further light on the role
of two standard axioms of choice consistency: Chernoff’s contraction property (α), and Sen’s
expansion property (γ). To clarify these links, we need the following notion:
Definition 4.38 A choice correspondence is elementary if it is single valued on all doubletons.
Then, we have [50]:
Corollary 4.39 For an elementary total choice correspondence c, we have:
(i) cis rationalizable by a total preorder if and only if (α)holds;
(ii) cis MMR if and only if it is eMMR if and only if (γ)holds.
In particular, we obtain a characterization of single valued MMR total choices:
Corollary 4.40 A total choice function is MMR if and only if axiom (γ)holds.
Corollary 4.40 is interesting in two different ways. First, it characterizes a class of total choice
functions, which are rationalizable is a wider sense than that prompted by the theory of revealed
preferences. Second, possibly more important, it sheds some light on the intrinsic semantics of
the standard axiom (γ) of expansion consistency, which is almost invariantly considered in
association to its dual counterpart, the standard axiom (α) of contraction consistency.
49This issue is connected to the problem of the lifting of choices having certain properties: see Cantone et al. [47]
for the general formulation of the problem and its logic-theoretic analysis in some special cases.
New trends in preference, utility, and choice: from mono to multi 51
4.4.2 Sequential multi-rationalization
We now switch to scenario (8b), that is, sequential types of multi-rationalizations.50 In this
stream of research, the literature is quite abundant: see, among many others, Apesteguia
and Ballester [15] for choices by sequential procedures, Manzini and Mariotti [172], Au and
Kawai [22], and Garcia and Alcantud [103] for sequentially rationalizable choices, Lleras et al. [162]
for consideration filters, Masatlioglu et al. [176] for attention filters, Masatlioglu and Naka-
jiama [175] for choices by iterative search, Manzini and Mariotti [173] for choices by lexicographic
semiorders, Rubinstein and Salant [222] for choices from lists, and Tyson [244] for general short-
listing procedures. To keep exposition compact, we shall limit our analysis to the two models
proposed by Rubinstein and Salant [222], and by Manzini and Mariotti [172].
In their model, Rubinstein and Salant [222] study choice functions from lists, where a list
is a sequence of distinct elements of a finite set. A choice function from lists singles out one
element from every list. They show that a certain class of choice functions from lists can be
characterized by two equivalent properties: Partition Independence (PI′) and List Independence
of Irrelevant Alternatives (LIIA). Property PI′extends to lists the classical condition of Path
Independence (PI) due to Plott [206]: it requires that arbitrarily dividing a list into several
sublists, choosing from each sublist, and finally choosing from the list of chosen elements yields
the same result as choosing from the original list. Property LIIA states that omitting unchosen
elements from a list does not alter the choice. The class Choice RS
seq(X) of choice functions from
lists in Xcharacterized by the satisfaction of PI′(equivalently, LIIA) is an enlargement of the
class of rationalizable choice functions on Xin the sense of revealed preference theory. In fact,
each function in Choice RS
seq(X) is parameterized by a preference relation %over Xand a labelling
of every ∼-indifference set by “First” or “Last”. Then, given a list, each such function identifies
the set of %-maximal elements within the list, and chooses the first or the last element among
them according to the label of the ∼-indifference set they belong to.
Rubinstein and Salant successively extend their approach to cases in which the order of
the elements in the list is not directly observable, e.g., when the list is virtual. Under such
circumstances, they analyze choice correspondences attaching to every set of alternatives all the
elements that are chosen for some ordering of that set. They prove the following interesting two
facts: (1) choice functions from lists that satisfy PI′induce choice correspondences that satisfy
WARP, and, conversely, (2) a choice correspondence satisfying WARP can be “explained” by a
choice function from lists that satisfies PI′. Thus, their results provide a new interpretation of
the notion of choice correspondence.
Finally, Rubinstein and Salant consider situations in which the decision maker determinis-
tically chooses from lists generated from sets by a random process. A random choice function
assigns to every set of alternatives a probability measure over the set, where the probability of
an element is the likelihood that it will be chosen from the set. Then they show that a choice
function from lists satisfies PI′if and only if the induced random choice function is monotone
(in the sense that the probability of choosing an element from a set weakly increases as the set
of available items shrinks).
Manzini and Mariotti [172] study choice functions that can be justified by maximizing more
than one preference relation in a given order. In the simplest case of two rationales, they
50Somehow in between approaches (8a) and (8b) lies the very interesting model constructed by Cherepanov,
Feddersen, and Sandroni [57], who provide a general framework for a formal and testable theory of rationalization,
in which a decision maker selects her preferred alternative from among those that she can rationalize.
52 A. Giarlotta
call this procedure “rational shortlist method”. The employed terminology is suggestive of the
procedure: intuitively, the first rationale identifies a shortlist of candidate alternatives from
which the second rationale selects a unique element. Below we recall the notion of 2-sequential
rationalizability in the general case of a total choice correspondence [103].
Definition 4.41 A choice correspondence c: 2X→2Xis 2-sequentially rationalizable (also
called a rational shortlist method,RSM) if there is an ordered pair (≻1,≻2) of asymmetric
relations on Xsuch that c(A) = max(max(A, ≻1),≻2) for all menus A∈2X.
A 2-sequentially rationalizable choice correspondence describes a decision procedure of an
agent, who goes through two sequential rounds of elimination to select the alternatives: in the
first round, she only retains those items that are maximal according to ≻1; in the second, she
only keeps the items that are maximal according to ≻2.RSMs naturally apply in several practical
situations: for instance, in a portfolio selection, a cautious investor can first eliminate all risky
alternatives, and then select the one(s) that give the maximum expected return. RSMs also fit the
scenario of sequential “noncompensatory” heuristics widely studied in the psychology literature
(see, e.g., Tversky’s [241] “elimination by aspects” procedure), as well as that of consumers’
“two-stage consideration and choice” decision procedures in the management literature [248].
A crucial feature of the rational shortlist method is that the order of application of the two
rationales is fixed for all menus. In their Theorem 1, Manzini and Mariotti [172] characterize
the 2-sequential rationalizability of choice functions by the satisfaction of two testable prop-
erties of choice consistency, namely, standard expansion (γ) and a weak form of WARP. More
recently, Garc´ıa-Sanz and Alcantud [103] obtain a partial characterization of the 2-sequential
rationalizability of choice correspondences, which holds under a mild condition, called Choice
Without Dominated Elements (CWDE). In this respect, it is worth noticing that the process
of resolution described in Section 3.5 does preserve the 2-sequential rationalizability of choice
correspondences under condition CWDE [48].
Manzini and Mariotti [172] also consider the natural generalization of the rational shortlist
method, which is a rationalization procedure for choice functions by means of more than two
sequential criteria. (Again, we describe the notion in the general case of a choice correspondence.)
Definition 4.42 For each integer n≥2, a choice correspondence c: 2X→2Xis n-sequentially
rationalizable whenever there is an ordered n-tuple (≻1,≻2,...,≻n) of asymmetric relations on
Xsuch that, for all menus A∈2X, if the sets Mi,i= 0,1,...,n, are recursively defined by
•M0(A) := A, and
•Mi(A) := max(Mi−1(A, ≻i)) for i= 1,...,n,
then c(A) = Mn(A) (that is, c(A) = max(max(...(max(A, ≻1),≻2),...),≻n)). A choice corre-
spondence is sequentially rationalizable if it is n-sequentially rationalizable for some n≥2.
Using the same terminology as for simultaneous multi-rationalizability, we introduce the
following natural notion:
Definition 4.43 The sequential multi-rationalizability number of a choice correspondence c,
denoted by rat↑(c), as the least integer psuch that cis sequentially p-rationalizable if there is
one, and ∞otherwise.
New trends in preference, utility, and choice: from mono to multi 53
Notice that there are choice functions cthat fail to be sequentially rationalizable, that is,
rat↑(c) = ∞. Manzini and Mariotti [172] obtain a partial characterizations of sequentially
rationalizable choice functions, and, by means of a recursion lemma, they also express the 3-
sequential rationalizability of a choice function in terms of the existence of a suitable choice
correspondence. On the other hand, a full characterization of the 3-sequential rationalizability
of a choice function –let aside that of a choice correspondence– is still unknown. More generally,
the following problem appears to be highly nontrivial:
Problem 4.44 Characterize sequentially rationalizable choice functions and correspondences.51
Further, similarly to the case of simultaneous multi-rationalizability, it appears of some
interest to obtain estimates of the values of rat↑(c) for choice functions/correspondences that
are sequentially rationalizable.
4.5 Multiple, iterated, and hierarchical resolutions of choices
In this final section, we suggest how to generalize the notion of choice resolution introduced
in Section 3.5 by using multiple and iterated resolutions first, and then combining them into
hierarchical resolutions. Technical details are barely sketched, because a deeper analysis would
be lengthy and complicated. Recall that the process of resolution of a base choice space (X, cX)
at a single point x∈Xinto a single fibre choice space (Yx, cYx) operates as follows:
(1) view the item xas a menu, which (potentially) opens up at Yx;
(2) to make a choice in the resolved space (Z, cZ) = (X, cX)⊗x(Yx, cYx), first select from X,
where one of the choices is the (closed) menu x:
(2.a) if xis not picked up, then leave the point-menu Yxclosed;
(2.b) if xis picked up, then open the point-menu Yx, and make there choices as well.
The natural directions in which the notion of resolution can be generalized are apparent:
- multiple (horizontal) resolution, obtained by simultaneously resolving the points of a
nonempty set X′⊆Xinto fibre choice spaces;
- iterated (vertical) resolution, obtained by sequentially resolving an ordered list of points
into fibre choice spaces, where each point in the list belongs to the fibre choice in which
the preceding point has been resolved;
- hierarchical (tree) resolution, obtained as an arbitrary combination of horizontal and ver-
tical resolutions, thus expanding a choice space into a tree-structured choice space.
To give an idea of the technicalities involved in a multiple and/or iterated resolution, below
we examine the two simplest cases: (1) a horizontal resolution at two base points, and (2) a
vertical resolution at two base points. For the sake of readability, we simplify the convoluted
notation by dropping some subscripts. We start with case (1).
51A very surprising answer to a closely related question is given by Mandler, Manzini, and Mariotti [171].
In their paper, the authors show that “fast and frugal” sequential procedures are not incompatible with utility
maximization. In fact, two rather unexpected facts hold: (1) any agent who uses the benchmark model of
quickly-executing checklists always has a utility function, and (2) any utility maximizer can make decisions with a
quickly-executing checklist (under suitable conditions on the domain). In Mandler et al.’s [171] words: “Checklists
are a fast and frugal way to maximize utility.”
54 A. Giarlotta
Definition 4.45 Let (X, c), (Y1, c1), (Y2, c2) be three complete choice spaces on disjoint ground
sets X,Y1,Y2, and let x1, x2be two distinct points of X. Set Z:= (X∪Y1∪Y2)\ {x1, x2}.
Define a surjective map π:Z→Xby
π(z) :=
zif z∈X\ {x1, x2}
x1if z∈Y1
x2if z∈Y2.
The multiple (horizontal) resolution of (X, c)at (x1, x2)into (Y1, c1),(Y2, c2), denoted by
(Z, cZ) = (X, c)⊗→
x1,x2(Y1, c1),(Y2, c2),
is the complete choice space on Zwhose choice correspondence cZ: 2Z→2Zis defined by
cZ(A) :=
c(π(A)) ∪c1(A∩Y1)∪c2(A∩Y2)\ {x1, x2}if x1, x2∈c(π(A))
c(π(A)) ∪c1(A∩Y1)\ {x1}if x1∈c(π(A)) and x2/∈c(π(A))
c(π(A)) ∪c2(A∩Y2)\ {x2}if x1/∈c(π(A)) and x2∈c(π(A))
c(π(A)) otherwise.
(X, c) is the base choice space,x1, x2are the two (simultaneous) base points, (Y1, c1),(Y2, c2) are
the two (simultaneous) fibre choices, and πis the (simultaneous) projection.
The extension of the notion of multiple resolution to the general case, where several (even all)
points of the base space are simultaneously resolved into fibre choices, is straightforward.
Next, we examine case (2), that is, iterated resolutions at two points in a sequence.
Definition 4.46 Let (X, c), (Y1, c1), (Y11 , c11) be three complete choice spaces on disjoint
ground sets X,Y1,Y11. Select x∈Xand y1∈Y1. Set
Z:= (X∪Y1∪Y11)\ {x, y1}and Z1:= (Y1∪Y11)\ {y1}.
Further, define two surjective maps π:Z→Xand π1:Z1→Y1by
π(z) := (zif z∈X\ {x}
xotherwise and π1(z1) := (z1if z1∈Y1\ {y1}
y1otherwise .
The iterated (vertical) resolution of (X, c)at (x, y1)into (Y1, c1),(Y11, c11 ), denoted by
(Z, cZ) = (X, c)⊗↑
x,y1(Y1, c1),(Y11, c11 ),
is the complete choice space on Zwhose choice correspondence cZ: 2Z→2Zis defined by
cZ(A) :=
c(π(A)) ∪c1(π1(A∩Y1)) ∪c11(A∩Y11)\ {x, y1}if x∈c(π(A)) and y1∈c1(π1(A∩Y1))
c(π(A)) ∪c1(A∩Y1)\ {x1}if x∈c(π(A)) and y1/∈c1(π1(A∩Y1))
c(π(A)) otherwise.
(X, c) is the base choice space,x, y1are the two (sequential) base points, (Y1, c1),(Y11, c11 ) are
the two (sequential) fibre choices, and π, π1are the two (sequential) projections.
New trends in preference, utility, and choice: from mono to multi 55
The extension of the notion of multiple resolution to the general case, where the vertical resolu-
tion keeps going up to a certain height, is not conceptually difficult but technically complicated.
The most general form of resolution is the hierarchical resolution, which is obtained by
resolving a base choice space by means of a rooted tree (a connected acyclic graph with a
distinguished node, called root). The formal description is technically complicated, so we avoid
presenting it here. However, to give an idea of how they work, Figure 5 provides a graphical
representation of a simple case of hierarchical resolution, which has width 4 and height 3 (width
and height are defined as for trees). The employed notation –which, however, can be simplified–
is supposed to suggest how a hierarchical resolution is formally defined.
(X, c)
x1x2
(Y1, c1) (Y2, c2)
y11 y12 y13 y21
(Y11, c11 ) (Y12 , c12) (Y13 , c13 ) (Y21, c21 )
y111 y112 y211
(Y111, c111 ) (Y112 , c112) (Y211 , c211)
Figure 5: A hierarchical resolution of width 4 and height 3
Possible applications of hierarchical resolutions of choices are apparent for, e.g., corporate
structures, investment portfolios, etc. For instance, imagine the case of a large multi-national
company, whose very articulated organization suggests the CEO to fully delegate decision author-
ity to either national branches (with their own hierarchical structure) or transversal departments
(with their own hierarchical structure). Then the possibility to detect the inner structure of the
corporation by just observing its choice behavior on projects may have a high strategic impact
in the decision making process of its competitors.
To conclude, we connect hierarchical processes to similar approaches in MCDA. Corrente
et al. study multiple criteria hierarchy processes within the ROR approach in [63], and within
ELECTRE and PROMETHEE methodologies in [64]. More recently, Angilella et al. [10] consider
ROR and SMAA (Stochastic Multiobjective Acceptability Analysis) in a multiple criteria hierar-
chy process for the Choquet integral preference model, whereas Corrente et al. [65] examine a
multiple criteria process for ELECTRE Tri methods. Further, Angilella et al. [8] evaluate sustain-
able development by means of composite indices using the hierarchical-SMAA-Choquet integral
56 A. Giarlotta
approach. Additional contributions in this field take into account hierarchical structures: see
Fujimoto et al. [102] for a theoretical analysis of the hierarchical decomposition of the Choquet
integral, Del Vasto-Terrientes et al. [74] for an outranking-based methodology with a hierarchy
of criteria, and Del Vasto-Terrientes et al. [75] for a hierarchical multi-criteria sorting approach.
5 Conclusion
In this survey we have discussed some recent approaches to the theories of preference modeling,
utility representation, and choice rationalization. These approaches are inspired by a multiple
criteria philosophy, since they take into account several “points of view” –preference relations,
utility functions, or binary rationales for choices– to explain an agent’s behavior. We hope to
have provided the reader with enough convincing arguments on the naturalness and feasibility
of a multi-approach to these theories, and its advantages over the classical mono-approach.
Acknowledgements
The author is very grateful to Jos´e Carlos R. Alcantud, Domenico Cantone, Jean-Paul Doignon,
and Stephen Watson for several fruitful suggestions and discussions.
Appendix
This section contains two figures, which summarize some results of this survey. Figure 6 describes
all implications between combinations of weak (m, n)-Ferrers properties. For instance, the arrow
from the box (3,2) (i.e., strong interval orders) to the the box (3,1)&(2,2) (i.e., semiorders) says
that any strong interval order is a semiorder, but the vice versa is false in general. Notice that
the very last segment of the picture –that is, going from a total quasi-preorder to a simple
preference– can be refined into an infinite hierarchy by using strict (m, 1)-Ferrers properties.
Figure 7 exhibits the meet-semilattice of all NaP-preferences on X={x, y, z}. For compact-
ness, we simplify the notation in Figure 4 to identify the comonotonic configurations (C1), (C2),
(C2)′, (C3), (C3)′, and (C4) as follows (configuration (C5) never appears for NaP-preferences):
x∼Ny
(C1)
x≻Py
(C2)
y≻Px
(C2)′
x≻Ry∧x∼Sy
(C3)
y≻Rx∧y∼Sx
(C3)′
x∼Gy
(C4)
xyxyxyxyxyxy
Observe that many configurations in Figure 7 are isomorphic to each other (where an isomor-
phism between bi-preferences is defined in the obvious way). For instance, at level 3 of the
meet-semilattice in Figure 7 (the root is a level 0), the isomorphism class of the NaP-preference
emphasized by a white background comprises six elements (the other five elements being iden-
tified by a white dot). A simple computation shows that the number of non-isomorphic NaP-
preferences on a 3-element set is 20. The following combinatorial problem appears nontrivial:
Problem 5.1 For any integer n≥3, determine the number of pairwise non-isomorphic (either
all or normalized) NaP-preferences on an n-element set.
Notice that, in the special case of NaP-preferences having a semiorder as a possible compo-
nent, the above problem is related to a possible generalization of the Catalan number [236].
New trends in preference, utility, and choice: from mono to multi 57
(3,3) total preorder (reflexive, complete, transitive)
(4,2)
(5,1) & (3,2)
(5,1) & (2,2) (4,1) & (3,2) strong semiorder
(5,1) (3,2) strong interval order
(4,1) & (2,2)
(4,1) (3,1) & (2,2) semiorder
(3,1) (2,2) interval order
(2,1) total quasi-preorder (reflexive, complete, quasi-transitive)
(1,1) simple preference (reflexive, complete)
extended preference
(it contains a linear order)
Figure 6: Implications among combinations of weak (m, n)-Ferrers properties
58 A. Giarlotta
yz
x
Figure 7: The meet-semilattice of the NaP-preferences on the set X={x, y, z}
New trends in preference, utility, and choice: from mono to multi 59
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