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 direction: 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 multiple 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.
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- 1.
Notice that, since x and y are distinct, this formulation of completeness does not imply reflexivity.
- 2.
See Sect. 3.3 of this survey for a discussion on this point in relation to the so-called (m, n)-Ferrers properties.
- 3.
In case \(\succsim \) is complete, then the following statements are equivalent: (i) \(\succsim \) is quasi-transitive; (ii) for each \(x,y,z \in X\), \(x \succ y \succsim z\) implies \(x \succsim z\); (iii) for each \(x,y,z \in X\), \(x \succsim y \succ z\) implies \(x \succsim z\).
- 4.
We follow the approach described in Bouyssou and Pirlot (2004), 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 (Nishimura 2018).
- 5.
The literature also examines weaker forms of representability of a single binary relation, e.g., the existence of (continuous, semicontinuous) Richter-Peleg utility functions (Alcantud et al. 2016; Peleg 1970; Richter 1966). We shall deal with this topic in Sect. 4.3, where we also discuss some shortcomings of this notion, and introduce multi-utility representations.
- 6.
A set \(Y \subseteq X\) is weakly order-dense in X if, for each \(x_1,x_2 \in X\) such that \(x_1 \succ x_2\), there is \(y \in Y\) with the property that \(x_1 \succsim y \succsim x_2\). Such a set is often called Debreu order-dense, and the existence of a countable Debreu order-dense is referred to as Debreu-separability (Bridges and Mehta 1995).
- 7.
A jump in an ordered space \((X,\succsim )\) is a pair \((a,b) \in X^2\) such that \(a \succ b\) and there is no point \(c \in X\) such that \(a \succ c \succ b\). The topology \(\tau _\succsim \) is the order topology induced by \(\succsim \). The topological space \((X,\tau _\succsim )\) is separable if it contains a countable set D that intersects each nonempty open set. See Munkres (2000) for topological notions.
- 8.
This is not the terminology originally used by the authors.
- 9.
An ordinal is a well-ordered set \((X,<)\) such that each \(x \in X\) is equal to its initial segment \(\{y \in X : y < x\}\). The finite ordinals are the natural numbers. The first infinite ordinal is the set \(\omega _0\) of all natural numbers, endowed with the usual order. The first uncountable ordinal is the set \(\omega _1\) of all countable ordinals, endowed with the natural order. The famous continuum hypothesis, formulated by George Cantor in 1878, says that the cardinality of \({\mathbb R}\) is equal to \(\omega _1\) (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 \(\vert {\mathbb R}\vert > \omega _1\) holds). See the classical textbook by Kunen (1980) for ZFC axiomatic set theory.
- 10.
Here we use the notion of continuity employed in some standard textbooks in microeconomic theory, such as Mas-Colell et al. (1995, p. 46). Other authors sometimes employ a weaker notion of continuity: see, e.g., Sect. 1.6 of Bridges and Mehta (1995). However, from the point of view of applications, the distinction between the various notions of continuity is often immaterial. See also Evren and Ok (2011, p. 555), and Gerasímou (2013, pp. 2–3).
- 11.
Herden and Pallack (2002) 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 \(T_1\) but not Hausdorff. On the topic, see also Gerasímou (2013), who characterizes continuity in terms of closed semicontinuity and a property of “local expansion” of transitivity (Theorem 1 in Gerasímou (2013)).
- 12.
The 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 \(\varOmega \) satisfies some rather mild closure properties (see Cantone et al. (2016), Eliaz and Ok (2006) for a justification of this assumption).
- 13.
Selected items are underlined: thus, \(\underline{x}\,y\,z\) means \(c(\{x,y,z\}) = \{x\}\), \(\underline{y}\,\underline{z}\) 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 \(a \in X\): thus, it suffices to indicate how choices are defined for menus of size at least two.
- 14.
By 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.
- 15.
We should also distinguish between purely ordinal codomains, and those which also have an algebraic structure. Among the latter, let us mention (without getting into details) representations that employ non-Archimedean ordered fields, introduced by Narens (1985).
- 16.
See Kunen (1980) for the undefined notions of regular cardinal and Souslin line.
- 17.
The canonical completion of an asymmetric relation transforms incomparability into indifference.
- 18.
The transitive closure of a binary relation \(\succsim \) is the smallest transitive relation \(\succsim _{\mathrm {tc}}\) containing \(\succsim \).
- 19.
The dimension of a strict semiorder \(\succ \) is the least number of strict linear orders whose intersection gives \(\succ \).
- 20.
This is a work in progress (Giarlotta and Watson 2018c).
- 21.
The 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 \(\succ \) in place of \(\succsim \).
- 22.
A linear 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.
- 23.
- 24.
On the other hand, Schick (1986) and McClennen (1990) argue against the possibility of a money-pump phenomenon, observing that, after transactions between indifferent alternatives, an economic agent may well refuse a transaction between strictly preferred alternatives. However, as Piper (2014) notes, the above solutions are based on the (unlikely) circumstance that the economic agent remembers the past and accordingly plans the future.
- 25.
For the formal notion of the transitive coherence of two binary relations, see Sect. 4.1 on bi-preferences.
- 26.
According to the notation employed in Definition 2.14, we should denote these choice space by \((2^X,c_X)\). However, for the sake of simplicity, here we prefer to use the more direct notation \((X,c_X)\).
- 27.
Two choice spaces \((X,c_X)\) and \((W,c_W)\) are isomorphic if there exists a bijection \(\sigma :X \rightarrow W\) that preserves the choice structure, i.e., the equality \(\sigma (c_X(A)) = c_W(\sigma (A))\) holds for each menu \(A \in 2^X\).
- 28.
Contractibility is a form of “outer indiscernibility”, in the sense that a contractible menu cannot be distinguished 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. (2018b) in the process of dealing with congruence relations (i.e., structure-preserving equivalence relations) on a choice space.
- 29.
A menu is proper if it contains more than one item and is different from the ground set.
- 30.
This notion was originally introduced by Giarlotta and Greco (2013) 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).
- 31.
- 32.
A choice is replaceable if the consistency properties \((\alpha )\) and \((\rho )\) hold for it.
- 33.
Configurations (2)\('\), (M2)\('\), and (M3)\('\) are dual to, respectively, (2), (M2), and (M3).
- 34.
This procedure is reminiscent of the rational shortlist method, a bounded rationality approach to individual choice recently introduced by Manzini and Mariotti (2007). 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 Sect. 4.4 of this survey.
- 35.
Configurations (2)\('\), (C2)\('\), and (C3)\('\) are dual to, respectively, (2), (C2), and (C3).
- 36.
Cf. with the notion of continuity given in Sect. 2.3.
- 37.
In fact, Giarlotta and Watson (2018b) 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 Sect. 3.5.).
- 38.
The 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.
- 39.
See Footnote 6.
- 40.
A 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.
- 41.
Meet-semilattice means that \(\left( \textsf {NaP}(X),\sqsubseteq \right) \) is a poset, and for each \(\left( \succsim ^N_1, \succsim ^P_1\right) ,\left( \succsim ^N_2, \succsim ^P_2\right) \in \textsf {NaP}(X)\), there is a greatest element \(\left( \succsim ^N_3, \succsim ^P_3\right) \) in \(\textsf {NaP}(X)\) such that \(\left( \succsim ^N_3, \succsim ^P_3\right) \sqsubseteq \left( \succsim ^N_1, \succsim ^P_1\right) \) and \(\left( \succsim ^N_3, \succsim ^P_3\right) \sqsubseteq \left( \succsim ^N_2, \succsim ^P_2\right) \).
- 42.
For 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 Sect. 2.3 in Alcantud et al. (2018).
- 43.
- 44.
This type of representation has already been mentioned: see (3) in Sect. 4.2.
- 45.
A quasi-choice function on X is a map \(c :\varOmega \rightarrow \varOmega \cup \{\emptyset \}\) such that \(c(A) \subseteq A\) and \(0 \le \vert c(A)\vert \le 1\) for all \(A \in \varOmega \). Thus, the agent selects either a single item or no item at all from each menu. Similarly, a quasi-choice correspondence on X is a map \(c :\varOmega \rightarrow \varOmega \cup \{\emptyset \}\) such that \(c(A) \subseteq A\) and \(0 \le \vert c(A)\vert \le \vert A \vert \) for all \(A \in \varOmega \).
- 46.
In 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 X is finite.
- 47.
“Luce and Raiffa’s dinner” mentioned in the quotation below will be described in Example 4.35.
- 48.
It 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 (1986, 1988a, b, 1990a, b, 1991a, b) and later on by his followers Angilella and Giarlotta (2009); Angilella et al. (2010a), Giarlotta (1998, 2001), Greco (1997, 2005).
- 49.
This issue is connected to the problem of the lifting of choices having certain properties: see Cantone et al. (2017) for the general formulation of the problem and its logic-theoretic analysis in some special cases.
- 50.
Somehow in between approaches (8a) and (8b) lies the very interesting model constructed by Cherepanov et al. (2013), 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.
- 51.
A very surprising answer to a closely related question is given by Mandler, Manzini, and Mariotti (2012). 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 (2012) words: “Checklists are a fast and frugal way to maximize utility.”
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Acknowledgements
The author is very grateful to José Carlos R. Alcantud, Domenico Cantone, Jean-Paul Doignon, and Stephen Watson for several fruitful suggestions and discussions.
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Appendix
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) and (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 compactness, we simplify the notation in Fig. 4 to identify the comonotonic configurations (C1), (C2), (C2)\('\), (C3), (C3)\('\), and (C4) as follows (configuration (C5) never appears for NaP-preferences):
Observe that many configurations in Fig. 7 are isomorphic to each other (where an isomorphism between bi-preferences is defined in the obvious way). For instance, at level 3 of the meet-semilattice in Fig. 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 identified 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 \(n \ge 3\), 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 component, the above problem is related to a possible generalization of the Catalan number (Stanley 1999).
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Giarlotta, A. (2019). New Trends in Preference, Utility, and Choice: From a Mono-approach to a Multi-approach. In: Doumpos, M., Figueira, J., Greco, S., Zopounidis, C. (eds) New Perspectives in Multiple Criteria Decision Making. Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-030-11482-4_1
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