Necessary and possible preference structures

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Abstract

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–the necessary preference N and the possible preference P–to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair (N,P) such that N is a partial preorder on A and P is an extension of N satisfying mixed properties of transitivity and completeness. We characterize a NaP-preference (N,P) by the existence of a nonempty set R of total preorders such that R=N and R=P. In order to analyze the representability of NaP-preferences via families of utility functions, we generalize the notion of a multi-utility representation of a partial preorder by that of a modal utility representation of a pair of binary relations. Further, we give a dynamic view of the family of all NaP-preferences on a fixed set A by endowing it with a relation of partial order, which is defined according to the stability of the information represented by each NaP-preference.

Introduction

A natural way to model an economic agent’s preferences on a set A of alternatives is by means of a reflexive binary relation on A satisfying suitable ordering axioms. In this case, two derived binary relations are typically associated to the weak preference , namely: its symmetric part , called indifference and defined by ab if ab and ba; its asymmetric part , called strict preference and defined by ab if ab and ¬(ba).

The classical approach to the topic assumes that is a total preorder, i.e. a reflexive relation on A satisfying two additional properties: (i) transitivity; (ii) completeness. (Antisymmetry of is usually not assumed, due to the possibility of ties between couples of distinct alternatives.) The mathematical amenability of this approach lies in the fact that, under suitable separability conditions, a total preorder is representable by means of a continuous real-valued utility function. This allows one to translate a problem of maximizing a Debreu-separable preference into the much easier one of maximizing a real-valued function: see the book by Bridges and Mehta (1995) for an overview of the topic.

On the other hand, the assumption that the relation modeling a preference on A is “fully loaded” with properties (i) and (ii) often lacks adherence to reality. Therefore, two alternative approaches have emerged in time. A first approach consists of dropping the completeness axiom, allowing an economic agent to be occasionally indecisive: see Aumann (1962), Bewley (1986) and Mandler (2006), as well as the considerations in von Neumann and Morgenstern’s seminal work (von Neumann and Morgenstern, 1944, pp. 19–20). In this case, a preference is modeled by a potentially incomplete partial preorder. The recent literature on the topic devotes special attention to the semicontinuity/continuity of its utility representation (Back, 1986, Dubra et al., 2004, Eliaz and Ok, 2006, Evren and Ok, 2011, Herden and Levin, 2012, Mandler, 2005, Ok, 2002, Peleg, 1970, Richter, 1966).

A second alternative approach is to require that only satisfies completeness, whereas transitivity holds for the strict preference , but may fail for the indifference (hence for as well). In fact, in many fields (e.g. extensive measurement in mathematical psychology (Krantz, 1967, Lehrer and Wagner, 1985), choice theory under risk (Fishburn, 1968), decision making under risk (Rubinstein, 1988)) intransitivity of indifference is a natural feature of the associated preference structure. In 1956 Luce (1956) introduced the notion of a semiorder to model a situation of intransitive indifference with a “threshold of discrimination or perception”. Later on, Fishburn, 1970, Fishburn, 1973 extended the notion of semiorder to that of an interval order, whose underlying idea is to assign an “interval of evaluations” to each alternative. As in the case of total and partial preorders, recent research on the topic is focused on representability issues: see, e.g. the books (Fishburn, 1985, Pirlot and Vincke, 1997, Aleskerov et al., 2007) for some related (yet not fully updated, since research in the field is rapidly evolving) results on interval orders and semiorders.

In this paper we combine these alternative approaches to preference modeling into one. To explicitly take into account both incompleteness and intransitivity of preferences, we use two binary relations, called the necessary preference N and the possible preference P, in place of a single ordering relation . The necessary preference N represents the core of the preferential information on A provided by an economic agent, and it is assumed to be a partial preorder. The possible preference P is a completion of the core information by means of additional information. If the two binary relations N and P on A are connected to each other by some mixed properties of completeness and transitivity, then the pair (N,P) is a NaP-preference (necessary and possible preference) on A. Note that the axioms of a NaP-preference (N,P) are designed in a way that transitivity and completeness hold jointly but not singularly: in fact, N satisfies (i) but not necessarily (ii), whereas P satisfies (ii) but not necessarily (i).

Necessary and possible preference relations were originally introduced in Robust Ordinal Regression (ROR), a methodology developed within the realm of Multiple Criteria Decision Analysis (MCDA). (See Figueira et al. (2005) for a recent state-of-the-art on MCDA, and Angilella et al. (2010) and Greco et al., 2008, Greco et al., 2010a, Greco et al., 2010b for ROR.) In a ROR approach all information provided by an economic agent on a set A of n-dimensional alternatives (i.e. in the presence of a set G={gi:i{1,,n}} of n2 evaluation criteria gi:AR) is used to build a set U of global value functions u:RnR, which are “compatible” with the model. In this multi-dimensional setting, two binary relations N and P on A naturally arise from the family U as follows: aNbdefuU(u(a)u(b))aPbdefuU(u(a)u(b)) for each a,bA. Then the pair (N,P) is a NaP-preference on A.

The realm of decisions under uncertainty offers another well suited environment to define a necessary and possible preference structure. For example, in an Anscombe–Aumann setting (Anscombe and Aumann, 1963), prototypes of a necessary preference N and a possible preference P are given, respectively, by Bewley’s Knightian preferences (Bewley, 1986, Ghirardato et al., 2004) and Lehrer–Teper’s justifiable preferences (Lehrer and Teper, 2011a). In fact, given a pre-determined set of priors, according to Knightian preferences, an act f is preferred to another act g if this preference holds for all priors, whereas, according to a model of justifiable preferences, f is preferred to g if this is true for at least one prior. (Observe that justifiable preferences are in general intransitive (Lehrer and Teper, 2011a), since there might be acts f,g,h such that f is preferred to g for a given prior p,g is preferred to h for another prior q, but there is no prior r for which f is preferred to h.) In a von Neumann–Morgenstern’s setting (von Neumann and Morgenstern, 1944), a further example of a necessary preference relation extendable via a possible preference is given by the incomplete preference relation modeled in Dubra et al. (2004), where the authors consider a set of utility functions U such that, given two lotteries p and q,p is preferred to q if the expected utility of p is not smaller than the expected utility of q for all maps in U.

In our approach we look at necessary and possible preference relations not separately, but intertwined in a single model. We are aware of the fact that the very idea of modeling a preference by means of two binary relations in place of one is not entirely new: indeed, the models proposed by Gilboa et al. (2010) and by Lehrer and Teper (2011b) are of this type, albeit their perspective is different from ours. Specifically, Gilboa et al. deal with decisions under uncertainty in an Anscombe–Aumann setting, and define in this context two types of preference relations:

  • an objective preference , which represents preferences such that the decision maker can convince everybody that he is right;

  • a subjective preference , which represents preferences such that the decision maker cannot be convinced by anybody that he is wrong.

The objective preference is a partial preorder, whereas the subjective preference is a complete preorder that extends . More precisely, there exists a set C of probabilities on the set of states of nature S such that for every two acts f and g,fg if the expected utility of f is not smaller than the expected utility of g for all pC, whereas fg if the minimal expected utility of f on C is not smaller than the minimal expected utility of f on C.

Note that is a Knightian preference and thus it is a type of necessary preference N. On the other hand, has a different flavor than a possible preference P, despite being a complete extension of the objective preference (in fact, is the maxmin expected utility of Gilboa and Schmeidler (1989)). Nevertheless, it seems possible to define a possible preference P in the same model, and interpret it as representing preferences such that “someone cannot be convinced that he is wrong”. Indeed, one can imagine that each probability pC can be rationally selected by some individual whose preferences are exactly those given by the expected utility with respect to p. Then, according to this interpretation, the necessary preferences given by are those which are true for all individuals because they hold for all pC, whereas the possible preferences are those such that there exists at least one pC for which they hold, since some individual might choose to decide according to pC.

For what concerns approach (Lehrer and Teper, 2011b), Lehrer and Teper consider the following two decision rules:

  • a prudent rule, which declares that an act f is preferred to g provided that there is some positive evidence in this direction;

  • a lenient rule, which declares that f is preferred to g if there is no evidence supporting a strict preference of g over f.

Since in an Anscombe–Aumann setting the prudent rule is modeled by a Knightian preference relation and the lenient rule by a justifiable preference relation, the connection with necessary and possible preference structures is apparent.

The paper is organized as follows. In Section 2 we introduce the notion of a NaP-preference, discuss the economic significance of its axioms and give some examples. In Section 3 we characterize a NaP-preference by means of the existence of a nonempty family of total preorders, which witness universally each pair in the necessary component and existentially each pair in the possible component. In Section 4 we analyze the representability of a NaP-preference by suitable families of utility functions, introducing the notion of a modal utility representation of a pair of binary relations. In Section 5 we give a dynamic view of the family of all NaP-preferences on a fixed set of alternatives, endowing it with an economically significant partial order.

Section snippets

Necessary and possible preferences

Here we define the main notion of the paper and discuss its motivating idea. To start, we introduce some basic notation. Henceforth, A denotes a nonempty set of alternatives, and a reflexive relation on A. For the sake of synthesis, we slightly abuse notation and write, e.g. abcd whenever {(a,b),(b,c),(c,d)}. Recall that is the disjoint union of its asymmetric (and irreflexive) part and its symmetric (and reflexive) part . Note that if is a partial preorder, then is a strict

Resolutions of NaP-preferences

In this section we characterize a NaP-preference (N,P) on A in terms of the existence of a nonempty set R of total preorders on A such that R=N and R=P. We also show that this set is unique under a maximality condition. To start, we introduce two weak forms of NaP-preferences.

Definition 3.1

A pair (,) of binary relations on A is called:

  • a NaP-preorder if (N)–(P)–(C) hold;

  • a partial NaP-preference if (N)–(P)–(NP)–(PN) hold.

As for NaP-preferences, the binary relation is the gap of (,).

The next

Modal utility representations of NaP-preferences

The analysis of the representability of preference relations (e.g. preorders, interval orders, semiorders, etc.) is ubiquitous in decision theory. Recall that a preorder on a set A is representable (in R) if there exists a function u:AR such that for each x,yA, we have xy if and only if u(x)u(y); the map u is called a utility representation of . It is well known that a total preorder on A is representable (hence continuously representable due to Debreu’s Open Gap Lemma (Debreu, 1954))

The NaP-preference structure of a set

As discussed in Section 2, the information summarized by a NaP-preference (N,P) is of two types: (i) “positive”, represented by the necessary preference N; (ii) “negative”, represented by the impossible preference I. It is natural to assume that a larger amount of both types of information induces a higher stability of a NaP-preference, i.e. the smaller its gap G is, the more stable the NaP-preference is. In an attempt to provide the family of all NaP-preferences on A with an order

Acknowledgments

The authors wish to thank two anonymous referees for some useful comments and suggestions, which improved the content of the paper as well as the quality of its presentation.

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