Journal of Mathematical Economics 49 (2013) 163–172

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Journal of Mathematical Economics

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Necessary and possible preference structuresAlfio Giarlotta ∗, Salvatore GrecoDepartment of Economics and Business, University of Catania, Catania 95129, Italy

a r t i c l e i n f o

Article history:Received 8 May 2012Received in revised form2 January 2013Accepted 3 January 2013Available online 16 January 2013

Keywords:Incomplete preferenceIntransitive indifferenceTotal preorderPartial preorderInterval orderSemiorderNaP-preferenceModal utility representationPreference resolution

a b s t r a c t

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive andcomplete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold inmany applications, preferences are often modeled by means of weaker binary relations, dropping eithercompleteness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders).We introduce analternative 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 familiesof utility functions, we generalize the notion of a multi-utility representation of a partial preorder by thatof amodal utility representation of a pair of binary relations. Further, we give a dynamic view of the familyof all NaP-preferences on a fixed set A by endowing it with a relation of partial order, which is definedaccording to the stability of the information represented by each NaP-preference.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Anaturalway tomodel an economic agent’s preferences on a setA of alternatives is by means of a reflexive binary relation % on Asatisfying suitable ordering axioms. In this case, two derived binaryrelations are typically associated to theweak preference %, namely:its symmetric part ∼, called indifference and defined by a ∼ b ifa % b and b % a; its asymmetric part ≻, called strict preference anddefined by a ≻ b if a % b and ¬(b % a).

The classical approach to the topic assumes that % is a totalpreorder, i.e. a reflexive relation on A satisfying two additionalproperties: (i) transitivity; (ii) completeness. (Antisymmetry of% is usually not assumed, due to the possibility of ties betweencouples of distinct alternatives.) The mathematical amenabilityof this approach lies in the fact that, under suitable separabilityconditions, a total preorder % is representable by means of acontinuous real-valued utility function. This allows one to translatea problem of maximizing a Debreu-separable preference into themuch easier one of maximizing a real-valued function: see thebook by Bridges and Mehta (1995) for an overview of the topic.

On the other hand, the assumption that the relation%modelinga preference on A is ‘‘fully loaded’’ with properties (i) and(ii) often lacks adherence to reality. Therefore, two alternative

∗ Corresponding author. Tel.: +39 095 7537511.E-mail addresses: giarlott@unict.it (A. Giarlotta), salgreco@unict.it (S. Greco).

0304-4068/$ – see front matter© 2013 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2013.01.001

approaches have emerged in time. A first approach consists ofdropping the completeness axiom, allowing an economic agentto be occasionally indecisive: see Aumann (1962), Bewley (1986)andMandler (2006), as well as the considerations in von NeumannandMorgenstern’s seminal work (von Neumann andMorgenstern,1944, pp. 19–20). In this case, a preference is modeled by apotentially incomplete partial preorder. The recent literature on thetopic devotes special attention to the semicontinuity/continuity ofits utility representation (Back, 1986; Dubra et al., 2004; Eliaz andOk, 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 % onlysatisfies completeness, whereas transitivity holds for the strictpreference ≻, but may fail for the indifference ∼ (hence for% as well). In fact, in many fields (e.g. extensive measurementin mathematical psychology (Krantz, 1967; Lehrer and Wagner,1985), choice theory under risk (Fishburn, 1968), decision makingunder risk (Rubinstein, 1988)) intransitivity of indifference is anatural feature of the associated preference structure. In 1956 Luce(1956) introduced the notion of a semiorder to model a situationof intransitive indifference with a ‘‘threshold of discrimination orperception’’. Later on, Fishburn (1970, 1973) extended the notionof semiorder to that of an interval order, whose underlying idea isto assign an ‘‘interval of evaluations’’ to each alternative. As in thecase of total and partial preorders, recent research on the topic isfocused on representability issues: see, e.g. the books (Fishburn,1985; Pirlot and Vincke, 1997; Aleskerov et al., 2007) for some

164 A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172

related (yet not fully updated, since research in the field is rapidlyevolving) results on interval orders and semiorders.

In this paper we combine these alternative approaches topreference modeling into one. To explicitly take into accountboth incompleteness and intransitivity of preferences, we use twobinary relations, called the necessary preference %N and the possiblepreference%P , in place of a single ordering relation%. The necessarypreference %N represents the core of the preferential informationon A provided by an economic agent, and it is assumed to bea partial preorder. The possible preference %P is a completionof the core information by means of additional information. Ifthe two binary relations %N and %P on A are connected to eachother 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 notnecessarily (ii), whereas %P satisfies (ii) but not necessarily (i).

Necessary and possible preference relations were originallyintroduced in Robust Ordinal Regression (ROR), a methodologydeveloped within the realm of Multiple Criteria Decision Analysis(MCDA). (See Figueira et al. (2005) for a recent state-of-the-arton MCDA, and Angilella et al. (2010) and Greco et al. (2008,2010a,b) for ROR.) In a ROR approach all information provided byan economic agent on a set A of n-dimensional alternatives (i.e. inthe presence of a set G = {gi : i ∈ {1, . . . , n}} of n ≥ 2 evaluationcriteria gi: A → R) is used to build a setU of global value functionsu:Rn

→ R, which are ‘‘compatible’’ with the model. In this multi-dimensional setting, two binary relations%N and%P on A naturallyarise from the family U as follows:

a%N b def

⇐⇒ ∀u ∈ U(u(a) ≥ u(b))

a%P b def

⇐⇒ ∃u ∈ U(u(a) ≥ u(b))

for each a, b ∈ A. Then the pair%N , %P

is a NaP-preference on A.

The realm of decisions under uncertainty offers anotherwell suited environment to define a necessary and possiblepreference structure. For example, in an Anscombe–Aumannsetting (Anscombe and Aumann, 1963), prototypes of a necessarypreference%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, accordingto Knightian preferences, an act f is preferred to another act gif this preference holds for all priors, whereas, according to amodel of justifiable preferences, f is preferred to g if this is truefor at least one prior. (Observe that justifiable preferences are ingeneral intransitive (Lehrer and Teper, 2011a), since there mightbe acts f , g, h such that f is preferred to g for a given prior p, gis preferred to h for another prior q, but there is no prior r forwhich f is preferred to h.) In a vonNeumann–Morgenstern’s setting(von Neumann and Morgenstern, 1944), a further example of anecessary preference relation extendable via a possible preferenceis given by the incomplete preference relation modeled in Dubraet al. (2004), where the authors consider a set of utility functionsU such that, given two lotteries p and q, p is preferred to q if theexpected utility of p is not smaller than the expected utility of q forall maps in U.

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

– an objective preference %∗, which represents preferences suchthat the decisionmaker can convince everybody that he is right;

– a subjective preference %∧ , which represents preferences such

that the decision maker cannot be convinced by anybody thathe is wrong.

The objective preference %∗ is a partial preorder, whereas thesubjective preference %

∧ is a complete preorder that extends %∗.More precisely, there exists a set C∗ of probabilities on the set ofstates of nature S such that for every two acts f and g, f %∗ g if theexpected utility of f is not smaller than the expected utility of g forall p ∈ C∗, whereas f %

∧ g if the minimal expected utility of f onC∗ is not smaller than the minimal expected utility of f on C∗.

Note that %∗ is a Knightian preference and thus it is a type ofnecessary preference %N . On the other hand, %

∧ has a differentflavor than a possible preference %P , despite being a completeextension of the objective preference (in fact, %

∧ is the maxminexpected utility of Gilboa and Schmeidler (1989)). Nevertheless,it seems possible to define a possible preference %P in the samemodel, and interpret it as representing preferences such that‘‘someone cannot be convinced that he is wrong’’. Indeed, one canimagine that each probability p ∈ C∗ can be rationally selectedby some individual whose preferences are exactly those givenby the expected utility with respect to p. Then, according to thisinterpretation, the necessary preferences given by %∗ are thosewhich are true for all individuals because they hold for all p ∈ C∗,whereas the possible preferences are those such that there exists atleast one p ∈ C∗ for which they hold, since some individual mightchoose to decide according to p ∈ C∗.

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

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

– a lenient rule, which declares that f is preferred to g if there isno evidence supporting a strict preference of g over f .

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

The paper is organized as follows. In Section 2 we introduce thenotion of a NaP-preference, discuss the economic significance ofits axioms and give some examples. In Section 3 we characterize aNaP-preference by means of the existence of a nonempty familyof total preorders, which witness universally each pair in thenecessary component and existentially each pair in the possiblecomponent. In Section 4 we analyze the representability of a NaP-preference by suitable families of utility functions, introducing thenotion of amodal 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 aneconomically significant partial order.

2. Necessary and possible preferences

Here we define the main notion of the paper and discussits motivating idea. To start, we introduce some basic notation.Henceforth, A denotes a nonempty set of alternatives, and % areflexive relation on A. For the sake of synthesis, we slightlyabuse notation and write, e.g. a % b % c % d whenever{(a, b), (b, c), (c, d)} ⊆ %. Recall that % is the disjoint union ofits asymmetric (and irreflexive) part ≻ and its symmetric (andreflexive) part ∼. Note that if % is a partial preorder, then ≻ is astrict partial order (i.e. asymmetric and transitive), whereas ∼ isan equivalence relation. Let % be a partial preorder on A. For eachx ∈ A, the closed rays ↑

%x and ↓

%x are the sets

↑%x := {a ∈ A : a % x} and ↓

%x := {a ∈ A : x % a}.

A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172 165

Whenever the partial preorder% is clear from context, we drop thesubscript and simply write ↑x and ↓x. An extension of % is a binaryrelation %′ on A that contains % . (Note that we do not require that≻

′ contains ≻.) If the containment is strict, then the extension isproper. The relation % is called the base of the extension %′.

Definition 2.1. A pair%N , %P

of binary relations on a set A is

a necessary and possible preference (a NaP-preference) on A if thefollowing proprieties hold for each x, y, z ∈ A:(N) %N is a partial preorder;(P) %P is an extension of %N ;(C) x%P y or y%N x ;

(NP) x%N y%P z implies x%P z ;(PN) x%P y%N z implies x%P z .

The relations %N and %P are called, respectively, the necessarypreference and the possible preference. The preference gap %G andthe impossible preference %I are the binary relations derived from%N and %P as follows:

%G

:= %P\ %

N and %I:= A2

\ %P.

A NaP-preference on A is said to be totally stable if its gap is empty,and totally unstable if its gap is equal to A2

\∆(A), where ∆(A) :=

{(a, a) : a ∈ A} is the diagonal of A.

The interpretation of a NaP-preference%N , %P

on a set A of

alternatives is quite natural. Axiom (N) requires that the base re-lation %N of the preference has a typical preference structure. Thebinary relation %N models the ‘‘positive’’ information contained inan agent’s scheme of preferences, representing the part of it thathe considers necessary. Therefore, %N is, in general, neither anti-symmetric nor complete.

Axiom (P) states that an economic agent might have furtherpieces of preferential information, disjoint from the necessarycomponent %N andmodeled by the binary relation %I . The contentof this additional information is ‘‘negative’’, since it summarizesthe part of an agent’s preference structure that is impossible fromhis point of view. For technical reasons, we prefer to represent %I

via its complement %P , which then becomes the possible part ofhis preference structure. In other words, since A2 is the disjointunion %N

∪ %G∪ %I , the total information provided by an agent is

modeled by %N∪ %I , whereas the gap %G is a sort of ‘‘gray area’’

separating necessity of a preference from its impossibility.The last three axioms connect the two components of a NaP-

preference. Specifically, (C) is a mixed form of completeness of%P with respect to %N , which ensures that the impossibility of apreference of x over y yields the necessity of a preference of y overx. Note that as a consequence of (C), if x, y ∈ A are such that neitherx%N y nor y%N x hold, then we must have x∼

P y; thus, in a NaP-preference, a ‘‘necessary incomparability’’ is always associated to a‘‘possible indifference’’. In particular, %P is a completion of %N .

Properties (NP)–(PN) are two mixed forms of transitivity, sincethe composition in any order of one necessary preference %N andone possible preference %P always yields a possible preference.As a consequence, the binary relation %P itself satisfies a form oftransitivity, in fact it is quasi-transitive, i.e. if x%P y%P z holds witheither x≻

P y or y≻P z, thenwe also have x%P z. Note that this form

of transitivity is also satisfied by interval orders and semiorders. Onthe other hand, properties (NP)–(PN) are definitively weaker thanthe full transitivity of %P . In fact, given a NaP-preference

%N , %P

on A, one can define further types of mixed transitivity properties,obtained as suitable finite compositions of necessary and possiblepreferences. These properties imply that the possible preference%P is, respectively, an interval order, a semiorder, a strong intervalorder and a strong semiorder (Giarlotta, submitted for publication).However, none of these properties guarantees the transitivity of%P , which is obtained only under an additional axiom of mixedtransitivity.

Remark 2.2. Let%N , %P

be a NaP-preference on A.

(i) Since the possible preference %P is complete and negativelytransitive (i.e. for each x, y, z ∈ A, if ¬(x%P y) and ¬(y%P z),then ¬(x%P z)), it follows that the impossible preference %I

is a strict partial order; in particular, %I is transitive.(ii) Under axioms (N)–(P), property (C) can be equivalently stated

by requiring that for each x, y ∈ A, if x%I y then y≻N x. On

the other hand, (C) provides no information on the reverseimplication, since if y≻

N x holds, then we may have bothx%P y and x%I y (cf. Fig. 1). In particular, observe that despite%N

⊆ %P , we have ≻P

⊆ ≻N .

(iii) The gap %G lies in between ∅ and A2\∆(A). Further, these

bounds can be attained: indeed, the pairs (%, %), with % beingany total preorder on A, and

∆(A), A2

are, respectively,

totally stable and totally unstable NaP-preferences on A.Totally stable NaP-preferences represent situations of fullinformation, since the equality %N

∪ %I= A2 holds. On

the other hand, the unique case of a totally unstable NaP-preference represents a situation of minimum information,being %N

∪ %I= ∆(A) ∪ ∅.

If%N , %P

is a NaP-preference on A, then for each couple

{x, y} ⊆ A of distinct alternatives, there are six admissibleconfigurations, as described in Fig. 1. The directed edges of thecorresponding digraphs are of two types: the thick arrow from xto y represents the case x%N y, whereas the thin arrow from x to yrepresents the case x%G y. On the other hand, the absence of anytype of arrow from x to y represents the case x%I y.

The semantic of each configuration represented in Fig. 1 is quitenatural within the realm of preference theory.• Case (i) is a ‘‘necessary–necessary’’ configuration, and describes

a situation of core indifference between two alternatives x andy. This a rather typical circumstance whenever an economicagent, who is asked to compare two courses of action, judgesthem comparable and preferentially equivalent. Therefore,the necessary component of a NaP-preference need not beantisymmetric.

• Cases (ii) and (ii′) are ‘‘necessary–impossible’’ configurations,and describe a situation of strong preference of, respectively, xover y, and y over x. This happens whenever an economic agentfeels that one alternative is necessarily better than the otherone, at the same time excluding the possibility of the reversepreference in any possible scenario. Note that the combinedaction of the axioms (N)–(P)–(C) of a NaP-preorder yields thefollowing chain of equivalences for each x, y ∈ A:

y≻I x ⇐⇒ y%

I x ⇐⇒ (y%I x) ∧ (x≻

N y).• Cases (iii) and (iii′) are ‘‘necessary–possible’’ configurations,

and describe intermediate situations of weak preference. Inthese cases, an economic agent declares that one alternative isnecessarily preferred to the other one, but without excludingthe possibility that the reverse preference may hold as well insome scenarios. Note that the axioms of a NaP-preorder entailthe following chain of equivalences for each x, y ∈ A:

y≻G x ⇐⇒ (y≻

G x) ∧ (x≻N y) ⇐⇒ (y%

G x) ∧ (x%N y).

• Case (iv) is a purely ‘‘possible–possible’’ configuration, anddescribes a situation of weak incomparability. In fact, this istypical of situations such that at the givenmoment an economicagent is unable or unwilling to decide what the relationbetween x and y is.From the point of view of their informative content, configu-

rations (i), (ii) and (ii′) are the most stable of all, since there is noroom for indecisiveness. On the other hand, configuration (iv) is theleast stable among the six, only containing weak information thatcan evolve in any way. The intermediate configurations (iii) and(iii′) are partially stable, since the possible preferences in them can‘‘eventually stabilize’’ into either necessary or impossible.

166 A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172

Fig. 1. Admissible relations between two alternatives for a NaP-preference%N , %P

.

In this respect, it is worth emphasizing that the idea beyond theconstruction of a NaP-preference is essentially dynamic and notstatic. In fact, we aim at exploring how an agent’s NaP-preferencestructure might evolve from an initial state of partially unstableinformation (that is, at the present time, or at an initial state ofnature, or for a non-cooperative group of decision makers, etc.)toward a final state of stable information. The initial configurationis represented by a partial preorder (or many partial preordersin the case of a group decision), which is completed by someadditional coherent information. On the other hand, the finalconfiguration appears in the form of a complete preorder, whereall purely possible information has disappeared.

In Section 5 we will further dwell on this point, explaininghow the family of all the NaP-preferences on a given set canbe arranged into a semilattice, according to the content/stabilityof each of its elements. This order-theoretic structure has theleast stable configuration at its bottom (where all preferencesbetween pairs of distinct alternatives are in the gray area of purepossibility) and all themost stable configurations at the top (whereall preferences between pairs of distinct alternatives are eithernecessary or impossible).

We end this section with an example, which shows that,under conditions (N)–(P), axioms (C)–(NP)–(PN) are mutuallyindependent and consistent.

Example 2.3. To prove independence, we provide three examplesof pairs

%N , %P

of binary relations on the set A := {a, b, c, d}

such that (N)–(P) are always verified, whereas only two of thethree remaining axioms hold. Furthermore, we prove consistencyof the set of axioms in Definition 2.1 by exhibiting a (non-limit)NaP-preference on A. All four pairs of binary relations on A arerepresented in Fig. 2, where we use the same notation as in Fig. 1.(For simplicity, all loops (x, x) are not represented.)

It is easy to check that the pair%

4, %′

4

is a NaP-preference on

A. For the first three examples, we preliminarily observe that theyare non-trivial in following sense: (i) the corresponding digraphis connected; (ii) transitivity of the necessary component is non-vacuously verified; (iii) properties (NP) and (PN) are, whenevertrue, non-vacuously verified. The pair

%

1, %′

1

does not satisfy

(PN), since we have c %′

1d%

1b but ¬(c %′

1b). On the other hand,

the pair%

2, %′

2

does not satisfy (NP), since we have c %

2d%′

2b

yet ¬(c %′

2b). Finally, (C) does not hold for the pair

%

3, %′

3

,

because we have ¬(b%3d) and ¬(d%′

3b).

3. Resolutions of NaP-preferences

In this section we characterize a NaP-preference%N , %P

on A

in terms of the existence of a nonempty setR of total preorders onA such that

R = %N and

R = %P . We also show that this set

is unique under amaximality condition. To start, we introduce twoweak 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 lemma is needed to prove the main results of thispaper (Theorems 3.4 and 4.3).

Lemma 3.2. Let%, %′

be a pair of binary relations on A such that

properties (N)–(P) hold. For each x, y ∈ A such that x%′ y, let %xy bethe extension of % defined by

%xy := % ∪

↑

%x × ↓

%y

.

(i) The relation %xy is a partial order.(ii) If

%, %′

is a partial NaP-preference, then %xy is a partial

preorder contained in %′.(iii) If

%, %′

is a NaP-preorder such that ¬(y % x), then %xy is a

partial preorder contained in %′.(iv) If

%, %′

is a NaP-preorder and %′′ is a partial preorder such

that % ⊆ %′′⊆ %′ and ¬(y%′′ x), then the pair

%′′

xy, %′is a

NaP-preorder.

Proof. Part (i) is a straightforward computation and is left to thereader. For (ii), assume that the pair

%, %′

satisfies properties

(NP)–(PN). By (i), it suffices to show that %xy is contained in %′.Assume by contradiction that there exists (a, b) ∈

%xy \ %′

. Since

%′ contains%, it follows that (a, b) ∈ ↑%x×↓

%y, hence a % x%′ y %

b. Now (NP)–(PN) yield a%′ b , which contradicts the hypothesis.To prove (iii), assume that

%, %′

satisfies property (C) and

that ¬(y % x). Toward a contradiction, assume that there exists(a, b) ∈

%xy \ %′

. It follows that a % x and y % b. Furthermore,

¬(a%′ b) implies b % a by (C), whence y % b % a % x. Transitivityof % yields y % x, a contradiction.

For (iv), observe that the hypothesis implies that the pair%′′, %′

is a NaP-preorder such that ¬(y%′′ x). Therefore, part (iii)

yields that%′′

xy, %′is a NaP-preorder, as claimed. (As before, the

notation %′′xy stands for %′′

∪↑%′′

x × ↓%′′

y.) �

Remark 3.3. The relation %xy defined in Lemma 3.2 has animmediate economic interpretation. Let % be a core preference onA (i.e. a partial preorder) and %′ an extension of % representingsome possible scenarios. Assume that two alternatives x, y ∈ Aare such that at the moment x is not preferred to y (i.e. ¬(x %y)), yet there exists a possible scenario under which this mighthappen (i.e. x%′ y). Then %xy is the smallest core preferenceon A that extends % and witnesses a preference of x over y(i.e. x%xy y) in a way that is compatible with %′. More technically,the binary relation %xy is the partial preorder on A obtained bytaking the transitive closure of the binary relation % ∪ {(x, y)}.(Note that if x % y then %xy =%.) Lemma 3.2(ii) states thatunder the condition that the pair (%, %′) also satisfies the twomixed transitivity properties (NP)–(PN), the core preference %xyis fully compatible with the set of possible scenarios providedby %′. A similar conclusion holds under the conditions that thepair (%, %′) satisfies the mixed completeness property (C) andthere is no reverse core preference of y over x, as stated byLemma 3.2(iii).

A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172 167

Fig. 2. Examples of pairs%, %′

of binary relations on A := {a, b, c, d} satisfying (N)–(P) and other properties.

Next we characterize a NaP-preference%N , %P

by means of a

family R of total preorders witnessing universally each pair in %N

and existentially each pair in %P . The idea of the proof consists inbuilding for each pair (x, y) in the gap %G , both (i) a total preorderthat witnesses the absence of (x, y) in %N , and (ii) a total preorderthat witnesses the presence of (x, y) in %P . To this aim, for each(x, y) ∈ %G , first we create two partial preorders %

1and %

2such

that ¬(x%1y), x%

2y and %N

⊆ %i⊆ %P for each i ∈ {1, 2}. Then

we extend %1and %

2to two total preorders that are still contained

in %P . Finally, we collect them in a set, eliminating repetitions ifany. In order to obtain these complete extensions for an arbitraryground set A, we need the Axiom of Choice (AC) in its equivalentform of Zorn’s Lemma.

Theorem 3.4 (AC). The following statements are equivalent for a pair%N , %P

of binary relations on A:

(a)%N , %P

is a NaP-preference on A;

(b) there exists a nonempty family R :=%h : h ∈ H

of distinct

total preorders on A such that for each x, y ∈ A, we have:(RN) x%N y ⇐⇒ ∀h ∈ H (x%h y);(RP) x%P y ⇐⇒ ∃h ∈ H (x%h y).

Proof. (b)⇒ (a). Assume that there exists a nonempty familyR :=

{%h : h ∈ H} of total preorders on A satisfying (RN)–(RP).We provethat the five properties (N)–(P)–(C)–(NP)–(PN) in Definition 2.1hold for the pair

%N , %P

.

Reflexivity of each preorder %h and property (RN) yield thereflexivity of %N . To prove transitivity of %N , let x, y, z ∈ A besuch that x%N y%N z. Then, for each h ∈ H , we have x%h y%h zby (RN), hence x%h z by transitivity of %h. Another applicationof (RN) yields x%N z, as claimed. This shows that %N is a partialpreorder, i.e. axiom (N) holds. Furthermore, properties (RN)–(RP)along with the fact that the family R is nonempty imply that %P

is an extension of %N , i.e. (P) holds. For (C), let x, y ∈ A be suchthat¬(x%P y). Property (RP) implies that¬(x%h y) for each h ∈ H ,hence y%h x by completeness of %h. Then (RN) yields y%N x, so (C)holds. To prove (NP), let x, y, z ∈ A be such that x%N y%P z. Thehypothesis yields that x%h y%h z for some h ∈ H , hence x%h zby transitivity of %h. Now another application of (RP) entailsx%P z, as claimed. The proof of (PN) is similar.

(a) ⇒ (b). Assume that%N , %P

is a NaP-preference on A. If

%G= ∅, then %N

= %P is a total preorder. In this case, itis immediate to check that the singleton R :=

%N

satisfies

properties (RN)–(RP). Next, assume that %G is nonempty. For each(x, y) in the gap, we shall define two total preorders %−

xy and %+xy

having the following properties:(i) %N

⊆ %−xy ⊆ %P and %N

⊆ %+xy ⊆ %P ;

(ii) ¬(x%−xy y) and x%+

xy y.

Once all total preorders %−xy and %+

xy have been defined, we set

R :=%σ

xy : (x, y) ∈ %G∧ σ ∈ {−, +}

and select a subset H of the index set %G

×{+, −} such that

R =%h : h ∈ H

and %h = %h′ for any two distinct h, h′∈ H . Then, to complete the

proof, we show that R satisfies properties (RN)–(RP).In order to obtain each couple of total preorders %−

xy and %+xy

in the family R, we proceed as follows. Let (x, y) ∈ %G . Define afamily of partial preorders on A by

P −

xy :=% partial preorder on A : %

N⊆ % ⊆ %

P and ¬(x % y)

and order it by extension. Note that the family P −xy is nonempty,

because it contains%N . Further, every chainC in the posetP −

xy , ⊆

has an upper bound in it, namely, the partial preorder % :=

C.Thus, the nonempty poset

P −

xy , ⊆satisfies the ascending chain

condition, and we can apply Zorn’s Lemma to obtain a maximalelement in it, say %−

xy. To prove that %−xy is a total preorder on

A satisfying (i) and (ii), it suffices to show completeness of %−xy.

Toward a contradiction, assume that there exist a, b ∈ A such that¬(a%−

xy b) and ¬(b%−xy a). We claim that

(x, y) ∈ (↑a × ↓b) ∩ (↑b × ↓a)

where all closed rays are taken in the partially preordered setA, %−

xy

. Indeed, each (p, q) ∈ A2 that belongs to the intersection

of the two Cartesian products is such that p ∈ ↑a and q ∈ ↓a,i.e. p%−

xy a%−xy q. Transitivity of %−

xy yields p%−xy q. Thus, the claim

holds because we have ¬(x%−xy y). By symmetry, we can assume

without loss of generality that (x, y) ∈ ↑a × ↓b. Define a binaryrelation %∗ on A by

%∗:= %−

xy ∪ (↑a × ↓b) .

Observe that %∗ is a proper extension of %−xy, because we have

(a, b) ∈%∗

\ %−xy

. Further, Lemma 3.2(iv) implies that %∗ is a

partial preorder on A such that %N⊆ %∗

⊆ %P . Since ¬(x%∗ y),it follows that %∗ belongs to P −

xy , contradicting the maximality of%−

xy inP −

xy , ⊆. This completes the definition of %−

xy.To define the total preorder %+

xy, consider the family

P +

xy :=% partial preorder on A : %

N⊆ % ⊆ %

P and x % y

and order it by extension. The family P +xy is nonempty, since

Lemma 3.2(ii) implies that the binary relation %Nxy on A defined by

%Nxy := %

N∪ (↑x × ↓y)

belongs to P +xy . (Here the closed rays are taken in the partially

preordered setA, %N

.) Further, it is easy to check that

P +

xy , ⊆

satisfies the ascending chain condition. Therefore, we can applyZorn’s Lemma to obtain a maximal element %+

xy. We claim that %+xy

is a total preorder on A satisfying (i) and (ii). It suffices to showcompleteness of %+

xy. To this aim, assume by contradiction thatthere exist a, b ∈ A such that ¬(a%+

xy b) and ¬(b%+xy a). Consider

the binary relation %∗ on A defined by

%∗:= %+

xy ∪ (↑a × ↓b)

where the closed rays are taken in the partially preordered setA, %+

xy

. Note that %∗ strictly contains %+

xy, because ¬(a%+xy b).

168 A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172

Moreover, by Lemma3.2(iv), the hypothesis¬(b%+xy a) implies that

%∗ is a partial preorder contained in %P . It follows that %∗ is aproper extension of %+

xy in P +xy , which contradicts the maximality

of %+xy. This completes the definition of %+

xy.Finally, we show that the family R = {%h : h ∈ H} defined

above satisfies (RN)–(RP). For (RN), if x%N y, then x%h y for allh ∈ H by the definition of each total preorder in R. To prove theconverse by contrapositive, assume that ¬(x%N y). If ¬(x%P y),then ¬(x%h y) for each h ∈ H = ∅. On the other hand, x%P yimplies ¬(x%−

xy y), hence there exists h ∈ H such that ¬(x%h y)by definition of H . This proves (RN). For (RP), if x%P y, then thetotal preorder %+

xy ∈ R is such that x%+xy y, hence x%h y for some

h ∈ H . Conversely, if ¬(x %P y), then the definition of R impliesthat ¬(x%h y) for each h ∈ H . �

Observe that the given proof depends on the Axiom of Choiceonly in the case that A has uncountable cardinality, whereas itholds truewith no additional set-theoretic assumptions in the casethat A is (finitely or infinitely) countable.

Definition 3.5. Let % be a preorder on a set A, and %′ anextension of % . A preference resolution of

%, %′

is a family R of

distinct total preorders on A that satisfies conditions (RN)–(RP)as in Theorem 3.4. Denote by Res

%, %′

the collection of all

preference resolutions of%, %′

ordered by set-inclusion. Then

Res%, %′

, ⊆

is called the resolution poset of

%, %′

.

By Theorem 3.4,Res

%, %′

, ⊆

is nonempty if and only if

%, %′is a NaP-preference.

Remark 3.6. A NaP-preference on A is totally stable if and only if ithas a resolution containing only one element. In fact, if % is a totalpreorder on A, then the singleton {%} is the unique resolution ofthe totally stable NaP-preference (%, %).

We end this section with a discussion about the existence ofmaximal andminimal resolutions for a NaP-preference. Recall thata poset is a join-semilattice (respectively, ameet-semilattice) if everydoubleton has a least upper bound (respectively, a greatest lowerbound).

Lemma 3.7. The resolution poset of a NaP-preference is a join-semilattice with a maximum element.

Proof. Let%N , %P

be a NaP-preference on A. It is easy to check

that for each nonempty subcollection Res′ ⊆ Res%N , %P

,

the family R′:=

Res′ of total preorders on A satisfies

properties (RN)–(RP) as in Theorem 3.4. Therefore, R′ is aresolution of

%N , %P

aswell. This implies that

Res

%N , %P

, ⊆

is a join-semilattice, whose unique maximal element is M :=

Res%N , %P

. �

As a consequence of Lemma 3.7, we can restate Theorem 3.4 asfollows.

Corollary 3.8. Let % be a partial preorder on A and %′ an extensionof %. The pair

%, %′

is a NaP-preference on A if and only if it has a

unique maximal resolution.

All NaP-preferences on a finite set A obviously have bothmaximal and minimal resolutions. On the other hand, NaP-preferences on an infinite set A have a (unique)maximal resolutionby Corollary 3.8, but they may lack a minimal resolution.

Example 3.9. Let %N:= {(x, y) ∈ R2

: x ≥ y} be the usual totalorder ≥ on R. Define two extensions %

P1 and %

P2 of %N as follows:

%P1 := %

N∪ {(x, y) ∈ R2

: |x − y| ≤ 1} and

%P2 := %

N∪ {(x, y) ∈ R2

: |x − y| < 1}.

One can show that both%N , %

P1

and

%N , %

P1

are NaP-

preferences on R, but they differ in the existence of a minimalresolution, since the first has one whereas the second hasnone (Giarlotta and Watson, 2012). Note that the two possiblepreferences %

P1 and %

P2 are typical semiorders on R, having 1 as a

‘‘discrimination threshold’’.

It is an open problem to characterize minimally resolvable NaP-preferences, i.e. those that possess a minimal resolution (Giarlottaand Watson, 2012). In this respect a topological approach mightbe appropriate, since it is well known (Steiner, 1966) that thereexists a complete lattice isomorphism between the lattice of theAlexandroff–Tucker topologies on a set A and the lattice of partialpreorders on A. (A topology is said to be Alexandroff–Tucker ifarbitrary intersections of open sets are open.)

4. Modal utility representations of NaP-preferences

The analysis of the representability of preference relations(e.g. preorders, interval orders, semiorders, etc.) is ubiquitous indecision theory. Recall that a preorder % on a set A is representable(inR) if there exists a function u: A → R such that for each x, y ∈ A,we have x % y if and only if u(x) ≥ u(y); the map u is called autility representation of %. It is well known that a total preorder% on A is representable (hence continuously representable dueto Debreu’s Open Gap Lemma (Debreu, 1954)) if and only if it isDebreu-separable, i.e. there exists a countable subset Z of A withthe property that if x ≻ y then x % z % y for some z ∈ Z .

A recent stream of research concentrates on the analysis ofthe continuous and semicontinuous representability of all partialpreorders. Obviously an incomplete preorder admits no classicalrepresentation by means of a single utility function, hence twoalternative approaches to the topic have been identified (Evren andOk, 2011), using either (i) a single utility function in aweaker form,or (ii) a family of utility functions as a universal witness.

In the first approach, a partial preorder % on a set A isrepresentable if there exists a map u: A → R such that for eachx, y ∈ A, the following two implications hold:

x % y H⇒ u(x) ≥ u(y) and x ≻ y H⇒ u(x) > u(y).

In this case, the function u is called a Richter–Peleg representationof %: see Aumann (1962), Peleg (1970), Richter (1966) for theseminal work on the topic, and Bridges and Mehta (1995) fora survey. In the second approach, a partial preorder % on A isrepresentable if there exists a set U of real-valued maps on A suchthat for each x, y ∈ A, the following equivalence holds:

x % y ⇐⇒ ∀u ∈ Uu(x) ≥ u(y)

.

In this case, the family U is a multi-utility representation of %. Thisapproach, which has been investigated in Evren and Ok (2011),Mandler (2006), Ok (2002), is used inmost of the recent preferencemodels that deal with a potentially incomplete preference (see,e.g. Dubra et al., 2004; Ghirardato et al., 2003; Gilboa et al.,2010; Ok et al., 2012). Note that the representability of a singlepreference relation by means of a family of utility functions isuniversal, because every partial preorder admits a multi-utilityrepresentation (Proposition 1 in Evren and Ok, 2011).

The main reason to prefer approach (ii) over (i) is pointedout in Majumdar and Sen (1976), where it is observed that aRichter–Peleg representation determines a loss of informationinsofar as one cannot recover the partial preference from itsrepresentation. In addition, as noted in Evren and Ok (2011), inmost applications the multi-utility representation enables one to

A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172 169

find the maximal elements of a partially preordered set by solvinga multi-objective optimization problem.

Our approach falls into category (ii), since we aim at obtaininga multi-utility representation U of a (possibly partial) NaP-preference. To this aim, below we extend the notion of a multi-utility representation to pairs

%, %′

of binary relations. The type

of representation here introduced witnesses both components ofthe pair

%, %′

simultaneously, yet with a distinct modality: in

fact, the family U of utility functions represents % globally and %′

locally.

Definition 4.1. Let%, %′

be a pair of binary relations on A. A

modal utility representation of%, %′

is a nonempty collection

U =ukh : h ∈ H ∧ k ∈ Kh

of utility functions uk

h: A → R such that the following propertieshold for each x, y ∈ A:

(M1) x % y ⇐⇒ ∀h ∈ H ∀k ∈ Khukh(x) ≥ uk

h(y);

(M2) x%′ y ⇐⇒ ∃h ∈ H ∀k ∈ Khukh(x) ≥ uk

h(y).

If%, %′

admits a modal utility representation U, then it is called

modally representable. In this case, H is the set of modes of U,whereas Kh is the extent of mode h ∈ H . In particular, U is uni-modal if |H| = 1, and simple if |Kh| = 1 for each h ∈ H .

Note that a modal representation U of a pair%, %′

of binary

relations on A is a (special) multi-utility representation of itsfirst component %: in fact, U has the property that for each(x, y) belonging to %′, there exists a mode h ∈ H of themodal representation which witnesses x%′ y with a multi-utilityrepresentation.

Remark 4.2. As we shall see later on (Corollaries 4.4 and 4.6), theexistence of a particular type of modal representation of the pair%, %′

requires that the binary relations % and %′ satisfy suitable

properties. For instance, a simple modal representation U = {u0h :

h ∈ H} of%, %′

satisfies a simplified version of properties (M1)

and (M2). In fact, U is simple if and only if for each x, y ∈ A, wehave:

(U1) x % y ⇐⇒ ∀u ∈ Uu(x) ≥ u(y)

;

(U2) x%′ y ⇐⇒ ∃u ∈ Uu(x) ≥ u(y)

.

Furthermore, a uni-modal representation U = {uk0 : k ∈ K0} of

%, %′does not distinguish between the two components of the

pair, in fact it is equivalent to having a multi-utility representationof the partial preorder % = %′. Finally,

%, %′

has a simple uni-

modal representation U = {u00} if and only if % = %′ is a

representable total preorder.

Every NaP-preference is modally representable. To prove thisfact, firstwe canuse Theorem3.4 to obtain a resolutionR for it, andsuccessively invoke the multi-utility representability of any totalpreorder to derive a modal representation as the disjoint unionof the multi-utility representations associated to each element ofR. In fact, the modal representability of a NaP-preference is aparticular case of the following general result.

Theorem 4.3. A pair%, %′

of binary relations on A is modally

representable if and only if it is a partial NaP-preference.

Proof. For necessity, assume that there exists a nonempty fam-ily U =

ukh : h ∈ H ∧ k ∈ Kh

of utility functions uk

h: A →

R such that for each x, y ∈ A, properties (M1)–(M2) holdfor the pair

%, %′

. We show that

%, %′

satisfies the ax-

ioms (N)–(P)–(NP)–(PN) of a partial NaP-preference. Since U isnonempty, it follows that % is reflexive. To prove transitivity of % ,

assume that x, y, z ∈ A are such that x % y % z. Condition (M1)yields uk

h(x) ≥ ukh(z) for each h ∈ H and k ∈ K , hence x % z. This

proves that (N) holds. For (P), note that sinceU = ∅, it follows thatfor each x, y ∈ A, if x % y then x%′ y, as claimed. To finish the proofof necessity, we show that (NP) holds, since the proof of (PN) issimilar. Toward a contradiction, assume that there exist x, y, z ∈ Asuch that x % y%′ z but ¬(x%′ z). Conditions (M1)–(M2) entail

∀h ∈ H ∀k ∈ Khukh(x) ≥ uk

h(y)

∃h ∈ H ∀k ∈ Khukh(y) ≥ uk

h(z)

∀h ∈ H ∃k ∈ Khukh(z) > uk

h(x)

which implies ukh(z) > uk

h(z) for some ukh ∈ U, a contradiction.

To prove sufficiency, assume that the pair%, %′

is a partial

NaP-preference. Denote by G := %′\ % the gap of

%, %′

, and

set H := {0} ∪ G. For h := 0 ∈ H , let U0 :=uk0 : k ∈ K0

be a

multi-utility representation of %, i.e. a nonempty family of utilityfunctions uk

0: A → R such that for each x, y ∈ A, we have x % yif and only if uk

0(x) ≥ uk0(y) for all k ∈ K0. On the other hand, if

h := (a, b) ∈ G, then Lemma 3.2(ii) yields that the binary relation

%h := % ∪

↑

%a × ↓

%b

is a partial preorder on A contained in %′. For each h ∈ G, letUh :=

ukh : k ∈ Kh

be a multi-utility representation of %h. Set

U :=

h∈H

Uh =ukh : h ∈ H ∧ k ∈ Kh

= ∅.

To finish the proof, we show that U is a modal utility representa-tion of

%, %′

.

Let x, y ∈ A. If x % y, then uk0(x) ≥ uk

0(y) for all k ∈ K0by construction. Furthermore, since each %h is an extension of% and Uh is a multi-utility representation of %h, we also haveukh(x) ≥ uk

h(y) for all h ∈ G and k ∈ Kh. Conversely, the inclusionU0 ⊆ U yields x % y. This shows that (M1) holds. For (M2),assume that x%′ y. It follows that there exists h ∈ H such thatukh(x) ≥ uk

h(y) for each k ∈ Kh, namely, h := 0 if x % y, andh := (x, y) if ¬(x % y). To prove the converse by contrapositive,assume that ¬(x%′ y). Since ¬(x % y), there exists uk

0 in K0 suchthat uk

0(y) > uk0(x). Furthermore, if h is an element of G, then we

have ¬(x%h y), because h = (x, y) and %h ⊆ %′ by construction.Therefore, for each h ∈ G, there exists k ∈ Kh such that uk

h(y) >

ukh(x). Summarizing, we obtain that for each h ∈ H , there exists

k ∈ K such that ukh(y) > uk

h(x). This proves the claim. �

The following are limit cases of Theorem 4.3:

Corollary 4.4. Let%, %′

be a pair of binary relations on A.We have:

(i)%, %′

has a uni-modal representation if and only if % = %′ is a

partial preorder;(ii)

%, %′

has a simple uni-modal representation if and only if % =

%′ is a representable total preorder.

Similarly to what has been done in Evren and Ok (2011) andOk (2002) for partial preorders, a natural problem is to determineunder which conditions a (partial or total) NaP-preference has afinite or a countablemodal representation. In this respect, it wouldbe useful to determine the structure of those NaP-preferences thatpossess a minimal resolution and a resolution of minimum size.

We end this section with a characterization of those NaP-preferences that admit a simple modal representation.

Definition 4.5. A pair%, %′

of binary relations on A is order-

separable if it has a resolution whose elements are order-separabletotal preorders.

170 A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172

By Theorem 3.4, a necessary condition for%, %′

to be order-

separable is that it is a NaP-preference on A. Note a binary relation% on A is an order-separable total preorder if and only if the pair(%, %) is an order-separable totally stable NaP-preference.

Corollary 4.6. The following statements are equivalent for a pair%N , %P

of binary relations on A:

(i)%N , %P

is an order-separable NaP-preference on A;

(ii)%N , %P

has a simple modal representation.

In particular, a NaP-preference on a countable set has a simple modalrepresentation.

Proof. (i)⇒ (ii). Assume that%N , %P

is an order-separable NaP-

preference on A. Thus, there exists a nonempty family R :=%h : h ∈ H

of distinct order-separable total preorders on A such

that

R = %N and

R = %P . For each h ∈ H , let uh: A → Rbe a utility function that represents %h. Set U := {uh : h ∈ H}.For each x, y ∈ A, the nonempty family U satisfies properties(U1)–(U2) in Remark 4.2. It follows that

%N , %P

has a simple

modal representation.(ii) ⇒ (i). Assume that

%N , %P

has a simple modal represen-

tation. Thus, there exists a nonempty family U := {uh : h ∈ H} ofutility functions on A such that properties (U1)–(U2) in Remark 4.2hold. For each h ∈ H , let %h be the binary relation on A definedby x%h y if and only if uh(x) ≥ uh(y) for each x, y ∈ A. Sinceeach binary relation %h is a representable total preorder, it fol-lows that (after elimination of possible duplications) the nonemptyfamily R :=

%h : h ∈ H

is an order-separable resolution of

%N , %P. �

5. The NaP-preference structure of a set

As discussed in Section 2, the information summarized bya 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 toassume that a larger amount of both types of information inducesa higher stability of a NaP-preference, i.e. the smaller its gap %G is,themore stable the NaP-preference is. In an attempt to provide thefamily of all NaP-preferences on Awith an order relation based onstability, a possible approach might be to use reverse inclusions ofgaps, thus regarding the NaP-preference

%N

2, %P

2

as being ‘‘more

stable’’ than the NaP-preference%N

1, %P

1

whenever %G

2⊆ %G

1.

Nevertheless, this order relation is definitively too rich, since itannihilates some relevant pieces of preferential information; infact, it allows a ‘‘compensation’’ between the positive part %N andthe negative part %I , which is unlikely to hold in concrete decisionproblems. Thus, in the next definition we endow the family of allNaP-preferences on A with a less rich partial order, which is stillbased on the total amount of (positive and negative) information,yet allows no compensation at all.

Definition 5.1. Denote byNaP(A) the family of all NaP-preferenceson a set A. For each

%N

1, %P

1

and

%N

2, %P

2

in NaP(A), we say

that%N

2, %P

2

is a refinement of

%N

1, %P

1

, and we denote it by

%N2, %P

2

⊒

%N

1, %P

1

, if the following two conditions hold:

(R1) %N2

⊇ %N1;

(R2) %P2

⊆ %P1

(i.e. %I2

⊇ %I1).

We call the poset (NaP(A), ⊒) the NaP-preference ordered structureof A.

The NaP-preference ordered structure (NaP(A), ⊒) is rathercomplicated, even when A is quite small. The next exampledescribes the simplest (non-trivial) NaP-preference structure.

Example 5.2. In Fig. 1 we gave a graphical representation of thesix admissible relations between two alternatives of Awhenever aNaP-preference is defined on A. The set of these relations can alsobe seen as the NaP-preference structure on a two-element set A =

{x, y}, whose partial order ⊒ is represented by the dashed lines inFig. 3. Note that the three NaP-preferences at the top are totallystable, whereas the unique NaP-preference at the bottom is totallyunstable. Further, observe that had we used the order relationon NaP(A) based on the reverse inclusion of gaps, then the threetotally stable NaP-preference at the top of Fig. 3 would have beenindifferent, thus annihilating their apparently distinctive features.

Next, we list some properties of the poset (NaP(A), ⊒).

Lemma 5.3. For each nonempty family F :=

%Ni , %P

i

: i ∈ I

of NaP-preferences on A, the pair

i∈I %N

i ,

i∈I %Pi

is a NaP-

preference on A.

Proof. Given a nonempty family F :=

%Ni , %P

i

: i ∈ I

⊆

NaP(A), set %N:=

i∈I %N

i and %P:=

i∈I %P

i . It is immediateto check that

%N , %P

satisfies axioms (N)–(P). Further, if x, y ∈ A

are such that¬(x%Pi y) for each i ∈ I , then property (C) of

%N

i , %Pi

yields y%N

i x for each i ∈ I . This proves that%N , %P

satisfies (C) as

well. To show that (NP) holds for it, assume that x, y, z ∈ A are suchthat x%N

i y for each i ∈ I and y%Pk z for some k ∈ I . In particular, we

have x%Nk y%P

k z, hence x%Pk z by property (NP) of

%N

k , %Pk

. This

proves (NP). Property (PN) can be verified in a similar way. �

Lemma 5.4. The poset (NaP(A), ⊒) satisfies the following proper-ties:

(i) its minimum element is∆(A), A2

;

(ii) its maximal elements are (%, %), where % is any total preorderon A;

(iii) each nonempty subfamily F ⊆ NaP(A) has a meet.

Therefore, (NaP(A), ⊒) is a meet-semilattice with a unique minimalelement.

Proof. Parts (i) and (ii) are an immediate consequence of thedefinition of the partial order ⊒, whereas part (iii) follows fromLemma 5.3. �

For each fixed NaP-preference%N , %P

on A, we can consider

the family C%N , %P

of all maximal chains (i.e. linearly ordered

sets) of (NaP(A), ⊒) containing%N , %P

. The initial segments of

the elements inC%N , %P

having

%N , %P

as theirmaximum are

all admissible ‘‘histories’’ of%N , %P

, whereas the final segments

that start at%N , %P

are its admissible ‘‘futures’’. Taking this point

of view, each maximal chain in (NaP(A), ⊒) is a dynamic processin the universe of NaP-preferences: everything is potential at thebeginning (i.e. at the minimum NaP-preference) and becomeseffective in the end (i.e. at a maximal NaP-preference). All stagesof this process (i.e. the intermediate NaP-preferences) are theadmissible configurations that lead to a total stabilization of thepreferential information. In accordance with this interpretation,next we define the notion of a graded NaP-system on a fixed setA of alternatives. (As usual, the notation

%N

i2, %P

i2

A

%N

i1, %P

i1

stands for

%N

i2, %P

i2

⊒

%N

i1, %P

i1

and

%N

i2, %P

i2

=

%N

i1, %P

i1

.)

Definition 5.5. A NaP-system on A is a nonempty family S =%N

i , %Pi

: i ∈ I

of NaP-preferences on A indexed on a linearly

ordered set (I, D) such that the following property holds for eachi1, i2 ∈ I:

(S1) i2 ◃ i1 implies%N

i2, %P

i2

A

%N

i1, %P

i1

.

A. Giarlotta, S. Greco / Journal of Mathematical Economics 49 (2013) 163–172 171

Fig. 3. The NaP-preference structure on A = {x, y}.

A NaP-system S =

%Ni , %P

i

: i ∈ I

is graded if there exists

a function c: S → [0, 1] assigning to each NaP-preference%N

i , %Pi

∈ S its credibility grade ci ∈ [0, 1] in a way such that

the following additional property holds for each i1, i2 ∈ I:

(S2) i2 ◃ i1 implies ci2 < ci1 .

Of course, a NaP-system S on A is nothing but a chain inNaP(A). Therefore, in the case of a single-agent decision, its naturaleconomic interpretation is that of a family of NaP-preferences ona set A of alternatives, which models a monotonic evolution of thepreferences of a decision maker with respect to a suitable variable(e.g. states of nature, time, etc.). For example, in the case that theindex set I represents the time of a decision, onemight require that(I, D) is a well-ordered set, whose minimum is the present time.

By definition, a NaP-system S is graded if it is endowed witha map assigning a credibility value to each of its elements in away such that each NaP-preference in S has a smaller credibilitythan its predecessors. The fact that the informative content of eachNaP-preference in S increases as one proceeds upwards in thesystem justifies axiom (S2). Note that the idea of a graded NaP-system has already been implemented in some applications thatuse the ROR approach: see Greco et al. (2008), where nested sets ofutility functions represent different degrees of credibility of NaP-preferences.

We conclude this section with a characterization of a NaP-system. In the next result, the maximal resolution of a NaP-preference

%N

i , %Pi

on A – whose existence and uniqueness is

guaranteed by Corollary 3.8 – is denoted by Mi.

Proposition 5.6. Let (I, D) be a linearly ordered set, S :=%N

i , %Pi

: i ∈ I

a family of NaP-preferences on A, and M := {Mi :

i ∈ I} the collection of all maximal resolutions of the elements of S.The following statements are equivalent:

(a) S is a NaP-system;(b) M is a nested strictly decreasing family (i.e. i2 ◃ i1 implies

Mi2 ⊂ Mi1 ).

In particular, if (I, D) is order-separable, then S is a graded NaP-system if and only if (b) holds.

Proof. (a) ⇒ (b). Assume that S is a NaP-system on A. Let i1, i2 ∈ Ibe such that i2 ◃ i1. The definition of NaP-system implies that oneof the following cases holds:

(1) %Ni1

⊆ %Ni2

and %Pi2

⊂ %Pi1;

(2) %Ni1

⊂ %Ni2

and %Pi2

⊆ %Pi1.

Let %hi2

be an element of Mi2 := {%hi2

: h ∈ Hi2}. Since Mi2 is a

resolution of%N

i2, %P

i2

, we have

Mi2 = %N

i2and

Mi2 = %P

i2,

hence the following chain of inclusions holds:

%Ni1 ⊆ %

Ni2 ⊆ %h

i2 ⊆ %Pi2 ⊆ %

Pi1 .

Maximality of the resolution Mi1 of%N

i1, %P

i1

yields that %h

i2∈

Mi1 . This shows that the inclusion Mi2 ⊆ Mi1 holds. Furthermore,since we have

Mi2 =

Mi1 in case (1), and

Mi2 =

Mi1 in

case (2), it follows that the inclusion is strict.(b) ⇒ (a). Assume that M is a nested strictly decreasing family.

Let i1, i2 ∈ I be such that i2 ◃ i1. By hypothesis, the maximalresolutions Mi1 and Mi2 of, respectively,

%N

i1, %P

i1

and

%N

i2, %P

i2

are such that Mi2 ⊂ Mi1 . Therefore, we have

%Ni1 =

Mi1 ⊆

Mi2 = %

Ni2 and

%Pi2=

Mi2 ⊆

Mi1 = %

Pi1

hence%N

i2, %P

i2

is a refinement of

%N

i1, %P

i1

. Next, we use the

maximality of the resolutions to prove that the refinement is strict.The hypothesis yields the existence of a total preorder %k

i1on A

such that %ki1

∈ Mi1\Mi2 . By the maximality of Mi2 , we haveeither (1) %N

i2⊆ %k

i1, or (2) %k

i1⊆ %P

i2. Since %N

i1⊆ %k

i1⊆ %N

i1,

it follows that %Ni1

⊂ %Ni2

in case (1), and %Pi2

⊂ %Pi1

in case (2).

In both cases, we obtain%N

i2, %P

i2

A

%N

i1, %P

i1

. This shows that

property (S1) in Definition 5.5 holds for each i1, i2 ∈ I , and theproof is complete. �

Acknowledgments

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

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