Denis Bouyssou

Thierry Marchant

Multiattribute preference models with reference points $

Keywords:

In the context of multiple attribute decision making, preference models making use of reference points in an ordinal way have recently been introduced in the literature. This text proposes an axiomatic analysis of such models, with a particular emphasis on the case in which there is only one reference point. Our analysis uses a general conjoint measurement model resting on the study of traces induced on attributes by the preference relation and using conditions guaranteeing that these traces are complete. Models using reference points are shown to be a particular case of this general model. The number of reference points is linked to the number of equivalence classes distinguished by the traces. When there is only one reference point, the induced traces are quite rough, distinguishing at most two distinct equivalence classes. We study the relation between the model using a single reference point and other preference models proposed in the literature.

Introduction

In a series of papers, [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008[START_REF] Rolland | Reference-based preferences aggregation procedures in multicriteria decision making[END_REF] (see also [START_REF] Perny | Reference-dependent qualtitative models for decision making under uncertainty[END_REF], in the related context of decision making under uncertainty) has suggested to use reference points 1 in an ordinal way to build $ Authors are listed alphabetically. They have equally contributed to this paper. We wish to thank Patrice Perny and Antoine Rolland for their very helpful comments on an earlier draft of this text. The usual caveat applies.

Email addresses: bouyssou@lamsade.dauphine.fr (Denis Bouyssou), thierry.marchant@UGent.be (Thierry Marchant) preference models for multiattributed alternatives. This idea can be traced back to Fargier and Perny (1999) and Dubois, Fargier, and Perny (2003, p. 247). In these models, the preference between alternatives x and y rests on a comparison in terms of "importance" of the sets of attributes for which x and y are above the reference points. Rolland has analyzed the interest of such models and has proposed axioms that could characterize them. Most of his axiomatic analysis supposes that the reference points are known beforehand2 . Including reference points in the primitives of the model is a strong hypothesis and raises observational questions. Moreover, he invokes conditions that seem to be quite specific to models using reference points. It is therefore not easy to use them in order to compare these models with other ones that have been proposed and characterized in the literature.

The aim of this text is to propose an axiomatic analysis of preference models with reference points using the traditional primitives of conjoint measurement, i.e., a preference relation on the set of alternatives. Our analysis uses a general conjoint measurement model resting on the study of traces induced on attributes by the preference relation and using conditions guaranteeing that these traces are complete. We show that preference models with reference points are a particular case of this general model. This will allow us to characterize preference models with reference points using conditions that will facilitate their comparison with other preference models proposed in the literature. We concentrate on preference models that use a single reference point. These models induce traces on each attribute that are quite rough, distinguishing at most two distinct equivalence classes. These models are then compared with other types of preference models introduced in the literature, most notably, with models based on a discrete Sugeno integral (Bouyssou, Marchant, andPirlot 2009, Greco, Matarazzo, and[START_REF] Greco | Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules[END_REF] and models based on the notion of concordance (Bouyssou andPirlot 2005b, 2007).

Our general strategy will be similar to the one used in Bouyssou andPir-lot (2005b, 2007) to analyze models based on the notion of concordance.

They have shown that such models could be seen as particular cases of the general conjoint measurement models developed in Bouyssou andPirlot (2002, 2004a) that generate complete traces on differences between levels in which these traces are "rough", i.e., only distinguishing a limited number of equivalence classes. We show here that models using reference points are a particular case of models inducing complete traces on levels developed in Bouyssou and Pirlot (2004b) in which these traces are "rough" (for a general overview of preference models based on different kinds of traces, we refer to Bouyssou and Pirlot 2005a).

The paper is organized as follows. Section 2 introduces our notation and setting. Section 3 formalizes preference models using a single reference point. Section 4 recalls the main ingredients of the general conjoint measurement models introduced in Bouyssou and Pirlot (2004b). Section 5 characterizes preference models using a single reference point. Section 6 studies the relations between concordance relations and preference models using a single reference point. Section 7 is devoted to the study of preference models using a single reference point that are weak orders. A final section discusses our findings. Appendix D extends our results to preference models using several reference points.

Background

Binary relations

A binary relation K on a set A is a subset of A × A. We often write a K b instead of (a, b) ∈ K. The symmetric (resp. asymmetric) part of K is the binary relation σ(K) (resp. α(K)) on A such that a σ(K) b iff A weak order is a complete (a K b or b K a) and transitive binary relation. When K is a weak order on A, it is clear that σ(K) is an equivalence. We often abuse terminology and speak of equivalence classes of the weak order K instead of the equivalence classes of σ(K). In this case we also speak of the first, second, . . . , last equivalence class of K. [START_REF] Aleskerov | Utility maximization, choice and preference[END_REF] that the relation

K • defined letting, for all a, b, c ∈ A, a K • b ⇔ b K c ⇒ a K c, c K a ⇒ c K b,
is a weak order.

Notation

In this paper will always denote a binary relation on a set X = n i=1 X i with n ≥ 2. Elements of X will be interpreted as alternatives evaluated on a set N = {1, 2, . . . , n} of attributes and as an "at least as good as" relation between these alternatives. We denote by (resp. ∼) the asymmetric (resp. symmetric) part of . A similar convention holds when is starred, superscripted and/or subscripted.

For any nonempty subset J of the set of attributes N , we denote by X J (resp. X -J ) the set i∈J X i (resp. i∈N \J X i ). When x, y ∈ X, with customary abuse of notation, (x J , y -J ) will denote the element w ∈ X such that w i = x i if i ∈ J and w i = y i otherwise. We sometimes omit braces around sets. For instance, when J = {i} we write X -i and (x i , y -i ).

We say that attribute i ∈ N is influential (for ) if there are x i , y i , z i , w i ∈ X i and a -i , b -i ∈ X -i such that (x i , a -i ) (y i , b -i ) and (z i , a -i ) (w i , b -i ) and degenerate otherwise. A degenerate attribute has no influence whatsoever on the comparison of the elements of X and may be suppressed from N . As in Bouyssou and Pirlot (2005b), in order to avoid unnecessary minor complications, we suppose henceforth that all attributes in N are influential.

Let J ⊂ N be a proper nonempty subset of attributes. We say that is independent for J if, for all x J , y J ∈ X J ,

(x J , z -J ) (y J , z -J ), for some z -J ∈ X -J ⇒ (x J , w -J ) (y J , w -J ), for all w -J ∈ X -J .
If is independent for all proper nonempty subsets of N , we say that is independent. It is clear that is independent iff is independent for N \{i}, for all i ∈ N .

Preference models with a single reference point

The model that we study was introduced by [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008,?),?). It has close connections with ELECTRE TRI (Roy and Bouyssou 1993, Chap. 6). Remember that ELECTRE TRI is a technique used to assign alternatives to ordered categories. Suppose that there are only two categories: A and U , A being the best category. The limit between these two categories is indicated by a profile p that is at the same time the lower limit of A and the upper limit of U . In the pessimistic version of ELECTRE TRI, an alternative x ∈ X belongs to category A iff this alternative is declared at least as good as p. The central originality of ELECTRE TRI lies in the definition of this "at least as good as" relation that is based on the notions of concordance and non-discordance. Ignoring here the nondiscordance condition, an alternative x ∈ X is "at least as good as" the profile p if a "sufficient majority" of attributes support this assertion. When preference and indifference thresholds are equal, this is done as follows. A semiorder T i is defined on each attribute. The set of attributes supporting the proposition that x ∈ X is at least as good as p is simply T (x) = {i ∈ N : x i T i p i }. A positive weight w i is assigned to each attribute. These weights are supposed to be normalized so that n i=1 w i = 1. The test for deciding whether the subset of attributes T (x) is "sufficiently important" is done comparing i∈T (x) w i to a majority threshold λ ∈ [0.5, 1]. We have:

x ∈ A ⇔ i∈T (x) w i ≥ λ.
Ordered partitions A , U of this type have been studied and characterized in Bouyssou and Marchant (2007a). For the sequel, it will be useful to note that the concordance condition for testing if x is "at least as good as" p only distinguishes two kind of attributes: the ones for which x i T i p i and the ones for which this is not true. It does not make further distinctions among the attributes and, in particular, does not make use of the preference difference between x i and p i . Hence, the assignment of an alternative mainly rests on "ordinal considerations".

The model defined below uses similar principles. The main difference with ELECTRE TRI is that the aim of this model is to compare alternatives rather than assigning alternatives to ordered categories. In this model, there is a semiorder S i on each attribute. In order to compare the alternatives

x and y, we first compare each of them to a "reference point" π only using "ordinal considerations". Hence, we compute the subsets of attributes S(x) = {i ∈ N : x i S i π i } and S(y) = {i ∈ N : y i S i π i }. The comparison of x and y is based on the comparison of the subsets S(x) and S(y). This comparison uses an "importance relation" that is only required to be monotonic w.r.t. inclusion. The following definition, inspired by [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008,?),?), formalizes this idea. • for all i ∈ N , there is a semiorder S i on X i (with symmetric part I i and asymmetric part P i ),

• there is an element π ∈ X,

• there is a binary relation on 2 N that is monotonic w.r.t. inclusion, i.e., for all A, B, C, D ⊆ N , where S(x) = {i ∈ N :

A B ⇒ C D,
x i S i π i }.
The above model uses three parameters: the reference point π, the semiorders S i , i = 1, 2, . . . , n, and the importance relation .

Although this presentation allows to easily grasp the intuition of the model, it is possible to reformulate it using less parameters. Indeed, define, for all i ∈ N ,

A i = {x i ∈ X i : x i S i π i }. It is clear that, for all x ∈ X, we have S(x) = {i ∈ N : x i ∈ A i }.
Hence, we can alternatively write the above model only using the following parameters: the sets A i , i = 1, 2, . . . , n, and the importance relation .

In this reformulation, alternative x is at least as good as alternative y if the subset of attributes for which x has an evaluation that is "acceptable" (i.e., attributes i such that x i ∈ A i ) is "more important" (according to the relation ) than the subset of attributes for which y has an evaluation that is acceptable.

In Appendix D, we generalize this model to include multiple reference points.

The following lemma shows that a RSRP has a unique representation in terms of the sets A i and the importance relation . It makes use of the fact that all attributes have been supposed influential.

Lemma 1. A RSRP has a unique representation in terms of the sets A i , i ∈ N , and the relation . In this representation we have, for all i ∈ N ,

∅ A i X i .
Proof. Suppose that is a RSRP having two distinct representations, i.e., that we have

x y ⇔ S(x) S(y) ⇔ S (x) S (y),

where S(x) = {i ∈ N :

x i ∈ A i } and S (x) = {i ∈ N : x i ∈ A i }.
Suppose that for some e i ∈ X i , we have e i ∈ A i and e i /

∈ A i . Because attribute i ∈ N is influential, there are x i , y i , z i , w i ∈ X i and a -i , b -i ∈ X -i such that (x i , a -i ) (y i , b -i ) and (z i , a -i ) (w i , b -i ).
Because e i ∈ A i and (x i , a -i ) (y i , b -i ), we must have (e i , a -i ) (y i , b -i ). Similarly, because e i / ∈ A i and (z i , a -i ) (w i , b -i ), we must have (e i , a -i ) (w i , b -i ).

Suppose that y i ∈ A i . Using (e i , a -i ) (y i , b -i ) and the monotonicity of , we obtain (e i , a -i ) (s i , b -i ), for all s i ∈ X i . This is contradictory since we know that (e i , a -i ) (w i , b -i ).

Suppose that y i ∈ A i . Using (e i , a -i ) (y i , b -i ) and the monotonicity of , we obtain (e i , a -i ) (s i , b -i ), for all s i ∈ X i . This is contradictory since we know that (e i , a -i ) (w i , b -i ).

Hence,

y i / ∈ A i and y i / ∈ A i . Since we have (e i , a -i ) (y i , b -i ) and (e i , a -i ) (w i , b -i ), we must have w i ∈ A i and w i ∈ A i . Because (e i , a -i ) (y i , b -i ), y i / ∈ A i and e i / ∈ A i , we obtain (e i , a -i ) (e i , b -i ). Similarly, because (e i , a -i ) (w i , b -i ), w i ∈ A i and e i ∈ A i , we obtain (e i , a -i ) (e i , b -i ), a contradiction.
Hence, we have shown that, for all i ∈ N , we have A i = A i . It is easy to show that this implies = .

Because attribute i ∈ N is influential, there are x i , y i , z i , w i ∈ X i and

a -i , b -i ∈ X -i such that (x i , a -i ) (y i , b -i ) and (z i , a -i ) (w i , b -i ). If A i = ∅ or A i = X i , we have S((x i , a -i )) = S((z i , a -i )) and S((y i , b -i )) = S((w i , b -i )). Because is a RSRP, this implies (x i , a -i ) (y i , b -i ) ⇔ (z i , a -i ) (w i , b -i ), a contradiction.
Lemma 2. Let be RSRP with representation A i , i ∈ N , and .

1.

is reflexive iff is reflexive, 2.

is complete iff is complete, 3.

is transitive iff is transitive, 4.

is independent iff, for all i ∈ N and all A, B ⊆ N such that i / ∈ A and i / ∈ B, A B iff A ∪ {i} B ∪ {i}.

Proof. We know that has a unique representation in which, for all i ∈ N , ∅ A i X i . Hence, for all A, B ⊆ N , there are x, y ∈ X such that S(x) = A and S(y) = B.

Part 1. Let x ∈ X. Since is reflexive, we know that S(x) S(x), so that x

x. Hence, is reflexive. Conversely, let A ⊆ N . We have S(x) = A, for some x ∈ X. Since x x, we obtain A A. Hence, is reflexive.

The proof of Parts 2 and 3 is similar. Part 4. Suppose that is not independent so that we have (x i , a -i ) (x i , b -i ) and (y i , a -i ) (y i , b -i ), for some i ∈ N , x i , y i ∈ X i , and a, b ∈ X. Since is a RSRP, it is impossible that x i , y i ∈ A i or x i , y i / ∈ A i . Letting A = {j ∈ N \ {i} : a j ∈ A j } and B = {j ∈ N \ {i} : b j ∈ A j }, we obtain either A B and A ∪ {i} B ∪ {i} or A B and A ∪ {i} B ∪ {i}.

Conversely, suppose that we have A B and A ∪ {i}

B ∪ {i}. Let a, b ∈ X be such that A = {j ∈ N \ {i} : a j ∈ A j } and B = {j ∈ N \ {i} : b j ∈ A j }. Take x i ∈ A i and y i / ∈ A i . We obtain (x i , a -i ) (x i , b -i ) and (y i , a -i ) (y i , b -i ). The case A B and A ∪ {i} B ∪ {i} is similar. A capacity on N is a real valued function µ on 2 N such that, for all A, B ∈ 2 N , A ⊇ B ⇒ µ(A) ≥ µ(B). The capacity µ on N is normalized if, furthermore, µ(∅) = 0 and µ(N ) = 1.
Suppose that is a RSRP that is a weak order. It follows from the above lemma that must be a weak order. Moreover, since is monotonic w.r.t. inclusion and N is finite, there is a normalized capacity µ on N such that A B ⇔ µ(A) ≥ µ(B). Hence, we have x y ⇔ µ(S(x)) ≥ µ(S(y)). This shows that a RSRP being a weak order has at most 2 n distinct equivalence classes.

Models using traces on levels

Our central tool for the analysis of RSRP will be the models introduced in Bouyssou and Pirlot (2004b) that induce complete traces on levels on each attribute (see also [START_REF] Greco | Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules[END_REF]). We recall here the essential elements of these models.

Definition 2. Let be a binary relation on a set X = n i=1 X i . We define the binary relations + i , - i and ± i on X i letting, for all x i , y i ∈ X i ,

x i + i y i ⇔ ∀a, b ∈ X, [(y i , a -i ) b ⇒ (x i , a -i ) b], x i - i y i ⇔ ∀a, b ∈ X, [a (x i , b -i ) ⇒ a (y i , b -i )], x i ± i y i ⇔ [x i + i y i and x i - i y i ].
The relations + i , - i and ± i are traces on the levels of attribute i ∈ N generated by the relation . It is easy to check that these relations are always reflexive and transitive.

As shown below, the traces on levels combine nicely with the relation .

Lemma 3 (Bouyssou and Pirlot 2004b, Lemma 2). For all i ∈ N and x, y, z, w ∈ X:

[x y, z i + i x i ] ⇒ (z i , x -i ) y, (1)
[x y, y i - i w i ] ⇒ x (w i , y -i ), (2)
[z i ± i x i , y i ± i w i ] ⇒    x y ⇒ (z i , x -i ) (w i , y -i ), and x y ⇒ (z i , x -i ) (w i , y -i ), (3)
[x i ∼ ± i z i , y i ∼ ± i w i for all i ∈ N ] ⇒    x y ⇔ z w, and x y ⇔ z w. (4)
The following conditions will imply the completeness of marginal traces on levels.

Definition 3. We say that satisfies:

AC 1 i if (x i , a -i ) c and (y i , b -i ) d    ⇒    (y i , a -i ) c or (x i , b -i ) d, AC 2 i if c (y i , a -i ) and d (x i , b -i )    ⇒    c (x i , a -i ) or d (y i , b -i ), AC 3 i if (x i , a -i ) c and d (x i , b -i )    ⇒    (y i , a -i ) c or d (y i , b -i ),
for all a, b, c, d ∈ X and all x i , y i ∈ X i . We say that satisfies AC 1

(resp. AC 2, AC 3) if it satisfies AC 1 i (resp. AC 2 i , AC 3 i ) for all i ∈ N .
Lemma 4 (Bouyssou and Pirlot 2004b, Lemma 3). We have:

1. AC 1 i ⇔ + i is complete ⇔ [Not[y i + i x i ] ⇒ x i + i y i ]. 2. AC 2 i ⇔ - i is complete ⇔ [Not[y i - i x i ] ⇒ x i - i y i ]. 3. AC 3 i ⇔ [Not[y i + i x i ] ⇒ x i - i y i ] ⇔ [Not[x i - i y i ] ⇒ y i + i x i ]. 4. [AC 1 i , AC 2 i and AC 3 i ] ⇔ ± i is complete. 5.
In the class of all semiorders on X, AC 1, AC 2 and AC 3 are independent conditions.

The following shows the consequences of having complete traces on each attribute in terms of numerical representation.

Proposition 1 (Bouyssou and Pirlot 2004b, Theorem 2). Let be a binary relation on a set X = n i=1 X i . Suppose that, for all i ∈ N , the set X i /∼ ± i is at most countably infinite. Then there are real-valued functions u i on X i and a real-valued function

F on [ n i=1 u i (X i )] 2 such that, for all x, y ∈ X: x y ⇔ F ([u i (x i )]; [u i (y i )]) ≥ 0, (M)
with F is increasing in its first n arguments and decreasing in its last n arguments iff satisfies AC 1, AC 2 and AC 3.

We refer to Bouyssou and Pirlot (2004b) for the analysis of model (M) in the general case, i.e., when the set X i /∼ ± i can be uncountable (this requires conditions guaranteeing that the relations ± i have a numerical representation). This will not be useful here. Indeed, a characteristic feature of the models studied here is that they generate complete traces ± i that are "rough", only distinguishing a very limited number of distinct equivalence classes.

Let us conclude with a brief analysis of the very particular situation in which is a weak order. Definition 4. Let be a weak order on X. Attribute i ∈ N is said to be weakly separable if (x i , a -i ) (y i , a -i ) for some x i , y i ∈ X i and some

a -i ∈ X -i implies (x i , b -i ) (y i , b -i ), for all b -i ∈ X -i .
The relation is said to be weakly separable if all attributes i ∈ N are weakly separable.

The following lemma show that for weak orders, the three conditions AC 1 i , AC 2 i , and AC 3 i are equivalent. Furthermore, they are equivalent to requiring that attribute i ∈ N is weakly separable.

Lemma 5 (Bouyssou and Pirlot 2004b, Lemma 5). Let be a weak order on X. Conditions AC 1 i , AC 2 i and AC 3 i are equivalent. They hold iff attribute i ∈ N is weakly separable.

When

is a weakly separable weak order, i.e., a weak order satisfying AC 1, AC 2 and AC 3, it is possible to further specify the numerical representation given by model (M). We have:

Proposition 2 (Bouyssou and Pirlot 2004b, Proposition 8). Let be a weak order on a set X = n i=1 X i such that the set X/∼ is at most countably infinite. Then there are real-valued functions u i on X i and a real-valued function U on n i=1 u i (X i ) such that, for all x, y ∈ X:

x y ⇔ U ([u i (x i )]) ≥ U ([u i (y i )]), (U)
with U is nondecreasing in each of its arguments decreasing in its last n arguments iff is weakly separable.

Results

A characterization of RSRP

Our analysis of RSRP is based on the following two lemmas.

Lemma 6. If is a RSRP, then, for all i ∈ N , the relation ± i is a weak order having two distinct equivalence classes.

Proof. Let be a RSRP. Lemma 1 has shown that it has a unique representation using the sets A i and the relation . Using the definition of a RSRP, it is easy to see that if

x i , y i ∈ A i or if x i , y i / ∈ A i , we have x i ∼ ± i y i . Moreover, if x i ∈ A i and y i /
∈ A i , the monotonicity of w.r.t. inclusion implies x i ± i y i . Hence, the relation ± i is a weak order having at most two distinct equivalence classes.

Because we have supposed that each i ∈ N is influential, there are

x i , y i , z i , w i ∈ X i and a, b ∈ X such that (x i , a -i ) (y i , b -i ) and (z i , a -i ) (w i , b -i ). If z i + i x i and y i - i w i , (x i , a -i ) (y i , b -i ) implies (z i , a -i ) (w i , b -i ), a contradiction. Hence we must have either x i + i z i or w i - i y i .
Either case implies that the weak order ± i has at least two distinct equivalence classes.

Remark 1. Lemmas 4 and 6 imply that a RSRP satisfies AC 1, AC 2 and AC 3. Furthermore all sets X i /∼ ± i are finite, having two distinct elements. Hence, if a RSRP also has a representation in (M). It is not difficult to figure out the particular form of model (M) that leads to a RSRP. Indeed, it suffices to consider, on each i ∈ N , a function u i such that, for all x i ∈ X i ,

u i (x i ) = 1 if x i ∈ A i , 0 if x i / ∈ A i .
With the above definition, we clearly have

u i (x i ) ≥ u i (y i ) ⇔ x i ± i y i .
We define the function F letting:

F ([u i (x i )]; [u i (y i )]) = + exp( n i=1 (u i (x i ) -u i (y i ))) if x y, -exp( n i=1 (u i (y i ) -u i (x i ))) otherwise.
Using the definition of the functions u i and the fact that is monotonic w.r.t. inclusion, it is easy to show, using (4), that F is well defined and increasing (resp. decreasing) in its first (resp. last) n arguments.

Lemma 7. If is a relation on X such that, for all i ∈ N , the relation ± i is a weak order having two distinct equivalence classes, then is a RSRP.

Proof. For all i ∈ N , define A i as the set of elements of X i in the first equivalence class of ± i . Define the relation letting, for all A, B ⊆ N , A B if x y, for some x, y ∈ X such that S(x) = A and S(y) = B. We have to show that, for all x, y ∈ X, x y ⇔ S(x) S(y) and that is monotonic w.r.t. inclusion. If x y, the definition of implies that S(x) S(y). Suppose now that S(x) S(y). By construction, this implies that, for some z, w ∈ X we have z w, S(z) = S(x) and S(w) = S(y). This implies that, for all i ∈ N , x i ∼ ± i z i and y i ∼ ± i w i . Using (4), we obtain x y.

It remains to prove that is monotonic w.r.t. inclusion.

Suppose that for some x, y, z, w ∈ X we have S(z) ⊇ S(x), S(y) ⊇ S(w), and S(x) S(w). By construction of the sets A i , S(z) ⊇ S(x) and S(y) ⊇ S(w) imply that, for all i ∈ N , we have z i ± i x i and y i ± i w i . Since S(x) S(y), we have x y. Using (3), we obtain z w, so that S(z) S(w).

In view of the above two lemmas, a characterization of RSRP will be at hand if we impose conditions guaranteeing that all relations ± i are weak orders having two distinct equivalence classes.

Definition 5 (Conditions AC 1 * , AC 2 * , AC 3 * , AC 4 * ). We say that satisfies:

AC 1 * i if (x i , a -i ) c and (y i , b -i ) d    ⇒    (y i , a -i ) c or (z i , b -i ) d, AC 2 * i if c (y i , a -i ) and d (z i , b -i )    ⇒    c (x i , a -i ) or d (y i , b -i ), AC 3 * i if (x i , a -i ) c and d (z i , b -i )    ⇒    (y i , a -i ) c or d (y i , b -i ), AC 4 * i if c (y i , a -i ) and (y i , b -i ) d    ⇒    c (x i , a -i ) or (z i , b -i ) d, for all a, b, c, d ∈ X and all x i , y i , z i ∈ X i . We say that satisfies AC 1 * (resp. AC 2 * , AC 3 * , AC 4 * ) if it satisfies AC 1 * i (resp. AC 2 * i , AC 3 * i , AC 4 * i ) for all i ∈ N .
Let us show that the above four conditions hold for a RSRP.

Lemma 8. If is a RSRP then it satisfies AC 1 * , AC 2 * , AC 3 * , and AC 4 * . Proof. Suppose that AC 1 * i is violated, so that we have (x i , a -i ) c, (y i , b -i ) d, (y i , a -i ) c, and (z i , b -i ) d, for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . Since (x i , a -i ) c and (y i , a -i ) c, we have Not[y i + i x i ]. In a RSRP, this can only happen if y i / ∈ A i . Since y i / ∈ A i , we have S((z i , a -i )) ⊇ S((y i , a -i )). Because (y i , b -i ) d, we know that S((y i , b -i )) S(d).
Using the monotonicity of w.r.t. inclusion, we obtain S((z i , b -i )) S(d), so that (z i , b -i ) d, a contradiction. The proof of the other parts is similar.

The consequences of the above four conditions on the relations + i and i are detailed below. Lemma 9. For all x i , y i , z i ∈ X i ,

1. AC 1 * i ⇔ [Not[y i + i x i ] ⇒ z i + i y i ], 2. AC 2 * i ⇔ [Not[y i - i x i ] ⇒ z i - i y i ], 3. AC 3 * i ⇔ [Not[y i + i x i ] ⇒ z i - i y i ], 4. AC 4 * i ⇔ [Not[y i - i x i ] ⇒ z i + i y i ].
Proof. We prove Part 1, the proof of the other parts being similar. Negating AC

1 * i is equivalent to supposing that (x i , a -i ) c, (y i , b -i ) d, (y i , a -i ) c, (z i , b -i ) d, for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . This is equivalent to saying that we have Not[y i + i x i ] and Not[z i + i y i ].
Combining Lemmas 4 and 9 proves the following.

Lemma 10. 1. AC 1 * i ⇒ AC 1 i , 2. AC 2 * i ⇒ AC 2 i , 3. AC 3 * i ⇒ AC 3 i , 4. AC 4 * i ⇒ AC 3 i . A crucial
Proof. [AC 1 * i , AC 2 * i , AC 3 * i , AC 4 * i ] ⇒ [ ± i
is a weak order having at most two distinct equivalence classes]. Using Lemma 10, we know that AC 1 i , AC 2 i and AC 3 i hold, so that, using Lemma 4, ± i is a weak order. Suppose that we have, for some

x i , y i , z i ∈ X i , x i ± i y i and y i ± i z i . By construction, x i ± i y i implies either [x i + i y i and x i - i y i ] or (5a) [x i + i y i and x i - i y i ]. (5b) Similarly, y i ± i z i implies either [y i + i z i and y i - i z i ] or (5c) [y i + i z i and y i - i z i ]. ( 5d
)
The combination of (5a) and (5c) violates Lemma 9.1 (i.e., Part 1 of Lemma 9) since it implies Not[y i

+ i x i ] and Not[z i + i y i ].
The combination of (5b) and (5d) violates Lemma 9.2 since it implies Not[y i -

i x i ] and Not[z i - i y i ].
The combination of (5a) and (5d) violates Lemma 9.3 since it implies Not[y i

+ i x i ] and Not[z i - i y i ].
Finally, the combination of (5b) and (5c) violates Lemma 9.4 since it implies Not[y i -

i x i ] and Not[z i + i y i ]. [ ± i is a weak order having at most two distinct equivalence classes] ⇒ [AC 1 * i , AC 2 * i , AC 3 * i , AC 4 * i ]. Suppose that (x i , a -i ) c, and (y i , b -i ) d. If (y i , a -i ) c, we know that x i + i y i , so that x i ± i y i . Since ±
i is a weak order having only two distinct equivalence classes, this implies that, for all

z i ∈ X i , z i ± i y i , so that z i + i y i . Hence, (y i , b -i ) d implies (z i , b -i ) d.
This shows that AC 1 * i holds. The proof for the other three conditions is similar.

The proof is completed observing that, since all attributes are influential, all relations ± i have at least two equivalence classes. Our first characterization of RSRP is as follows.

Theorem 1. Let be a binary relation on a set X = n i=1 X i . The relation is a Relation with a Single Reference Point iff it satisfies AC 1 * , AC 2 * , AC 3 * , and AC 4 * .

Proof. Necessity results from Lemma 8. Sufficiency results from Lemmas 7 and 11.

The conditions used in the above theorem are independent, even when attention is restricted to a class of well behaved relations on X.

Lemma 12. In the class of all semiorders on X, conditions AC 1 * , AC 2 * , AC 3 * , and AC 4 * are independent.

Proof. See Appendix A.

An alternative characterization of RSRP

We have seen above a RSRP is characterized by the conjunction of conditions AC 1 * i , AC 2 * i , AC 3 * i and AC 4 * i . A drawback of this result is that it does not explicitly use conditions AC 1 i , AC 2 i and AC 3 i that characterize model (M). We show here how to weaken conditions AC 1 * i , AC 2 * i , AC 3 * i , AC 4 * i so as to make them independent from conditions AC 1 i , AC 2 i and AC 3 i . This will allow us to exactly state what must be added to the conditions characterizing model (M) to obtain RSRP.

Definition 6 (Conditions AC 1 * * , AC 2 * * , AC 3 * * , AC 4 * * ). We say that satisfies:

AC 1 * * i if (x i , a -i ) c and (y i , b -i ) d and (x i , b -i ) d            ⇒    (y i , a -i ) c or (z i , b -i ) d, AC 2 * * i if c (y i , a -i ) and d (z i , b -i ) and c (z i , a -i )            ⇒    c (x i , a -i ) or d (y i , b -i ), AC 3 * * i if (x i , a -i ) c and d (z i , b -i )    ⇒            (y i , a -i ) c or d (y i , b -i ) or d (x i , b -i ), AC 4 * * i if c (y i , a -i ) and (y i , b -i ) d and (x i , b -i ) d            ⇒    c (x i , a -i ) or (z i , b -i ) d,
for all a, b, c, d ∈ X and all x i , y i , z i ∈ X i . We say that satisfies AC 1 * * (resp. AC 2 * * , AC 3 * * , AC 4 * * ) if it satisfies AC 1 * * i (resp. AC 2 * * i , AC 3 * * i , AC 4 * * i ) for all i ∈ N .

Lemma 13. Let be a binary relation on X. We have:

1. satisfies AC 1 * i iff it satisfies AC 1 i and AC 1 * * i , 2.
satisfies AC 2 * i iff it satisfies AC 2 i and AC 2 * * i , 3.

satisfies AC 3 * i iff it satisfies AC 3 i and AC 3 * * i , 4.

satisfies AC 4 * i iff it satisfies AC 3 i and AC 4 * * i .

Proof. Part 1. It is clear that AC 1 * i implies AC 1 * * i (since the latter condition amount to add a premise to the former condition) and AC 1 i (in view of Lemma 10). Let us show that the reverse implication holds. Suppose that AC 1 * i is violated, so that we have:

(x i , a -i ) c, (y i , b -i ) d, (y i , a -i ) c, (z i , b -i )
d, for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . Since AC 1 i holds, we know that + i is a weak order. In view of the fact that (x i , a -i ) c and (y i , a -i ) c, we must have

x i + i y i . Since (y i , b -i ) d, we obtain (x i , b -i ) d. Hence we have: (x i , a -i ) c, (y i , b -i ) d, (x i , b -i ) d, (y i , a -i ) c, (z i , b -i ) d, violating AC 1 * * i . Part 2. It is clear that AC 2 * i implies AC 2 * *
i (since the latter condition amount to add a premise to the former condition) and AC 2 i (in view of Lemma 10). Let us show that the reverse implication holds. Suppose that AC 2 * i is violated, so that we have:

c (y i , a -i ), d (z i , b -i ), c (x i , a -i ), d (y i , b -i ),
for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . Since AC 2 i holds, we know that - i is a weak order. In view of the fact that d

(z i , b -i ) and d (y i , b -i ), we must have y i - i z i . Since c (y i , a -i ), we obtain c (z i , a -i ). Hence we have: c (y i , a -i ), d (z i , b -i ), c (z i , a -i ), c (x i , a -i ), d (y i , b -i ), violating AC 2 * * i . Part 3. It is clear that AC 3 * i implies AC 3 * *
i (since the latter condition amount to add a possible conclusion to the former condition) and AC 3 i (in view of Lemma 10). Let us show that the reverse implication holds. Suppose that AC 3 * i is violated, so that we have:

(x i , a -i ) c, d (z i , b -i ), (y i , a -i ) c, d (y i , b -i ), for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . Since (x i , a -i )
c and (y i , a -i ) c, it is impossible that y i + i x i . Using AC 3 i and Lemma 4 implies that we must have x i i y i . Hence, because d (y i , b -i ), we must have d (x i , b -i ). Hence we have:

(x i , a -i ) c, d (z i , b -i ), (y i , a -i ) c, d (y i , b -i ), d (x i , b -i ), violating AC 3 * * i . Part 4. It is clear that AC 4 *
i implies AC 4 * * i (since the latter condition amount to add a premise to the former condition) and AC 3 i (in view of Lemma 10). Let us show that the reverse implication holds. Suppose that AC 4 * i is violated, so that we have:

c (y i , a -i ), (y i , b -i ) d, c (x i , a -i ), (z i , b -i )
d, for some a, b, c, d ∈ X and some x i , y i , z i ∈ X i . Since c (y i , a -i ) and c (x i , a -i ), it is impossible that y i i x i . Using AC 3 i and Lemma 4 implies that we must have

x i + i y i . Since (y i , b -i ) d, we therefore obtain (x i , b -i ) d. Hence we have: c (y i , a -i ), (y i , b -i ) d, (x i , b -i ) d, c (x i , a -i ), (z i , b -i ) d, violating AC 4 * * i .
Combining the above lemma with Theorem 1 proves the following:

Theorem 2. Let be a binary relation on a set X = n i=1 X i . The relation is a Relation with a Single Reference Point iff it satisfies AC 1, AC 2, AC 3, AC 1 * * , AC 2 * * , AC 3 * * , and AC 4 * * .

The following lemma shows that the conditions used in the above result are independent even when attention is restricted to a class of well behaved relations on X.

Lemma 14. In the class of all semiorders on X, conditions AC 1 i , AC 2 i , AC 3 i , AC 1 * * , AC 2 * * , AC 3 * * , and AC 4 * * are independent.

Proof. See Appendix B.

Relation to concordance relations

In view of Lemma 2.4, it is easy to build examples showing that a RSRP does not have to be independent. It is well known that concordance relations are independent (Bouyssou andPirlot 2005b, 2007). Using this observation, [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006a[START_REF] Rolland | Procédures d'agrégation ordinale de préférences avec points de référence pour l'aide à la décision[END_REF] has concluded that models using reference points were more "flexible" than concordance relations. This section is devoted to a precise analysis of the links between concordance relations and RSRP.

We first recall the definition of a concordance relation (Bouyssou andPirlot 2005b, 2007) when is reflexive (since a concordance relation is independent, it is easy to check that it is either reflexive or irreflexive).

Definition 7. Let be a reflexive binary relation on X. We say that is a concordance relation with attribute transitivity (or, more briefly, that is a CR-AT) if there are:

• a semiorder T i on each X i (i = 1, 2, . . . , n),

• a binary relation C between subsets of N having N for union that is monotonic w.r.t. inclusion, i.e., for all A, B, C, D ⊆ N such that

A ∪ B = N and C ∪ D = N , [A C B, C ⊇ A, B ⊇ D] ⇒ C C D, (6)
such that, for all x, y ∈ X,

x y ⇔ T (x, y) C T (y, x),

where T (x, y) = {i ∈ N :

x i T i y i }.
Hence, when is a CR-AT, the preference between x and y only depends on the subsets of attributes favoring x or y in terms of the semiorder T i . It does not depend on preference differences between the various levels on each attribute besides the distinction between levels indicated by T i .

Definition 8. Let be a binary relation on a set X = n i=1 X i . We define the binary relations * i and * * i on X 2 i letting, for all x i , y i , z i , w i ∈ X i ,

(x i , y i ) * i (z i , w i ) ⇔ [for all a -i , b -i ∈ X -i , (z i , a -i ) (w i , b -i ) ⇒ (x i , a -i ) (y i , b -i )], (x i , y i ) * * i (z i , w i ) ⇔ [(x i , y i ) * i (z i , w i ) and (w i , z i ) * i (y i , x i )].
It is clear that the relations * i and * * i are always reflexive and transitive. Bouyssou andPirlot (2005b, 2007) have shown that CR-AT are reflexive relations that are characterized by the fact that, for all i ∈ N , the relation * * i is a weak order having at most three distinct equivalence classes and ± i is a weak order. They have given necessary and sufficient conditions on ensuring that this happens.

We are now in position to analyze the relations between CR-AT and RSRP. We start with a result showing that there are CR-AT that are not RSRP. This result exploits the fact that in a CR-AT the weak orders ± i may have more than two distinct equivalence classes. This cannot be the case in a RSRP.

Lemma 15. A CR-AT may fail to be a reflexive RSRP.

Proof. This is obvious since a CR-AT does not impose a limit on the number of equivalence classes of the relations ± i . For instance, let n = 3, X 1 = X 2 = X 3 = R, and X = n i=1 X i . For all i ∈ N , let T i be identical to ≥. Consider the relation on X such that

x y ⇔ |T (x, y)| ≥ |T (y, x)|
By construction, this relation is a CR-AT. It is not a RSRP. Indeed, it is easy to see that for all i ∈ N , we have ± i = ≥. But we know that with a RSRP, all relations ± i have two distinct equivalence classes. Our next result shows that a reflexive RSRP that is independent is a CR-AT.

Lemma 16. A reflexive RSRP that is independent is a CR-AT.

Proof. We know from Lemma 6 that in a RSRP all relations ± i are weak orders. Hence, it is enough to show that all relations * * i are weak orders having at most three equivalence classes.

It is easy to show that, for all i ∈ N and all (Bouyssou and Pirlot 2005a, Lemma 3.8). In particular, using the reflexivity of * * i , if

x i , x i , y i , y i , z i , z i , w i , w i ∈ X i , (x i , y i ) * * i (z i , w i ), x i ± i x i , y i ± i y i , z i ± i z i , w i ± i w i , imply (x i , y i ) * * i (z i , w i )
x i ∼ ± i z i and y i ∼ ± i w i we have (x i , y i ) ∼ * * i (z i , w i ). Let a i , b i ∈ X i .
For the ordered pair (a i , b i ), it is clear that we have one of the following four situations:

1. a i ∈ A i and b i ∈ A i , 2. a i ∈ A i and b i / ∈ A i , 3. a i / ∈ A i and b i ∈ A i , 4. a i / ∈ A i and b i / ∈ A i .
Consider two ordered pairs (x i , y i ) and (z i , w i ) of elements in X i . If these two ordered pairs are in the same situation, we have x i ∼ ± i z i and y i ∼ ± i w i , so that (x i , y i ) ∼ * * i (z i , w i ). All ordered pairs (x i , y i ) in the second situation are above all ordered pairs (z i , w i ) in the first situation in terms of * * i . Indeed, in this case, we know that [x i ∼ ± i z i ∼ ± i w i ] ± i y i . Since * * i is reflexive, we have (z i , w i ) * * i (z i , w i ). Using the fact that x i ± i z i and w i ± i y i , we obtain (x i , y i ) * * i (z i , w i ). A similar reasoning shows that all ordered pairs (x i , y i ) in the second situation are above all ordered pairs (z i , w i ) in the in the fourth situation in terms of * * i . Similarly, it is easy to check that all ordered pairs (x i , y i ) in the first situation are above all ordered pairs (z i , w i ) in the in the third situation in terms of * * i and that all ordered pairs (x i , y i ) in the fourth situation are above all ordered pairs (z i , w i ) in the in the third situation in terms of * * i . Hence, the relation * * i will be complete iff the elements in the first and fourth situations are equivalent in terms of * * i . Using Lemma 2.4, it is easy to see that this is equivalent to requiring that is independent. Observe that, when this is the case, * * i has at most three distinct equivalence classes. Observe finally that an independent relation being a RSRP is a very particular CR-AT since, in this case, all relations ± i have only two distinct equivalence classes. 20 Summarizing, there are CR-AT that are not reflexive RSRP (the ones in which the relations ± i have more than two equivalence classes) and there are reflexive RSRP that are not CR-AT (the ones that are not independent).

The case of weak orders

Results

Although all the conditions needed to characterize RSRP are independent in the class of all semiorders on X, the situation is vastly different when we turn to weak orders.

We know from Lemma 5 that conditions AC 1 i , AC 2 i , and AC 3 i are equivalent for weak orders. The same is true for conditions AC 1 * i , AC 2 * i , AC 3 * i and AC 4 * i .

Lemma 17. Let be a weak order on a set X. Then conditions AC

1 * i , AC 2 * i , AC 3 * i and AC 4 * i are equivalent. Proof. [AC 2 * i ⇒ AC 1 * i ] Suppose that AC 1 * i is violated, so that (x i , a -i ) c, (y i , b -i ) d, (y i , a -i ) c, (z i , b -i ) d.
Using the fact that is a weak order, we obtain (x i , a -i ) (y i , a -i ) and (y

i , b -i ) (z i , b -i ). Since is reflexive, we have (y i , a -i ) (y i , a -i ) and (z i , b -i ) (z i , b -i ). Applying AC 2 * i , we obtain (y i , a -i ) (x i , a -i ) or (z i , b -i ) (y i , b -i ), a contradiction. [AC 3 * i ⇒ AC 2 * i ] Suppose that AC 2 * i is violated, so that c (y i , a -i ), d (z i , b -i ), c (x i , a -i ), and d (y i , b -i ).
Using the fact that is a weak order, we obtain (x i , a -i ) (y i , a -i ) and (y

i , b -i ) (z i , b -i ). Since is reflexive, we have (x i , a -i ) (x i , a -i ) and (z i , b -i ) (z i , b -i ). Applying AC 3 * i , we obtain (y i , a -i ) (x i , a -i ) or (z i , b -i ) (y i , b -i ), a contradiction. [AC 4 * i ⇒ AC 3 * i ] Suppose that AC 3 * i is violated, so that (x i , a -i ) c, d (z i , b -i ), (y i , a -i ) c, d (y i , b -i ).
Using the fact that is a weak order, we obtain (y

i , b -i ) (z i , b -i ) and (x i , a -i ) (y i , a -i ). Since is reflexive, we have (y i , a -i ) (y i , a -i ) and (y i , b -i ) (y i , b -i ). Applying AC 4 * i , we obtain (y i , a -i ) (x i , a -i ) or (z i , b -i ) (y i , b -i ), a contradiction. [AC 1 * i ⇒ AC 4 * i ] Suppose that AC 4 * i is violated, so that c (y i , a -i ), (y i , b -i ) d, c (x i , a -i ), (z i , b -i ) d.
Using the fact that is a weak order, we obtain (x i , a -i ) (y i , a -i ) and (y

i , b -i ) (z i , b -i ). Since is reflexive, we have (x i , a -i ) (x i , a -i ) and (y i , b -i ) (y i , b -i ). Applying AC 1 * i , we obtain (y i , a -i ) (x i , a -i ) or (z i , b -i ) (y i , b -i ), a contradiction.
We have shown above that AC 1 * i (resp. AC 2 * i , AC 3 * i , AC 4 * i ) was equivalent to the conjunction of AC 1 i and AC 1 * * i (resp. AC 2 i and AC 2 * * i , AC 3 i and AC 3 * * i , AC 3 i and AC 4 * * i ). In the class of weak orders, conditions AC 1 i and AC 1 * * i are independent.

Lemma 18. In the class of all weak orders on X, conditions AC 1 i and AC 1 * * i are independent.

Proof. See Appendix C.

Summarizing the above observations, we have:

Proposition 3. Let be a weak order on X. It is a Relation with a Single Reference Point iff it satisfies AC 1 * iff it satisfies AC 1 and AC 1 * * .

On the basis of Lemmas 5, 13 and 17 and building obvious variations of Lemma 18, it is possible to formulate many alternative equivalent results for the case of weak orders. We leave the details to the interested reader.

Relation to models based on a discrete Sugeno integral

We have shown in Section 3 that a RSRP that is a weak order has a numerical representation such that:

x y ⇔ µ(S(x)) ≥ µ(S(y))
where µ is a normalized capacity on N .

This representation is reminiscent of representation of weak orders based on a discrete Sugeno integral studied in [START_REF] Bouyssou | A conjoint measurement approach to the discrete Sugeno integral[END_REF] and [START_REF] Greco | Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules[END_REF]. Indeed, as shown below, a RSRP always has a representation using a discrete Sugeno integral.

The following definitions are taken from [START_REF] Bouyssou | A conjoint measurement approach to the discrete Sugeno integral[END_REF].

Definition 9. Let β = (β 1 , β 2 , . . . , β p ) ∈ [0, 1] p .
The discrete Sugeno integral of the vector (β 1 , β 2 , . . . , β p ) ∈ [0, 1] p w.r.t. the normalized capacity ν on P = {1, 2, . . . , p} is defined by:

Sug ν [β] = T ⊆P ν(T ) ∧ i∈T β i .
Definition 10. A weak order on X has a representation in the discrete Sugeno integral model if there are a normalized capacity µ on N and functions u i : X i → [0, 1] such that, for all x, y ∈ X,

x y ⇔ Sug µ [(u 1 (x 1 ), u 2 (x 2 ), . . . , u n (x n ))] ≥ Sug µ [(u 1 (y 1 ), u 2 (y 2 ), . . . , u n (y n ))].
Definition 11. The relation on X is said to be strongly 2-graded on at-

tribute i ∈ N (condition 2 * -graded i ) if, for all a, b, c, d ∈ X and all x i , y i , z i ∈ X i , (x i , a -i ) c and (y i , b -i ) d and d c            ⇒    (y i , a -i ) c or (z i , b -i ) d,
The relation on X is said to be 2-graded on attribute i ∈ N (condition 2-graded i ) if, for all a, b, c, d ∈ X and all x i , y i , z i ∈ X i ,

(x i , a -i ) c and (y i , b -i ) d and (x i , b -i ) d and d c                    ⇒    (y i , a -i ) c or (z i , b -i ) d,
A binary relation is said to be strongly 2-graded (condition 2 * -graded) if it is strongly 2-graded on all attributes i ∈ N . Similarly, a binary relation is said to be 2-graded (condition 2-graded) if it is 2-graded on all attributes i ∈ N . Bouyssou et al. (2009, Lemma 1) have shown that condition 2 * -graded i holds iff conditions 2-graded i and AC 1 i hold.

It is clear that condition 2 * -graded i is a weakening of AC 1 * i since it amounts to adding a premise to this condition. Similarly, condition 2-graded i is a weakening of AC 1 * * i . [START_REF] Bouyssou | A conjoint measurement approach to the discrete Sugeno integral[END_REF] and [START_REF] Greco | Axiomatic characterization of a general utility function and its particular cases in terms of conjoint measurement and rough-set decision rules[END_REF] have shown the following (this result can be extended to cover sets of arbitrary cardinality adding a condition imposing that the weak order has a numerical representation. This will not be useful here).

Proposition 4. Let be a weak order on X such that X/∼ is at most countably infinite. Then has a representation in the discrete Sugeno integral model iff it satisfies condition 2 * -graded iff it satisfies conditions AC 1 and 2-graded.

The following lemma is a direct consequence of the fact that condition 2 * -graded i is a weakening of AC 1 * i . Lemma 19. Let be a weak order. If it is a RSRP, then it has a representation in the discrete Sugeno integral model.

Proof. We know that a RSRP that is weak order only has a finite number of distinct equivalence classes. Hence, the proof follows from Proposition 4 and the fact that AC 1 * implies 2 * -graded.

The above lemma has shown that, for weak orders, a RSRP always has a representation in the discrete Sugeno integral model. Suppose that the weak order is represented as a RSRP using sets A i and an importance relation . We know from Lemma 2 that is a weak order on N , so that it can be represented by a normalized capacity µ on N . It is then easy to devise a representation of this weak order in the discrete Sugeno integral model.

Define, for all i ∈ N and all x i ∈ X i ,

u i (x i ) = 1 if x i ∈ A i , 0 otherwise.
Using such functions u i , it is easy to see that, for all x ∈ X, and all J ⊆ N , we have:

µ(J) ∧ i∈J u i (x i ) = µ(J) if x i ∈ A i for all i ∈ J, 0 otherwise. Hence, when S(x) = J we have Sug µ [(u 1 (x 1 ), u 2 (x 2 ), . . . , u n (x n ))] = µ(J), so that x y ⇔ S(x) S(y) ⇔ µ(S(x)) ≥ µ(S(y)) ⇔ Sug µ [(u 1 (x 1 ), u 2 (x 2 ), . . . , u n (x n ))] ≥ Sug µ [(u 1 (y 1 ), u 2 (y 2 ), . . . , u n (y n ))].
We will see in Appendix D that the situation is different when considering relations using several reference points: such relations may not have a representation in the discrete Sugeno integral model.

Discussion

This paper has shown how to use the general model developed in Bouyssou and Pirlot (2004b) to characterize preference models using a single reference point introduced by [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008)). This analysis is extended to the case of multiple reference points in Appendix D. Basically, models using reference points are particular cases of the model inducing complete traces on the levels of each attribute. The number of reference points is linked to the number of distinct equivalence classes generated by these traces. Models using a single reference point generate traces that are quite rough. We have proposed a complete characterization of these models using a traditional conjoint measurement setting in which the only primitive is a preference relation on X. This analysis leads to conditions that could be the subject of empirical tests. Moreover these conditions allow to better position models using reference points among other preference models for multiattributed alternatives. In particular, we have shown that preference models using a single reference point that are weak orders are a particular case of a model based on the discrete Sugeno integral. Moreover we have analyzed the relations between preference models using a single reference point and concordance relations. None of these two models is a subclass of the other.

We conclude with the indication of directions for future research on the subject. The first is to analyze models using reference points that would include an idea of "non-discordance" as in the ELECTRE TRI model. Using the results in Bouyssou and Marchant (2007a,b), this should not be overly difficult. The second is to pursue the analysis of preference models using several reference points. Indeed, [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008) ) has proposed several particular cases of the general model that we study in Appendix D. Finally, it is clearly of vital interest to investigate elicitation techniques that would lead to specify the parameters of the models that we have studied.

Appendices

In all examples in Sections A, B, and C, we take n = 3 and X 1 = {x, y, z}, X 2 = {a, b} and X 3 = {p, q}. To save space, we often write xap instead of (x, a, p).

is (y 1 , b -1 ) d. If y 1 =
x then since all elements of X 1 are above x according to + 1 , we will have that (z 1 , b -1 ) d, for all z 1 ∈ X 1 .

Similarly, the first premise of AC 1 * * 1 is (x 1 , a -1 ) c. If x 1 = x then since all elements of X 1 are above x according to + 1 , we will have that (y 1 , a -1 ) c, for all y 1 ∈ X 1 . This shows that the only possible violations of AC 1 * * 1 will occur if either x 1 = y and y 1 = z or x 1 = z and y 1 = y.

Let us consider the first case with x 1 = y and y 1 = z. The premises of AC 1 * * 1 state that (y, a 1 is satisfied. Hence AC 1 * * 1 holds. Transposing the boolean matrix in the above example, one easily obtain an example satisfying all our conditions except AC 2 on one attribute. This is detailed below. Example 6. Let on X be defined by the following table :  Let us consider the first case with y 1 = y and z 1 = z. The premises of AC 2 * * 1 state that c (y, a -1 ), d (z, b -1 ), and c (z, a -1 ). It is easy to check that whenever we have both c (y, a -1 ) and c (z, a -1 ) we also have c (x, a -1 ). Hence, the first conclusion of AC 2 * * 1 will hold. Let us now consider the second case in which y 1 = z and z 1 = y. The premises of AC 2 * * 1 state that c (z, a -1 ), d (y, b -1 ), and c (y, a -1 ). As observed above, whenever we have both c (z, a -1 ) and c (y, a -1 ), we also have that c (x, a -1 ). Hence the first conclusion of AC 2 The second premise of AC 1 * * 1 is (y 1 , b -1 ) d. If y 1 = x then, since all elements of X 1 are above x according to + 1 , we will have that (z 1 , b -1 ) d, for all z 1 ∈ X 1 .

Similarly, the first premise of AC 1 * * 1 is (x 1 , a -1 ) c. If x 1 = x then, since all elements of X 1 are above x according to + 1 , we will have that (y 1 , a -1 ) c, for all y 1 ∈ X 1 . This shows that the only possible violations of AC 1 * * 1 will occur if either x 1 = y and y 1 = z or x 1 = z and y 1 = y.

Let us consider the first case with x 1 = y and y 1 = z. The premises of AC 1 * * 1 state that (y, a

D. Multiple reference points

D.1. Definition

The model defined below, building on [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006aRolland ( ,b, 2008[START_REF] Rolland | Reference-based preferences aggregation procedures in multicriteria decision making[END_REF], extends Definition 1 to deal with the case of multiple reference points. In this model, there is a semiorder R i on each attribute. In order to compare the alternatives x and y, we first compare each of them to a number of "reference points" π 1 , π 2 , . . . , π only using "ordinal considerations". For each profile π k , we compute the subsets of attributes R k (x) = {i ∈ N :

x i R i π k i } and R k (y) = {i ∈ N : y i R i π k i }.
The comparison of x and y is based on the two -tuples (R 1 (x), R 2 (x), . . . , R (x)) and (R 1 (y), R 2 (y), . . . , R (y)). This comparison uses an "importance relation" that will only be required to be monotonic w.r.t. inclusion. Definition 12. A binary relation is a Relation with Multiple Reference Points (or more briefly is a RMRP) if:

• for all i ∈ N , there is a semiorder R i on X i ,

• there are ∈ N + elements of X, π 1 , π 2 , . . . , π , interpreted as "reference points",

• there is a binary relation L on (2 N ) that is monotonic w.r.t. inclusion, i.e., for all A

1 , B 1 , C 1 , D 1 , . . . , A , B , C , D ⊆ N , (A 1 , . . . , A ) L (B 1 , . . . , B ) ⇒ (C 1 , . . . , C ) L (D 1 , . . . , D ), whenever, for all k ∈ L = {1, 2, . . . , }, C k ⊇ A k , B k ⊇ D k , and there are x, y, z, w ∈ X such that R L (x) = (A 1 , . . . , A ), R L (y) = (B 1 , . . . , B ), R L (z) = (C 1 , . . . , C ), and R L (w) = (D 1 , . . . , D ),
such that, for all x, y ∈ X,

x y ⇔ (R 1 (x), R 2 (x), . . . , R (x)) L (R 1 (y), R 2 (y), . . . , R (y)), (RMRP) where R k (x) = {i ∈ N : x i R i π k i } and R L (x) = (R 1 (x), R 2 (x), . . . , R (x)).
We write R L (x) R L (y) to mean that, for all k ∈ L, R k (x) ⊇ R k (y).

D.2. Results

We start by showing that a RMRP is a particular case of model (M).

Lemma 20. If a binary relation on X is a RMRP with reference points, it satisfies AC 1, AC 2 and AC 3, so that all relations ± i are weak orders. Furthermore, for all i ∈ N , the weak order ± i has at most + 1 distinct equivalence classes.

Proof. Let us show that AC 1 holds. Suppose that (x i , a -i ) c and (y i , b -i ) d. Because R i is a semiorder, we know (see the end of Section 2.1) that the relation R • i is a weak order. By construction, we have either

x i R • i y i or y i R • i x i . Suppose that x i R • i y i . By construction, for all k ∈ L, y i R i p k i implies x i R i p k i . This implies R L ((x i , b -i )) R L ((y i , b -i )). Since (y i , b -i ) d, we know that R L ((y i , b -i )) L R L (d). Using the fact that L is monotonic w.r.t. inclusion, we obtain R L ((x i , b -i )) L R L (d), so that (x i , b -i ) d. A similar proof shows that if y i R • i x i then we have (y i , a -i ) c. Hence, AC 1 holds.
The proof for AC 2 and AC 3 is similar.

Let us show that the weak order ± i has at most +1 distinct equivalence classes. For all k ∈ L, define

A k i = {x i ∈ X i : x i R i π k i }. Because R i is a semiorder, the relation R • i is a weak order. Let k, k ∈ L. We have either π k i R • i π k i or π k i R • i π k i . Hence, we have either A k i ⊆ A k i or A k i ⊆ A k i . This
shows that, for all i ∈ N , the sets A k i , k ∈ L are nested. Moreover, it is clear that if x i , y i ∈ X i belong exactly to the same subsets A k i , k ∈ L, we must have x i ∼ ± i y i . This completes the proof since the sets A k i are nested. The above lemma shows that a RMRP satisfies AC 1, AC 2 and AC 3 and is such that, for all i ∈ N , X i /∼ ± i is finite. Hence, it has a representation in model (M). We show below that the converse is true, as soon as all sets X i /∼ ± i are finite.

Lemma 21. Let be a binary relation on X. If satisfies AC 1, AC 2, and AC 3 and, for all i ∈ N , the sets X i /∼ ± i are finite then is a RMRP.

Proof. For all i ∈ N , we know that ± i is a weak order and that X i /∼ ± i is finite. Let i be the number of distinct equivalence classes of X i /∼ ± i . Let = max i∈N i . The proof will be complete if we show that a RMRP.

By construction, there at least one i ∈ N such that ± i has exactly distinct equivalence classes. On all such attributes i ∈ N , we define E k i , for k = 1, 2, . . . , , as the set containing all elements in the kth equivalence class of ± i . Consider now an attribute j ∈ N such that ± j has j distinct equivalence classes with j < . For all such attributes j ∈ N , we define E k j , for k = 1, 2, . . . , j , as the set containing all elements in the kth equivalence class of ± j . For k = j + 1, j + 2, . . . , , we define E k j = E j j . For all i ∈ N , let us take R i = ± i . We use profiles π 1 , π 2 , . . . , π that are build as follows. For all i ∈ N and k ∈ L, let π k i be any element belonging to E -k+1 i . We define the binary relation L on (2 N ) letting (A 1 , . . . , A ) L (B 1 , . . . , B ) whenever there are x, y ∈ X such that x y, R L (x) = (A 1 , . . . , A ), and R L (y) = (B 1 , . . . , B ).

We claim that the family of reference points π 1 , π 2 , . . . , π together with the weak orders R i = ± i and the relation L is a representation of in model (RMRP).

Suppose first that x y. Then, by construction, we have R L (x) L R L (y).

Suppose now that (A 1 , . . . , A ) L (B 1 , . . . , B ) and consider x, y ∈ X such that R L (x) = (A 1 , . . . , A ), and R L (y) = (B 1 , . . . , B ). Because (A 1 , . . . , A ) L (B 1 , . . . , B ), we know that there are z, w ∈ X such that R L (z) = (A 1 , . . . , A ), R L (w) = (B 1 , . . . , B ), and z w. Given the construction of the profiles π 1 , π 2 , . . . , π , R L (x) = R L (z) implies that, for all i ∈ N , x i ∼ ± i z i . Similarly, R L (y) = R L (w) implies that, for all i ∈ N , y i ∼ ± i w i . Using (4), we obtain x y.

It remains to show that L is monotonic w.r.t. inclusion. Suppose that (A 1 , . . . , A ) L (B 1 , . . . , B ). This implies that x y, for some x, y ∈ X such that R L (x) = (A 1 , . . . , A ), and R L (y) = (B 1 , . . . , B ). Take any (C 1 , . . . , C ) and (D 1 , . . . , D ) such that, k ∈ L, C k ⊇ A k , B k ⊇ D k , and there are z, w ∈ X such that R L (z) = (C 1 , . . . , C ), and R L (w) = (D 1 , . . . , D ). By construction of the family of reference points π 1 , π 2 , . . . , π , this implies that z i ± i x i and y i ± i w i , for all i ∈ N . Using, (3), we obtain z w, so that (C 1 , . . . , C ) L (D 1 , . . . , D ), as required.

Lemmas 20 and 21 show that, whenever X i /∼ ± i is finite, for all i ∈ N , the Model with Multiple Points of Reference is equivalent to model (M). Moreover:

• if has a representation in the Model with Multiple Points of Reference, it also has a representation in the Model with Multiple Points of Reference in which all relations R i are weak orders,

• if has a representation in the Model with Multiple Points of Reference, it also has a representation in which reference points dominates each other according to the weak orders R i , i.e., for all k ∈ {2, 3, . . . , } and all i ∈ N , π k i R i π k-1 i . In such a representation, for all x ∈ X, with R L (x) = (A 1 , . . . , A ) we have A 1 ⊇ A 2 ⊇ • • • ⊇ A (this was already observed in Rolland 2008, p. 51).

Indeed, the reference points π 1 , π 2 , . . . , π and the relations R i that are built in the proof of Lemma 21 have these two properties.

Our findings are summarized below.

Theorem 3. A binary relation on X is a RMRP iff it satisfies AC 1, AC 2 and AC 3 and, for all i ∈ N , the set X i /∼ ± i is finite. If a relation is a RMRP, it always has a representation in which all relations R i are weak orders and in which the points of reference are such that for all k ∈ {2, 3, . . . , } and all i ∈ N , π k i R i π k-1 i .

The uniqueness of the representation of RMRP is obviously quite weak. Since its study is cumbersome and does not appear to be particularly informative, we do not detail this point.

The above result shows that in a reflexive RMRP all relations ± i have a finite number of equivalence classes. We have seen above (see the proof of Lemma 15) that there are CR-AT in which the relations ± i have infinitely many equivalence classes. Hence, there are CR-AT that are not reflexive RMRP. The converse is also true since a reflexive RMRP may fail to independent whereas we know that a CR-AT always is. The claim of [START_REF] Rolland | Agrégation de préférences et point de référence[END_REF]Rolland ( , 2006a[START_REF] Rolland | Procédures d'agrégation ordinale de préférences avec points de référence pour l'aide à la décision[END_REF] that models using reference points are more "flexible" than concordance relations is nevertheless clearly true if attention is restricted to the case of finite sets X. Indeed, a CR-AT always satisfies conditions AC 1, AC 2 and AC 3. Moreover, on finite sets, the condition stating that, for all i ∈ N , the set X i /∼ ± i is finite trivially holds. Hence, in this particular but important case, all CR-AT are reflexive RMRP.

Using Lemma 5, it is easy to formulate a result characterizing model (RMRP) when is a weak order. In this case, we know that the three conditions AC 1, AC 2 and AC 3 are equivalent and may be replaced by weak separability. For the record, we state the following: Proposition 5. Let be a weak order on X. It is a Relation with Multiple Reference Points iff it satisfies AC 1 and, for all i ∈ N , the set X i /∼ ± i is finite.

Since the relation L is defined on the finite set (2 N ) , it is simple to show that a RMRP that is a weak order can only have a finite number of equivalence classes.

Consider now a relation defined on a finite set X. It will be a RMRP as soon as it satisfies AC 1. In view of Proposition 4, it will have a representation in the discrete Sugeno integral model if, furthermore, it is 2-graded. Clearly, a relation satisfying AC 1 does not have to be 2-graded. This shows that, contrary to the situation with RSRP, a RMRP may not have a representation in the discrete Sugeno integral model. This was already noted in Rolland (2008, Example 61, p. 138).

  [a K b and b K a] (resp. a α(K) b iff [a K b and Not[b K a]]). An equivalence is a reflexive (a K a), symmetric (a K b ⇒ b K a) and transitive ([a K b and b K c] ⇒ a K c) binary relation on A. An equivalence relation partitions A into equivalence classes. The set of equivalence classes induced by the equivalence K is denoted by A/K.

A

  semiorder is a reflexive (a K a), Ferrers ([a K b and c K d] imply [a K d or c K b]) and semitransitive ([a K b and b K c] imply [a K d or d K c]) binary relation. If K is a semiorder, it is well known

  whenever, C ⊇ A, B ⊇ D, and there are x, y, z, w ∈ X such that S(x) = A, S(y) = B, S(z) = C, and S(w) = D, such that, for all x, y ∈ X, x y ⇔ S(x) S(y), (RSRP)

  -1 ) c, (z, b -1 ) d, and (y, b -1 ) d. It is easy to check that whenever we have both (z, b -1 ) d and (y, b -1 ) d, we also have that (x, b -1 ) d. Hence the second conclusion of AC 1 * * 1 is satisfied. Let us now consider the second case in which x 1 = z and y 1 = y. The premises of AC 1 * * 1 state that (z, a -1 ) c, (y, b -1 ) d, and (z, b -1 ) d. As observed above, whenever we have both (y, b -1 ) d and (z, b -1 ) d, we also have that (x, b -1 ) d. Hence the second conclusion of AC 1 * *

  -1 ) c, (z, b -1 ) d, and (y, b -1 ) d. It is easy to check that whenever we have both (z, b -1 ) d and (y, b -1 ) d, we also have that (x, b -1 ) d. Hence the second conclusion of AC 1 * * 1 is satisfied. Let us now consider the second case in which x 1 = z and y 1 = y. The premises of AC 1 * * 1 state that (z, a -1 ) c, (y, b -1 ) d, and (z, b -1 ) d. As observed above, whenever we have both (y, b -1 ) d and (z, b -1 ) d, we also have that (x, b -1 ) d. Hence the second conclusion of AC 1 * * 1 is satisfied. Hence AC 1 * * 1 holds.

  consequence of the combination of AC 1 * , AC 2 * , AC 3 * and AC 4 * is presented below.

	Lemma 11. The relation satisfies AC 1 * i , AC 2 * i , AC 3 * i , and AC 4 * i iff the binary relation ±

i is a weak order having two distinct equivalence classes.

The notion of "reference point" is unfortunately used in the literature with many different meanings. The interpretation of the reference points in the models studied in this paper is discussed below. These reference points have little to do with the reference point used in prospect theory to distinguish gains from losses(Kahneman and[START_REF] Kahneman | Prospect theory: An analysis of decision under risk[END_REF] Kahneman 1992) or from the reference points used as a crucial element in the framing of decisions[START_REF] Tversky | Rational choice and the framing of decisions[END_REF]).

For exceptions, seeRolland (2003, Th.

3) andRolland (2008, Section 2.1.2).

b, a - 2 b, q +

3 p and q - 3 p. Using Lemma 9 shows that AC 1 * i , AC 2 * i , AC 3 * i , and

AC 4 * i hold for i = 2, 3.

a, b - 2 a, q +

3 p and q ∼ - 3 p. Using Lemma 9 shows that AC 1 * i , AC 2 * i , AC 3 * i , and

AC 4 * i hold for i = 2, 3. On attribute 1, we have [x ∼ +

A. Proof of Lemma 12

We have to show that in the class of all semiorders on X, conditions AC 1 * , AC 2 * , AC 3 * , and AC 4 * are independent.

We provide below the required four examples. In all these examples, we define by its boolean matrix, so that it is easy to check that the relations considered below are semiorders. Indeed, if it is possible to arrange the rows and columns of the boolean matrix in the same order so that the boolean matrix is stepped, we know that the relation is a semiorder [START_REF] Aleskerov | Utility maximization, choice and preference[END_REF].

Example 1. Let on X be defined by the following table :   xaq yaq zaq xap yap zap ybq xbp xbq ybp zbp zbq xaq

It is clear that this relation is a semiorder on X.

It is easy to check that we have a + 2 b, a - 2 b, q + 3 p and q - 3 p. Using Lemma 9 shows that AC 1 * i , AC 2 * i , AC 3 * i , and AC 4 * i hold for i = 2, 3. On attribute 1, we have x ∼ - 1 y ∼ - 1 z and x + 1 y + 1 z. Using Lemma 9 shows that AC 2 * 1 , AC 3 * 1 and AC 4 * 1 , while AC 1 * 1 is clearly violated. Transposing the boolean matrix in the above example, one easily obtain an example satisfying all our conditions except AC 2 on one attribute. Indeed, on each attribute the consequence of this transposition is is to interchange the roles of + i and - i and to reverse them. This is detailed below. Example 2. Let on X be defined by the following table :  zbq zbp ybp xbq xbp ybq zap yap xap zaq yaq xaq zbq

It is clear that this relation is a semiorder on X.

It is easy to check that we have b + 2 a, b - 2 a, p + 3 q and p - 3 q. Using Lemma 9 shows that AC 1 * i , AC 2 * i , AC 3 * i , and AC 4 * i hold for i = 2, 3. On attribute 1, we have x ∼ + 1 y ∼ + 1 z and z - 1 y - 1 x. Using Lemma 9 shows that AC 1 * 1 , AC 3 * 1 and AC 4 * 1 , while AC 2 * 1 is clearly violated. Example 3. Let on X be defined by the following table :   xaq yaq xap xbp xbq yap ybq zap zaq zbq ybp zbp xaq

B. Proof of Lemma 14

We have to show that in the class of all semiorders on X, conditions

i trivially holds. If x i = y i then the second possible conclusion of AC 1 * * i will trivially hold since z i must be either x i or y i . A similar reasoning applies to AC 2 * * i and AC 4 * * i . The premises of AC 3

then the third conclusion of AC 3 * * i trivially holds. If x i = z i , then the y i in the conclusion of AC 3 * * i must be either x i or z i . In the first (resp. second) case, the first (resp. second) conclusion trivially holds.

Example 5. Let on X be defined by the following table:

It is clear that this relation is a semiorder. It is easy to check that the traces are as follows.

On attribute 2, we have b + 2 a and b - 2 a. Hence, AC 1 2 , AC 2 2 , and AC 3 2 hold. Since X 2 has only two elements, AC 1 * * 2 , AC 2 * * 2 , AC 3 * * 2 , and AC 4 * * 2 trivially hold. On attribute 3, we have p + 3 q and p - 3 q. Hence, AC 1 3 , AC 2 3 , and AC 3 3 hold. Since X 3 has only two elements, AC 1 * * 3 , AC 2 * * 3 , AC 3 * * 3 , and AC 4 * * 3 trivially hold.

On attribute 1, - 1 is a clique. Furthermore, we have y + 1 x and z + 1 x, but neither y + 1 z nor z + 1 y. It easy to check that this implies that AC 2 1 and AC 3 1 hold, while AC 1 1 is violated.

xaq zaq yaq xap yap zap xbq zbq ybq xbp ybp zbp xaq

It is clear that this relation is a semiorder. It is easy to check that the traces are as follows.

On attribute 2, we have a + 2 b and a - 2 b. Hence, AC 1 2 , AC 2 2 , and AC 3 2 hold. Since X 2 has only two elements, AC 1 * * 2 , AC 2 * * 2 , AC 3 * * 2 , and AC 4 * * 2 trivially hold. On attribute 3, we have q + 3 p and q - 3 p. Hence, AC 1 3 , AC 2 3 , and AC 3 3 hold. Since X 3 has only two elements, AC 1 * * 3 , AC 2 * * 3 , AC 3 * * 3 , and AC 4 * * 3 trivially hold.

On attribute 1, + 1 is a clique. Furthermore, we have x - 1 y and x - 1 z, but neither y - 1 z nor z - 1 y. It easy to check that this implies that AC 1 1 and AC 3 1 hold, while AC 2 1 is violated.

It remains to check that AC 1 * * 1 , AC 2 * * 1 , AC 3 * * 1 , and AC 4 * * 1 are satisfied. Using Lemma 9, it is easy to check that AC 1 * 1 , AC 3 * 1 , and AC 4 * 1 are satisfied. Indeed, since + 1 is a clique, the premise of AC 1 * 1 and AC 3 * 1 is never satisfied while the conclusion AC 4 * 1 always holds. Let us check that AC 2 * * 1 holds. The second premise of

Similarly, the first premise of AC 2 * * 1 is c (y 1 , a -1 ). If y 1 = x then, since all elements of X 1 are below x according to - 1 , we will have that c (x 1 , a -1 ), for all x 1 ∈ X 1 . This shows that the only possible violations of AC 2 * * 1 will occur if either y 1 = y and z 1 = z or y 1 = z and z 1 = y. On attribute 2, we have a + 2 b and a - 2 b. Hence, AC 1 2 , AC 2 2 , and AC 3 2 hold. Since X 2 has only two elements, AC 1 * * 2 , AC 2 * * 2 , AC 3 * * 2 , and AC 4 * * 2 trivially hold.

On attribute 1, we have [x

, and AC 4 * * 1 hold. On attribute 3, we have p + 3 q and q - 3 p. Hence, AC 1 3 and AC 2 3 hold but AC 3 3 is violated. Since X 3 has only two elements, AC 1 * * 3 , AC 2 * * 3 , AC 3 * * 3 , and AC 4 * * 3 trivially hold.

C. Proof of Lemma 18

We have to show that in the class of all weak orders on X, conditions AC 1 and AC 1 * * are independent. We need two examples. The first gives an example of a weak order satisfying AC 1 and AC 1 * * i and on all but one of the attributes. The second gives an example of a weak order satisfying AC 1 * * and AC 1 i on all but one of the attributes.

Example 8. Let on X be the weak order obtained using an additive value function model with the following value functions:

It is clear that this relation is the weak order on X such that: [xbp, xbq, ybp, ybq, zap, zbp, zbq] [xap, yap, yaq] [xaq, zaq] It is easy to check that we have b + 2 a, and q + 3 p. Using Lemma 9 shows that AC 1 * i , and, hence, AC 1 i and AC 1 * * i hold for i = 2, 3. On attribute 1, we have y + 1 x, z + 1 x, but neither y + 1 z nor z + 1 y, so that AC 1 1 is violated.

Let us check that AC 1 * * 1 holds.

A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4): 297-323, 1992.


Béatrice Blondel
email: beatrice.blondel@inserm.fr

Nicolas Drewniak
email: nicolas.drewniak@inserm.fr

Hugo Pilkington
email: hugo.pilkington@inserm.fr

Jennifer Zeitlin
email: jennifer.zeitlin@inserm.fr

Out-of-hospital births and the supply of maternity units in France

Keywords: health services, distance, maternity units, supply, out-of-hospital birth, access, social class

Maternity unit closures in France have increased distances that women travel to deliver in hospital. We studied how the supply of maternity units influences the rate of out-of-hospital births using birth certificate data. In 2005-6, 4.3 per 1000 births were out-of-hospital. Rates were more than double for women living 30 km or more from their nearest unit and were even higher for women of high parity. These associations persisted in multilevel analyses adjusting for other maternal characteristics. Long distances to maternity units should be a concern to health planners because of the maternal and infant health risks.

Introduction

There is a trend in many countries towards fewer maternity units and the centralisation of births in larger units [START_REF] Neto | Perinatal care in Portugal: effects of 15 years of a regionalized system[END_REF][START_REF] Pilkington | Impact of maternity unit closures on access to obstetrical care: the French experience between 1998 and 2003[END_REF]Zeitlin and Mohangoo, 2008). Several interrelated reasons underlie this trend, such as medical safety, financial pressures and the need to efficiently use equipment and medical staff. The decreasing number of units can lead to a loss of services in certain areas and increase travel distances to hospital for delivery [START_REF] Nesbitt | Access to maternity care in rural Washington: its effect on neonatal outcomes and resource use[END_REF][START_REF] Sutherns | Accessing maternity care in rural Canada: there's more to the story than distance to a doctor[END_REF][START_REF] Tucker | Sustainable maternity services in remote and rural Scotland? A qualitative survey of staff views on required skills, competencies and training[END_REF]. Studies have shown that long travel time to the maternity unit [START_REF] Ravelli | Travel time from home to hospital and adverse perinatal outcomes in women at term in the Netherlands[END_REF] and poor access to obstetric care [START_REF] Nesbitt | Access to maternity care in rural Washington: its effect on neonatal outcomes and resource use[END_REF] are associated with higher neonatal mortality and morbidity.

Childbirth has particular features which accentuate the importance of the geographical accessibility of health services for deliveries planned in hospital and at home. The onset of labour is unpredictable and it can progress quickly, leading to unexpected delivery at home or en route to hospital. Accidental out-of-hospital births often occur without medical assistance and are more common before term, leading to a higher risk of mortality [START_REF] Jones | Mortality in out-of-hospital premature births[END_REF][START_REF] Sheiner | Accidental out-of-hospital delivery as an independant risk factor for perinatal mortality[END_REF]. In Finland, the perinatal mortality was 2.5 times higher for babies born accidentally out-of-hospital than for babies born in hospital [START_REF] Viisainen | Accidental out-ofhospital births in Finland: incidence and geographical distribution 1963-1995[END_REF]. For planned home births, the accessibility of maternity units is essential to ensure timely emergency transfer for complications during labour or delivery; these transfers occur in 15 to 30% of cases [START_REF] Evers | Perinatal mortality and severe morbidity in low and high risk term pregnancies in the Netherlands: prospective cohort study[END_REF][START_REF] Hutton | Outcomes associated with planned home and planned hospital births in low-risk women attended by midwives in Ontario, Canada, 2003-2006: a retrospective cohort study[END_REF].

In France, the number of maternity units declined dramatically from 1128 in 1981 to 816 in 1995 and 585 in 2006. Studies have assessed the impact of these closures on women's choices of maternity unit [START_REF] Combier | Choosing where to deliver: decision criteria among women with low-risk pregnancies in France[END_REF] and on the distance that women must travel to deliver [START_REF] Pilkington | Impact of maternity unit closures on access to obstetrical care: the French experience between 1998 and 2003[END_REF]. There have been no studies, however, on difficulties women face getting to maternity units or on out-of-hospital deliveries.

Our objectives were to calculate the incidence of out-of-hospital births and to determine whether this incidence varied according to distance to the closest maternity unit and recent maternity closures. A secondary objective was to assess whether specific sociodemographic groups of women were more vulnerable to the effects of distance and at greater risk for out-of-hospital delivery.

Population and method

Data were obtained from vital statistics. Birth certificates record information about whether delivery occurred in or out of hospital, but do not distinguish between accidental out-of-hospital births and planned home births. Certificates also provide data on maternal age, parity, maternal and paternal occupation, municipality of residence and of birth. The municipality is the smallest administrative division in France.

Analyses were done on singleton live births in 2005-6 in metropolitan France (N=1,517,599). The birth certificate was revised in 1998, including the coding of the question on place of delivery. By 2006, the new coding rules had not yet been fully implemented in some municipalities. We therefore developed rules to exclude births in these areas. We first excluded municipalities if more than 8% of birth certificates had missing data about place of delivery and if the total number of births was greater than 10. We then excluded municipalities which had unrealistically high out-of-hospital birth rates (higher than 2% and more than 40 births). This rate is four times higher than the known rates before the introduction of the new coding rules [START_REF] Faur | La situation démographique en 1986[END_REF]. The 40 delivery cut-off corresponds to approximately two and half times the average annual number of home births carried out by a midwife who performs home deliveries (unpublished data). Finally we excluded all departments (the next administrative level up from the municipality), where more than 20% of births were excluded in the previous stages, in order to carry out our analyses as much as possible on geographically complete areas. In all, 11.1% of births were excluded and the final analysis sample included 1,349,751 births. We carried out a sensitivity analysis to test the impact of these exclusions by comparing included and excluded births using an alternative indicator of out-of-hospital births, defined as the proportion of births occurring in municipalities without a maternity unit.

The location of maternity units was available from Ministry of Health annual statistics.

We computed distances between places of residence and maternity units after placing each hospital and each residence on the map at the centre of the municipality. Distances were calculated using the road network with geographic information system software ArcView 9.3.1 (ESRI, Redlands, CA, USA). Supply characteristics were the mother's distance to her closest maternity unit, whether a maternity unit had closed between 2003 and 2006 within a 15-km radius of her home, and urban, peri-urban or rural residence. We chose a 15-km radius because threequarters of French women give birth within 15 km of their home [START_REF] Doisneau | Une femme accouche en moyenne à 14 km de son domicile[END_REF].

We studied the association between out-of-hospital births and maternal and supply characteristics using a multilevel model with the population average option, in order to estimate the impact of individual and municipal level determinants. The descriptive analyses were done with SAS (SAS Institute Inc., Cary, NC, USA) and the multilevel analyses with HLM version 6 (Scientific Software International Inc, Lincolnwood, IL, USA).

Results

In 2005-6, 5740 women delivered out of hospital (4.3 per 1000 births). During this period, 2.8 per 1000 births took place in municipalities without a maternity unit; the rate was similar, 2.7 per 1000, for births excluded from the analysis sample.

The out-of-hospital birth rate was lowest among women aged 25-29 years (Table 1).

The rate rose steeply with parity from 2.8 per 1000 among primiparas to 14.3 per 1000 for women who had at least five deliveries. The relation with social class had a J shape with a minimum rate in the shop assistant and service worker group. Out-ofhospital births occurred more often in rural areas and were strongly associated with distance to the closest maternity unit. Out-of-hospital births were not more frequent when a maternity unit had been closed within a 15-km radius.

The association between distance to the closest maternity unit and out-of-hospital births varied according to parity (Figure 1). Among primiparas, rates increased from 2.3 per 1000 when the minimum distance was under five km to 7.5 per 1000 when the minimum was 45 km or more; these rates were 5.4 and 26.2 per 1000, respectively, among women of parity four or higher.

In the multivariate analysis, this interaction between parity and distance was significant (p<0.001). The odds ratio for out-of-hospital births for women who lived at least 45 km from the closest maternity unit was 2.47 among women of parity one or two and 6.46 among women of parity three or higher, when the reference group was women of parity one or two living at less than five km from the closest maternity unit (Table 2). The risk of out-of-hospital birth was also higher among older women, women of the higher and lower social classes, and among those who lived in rural areas. The association with the recent closure of a maternity unit near-by was not significant.

Discussion

Our study showed that the risk of out-of-hospital delivery was strongly associated with parity and the distance to the closest maternity unit. Closure of a unit near to a woman's home in the recent past did not increase this risk, however.

Our study has several limitations. First, we had to exclude 11% of births to ensure good data quality. Our sensitivity analysis of births that occurred in municipalities without a maternity unit suggests, however, that the exclusions did not affect our estimate of national incidence. Secondly, the data available for analysis were limited.

French birth certificates do not include medical data, so we could not study the medical complications associated with out-of-hospital births. The certificate also does not document whether the delivery was initially planned at home or in hospital.

Finally, distance was computed from the centre of each municipality and not from each mother's home and hospital address; while this adds imprecision to our measures, there is no reason to expect a systematic bias favouring hospital or out-ofhospital births.

Rates of out-of-hospital births in France are low, similar to those in European countries where planned home births are not common (Zeitlin and Mohangoo, 2008).

Rates are between 1 and 2 percent in other countries where home delivery is accepted as a part of maternity care provision, such as the USA (0.9%), Canada (1.2%) or the United Kingdom (2.3% in England and 1.2% in Scotland) [START_REF] Macdorman | Trends and characteristics of home and other out-of-hospital births in the United States, 1990-2006[END_REF]Statistics Canada, 2007;Zeitlin and Mohangoo, 2008). The highest rate in Europe is in the Netherlands (30%) where home deliveries are an integral component of maternity care.

Our results on the role of maternal characteristics corroborate those from previous studies. Parity has been consistently associated with the incidence of out-of-hospital births [START_REF] Declercq | Characteristics of planned and unplanned home births in 19 States[END_REF][START_REF] Macdorman | Trends and characteristics of home and other out-of-hospital births in the United States, 1990-2006[END_REF][START_REF] Viisainen | Accidental out-ofhospital births in Finland: incidence and geographical distribution 1963-1995[END_REF]. The progression of labour is faster in multiparas, especially in the first stage [START_REF] Vahratian | The impact of parity on course of labor in a contemporary population[END_REF], which may increase the risk of accidental out-of-hospital delivery. Planned deliveries at home are also more common among multiparas [START_REF] Anthony | Maternal factors and probability of a planned home birth[END_REF][START_REF] Declercq | Characteristics of planned and unplanned home births in 19 States[END_REF], primarily because previous low-risk delivery is a key criterion for selecting eligible women for home birth. Moreover, transfers to hospital are less frequent for multiparas planning home births [START_REF] Hutton | Outcomes associated with planned home and planned hospital births in low-risk women attended by midwives in Ontario, Canada, 2003-2006: a retrospective cohort study[END_REF].

Accidental out-of-hospital births are more frequent among women of lower social class [START_REF] Viisainen | Accidental out-ofhospital births in Finland: incidence and geographical distribution 1963-1995[END_REF], which may reflect difficulties in use of healthcare services during pregnancy [START_REF] Declercq | Characteristics of planned and unplanned home births in 19 States[END_REF][START_REF] Viisainen | Accidental out-ofhospital births in Finland: incidence and geographical distribution 1963-1995[END_REF]. In contrast, the slightly higher rates of out-of-hospital births in the higher social classes in our study may be due to planned home births. Home birthsand other alternative approaches to childbirthare more frequently chosen by more educated and affluent women [START_REF] De Jonge | Are there inequalities in choice of birthing position? Sociodemographic and labour factors associated with the supine position during the second stage of labour[END_REF][START_REF] Declercq | Characteristics of planned and unplanned home births in 19 States[END_REF].

In our study, the risk of out-of-hospital delivery for women living 30 km or more from the closest maternity unit was over twice as high as for women living less than five km away. This group accounts for 7% of all pregnant women in France. Unplanned out-of-hospital births have also been associated with living in remote areas in both Finland and Norway [START_REF] Viisainen | Accidental out-ofhospital births in Finland: incidence and geographical distribution 1963-1995[END_REF][START_REF] Schmidt | Deliveries in maternity homes in Norway: results from a 2-year prospective study[END_REF]. For high risk infants, long distances have also been found to restrict access to tertiary care facilities at delivery [START_REF] Pilkington | Distribution of maternity units and spatial access to specialised care for women delivering before 32 weeks of gestation in Europe[END_REF][START_REF] Samuelson | Maternal characteristics associated with place of delivery and neonatal mortality rates among very-lowbirthweight infants, Georgia[END_REF]. However, we found that recent maternity closures were not associated with more out-of-hospital births. At the beginning of the 2000s, closures took place primarily in areas where supply was relatively abundant and they did not result in increased average travel time for women [START_REF] Pilkington | Impact of maternity unit closures on access to obstetrical care: the French experience between 1998 and 2003[END_REF].

The reduction in the supply of maternity units is a long-term trend observed in many countries. Consequently, the proportion of women living far away from a maternity unit will probably grow, especially in rural areas. Unplanned births out of hospital are a useful indicator for monitoring potentially adverse effects of these closures.

Perinatal health information systems should be able to report on the place of delivery and distinguish between planned and unplanned out-of-hospital births, as recommended in England and Wales [START_REF] Mori | An estimation of intrapartum-related perinatal mortality rates for booked home births in England and Wales between 1994 and 2003[END_REF] and Sweden [START_REF] Hildingsson | Characteristics of women giving birth at home in Sweden: a national register study[END_REF]. However, even without this distinction, monitoring out-of-hospital births in geographically remote areas supplies essential information, since planned home births in these areas are not advisable.

Conclusion

Out-of-hospital births are rare in France. However, in underserved areas, rates are higher, especially for women of higher parity. It is important to inform women, especially multiparas living far from their maternity unit, about the risks associated with accidental births out of hospital. Particular attention should be given to the organization of health services in remote areas where women have to travel a long distance for childbirth in order to guarantee a minimum level of safety in emergency situations. (2) Household classification.

(3) Reference group: women with parity 1 or 2 living at less than 5 km from a maternity unit.

Fig. 1

 1 Fig. 1 Rate of out-of-hospital births by distance to the closest maternity unit and parity

Table 1 .

 1 Out-of-hospital births according to maternal characteristics, supply of medical services and geographical characteristics

	N	Rate per
		1000 births (1)

(1) Chi-square test of association with out-of-hospital births: p<0.001 for each variable; (2) Household classification; (3) Classification of the National Institute of Statistics and Economic Studies.

Table 2 .

 2 Odds ratios for out-of-hospital birth by maternal characteristics, supply of Multilevel analysis; OR adjusted for all variables included in the table; N=1,349,751.

	medical services and geographical characteristics
		aOR (1)	95% CI
	Maternal age
	< 25 yrs	1.05	0.98,1.13
	25-29	1
	30-34	1.13	1.07,1.20
	35-39	1.20	1.12,1.29
	≥ 40	1.63	1.47,1.81
	Occupation category (2)
	Professional/managerial	1.31	1.21, 1.43
	Intermediate	1.14	1.06,1.23
	Administrative, self-employed	1
	Shop assistant, service workers	0.98	0.89,1.09
	Skilled manual	1.29	1.18,1.40
	Unskilled manual	1.56	1.42,1.72
	No occupation	1.96	1.82, 2.12
	Women with parity 1 or 2
	Distance to closest maternity unit (3)
	<5 km	1
	5-14	1.14	1.03,1.27
	15-29	1.39	1.24,1.57
	30-44	1.78	1.55,2.05
	45 +	2.47	2.02,3.02
	Women with parity 3 +
	Distance to closest maternity unit (3)
	<5 km	1.73	1.57, 1.90
	5-14	2.32	2.04, 2.63
	15-29	3.25	2.84, 3.71
	30-44	3.71	3.13, 4.41
	45 +	6.46	4.92, 8.48
	Maternity unit closure in a
	15-km radius
	No	1
	Yes	0.91	0.84,1.00
	Rurality
	Urban	1
	Periurban	1.09	0.99,1.19
	Rural	1.43	1.29,1.58
	(1)

Funding

This study was financed by a grant from the French Institute for Public Health Research under the programme 'Health Services Research 2008'. The Institute was not involved in any stage of the study.

Disclosure of interests

The authors have no conflicts of interest to declare