On Egalitarian Belief Merging

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On Egalitarian Belief Merging

Patricia Everaere, Sébastien Konieczny, Pierre Marquis

To cite this version:

Patricia Everaere, Sébastien Konieczny, Pierre Marquis. On Egalitarian Belief Merging. Principles of

Knowledge Representation and Reasoning: Proceedings of the Fourteenth International Conference-

KR’14, Jul 2014, Vienna, Austria. pp.121-130, �10.13140/2.1.3326.9761�. �hal-01113341�

On Egalitarian Belief Merging

Patricia Everaere

LIFL-CNRS, Universit´e Lille 1, France

patricia.everaere@univ-lille1.fr

S´ebastien Konieczny Pierre Marquis CRIL-CNRS, Universit´e d’Artois, France

{konieczny,marquis}@cril.fr

Abstract

Belief merging aims at defining the be- liefs of a group from the beliefs of each member of the group. It is re- lated to more general notions of ag- gregation from economy (social choice theory). Two main subclasses of belief merging operators exist: majority oper- ators which are related to utilitarianism, and arbitration operators which are re- lated to egalitarianism. Though utilitar- ian (majority) operators have been ex- tensively studied so far, there is much less work on egalitarian operators. In or- der to fill the gap, we investigate pos- sible translations in a belief merging framework of some egalitarian proper- ties and concepts coming from social choice theory, such as Sen-Hammond eq- uity, Pigou-Dalton property, median, and Lorenz curves. We study how these prop- erties interact with the standard rational- ity conditions considered in belief merg- ing. Among other results, we show that the distance-based merging operators sat- isfying Sen-Hammond equity are mainly those for which leximax is used as the aggregation function.

Introduction

The aim of belief merging is to define a coherent belief base from a set of jointly incoherent belief bases, representing the beliefs of a group of agents. The rationality properties of be- lief merging operators have been studied in [Revesz, 1997;

Lin and Mendelzon, 1999; Konieczny and Pino P´erez, 2002a;

Konieczny, Lang, and Marquis, 2004; Everaere, Konieczny, and Marquis, 2010b]. Especially, in [Konieczny and Pino P´erez, 2002a], a number of postulates characterizing the so called IC merging operators have been identified. At the same time, many definitions of propositional belief merg- ing operators have been pointed out. Most of these opera- tors are distance-based ones, which means that they can be defined using a distance between interpretations and an ag- gregation function [Revesz, 1997; Lin and Mendelzon, 1999;

Konieczny, Lang, and Marquis, 2004; Everaere, Konieczny, and Marquis, 2010a].

Two main subclasses of IC belief merging operators have been defined in [Konieczny and Pino P´erez, 2002a]: majority operators, which solve conflicts using majority, and arbitra- tion operators, which try to find a consensual result. However, while many distance-based majority merging operators have been defined in the literature, very few arbitration operators have been identified so far. To be more precise, the only ar- bitration IC merging operators we are aware of are distance- based operators usingleximaxas aggregation function.

Majority merging operators are closely related to the util- itarian social welfare approaches, where the aim is to de- termine solutions with the best aggregated utility [Harsanyi, 1955; Moulin, 1988; Sen, 2005]. On the other hand, arbi- tration operators are related to the egalitarian social wel- fare approaches, where the objective is to find solutions which are as fair as possible; this usually means that they give as much as possible to the poorest agents. To this ex- tent, poverty measures [Rawls, 1971; Gini, 1921; Sen, 1973;

Dutta, 2002] are relevant to the design of egalitarian ap- proaches.

The aim of this paper is to introduce and study new egali- tarian operators, by exhibiting other fairness conditions than arbitration and by pointing out belief merging operators sat- isfying them. Our methodology to reach our goal consists in investigating equity conditions considered in social choice theory [Arrow, Sen, and Suzumura, 2002] in order to deter- mine if they can be reasonably imported in the belief merg- ing setting. Thus, in the following, we translate to the be- lief merging framework two egalitarian conditions coming from social choice theory: Sen-Hammond equity, and Pigou- Dalton property. We show that the distance-based merging operators satisfying Sen-Hammond equity are mainly those for which leximax is used as the aggregation function. We also introduce two new families of belief merging operators, based respectively on the median and on an aggregated sum (Lorenz curves). We identify the rationality properties satis- fied by these operators and study in particular their egalitarian behaviour.

The rest of the paper is organized as follows. First we give some preliminaries on propositional belief merging, focusing on IC merging operators and distance-based operators. Then we show how Sen-Hammond equity condition, and Pigou- Dalton property can be expressed in the belief merging set-

ting. Since we want to define other egalitarian distance-based merging operators than those based onleximax, some IC pos- tulates must be relaxed; we define a general family of be- lief merging operators, called pre-IC merging operators; they are obtained by relaxing two IC postulates. On this ground, we define the family of median distance-based merging op- erators. We show that the operators of this family based on theleximed^kaggregation functions are pre-IC operators, and those for whichk ≥ 0.5 satisfy also the arbitration postu- late(Arb), but not the Pigou/Dalton property. Finally we in- troduce the family of cumulative sum distance-based merg- ing operators; we identify in this family some pre-IC opera- tors, and among them an operator satisfying the Pigou-Dalton property. Some proofs are omitted for space reasons.

On Propositional Belief Merging

We consider a propositional languageLdefined from a finite set of propositional variablesP and the usual connectives.

An interpretation (or state of the world)ωis a total function fromP to{0,1}.Ωis the set of all interpretations. An inter- pretation is usually denoted by a bit vector whenever a strict total order onP is specified. An interpretationωis a model of a formulaφ∈ Lif and only if it makes it true in the usual truth functional way.|=and≡denote logical entailment and equivalence, respectively.[φ]denotes the set of models of a formulaφ, i.e.,[φ] ={ω∈Ω|ω|=φ}.

AbaseKdenotes the set of beliefs of an agent, it is a finite set of propositional formulae, interpreted conjunctively (i.e., viewed as the conjunction of its elements).

AprofileEdenotes a group ofnagents that are involved in the merging process; formally E is given by a muti-set {K₁, . . . , K_n}of bases. VE denotes the conjunction of all elements ofE, andtdenotes the multi-set union. Two multi- setsE={K1, . . . , Kn}andE⁰={K₁⁰, . . . , K_n⁰}are equiv- alent, noted E ≡ E⁰, iff there exists a permutation πover {1, . . . , n}s.t. for eachi∈1, . . . , n, we haveKi≡K_π(i)⁰ .

Anintegrity constraintµis a formula restricting the possi- ble results of the merging process.

Amerging operator4is a function which associates with a profileEand an integrity constraintµamergedbase4µ(E).

The logical properties given in [Konieczny and Pino P´erez, 2002a] for characterizing IC belief merging operators are:

Definition 1 A merging operator4is anIC merging opera- toriff it satisfies the following properties:

(IC0) 4_µ(E)|=µ

(IC1) Ifµis consistent, then4µ(E)is consistent (IC2) IfV

Eis consistent withµ, then4µ(E)≡V E∧µ (IC3) IfE₁≡E₂andµ₁≡µ₂, then4_µ1(E₁)≡ 4_µ2(E₂) (IC4) IfK1|=µandK2|=µ, then4µ({K1, K2})∧K1is

consistent if and only if4µ({K1, K2})∧K2is consis- tent

(IC5) 4µ(E1)∧ 4µ(E2)|=4µ(E1tE2) (IC6) If4_µ(E₁)∧ 4_µ(E₂)is consistent,

then4_µ(E₁tE₂)|=4_µ(E₁)∧ 4_µ(E₂) (IC7) 4µ₁(E)∧µ2|=4µ₁∧µ₂(E)

(IC8) If4µ1(E)∧µ2is consistent, then4µ1∧µ2(E)|=4µ1(E)

See [Konieczny and Pino P´erez, 2002a] for explanations on these properties. Two subclasses of IC merging operators are also defined in [Konieczny and Pino P´erez, 2002a]:

Definition 2 AnIC majority operatoris an IC merging oper- ator which satisfies the following majority property:

(Maj) ∃n 4µ(E1tE2t. . .tE2

| {z }

n

)|=4µ(E2)

AnIC arbitration operatoris an IC merging operator which satisfies the following arbitration property:¹

(Arb) If

4µ₁(K1)≡ 4µ₂(K2)

4µ₁↔¬µ2({K1, K2})≡(µ1↔ ¬µ2) µ16|=µ2

µ26|=µ1

then4µ₁∨µ2({K1, K2})≡ 4µ₁(K1)

Majority operators solve conflicts using majority. (Maj) says that if one duplicates sufficiently many times a profile E₂, then the result of the merging of E₁ with the E₂ du- plications will obey the choices of the profile E₂. Arbitra- tion operators try to find a consensual result. See the corre- sponding condition 8 in Definition 3 and Example 1 which illustrates how this property gives a preference to consensual (“median”) choices of interpretations.

The representation theorems enable to interpret these log- ical properties as constraints on the choice of interpretations for defining the models of the resulting belief base:

Definition 3 A syncretic assignmentis a function mapping each profileEto a total pre-order≤E2overΩsuch that for any profilesE, E1, E2and for any belief basesK, K⁰the fol- lowing conditions hold:

1. Ifω|=V

Eandω⁰ |=V

E, thenω'Eω⁰ 2. Ifω|=V

Eandω⁰ 6|=V

E, thenω <Eω⁰ 3. IfE1≡E2, then≤E₁=≤E₂

4. ∀ω|=K∃ω⁰|=K⁰ω⁰ ≤_{K,K⁰_}ω

5. Ifω≤E₁ ω⁰andω≤E₂ ω⁰, thenω≤E₁tE2ω⁰ 6. Ifω <E₁ ω⁰andω≤E₂ ω⁰, thenω <E₁tE2ω⁰

A majority syncretic assignment is a syncretic assignment which satisfies the following condition:

7. Ifω <E₂ ω⁰, then∃n ω <E₁tEⁿ₂ ω⁰

Afair syncretic assignmentis a syncretic assignment which satisfies the following condition:

8. Ifω <_K1 ω⁰,ω <_K2 ω⁰⁰, and ω⁰ '_{K1,K2} ω⁰⁰, then ω <_{K1,K2}ω⁰

Proposition 1 ([Konieczny and Pino P´erez, 2002a]) A me- rging operator4 is an IC merging operator (resp. an IC majority, an IC arbitration operator) iff there exists a syn- cretic assignment (resp. a majority syncretic assignment, a fair syncretic assignment) that maps each profileEto a total pre-order≤EoverΩsuch that[4µ(E)] = min([µ],≤E).

1WhenE={K}we note4µ(K)instead of4µ({K}).

2For every pre-order≤,<denotes its strict part and'the cor- responding indifference relation. Furthermore, we will use≤Kas a short for≤{K}, and≤ωas a short for any≤{K_ω}whereωis the unique model ofKω.

Let us now give some examples of IC merging opera- tors using the familly of distance-based merging operators [Konieczny, Lang, and Marquis, 2004]:

Definition 4 Adistance³ between interpretations is a func- tiond: Ω×Ω→IR⁺such that for anyω1,ω2∈Ω:

• d(ω₁, ω₂) =d(ω₂, ω₁)

• d(ω1, ω2) = 0iffω1=ω2

Usual distances considered in merging are the Hamming distancedH:dH(ω1, ω2)is the number of propositional let- ters on which the two interpretations differ (this corresponds to the 1-norm distance, also referred to as the Manhattan dis- tance) and the drastic distancedD, defined asdD(ω1, ω2) = 0 if ω1 = ω2, and= 1otherwise (this corresponds to the infinity-norm distance, also known as Chebyshev distance).

Definition 5 Anaggregation functionis a mapping⁴ f from R^mtoR, which satisfies:

• ifxi ≥x⁰_i, then (non-decreasingness) f(x1, ..., xi, ..., xm)≥f(x1, ..., x⁰_i, ..., xm)

• f(x1, . . . , xm) = 0if∀i, xi= 0 (minimality)

• f(x) =x (identity)

• Ifσis a permutation over{1, . . . , m}, then

f(x₁, . . . , x_m) =f(x_σ(1), . . . , x_σ(m)) (symmetry) Some additional properties can be considered forf, espe- cially:

• ifxi > x⁰_i, then (strict non-decreasingness) f(x1, ..., xi, ..., xm)> f(x1, ..., x⁰_i, ..., xm)

Definition 6 Letdandf be respectively a distance between interpretations and an aggregation function. The distance- based merging operator 4^d,f is defined by [4^d,f_µ (E)] = min([µ],≤E), where the total pre-order≤E onΩis defined in the following way (withE={K1, . . . , Kn}):

• ω≤Eω⁰iffd(ω, E)≤d(ω⁰, E)

• d(ω, E) =f(d(ω, K₁), . . . , d(ω, K_n))

• d(ω, K) = min_ω⁰_|=Kd(ω, ω⁰)

For usual aggregation functions, whatever the chosen dis- tance, the corresponding distance-based operators exhibit good logical properties:

Proposition 2 ([Konieczny and Pino P´erez, 2002a]) For any distanced:

• iff is thesum Σ,leximin⁵, orΣⁿ(the sum of then^th powers), then4^d,fis an IC majority operator.

• iff isleximax, then4^d,fis an IC arbitration operator.

3Formally, we work with pseudo-distances since triangular in- equality is not required, but for the sake of simplicity we will abuse words in this way.

4Strictly speaking, it is a family of mappings, one for eachm.

5The leximin (resp. leximax) aggregation function selects the in- terpretations that are minimal for the lexicographic order, once the distances are sorted into the increasing (resp. decreasing) order.

Conditions for Egalitarian Merging

The only egalitarian property that has been proposed so far for belief merging is thearbitrationproperty, represented by the(Arb)postulate (or the corresponding semantic condition 8 on syncretic assignments). So a key issue we would like to address is to determine whether other egalitarian properties are possible in the belief merging framework, and, if so, how they relate with arbitration.

If one looks closely at condition 8 in Definition 3, it is clear that the arbitration property only imposes some constraints on the merging of profiles consisting of two belief bases. Then the IC properties(IC5)and(IC6)ensure the propagation of the constraints on profiles of any size.

We now propose a first alternative condition, coming from social choice theory, for characterizing egalitarian behaviour in belief merging. This condition, proposed by Hammond in [Hammond, 1976] is known in the literature as the Sen- Hammond equity condition [Sen, 1997; Suzumura, 1983].

This condition in this setting is expressed in the following way:

Definition 7 (Sen97) If person i is worse off than person j both in x and in y, and if i is better off himself in x than in y, while j is better off in y than in x, and if furthermore all others are just as well off in x as in y, then x is socially at least as good as y.

It is translated in the belief merging setting as constraints on the total pre-orders associated with the input profiles. These constraints concern profiles of arbitrary size, and not only those consisting of two bases. Before doing it, we first need to define a notion of the respective “satisfaction” of two bases given an interpretation:

Definition 8 Given a merging operator 4 defining an as- signment which maps every profileEto a total pre-order≤E

overΩ, given an interpretationω, and two basesK₁andK₂, we say thatK₁isbetter thanK₂givenω, denotedK₁<_ωK₂, iff∃ω1|=K₁,∀ω2|=K₂, ω₁<_ωω₂.

We can now give the translation of the Sen-Hammond Eq- uity (SHe) condition:

Definition 9 (Condition (SHe)) LetE={K1, . . . , Kn}.

ω <K₁ ω⁰ ω⁰<K₂ω

∀i6= 1,2ω'K_i ω⁰ K1<ωK2

K1<ω⁰ K2











=⇒ ω⁰≤Eω

When distance-based merging operators are considered, this condition is equivalent to:

Definition 10 (Condition (SHE))

d(ω, K₁)< d(ω⁰, K₁)< d(ω⁰, K₂)< d(ω, K₂)

∀i6= 1,2d(ω, K_i) =d(ω⁰, K_i)

=⇒

f(d(ω⁰, K1), d(ω⁰, K2), . . . , d(ω⁰, Kn))≤ f(d(ω, K₁), d(ω, K₂), . . . , d(ω, K_n))

Proposition 3 A distance-based merging operator∆^d,f sat- isfies(SHE)if and only if it satisfies(SHe).

ω1

ω2

ω₃ ω₄

≤K₁

ω3

ω2

ω₁

≤K₂

ω4

Figure 1: Egalitarian behaviour - (Arb)

Proof: (SHE) is exactly the translation of (SHe) when the merging operator is defined from a distance and an ag- gregation function. Indeed, in this case, K₁ <_ω K₂ iff

∃ω₁ |= K₁,∀ω₂ |= K₂, ω₁ <_ω ω₂. Then ∃ω₁ |= K1,∀ω2 |= K2, d(ω, ω1) < d(ω, ω2): ∃ω1 |= K1, d(ω, ω1)< min_ω2|=K2d(ω, ω2) =d(ω, K2). Hence we have d(ω, K1)< d(ω, K2).

ω <K₁ ω⁰⇔d(ω, K1)< d(ω⁰, K1) ω⁰<K₂ω⇔d(ω⁰, K2)< d(ω, K2)

∀i6= 1,2ω'K_i ω⁰ ⇔ ∀i6= 1,2d(ω, Ki) =d(ω⁰, Ki) K1<ωK2⇔d(ω, K1)< d(ω, K2)

K1<ω⁰ K2⇔d(ω⁰, K1)< d(ω⁰, K2)

Using transitivity, we get: d(ω, K1) < d(ω⁰, K1) <

d(ω⁰, K2) < d(ω, K2)and∀i 6= 1,2ω 'K_i ω⁰ ⇔ ∀i 6=

1,2d(ω, Ki) =d(ω⁰, Ki).Henceω⁰ ≤E ω⇔f(d(ω⁰, K1), d(ω⁰, K2), . . . , d(ω⁰, Kn)) ≤ f(d(ω, K1), d(ω, K2), . . . ,

d(ω, Kn)). 2

This Sen-Hammond Equity condition expresses the follow- ing idea: compare two “situations”ω1andω2that are equally good for all the agents except two of them. For these two agents one of them (K1) is in the two situations better than the other agent (K2). Then the fairer situation is the one that gives the more to the less satisfied agent (K2).

This condition looks close to the arbitration condition (compare(SHe)with condition 8 of Definition 3), but the two conditions are logically independent. Let us now illustrate on a simple example how they differ:

Example 1 Consider a propositional language over two variables and a distance-based merging operator4^d,f. Sup- pose that the distancedon which4^d,fis built is the shortest path distance on the following graph:

ω1 1 ω2

1 ω3 1 ω4.

Figure 1 illustrates the case when the unique model ofK₁ isω₁and the unique model ofK₂isω₃. It is clear here that ω1andω3play symmetrical roles, so that they cannot be dis- tinguished when we mergeK1andK2. However, on this ex- ample,ω2appears as a more consensual choice thanω1and ω3 for this merging. Accordingly,(Arb) imposes thatω2 is strictly preferred toω1andω3in≤_{K1,K2}.

Suppose now that we want to mergeE⁰={K₁⁰, K₂⁰}where [K₁⁰]={ω1, ω3} and [K₂⁰] ={ω4}, and that the choice to be made is betweenω1andω2(i.e., the integrity constraintµsat- isfies[µ] = {ω1, ω2}). This is illustrated by Figure 2.(Arb) imposes no constraint on this choice. But clearly, whatever the choice between{ω1},{ω2}, or{ω1, ω2}, the result will be closer toK₁⁰ than toK₂⁰. So a simple equity argument is to

ω1 ω3

ω2 ω4

≤_K⁰

1

ω4

ω3

ω₂ ω₁

≤_K⁰

2

Figure 2: Egalitarian behaviour - (SHE)

commit to the choice that is the best for the farest baseK₂⁰, so to consider thatω₂is strictly preferred toω₁in≤E⁰. This is what(SHe)gives.

Unfortunately, the family of distance-based IC operators satisfying condition(SHE)looks rather limited:

Proposition 4 Let d be any distance andf be any aggre- gation function satisfying strict non-decreasingness. The IC merging operator∆^d,fsatisfies condition(SHE)if and only iff =leximax.

Proof: We know that∆^d,leximaxis an IC operator such that leximax satisfies strict-decreasingness. It is easy to check that∆^d,leximaxsatisfies(SHE), so we have mainly to prove the converse implication.

Consider a distancedand an aggregation functionf satisfy- ing strict non-decreasingness, and suppose that∆^d,fsatisfies (SHE).

LetX= (x1, . . . , xn)andX⁰ = (x⁰₁, . . . , x⁰_n)be two vectors of distances to a profileE, ordered in the descending way (∀i, xi = d(ω, Ki) andxi ≥ xi+1; x⁰_i = d(ω⁰, Ki)and x⁰_i≥x⁰_i+1). We have to show that:

X <leximaxX⁰⇔f(X)< f(X⁰) (1) and

X 'leximaxX⁰⁶⇔f(X) =f(X⁰) (2) We start with statement (1). Suppose thatX <_leximax X⁰. We know that∃ls.t.∀i < l, xi=x⁰_iandx_l< x⁰_l.

Case 1: l = n or ∀i > l, xi = x⁰_i: using strict non- decreasingness we getf(X)< f(X⁰).

Case 2: Suppose that∃k > l, xk 6=x⁰_k and∀i 6=k, l, xi = x⁰_i.

• If x_k ≤ x⁰_k, then using strict-decreasingness (as x_l< x⁰_l), we getf(X)< f(X⁰).

• Ifxk > x⁰_k, then we have x⁰_l > xl ≥ xk > x⁰_k. Consider a vectorY = (y₁, . . . , y_n)s.t.∀i 6=k, l, y_i=x⁰_iandy_k, y_lsuch thatx⁰_l> y_l> y_k> x_l. As we have∀i6=k, l,y_i=x⁰_iandx⁰_l> y_l> y_k > x⁰_k, because∆^d,fsatisfies(SHE), we can conclude that f(Y)≤f(X⁰). Furthermore∀i6=k, l,yi=xiand yl > yk > xl ≥ xk, using strict decreasingness, we get f(Y) > f(X). By transitivity, we obtain f(X)< f(X⁰).

6IfX'leximaxX⁰, then∀i, xi=x⁰i, so X=X’.

Case 3: Let us consider now the general case.∀i < l, xi = x⁰_i, xl < x⁰_l andl < n. We definen−l+ 1 vectors Y^r= (y^r₁, . . . , y_n^r), forr= 0ton−l:

y_i^r=







x_iifi < l x_l+_n−l^r (x⁰_l−x_l)ifi=l

xiifl+ 1≤i≤n−r x⁰_iifn−r+ 1≤i≤n

We haveY⁰ = X andY^n−l = X⁰, and any two con- secutive vectorsY^randY^r+1 differ only on two com- ponents, namelyy_l^randy_l^r+1on the one hand, andy_n−r^r andy^r+1_n−ron the other hand. Furthermore,Y^r<_leximax Y^r+1, because∀i < l, y_i^r = y^r+1_i ,∀i ≤l, y_i^r ≥y^r_i+1,

∀i ≤ l, y_i^r+1 ≥ y^r+1_i+1 andy^r_l < y_l^r+1. Using Case 2, we can conclude thatf(Y^r)< f(Y^r+1). By transitivity we getf(Y⁰) = f(X) < f(Y^n−l) = f(X⁰), and the conclusion follows.

Suppose now thatf(X)< f(X⁰). IfX 'leximax X⁰, then X =X⁰andf(X) = f(X⁰): contradiction. IfX >_leximax X⁰, then taking advantage of the first part of the proof, we get f(X)> f(X⁰): contradiction. SoX <_leximax X⁰.

Consider now statement (2).

IfX'leximaxX⁰, thenX =X⁰andf(X) =f(X⁰).

Suppose now f(X) = f(X⁰). If X >leximax X⁰, then taking advantage of the first part of the proof, we get f(X) > f(X⁰): contradiction. If X <leximax X⁰, then taking advantage of the first part of the proof, we get f(X)< f(X⁰): contradiction. SoX =leximax X⁰. 2 Let us stress here that the condition of strict non- decreasingness is quite natural and not very demanding. Ac- tually all the aggregation functions giving rise to IC merging operators we are aware of (includingΣ,leximax,leximin, Σⁿ, etc.) satisfy non-decreasingness.

Given this proposition, defining other egalitarian distance- based merging operators requires to focus on other equity principles, or to weaken some IC postulates. We explore both ways in the following.

Thus, we first focus on another egalitarian condition from the social choice literature, namely Pigou-Dalton transfer principle [Dalton, 1920]. The idea underlying it is that ev- ery transfer from the most satisfied agent to the least satisfied one decreases the inequalities:

Definition 11 Let f be an aggregation function. f satis- fies the Pigou-Dalton condition if for all vectors X = (x1, . . . , xn)andX⁰= (x⁰₁, . . . , x⁰_n), ifx1 < x⁰₁≤x⁰₂< x2

and x⁰₁ −x1 = x2 −x⁰₂ and ∀i 6= 1,2, xi = x⁰_i then f(X)< f(X⁰).

This principle states that ifX⁰can be obtained fromX by just making some satisfaction transfer from a well-satisfied agent to a less satisfied one, without changing the fact that the first one is still more satisfied than the second one, then X⁰is fairer thanX.

This principle can be translated as follows for distance- based merging:

Definition 12 (Condition (PD)) If∃kandls.t.d(ω, Kk)<

d(ω⁰, Kk) ≤ d(ω⁰, Kl) < d(ω, Kl) and d(ω⁰, Kk) − d(ω, K_k) = d(ω, K_l)−d(ω⁰, K_l) and ∀i 6= k and i 6=

l, d(ω, K_i) =d(ω⁰, K_i)thenω⁰<_Eω.

Of course not all distance-based IC operators satisfy the (PD) condition. However, it is satisfied by the well-known arbitration operators based onleximax:

Proposition 5 Letdbe any distance.

• ∆^d,leximax satisfies the(PD)condition.

• ∆^d,Σand∆^d,leximin do not satisfy the(PD)condition.

Pre-IC Operators

We now define a general family of belief merging operators, called pre-IC merging operators, obtained by relaxing the two postulates(IC5)and(IC6)into two natural conditions used in other aggregation theories contexts.

Definition 13 A merging operator4ispre-IC merging op- erator iff it satisfies(IC0) to(IC4),(IC7)to(IC8)and the following properties:

(IC5b) 4µ(K1) ∧ 4µ(K2) ∧ . . . ∧ 4µ(Kn) |= 4µ({K1, K2, . . . , Kn})

(IC6b) If4µ(K1)∧ 4µ(K2)∧. . .∧ 4µ(Kn)is consistent, then4µ({K1, K2, . . . , Kn})|=4µ(K1)∧ 4µ(K2)∧ . . .∧ 4µ(K_n)

Thus, switching from IC operators to pre-IC ones simply consists in replacing the postulates (IC5) and(IC6) postu- lates by the weaker postulates(IC5b)and(IC6b). Indeed, it is easy to prove that(IC5b)(resp.(IC6b)) is implied by(IC5) (resp.(IC6)). As a consequence, we have:

Proposition 6 Every IC merging operator is a pre-IC merg- ing operator.

Let us now present a representation theorem suited to the pre-IC family:

Definition 14 Apre-syncretic assignmentis a function map- ping each profileEto a total pre-order≤EoverΩsuch that for any profilesE,E1,E2and for any belief basesK, K⁰, the following conditions hold:

1. Ifω|=Eandω⁰|=Ethenω≡_Eω⁰ 2. Ifω|=Eandω⁰6|=Ethenω <Eω⁰ 3. IfE₁≡E₂then≤E1=≤E2

4. ∀ω|=K,∃ω⁰ |=K⁰ω⁰≤_{K,K0}ω 5b. If∀i ω≤K_iω⁰thenω≤_{K1,...,Kn}ω⁰ 6b. If∀i ω≤Kiω⁰and∃k ω <Kk ω⁰

thenω <_{K1,...,Kn}ω⁰

Conditions 5b and 6b are direct translations of Pareto con- ditions, which are usual conditions in social choice, multi- criteria decision making, etc. So they should be considered as minimal aggregation conditions to be satisfied. Conditions 5 and 6 (Definition 3) are much more demanding, since they constraint all unions of two profiles.

Proposition 7 A merging operator∆is a pre-IC merging op- erator iff there exists a pre-syncretic assignment that maps each profileEto a total pre-order≤EoverΩsuch that

[∆µ(E)] =min([µ],≤E).

Proof:

Only If Let∆ be a pre-IC merging operator. We associate with it a pre-syncretic assignment as follows: for any profileE={K1, K₂, . . . , K_n}, we define≤Ebyω≤E

ω⁰ if and only ifω |= ∆_µ(E)where[µ] = {ω, ω⁰}.

From [Konieczny and Pino P´erez, 2002b], we know that

≤_Eis a pre-order; furthermore, as∆satisfies (IC0-IC4) and (IC7-IC8), conditions1to4are satisfied by the as- signment. We have then only to check conditions 5’ and 6’.

Suppose that ∀i ∈ {1, . . . , n}, ω ≤K_i ω⁰. Then

∀i ∈ {1, . . . , n}, ω |= ∆_{ω,ω⁰_}(Ki). Let µ such that [µ] = {ω, ω⁰}: ∀i ∈ {1, . . . , n}, ω |= ∆µ(Ki), so ω |= ∆µ(K1) ∧ ∆µ(K2) ∧ . . .∆µ(Kn). From (IC5’), we know that ∆µ(K1) ∧

∆µ(K2) ∧ . . .∆µ(Kn) |= ∆µ({K1, K2, . . . , Kn}), so ω |= ∆µ({K1, K2, . . . , Kn}). Therefore, ω |=

∆µ({K1, K2, . . . , Kn}):ω ≤E ω⁰, hence condition 5’

is satisfied.

Suppose that∃k, ω <_Kk ω⁰ and∀i 6= k, ω ≤_Ki ω⁰. Then ω |= ∆_µ(K_k), ω⁰ 6|= ∆_µ(K_k) and

∀i 6= k, ω |= ∆_µ(K_i). We have ω |= ∆_µ(K_k), ω 6|= ∆µ(Kk) and ∀i 6= k, ω |= ∆µ(Ki), so ω |= ∆µ(K1) ∧ ∆µ(K2) ∧ . . . ∧ ∆µ(Kn) and ω⁰ 6|= ∆µ(K1) ∧ ∆µ(K2) ∧ . . . ∧ ∆µ(Kn):

[∆µ(K1)∧∆µ(K2)∧. . . ∧∆µ(Kn)] = {ω}. From (IC6’), and as∆µ(K1)∧∆µ(K2)∧. . .∧∆µ(Kn)is consistent, we know that ∆µ({K1, K2, . . . , Kn}) |=

∆µ(K1) ∧ ∆µ(K2) ∧ . . . ∧ ∆µ(Kn), so {ω} = [∆µ({K1, K2, . . . , Kn})] and ω⁰ 6|= ∆µ({K1, K2, . . . , Kn}). Therefore, ω |= ∆µ({K1, K2, . . . , Kn}) and ω⁰ 6|=

∆µ({K1, K2, . . . , Kn}): ω <E ω⁰: so condition 6’ is satisfied.

If Consider a pre-syncretic assignment mapping each belief setEto a total pre-order≤Eover interpretations. We de- fine an operator∆as follows:[∆µ(E)] =min([µ],≤E

).

From [Konieczny and Pino P´erez, 2002b], we know that

∆satisfies (IC0-IC4) and (IC7-IC8). Let us show that∆ satisfies (IC5’-IC6’).

Suppose that∆µ(K1)∧∆µ(K2)∧. . .∆µ(Kn)is con- sistent (if not, (IC5’-IC6’) is trivially satisfied).

Let ω |= ∆_µ(K₁) ∧ ∆_µ(K₂) ∧ . . . ∧ ∆_µ(K_n).

So ∀ω⁰ |= µ, ω ≤_Ki ω⁰. From 5’, we know that ∀ω⁰ |= µ, ω ≤_{{K1,K2,...,Kn}} ω⁰: then ω |= ∆_µ({K₁, K₂, . . . , K_n}). We can con- clude that ∆µ(K1) ∧ ∆µ(K2) ∧ . . . ∧ ∆µ(Kn) |=

∆µ({K1, K2, . . . , Kn})and (IC5’) is satisfied.

To show (IC6’), suppose that ∆µ({K1, K2, . . . , Kn}) 6|= ∆µ(K1)∧∆µ(K2)∧ . . . ∧∆µ(Kn). So ∃ω s.t.

ω |= ∆µ({K1, K2, . . . , Kn}) and ω 6|= ∆µ(K1) ∧

∆µ(K2)∧. . .∧∆µ(Kn). As∆µ(K1)∧∆µ(K2)∧. . .∧

∆µ(Kn)is consistent, there is an interpretation ω⁰ |=

∆µ(K1)∧∆µ(K2)∧. . .∧∆µ(Kn). So∀i ω⁰ ≤Ki ω and asω6|= ∆µ(K1)∧∆µ(K2)∧. . .∧∆µ(Kn),∃ks.t.

ω6|= ∆_µ(K_k):ω⁰<_Kkω.

With condition 6’, we know that

ω⁰ <_{K1,K₂,...,K_n} ω: contradiction with the fact that ω |= ∆µ({K1, K2, . . . , Kn}). Hence,

∆_µ({K1, K₂, . . . , K_n})|= ∆_µ(K₁)∧∆_µ(K₂)∧. . .∧

∆_µ(K_n): (IC6’) is satisfied.

2 As one can expect, it is much easier to satisfy the pre-IC merging conditions than IC merging ones. We can for in- stance show that:

Proposition 8 Ifdis any distance andf is any aggregation function satisfying strict non-decreasingness, then the merg- ing operator∆^d,fis a pre-IC merging operator.

One can compare this result with a corresponding one about distance-based IC merging operators, reported in [Konieczny, Lang, and Marquis, 2004], and showing that the aggregation function f has to satisfy two additional condi- tions in order to guarantee that the distance-based merging operators given bydandfare IC merging operators.

Median Operators

In this section we define a new family of merging opera- tors using generalized median aggregation functions. Inter- estingly, some operators of this family are pre-IC merging op- erators and they satisfy(Arb). The idea of using the median value is very motivated by trying to be as fair as possible. In- stead of focusing on a unique aggregation function, we study a full family ofk-median aggregation functions, inspired by the phantom voters voting rules of [Moulin, 1988].

Definition 15 Letk∈]0,1]be a real number, thek−median med^k({x1, . . . , xn})of a multi-setX={x1, . . . , xn}of val- ues from a totally ordered set, is the valuem^k=x_σ(dn∗ke)of X, whereσis a permutation ofX where thexiare sorted in ascending order (d.edenotes the ceiling function).

Fork= 0.5, the usual notion of median is retrieved.

In many cases thesek−median functionsmed^kare not dis- criminative enough, just likeminandmaxfunctions. So, we can define k-leximedian operators, noted k-leximed, which are to med^k whatleximin (resp.leximax) is tomin (resp.

max).

Definition 16 LetL1andL2be two multi-sets consisting of nelements from a totally ordered set:

L1≤^k_leximed L2iff

• med^k(L₁)<med^k(L₂)or

• med^k(L1) =med^k(L2)and

L1\ {med^k(L1)} ≤^k_leximed L2\ {med^k(L2)}

Let us define successively thek-median merging operators, and thek-leximedian ones:

Definition 17 LetE = {K1, ..., Kn} be a profile,da dis- tance between interpretations and k ∈]0,1] a real num- ber. Let d^d,k_med(ω, E) = med^k(d(ω, K1), d(ω, K2), . . . ,

K₁ K₂ K₃ distance vector med^0.5

000 0 1 1 (0,1,1) 1

001 1 0 2 (0,1,2) 1

010 1 2 2 (1,2,2) 2

011 2 1 3 (1,2,3) 2

100 0 2 0 (0,0,2) 0

101 1 1 1 (1,1,1) 1

110 1 3 1 (1,1,3) 1

111 2 2 2 (2,2,2) 2

Table 1: Merging with∆^dH,med0.5

d(ω, K_n)).We define[∆^d,med_µ ^k(E)]by

[∆^d,med_µ ^k(E)] ={ω|=µ|d^k_med(ω, E)is minimal}.

Here is an example illustrating the behaviour ofk-median operators:

Example 2 We consider a profileEof three bases such that [K₁] ={000,100},[K₂] ={001}and[K₃] ={100}. There is no integrity constraint (µ=>), we use the Hamming dis- tanced_H and the valuek= 0.5for the “standard” median.

The computations are presented in Table 1.

We get[∆^d_>^H,med0.5(E)] ={100}. The only selected inter- pretation is 100, because the best vector is(0,0,2), with a median value of0.

Some specific values ofklead to well-known merging op- erators: given a profileE ofnbases,∆^med, with ∈]0,_n¹] corresponds to the min operator, and ∆^medα, with α ∈ ]ⁿ⁻¹_n ,1]to themaxoperator.

Let us now make precise the rationality postulates satisfied by these operators:

Proposition 9 For any real number k ∈]0,1]and any dis- tanced,∆^d,medksatisfies(IC0),(IC1),(IC3),(IC4),(IC7), (IC8)and (IC5b).(IC2),(IC5),(IC6), (IC6b),(Maj)and (Arb)are not satisfied in general.

Proof: For space reasons, we provide only the less obvious proofs.

(IC2) Suppose k = 0.5, d = d_H and [K₁] = {00}, [K₂] = {00, 01} and [K₃] = {00, 01}. We have d^d_m^H,0.5(00,{K₁, K₂, K₃}) = 0 andd^d_m^H,0.5(01,{K₁, K₂, K₃}) = 0: (IC2) is not satisfied. Note that for k ∈]ⁿ⁻¹_n ,1]and any distanced,(IC2)is satisfied (the aggregation functionmaxis retrieved).

(IC4) SupposeK1|=µ,K2|=µand∆^d,med_µ ^k({K1, K2})∧

K₁ consistent. Let ω₁ |= ∆^d,med_µ ^k({K₁, K₂})∧K₁. d^d,k_m (ω₁, {K₁, K₂}) = med^k(d(ω₁, K₁), d(ω₁, K₂))

= med^k(0, d(ω1, K2)) = med^k(0, d(ω1, ω2)), where ω2 |= K2 s.t. d(ω1, K2) = d(ω1, ω2). Then d(ω2, K1) ≤ d(ω1, ω2): d^d,k_m (ω2,{K1, K2}) = med^k(d(ω2, K1),0)≤ d^d,k_m (ω1,{K1, K2}). As a con- sequenceω2is selected.

(IC5) and (IC6) Suppose k = 0.5, and ω1 s.t.

d^d,0.5_m (ω1, E1) = med^0.5(0,0) = 0,d^d,0.5_m (ω1, E2) = med^0.5(3,4,4) = 4 and accordingly d^d,0.5_m (ω1, E1 t E2) = med^0.5(0,0,3,4,4) = 3. Suppose ω2 s.t.

d^d,0.5_m (ω2, E1) = med^0.5(1,1) = 1,d^d,0.5_m (ω2, E2) = med^0.5(2,6,7) = 6 and d^d,0.5_m (ω₂, E₁ t E₂) = med^0.5(1,1,2,6,7) = 2. Then with [µ] = {ω1, ω₂}, [∆^d,med_µ ^0.5(E₁)] = {ω1}and[∆^d,med_µ ^0.5(E₂)] = {ω1} (and then∆^d,med_µ ^0.5(E₁)and∆^d,med_µ ^0.5(E₂)are jointly consistent), whereas[∆^d,med_µ ^0.5(E₁tE2)] ={ω2}. This example shows that neither(IC5)nor(IC6)is satisfied.

(Arb) Suppose0 < k < 0.5(so that∆^d,medk is equivalent to ∆^d,min). The following example shows that (Arb) is not satisfied. Let ω1,ω2,ω3 be three interpretations s.t. d(ω1, ω3) = 1,d(ω1, ω2) = 1andd(ω2, ω3) = 2.

The basesK1andK2are defined byK1 = {ω2}and K2 = {ω3}; and the constraints µ1 and µ2 are de- fined byµ1={ω1, ω3},µ2 ={ω1, ω2}. Then we have:

∆µ₁(K1)≡∆µ₂(K2)≡ {ω1}

∆µ₁↔¬µ2({K1, K2}))≡µ1↔ ¬µ2≡ {ω2, ω3} µ16|=µ2andµ16|=µ2

)

but ∆µ₁∨µ2({K1, K2})) ≡ {ω2, ω3} and∆µ₁(K1) ≡

{ω1}: contradiction . 2

It turns out that k-median operators satisfy some but not all the expected rationality properties. In particular one very natural postulate(IC2), asking that the result of the merging is just the conjunction of the bases when this conjunction is consistent, is not satisfied (except for values ofkmaking the k-median identical tomax). We also have:

Proposition 10 Ifk≥0.5, then∆^d,medksatisfies(Arb).

No other equity condition is satisfied by such operators:

Proposition 11 Whateverk,∆^d,medk does not satisfy(PD) or(SHE).

Proof: Consider the following counter-example, fork <

0.5: med^k(1,3) = 1 and med^k(2,2) = 2: (1,3) <_medk

(2,2): none of(PD)or(SHE)is satisfied.

Consider the following counter-example, for 0.5 ≤ k <

1: med^k(0,4,4,7) = 4 and med^k(0,4,5,6) = 5 (0,4,4,7)<_medk (0,4,5,6): none of(PD)or(SHE)is sat- isfied.

Consider the following counter-example, for k = 1:

med^k(0,4,4,7) = 7 and med^k(0,3,4,8) = 8 (0,4,4,7) <_medk (0,3,4,8): none of (PD) or (SHE)

is satisfied. 2

Let us now turn to thek-leximedian operators, that satisfy more expected properties:

Definition 18 Let E = {K₁, ..., K_n} be a profile, µ an integrity constraint, d a distance between interpretations and k ∈]0,1] a real number. Define d^k_leximed(ω, E) = leximed^k(d(ω, K1), d(ω, K2), . . . , d(ω, Kn)).Then

[∆^d,leximed_µ ^k(E)] ={ω|=µ|d_leximedk(ω, E)is minimal⁷}

7w.r.t. the lexicographic order.

Again, some standard operators are recovered by consid- ering specific values ofk:∆^d,leximedk withk ∈]0,_n¹[corre- sponds to theleximin operator∆^d,leximin, and∆^d,leximed1 to theleximax operator∆^d,leximax.

As expected∆^d,leximedkoperators satisfy more interesting properties than∆^d,medkones:

Proposition 12 For any distance d and any k ∈]0,1],

∆^d,leximedkis a pre-IC merging operator.

We do not have better than that. In particular, these opera- tors are not IC merging ones in the general case:

Proposition 13 ∆^d,leximedk does not satisfy any of (IC5), (IC6),(Maj)and(Arb)in general.

Concerning egalitarian properties, we reach the arbitration property, but not the other ones:

Proposition 14 Ifk ≥ 0.5, then∆^d,leximedk satisfies(Arb) but does not satisfy(PD)or(SHE)in general.

It is easy to explain why the conditionk≥0.5is needed.

Indeed, we know that whenk goes towards0theleximed^k function goes towardsleximin, and that whenkgoes towards 1leximed^kgoes towardsleximax. So it is natural to obtain an egalitarian behaviour for values between “classical” median (k=0.5) andleximax.

Considering Proposition 4, Proposition 14 shows that re- laxing some IC postulates is a way for escaping from the leximax-based operators while satisfying some egalitarian condition.

Cumulative Sum Merging Operators

A very convenient representation of the inequalities of a dis- tribution of income is the Lorenz curve [Lorenz, 1905]. The principle is to focus on the poorest (least satisfied) agents, by looking first as the utility of the poorest one, then at the sum of the utilities of the two poorest ones, etc. To be more precise, on a Lorenz curve, each elementkof thex-axis corresponds to thekpoorest agents and the value associated with it on the y-axis is the sum of the utilities of those agents.

This curve can be interpreted in different ways to measure how much a distribution is fair. In particular, the fairest dis- tribution is when for anyn, then%poorest agents ownn%

of the income. The well-known Gini coefficient [Gini, 1921;

Sen, 1973; Dutta, 2002], one of the main inequality measures, is the (double of the) area between the Lorenz curve and the fairest distribution.

In the following, we adhere to the notion of cumulative sum for defining a new family of merging operators. We translate the distance between a base and an interpretation into a sat- isfaction value by reversing the scale. Then we compute the cumulative satisfaction vector and take advantage of an ag- gregation function on it.

Let us define formally the cumulative sum merging opera- tors:

Definition 19 Letdbe a distance between interpretations,f an aggregation function,Ea profile andµan integrity con- straint. LetM =max({d(ω, ω⁰)|ω, ω⁰ ∈Ω}). For an inter- pretationω, we consider the vector(wσ(1), wσ(2), . . . , wσ(n))

K₁ K₂ K₃ Cum. Sat. Σ

000 3 2 2 (2,4,7) 13

001 2 3 1 (1,3,6) 10

010 2 1 1 (1,2,4) 7

011 1 2 0 (0,1,3) 4

100 3 1 3 (1,4,7) 12 101 2 2 2 (2,4,6) 12

110 2 0 2 (0,2,4) 6

111 1 1 1 (1,2,3) 6

Table 2: Merging with∆^CS(dH,Σ)

where w_i = M − d(ω, K_i) is the satisfaction value of agent i for the interpretation ω, and σ is the permutation of{1, . . . , n}sorting thew_i in ascending order (the less the least satisfied). Then we define the vector of cumulated satis- faction ofµ,W_d(ω, E) = (W₁, W₂, . . . , W_n), whereW_i = Σⁱ_k=1w_σ(k). Finally, the selected interpretations are the ones which maximize the cumulated satisfactionWd(ω, E):

[∆^CS(d,f)_µ (E)] ={ω|=µ|f(W_d(ω, E))is maximal}.

Let us illustrate the behavior of a cumulative sum merging operator on a simple example:

Example 3 We step back to Example 2. The computations are presented in Table 2, in which the values represent satis- faction values of the bases (the more the best), and not any- more distances (the less the best).

We get[∆^CS(d_> ^H,Σ)(E)] = {000}. The selected interpre- tation is000, because the sum of the cumulative satisfaction vector gives the maximal value of13.

We recover some existing operators as cumulative sum ones:

Proposition 15 Let d be any distance.

• ∆CS(d,leximin)= ∆^d,leximax

• ∆CS(d,leximax)= ∆^d,leximin

Let us finally turn to the logical properties:

Proposition 16 Ifdis any distance, andf is an aggregation function satisfying strict non-decreasingness, then∆^CS(d,f) is a pre-IC merging operator.

• ∆^CS(d,f) does not satisfy(Arb)and(Maj)in the gen- eral case.

• ∆^CS(d,f)does not satisfy(IC5)and(IC6)in the general case.

While(Arb)is not satisfied in general, the Pigou-Dalton condition is ensured for instance when any sum of the m- powers,mvarying, is used as aggregation function:

Proposition 17 For all integerm,∆^CS(d,Σm)satisfies(PD), but(SHE)is not satisfied.

Proof:

∆^CS(d,Σm)satifies Pigou-Dalton property.

Considerωandω⁰s.t.∃kandls.t.d(ω, Kk)< d(ω⁰, Kk)≤ d(ω⁰, Kl) < d(ω, Kl) and d(ω⁰, Kk) − d(ω, Kk) = d(ω, Kl)−d(ω⁰, Kl)and∀i 6= k and i 6= l, d(ω, Ki) = d(ω⁰, Ki). We notegthe valued(ω⁰, Kk)−d(ω, Kk), so we