- Published 2004 in NMR

We propose in this paper a new family of belief merging operators, that is based on a game between sources : Until a coherent set of sources is reached, at each round a contest is organized to find out the weakest sources, then those sources has to concede (weaken their point of view). This idea leads to numerous new interesting operators (depending of the exact meaning of “weakest” and “concede”, that gives the two parameters for this family) and opens new perspectives for belief merging. Some existing operators are also recovered as particular cases. Those operators can be seen as a special case of Booth’s Belief Negotiation Models (Booth 2002), but the achieved restriction forms a consistent family of merging operators that worths to be studied on its own. Introduction The problem of (propositional) belief merging (Revesz 1997; Lin & Mendelzon 1999; Liberatore & Schaerf 1998; Konieczny & Pino Pérez 1999; 2002a; Konieczny, Lang, & Marquis 2004) can be summarized by the following question: given a set of sources (propositional belief bases) that are mutually inconsistent, how to reach a coherent belief base reflecting the beliefs of the set ? The idea here is that some/each sources has to concede on some points in order to solve the conflicts. If one has some notion of relative reliability between sources, it is enough and sensible to force the less reliable ones to give up first. There is a lot of different means to do that, which has provided a large literature, e.g. (Cholvy 1993; 1995; 1998; Benferhat et al. 1998; Benferhat, Dubois, & Prade 1998). But often we do not have such information, and even if we get it, it remains the more fundamental problem of how to merge sources of equal reliability (Konieczny & Pino Pérez 1999; 2002a). In this paper we will investigate the merging methods based on a notion of game between the sources. The intuitive idea is simple: when trying to impose its wish, each source will try to form some coalition with near-minded sources. So the source that is the “furthest” from the other ones will certainly be the weakest one. And it will be that source that have to concede first. In this work, we will not focus on how the coalitions form, we only take this idea to designate the weakest ones. So the merging is based on the following game: Until a coherent set of sources is reached, at each round a contest is organized to find out the weakest sources, then those sources has to concede (weaken their point of view). We can state several intuitions and justifications for the use of such operators. We have already given the first one: coalition with near-minded sources. In a group decision process between rational sources, it can be sensible to expect the sources to look for near-minded sources in order to find help to defend their view, so the “furthest” source is the more likely to have to concede on its view. A second intuition is the one given by a social pressure on the sources. When confronting several points of view, usually people that have the more exotic views try to change their opinion in order to be accepted by the other members of the group, so opinions that are defended by the least number of sources are usually given up more easily in the negotiation process. A last intuition that gives the main rationale for that kind of operators is Condorcet’s Jury theorem. This theorem states that if all the members of a jury are reliable (in the sense that they have more than a half of chances to find the truth), then listen to the majority is the more rational choice. After stating some useful definitions and notations in the following section, we will define the new family of operators we propose. The definition will use a notion of weakening and choice functions. We will explore those notions in a subsequent section and we will give some examples of specific operators in order to illustrate their behaviour. We will then look at the logical properties of those operators. Finally, we will look at the links between this work and related works (especially Booth’s proposal (Booth 2001; 2002)), before concluding with some open issues and perspectives of this work. Preliminaries We consider a propositional language over a finite alphabet of propositional symbols. An interpretation is a function from to . The set of all the interpretations is denoted . An interpretation is a model of a formula , noted , if and only if it makes it true in the usual classical truth functional way. Let be a formula, denotes the set of models of , i.e. ! " . Conversely, let # be a set of interpretations, $% '& ( )#* denotes the formula (up to logical equivalence) whose set of models + is # . A belief base is a consistent propositional formula (or, equivalently, a finite consistent set of propositional formulae considered conjunctively). Let ,/././.0 / 21 be 3 belief bases (not necessarily different). We call belief profile the multi-set 4 consisting of those 3 belief bases: 45 5 ,/././.0 / 21% (i.e. two sources can have the same belief base). We note 4 the conjunction of the belief bases of 4 , i.e. 46 7 ,98 :/:/:;8 21 . We say that a belief profile is consistent if 4 is consistent. The multi-set union will be noted < and the multi-set inclusion will be noted = . The cardinal of a finite (multi-)set > is noted ?@ A>B (the cardinal of a finite multi-set is the sum of the numbers of occurrences of each of its elements). Let C be the set of all finite belief profiles. Two belief profiles 4D, and 4FE are said to be equivalent ( 4D,HGI4JE ) if and only if there is a bijection between

@inproceedings{Konieczny2004PropositionalBM,
title={Propositional belief merging and belief negotiation model},
author={S{\'e}bastien Konieczny},
booktitle={NMR},
year={2004}
}