A collective choice method based on individual preferences relational systems (p.r.s.)

In group decision-making literature, several procedures are proposed in order to establish a collective preference from the different individual ones. The majority of these procedures, however, reveal that the individual preferences are always expressed in total pre-orders (or ranking). Indeed, until now very few have considered individual preferences which are expressed in partial pre-orders or, more generally, in preferences relational systems (p.r.s.). Moreover, many of these procedures generate collective preferences which are expressed in total pre-orders (ranking decision-making problematic). The efforts reported in the literature to develop procedures which treat other decision-making problematics—such as choice problematic—remain insufficient. In this paper, we propose a method which would determine from individual p.r.s. at least one collective subset containing the “best” alternatives. Each of these collective subsets results from the exploitation—according to the choice problematic—of a collective p.r.s. obtained from the aggregation of the individual p.r.s. Furthermore, each collective p.r.s. has two main characteristics: (i) it is at a minimum distance from all individual p.r.s. and (ii) it takes into account the members’ relative importance.     

Quasi-kernels and quasi-sinks in infinite graphs

Given a directed graph G=(V,E) an independent set A⊂V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed graph has a quasi-kernel. The plain generalization for infinite graphs fails, even for tournaments. We study the following conjecture: for any digraph G=(V,E) there is a a partition (V₀,V₁) of the vertex set such that the induced subgraph G[V₀] has a quasi-kernel and the induced subgraph G[V₁] has a quasi-sink.