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Dominance plausible rule and transitivity

Abstract:

In qualitative decision theory, a very natural way for defining preference relations over policies (acts) -functions from a set S of states to a set X of consequences- is by using the so called Dominance Plausible Rule. In this context we need a relation > over X and a relation ≤ over P(S) (the subsets of S). Then we define ≥ as follows: f ≥ g ↔ [f > g] ≤ [g > f], where [f > g] denotes the set {s ∈ S : f(s) > g(s)}. In many cases > is a modular relation and ≤ is a total preorder. A quite rational and desirable property for the relation over policies is transitivity. In general, the relation ≥ defined by the Dominance Plausible Rule is not transitive in spite of the transitivity of ≤. In this work we characterize the properties of the relation > forcing the relation ≥ over policies to be transitive. All this under the hypothesis of modularity of the relation >. © 2011 Lavoisier, Paris.