Axiomatizations and Factorizations of Sugeno Utility Functions

Abstract

In this paper we consider a multicriteria aggregation model where local utility functions of different sorts are aggregated using Sugeno integrals, and which we refer to as Sugeno utility functions. We propose a general approach to study such functions via the notion of pseudo-Sugeno integral (or, equivalently, pseudo-polynomial function), which naturally generalizes that of Sugeno integral, and provide several axiomatizations for this class of functions.Moreover, we address and solve the problem of factorizing a Sugeno utility function as a composition q(ϕ 1 (x 1 ), . . . , ϕn(xn)) of a Sugeno integral q with local utility functions ϕ i , if such a factorization exists.

Pseudo-polynomial Functions over Finite Distributive Lattices

“…This theorem appears in [7,8] for the special case of chains; for the sake of self-containedness, we reproduce the proof here. We use the notation 1 I for the characteristic vector of I ⊆ [n] in i∈[n] X i , i.e., 1 I ∈ i∈[n] X i is the n-tuple whose i-th component is 1 Xi if i ∈ I, and 0 Xi otherwise.…”

“…(5) and (7) could be reversed as the choice of 0 X k and 1 X k is arbitrary. Hence, the current notion of boundary condition is not more restrictive than the one used in [8].…”

“…Now we can prove that pseudo-median decomposability actually characterizes pseudo-polynomial functions (see [6,8] for the case of chains, where the proof is slightly simpler).…”

“…With this notation (8) reads as u (z) = med (u (a) , z, u (b)). In order to verify this equality, we write u in disjunctive normal form as in Remark 2: u (y) = s ∨ (t ∧ y), where s ≤ t. Now the proof is a straightforward computation, making heavy use of distributivity:…”

“…Theorem 11 is of different nature than Theorem 6 and the various characterizations obtained in [8]: here the necessary and sufficient condition for f being a pseudo-polynomial function is given solely in terms of f itself, without referring to the existence of certain functions ϕ k .…”