Sugeno Utility Functions I: Axiomatizations

Pseudo-polynomial Functions over Finite Distributive Lattices

“…In this paper we extend the authors' previous works [6,7] by considering an aggregation model f : X 1 × · · · × Xn → Y for arbitrary sets X 1 , . .…”

“…We refined the axiomatization given in [6,8] by providing necessary and sufficient conditions for a function defined on a Cartesian product of arbitrary underlying sets and valued in a finite distributive lattice, to be factorizable as a pseudo-polynomial function (6). Moreover, in doing so, we were able to furnish all possible factorizations, if such a factorization exists, and we proposed a new procedure for constructing them, which subsumes that of [7,8] in the case when the codomain is a finite chain.…”

“…, ϕ n (x n )), where p : Y n → Y is a Sugeno integral, and each ϕ k : X k → Y is an order-preserving map. Such functions were referred to as Sugeno utility functions in [6]. Pseudopolynomial functions were defined as functions of the form (2), where p is an arbitrary (possibly non-idempotent) lattice polynomial function, and each ϕ k satisfies a certain boundary condition (which is weaker than order-preservation).…”

“…Note that every quasi-polynomial function (1) can be regarded as a pseudo-polynomial function, where X 1 = · · · = X n = X and ϕ 1 = · · · ϕ n = ϕ. Moreover, pseudopolynomial functions naturally subsume Sugeno utility functions, and several axiomatizations were established for this function class in [6]. The question of factorizing a given Sugeno utility function into a composition (2) was addressed in [7], where a method for producing such a factorization was presented.…”

“…For further background on aggregation functions and their use in decision making, we refer the reader to [2,13]; for basics in the theory of lattices, see [10,14]. In Section 3 we develop a general framework used to derive an axiomatization of pseudo-polynomial functions of somewhat different nature than those proposed in [4,6,7], and which will provide tools for determining all possible factorizations of given pseudo-polynomial functions in Section 4. These results are then illustrated in Section 5 by means of a concrete example, and in Section 6 we show how this new procedure can be applied to derive the algorithm provided in [7,8].…”