A Quantile Approach to Integration with Respect to Non-additive Measures

Abstract

Abstract. The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. We focus on the notion of quantile of a lattice-valued function which have several properties of its real-valued counterpart and we study a class of aggregation functionals that generalizes Sugeno integrals to the setting of complete lattices. Then we introduce in the real-valued case some classes of aggregation functionals that extend Choquet and Sugeno integrals by considering a multiple quantile model generalizing the approach proposed in [3].

Quantiles in Abstract Convex Structures

“…It appears that there are many connections between the work presented here with the results of [3,4,[7][8][9][10][11][12]. Applications of these types of results can be found in in [4,13,14].…”

“…As it is well known the quantile is a generalization of the concept of median and it plays an important role in statistical and economic literature. We study quantile in an ordinal framework and we consider an axiomatic representation of quantiles as in [1,7,8]. Here we provide a definition and characterization of quantiles for lattice-valued operators.…”

“…By Proposition 1 in [7] if L is a completely distributive lattice an aggregation functional F : L A → L is a lattice-valued quantile with respect to a non-additive measure m : A → L if and only if there exists a upper set U such that…”

Risk Perception and Ambiguity in a Quantile Cumulative Prospect Theory

“…Crucially, they introduced a reference set, that is, an interval of outcomes that the decision-maker feels more familiar with than other possible outcomes, instead of assuming the existence of a unique reference point. 14 Basili et al [7] introduced a characterization of the precautionary principle 15 for a decision making process under uncertainty with catastrophic losses and/or windfall gains. Unlike Basili et al [6], where it is assumed that people act as in stan- 13 Assumption that decision-makers are affected by familiarity bias is confirmed in empirical researches, such as de Lara Resende and Wu (2010) who find that "the decision weighting function parameters estimated for losses show an elevation parameter for less familiar domains higher than the elevation parameter for more familiar domain".…”

“…135 ] put in evidence that even "identical (convex hulls of) possible priors can be treated differently by the same individual depending on the source of uncertainty". 15 Precautionary Principle is considered the rational guide to policy making in situations characterized by scientific uncertainty, irreversibility, and catastrophic events. dard cumulative prospect theory, they suppose that the decision maker's behavior is characterized by the opposite ambiguity attitude with respect to extremely negative events, indeed pessimism in place of optimism (since the latter behavior could induce dissipative choices instead of conservative ones) and optimism instead of pessimism towards windfall gains.…”