Measure and integral with purely ordinal scales

TL;DR

This paper introduces a purely ordinal framework for aggregation functionals on lattice-valued functions, including models for decision behavior, by exploring reflection lattices and their properties.

Contribution

It develops a novel ordinal model for aggregation, introduces reflection lattices for modeling psychological effects, and investigates the structure of interval-valued functions on lattices.

Findings

Includes quantiles, Ky Fan metric, and Sugeno integral as special cases

Introduces reflection lattices for modeling decision effects

Analyzes the lattice of non-void intervals and monotone interval-valued functions

Abstract

We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 0 on the real interval $[-1,1]$. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and their inner product.