Beyond Scalars: Zonotope-Valued Utility for Representation of Multidimensional Incomplete Preferences(Incomplete Version)

TL;DR

This paper introduces a novel framework using zonotope-valued utilities to represent complex multidimensional incomplete preferences, capturing trade-offs and incomparabilities more effectively than traditional scalar models.

Contribution

It proposes a new geometric approach to preference modeling with zonotopes, extending existing methods to handle multidimensional incompleteness and uncertainty.

Findings

Zonotope-valued utilities effectively model multidimensional preferences.

The framework captures incomparability and trade-offs among alternatives.

Axiomatization provides a rigorous foundation for the approach.

Abstract

In this paper, I propose a new framework for representing multidimensional incomplete preferences through zonotope-valued utilities, addressing the shortcomings of traditional scalar and vector-based models in decision theory. Traditional approaches assign single numerical values to alternatives, failing to capture the complexity of preferences where alternatives remainmain incomparable due to conflicting criteria across multiple dimensions. Our method maps each alternative to a zonotope, a convex geometric object in \(\mathbb{R}^m\) formed by Minkowski sums of intervals, which encapsulates the multidimensional structure of preferences with mathematical rigor. The set-valued nature of these payoffs stems from multiple sources, including non-probabilistic uncertainty, such as imprecise utility evaluation due to incomplete information about criteria weights, and probabilistic uncertainty arising from stochastic decision environments. By decomposing preference relations into interval orders and utilizing an extended set difference operator, we establish a rigorous axiomatization that defines preference as one alternative's zonotope differing from another's within the non-negative orthant of \(\mathbb{R}^m\). This framework generalizes existing representations and provides a visually intuitive and theoretically robust tool for modeling trade-offs among each dimension, while preferences are incomparable.