标题： 超越标量：区间值效用用于多维不完全偏好的表示（不完整版本） 显示英文标题

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

摘要： 在本文中，我提出了一种新的框架，通过超柱体值效用来表示多维不完全偏好，解决了传统标量和向量模型在决策理论中的不足。 传统方法为每个备选方案分配单一数值，无法捕捉到由于多个维度上的冲突标准而导致备选方案仍然不可比较的偏好复杂性。 我们的方法将每个备选方案映射到一个超柱体，这是一个由区间Minkowski和形成的凸几何对象，在\(\mathbb{R}^m\)中，它以数学严谨的方式概括了偏好的多维结构。 这些收益的集值性质来源于多个来源，包括非概率不确定性，例如由于对标准权重的不完整信息导致的不精确效用评估，以及来自随机决策环境的概率不确定性。 通过将偏好关系分解为区间序，并利用扩展的集差运算符，我们建立了一个严格的公理化体系，将偏好定义为一个备选方案的超柱体在\(\mathbb{R}^m\)的非负象限内与另一个备选方案的超柱体存在差异。 该框架推广了现有的表示方法，并提供了一个视觉直观且理论稳健的工具，用于建模每个维度之间的权衡，同时偏好是不可比较的。 显示英文摘要

摘要： 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.