Despite having the same basic prophet inequality setup and model of loss aversion, conclusions in our multi-dimensional model differs considerably from the one-dimensional model of Kleinberg et al. For example, Kleinberg et al. gives a tight closed-form on the competitive ratio that an online decision-maker can achieve as a function of $\lambda$, for any $\lambda \geq 0$. In our multi-dimensional model, there is a sharp phase transition: if $k$ denotes the number of dimensions, then when $\lambda \cdot (k-1) \geq 1$, no non-trivial competitive ratio is possible. On the other hand, when $\lambda \cdot (k-1) < 1$, we give a tight bound on the achievable competitive ratio (similar to Kleinberg et al.). As another example, Kleinberg et al. uncovers an exponential improvement in their competitive ratio for the random-order vs. worst-case prophet inequality problem. In our model with $k\geq 2$ dimensions, the gap is at most a constant-factor. We uncover several additional key differences in the multi- and single-dimensional models.

翻译：尽管采用相同的基本先知不等式设定和损失厌恶模型，我们的多维度模型与Kleinberg等人提出的单维度模型得出的结论存在显著差异。例如，Kleinberg等人给出了在线决策者关于任一非负 λ 所能达到的竞争比的紧确闭式表达式。而在我们的多维度模型中，存在尖锐的相变现象：若 k 表示维度数，则当 λ·(k-1) ≥ 1 时，任何非平凡竞争比均不可实现；反之，当 λ·(k-1) < 1 时，我们给出了可达竞争比的紧确界（与Kleinberg等人的结论类似）。另一个例子是，Kleinberg等人发现其随机顺序与最坏情况先知不等式问题的竞争比存在指数级改进。而在我们的 k≥2 维度模型中，该差距至多仅为常数因子。此外，我们揭示了多维度与单维度模型间的若干关键差异。