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：尽管具有相同的基本先知不等式设定和损失厌恶模型，我们的多维度模型得出的结论与Kleinberg等人的单维度模型存在显著差异。例如，Kleinberg等人给出了在线决策者能实现的竞争比关于参数λ的紧致闭式解（对任意λ≥0成立）。但在我们的多维度模型中，存在一个尖锐的相变：若k表示维度数，则当λ·(k-1) ≥ 1时，无法实现任何非平凡竞争比；反之，当λ·(k-1) < 1时，我们给出了可实现竞争比的紧致上界（类似于Kleinberg等人的结果）。另一个例子是，Kleinberg等人在随机顺序与最坏情况先知不等式问题中发现竞争比呈指数级提升，而我们的k≥2维度模型中，该差距最多为常数因子。我们进一步揭示了多维度与单维度模型之间若干关键差异。