Optimal Stopping with Multi-Dimensional Comparative Loss Aversion

Linda Cai, Joshua Gardner, S. Matthew Weinberg

Abstract:

Motivated by behavioral biases in human decision makers, recent work by Kleinberg et al. explores the effects of loss aversion and reference dependence on the prophet inequality problem, where an online decision maker sees candidates one by one in sequence and must decide immediately whether to select the current candidate or forego it and lose it forever. In their model, the online decision-maker forms a reference point equal to the best candidate previously rejected, and the decision-maker suffers from loss aversion based on the quality of their reference point, and a parameter $\lambda$ that quantifies their loss aversion. We consider the same prophet inequality setup, but with candidates that have \emph{multiple features}. The decision maker still forms a reference point, and still suffers loss aversion in comparison to their reference point as a function of $\lambda$, but now their reference point is a (hypothetical) combination of the best candidate seen so far \emph{in each feature}.

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 $\lambda \cdot (k-1) \geq 1$, \emph{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.