Lookback Prophet Inequalities

TL;DR

This paper extends prophet inequalities by incorporating lookback and decay functions, allowing for revisiting rejected items and recovering part of their value, thus providing a more realistic online selection model.

Contribution

It introduces a generalized model with decay functions for revisited items and refines the analysis of competitive ratios under this framework.

Findings

Competitive ratios are improved with lookback under mild conditions.

The problem reduces to a single decay function case with parameter gamma.

Bounds on competitive ratios are expressed as functions of gamma.

Abstract

Prophet inequalities are fundamental optimal stopping problems, where a decision-maker observes sequentially items with values sampled independently from known distributions, and must decide at each new observation to either stop and gain the current value or reject it irrevocably and move to the next step. This model is often too pessimistic and does not adequately represent real-world online selection processes. Potentially, rejected items can be revisited and a fraction of their value can be recovered. To analyze this problem, we consider general decay functions $D_1,D_2,\ldots$, quantifying the value to be recovered from a rejected item, depending on how far it has been observed in the past. We analyze how lookback improves, or not, the competitive ratio in prophet inequalities in different order models. We show that, under mild monotonicity assumptions on the decay functions, the problem can be reduced to the case where all the decay functions are equal to the same function $x \mapsto \gamma x$, where $\gamma = \inf_{x>0} \inf_{j \geq 1} D_j(x)/x$. Consequently, we focus on this setting and refine the analyses of the competitive ratios, with upper and lower bounds expressed as increasing functions of $\gamma$.