I.i.d. Prophet Inequalities with Discounted Rewards: As Hard as the Non-i.i.d. Case

2026-06-29

AI Summary

This study investigates the Prophet inequality problem under discounted rewards, revealing that temporal discounting substantially degrades performance even when rewards are i.i.d. The authors analyze how discounting affects the competitive ratio for single-threshold quantile policies and arbitrary stopping rules, introducing extensions such as effective time window calibration, heterogeneous decay factors, and unequal-length phases. They establish the first tight lower bound of $1/2$ on the competitive ratio under common discounting schemes—down from the classical $1 - 1/e$—and show this bound holds universally across all stopping rules, indicating a difficulty level comparable to the non-i.i.d. setting. Furthermore, in an infinite-horizon continuous-discount model, they prove that any arbitrarily weak decay reduces the benchmark from 1 to $1/2$, and propose a novel strategy that achieves this bound.

Abstract

We study prophet inequalities with discounted rewards, where i.i.d. base rewards are multiplicatively discounted over time. Our main message is that even this structured and arbitrarily weak form of nonstationarity can erase the classical advantage of the stationary i.i.d. setting. Focusing on single-quantile threshold policies, we show that the competitive ratio transitions from the classical $1-1/e$ guarantee to a fundamental $1/2$ barrier as discounting accumulates over many phases in a canonical regime with a common-decay factor and equal-length phases. We further show that, in the same regime, the $1/2$ barrier persists even for arbitrary stopping rules. Consequently, i.i.d. base rewards under discounting can be as hard as the fully non-i.i.d. case. On the algorithmic side, we design single-quantile threshold rules that attain the tight bounds by calibrating acceptance decisions to an effective horizon induced by discounting, and we extend this calibration to heterogeneous decay factors and unequal phase lengths. We further show that a similar discontinuous breakdown persists in an infinite-horizon continuous-decay benchmark, where arbitrarily weak decay collapses the stationary benchmark from $1$ to $1/2$.

Problem

• prophet inequalities
• discounted rewards
• i.i.d.
• nonstationarity
• competitive ratio

Innovation

• prophet inequalities
• discounted rewards
• competitive ratio
• single-quantile threshold
• nonstationarity