# Constrained-Order Prophet Inequalities

Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least 0.669… times as great as that of the prophet. In fact, there exists a threshold stopping rule which guarantees a gambler-to-prophet ratio of at least 1-1/e=0.632…. In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is 1/2. In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed—namely, the forward and reverse orderings—the gambler-to-prophet ratio improves to φ^-1=0.618…, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from 0.5 to φ^-1, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed φ^-1+o(1) until the number of allowed permutations grows to O(log n). The ratio reaches 1-1/e-ε for a suitably chosen set of O(poly(ε^-1)·log n) permutations and does not exceed 1-1/e even when the full set of n! permutations is allowed.

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11/14/2018

### Prophet Inequalities for Independent Random Variables from an Unknown Distribution

A central object in optimal stopping theory is the single-choice prophet...
04/21/2020

### Competing Optimally Against An Imperfect Prophet

Consider a gambler who observes the realizations of n independent, non-n...
11/12/2019

### On optimal ordering in the optimal stopping problem

In the classical optimal stopping problem, a player is given a sequence ...
08/29/2021

### Tight Guarantees for Static Threshold Policies in the Prophet Secretary Problem

In the prophet secretary problem, n values are drawn independently from ...
10/20/2017

### Uniformly bounded regret in the multi-secretary problem

In the secretary problem of Cayley (1875) and Moser (1956), n non-negati...
01/28/2020

### Prophet Inequalities with Linear Correlations and Augmentations

In a classical online decision problem, a decision-maker who is trying t...