On the Sn/n-Problem

TL;DR

This paper provides tight bounds and a comprehensive analysis of the Chow-Robbins game, a classical stopping problem, using continuous approximations and applies these results to determine optimal strategies for large n.

Contribution

It introduces tight bounds for the value function of the Chow-Robbins game with subgaussian increments and establishes the existence of optimal stopping times using continuous process analogies.

Findings

Derived tight upper bounds for the value function with subgaussian increments.

Proved the existence of optimal stopping times in the discrete case.

Computed the continuation and stopping sets for n up to 10^5.

Abstract

The Chow-Robbins game is a classical still partly unsolved stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide if you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads \[V(n,x) = \sup_{\tau }\operatorname{E} \left [ \frac{x + S_\tau}{n+\tau}\right]\] where $S$ is a random walk. We give a tight upper bound for $V$ when $S$ has subgassian increments. We do this by usinf the analogous time continuous problem with a standard Brownian motion as the driving process. From this we derive an easy proof for the existence of optimal stopping times in the discrete case. For the Chow-Robbins game we as well give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 10^{5}$.