Exact Solution to the Chow-Robbins Game for almost all n, using the Catalan Triangle

Abstract: The payoff in the Chow-Robbins coin-tossing game is the proportion of heads when you stop. Knowing when to stop to maximize expectation was addressed by Chow and Robbins(1965), who proved there exist integers ${k_n}$ such that it is optimal to stop when heads minus tails reaches this. Finding ${k_n}$ exactly was unsolved except for finitely many cases by computer. We show ${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta ( - 1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil$ for almost all n, where $\alpha $ is the Shepp-Walker constant.This comes from our estimate ${\beta_n} = \alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta ( - 1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}} + O\left( {{n^{ - 7/24}}} \right)$ of real numbers defined by Dvoretzky(1967) for a more general Value function which is continuous in its first argument and easier to analyze. An $O({n^{ - 1/4}})$ dependence was conjectured by Christensen and Fischer(2022) from numerical evidence. Our proof uses moments involving Catalan and Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of H\"aggstr\"om and W\"astlund(2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod's embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in a different way. We use them first for many more examples, but the new idea is to use them algebraically in the tree, with feedback to get a sharper Value estimate near the border, to settle almost all n.

• The paper derives an exact formula for the stopping boundary k R n ~ a n for almost all n, solving a long-standing optimal stopping problem in the Chow-Robbins game.
• Methodology involves backward induction using Catalan numbers, Skorohod's embedding for Brownian motion analogy, and numerical strategies to extend solved cases.
• The findings provide theoretical support for previous empirical results, improve boundary precision, and have implications for optimal stopping problems in statistics and finance.

Insights into the Exact Solution to the Chow-Robbins Game

John H. Elton’s paper explores the long-standing challenge of finding an exact solution for the optimal stopping problem in the Chow-Robbins game, using advanced mathematical constructs such as the Catalan triangle. The Chow-Robbins game presents itself as a stochastic process where you are required to stop a symmetric random walk optimally—that is, at a point that maximizes the expected proportion of heads minus tails over a series of coin tosses.

Elton’s significant advancement lies in his derivation of a formula for the stopping boundary: $$k_n = \alpha \sqrt{n} - 1/2 + \frac{(-2\zeta(-1/2)) \alpha}{\pi} n^{-1/4} + O(n^{-7/24})$$ for almost all $n$, where $\alpha$ is the Shepp-Walker constant. This development builds on Dvoretzky’s earlier approach and resolves the unsolved challenge of finding the exact integer bounds for large $n$.

Methodological Contributions

Backward Induction with Catalan Numbers:

The paper employs backward induction with input from Catalan triangle numbers, deriving moments that lead to sharper value approximations. This framework hinges on the use of tree structures populated with weights that are directly derived from Catalan numbers, providing insights into optimal decision-making boundaries.

Skorohod's Embedding:

A notable methodological contribution is integrating Skorohod's embedding, which approximates the random walk game using Brownian motion analogies. The alignment with Shepp’s results, concerning continuous-time processes, facilitates deriving an asymptotic understanding of the stopping rule.

Numerical Strategies:

The study involves numerical exploration beyond known empirical bounds, extending the domain of solved cases significantly. Here, the pivotal function of the Catalan triangle in approximating and bounding real solutions stands as a cornerstone for both computational efficiency and analytical clarity.

Theoretical Implications

Elton’s formula facilitates the rigorous establishment of stop rules that had been previously bounded only numerically or conjecturally for large $n$. A critical result is showing that earlier empirical findings by Christensen and Fischer—specifically, the $n^{-1/4}$ dependence—observes theoretical backing while improving boundary precision.

Piecewise Linear Approximation:

A novel approximation approach is observing piecewise linear behavior in the function $V(u, b)$ near the decision boundary, tangentially aligning with previously established quadratic approximations for boundary discrepancies.

Future Directions

The implications of Elton’s formulation superficially transcend beyond simple coin toss games, casting new light on stopping problems in statistical testing and financial decision-making. The approach signals potential in extending backward induction models paired with tree-based Catalan weighting for computational scenarios involving deeper decision sequences in asymptotically complex environments.

In advancing the optimal stopping discourse, further exploration could focus on extending the framework to arbitrary martingales and potentially leveraging the Catalan matrix framework in recursive influencing domains, such as strategic game-theory models with stochastic elements. Elton’s closing remarks on computational efficiency promise an enduring impact on both theoretical explorations and practical applications in fields requiring precise stopping-time algorithms.