Guessing Strategies for Shuffling Machines

TL;DR

This paper analyzes guessing strategies for a specific shuffling process, deriving optimal strategies and expected rewards, and confirming a conjecture in a special case.

Contribution

It provides explicit position matrices, proves optimal guessing strategies for no-feedback and complete-feedback cases, and confirms a conjecture in a special case.

Findings

Optimal no-feedback guessing strategies are established in some cases.

A conjectured general no-feedback strategy is proposed and asymptotics are analyzed.

An optimal and unique guessing strategy for the complete-feedback case is proved, along with its expected reward.

Abstract

We investigate a one-time single shelf shuffle by establishing the position matrix explicitly. In some cases, we prove a no-feedback optimal guessing strategy. A general no-feedback strategy is conjectured, and asymptotics for the expected reward are given. For the complete-feedback case, we give a guessing strategy, prove that it is optimal and unique, and find the expected reward. Our results prove a conjecture of Diaconis, Fulman, and Holmes in a special case.