Card guessing after an asymmetric riffle shuffle

Abstract

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in (0,1)\setminus \{\frac{1}{2}\}$, the riffle shuffle is assumed to be unbalanced, such that the cut is expected to happen at position $p\cdot n$. The goal of the game is to maximize the number of correct guesses of the cards: one after another a single card is drawn from the top, and shown to the guesser until no cards remain. We provide a detailed analysis of the optimal guessing strategy and study the distribution of the number of correct guesses.

20 pages, 3 figures. Refernces updated. Extends results of arXiv:2306.16079