Card guessing after an asymmetric riffle shuffle

TL;DR

This paper analyzes an optimal card guessing strategy after a single asymmetric riffle shuffle, focusing on maximizing correct guesses and understanding the distribution of correct guesses in a game with complete feedback.

Contribution

It introduces a detailed analysis of the optimal guessing strategy for an asymmetric riffle-shuffled deck and characterizes the distribution of correct guesses.

Findings

Optimal guessing strategy derived for asymmetric riffle shuffle

Distribution of correct guesses characterized

Insights into the impact of asymmetry on guessing performance

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 $p\in(0{,}1)\setminus\{\frac12\}$, 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.