Optimal no-feedback card guessing after a fixed number of riffle shuffles

Abstract

An ordered deck of $n$ cards is subjected to $k$ independent Gilbert--Shannon--Reeds riffle shuffles. A player then guesses each card without feedback, with repeated guesses allowed. For every fixed $k\ge 1$, we prove that, once $n$ is sufficiently large, an optimal guess at position $i$ in the top half of the deck is $\lfloor i/2^k\rfloor +1$; the bottom-half strategy is obtained by reflection. We also show that the optimal expected number of correct guesses is $2\sqrt{n/((2^k-1)\pi )}+O_k(1)$. Our starting point is the result of our earlier work that the label at each position is distributed as an equally weighted mixture of $2^k$ components, each a shifted sum of two independent binomial random variables (a Poisson-binomial distribution). The proof of the optimal strategy combines three ingredients: a uniform concentration estimate that separates the mixture components, exact binomial inequalities that control the gaps between competing peaks, and a coefficient comparison that resolves modal ties. The expected-score asymptotic is proved independently of the strategy, using only the peak heights of the mixture components.