Extremal expectations in card guessing with partial feedback: proofs of two conjectures of Diaconis-Graham-Spiro

Abstract

We study the minimum and maximum expected scores over all adaptive strategies in sequential card guessing with yes/no feedback. A deck contains $n$ labels, each appearing $m$ times, and is shuffled uniformly. The cards are guessed one at a time, and after each guess the player is told only whether it was correct. Always guessing the same label gives exactly $m$ points. How far can the expected score depart from $m$ if all previous feedback is used either to seek or to avoid correct guesses? Diaconis, Graham, and Spiro conjectured that the minimum expectation is $m-o(m)$ as $m\to \infty$ with $n$ sufficiently large in terms of $m$, and that, for each fixed $m$, the limit superior of the maximum expectation as $n\to \infty$ is at most $(e-1)m$. Writing these extrema as $P_{m,n}^{-}$ and $P_{m,n}^{+}$, we prove \[ 0\le m-P^-_{m,n}\le3m^{3/4} \qquad(n\ge m\ge1), \] and the finite-parameter bound \[ \frac{P^+_{m,n}}m\le\sum_{k=1}^{n}\frac1{k!}<e-1 \] for all positive integers $m,n$. The first bound gives a first-order asymptotic uniform in $n\ge m$; the universal constant $e-1$ in the second cannot be reduced. Thus both conjectures follow. The lower bound combines a switching comparison for restricted permutations with martingale second-moment estimates. For the upper bound, we represent the repeated cards as a random interleaving of independent decks of distinct cards and prove that the optimal expectation is additive when the component-deck labels are revealed.

arXiv comment18 pages