Guessing cards with complete feedback

TL;DR

This paper analyzes a card guessing game with complete feedback, determining the expected number of correct guesses achievable by optimal strategies and revealing how this expectation scales with deck parameters.

Contribution

It establishes the first-order correction to the expected correct guesses for large deck sizes, extending previous results and answering a question posed by Diaconis.

Findings

Optimal guessing strategy is nearly as good as always guessing the same card type.

Expected correct guesses slightly exceed the baseline of $m$, the number of each card type.

The results differ from cases with fixed $m$ or fixed $n$, showing new asymptotic behavior.

Abstract

We consider the following game that has been used as a way of testing claims of extrasensory perception (ESP). One is given a deck of $mn$ cards comprised of $n$ distinct types each of which appears exactly $m$ times: this deck is shuffled and then cards are discarded from the deck one at a time from top to bottom. At each step, a player (whose psychic powers are being tested) tries to guess the type of the card currently on top, which is then revealed to the player before being discarded. We study the expected number $S_{n,m}$ of correct predictions a player can make: one could always guess the exact same type of card which shows that one can achieve $S_{n,m}>m$. We prove that the optimal (non-psychic) strategy is just slightly better than that and find the first order correction when $n, m$ grows at suitable rates. This is very different from the case where $m$ is fixed and $n$ is large (He & Ottolini) and similar to the case of fixed $n$ and $m$ is large (Graham & Diaconis). The case $m=n$ answers a question of Diaconis.