Guessing cards with complete feedback

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.