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 Sn,m of correct predictions a player can make: one could always guess the exact same type of card which shows that one can achieve Sn,m ě m. We prove that the optimal (non-psychic) strategy is just slightly better than that and Sn,m " m `π ? 2 ? m ln n `op ? m ln nq whenever pln nq 3`ε ! m. This is very different from the case where m is fixed and n Ñ 8 (He & Ottolini) and similar to the case of fixed n and m Ñ 8 (Graham & Diaconis). The case m " n answers a question of Diaconis.

Central limit theorem in complete feedback games

“…It is worth pointing out that, while there is hope for a CLT to hold even when m grows much faster than n, the asymptotic of the expected value (first part of Theorem 1.1) eventually breaks down, as shown in [OS22].…”

“…Their main tool is the analysis of a certain variation of the birthday problem via Stein's methods, which will be our main tool here as well. In the case m i ≡ m where both m and n are growing, the asymptotic for the expected optimal score was obtained by the first author and Steinerberger in [OS22], covering a variety of regimes that include the case n = m (the original Zener's setting).…”