Card Guessing with Partial Feedback

TL;DR

This paper studies a card guessing game with partial feedback, showing that the expected number of correct guesses is at most m + O(m^{3/4} log m), resolving a long-standing open problem.

Contribution

It introduces and analyzes a partial feedback model for card guessing, providing tight bounds on the expected correct guesses and solving a question from 1981.

Findings

Expected correct guesses are at most m + O(m^{3/4} log m)

The result holds uniformly in n, the number of card types

Resolves the open problem posed by Diaconis and Graham in 1981

Abstract

Consider the following experiment: a deck with $m$ copies of $n$ different card types is randomly shuffled, and a guesser attempts to guess the cards sequentially as they are drawn. Each time a guess is made, some amount of "feedback" is given. For example, one could tell the guesser the true identity of the card they just guessed (the complete feedback model) or they could be told nothing at all (the no feedback model). In this paper we explore a partial feedback model, where upon guessing a card, the guesser is only told whether or not their guess was correct. We show in this setting that, uniformly in $n$, at most $m+O(m^{3/4}\log m)$ cards can be guessed correctly in expectation. This resolves a question of Diaconis and Graham from 1981, where even the $m=2$ case was open.