Online Card Games

TL;DR

This paper analyzes an adversarial version of an online card guessing game, proving the optimal shuffling strategy and establishing bounds on the guesser's success rate.

Contribution

It introduces an adversarial setting for the game, identifies the greedy shuffler as optimal, and bounds the guesser's expected correct guesses asymptotically.

Findings

Greedy shuffling strategy is uniquely optimal for the shuffler.

The guesser can achieve at most logarithmic success rate against the optimal shuffler.

Asymptotic bounds are established for fixed number of card types.

Abstract

Consider the following one player game. A deck containing $m$ copies of $n$ different card types is shuffled uniformly at random. Each round the player tries to guess the next card in the deck, and then the card is revealed and discarded. It was shown by Diaconis, Graham, He, and Spiro that if $m$ is fixed, then the maximum expected number of correct guesses that the player can achieve is asymptotic to $H_m \log n$, where $H_m$ is the $m$th harmonic number. In this paper we consider an adversarial version of this game where a second player shuffles the deck according to some (possibly non-uniform) distribution. We prove that a certain greedy strategy for the shuffler is the unique optimal strategy in this game, and that the guesser can achieve at most $\log n$ expected correct guesses asymptotically for fixed $m$ against this greedy strategy.