A Better Way to Deal the Cards

TL;DR

This paper analyzes how different shuffling and dealing methods affect the randomness of card decks, revealing that strategic arrangements and dealing sequences can significantly improve randomness with fewer shuffles.

Contribution

It introduces new insights into optimizing card shuffling and dealing strategies to enhance randomness, using the Gilbert-Shannon-Reeds model and variation distance.

Findings

Matching top and bottom cards improves randomness with fewer shuffles

Alternative dealing methods can match the effect of multiple extra shuffles

Cutting strategies in poker influence deck randomness

Abstract

This paper considers the effect of riffle shuffling on decks of cards, allowing for some cards to be indistinguishable from other cards. The dual problem of dealing a game with hands, such as bridge or poker, is also considered. The Gilbert-Shannon-Reeds model of card shuffling is used, along with variation distance for measuring how close to uniform a deck has become. The surprising results are that for a deck with only two types of cards (such as red and black), the shuffler can greatly improve the randomness of the deck by insuring that the top and bottom cards are the same before shuffling. And in the case of dealing cards for a game with "hands", such as bridge or poker, the normal method of dealing cyclically around the table is very far from optimal. In the case of a well-shuffled bridge deck, changing to another dealing method is as good as doing 3.7 extra shuffles. How the deck is cut in poker affects its randomness as well.