Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special

TL;DR

This paper explores how poker hand rankings change with decks of varying sizes, revealing unique phenomena and stability thresholds, especially highlighting the special status of the standard 52-card deck.

Contribution

It provides a comprehensive analysis of poker hand rankings across all deck sizes and uncovers phenomena like the stability threshold at 761 ranks and frequency discrepancies.

Findings

Hand rankings vary with deck size, with notable phenomena at specific thresholds.

The standard 52-card deck is the smallest avoiding frequency discrepancies.

Hand ranking stability is not achieved until a deck size of 761 ranks.

Abstract

We study poker hand rankings in the partially generalised setting of a deck with $r$ ranks, rather than the typical 13 ranks. We provide the hand rankings for all $r$ and observe some interesting phenomena such as the smallest $r$ such that flushes rank below one-pair hands. Perhaps surprisingly, as $r$ grows without bound, the hand ranking is not stable until $r=761$ (a 3044 card deck). We consider showdown frequency, which is the frequency that a given type of hand is declared by a player at showdown, and make note of counterintuitive instances in which a hand with lower absolute frequency than some other hand nonetheless has a higher showdown frequency. This can be interpreted as a form of Gadbois paradox but in the typical setting of poker without wild cards. Conveniently, the standard deck with 13 ranks turns out to be the smallest deck that avoids a discrepancy between absolute frequency and showdown frequency for all hand types other than having a high card.