Extremal poker hand rankings: why the standard 52 card deck and a 3044 card deck are special
Christopher Williamson
TL;DR
This work generalizes poker hand rankings to decks with $r$ ranks (totaling $4r$ cards) and computes frequency-based rankings for all $r$, revealing when (and how) the usual hierarchy changes as the deck scales. It introduces the concept of showdown frequency to capture how players declare hands at showdown under a fixed frequency ranking, exposing Gadbois-type phenomena even without wild cards. The study shows that flushes drop below one-pair only at $r\geq 307$ and that straights become the rarest non-SF hand only at $r\geq 761$, with the frequency ranking stabilizing for $r\geq 761$, while the smallest $r$ where frequency and showdown rankings align (including high-card) is $r=23$. The results illuminate why the standard 52-card deck is a special, relatively stable regime and provide a framework to analyze variant decks and their strategic implications in showdowns.
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.