Poker with Wildcards

Probability

Overview

Some card game rules include a joker in the deck or treat specific cards as wild cards. While these are generally used in draw poker and video poker (games played on gaming machines under specific rules), it is also possible to consider them in the context of flop poker games such as Texas Hold’em and Omaha. To examine these flop poker games, let’s briefly summarize the representative rules. This article explains the rules for the following games.

  • Joker Wild
  • Joker as a Bug
  • Deuces Wild
  • One-Eyed Jacks

Joker Wild

Joker Wild is a poker variant in which one or two Jokers are added to a standard 52-card deck, allowing the Jokers to be used as wild cards that can substitute for any other card. Combining four of a kind with a Joker results in a hand called “Five of a Kind,” which is treated as a hand stronger than a Royal Straight Flush. However, note that, with the exception of Five of a Kind, you cannot substitute a Joker for a card in your hand when forming a hand. For example, if a Flush consists of A, K, T, 9 + Joker, it is treated as equivalent to a Flush of A, K, Q (Joker), T, 9. In a showdown between two Flushes, such as A, J, T, 7 + Joker vs. K, Q, T, 7 + Joker, the latter would be declared the winner (since the Joker counts as a K in the former and as an A in the latter).

 Under these rules, the Joker is extremely powerful, and a hand containing a Joker plus any other cards might actually have a higher win rate than AA (I plan to analyze this in the future and post the results on my blog). If you want to enjoy high-stakes hands, I think you should use two Jokers, but in that case, the game might boil down to whether you can draw a Joker. If you simply want to add a Joker as a bit of flair, limiting it to just one will result in a game that feels more like traditional poker.

Joker as a Bug

In Joker Wild, the Joker is too powerful, which means that having a Joker in your hand creates too much of a disparity regardless of the combination of cards you hold. “Joker as a Bug” is a variant that mitigates this effect by restricting how the Joker can be used.

 In “Joker as a Bug,” the Joker is treated as an Ace in rank and cannot be used to substitute for other ranks when forming pairs or three-of-a-kind. It can only be treated as a wild card when making a flush or a straight. Additionally, since a five-of-a-kind can only consist of Aces, it might be possible to simply eliminate it as a valid hand (though you’d need to determine whether four Aces plus a Joker is stronger than a straight flush—a scenario with an astronomically low probability of occurring). In “Joker as a Bug,” players should be aware that there is a significant chance of losing even if they hold two Jokers in their hand.

Deuces Wild

Deuces Wild is a poker variant in which all cards ranked as 2 are treated as wild cards. There are also rules where, in addition to 2s, 3s and 9s are treated as wild cards; this variant is apparently known as “Baseball Poker.” In Deuces Wild draw poker, a Five of a Kind is the strongest hand. However, in games like Texas Hold’em where hands are formed using two hole cards and five community cards, a Straight Flush is less likely to occur, so the general rule seems to be Straight Flush > Five of a Kind.

 Since there are as many as four wild cards, it’s very easy to form extremely strong hands, making it highly likely that players will go all-in early in the game. It might be better to play with short stacks of around 40 big blinds, similar to Short Deck Hold’em. In Deuces Wild, if a player holds two wild cards in their hand (a pocket pair of twos), they are guaranteed to win. However, as a special rule, if a player declares this immediately upon being dealt their hand, they receive a walkover victory and can collect the blinds and the ante.

 Conversely, if you do not declare this and proceed to the showdown, you are apparently considered to have lost. The rules seem to be set up so that you cannot increase your bet preflop when victory is already guaranteed.

One-Eyed Jacks

One-Eyed Jacks refers to the Jacks with only one eye depicted on the card—specifically, the Jack of Spades and the Jack of Hearts. The poker variant known as One-Eyed Jacks treats these two cards as wild cards. In addition, there appear to be variations that also treat the Suicide King (the King of Hearts) as a wild card. Since the probability of forming a hand varies by suit, it becomes impossible to group suited hands—such as KJ—together as in standard Texas Hold’em, which is likely to make pre-flop strategy particularly complex.

 Even aside from wild cards, it might be possible to devise interesting rules by introducing asymmetry based on suit or by establishing a hierarchy among the suits in a flush (though, under rules similar to Texas Hold’em, where there’s only one possible flush color, the number of hole cards or community cards would need to be adjusted).

Summary

In this article, I’ve explained some representative rules for poker games that include wild cards. Since Deuces Wild and One-Eyed Jacks are not included in the standard games on my poker app, “Exotic Poker,” I plan to add them soon and implement NPC algorithms that support those rules. I also intend to prepare an article discussing hand strength in the near future.

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