Anything's Wild

Anything's Wild is a game where any of the thirteen ranks can be selected as the wild rank. I actually prefer to think of it as a "category", which games such as Deuces Wild and Sevens Wild belong to. In this article I'm going to compare the differences in the distribution of the 2,598,960 hands between each wild rank. These numbers, for each type of hand, are shown below in a visual format. Because Aces are both low and high, I have included them on both sides of the charts, for reasons which will soon become obvious:

Observations

The following observations can be made:

1. The number of Full House, Five of a Kind, Four Wilds and Natural Royal Flush hands remains constant regardless of the wild rank.
   
   
2. Fives through Nines, all of which are middle ranks, have the same distribution of hands.
   
   
3. When the wild rank is a Ten or higher there are 276 fewer Wild Royal Flushes, but 276 more Straight Flushes.
   
   
4. Except for Straight Flushes and Wild Royal Flushes, Tens have the same numbers as Fives through Nines because it is also a middle rank.
   
   
5. Except for Straight Flushes and Wild Royal Flushes, all hands have a perfectly symmetrical distribution across the A23456789TJQKA spectrum (the numbers for A234567 are a perfect mirror image of the numbers for AKQJT98).
   
   
6. If we eliminate the shuffling of the 276 Wild Royal Flushes and Straight Flushes by grouping all Straight Flushes, Wild Royal Flushes, and Natural Royal Flushes together (because all of these hands are a subset of Straight Flushes), these totals are also perfectly symmetrical:
   
   
   
   
   
7. Aces are in a class all their own.

Wild Royal Flush Discrepancy

It probably makes sense that there are 276 fewer Wild Royal Flushes when the wild rank is a Ten or higher, but the reason why this is the case might not be immediately obvious. To figure out why, let's compare Wild Royal Flush hands in Deuces Wild and Aces Wild.

Using one wild card, in Aces Wild the only possible combination of natural cards is KQJT (otherwise the hand would have 2 wild cards), and the suit of the KQJT must be different than the suit of the Ace (otherwise it would be a Natural Royal Flush). This means that there are Combin(4,1) × Combin(4,4) × Combin(4,3) = 12 possible Wild Royal Flushes that use exactly one wild card in Aces Wild. In Deuces Wild, the natural cards can be AKQJ, AKQT, AKJT or KQJT, in any of the four suits, and the suit of the Deuce does not matter. This results in Combin(4,1) × Combin(5,4) × Combin(4,1) = 80 possible Wild Royal Flushes that use exactly one Deuce.

For hands with 2 wild cards, in Aces Wild the only possible natural cards are KQJ, KQT, KJT or QJT, in any of the four suits, resulting in Combin(4,2) × Combin(4,1) × Combin(4,3) = 96 possible Wild Royal Flushes. In Deuces Wild the natural cards can be AKQ, AKJ, AKT, AQJ, AQT, AJT, KQJ, KQT, KJT or QJT, which results in Combin(4,2) × Combin(4,1) × Combin(5,3) = 240 Wild Royal Flushes.

With 3 wild cards, in Aces Wild the only possible natural cards are KQ, KJ, KT, QJ, QT or JT, which also results in 96 possible Wild Royal Flushes. In Deuces Wild there are 160 possibilities. The total number of Wild Royal Flushes for Deuces Wild is 80 + 240 + 160 = 480, and for Aces Wild is 12 + 96 + 96 = 204. The difference between 480 and 204 is 276 hands, which end up being scored as Straight Flushes. The reason why these hands are counted as Straight Flushes is because without a low rank (2 through 9) as the wild card, more low-end Straight Flushes can be completed, in the same way that having a low wild card allows more Wild Royal Flushes (high-end Straight Flushes) to be completed.

The Case With The Ace

Aces are unique from all other ranks, for one very specific reason: they are both the highest and lowest rank in the game. This is why Deuces and Kings, Threes and Queens, and Fours and Jacks share the same distribution of possible hands: each rank is equidistant from the nearest terminus. The fact that Aces are both high and low leads to an interesting side effect when Aces are wild: Five-high Straights and Straight Flushes ("wheel" hands) are virtually eliminated from the game.

In Deuces Wild, it is possible to have four cards to a Straight or Straight Flush with a gap that only a Deuce can fill. This hand is A345 (which is usually never played unless all four cards are suited, but that's a different matter). There is an equivalent hand in Threes Wild, Fours Wild, and so on, up to and including Kings Wild. However, there is no such hand in Aces Wild. The only two Straights which normally use an Ace are A2345 and AKQJT. If either KQJT or 2345 occur in Aces Wild, there is no gap that only an Ace can fill. KQJT can be completed by a Nine, and 2345 can be completed by a Six. To prove the point, let's compare all possible Four to a Straight combinations (with or without a gap) in Deuces Wild and Aces Wild. The hands in bold are ones which differ between the two games:


There are 34 combinations in Deuces Wild, but only 33 combinations in Aces Wild. It is this missing Straight/Straight Flush combination which makes Aces Wild unique. It could legitimately (and pointlessly) be debated whether the wild card in W2345 is taking the place of an Ace or a Six since both have the same value. However, if you always treat a wild card as completing the high end of a Straight or Straight Flush in hands which consist of four consecutive natural cards and one wild card, then A2345 is virtually eliminated as a possible hand in Aces Wild because the Ace would always act as a Six.

Rank Effect on Long-Term Return

It makes sense that Deuces are the most valuable rank, because they are near an extreme end and cannot be used to complete a Natural Royal Flush. Likewise, it makes sense that Tens are the least valuable because they are a middle rank and can complete a Natural Royal Flush.

But why don't Fives, Sixes, Sevens, Eights and Nines all have the same long-term return since they share the exact same distribution of possible hands? Honestly, I'm not entirely sure. I believe it has to do with with how close the wild rank is to the Royal Flush ranks, but I can't explain why. Until I can prove or disprove this theory, the actual reason will remain a bit of a mystery to me.