Wild card poker paradox

Wild card poker is a variation of poker that implements the use of at least one “wild card.” A wild card is a designated card that a player can assign any value and suit. A wild card livens the action by making stronger hands easier to complete.

This post is about the following question. In wild card poker, which hand is more valuable: three of a kind or two pair?

The mathematics are peculiar and the question is something of a paradox. Before I get into this question, it is useful to understand the logic of poker hand rankings.

What makes a poker hand valuable?

Poker hand rankings could be arbitrary and a rule based on tradition.

But, as everyone knows, there is a clever logic to poker rankings. The value of a hand is determined by its mathematical probability of occurrence, or frequency. A hand is ranked more valuable if it is less likely or harder to make.

This is why a flush has a higher ranking than a straight: it is harder and less probable to complete five unsequenced cards of a suit (flush) than to get five sequenced but unsuited cards (straight).

It is a standard probability exercise to verify that the poker hand rankings are based on their frequency. Here is a table that summarizes the rankings and probabilities (detailed math here):

RANKING OF POKER HAND VS PROBABILITY
Poker hand Number of ways Probability
Straight Flush 40 0.0015%
Four of a Kind 624 0.024%
Full House 3,744 0.14%
Flush 5,108 0.20%
Straight 10,200 0.39%
Three of a Kind 54,912 2.1%
Two Pair 123,552 4.75%
One Pair 1,098,240 42%
High Card 1,302,540 50.1%
Total 2,598,960 100%

In a way, the frequency ranking makes the game of poker fair and logical. A three of a kind is more valuable than a two pair exactly because it is harder to achieve that hand by more than two percentage points.

This ranking is fundamental to standard games of poker.

But an extension is to ask: what happens when a wild card is introduced?

Wild card poker effects

Poker can also be played with wild cards to liven up the action. One common variation is to add a joker card to the deck and allow it to be wild and represent any card.

The addition of a wild card does three things.

1. It creates a new hand “five of a kind” where all five cards are the same value, like five aces or five kings.

2. It affects the probability of getting a hand. It is now easier to make stronger hands because the wild card can be designated to complete a full house or a pair.

3. It forces a player to make a choice about the wild card. Which card should the wild card represent? The answer is obviously to choose to card that will make the best possible hand given the other cards the player is holding.

This third effect creates a new choice for the player in declaring a hand. If a player holds a pair and also a wild card, then there are two possible ways to declare the hand. The hand can be called a three of a kind if the player declares the wild card to match the pair. Alternately the hand can be called a two pair if the player declares the wild card to match an unpaired card to go along with the pair.

(Say a player has 4,4,2,6, and a joker. The joker can either be a 4 to complete a three of a kind, or it can be a 6 to complete two pairs of 44 and 66)

Which hand should be chosen? This is the source of the paradox.

The wild card poker paradox

For the moment we will assume the standard ranking that three of a kind is more valuable than a two pair.

We can then calculate the probability of making each hand in wild card poker. We will make the assumption that a player holding a pair and a wild card will choose to make this hand a three of a kind rather than a two pair.

The resulting probabilities are (taken from curiouser.co.uk):

SINGLE WILD CARD POKER
(THREE OF A KIND RANKED ABOVE TWO PAIR)
Poker hand Number of ways Probability
Five of a kind 13 0.00045%
Straight Flush 184 0.0064%
Four of a Kind 3,120 0.11%
Full House 6,552 0.23%
Flush 7,804 0.20%
Straight 20,532 0.27%
Three of a Kind 137,280 4.8%
Two Pair 123,552 4.75%
One Pair 1,268,088 44%
High Card 1,302,560 45%
Total 2,869,685 100%

Most of the frequencies are in line with the standard ranking, but there is one glaring exception. The three of a kind occurs with a slightly higher frequency than two pair! This violates the principle that less probable hands are ranked as more valuable!

This table suggests that ranking three of a kind higher than two pair is a mistake. One attempt to fix this problem is to reverse the rankings: instead make two pair more valuable than a three of a kind.

The problem with this approach is that players will respond to the rule change. If a player is holding a pair and a wild card, then the player will declare the wild card to complete the higher ranked two pair rather than the three of a kind. Therefore, we have to recalculate the probabilities.

Under this new rule, the probabilities become (again from curiouser.co.uk)

SINGLE WILD CARD POKER
(TWO PAIR RANKED ABOVE THREE OF A KIND)
Poker hand Number of ways Probability
Five of a kind 13 0.00045%
Straight Flush 184 0.0064%
Four of a Kind 3,120 0.11%
Full House 6,552 0.23%
Flush 7,804 0.20%
Straight 20,532 0.27%
Two Pair 205,920 7.2%
Three of a Kind 54,912 1.9%
One Pair 1,268,088 44%
High Card 1,302,560 45%
Total 2,869,685 100%

Again, the table is fine except for the exception: now that the two pair is ranked as more valuable, the frequency of its occurrence is higher than three of a kind! The poker rankings again violate the principle of more valuable hands occurring less frequently.

As one can imagine, the ranking problem persists when playing wild card poker with two jokers or other rules like “deuces wild.” The situation was studied by mathematical researchers John Emert and Dale Umbach of Ball State University in a 1996 Chance article who explained:

When wild cards are allowed, there is no ranking of the hands that can be formed for which more valuable hands occur less frequently

The paradox is not going away, and it is fruitless to pursue a completely logical frequency ranking in wild card poker.

Resolving the paradox: inclusion frequency

Emert and Umbach did take a stab at fixing the wild card poker paradox. Since frequency ranking could not work, it would have to be abandoned and a new standard established.

The idea is called inclusion frequency, which they describe as the following (quoted from here)

We propose a ranking that, rather than partitioning the ranked hands into disjoint categories, acknowledges that certain hands can be labeled in several ways. For example, any hand that could be labeled a full house could also be considered as two pair, three of a kind, or even one pair. A hand such as (Waaab) could be declared to be any of these types as well as a four of a kind”¦

We define “inclusion frequency” for each type of hand to be the number of five card hands that may be declared as such. The inclusion frequency ranking is determined by ranking the hands so that those type of hands that have smaller inclusion frequencies are more valuable.

This idea will always make a full-house more valuable than a three of a kind or a two pair, for instance, because the full house could be included in the count of three of a kind with two extra cards or in the count of two pairs.

Similarly, a three of a kind using a wild will also be included in the count of two pair, making the inclusion frequency of a two pair necessarily larger than three of kind. A three of kind will therefore always be more valuable than a two pair.

Now there is no paradox about which hand is more valuable. The only peculiar thing is that the flush becomes more valuable than a full house with one or more wilds. With four wilds it can be shown that a flush is more valuable than even a four of a kind!

You can see the inclusion ranking of the hands here.

While this new ranking removes the paradox, it may come at a high price of changing the rules and potentially confusing players.

And so wild card poker remains something of a mystery with its paradoxical card rankings.



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  • Scott

    Since any hand with a pair and a wild card could be made into a Three of a Kind or Two Pair, and since any person will naturally choose the better hand, wouldn’t a solution be to simply get rid of one or the other? Simply eliminate two pairs from the game. After all, when playing with Three of a Kind as higher ranked, Two Pairs will be eliminated anyway – no one will choose them.

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    I at first thought the same thing, but I was overlooking the “pure” two pairs that occur in a hand without a wild card, like 4455A. We could just call those pairs, but I would bet the game would have a lot of ties, or the person with the two pair would lose a single pair with a higher kicker and that seems really unfair.

  • Scott

    With that in mind there is an easier way to resolve the paradox: do nothing.

    If we get technical, the primary premise – higher ranked hands have lower probabilities – isn’t universally true.

    It is true for the general classification of hands (royal flush vs. pair) but the actual ranking of hands is more granular.

    A pair of aces is higher than a pair of twos, despite the odds being the same. Even more arbitrary is the valuation of one suit higher than another. Though arbitrary, they are necessary to resolve ties.

    While some versions will treat all suits as the same, splitting the pot on tied hands, all versions of poker treat the face value of the cards as being different values, despite the fact that they have no inherently different value based on probabilities.

    This brings us to our solution: when 3 of a Kind is ranked higher than 2 Pair, the odds are sufficiently close that we can effectively treat them as being equal. However, this proximity in probability only arises when the 3 of a Kind is treated as better than 2 Pair. So to maintain their similary probabilities, we still need to place more weight on the 3 of a Kind. We do that by saying that 3 of a Kind and 2 Pair are equal ranking, but 3 of a Kind wins the “tie” between it and a Two Pair – much in the same manner that Aces would beat Twos in a “tie” between pairs.

    This is a simply reclassification, placing the difference between 3 of a Kind and 2 Pair on a “lower” level, the same level that we evaluate card face values. In the end it results in no change in game play, so rather than confuse the players with this “meta”-change, things can simply be left alone.

  • William

    Why are the probabilities dependent on what the player chooses? Why not just calculate the pure probability of getting a set of cards when a wild card is included, and then let the player choose which one they want to declare? The players choice shouldn’t influence the chance of getting certain cards.

  • Scott

    William: “The players choice shouldn’t influence the chance of getting certain cards.”

    It does when you involve wild cards. A wild card is what the player chooses it to be. We are, of course, assuming that a player will always choose to make their hand the best possible.

  • E

    In practice, a player does not need to declare what the wild card is. The rule in all casinos that I know of is “the cards speak”. In every poker game in any casino, you can just show your cards silently and it is the dealer’s job to determine who won.

    I have never seen wild cards in a real poker room but in the game of Omaha, players are often confused as to what they hold. Even if you announce it wrong, so long as you show your cards, the dealer should be figuring it out correctly for you.

  • Ron

    In our poker group we have decided not to use wild cards in our poker games. The reason is that there is not a definitive ruling on the ranking of a five-card hand. For example would a hand of five eights beat a Straight Flush and/or a Royal Flush?
    I would think it would depend on the odds of receiving a five-card hand.
    Thanks
    Ron

  • Michael

    Ron you bring up a good point. The odds are getting 5 of a kind of lower than that of a straight flush, because there are only 13 ways to make 5 of a kind, compared to 40 ways to make a straight flush. This ignores wild cards (although one is necessary to make 5 of a kind) because the point is that 5 of a kind is rare strictly because we have combined all ways to make a straight flush into a single category, so we have not only A K Q J 10 of each suit, but also K Q J 10 9, etc. Some types of hands lend themselves to being formed in more ways than others: 4 or 5 of a kind, for instance, can only be formed one way for each rank of card, while a straight flushes can be formed for each suit. Why group the hands the way they are done? In some respects it seems pretty arbitrary, other than for the sake of compactness: it could get unwieldy, say, to have separate flushes for each suit. But there is nothing (except perhaps dislike of inconsistency and a fear of descending down the road to anarchy) preventing us from arbitrarily designating that “Royal Flush” is a separate type of hand from a straight flush, and therefore, since there can only be 4 royal flushes whereas there can be 13 five-of-a-kinds the royal flush ranks higher.

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