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Bob Dancer writes a video poker column for beginners to experts. He also writes a column with Jeffrey Compton, "Player's Edge", featuring information on promotions at various Las Vegas casinos. Player's Edge is published each Friday in the Neon section of the Las Vegas Review-Journal. Click here to send Bob Dancer an e-mail.

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April 24, 2007

A Double Bonus Deuces Wild Problem

I was playing Double Bonus Deuces Wild at Hooter's during their bartop Card of the Day Bingo promotion. (I've heard reports that this promotion is now over. I haven't been there recently to verify this.) While playing I came across a strategy problem that I hadn't considered so I went home and figured it out. The solution was unusual, so I thought I'd share it with you.

First, the game. I've never written about this game, other than in passing, and so first let's look at the pay schedule.

Royal Flush	250	4,000
4 Deuces with Ace	400	2,000
4 Deuces	200	1,000
Wild Royal	25	125
5 Aces	160	800
5 3s, 4s, or 5s	50	250
5 6s - Ks	20	100
Straight Flush	12	60
4 of a Kind	4	20
Full House	3	15
Flush	2	10
Straight	1	5
3 of a Kind	1	5
Return		99.81%

Not every DBDW game has this pay schedule, of course. In "Video Poker for Winners," we identify this one as "25/12" which are the returns for the Wild Royal and Straight flush respectively. The "next best" pay schedule is 12/9, where the straight flush is reduced from 12-for-1 to 9-for-1. That single change reduces the return by 1.2%. This would change whether I found the game playable, but would not affect the answer to the strategy problem I was wondering about.

Now let's look at the promotion. You get a "Card of the Day" bonus of 40 coins for a natural four-of-a-kind (i.e. without wild cards) for all eligible quads. (A five-of-a-kind including exactly one deuce also receives the bonus.) To be "eligible," a quad must not have been hit by anyone at the bar since the round started. The round starts with 13 eligible quads and the number decreases by one until at last all have been hit, when a new round starts and all 13 become eligible again.

The hand I was musing about was one containing two pair, such as 5577K. Since the full house returns 3-for-1 in this version of a type of Deuces Wild, you know the correct strategy would be to hold either the pair of fives OR the pair of sevens, not both. With the different returns for five-of-a-kinds, where five 5s give you 250 coins and five 7s give you "only" 100", the usual correct play on this hand would be to hold the fives by themselves.

Now we go back to the promotion. What if 7s are eligible (i.e. four natural sevens get paid 60 coins rather than 20 coins) and 5s are not eligible? Would this be enough to shift the correct play to holding the 7s instead of the 5s? I strongly suggest you try to work out the problem yourself before reading the rest of this article. If you end up with the same answer I do, great. (If you could have figured it out at the machine you're far better at this than I am.) If you determine you don't know how to figure it out, then you'll learn something by the article.

There is no software on the market that addresses this question directly, although you can use WinPoker to help you. Although I don't sell WinPoker anymore, it remains a good analytical tool. WinPoker includes the analysis for the Deuces Deluxe game. (Frugal VP software includes this game, but does not allow the type of analysis needed to help us on this problem.) Deuces Deluxe is a game that allows different pay schedule amounts for natural quads and quads with wild cards.

So, with any Deuces Deluxe pay schedule, I enter in the hand 5577K. Holding either pair, it tells me that of the 16,215 times you draw to the pair you end up with a natural 4-of-a-kind 41 times and a 5-of-a-kind 20 times. The software doesn't differentiate between 5-of-a-kinds with one deuce, with two deuces, and with three deuces, but it's easy to see that there are only four of these 5-of-a-kinds with one deuce. These are all four fives with the deuce of clubs, all four fives with the deuce of diamonds, all four fives with the deuce of hearts, and all four fives with the deuce of spades. These are the key numbers I need to work with.

Let's continue with 5577K. The number of 3-of-a-kinds, full houses, and wild 4-of-a-kinds are going to be identical whether we hold 55 or 77, so all we need to concentrate on are the natural quads and the quints with exactly one deuce.

So holding the fives, we get 20 coins for the natural quads 41 times, 250 for the quints with one, two or three deuces 20 times. This multiplies out to 41*20 + 20*250 = 820 + 5,000 = 5,820. We don't need to differentiate between whether the quints are "natural" because we are assuming we get no extra bonus either way. Also we aren't including the "quads including wild cards" as the number of times these occur, and the payout, will be the same both ways.

Holding the sevens, we get 60 coins for the natural quads 41 times, 100 for the quints with two or three deuces 16 times, and 140 for the quints with exactly one deuce 4 times. This multiplies out to 41*60 + 16*100 + 4*140 = 2,460 + 1,600 + 560 = 4,620.

Since this difference of 1,200 coins is spread out over 16,215 hands, playing for dollars holding 55 remains better than 77 by $1,200/16,215 = 7.4¢ the sevens are eligible for the 40-coin bonus and the fives aren't. After seeing this math once, it's easy to see that if the hand were AA553, the bonus for five aces would swamp whether or not you got the 40 coin bonus a total of 46 times on the fives.

This assumed it was one of the first twelve quads hit per round. For the thirteenth quad, there is a $500 bonus in addition to the regular 40 coins. This is a 2,040-coin bonus if you're playing for quarters but only a 140-coin bonus if you're playing for five dollars. If you're going to play this game during this promotion, I suggest you look at the two pair problem for stakes you plan to play. This article tells you how to do it.

This turned out to be not a particularly close play, but I didn't know for sure while at the casino. I donÕt know what the next promotion will be, but the more you go through exercises like this, the easier it will be for you to figure out how to play the next promotion.