Competent Double Double Bonus players know that from a hand like A♠ A♥ A♣ 4♦ 7♠, the correct play is AAA and not AAA4. Players might be tempted to hold AAA4 because four aces with a 2, 3, or 4 (a kicker) in this game receive 2,000 coins and four aces with any other fifth card “only” receive 800 coins.
Sometimes, however, there are one or more progressives on this game. If there is only one progressive, it’s usually on the royal flush. The second progressive goes on four aces with a kicker (AWAK). There can also be progressives on aces without a kicker, and sometimes other hands as well.
If the progressive on AWAK gets high enough, it becomes correct to hold the kicker. Today’s article looks at how high this progressive must be to make this play the right one.
There are three separate types of hands that are affected:
- One kicker, e.g. A♠ A♥ A♣ 4♦ 7♠
- Two paired kickers, e.g. A♠ A♥ A♣ 4♦ 4♠
- Two unpaired kickers, e.g. A♠ A♥ A♣ 4♦ 3♠
The level of AWAK that causes you to hold the kicker is different in all three of these examples.
One option is to hold the kicker. There are 47 possible draws. You can end up with 3-of-a-kind, a full house, or AWAK. How many of each varies depending on which hand type you have:
| Hand 1 | Hand 2 | Hand 3 | |
| 3-of-a-kind | 43 | 44 | 43 |
| Full House | 3 | 2 | 3 |
| AWAK | 1 | 1 | 1 |
Another option is to only hold AAA by itself. There are 1,081 possible combinations of cards you can draw. This adds the possibility of drawing AAAA without a kicker. The distribution of draws is:
| Hand 1 | Hand 2 | Hand 3 | |
| 3-of-a-kind | 969 | 968 | 969 |
| Full House | 66 | 67 | 66 |
| AWAK | 11 | 10 | 11 |
| AAAA with no kicker | 35 | 36 | 36 |
The stakes we play for make no difference, but to make things easier to talk about, I’m assuming we play for dollars, five coins at a time.
Three-of-a-kind always returns $15.
For the sake of this paper, I’m assuming full houses return $45. That is, either the 9/6 game or the 9/5 game. If you’re actually playing an 8/5 game or worse, you’ll be able to apply the same methodology to figure it out yourself.
The amount you get for aces without a kicker matters — so for each of the hands, I’m going to figure it out for this hand returning both $800 and $900. The $800 figure could mean there is no progressive on this hand, or it could be that there is a progressive, but it was just hit. If the actual progressive for this is at, say $833, then the number you should change for your strategy is one third of the way between $800 and $900.
I will work out, in detail, the magic number for AWAK assuming AAAA returns $800. Once you have seen how to do it and can duplicate it for yourself, I’ll just give you the numbers.
Let AK stand for the number where it doesn’t matter if we hold the kicker or not. We get this number by setting the EV (expected values) of the two possible draws to be equal to each other.
(969*$15 + 66*$45 + 11*AK + 35*$800) / 1,081 = (43*$15 + 3*$45 + 1*AK)/47
Since 1,081 is exactly 23 times as large as 47, we multiply the right side of the equation by 23/23. This doesn’t change its value. It merely makes the denominator on both sides equal.
(969*$15 + 66*$45 + 11*AK + 35*$800)/1,081 = (23*43*$15 + 23*3*$45 + 23*AK)/1,081
We now multiply both sides by 1,081 to remove the denominator. Multiplying all numbers out we get:
$14,535 + $2,970 + 11*AK + $28,000 = $14,835 + $3,105 + 23*AK
Moving the dollar amounts to the left side and the AK amounts to the right, we get:
$27,565 = 12*AK
$2,297.1 = AK
My final table summarizes all three types of hands, with the value of AAAA at 800 and also at 900. The fourth column is the average weighted by how often each hand occurs. The fifth column is how much the weighted average increases per units of 10 on AAAA.
| Hand 1 | Hand 2 | Hand 3 | Average | Incr | |
| AAAA = 800 | 2297.1 | 2237.3 | 2182.1 | 2284.6 | |
| AAAA = 900 | 2588.7 | 2514.2 | 2459.0 | 2573.8 | 28.6 |
This may be useful to some of you. The article was inspired by me playing next to somebody who had just hit aces with a kicker and commented that it was only because he had misplayed the hand. The progressive was at 2283 and he had memorized the number of 2280 at which to make the strategy change.
The number he was using was approximately correct, if the progressive for AAAA were at 800. As I recall, it was quite a bit higher than that, so he’ll be glad to know he actually made the correct play.
On a practical basis, anywhere around the correct break point number, it doesn’t really matter which of the two plays you make. The EV is very close to being identical either way.
Makes sense
it seems like between 4-6 pm in atlantic city casinos the machines get hot as people get off work and come in.other times they are average.can casinos do this?
We are regular costumers at the Borgata, I find video poker machines are more liberal late at night.
Bob, interesting article. So in simple terms, can you give us the thresholds on aces for holding the kicker ?.
A simplified way to apply this concept is to hold a kicker whenever AAAA with a kicker pays at least 2.85 times the payout for AAAA with no kicker. This is not as precise as Bob’s calculation, but comes close to the “average” numbers in his chart. It has the advantage of requiring you to memorize only one number, and can be applied easily, particularly where both AAAA payouts are on progressive meters.
I think the same number would work rough justice as applied to four 2s, 3s and 4s with and without a kicker. However, the strategy adjustment would be required very infrequently because, in the case of 2s, 3s and 4s, the base payout with a kicker ($800) is 2x the payout without the kicker ($400), as opposed to 2.5x for AAAA ($2,000 vs $800). Accordingly, the payout with the kicker would need to be about $1140 before an adjustment is needed (assuming the payout without the kicker is at $400).