Kelly and Acey Deucey

Finding the below ask-the-W was very helpful; in dealer's choice home poker games, Acey Duecey is a favorite with some and it happens to be a game you need to know something about.

The link answers the basic question, but I think it is very obvious that many players, myself included, get tripped up on not so much on the strategy of when to bet, but on the Kelly Criterion. In other words, I see players get wiped out all the time even when the bet they took was correct otherwise.

If you want to call the bankroll the amount a player is willing to lose in an evening, can anybody use Kelly to determine how much is smart to bet? All I've got to go on now is a notion that Kelly rarely exceeds 5% of bankroll [which may be incorrect]

https://wizardofodds.com/ask-the-wizard/228/

An approximation for the Kelly fraction is advantage divided by variance, but the Kelly fraction can be calculated exactly. Below are my numbers for the first hand from a single deck by number of ranks of cards between the initial two cards. You can see how the approximate Kelly fraction EV/var. is not good at high player advantages.

This calculation assumes that the player loses twice the bet if the third card is equal in rank to one of the initial cards.

My Kelly calculation ignores the effect on the bankroll of paying antes. However, I'm sure a correct Kelly analysis would have to consider the antes, too.

Ranks between	Cards in	Cards same	Cards out	EV	var.	EV/var.	Exact Kelly
7	28	6	16	0.00	1.36	0.00	0.00
8	32	6	12	0.16	1.33	0.12	0.11
9	36	6	8	0.32	1.26	0.25	0.20
10	40	6	4	0.48	1.13	0.42	0.27
11	44	6	0	0.64	0.95	0.67	0.32

See above.

Thanks a bunch! so to see "exact Kelly" as 0.32 means it is best practice to bet about 32% of bankroll, no more, given the situation [and with the proviso about the ante]?

Quote: odiousgambit

Thanks a bunch! so to see "exact Kelly" as 0.32 means it is best practice to bet about 32% of bankroll, no more, given the situation [and with the proviso about the ante]?

Yes.

Betting the exact Kelly fraction of your bankroll produces the maximum growth rate for your bankroll. Betting a little more than Kelly or a little less than Kelly would cause the growth rate to be less.

To calculate my "exact Kelly" numbers, I set up a table in Excel varying the fraction f to maximize the bankroll growth. The formula I used for bankroll growth is: ( 1 + f ) ^ (prob. inside) * ( 1 - f ) ^ (prob. outside) * ( 1 - 2f ) ^ (prob. same). (This formula would be exact if someone else were paying your antes.)

To make the bet sizing easy, I would bet 1/10 of my current bankroll with 2-J, for example, 1/5 with 2-Q, 1/4 with 2-K, and 1/3 with 2-A.

I am surprised the variance is low

Quote: odiousgambit

I am surprised the variance is low

The variance is on a per-bet basis. I was surprised by the 0.95 variance for betting on 2-A, but then I realized you would win 1:1 most of the time and lose 2:1 only occasionally.

For the 2-A bet, the EV for the first bet from a single deck is: 1 * 44/50 - 2 * 6/50 = 0.64. Then the variance is 44/50 * ( 1 - 0.64 )^2 + 6/50 * ( -2 - 0.64 )^2 = 0.95.

Quote: ChesterDog

The variance is on a per-bet basis. I was surprised by the 0.95 variance for betting on 2-A, but then I realized you would win 1:1 most of the time and lose 2:1 only occasionally.

For the 2-A bet, the EV for the first bet from a single deck is: 1 * 44/50 - 2 * 6/50 = 0.64. Then the variance is 44/50 * ( 1 - 0.64 )^2 + 6/50 * ( -2 - 0.64 )^2 = 0.95.

Your chart and the wizard page do not seem to be a perfect fit. Your chart shows break-even at 7 cards in between, and an advantage at 8, while none of the Wizard examples have that exact thing. But I follow your chart OK, I think I can use it to solve for third-card-match being a 1x loss. That actually is the only rule I have ever seen in effect in games I have been in. If I miscalculate the variance a bit it won't matter much.

So, thanks again!

returning to this,

the game I see the most is where 'ties' simply lose. I now realize this is extremely easy to break down for the question of whether you should bet or not.

for example,

6 in between = 24 win, 26 lose. Not good.
7 in between = 28 win, 22 lose

>>>>

for 7 in between,

EV = 6/50= 0.12

variance approx 1.3
ev/var =0.12/1.3 = 0.09 for Kelly approx.

this of course all increasing with more in between

to play,

giving ace a value of 14, etc., add 8 to the low card; then highest low card favorable is a six [to ace, 14]; with a 2 you need to see a 10+ on the other end . I like to look first for the low card to make a quick decision whether to at least consider a bet. Or you could look for the high card and want to see at least a Ten. In fact I think I just convinced myself that is better.

note that for a 6 card you have this for the in between cards,
6... 7 8 9 T J Q K ... A
meaning 7, 8, 9, T, J, Q, and K are the in-betweeners

continuing,
5...6 7 8 9 T J Q... K
4... 5 6 7 8 9 T J...Q
3... 4 5 6 7 8 9 T... J
2... 3 4 5 6 7 8 9... T

Usually the deck is exhausted before another deck is used, so clearly you do not want to see 7s 8s and 9s, the cards that are always good, come out early. I can guess 6 thru 10 might be a good expansion of that. edit: just realized I have that backwards. *

if losing track of that, early failure is probably just generally good to see and vice versa

In these home poker games that I am familiar with anyway, the money you might win or lose playing poker is insignificant if Acey Deucey and similar games are allowed, which will be where you win or lose for the evening. The poker games will be regimented as to how many raises are allowed, max bet set low, etc. In Acey Deucy, especially, all that goes out the window. I think this shows players are unconfident about their poker skills but like games of luck where they can remember scoring big without skill. Those that have been burned hate Acey Deucey, on the other had. Personally, I like a games where "poker only" is the house rule, but otherwise dealer's choice, if poker, is ok.

Your mileage may vary.

* No, I guess that is right, maybe I need more coffee this morning