A-10 Hold'em, 20 card deck

I have noticed and played (for play money) a new online multi-player game called 10-A Texas Holdem. The rules are identical to conventional NL Texas Hold'em with the following exceptions.

1. 20 card deck with only 5 ranks: T,J,Q,K,A !!!!!!
2. 5 players max.

Here are some some aspects of the game.

1. The only flush is a Royal Flush.

2. You have 7 cards with which to make a 5 card poker hand and there are only 5 ranks. Lowest possible hand is two pair. A Full House seems to be the most common winning hand.

3. There are only 25 starting 2-card hand categories: 5 pairs, 10 suited unpaired hands, and 10 unsuited ("off") unpaired hands.

4. A Royal is sufficiently rare such that suited and unsuited hands seem nearly identical in game play.

5. If the board comes as a straight (5 ranks) then all players still in the hand split the pot except in the rare case when a player has a Royal. With no pair on the board, it is impossible for any player to make a better hand than the board, except for a RF.

6. I am interested in calculating the EV at showdown of each of the 25 starting 2-card hands in a heads-up game, and in a game with 4 opponents. Because the deck is so small and the possibile outcomes are unusually limited, I am taking awhile to consider analytic approaches to analyzing the game.

Any thoughts, comments, ideas?

Head-to-head would seem to be going through all the 25 hands and working out the possible hands for the other player and how often they can occur, e.g. AsAh can occur 6 times, opposite it can be AA (dc) KK (sh 1, sd 4, dc 1) etc. Then work out all the results for the remaining ways of getting 5 cards from the 16 left. Best of luck.

I actually think I am going to do something different.

First, I will ignore suits when evaluating hands because:

1. The only hand where suits matter is Royal Flush, and it occurs rarely.

2. The prospects for making a Royal Flush are very easy to analyze and can be added on later.

3. The prospects for making a Royal Flush is not much of a discriminator - RF prospects are identical for all off-suit hands, e.g., ATo and KQo will make a Royal with the same frequency. Similarly, RF probabilities are also identical for all suited 2-card hands, and further, all paired 2-card hands would also have an identical probability to make a Royal.

So, I am planning on looking at all possible 5-card Boards, because when considering ranks there are only 121 of them in this wacky game:

20 4oak
20 Full Houses
30 3oak+ 2 singletons
30 2pair+1singleton
20 1 pair+3singletons
1 Straight (no pair)

Then for each of the 121 boards I'll calculate for each of the 25 different 2-card hands the frequency of occurrence and hand ranking at showdown. That info can be straightforwardly applied to generate heads-up EVs and rankings for the 25 cards initial hands categories, and should be usable to eventually generate more complex information such as EV in a 5-handed game. It will also be easy to modify the answers for the effects of the ability to make a royal flush.

Its fun to evaluate a game where there are so few cards and possible hand categories.

P.S. Monte carlo simulation might be attractive for analyzing multi-player games, even with its inherent lack of precision due to statistical variance.

Quote: gordonm888

I actually think I am going to do something different.

First, I will ignore suits when evaluating hands because:

1. The only hand where suits matter is Royal Flush, and it occurs rarely.

2. The prospects for making a Royal Flush are very easy to analyze and can be added on later.

3. The prospects for making a Royal Flush is not much of a discriminator - RF prospects are identical for all off-suit hands, e.g., ATo and KQo will make a Royal with the same frequency. Similarly, RF probabilities are also identical for all suited 2-card hands, and further, all paired 2-card hands would also have an identical probability to make a Royal.

So, I am planning on looking at all possible 5-card Boards, because when considering ranks there are only 121 of them in this wacky game:

20 4oak
20 Full Houses
30 3oak+ 2 singletons
30 2pair+1singleton
20 1 pair+3singletons
1 Straight (no pair)

Then for each of the 121 boards I'll calculate for each of the 25 different 2-card hands the frequency of occurrence and hand ranking at showdown. That info can be straightforwardly applied to generate heads-up EVs and rankings for the 25 cards initial hands categories, and should be usable to eventually generate more complex information such as EV in a 5-handed game. It will also be easy to modify the answers for the effects of the ability to make a royal flush.

Its fun to evaluate a game where there are so few cards and possible hand categories.

P.S. Monte carlo simulation might be attractive for analyzing multi-player games, even with its inherent lack of precision due to statistical variance.

Can you remember this SHORT DECK POKER( 6 to T, J, Q, K, Ace, total 36 cards) simulation program for two players ? I think it can be modified to analyse your new game ? see image below.

https://ibb.co/52pvvJz

You can input any no of player's cards, board cards or opponent's cards.
To generate EV in a 5-handed game will be quite challenging.

I haven't checked these, other than they add up and in total each hand wins the same as it loses, but it seems reasonable that AA would be the best hand to start with and JT the worst.

Hand	Win	Lose	Tie	EV	Rank
AsAh	2 565 720	976 464	467 640	.396 341	1
KsKh	2 294 928	1 220 616	494 280	.267 920	2
QsQh	2 010 888	1 468 512	530 424	.135 262	3
JsJh	1 681 344	1 736 472	592 008	-.013 748	13
TsTh	1 460 664	1 931 952	617 208	-.117 533	23
AsKs	1 164 096	870 048	639 072	.109 998	4
AsQs	1 097 352	924 696	651 168	.064 587	6
AsJs	1 036 944	975 312	660 960	.023 055	8
AsTs	995 832	1 015 848	661 536	-.007 488	12
KsQs	1 036 416	984 288	652 512	.019 500	9
KsJs	976 008	1 034 904	662 304	-.022 032	15
KsTs	934 896	1 075 440	662 880	-.052 575	17
QsJs	911 760	1 098 480	662 976	-.069 848	19
QsTs	870 648	1 139 016	663 552	-.100 391	21
JsTs	808 272	1 212 624	652 320	-.151 261	24
AsKh	3 442 788	2 629 224	1 947 636	.101 446	5
AsQh	3 242 556	2 793 168	1 983 924	.056 036	7
AsJh	3 061 332	2 945 016	2 013 300	.014 504	10
AsTh	2 937 996	3 066 624	2 015 028	-.016 039	14
KsQh	3 059 748	2 971 944	1 987 956	.010 949	11
KsJh	2 878 524	3 123 792	2 017 332	-.030 583	16
KsTh	2 755 188	3 245 400	2 019 060	-.061 126	18
QsJh	2 685 780	3 314 520	2 019 348	-.078 400	20
QsTh	2 562 444	3 436 128	2 021 076	-.108 943	22
JsTh	2 375 316	3 656 952	1 987 380	-.159 812	25

Quote: charliepatrick

I haven't checked these, other than they add up and in total each hand wins the same as it loses, but it seems reasonable that AA would be the best hand to start with and JT the worst.

Hand	Win	Lose	Tie	EV	Rank
AsAh	2 565 720	976 464	467 640	.396 341	1
KsKh	2 294 928	1 220 616	494 280	.267 920	2
QsQh	2 010 888	1 468 512	530 424	.135 262	3
JsJh	1 681 344	1 736 472	592 008	-.013 748	13
TsTh	1 460 664	1 931 952	617 208	-.117 533	23
AsKs	1 164 096	870 048	639 072	.109 998	4
AsQs	1 097 352	924 696	651 168	.064 587	6
AsJs	1 036 944	975 312	660 960	.023 055	8
AsTs	995 832	1 015 848	661 536	-.007 488	12
KsQs	1 036 416	984 288	652 512	.019 500	9
KsJs	976 008	1 034 904	662 304	-.022 032	15
KsTs	934 896	1 075 440	662 880	-.052 575	17
QsJs	911 760	1 098 480	662 976	-.069 848	19
QsTs	870 648	1 139 016	663 552	-.100 391	21
JsTs	808 272	1 212 624	652 320	-.151 261	24
AsKh	3 442 788	2 629 224	1 947 636	.101 446	5
AsQh	3 242 556	2 793 168	1 983 924	.056 036	7
AsJh	3 061 332	2 945 016	2 013 300	.014 504	10
AsTh	2 937 996	3 066 624	2 015 028	-.016 039	14
KsQh	3 059 748	2 971 944	1 987 956	.010 949	11
KsJh	2 878 524	3 123 792	2 017 332	-.030 583	16
KsTh	2 755 188	3 245 400	2 019 060	-.061 126	18
QsJh	2 685 780	3 314 520	2 019 348	-.078 400	20
QsTh	2 562 444	3 436 128	2 021 076	-.108 943	22
JsTh	2 375 316	3 656 952	1 987 380	-.159 812	25

Very interesting and many thanks. Can I ask how you calculated this? Combinations? Looping code? Simulation?

So, being suited is worth about 0.009 in EV - less than 1%. And based on my time playing this game I was speculating that a TT pair was a bad hand, but I did not realize that it was the third worst hand (out of 25.)

In fact, the only starting hands that are positive EV are AJ-AK (suited and off) and QQ, KK and AA.

Quote: ssho88

Can you remember this SHORT DECK POKER( 6 to T, J, Q, K, Ace, total 36 cards) simulation program for two players ? I think it can be modified to analyse your new game ? see image below.

https://ibb.co/52pvvJz

You can input any no of player's cards, board cards or opponent's cards.
To generate EV in a 5-handed game will be quite challenging.

Thanks for pointing this out.

I try to input AsAh as player's first two cards, the sim results(10 million rounds) :-

EV = (6398335-2433259)/10,000,000 = 0.3965096, which is quite close to charliepatrick's results.

See attached image for the breakdowns.

https://ibb.co/X8Cp9Ld

The sim results shown that there are not possible to form THREE OF A KIND when your initial two cards is AsAh.( Is it true ??).

Yes I just used looping code. For the first bit, the four cards of the two hands using analysis (e.g. Player1=AsAh represents 6 hands), then all hands that it could marry with (Player2=AdAc=1 hand, KsKh=1, KsKd=2,....). For the final part just looped through all the ways to pick five cards from the remaining 16 - this is OK for such a small deck but wouldn't work for a standard deck! You could do the first part by brute force as it probably takes longer to think about it than just run the code. (Also I didn't optimize for the deck size, so it did check for all possible poker hands including One Pair, Flushes etc. as I already had that code.)

Quote: ssho88

I try to input AsAh as player's first two cards, the sim results(10 million rounds) :-

EV = (6398335-2433259)/10,000,000 = 0.3965096, which is quite close to charliepatrick's results.

See attached image for the breakdowns.

https://ibb.co/X8Cp9Ld

The sim results shown that there are not possible to form THREE OF A KIND when your initial two cards is AsAh.( Is it true ??).

Thanks for this.

I didn't investigate what hands won and lost, but if you start with a pair AA you could land up with two pair (KKQQJ - and hence rarely Tie, but mostly Lose). If there's an Ace on the board then either the board has a pair or better (AKKQJ/AKKQQ/KKKQJ/KKKQQ - you make Full House, AAKQJ/AKKKK/KKKKQ - Quads) or a straight (AKQJT - you play the board or make a Straight Flush). So it does look as if you can't make Trips starting with a Pair.

Quote: charliepatrick

Thanks for this.

I didn't investigate what hands won and lost, but if you start with a pair AA you could land up with two pair (KKQQJ - and hence rarely Tie, but mostly Lose). If there's an Ace on the board then either the board has a pair or better (AKKQJ/AKKQQ/KKKQJ/KKKQQ - you make Full House, AAKQJ/AKKKK/KKKKQ - Quads) or a straight (AKQJT - you play the board or make a Straight Flush). So it does look as if you can't make Trips starting with a Pair.

Yes, exactly right.

While playing the game, I have learned to raise when a straight is made on the board. The other players think I have made a better hand and usually fold -not realizing that when a straight is on the board that it is impossible for any player to have a better hand. LOL, the game is so new that many players do not understand it.

Quote: gordonm888

Yes, exactly right.

While playing the game, I have learned to raise when a straight is made on the board. The other players think I have made a better hand and usually fold -not realizing that when a straight is on the board that it is impossible for any player to have a better hand. LOL, the game is so new that many players do not understand it.

That’s surprising to me unless there’s 3 to a flush on the board. Broadway on the board is no different in this game and hold em.

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Quote: unJon

That’s surprising to me unless there’s 3 to a flush on the board. Broadway on the board is no different in this game and hold em.

Exactly, when a straight is made on the board(with 3 to a flush or 4 to a flush), there is still possible to have a better hand, i.e straight flush.