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?

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:gordonm888I 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.

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

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.Quote:ssho88I 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 ??).

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.