The Triton poker series has popularized short deck hold-em. I am sure that everyone knows by now that Short Deck is played with a 36-card deck since all the 2's, 3's, 4's, and 5's have been removed (of course, this is why Short Deck is also called Six-Plus poker).

Short Deck is a wildly fast-moving game with frequent large pots and tons of action. Pre-flop equities run much tighter than in Long Deck (regular 52-card NLHE poker). Similar to PLO, the in-hand swings in Short Deck are wild it being common for each card (flop, turn, river) to greatly change who is ahead in the hand.

In previous threads, I looked at how often does a Flush occur in Short Deck vs Long Deck, how often do Quads occur in Short Deck vs Long Deck, and how often do Full Houses occur in Short Deck vs Long Deck.

Here I want to take a more "high level" view. I want to investigate the distribution of winning hands in Short Deck vs Long Deck. As in the previous threads, I looked at 6-max Short Deck vs 6-max Long Deck. Unlike the previous analyses, this one does not seem amenable to analytical (combinatoric) solutions so I ran 10,000,000 deals of each poker variant. As was previously assumed, on each deal all hands go to showdown.

The table below presents the respective distributions of the category of winning hands. Following the adjusted rules of Short Deck, the table reflects that a Flush beats a Full House in Short Deck (in this and all other of my recent Short Deck threads a Straight beats Three of a Kind).

Long Deck Category	Six-Max Win Percentage	__________	Short Deck Category	Six-Max Win Percentage
Royal Flush	0.02%	.	Royal Flush	0.12%
Straight Flush	0.15%	.	Straight Flush	0.57%
Four of a Kind	0.89%	.	Four of a Kind	2.84%
Full House	10.95%	.	Flush	8.52%
Flush	10.83%	.	Full House	25.37%
Straight	16.56%	.	Straight	35.25%
Three of a Kind	13.89%	.	Three of a Kind	11.68%
Two Pair	30.50%	.	Two Pair	14.46%
One Pair	16.15%	.	One Pair	1.19%
High Card	0.06%	.	High Card	0.00%
.
TOTAL	100.00%	.	TOTAL	100.00%

You will see immediately the winning hands are generally better in Short Deck compared to Long Deck due to the number of ranks being compressed from 13 to 9. The other salient findings are that Full House and Flush have dramatically different frequencies in the two variants. A flush and full house have very similar winning frequencies in 6-max Long Deck (both around 11%). However, due to the rank compression, full houses are much more common than flushes in Short Deck. Of course, for the same reason Straights are much more common in Short Deck than in Long Deck as well. Note that the results in the table depend upon the hand rankings utilized (i.e., the Short Deck percentages would change if a different hand ranking was used).

One final peculiarity. In the 10 million short deck 6-max deals, High Card was never the winning hand! I wonder if this result was guaranteed or if High Card winning is so rare that 10 million trials was not sufficient for this phenomenon to occur.

I think high card can win in short deck but it's exceedingly rare.

The two hands have to share a card and kickers come into play.

So like if the board runout is 678JQ and you have AK against AT or KT against K9 you can have a high card win. But the prevalence of straights with 5 unpaired cards on the board make it almost impossible to be in that situation.

Here is more information on winning hands in Short Deck vs Long Deck (6-max). I ran another set of 10,000,000 deals of each variant and kept track of which hole-card combo won each deal, where examples of hole-card combos are AQs, KJo, 99, etc. As before, every hand goes to showdown on every deal.

The table below presents the tally (pct) of winning hole-card hand combos in 6-max Long Deck. In the case of any deal resulting in a split pot between N players, I awarded each player 1/N.

Table 1: Tally of 6-Max Long Deck Winning Hands by Starting Hand Combo

100*Pct	__A__	__K__	__Q__	__J__	__T__	__9__	__8__	__7__	__6__	__5__	__4__	__3__	__2__
A	133	151	140	133	124	110	103	99	95	98	95	91	89
K	57	117	134	126	119	104	93	91	86	83	79	78	74
Q	55	52	103	123	118	103	91	80	78	74	71	70	67
J	51	48	48	90	115	102	91	81	71	69	67	64	62
T	48	47	46	46	81	101	91	81	72	64	63	60	57
9	45	43	41	42	41	70	91	81	72	64	57	55	52
8	43	40	39	38	37	37	64	83	74	67	58	51	49
7	41	38	35	34	35	35	35	58	75	69	61	54	46
6	40	37	34	32	32	32	33	33	53	71	65	57	48
5	41	36	34	30	29	30	30	31	31	49	69	61	54
4	40	35	33	30	29	27	27	29	30	30	46	58	52
3	39	34	32	30	29	26	25	26	27	28	28	43	48
2	38	34	31	29	27	26	25	24	24	26	25	24	41

Following poker convention, the "lower triangular" portion of the table consists of suited hands, the "upper triangular" portion of the table consists of unsuited hands, and the "main diagonal" (in bold) of the table consists of the pairs.

The table's first entry of 133 means that 1.33% of all 6-max Long Deck deals were won by AA. Also, the 57 entry just below the 133 means that 0.57% of all 6-max Long Deck deals were won by AKs. Finally, the 151 entry just to the right of the 133 means that 1.51% of all 6-max Long Deck deals were won by AKo.

While these tallies contain a bunch of information, it is important to remember that not all cells in the table are created equal. Of course, there are more unsuited starting hands than suited starting hands which serves to give unsuited hands an "advantage" in the table's tally.

The fact reflected in the table that more hands are won with AKo than AKs is undoubtedly due to there being a lot more AKo starting hands than AKs starting hands over the course of the 10 million deals in the simulation. Everybody surely knows that there are 12 combos for each unsuited element (such as AKo), 4 combos for each suited element (such as AKs), and 6 combos for each paired element (such as AA).

In the next post I will use these combo counts to adjust the above tallies to be more illuminating.

Here is the follow-up to the previous post.

The table below presents the tally (pct) of specific winning hole-card hands in 6-max Long Deck. Examples of specific hole-card hands are AdQd, KhJc, 9d9c, etc.

As explained in the previous post, the entries of this table (for a specific hand such as KhJc) are derived directly from the previous table (for the corresponding combo such as KJo). Every unsuited entry of the previous table was divided by 12, every suited entry of the previous table was divided by 4, and every paired entry of the previous table was divided by 6.

Table 2: Tally of 6-Max Long Deck Winning Hands by Specific Starting Hand

1000*Pct	__A__	__K__	__Q__	__J__	__T__	__9__	__8__	__7__	__6__	__5__	__4__	__3__	__2__
A	222	126	117	110	103	91	86	83	79	82	79	76	74
K	142	195	112	105	99	87	78	76	72	69	66	65	62
Q	137	129	171	103	98	85	76	67	65	62	59	59	56
J	128	120	120	150	96	85	75	67	59	58	56	53	51
T	121	118	116	114	134	84	76	67	60	54	52	50	47
9	112	108	103	104	102	117	76	68	60	53	48	46	44
8	107	100	97	96	93	92	106	69	61	56	48	43	41
7	101	95	88	85	89	87	87	97	62	57	51	45	38
6	99	92	84	80	81	80	82	83	88	59	54	47	40
5	103	91	84	76	73	75	76	77	78	81	57	51	45
4	99	88	81	76	73	67	68	73	74	75	76	48	44
3	99	85	81	75	72	66	64	66	67	70	70	71	40
2	94	84	77	72	69	64	61	59	60	66	63	60	68

To repeat from above, following poker convention, the "lower triangular" portion of the table consists of suited hands, the "upper triangular" portion of the table consists of unsuited hands, and the "main diagonal" (in bold) of the table consists of the pairs.

The table's first entry of 222 means that 0.222% of all 6-max Long Deck deals were won by a specific AA hand such as AhAc. Also, the 142 entry just below the 222 means that 0.142% of all 6-max Long Deck deals were won by a specific AKs hand such as AdKd. Finally, the 126 entry just to the right of the 222 means that 0.126% of all 6-max Long Deck deals were won by a specific AKo hand such as AhKc.

Presenting the tallies (pcts) at the specific hand level rather than at the combined combo level allows us to see "expected" results such as AKs is preferred to AKo. In addition, pairs are now placed on equal footing with both suited and unsuited starting hands.

The table shows that, according to this simulation, the ten best starting hands in 6-max Long Deck poker are AA, KK, QQ, JJ, AKs, AQs, TT, KQs, AJs, and AKo. Of course, in this simulation every hand goes to showdown on every deal, so "playability" and other such factors are not taken into account.

As Long Deck poker is the poker that everybody is most familiar with, I won't spend any time discussing the results shown in this table since everybody is probably very familiar with the relative strengths of 6-max Long Deck starting hands.

The next two posts will be the parallel posts for Short Deck. As before, the Short Deck rules I use are that Flush beats Full House (but Straight still beats Three of a Kind).

Here is more information on winning hands in Short Deck vs Long Deck (6-max). I ran another set of 10,000,000 deals of each variant and kept track of which hole-card combo won each deal, where examples of hole-card combos are AQs, KJo, 99, etc. As before, every hand goes to showdown on every deal.

The table below presents the tally (pct) of winning hole-card hand combos in 6-max Short Deck. In the case of any deal resulting in a split pot between N players, I awarded each player 1/N.

Table 3: Tally of 6-Max Short Deck Winning Hands by Starting Hand Combo

100*Pct	__A__	__K__	__Q__	__J__	__T__	__9__	__8__	__7__	__6__
A	176	245	229	220	212	186	175	162	149
K	94	156	239	230	226	173	150	140	129
Q	87	90	139	240	236	181	155	131	123
J	86	88	91	130	253	197	168	143	120
T	83	86	90	95	123	217	185	157	133
9	74	70	72	77	82	106	177	158	135
8	70	63	64	67	73	71	97	144	124
7	66	59	57	60	65	65	60	89	109
6	63	56	54	53	57	57	54	49	82

To repeat from above, following poker convention, the "lower triangular" portion of the table consists of suited hands, the "upper triangular" portion of the table consists of unsuited hands, and the "main diagonal" (in bold) of the table consists of the pairs.

The table's first entry of 176 means that 1.76% of all 6-max Short Deck deals were won by AA. Also, the 94 entry just below the 176 means that 0.94% of all 6-max Short Deck deals were won by AKs. Finally, the 245 entry just to the right of the 176 means that 2.45% of all 6-max Short Deck deals were won by AKo.

While these tallies contain a bunch of information, it is important to remember that not all cells in the table are created equal. Of course, there are more unsuited starting hands than suited starting hands which serves to give unsuited hands an "advantage" in the table's tally.

The fact reflected in the table that more hands are won with AKo than AKs is undoubtedly due to there being a lot more AKo starting hands than AKs starting hands over the course of the 10 million deals in the simulation. Everybody surely knows that there are 12 combos for each unsuited element (such as AKo), 4 combos for each suited element (such as AKs), and 6 combos for each paired element (such as AA).

In the next post I will use these combo counts to adjust the above tallies to be more illuminating.

Here is the follow-up to the previous post on Short Deck.

The table below presents the tally (pct) of specific winning hole-card hands in 6-max Short Deck. Examples of specific hole-card hands are AdQd, KhJc, 9d9c, etc.

As explained in the previous post, the entries of this table (for a specific hand such as KhJc) are derived directly from the previous table (for the corresponding combo such as KJo). Every unsuited entry of the previous table was divided by 12, every suited entry of the previous table was divided by 4, and every paired entry of the previous table was divided by 6.

Table 4: Tally of 6-Max Short Deck Winning Hands by Specific Starting Hand

1000*Pct	__A__	__K__	__Q__	__J__	__T__	__9__	__8__	__7__	__6__
A	294	204	191	183	177	155	146	135	124
K	234	259	200	192	188	144	125	117	108
Q	218	225	232	200	197	151	129	109	102
J	214	219	227	217	211	164	140	119	100
T	208	216	225	236	205	181	154	131	111
9	184	174	180	193	205	177	148	132	113
8	175	158	160	169	182	176	162	120	104
7	166	148	141	149	162	163	150	149	91
6	158	140	135	133	143	143	136	123	137

To repeat from above, following poker convention, the "lower triangular" portion of the table consists of suited hands, the "upper triangular" portion of the table consists of unsuited hands, and the "main diagonal" (in bold) of the table consists of the pairs.

The table's first entry of 294 means that 0.294% of all 6-max Short Deck deals were won by a specific AA hand such as AhAc. Also, the 234 entry just below the 294 means that 0.234% of all 6-max Short Deck deals were won by a specific AKs hand such as AdKd. Finally, the 204 entry just to the right of the 294 means that 0.204% of all 6-max Short Deck deals were won by a specific AKo hand such as AhKc.

Presenting the tallies (pcts) at the specific hand level rather than at the combined combo level allows us to see "expected" results such as AKs is preferred to AKo. In addition, pairs are now placed on equal footing with both suited and unsuited starting hands.

The table shows that, according to this simulation, the ten best starting hands in 6-max Short Deck poker are AA, KK, JTs, AKs, QQ, QJs, QTs, KQs, KJs, and AQs. Of course, in this simulation every hand goes to showdown on every deal, so "playability" and other such factors (which may be less critical in Short Deck?) are not taken into account.

The above table illuminates many fascinating elements of Short Deck poker. In general, you can see that pre-flop hand equities are much tighter than in Long Deck. Since ranks are compressed in Short Deck, virtually every hand has a viable chance of making a straight, trips, two pair, or even a full house. Also, since flushes are elevated above full houses in Short Deck, suited connectors take on added value. We saw above that JTs (which I think is the 18th best starting hand in 6-max Long Deck) shoots up to become the 3rd best starting hand in 6-max Short Deck, behind only AA and KK. JTo is also a very good hand in Short Deck due to its many straight possibilities. Other suited connectors which are very strong starting hands in Short Deck include QJs, QTs, KQs, KJs, KTs, and T9s.

I finally got around to doing a comprehensive NLHE Short Deck starting hand ranking. I pitted each of the 81 starting hands vs five other random hands, with random boards and all hands going to showdown on all deals. (Of course, this is just one way to rank the starting hands.)

I ran 81 separate simulations of one million deals each. In these NLHE Short Deck simulations, a flush beats a full house and a straight beats three of a kind. The results are quite similar to those appearing in the previous post from a simulation of 10 million total deals (with no "control" over which hands occur in any deal).


Table 5: Ranking of 6-Max NLHE Short Deck Starting Hands (vs five random hands)

__Rank__	Starting Hand	___Tally___
1	AA	310,305
2	KK	272,042
3	QQ	247,000
4	JTs	246,191
5	AKs	240,625
6	QJs	237,712
7	KQs	236,232
8	QTs	233,530
9	KJs	229,397
10	AQs	229,391
11	JJ	228,455
12	KTs	225,697
13	AJs	223,029
14	JTo	221,474
15	ATs	217,509
16	T9s	216,207
17	TT	214,347
18	AKo	214,141
19	QJo	211,283
20	KQo	209,638
21	QTo	207,259
22	KJo	202,643
23	J9s	201,425
24	AQo	200,887
25	KTo	197,548
26	A9s	194,267
27	AJo	193,247
28	T8s	192,857
29	Q9s	190,101
30	T9o	188,192
31	99	187,223
32	ATo	186,806
33	98s	186,266
34	A8s	185,050
35	K9s	182,544
36	J8s	178,943
37	A7s	173,920
38	J9o	172,840
39	97s	170,923
40	T7s	170,368
41	88	170,034
42	Q8s	167,745
43	A6s	163,953
44	K8s	163,504
45	T8o	162,618
46	A9o	161,968
47	Q9o	158,649
48	87s	157,542
49	J7s	157,398
50	K7s	156,105
51	98o	155,872
52	77	155,095
53	96s	152,514
54	A8o	151,680
55	K9o	151,017
56	T6s	149,715
57	K6s	149,548
58	Q7s	149,434
59	J8o	147,332
60	66	143,811
61	86s	142,782
62	Q6s	141,473
63	A7o	141,245
64	97o	139,142
65	J6s	138,323
66	T7o	137,584
67	Q8o	135,630
68	K8o	130,038
69	76s	129,726
70	A6o	129,453
71	J7o	124,595
72	87o	124,036
73	K7o	122,030
74	96o	118,591
75	T6o	116,911
76	Q7o	115,497
77	K6o	113,949
78	86o	108,223
79	Q6o	107,746
80	J6o	104,025
81	76o	94,710

The tally for a starting hand reported in the table is the number of the one million NLHE deals (vs five other random hands) which the starting hand won plus 1/N for each deal which the hand is involved with an N-way chop.

One comforting result of this set of 81 simulations is that a starting hand wins "on average" 1/6 of the time vs 5 other random hands. That is, if you properly weight each of the above 81 starting hands (6 for pairs, 4 for suited, and 12 for unsuited), the average equity is 16.6666%, which of course is 1/6.

Now let's arrange these equities into the familiar 9x9 tableau.


Table 6: Equities of 6-Max NLHE Short Deck Starting Hands (vs five random hands)

10*Pct	__A__	__K__	__Q__	__J__	__T__	__9__	__8__	__7__	__6__
A	310	214	201	193	187	162	152	141	129
K	241	272	210	203	198	151	130	122	114
Q	229	236	247	211	207	159	136	115	108
J	223	229	238	228	221	173	147	125	104
T	218	226	234	246	214	188	163	138	117
9	194	183	190	201	216	187	156	139	119
8	185	164	168	179	193	186	170	124	108
7	174	156	149	157	170	171	158	155	95
6	164	150	141	138	150	153	143	130	144

The table above follows poker convention so that the "lower triangular" portion of the table consists of suited hands, the "upper triangular" portion of the table consists of unsuited hands, and the "main diagonal" (in bold) of the table consists of the pairs.

The first entry in the table (310 for AA) reflects that AA has a 31.0% equity vs five other random hands. 272 for KK means that KK has a 27.2% equity vs five other random hands. 241 for AKs means that AKs has a 24.1% equity vs five other random hands, and so on. From a statistical point of view, since they are based upon a simulation of one million deals these equities should be accurate to within 0.1% (so within 1 using the figures in the table).

We have already discussed how Short Deck "rewards" suited connectors such as JTs, QJs, KQs, QTs, KJs, etc. so I won't belabor the point again.