Ultimate Texas Hold 'Em: Face up cards?

Quote: jml24

Since UTH is my favorite casino game and I am a software developer, I have found this thread very interesting. It seems to confirm that the casinos are correct to be lax about sharing card information. Sometimes there are signs stating not to show your cards to other players but I see people share their cards all the time without being chastised by the dealers. Compare this with how aggressively dealers enforce the rule against looking at your phone and you can see what the casinos consider important.

I would be more interested in testing some obvious cases where preflop strategy might change. For example, I have an ace and I know two other players also have an ace. My gut says it would still make sense to raise preflop here even though my odds of making a pair of aces has dramatically reduced. It looks like there is not really a good AP opportunity through knowing exposed cards but some tweaks to basic strategy for specific cases to gain a small improvement would still be cool.
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Quote: aceside

Quote your above “If the player can see one other hand, this counts as two cards exposed, so the player sees his two hole cards plus 2 others (Nexp=2). This introduces randomness to the calculation, since I randomly choose two exposed cards and then do the exact calculation using the optimal strategy taking into account those four cards.”

Can you use this particular player hand of a pocket pair of 3’s to illustrate the above description? I don’t understand the randomness you mention. How much gain can a player obtain from 10-card-exposure collusion for a pocket pair of 3’s, exactly?
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Quote: Mental

EDIT: My hunch was wrong. Ac2d is a tiny bit better EV to raise than check when you know that your friend has an Ah3h. Your ace still has value as a kicker or high card.
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Quote: gordonm888

Below is a summary table looking at the advantage gained in EV when flipping the "Raise4x or Check" decision upon seeing the four cards listed in the first column, labeled Observed Cards.

Quote: aceside

Quote: gordonm888

Below is a summary table looking at the advantage gained in EV when flipping the "Raise4x or Check" decision upon seeing the four cards listed in the first column, labeled Observed Cards.

This research is very scientific. Wonderful work! I’d like to see the same analysis on these four hands too:

A2o, 33, K6o, JTo.

If possible. Player’s suited hands are rather rare and thus less important. I believe.
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Quote: gordonm888

Below is a summary table looking at the advantage gained in EV when flipping the "Raise4x or Check" decision upon seeing the four cards listed in the first column, labeled Observed Cards. In the Observed Cards column, H= a card with the same rank as the higher of the two player cards. Similarly, L = a card with the same rank as the Lower of the two player cards. So, the Observed Cards HHL-x for Q8o means that you have observed QQ8-x as the other four cards.

In the other columns heade by K5o, Q8o, K2s, Q6s, J8s and T9s, the entries are the improvement in EV gained by changing the "Raise4x or Check" decision away from the decision called for in basic strategy. In the last row of the last column I show that the cumulative gain from changing all these "Raise4x or Check" decisions based upon 4 cards is only about 0.06% in player EV.

These calculations do include the possibility of these hands making straights or flushes. However, note that I have NOT taken into account the effect of flush blockers or straight blockers in the kickers of the observed hands; I've simply considered the effect of observing "pair blockers." Therefore, if one were to take into account the effect of the observed 4 cards on the probability of making straights and flushes, then the values below would be different.

Player Hand		K5o	Q8o	K2s	Q6s	J8s	T9s	Sum
Probability of Hands		0.009049774	0.009049774	0.003016591	0.003016591	0.003016591	0.003016591
Observed Cards	Prob of ObsCards
NONE (xxxx)	0.589452888						0.019684
High-xxx	0.172522796	0.054482	0.056866	0.082587	0.057954	0.060932
Low-xxx	0.172522796		0.024866		0.009954	0.036932
HH-xx	0.012323057	0.162482	0.176866	0.190587	0.177954	0.172932
LL-xx	0.012323057	0.042482	0.112866	0.022587	0.079954	0.124932
HL-xx	0.036969171	0.112482	0.124866	0.116587	0.109954	0.128932
HHH-x	0.000191055	0.260482	0.296866	0.288587	0.297954	0.284932
HHL-x	0.001719496	0.120482	0.264866	0.124587	0.249954	0.260932
HLL-x	0.001719496	0.170482	0.232866	0.150587	0.201954	0.236932
LLL-x	0.000191055	0.100482	0.198866	0.056587	0.151954	0.210932
HHHL	1.30265E-05	0.306482	0.360866	0.310587	0.353954	0.348932
HHLL	3.90795E-05	0.166482	0.342866	0.146587	0.311954	0.338932
LLLH	1.30265E-05	0.228482	0.320866	0.184587	0.273954	0.324932
Advantage		0.016666287	0.02326005	0.021736507	0.019842183	0.026293711	0.011602791
Adv*Hand Prob		0.000150826	0.000210498	6.55702E-05	5.98558E-05	7.93174E-05	3.50009E-05	0.000601068

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Quote:

"These calculations do include the possibility of these hands making straights or flushes."

Quote: aceside

Quote: gordonm888

Below is a summary table looking at the advantage gained in EV when flipping the "Raise4x or Check" decision upon seeing the four cards listed in the first column, labeled Observed Cards.

This research is very scientific. Wonderful work! I’d like to see the same analysis on these four hands too:

A2o, 33, K6o, JTo.

If possible. Player’s suited hands are rather rare and thus less important. I believe.
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Quote: Mental

The hands with all interfering cards are easy for me to do. I don't have to average over all the other cards that are not H or L.

A2o HHHL R: -1.237798 C: -0.817160 D: -0.420638
A2o LLLH R: -0.842980 C: -0.538027 D: -0.304953

But these results are not very useful. When all the exposed cards match your hand, you will of course check instead of raising marginal hands.

Quote: aceside

Quote: Mental

The hands with all interfering cards are easy for me to do. I don't have to average over all the other cards that are not H or L.

A2o HHHL R: -1.237798 C: -0.817160 D: -0.420638
A2o LLLH R: -0.842980 C: -0.538027 D: -0.304953

But these results are not very useful. When all the exposed cards match your hand, you will of course check instead of raising marginal hands.

Actually, your calculation is very useful. We can extrapolate to the unknown territories. If there are four duplicate cards in the exposed cards, the average EV gain is
(0.42+0.30)/2 + 0.12 = 0.48.

For each duplicate card, the EV gain is
0.48/4= 0.12.

Therefore, we should check A2o whenever we see one duplicate.
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Quote: gordonm888

I expect A2 will play the same way. If you observe Axxx then check; if you observe 2xxx then Raise 4.

Quote: gordonm888

Yes, I agree with Mental. I advise being wary of downgrading to a CHECK when seeing a duplicate rank of your low card, if the rank of the card is quite low.

On the cases I posted, notice that there are no advantages to flipping the initial decision from Raise 4 to check for the hands for the hand Lxxx for K5o and K2o. Simply, seeing a matching 5 or a matching 2 in those cases does not reduce your equity (EV) sufficiently to warrant just Checking your hand. The majority of your prospects for winning those hands come from pairing your King.

I expect A2 will play the same way. If you observe Axxx then check; if you observe 2xxx then Raise 4.

On a hand like JTo, seeing either Jxxx or Txxx when you peek at neighbor's cards is sufficient to downgrade this hand to a check.
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Quote: aceside

Quote Mental "Even exposed 28o would cause you to check 33."
I believe you mean to say, even exposed 22 would cause you to check 33. This is cool!
Gordon already pointed this out.
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Quote: gordonm888

A deuce is a poison pill for a pair of threes in most poker games that are played against a dealer hand. I have seen this in Pai Gow Poker, 4 card poker, etc. A hand with a 33 pair beats only a hand with a pair of twos, and high card hands. Thus, a deuce kicker in your hand is significantly emasculates a 33 pair* because there are now only 3 combinatorial ways for dealer to make a 22 pair.

Minor effect: A deuce also does a poor job of blocking straights.

Similarly, a hand like K-5 is really improved by observing an Ace in the exposed cards.
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Quote: psiguy686

I stumbled upon this thread, I have a loose cheat cheat of basic collusion decisions which I’ve more or less committed to memory.

The play button of course has no house edge so anytime you can make a correct decision based on holds cards exposed would add to your EV. Might not beat the game but certainly makes it more fun.

Google sheet link

/spreadsheets/u/0/d/1rN7dst-b9OEdkp1U0aLKlLZuSWghSRgUF_qmdKhERKg/htmlview
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Quote: aceside

Here is my calculation of the EV gain of pocket 33 from collusion, based on your results. We assume all the gain comes from seeing one or two remaining 3s.

A pocket 33 hand happens once in every 221 hands. If it happens, the probability of seeing at least one 3 in your neighbor’s two cards is (2x48+1)/1225=7.9%, where 1225=C(50,2). If both events happen, player gains an edge (saved a loss) of 0.15+0.08=0.23. Therefore, a collusion player gets an advantage of

(1/221)x7.9%x0.23 = 0.008%.
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3c 3d 91
PP   -0.4725 /   1 =  -0.4725     -0.4725 -  -0.4725
PU   -0.2602 /   1 =  -0.2602     -0.2602 -  -0.2602
UU   -0.0710 /   1 =  -0.0710     -0.0710 -  -0.0710
PO   -1.6606 /  11 =  -0.1510     -0.1675 -  -0.1380
UO    0.3952 /  11 =   0.0359      0.0232 -   0.0469
OO    8.3462 /  66 =   0.1265      0.0811 -   0.1501

Quote: psiguy686

I stumbled upon this thread, I have a loose cheat cheat of basic collusion decisions which I’ve more or less committed to memory.

The play button of course has no house edge so anytime you can make a correct decision based on holds cards exposed would add to your EV. Might not beat the game but certainly makes it more fun.

Google sheet link

/spreadsheets/u/0/d/1rN7dst-b9OEdkp1U0aLKlLZuSWghSRgUF_qmdKhERKg/htmlview
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Quote: aceside

Great work! This is a lot of work, so I am thinking you may restrict your consideration of over cards to a pair of Aces, or an Ace and an unsuited King, because these two high cards are often easily exposed.
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Quote: aceside

Let me use your results to calculate the collusion EV gain on this player hand, Q7o. We consider three contributions from different neighbor’s two-card cases:
#1. two over-cards
#2. one over-card and one mid-card,
#3. one over card and one under-card.

A player hand Q7o happens twice in every 221 hands. If it happens, the probability of case #1 is C(8,2)/C(50,2)=28/1225, the probability of case #2 is 128/1225, and the probability of #3 is 160/1225. The player EV gains are 0.0435+0.0682, 0.0174+0.0682, and 0.0074+0.0682, respectively. Therefore, the total collusion advantage gain from the above three cases is:

(2/221) x [(28/1225) x 0.1117 + (128/1225) x 0.0856 + (160/1225) x 0.0756] = 0.02%,

which is 2.5 times greater than that on the player hand of pocket pair 33.
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Quote: aceside

For the hand Q8o, it’s actually very simple to collude. Whenever you see a duplicate in your neighbor’s hand, you check, instead of basic strategy raise 4x.

I want an edge of a lot more! That’s why I keep bugging you.
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Quote: Mental

Quote: aceside

For the hand Q8o, it’s actually very simple to collude. Whenever you see a duplicate in your neighbor’s hand, you check, instead of basic strategy raise 4x.

I want an edge of a lot more! That’s why I keep bugging you.
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You cannot get blood from a stone. The UTH game is not beatable with 10 exposed cards of other players. You can reduce the house edge, but not by any appreciable amount with only two cards exposed.

Analysis of 78 offsuit hands

Kc,Ad,91,,
HH,1,0.3448,0.3448,0.3448
LH,1,0.3466,0.3466,0.3466
LL,1,0.333,0.333,0.333
HU,11,0.5133,0.4937,0.5264
LU,11,0.5081,0.4872,0.5207
UU,66,0.6485,0.6015,0.6678
Qc,Ad,91,,
HH,1,0.317,0.317,0.317
LH,1,0.3239,0.3239,0.3239
LL,1,0.3131,0.3131,0.3131
HM,1,0.4718,0.4718,0.4718
LM,1,0.4659,0.4659,0.4659
MM,1,0.5841,0.5841,0.5841
HU,10,0.4807,0.4614,0.4937
LU,10,0.4781,0.4566,0.4911
UM,10,0.6083,0.593,0.6185
UU,55,0.6106,0.565,0.6296
Jc,Ad,91,,
HH,1,0.2916,0.2916,0.2916
LH,1,0.3032,0.3032,0.3032
LL,1,0.2955,0.2955,0.2955
HM,2,0.4406,0.4402,0.441
LM,2,0.437,0.4367,0.4373
MM,3,0.5538,0.5462,0.5668
HU,9,0.4496,0.4304,0.4626
LU,9,0.4495,0.4271,0.4633
UM,18,0.5715,0.5563,0.5824
UU,45,0.5738,0.529,0.5914
Tc,Ad,91,,
HH,1,0.2681,0.2681,0.2681
LH,1,0.2845,0.2845,0.2845
LL,1,0.2801,0.2801,0.2801
HM,3,0.4109,0.4102,0.4115
LM,3,0.4096,0.409,0.4102
MM,6,0.5215,0.5095,0.5322
HU,8,0.4204,0.4109,0.4291
LU,8,0.423,0.4133,0.4336
UM,24,0.5361,0.5271,0.5449
UU,36,0.5389,0.5113,0.5531
9c,Ad,91,,
HH,1,0.1841,0.1841,0.1841
LH,1,0.2035,0.2035,0.2035
LL,1,0.1962,0.1962,0.1962
HM,4,0.3404,0.3397,0.3411
LM,4,0.3382,0.3368,0.3395
MM,10,0.4613,0.4483,0.4696
HU,7,0.3332,0.3257,0.3395
LU,7,0.3344,0.3231,0.3421
UM,28,0.4611,0.4534,0.4678
UU,28,0.4479,0.4227,0.4596
8c,Ad,91,,
HH,1,0.1519,0.1519,0.1519
LH,1,0.1718,0.1718,0.1718
LL,1,0.1641,0.1641,0.1641
HM,5,0.3055,0.3049,0.3073
LM,5,0.3025,0.3003,0.3049
MM,15,0.4246,0.4109,0.4318
HU,6,0.2936,0.2889,0.297
LU,6,0.2938,0.285,0.2984
UM,30,0.4199,0.4142,0.4239
UU,21,0.4022,0.3815,0.4107
7c,Ad,91,,
HH,1,0.1079,0.1079,0.1079
LH,1,0.1295,0.1295,0.1295
LL,1,0.122,0.122,0.122
HM,6,0.2603,0.2549,0.2635
LM,6,0.2574,0.251,0.2617
MM,21,0.3798,0.356,0.3884
HU,5,0.2418,0.2409,0.2429
LU,5,0.2417,0.2385,0.2443
UM,30,0.3683,0.3624,0.3715
UU,15,0.3436,0.3309,0.3501
6c,Ad,91,,
HH,1,0.0501,0.0501,0.0501
LH,1,0.0759,0.0759,0.0759
LL,1,0.071,0.071,0.071
HM,7,0.2021,0.1943,0.208
LM,7,0.2016,0.1924,0.2089
MM,28,0.322,0.2933,0.3343
HU,4,0.1796,0.1791,0.1799
LU,4,0.1814,0.1796,0.1832
UM,28,0.3062,0.2984,0.3118
UU,10,0.2774,0.2662,0.2845
5c,Ad,91,,
HH,1,0.0484,0.0484,0.0484
LH,1,0.0815,0.0815,0.0815
LL,1,0.0875,0.0875,0.0875
HM,8,0.201,0.1907,0.21
LM,8,0.2086,0.2005,0.2189
MM,36,0.3219,0.2889,0.3381
HU,3,0.1614,0.1611,0.1616
LU,3,0.1703,0.1691,0.1715
UM,24,0.2891,0.2793,0.2976
UU,6,0.2441,0.2335,0.2542
4c,Ad,91,,
HH,1,-0.0086,-0.0086,-0.0086
LH,1,0.026,0.026,0.026
LL,1,0.033,0.033,0.033
HM,9,0.1451,0.1248,0.1597
LM,9,0.1533,0.129,0.1688
MM,45,0.2672,0.2149,0.288
HU,2,0.105,0.1049,0.1051
LU,2,0.1156,0.1149,0.1162
UM,18,0.2341,0.2172,0.2475
UU,3,0.1851,0.1783,0.1984
3c,Ad,91,,
HH,1,-0.0644,-0.0644,-0.0644
LH,1,-0.0258,-0.0258,-0.0258
LL,1,-0.0151,-0.0151,-0.0151
HM,10,0.092,0.0709,0.1064
LM,10,0.1034,0.0795,0.1189
MM,55,0.2168,0.1615,0.2419
HU,1,0.0507,0.0507,0.0507
LU,1,0.0652,0.0652,0.0652
UM,10,0.1825,0.165,0.1955
UU,1,0.1249,0.1249,0.1249
2c,Ad,91,,
HH,1,-0.1222,-0.1222,-0.1222
LH,1,-0.0796,-0.0796,-0.0796
LL,1,-0.065,-0.065,-0.065
HM,11,0.0369,0.015,0.0521
LM,11,0.0518,0.0282,0.0676
MM,66,0.1645,0.1067,0.1916
Qc,Kd,91,,
HH,1,0.1049,0.1049,0.1049
LH,1,0.1145,0.1145,0.1145
LL,1,0.1063,0.1063,0.1063
HO,1,0.3295,0.3295,0.3295
LO,1,0.3302,0.3302,0.3302
OO,1,0.512,0.512,0.512
HU,10,0.2998,0.2663,0.3157
LU,10,0.3,0.2658,0.3163
UO,10,0.4921,0.4657,0.5065
UU,55,0.4587,0.3859,0.4864
Jc,Kd,91,,
HH,1,0.0788,0.0788,0.0788
LH,1,0.092,0.092,0.092
LL,1,0.0855,0.0855,0.0855
HO,1,0.2958,0.2958,0.2958
LO,1,0.2987,0.2987,0.2987
OO,1,0.4713,0.4713,0.4713
HM,1,0.2445,0.2445,0.2445
LM,1,0.2439,0.2439,0.2439
MO,1,0.4375,0.4375,0.4375
MM,1,0.3687,0.3687,0.3687
HU,9,0.2678,0.233,0.282
LU,9,0.2694,0.2329,0.2849
UO,9,0.4528,0.4259,0.466
UM,9,0.4073,0.3816,0.4195
UU,45,0.4206,0.3466,0.4463
Tc,Kd,91,,
HH,1,0.0571,0.0571,0.0571
LH,1,0.0743,0.0743,0.0743
LL,1,0.0703,0.0703,0.0703
HO,1,0.2661,0.2661,0.2661
LO,1,0.2715,0.2715,0.2715
OO,1,0.4343,0.4343,0.4343
HM,2,0.2144,0.2144,0.2145
LM,2,0.2156,0.2149,0.2162
MO,2,0.4007,0.4005,0.4009
MM,3,0.3395,0.3308,0.356
HU,8,0.2403,0.2181,0.253
LU,8,0.2438,0.219,0.2585
UO,8,0.4174,0.3981,0.4286
UM,16,0.3714,0.3533,0.3835
UU,36,0.3871,0.3353,0.4062
9c,Kd,91,,
HH,1,-0.0235,-0.0235,-0.0235
LH,1,-0.0059,-0.0059,-0.0059
LL,1,-0.0151,-0.0151,-0.0151
HO,1,0.1975,0.1975,0.1975
LO,1,0.2011,0.2011,0.2011
OO,1,0.3741,0.3741,0.3741
HM,3,0.1431,0.143,0.1431
LM,3,0.1415,0.1399,0.143
MO,3,0.3358,0.3356,0.336
MM,6,0.2782,0.2663,0.2897
HU,7,0.1555,0.1482,0.1631
LU,7,0.1554,0.1442,0.1652
UO,7,0.3432,0.3366,0.3499
UM,21,0.2945,0.2862,0.3022
UU,28,0.297,0.2707,0.3121
8c,Kd,91,,
HH,1,-0.1072,-0.1072,-0.1072
LH,1,-0.086,-0.086,-0.086
LL,1,-0.0966,-0.0966,-0.0966
HO,1,0.1097,0.1097,0.1097
LO,1,0.1132,0.1132,0.1132
OO,1,0.2827,0.2827,0.2827
HM,4,0.0708,0.0703,0.0717
LM,4,0.069,0.0666,0.0715
MO,4,0.2589,0.2587,0.2591
MM,10,0.215,0.2034,0.2229
HU,6,0.0661,0.0573,0.0705
LU,6,0.066,0.0534,0.073
UO,6,0.2506,0.2429,0.2544
UM,24,0.2157,0.207,0.2202
UU,21,0.2019,0.1731,0.2132
7c,Kd,91,,
HH,1,-0.1385,-0.1385,-0.1385
LH,1,-0.116,-0.116,-0.116
LL,1,-0.1265,-0.1265,-0.1265
HO,1,0.0736,0.0736,0.0736
LO,1,0.077,0.077,0.077
OO,1,0.2427,0.2427,0.2427
HM,5,0.0369,0.0357,0.0392
LM,5,0.035,0.0323,0.0383
MO,5,0.22,0.2196,0.2204
MM,15,0.1782,0.1652,0.1848
HU,5,0.0266,0.0214,0.028
LU,5,0.0261,0.0175,0.0306
UO,5,0.2069,0.2031,0.208
UM,25,0.1733,0.1679,0.1754
UU,15,0.154,0.1328,0.1616
6c,Kd,91,,
HH,1,-0.1832,-0.1832,-0.1832
LH,1,-0.1582,-0.1582,-0.1582
LL,1,-0.1672,-0.1672,-0.1672
HO,1,0.0254,0.0254,0.0254
LO,1,0.0302,0.0302,0.0302
OO,1,0.1927,0.1927,0.1927
HM,6,-0.0079,-0.0145,-0.0036
LM,6,-0.0086,-0.0162,-0.0034
MO,6,0.1719,0.1667,0.1745
MM,21,0.1332,0.1089,0.1424
HU,4,-0.0267,-0.0269,-0.0265
LU,4,-0.0264,-0.0291,-0.0238
UO,4,0.1508,0.1507,0.1509
UM,24,0.1203,0.1147,0.123
UU,10,0.0927,0.0818,0.1001
5c,Kd,91,,
HH,1,-0.2344,-0.2344,-0.2344
LH,1,-0.2025,-0.2025,-0.2025
LL,1,-0.2053,-0.2053,-0.2053
HO,1,-0.0312,-0.0312,-0.0312
LO,1,-0.0236,-0.0236,-0.0236
OO,1,0.1313,0.1313,0.1313
HM,7,-0.0569,-0.0663,-0.0493
LM,7,-0.0522,-0.0626,-0.0439
MO,7,0.1168,0.1088,0.1227
MM,28,0.0861,0.056,0.0998
HU,3,-0.0828,-0.0831,-0.0826
LU,3,-0.0778,-0.0797,-0.0759
UO,3,0.0904,0.0902,0.0906
UM,21,0.0659,0.0577,0.0719
UU,6,0.0308,0.0211,0.0406
4c,Kd,91,,
HH,1,-0.2992,-0.2992,-0.2992
LH,1,-0.2661,-0.2661,-0.2661
LL,1,-0.2681,-0.2681,-0.2681
HO,1,-0.0947,-0.0947,-0.0947
LO,1,-0.0873,-0.0873,-0.0873
OO,1,0.0695,0.0695,0.0695
HM,8,-0.1184,-0.1299,-0.1068
LM,8,-0.1137,-0.1262,-0.1018
MO,8,0.0561,0.0462,0.066
MM,36,0.0275,-0.0071,0.0475
HU,2,-0.1461,-0.1463,-0.1459
LU,2,-0.1395,-0.1405,-0.1384
UO,2,0.0285,0.0284,0.0287
UM,16,0.0057,-0.0043,0.0156
UU,3,-0.0337,-0.0401,-0.0212
3c,Kd,91,,
HH,1,-0.3629,-0.3629,-0.3629
LH,1,-0.3254,-0.3254,-0.3254
LL,1,-0.3228,-0.3228,-0.3228
HO,1,-0.155,-0.155,-0.155
LO,1,-0.1442,-0.1442,-0.1442
OO,1,0.0127,0.0127,0.0127
HM,9,-0.177,-0.1903,-0.1608
LM,9,-0.1687,-0.1812,-0.1521
MO,9,0.0003,-0.0112,0.0148
MM,45,-0.0264,-0.0646,-0.0042
HU,1,-0.2068,-0.2068,-0.2068
LU,1,-0.1958,-0.1958,-0.1958
UO,1,-0.0292,-0.0292,-0.0292
UM,9,-0.0506,-0.0619,-0.0363
UU,1,-0.0982,-0.0982,-0.0982
2c,Kd,91,,
HH,1,-0.4277,-0.4277,-0.4277
LH,1,-0.3858,-0.3858,-0.3858
LL,1,-0.3785,-0.3785,-0.3785
HO,1,-0.2163,-0.2163,-0.2163
LO,1,-0.2022,-0.2022,-0.2022
OO,1,-0.0457,-0.0457,-0.0457
HM,10,-0.2372,-0.2519,-0.2236
LM,10,-0.225,-0.2373,-0.2109
MO,10,-0.0571,-0.0698,-0.0447
MM,55,-0.0823,-0.1235,-0.0584
Jc,Qd,91,,
HH,1,-0.0964,-0.0964,-0.0964
LH,1,-0.0871,-0.0871,-0.0871
LL,1,-0.0963,-0.0963,-0.0963
HO,2,0.1311,0.1226,0.1396
LO,2,0.1312,0.1227,0.1396
OO,3,0.3244,0.3049,0.335
HU,9,0.1083,0.0598,0.1284
LU,9,0.1078,0.0584,0.1283
UO,18,0.306,0.261,0.3306
UU,45,0.2769,0.1759,0.3123
Tc,Qd,91,,
HH,1,-0.119,-0.119,-0.119
LH,1,-0.1065,-0.1065,-0.1065
LL,1,-0.1142,-0.1142,-0.1142
HO,2,0.0992,0.0905,0.1079
LO,2,0.1009,0.0922,0.1095
OO,3,0.284,0.2641,0.295
HM,1,0.0405,0.0405,0.0405
LM,1,0.0391,0.0391,0.0391
MO,2,0.2402,0.2337,0.2467
MM,1,0.1603,0.1603,0.1603
HU,8,0.0809,0.0449,0.0976
LU,8,0.0814,0.0439,0.0997
UO,16,0.2687,0.2313,0.2912
UM,8,0.2155,0.1882,0.2305
UU,36,0.2437,0.1659,0.2698
9c,Qd,91,,
HH,1,-0.2009,-0.2009,-0.2009
LH,1,-0.1894,-0.1894,-0.1894
LL,1,-0.2038,-0.2038,-0.2038
HO,2,0.0271,0.0179,0.0364
LO,2,0.0262,0.017,0.0355
OO,3,0.2189,0.199,0.2323
HM,2,-0.0335,-0.034,-0.033
LM,2,-0.0386,-0.0399,-0.0373
MO,4,0.1716,0.1643,0.179
MM,3,0.1024,0.0929,0.1206
HU,7,-0.005,-0.0277,0.0047
LU,7,-0.0093,-0.0344,0.0026
UO,14,0.1917,0.1618,0.2086
UM,14,0.1362,0.1176,0.1456
UU,28,0.1527,0.0984,0.1718
8c,Qd,91,,
HH,1,-0.2811,-0.2811,-0.2811
LH,1,-0.2673,-0.2673,-0.2673
LL,1,-0.2844,-0.2844,-0.2844
HO,2,-0.0505,-0.0515,-0.0495
LO,2,-0.0521,-0.0524,-0.0519
OO,3,0.1436,0.1394,0.1505
HM,3,-0.1036,-0.1043,-0.1031
LM,3,-0.1101,-0.1121,-0.1079
MO,6,0.1007,0.0991,0.1023
MM,6,0.0423,0.0298,0.0547
HU,6,-0.0891,-0.0952,-0.0849
LU,6,-0.095,-0.1048,-0.0879
UO,12,0.1098,0.1036,0.1138
UM,18,0.06,0.0531,0.0647
UU,21,0.0637,0.0403,0.0741
7c,Qd,91,,
HH,1,-0.3713,-0.3713,-0.3713
LH,1,-0.353,-0.353,-0.353
LL,1,-0.37,-0.37,-0.37
HO,2,-0.1457,-0.1492,-0.1422
LO,2,-0.1462,-0.1491,-0.1433
OO,3,0.0435,0.0344,0.0505
HM,4,-0.1808,-0.182,-0.1794
LM,4,-0.1867,-0.1892,-0.1838
MO,8,0.0174,0.014,0.021
MM,10,-0.0243,-0.0365,-0.016
HU,5,-0.1867,-0.193,-0.1845
LU,5,-0.1916,-0.2012,-0.1862
UO,10,0.0074,-0.0012,0.0117
UM,20,-0.0255,-0.0315,-0.023
UU,15,-0.0416,-0.0642,-0.0333
6c,Qd,91,,
HH,1,-0.4011,-0.4011,-0.4011
LH,1,-0.3798,-0.3798,-0.3798
LL,1,-0.3946,-0.3946,-0.3946
HO,2,-0.1797,-0.1834,-0.1759
LO,2,-0.1778,-0.1813,-0.1743
OO,3,0.0055,-0.0041,0.0124
HM,5,-0.2117,-0.2135,-0.2089
LM,5,-0.2157,-0.2182,-0.2121
MO,10,-0.0185,-0.0222,-0.0145
MM,15,-0.0572,-0.0707,-0.0502
HU,4,-0.2257,-0.2266,-0.225
LU,4,-0.2294,-0.2326,-0.2262
UO,8,-0.0351,-0.0385,-0.0318
UM,20,-0.0659,-0.067,-0.0647
UU,10,-0.0896,-0.1008,-0.0819
5c,Qd,91,,
HH,1,-0.4502,-0.4502,-0.4502
LH,1,-0.4223,-0.4223,-0.4223
LL,1,-0.4317,-0.4317,-0.4317
HO,2,-0.2327,-0.2336,-0.2317
LO,2,-0.2264,-0.2268,-0.2261
OO,3,-0.0518,-0.0559,-0.045
HM,6,-0.2579,-0.2658,-0.2527
LM,6,-0.2569,-0.2646,-0.2516
MO,12,-0.0695,-0.0746,-0.0664
MM,21,-0.1013,-0.1263,-0.092
HU,3,-0.2812,-0.2817,-0.2807
LU,3,-0.2804,-0.2825,-0.2783
UO,6,-0.0935,-0.0945,-0.0925
UM,18,-0.1188,-0.1253,-0.1153
UU,6,-0.1519,-0.162,-0.142
4c,Qd,91,,
HH,1,-0.5181,-0.5181,-0.5181
LH,1,-0.4896,-0.4896,-0.4896
LL,1,-0.4997,-0.4997,-0.4997
HO,2,-0.2989,-0.2998,-0.2979
LO,2,-0.2929,-0.2933,-0.2925
OO,3,-0.1168,-0.1207,-0.11
HM,7,-0.3217,-0.333,-0.313
LM,7,-0.3214,-0.3321,-0.3129
MO,14,-0.1324,-0.1405,-0.1261
MM,28,-0.1617,-0.1936,-0.1472
HU,2,-0.3476,-0.3479,-0.3474
LU,2,-0.3457,-0.3468,-0.3447
UO,4,-0.1583,-0.1589,-0.1576
UM,14,-0.1815,-0.1907,-0.1746
UU,3,-0.2196,-0.2263,-0.207
3c,Qd,91,,
HH,1,-0.5847,-0.5847,-0.5847
LH,1,-0.5524,-0.5524,-0.5524
LL,1,-0.5587,-0.5587,-0.5587
HO,2,-0.3616,-0.3625,-0.3608
LO,2,-0.3528,-0.3532,-0.3524
OO,3,-0.1763,-0.1801,-0.1694
HM,8,-0.3827,-0.3966,-0.3695
LM,8,-0.3792,-0.3903,-0.3662
MO,16,-0.1903,-0.2007,-0.1795
MM,36,-0.2174,-0.2546,-0.1958
HU,1,-0.4113,-0.4113,-0.4113
LU,1,-0.4055,-0.4055,-0.4055
UO,2,-0.2185,-0.2189,-0.2182
UM,8,-0.24,-0.2511,-0.229
UU,1,-0.2872,-0.2872,-0.2872
2c,Qd,91,,
HH,1,-0.6525,-0.6525,-0.6525
LH,1,-0.616,-0.616,-0.616
LL,1,-0.6184,-0.6184,-0.6184
HO,2,-0.4254,-0.4263,-0.4246
LO,2,-0.4136,-0.4141,-0.4132
OO,3,-0.2368,-0.2406,-0.2299
HM,9,-0.4452,-0.4613,-0.4271
LM,9,-0.4383,-0.4511,-0.4198
MO,18,-0.2497,-0.2621,-0.234
MM,45,-0.2751,-0.3167,-0.2502
Tc,Jd,91,,
HH,1,-0.2803,-0.2803,-0.2803
LH,1,-0.2716,-0.2716,-0.2716
LL,1,-0.2819,-0.2819,-0.2819
HO,3,-0.0517,-0.0685,-0.0345
LO,3,-0.0524,-0.0692,-0.0351
OO,6,0.1455,0.1087,0.1683
HU,8,-0.0655,-0.114,-0.043
LU,8,-0.0669,-0.1161,-0.0434
UO,24,0.1316,0.0791,0.1667
UU,36,0.1105,0.008,0.1508
9c,Jd,91,,
HH,1,-0.369,-0.369,-0.369
LH,1,-0.3623,-0.3623,-0.3623
LL,1,-0.3803,-0.3803,-0.3803
HO,3,-0.1305,-0.1479,-0.1125
LO,3,-0.1348,-0.1521,-0.1168
OO,6,0.074,0.0378,0.1007
HM,1,-0.2009,-0.2009,-0.2009
LM,1,-0.2087,-0.2087,-0.2087
MO,3,0.0166,0.0049,0.0295
MM,1,-0.0735,-0.0735,-0.0735
HU,7,-0.1576,-0.1947,-0.1432
LU,7,-0.1649,-0.2035,-0.1488
UO,21,0.0488,0.0023,0.0776
UM,7,-0.0154,-0.0433,-0.0034
UU,28,0.0139,-0.0675,0.0409
8c,Jd,91,,
HH,1,-0.4492,-0.4492,-0.4492
LH,1,-0.4418,-0.4418,-0.4418
LL,1,-0.4639,-0.4639,-0.4639
HO,3,-0.2072,-0.2191,-0.2003
LO,3,-0.2134,-0.2248,-0.2076
OO,6,0.0004,-0.0252,0.0158
HM,2,-0.2723,-0.2729,-0.2717
LM,2,-0.2828,-0.2836,-0.2819
MO,6,-0.0551,-0.0633,-0.0501
MM,3,-0.1291,-0.1389,-0.11
HU,6,-0.2411,-0.2641,-0.234
LU,6,-0.2514,-0.2767,-0.2417
UO,18,-0.0316,-0.0639,-0.0201
UM,12,-0.0926,-0.1103,-0.0865
UU,21,-0.0739,-0.1286,-0.0586
7c,Jd,91,,
HH,1,-0.5364,-0.5364,-0.5364
LH,1,-0.5266,-0.5266,-0.5266
LL,1,-0.5508,-0.5508,-0.5508
HO,3,-0.295,-0.2983,-0.2915
LO,3,-0.3017,-0.3052,-0.2994
OO,6,-0.0873,-0.0997,-0.0786
HM,3,-0.3484,-0.3494,-0.3473
LM,3,-0.3599,-0.3616,-0.3581
MO,9,-0.1338,-0.1372,-0.1308
MM,6,-0.193,-0.2055,-0.1807
HU,5,-0.3336,-0.3365,-0.3316
LU,5,-0.345,-0.3511,-0.3396
UO,15,-0.1246,-0.1307,-0.1212
UM,15,-0.1754,-0.179,-0.1735
UU,15,-0.1721,-0.1883,-0.1648
6c,Jd,91,,
HH,1,-0.6321,-0.6321,-0.6321
LH,1,-0.6168,-0.6168,-0.6168
LL,1,-0.6395,-0.6395,-0.6395
HO,3,-0.3945,-0.3988,-0.3887
LO,3,-0.3986,-0.4019,-0.3935
OO,6,-0.1907,-0.2018,-0.1825
HM,4,-0.4286,-0.4302,-0.4265
LM,4,-0.438,-0.44,-0.4353
MO,12,-0.2191,-0.2223,-0.214
MM,10,-0.2602,-0.2724,-0.2523
HU,4,-0.4382,-0.4399,-0.4367
LU,4,-0.4473,-0.4513,-0.4434
UO,12,-0.2328,-0.2368,-0.2275
UM,16,-0.2651,-0.2665,-0.2637
UU,10,-0.2852,-0.2976,-0.2769
5c,Jd,91,,
HH,1,-0.6654,-0.6654,-0.6654
LH,1,-0.6433,-0.6433,-0.6433
LL,1,-0.6589,-0.6589,-0.6589
HO,3,-0.4326,-0.4348,-0.4307
LO,3,-0.4316,-0.4321,-0.4307
OO,6,-0.2336,-0.2406,-0.2274
HM,5,-0.4602,-0.4626,-0.4569
LM,5,-0.464,-0.4662,-0.4607
MO,15,-0.2565,-0.258,-0.2552
MM,15,-0.2914,-0.3047,-0.2844
HU,3,-0.4782,-0.4793,-0.4771
LU,3,-0.4822,-0.4849,-0.4795
UO,9,-0.2765,-0.2786,-0.2746
UM,15,-0.3041,-0.3056,-0.302
UU,6,-0.3322,-0.3429,-0.3224
4c,Jd,91,,
HH,1,-0.7353,-0.7353,-0.7353
LH,1,-0.7121,-0.7121,-0.7121
LL,1,-0.7281,-0.7281,-0.7281
HO,3,-0.5014,-0.5036,-0.4996
LO,3,-0.5004,-0.5009,-0.4994
OO,6,-0.302,-0.3089,-0.2958
HM,6,-0.5247,-0.5341,-0.5191
LM,6,-0.529,-0.5364,-0.5241
MO,18,-0.321,-0.3268,-0.3178
MM,21,-0.3515,-0.377,-0.3418
HU,2,-0.5462,-0.5467,-0.5457
LU,2,-0.549,-0.5503,-0.5477
UO,6,-0.3432,-0.3447,-0.342
UM,12,-0.3672,-0.3744,-0.3631
UU,3,-0.4011,-0.4077,-0.3893
3c,Jd,91,,
HH,1,-0.8037,-0.8037,-0.8037
LH,1,-0.7761,-0.7761,-0.7761
LL,1,-0.7882,-0.7882,-0.7882
HO,3,-0.566,-0.5682,-0.5644
LO,3,-0.5614,-0.5619,-0.5604
OO,6,-0.3634,-0.3703,-0.3573
HM,7,-0.5861,-0.5991,-0.577
LM,7,-0.5868,-0.5953,-0.5787
MO,21,-0.3795,-0.3884,-0.3728
MM,28,-0.4066,-0.4392,-0.3917
HU,1,-0.6109,-0.6109,-0.6109
LU,1,-0.6098,-0.6098,-0.6098
UO,3,-0.4046,-0.4056,-0.4039
UM,7,-0.426,-0.4361,-0.4187
UU,1,-0.469,-0.469,-0.469
2c,Jd,91,,
HH,1,-0.8734,-0.8734,-0.8734
LH,1,-0.8412,-0.8412,-0.8412
LL,1,-0.8491,-0.8491,-0.8491
HO,3,-0.6317,-0.6339,-0.63
LO,3,-0.6235,-0.624,-0.6225
OO,6,-0.426,-0.4328,-0.4199
HM,8,-0.6494,-0.6653,-0.636
LM,8,-0.6464,-0.6583,-0.6337
MO,24,-0.4399,-0.4511,-0.4288
MM,36,-0.4644,-0.5027,-0.4427
9c,Td,91,,
HH,1,-0.5199,-0.5199,-0.5199
LH,1,-0.5171,-0.5171,-0.5171
LL,1,-0.5379,-0.5379,-0.5379
HO,4,-0.2697,-0.2976,-0.2415
LO,4,-0.2767,-0.3047,-0.2484
OO,10,-0.0544,-0.112,-0.0115
HU,7,-0.2965,-0.3469,-0.2696
LU,7,-0.3058,-0.3569,-0.2793
UO,28,-0.0802,-0.1448,-0.0309
UU,28,-0.1147,-0.2236,-0.0681
8c,Td,91,,
HH,1,-0.6047,-0.6047,-0.6047
LH,1,-0.6034,-0.6034,-0.6034
LL,1,-0.6305,-0.6305,-0.6305
HO,4,-0.3518,-0.3754,-0.3369
LO,4,-0.3626,-0.3857,-0.3488
OO,10,-0.1339,-0.1827,-0.1057
HM,1,-0.4302,-0.4302,-0.4302
LM,1,-0.4447,-0.4447,-0.4447
MO,4,-0.2023,-0.218,-0.1922
MM,1,-0.3013,-0.3013,-0.3013
HU,6,-0.3843,-0.4222,-0.3676
LU,6,-0.3985,-0.4377,-0.3831
UO,24,-0.1656,-0.2174,-0.1383
UM,6,-0.2378,-0.2654,-0.2251
UU,21,-0.2063,-0.2899,-0.1759
7c,Td,91,,
HH,1,-0.6924,-0.6924,-0.6924
LH,1,-0.6909,-0.6909,-0.6909
LL,1,-0.7221,-0.7221,-0.7221
HO,4,-0.4389,-0.4539,-0.4302
LO,4,-0.4518,-0.4656,-0.4447
OO,10,-0.2188,-0.2507,-0.2027
HM,2,-0.508,-0.5087,-0.5074
LM,2,-0.5252,-0.526,-0.5245
MO,8,-0.2808,-0.2904,-0.2748
MM,3,-0.359,-0.3696,-0.3385
HU,5,-0.4761,-0.4968,-0.4685
LU,5,-0.4933,-0.516,-0.4848
UO,20,-0.2562,-0.2889,-0.243
UM,10,-0.3211,-0.337,-0.315
UU,15,-0.3026,-0.3526,-0.2859
6c,Td,91,,
HH,1,-0.7856,-0.7856,-0.7856
LH,1,-0.781,-0.781,-0.781
LL,1,-0.813,-0.813,-0.813
HO,4,-0.5327,-0.5368,-0.5263
LO,4,-0.5449,-0.5487,-0.5399
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HM,3,-0.5877,-0.5891,-0.5862
LM,3,-0.6048,-0.6061,-0.6034
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HU,4,-0.5753,-0.5773,-0.5728
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UM,12,-0.4082,-0.4102,-0.4056
UU,10,-0.4075,-0.4213,-0.3983
5c,Td,91,,
HH,1,-0.8906,-0.8906,-0.8906
LH,1,-0.8761,-0.8761,-0.8761
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LO,4,-0.6463,-0.6481,-0.6447
OO,10,-0.4232,-0.4334,-0.4168
HM,4,-0.6725,-0.6746,-0.6699
LM,4,-0.6833,-0.6849,-0.6812
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UM,12,-0.5006,-0.5025,-0.4985
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4c,Td,91,,
HH,1,-0.946,-0.946,-0.946
LH,1,-0.9303,-0.9303,-0.9303
LL,1,-0.9545,-0.9545,-0.9545
HO,4,-0.6951,-0.699,-0.6922
LO,4,-0.7007,-0.7025,-0.6991
OO,10,-0.4777,-0.488,-0.4714
HM,5,-0.7233,-0.7264,-0.7194
LM,5,-0.734,-0.7361,-0.7313
MO,20,-0.5036,-0.5056,-0.5012
MM,15,-0.5391,-0.5529,-0.5309
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It is so easy to find +EV games. Why beat your head against a game with a big house edge?
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Quote: aceside

That third column from the last is the average DeltaEV = EV_Check - EV_Raise. The last two are the minimum and maximum DeltaEV respectively in that category. The average DeltaEV is averaged over this range. This is my understanding of Mental.
link to original post

Quote: aceside

Hi, I’ve been thinking about how to calculate a collusion strategy for player’s suited hands. Firstly, suited hands are rarer and thus less important. Secondly, their EV gains from collusion may need us to consider card suit, instead of card rank. Luckly, there are only seven suited hands to consider, lying at the border line of the raise/check decision:

K2, K3, K4, Q6, Q7, J8, J9.

Is it possible you do a similar calculation as yours earlier on these seven hands? In this calculation, we fix the suit of both these two exposed cards to the same player’s suit. For example, when the player holds a suited hand Qc6c, there are C(11,2)=55 ways of choosing two exposed cards, because there are eleven cards of clubs in the remaining deck. So, you only need to do 55 calculations to estimate this part of EV gain.

In other words, the two exposed cards are always clubs and the player’s two cards are always clubs too. Is that possible you do such a calculation?

Thank you in advance!
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Quote: Mental

I think the best way to see the effect of exposed suited cards for collusion is to keep the ranks of the exposed cards the same and just vary the number that are suited. I am running calculations now that choose four ranks for the exposed cards that do not match the ranks of the hole cards. I do five calculations for each set of suited hole cards with 0 to 4 exposed cards matching that suit.

For example, for 8c,Qc hole cards:
3d,6h,Th,As
3c,6h,Th,As
3c,6c,Th,As
3c,6c,Tc,As
3c,6c,Tc,Ac
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