Late Position EV for Suited Broadways

Late Position EV for Suited Broadways (100K Trials)
Starting Hand	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	Reraise
AKs	2.84	3.59	3.21	4.95	2.89	3.66
AQs	2.12	2.85	2.35	4.03	1.16	1.46
AJs	1.59	2.31	1.68	3.26	0.30	0.13
A10s	1.38	1.96	1.54	2.50	-0.48	-0.32
KQs	1.08	1.54	1.32	2.30	-0.41	-0.73
KJs	1.09	1.27	1.18	1.93	-0.58	-0.98
K10s	0.83	0.89	0.88	1.19	-0.84	-1.64
QJs	0.88	1.11	1.04	1.47	-0.87	-1.22
Q10s	0.65	0.74	0.55	0.59	-1.03	-1.45
J10s	0.52	0.56	0.45	0.47	-1.25	-1.46

Middle Position EV for Suited Broadways

(This will be the last post today. Late Position and subsequent trials will post tomorrow.)

Middle Position EV for Suited Broadways (100K Trials)
Starting Hands	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	3bet Raiser
AKs	3.62	3.49	4.03	4.49	2.51	2.96
AQs	2.46	2.59	2.94	3.54	0.81	0.98
AJs	1.27	2.00	1.76	2.91	-0.39	-0.36
A10s	0.95	1.57	1.46	2.45	-0.86	-1.23
KQs	0.92	1.06	1.44	1.86	-0.90	-1.03
KJs	0.81	1.01	1.35	1.47	-0.96	-1.72
K10s	0.45	0.52	0.74	1.01	-1.41	-2.15
QJs	0.49	0.62	0.95	1.34	-1.31	-1.96
Q10s	0.19	0.24	0.59	0.53	-1.48	-2.42
J10s	0.13	0.08	0.37	0.44	-1.57	-2.26

Early Position EV for Suited Broadways

Early Position EV for Suited Broadways (100K Trials)
Starting Hands	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	3bet Raiser
AKs	3.53	3.34	3.64	4.29	1.87	2.17
AQs	1.72	2.37	2.26	3.28	-0.42	-0.85
AJs	1.44	1.88	1.58	2.54	-1.40	-2.55
A10s	1.08	1.38	1.27	2.06	-1.61	-3.07
KQs	0.68	0.91	1.02	1.37	-1.99	-3.54
KJs	0.84	0.74	0.85	0.92	-1.36	-2.78
K10s	0.22	0.24	0.25	0.29	-1.60	-2.89
QJs	0.39	0.36	0.38	0.66	-1.64	-2.89
Q10s	0.05	0.01	-0.05	-0.10	-1.69	-3.04
J10s	0.05	-0.09	-0.23	-0.52	-1.41	-2.77

UTG Expected Value for Suited Broadways

Let's move along to suited Broadways: any two unpaired cards ten or higher, such as Ace-King or Jack-Ten, with matching suits.

We will begin our trials from under the gun, and proceed around the horn.

Under the Gun EV for Suited Broadways (100K Trials)
Starting Hands	Open Call	Open Raise
AKs	3.41	3.18
AQs	1.49	2.11
AJs	1.15	1.36
A10s	0.81	1.05
KQs	0.43	0.73
KJs	0.49	0.53
K10s	0.16	0.03
QJs	0.32	0.33
Q10s	-0.78	-0.19
J10s	-0.37	-0.06

EV for Pocket Pairs from the Big Blind when raised

Since we can check and see the flop in unraised pots from the Big Blind, we will only concern ourselves with hands where we're raised. Here, as before with the small blind, our baseline is not 0.00, since we have $3 invested before the cards are dealt. Thus our baseline is -3.00. To play a hand, the EV must be greater than -3.00.

EV for Pocket Pairs Big Blind (100k Trials)
Starting Hand	Cold Call	3-Bet Raiser
AA	12.49	17.28
KK	6.73	9.36
QQ	3.45	5.49
JJ	1.06	2.18
TT	-0.42	0.24
99	-1.48	-1.04
88	-2.08	-1.74
77	-2.24	-2.45
66	-2.51	-2.75
55	-2.71	-3.01
44	-2.90	-3.25
33	-2.99	-3.72
22	-3.17	-3.92

EV for Pocket Pairs in the Small Blind

Keep in mind here that, unlike the other positions, where we can fold without investing a single dollar in the pot, we are forced to invest $1 in the small blind every time in a 3/6 game. Therefore we are looking not for hands whose EV is above 0.00, but whose EV is above -1.00, the net result when we fold our hand in the small blind.

EV for Pocket Pairs Small Blind (100k Trials)
Starting Hand	Limp Behind	Raise Limpers	Cold Call	3-Bet Raiser
AA	8.09	12.51	13.24	15.81
KK	5.07	8.70	7.44	9.00
QQ	3.67	6.69	4.43	5.62
JJ	2.23	4.87	2.18	2.63
TT	1.22	3.34	0.56	1.02
99	0.08	1.73	-0.55	-0.33
88	-0.59	0.78	-1.17	-1.04
77	-1.12	-0.02	-1.77	-1.68
66	-1.52	-1.04	-2.12	-1.98
55	-1.87	-1.43	-2.56	-2.40
44	-2.13	-1.93	-1.89	-2.49
33	-2.51	-2.22	-1.94	-2.67
22	-2.78	-2.58	-2.08	-3.00

EV for Pocket Pairs on the Button

EV for Pocket Pairs on the Button (100k Trials)
Starting Hand	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	3-Bet Raiser
AA	6.49	7.04	9.98	14.15	14.86	18.32
KK	4.30	4.95	6.86	10.47	8.10	9.80
QQ	3.37	3.91	5.12	7.91	4.52	5.24
JJ	2.65	3.10	3.52	6.06	2.08	2.23
TT	1.95	2.36	2.58	4.27	0.75	0.82
99	1.33	1.67	1.42	2.76	-0.10	-0.47
88	0.83	0.99	0.82	1.87	-0.63	-1.34
77	0.31	0.38	0.29	1.15	-0.98	-1.58
66	-0.09	0.08	0.03	0.66	-1.14	-1.93
55	-0.45	-0.30	-0.29	0.13	-1.44	-2.35
44	-0.78	-0.59	-0.54	-0.35	-1.67	-2.52
33	-1.04	-0.91	-0.68	-0.56	-1.80	-2.78
22	-1.27	-1.10	-0.82	-0.79	-1.96	-2.67

Late Position EV for Pocket Pairs

Again, cumulative notes to follow later.

Late Position EV for Pocket Pairs (100K Trials)
Starting Hand	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	3-Bet Raiser
AA	8.58	8.96	11.40	14.27	14.91	17.70
KK	5.57	6.27	7.30	10.52	8.45	10.04
QQ	4.23	4.84	5.31	8.03	4.94	5.87
JJ	3.03	3.50	3.53	5.76	2.81	3.11
TT	2.02	2.51	2.42	3.99	1.05	1.46
99	1.34	1.49	1.16	2.37	0.04	-0.02
88	0.52	0.76	0.57	1.37	-0.60	-0.66
77	-0.02	0.21	-0.01	0.69	-0.87	-1.33
66	-0.52	-0.13	-0.08	0.39	-1.07	-1.54
55	-0.89	-0.67	-0.32	-0.17	-1.39	-2.08
44	-1.22	-0.99	-0.69	-0.49	-1.64	-2.15
33	-1.35	-1.08	-0.85	-0.75	-1.85	-2.41
22	-1.66	-1.22	-1.02	-0.88	-1.88	-2.49

Middle position EV trials

Notes to follow at a later date.

Middle Position EV for Pocket Pairs (100K Trials)
Starting Hand	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	Reraise
AA	11.95	10.76	13.86	14.75	16.58	17.89
KK	6.82	7.37	8.33	10.41	9.10	10.21
QQ	4.73	5.13	6.16	7.76	4.88	5.71
JJ	3.34	3.55	3.86	5.25	2.19	2.72
TT	1.95	2.41	2.26	3.40	0.63	0.49
99	0.85	1.39	1.06	1.73	-0.29	-0.81
88	-0.09	0.55	0.38	0.88	-0.92	-1.32
77	-0.59	-0.46	-0.18	0.30	-1.43	-2.04
66	-1.07	-0.73	-0.30	-0.07	-1.56	-2.27
55	-1.48	-1.19	-0.74	-0.42	-1.86	-2.58
44	-1.74	-1.49	-0.85	-0.92	-2.01	-2.74
33	-2.05	-1.70	-1.11	-1.07	-2.09	-2.88
22	-2.22	-2.04	-1.43	-1.54	-2.17	-2.97

Early position trial results with notes

Early Position Expected Values for Pocket Pairs (100,000 trials)
Starting Hand	Open Call	Open Raise	Limp Behind	Raise Limpers	Cold Call	3-bet vs Raise
AA	12.88	11.29	14.57	14.89	17.97	18.91
KK	7.07	7.56	8.56	10.56	9.72	10.36
QQ	4.88	5.16	5.93	7.26	4.89	4.83
JJ	3.08	3.37	3.40	4.86	1.92	1.83
TT	1.67	2.08	1.74	2.69	0.00	-0.39
99	0.26	0.86	0.98	1.49	-1.08	-2.07
88	-0.16	0.12	0.36	0.83	-1.79	-3.14
77	-0.77	-0.33	-0.11	0.07	-2.19	-3.64
66	-1.14	-0.75	-0.40	-0.41	-2.15	-3.82
55	-1.43	-1.25	-0.74	-0.48	-2.37	-3.98
44	-1.66	-1.33	-0.81	-0.89	-2.60	-4.32
33	-1.86	-1.57	-1.03	-1.08	-2.54	-4.34
22	-2.03	-2.02	-1.18	-1.20	-2.70	-4.37

- Once again, open limping AA carries greater value than open raising, but once we see action in front, AA once again profits most from raising. It appears the key to AA's success is to ensure you have some action, which open limping does.

- Open raising remains the supreme play for all other playable pairs when facing one bet.

- Open raising is profitable until we get down to the marginally profitable 88. The smaller the profit, the smaller the margin for error. Playing 88 the slightest bit unprofitably can turn it into an unprofitable hand.

- Likewise, against limpers, pairs are profitable when raised down to 77, where the profit margin is even smaller. A novice could do well to not play these hands at all. We can explore the best way to play hands like these once we explore postflop strategies.

- Acting when facing a raise presents some interesting data. AA and KK, of course, are most profitable when reraised (3-betting). But by a thin margin, QQ and JJ are most profitable when calling two bets. The margin is so thin, however, that you could just reraise those hands as well and not see a significant dropoff. In fact, since win percentages increase when you raise over calling and reduce your variance with these hands as a result, it can be argued that reraising may be worth the slight loss of profitability since you'll win more often by pricing out weaker hands that may outdraw you.

Early position pocket pair recommendations:

When 1st to act, limp in with AA and raise with KK-88.

Against limpers, raise with AA-77.

Against a raise, reraise with AA and KK, but call with QQ and JJ.

Raw UTG preflop data with notes

UTG Expected Values: Pocket Pairs Over 100,000 trials
Starting Hand	Open Call	Open Raise
AA	13.48	11.87
KK	7.60	7.70
QQ	4.93	5.14
JJ	2.96	3.07
TT	1.41	1.88
99	0.47	0.51
88	-0.19	0.00
77	-0.80	-0.65
66	-1.21	-1.04
55	-1.40	-1.46
44	-1.62	-1.66
33	-1.75	-1.86
22	-2.00	-2.06

- Each hand was tested 100,000 times at a 9 handed 3/6 table with a general profile, while open calling one small bet and 100,000 times while open raising to 2 bets. Obviously, being 1st to act, there's no need to test hands against limpers or raisers. The general table profile has no more than two aggressive players, no more than 2-3 loose players and generally sees 3.6 players per flop. This is not a juicy game, but it's consistent with the play in most cardrooms, in line with the research goal to determine the optimal strategy for such a game.

- If raised by someone acting after us, the simulator automatically calls for one extra bet, but only calls for 2 or more extra bets with tens or better, reraising with Queens or better if possible. Tens or better are very strong hands that consistently profit even against top hands. Any pocket pair weaker than this is thrown away when facing two more bets (and previous research confirmed that nines and below are big losers when played for three or more bets preflop. I can simulate and outline this scenario at a later date to illustrate this).

- Most pocket pairs need to flop a set or a disguised open ended straight draw to continue playing on the flop... or they must be folded. Stronger pairs can often win on their own provided that pair's rank is either higher than the high card on the flop, or sometimes when right below it. Middle and low pairs almost always need to flop a set to have value, and even then they run the costly danger of another player flopping a better set or making a flush or straight to beat them.

- The surprising stat from above: pocket aces turns a bigger profit when open limped than when open raised, over $1.50 more per hand. All other hands turned a better profit when raised. AA's result was such a baffling outlier that I ran 100,000 more trials for each scenario with AA to make sure, but the 2nd results were no greater than 4 to 5 cents off the original results.

The most likely cause is that when someone raises after we limp, which happens rather often, AA reraises or calls a full number of bets, and is so far ahead of all other hands that it wins most of the resulting larger pots. All other pairs run the risk of being dominated or outdrawn in such a scenario.

- Due to the parameters of Wilson Software's Turbo Texas Hold'Em simulator, I cannot set the simulator to raise nines through Kings, then limp Aces, if I were to test an overall strategy. The simulator only allows you to set a minimum bar for raising pocket pairs, with no more specific definition.

However, since you only see pocket Aces roughly once every 221 hands, the difference in profitability overall would add up to about a penny per hand over a lengthy simulation. And of course, AA is still immensely profitable when raised instead. We'll note reality, but in subsequent simulations we'll let it slide and raise Aces preflop UTG.

- Clearly, as noted a while back, the dividing line for profitable pocket pairs under the gun is nines or better. Eights break even when raised, though the actual return was a very small profit, so small that the average profit per hand was 0.00. Not playing them under the gun at all may be the better play in order to reduce variance.

- Conclusion: Under the gun before the flop, raise with pocket pairs nine or better, except for Aces, which can be limped.

Re-evaluating the testing process

Pardon the absence during a busy time for me. Normal posting will resume shortly.

I have run multiple test trials over the last few weeks, lately gravitating to more specific scenarios in an attempt to more accurately gauge where hands make their money using a conventionally solid postflop strategy. However, I'm running into a problem: I've so expanded the scope of this project as a result that to ultimately complete the task strikes me as overwhelming. Each hand in each situation requires a minimum of 50,000 trials to minimize the variance so we get a useful net profit number.

If I run too few trials, the variance ends up so high that the data is useless. The variance over 25,000 trials can get into the $0.30 range, rendering most trials useless. You need at least 50,000 to get variance within $0.15 and you would prefer 100,000. If I don't run trials by position, an overall number for all positions of a given hand doesn't tell me very much about the hand's profitability (is it profitable when open raised but not when calling other players? Does it profit against multiple players? Is it profitable in late position?). A single number over a large span of cumulative trials won't educate me about how or where a hand is best played before the flop.

Also, we eventually need to transition to evaluating postflop play to see if we can improve on conventionally solid strategy, and there are so many variables there (Yes, I do have a framework from which I will run tests). But one step at a time: if we can simply get a reliable barometer on profitable preflop hands, that is a big step forward.

To make the task less overwhelming, I now plan to do these tests by the sets listed. I will start with pocket pairs, then the suited broadways, the offsuit broadways, the suited Aces, Kings, Queens, then Jacks, then the suited connectors, and then I shall add other hands as low-end hands show the potential for profitable play. I previously hadn't tested the top hands like AA and AK, but I will now include them to provide general barometers of how hands improve in profitability with position.

I will test all groups in the following circumstances:

Open (1st in) call, Open (1st in) raise, limp behind, raise limpers, call raise, reraise

I will test all the above circumstances for the following positions:

UTG, EP, MP, LP, Button, SB, BB

Already, we face a daunting task. Now, the groups:

(AA, KK, QQ) JJ, TT, 99, 88, 77, 66, 55, 44, 33, 22

AKs, AQs, AJs, A10s, KQs, KJs, K10s, QJs, Q10s, J10s

AKo, AQo, AJo, A10o, KQo, KJo, K10o, QJo, Q10o, J10o

A9s, A8s, A7s, A6s, A5s, A4s, A3s, A2s

K9s, K8s, K7s, K6s, K5s, K4s, K3s, K2s

Q9s, Q8s, Q7s, Q6s, Q5s, Q4s, Q3s, Q2s

J9s, J8s, J7s, J6s, J5s, J4s, J3s, J2s

T9s, 98s, 87s, 76s, 65s, 54s

For each group I will do the following:

1. Program the simulator to run trials for each respective hand,
setting button positions for each hand.

2. Run 1st in trials for calls and raises (26 50K trials for each position)
- Button
- LP
- MP
- EP
- UTG

3. Run limper trials for calls and raises (26 50K trials for each position)
- SB
- Button
- LP
- MP
- EP

4. Run raiser trials for calls and raises (26 50K trials for each position)
- BB
- SB
- Button
- LP
- MP
- EP

5. Rip my hair out.

Now, to wit: the simulator can run about 3.3 million hands an hour, give or take depending on what scenario you research. Running 5 million trials takes about an hour and a half. 13 separate 100K trials adds up to 1.3 million trials. Run any of the above listed scenarios for every position and you get either 6.5 million trials or 7.8 million trials. That's about 2 hours for each. Not including time to manually notate the results, that's 6 hours to run trials on one set of hands. And that assumes we can set the simulator up to quickly run them all at once, without any manual manipulation inbetween.