Applying Elindaur's Concept

Ok, so let's assume that you raise 100% of the time from the hijack (hijack means 2 off the button?) with

66+
Axs
A9o+
K9s+
KTo+
Q9s+
QJo+
JTs+

and you raise 0% of the time from the hijack with all other hands.

We need to find a mixed strategy to counter the donkbet.

The above range of hands comprises:

PAIRS + SUITED ACES + OFFSUIT ACES +
(6*7 + 3*2) + (4*9 + 3*3) + (9 + 12*4) +

SUITED KINGS + OFFSUIT KINGS + SUITED QUEENS
(3 + 4*3) + 36 + (3 + 4*2) +

12 + 4 =

42 + 6 + 36 + 9 + 9 + 48 + 3 + 12 + 36 + 3 + 8 + 12 + 4 =

228 Total hands.

We'll have top pair or better with 77, 99, TT, JJ, QQ, KK, AA, A9s, A9o, K9s, Q9s, which is 3+3+6+6+6+6+6+3+9+3+3 = 108 hands.

We'll be have two overcards and a flush draw with ATc, AJc, AQc, AKc, KTc, KJc, KQc, QTc, QJc, JTc, which is 10 hands.

We'll have a pair, an overcard, and a flush draw with A4c.

We have a flush draw and one overcard wtih A2c, A3c, A5c, A6c, and A8c, which is 5 hands.
That puts our premium holdings on the flop at 108 + 10 + 1 + 5 = 124 hands.

If we folded everything else, our opponent would win 4.5 SB 109/228 = 45.6% of the time.

The other 54.4% of the time, we will sometimes win and sometimes lose.

So clearly we need to be continuing with a lot more hands than those above.

I haven't gotten to the construction of the mixed strategy that maximizes our EV. But I'm going to stop here because this post is taking too long.

I hope I was barking up the right tree.

[elindauer]

Outstanding start vkh. edit: except for getting all the math wrong. doh.

I'll assume your numbers are correct and try to push this forward a little more. edit: your numbers are off. doh. I've gone back and reworked the math below...


Total hands:
pairs: AA, KK, QQ, JJ, TT, 88, 66 = 42
sets: 99, 77 = 6
Axs: 12*4 - 3 = 45 (48 preflop, but 3 are not possible given the flop)
AK-AT 64
A9 12
KQ-KT 48
K9s 3
QJ 16
JTs 4

That's 198 possiblities. Someone correct me if I made a mistake.


Of those, the hands that are strong on the flop are:

pairs: AA, KK, QQ, JJ, TT = 30
sets: 99, 77 = 6
Axc: 10 (A9c and A7c are impossible)
AKc - ATc = 4
A9 12
KQc-KTc 3
K9s 3
QJc 1
JTs 4


That's a total of 73 hands. Someone again correct me if I've missed anything.

73/198 = 36% of our hand range really looks good on this flop.

We're going to need to estimate our equity in the pot on the flop with those premium hands, the ones we hold 54% of the time. Someone handy with pokerstove could plugin in our range {77, 99-AA, A9, K9s, Q9s} vs, say, a random hand to give us a start. Also interesting for this analysis would be our range vs the mindless' suggested BB range of AA, KK, 22-99, A8-AT, Axs, KT-KJ, Q9-QJ and suited connectors down to 45s.


In the meantime, I'll stab at this prematurely. The big blind is getting 4:1 to bluff, so we're going to need to continue quite often to stop him from making more money than he deserves from a bet-every-time strategy. Way more than the 36% a weak-tight would play.

If the BB didn't have any equity in the pot, we'd need to continue at least 80% of the time to stop him from bluffing. Since he's likely to have some kind of draw as well, we'll have to play even more, and probably do a lot of raising.

No surprise here that many players default strategy against this bet is to raise every time. This isn't a bad strategy at all, as the above analysis shows. We can probably do a little better though.

Let's say we continue with 90% of our hand range, and we raise 70%. That's going to be hard for him to handle I think. Of these times, we're frequently going to have a hand that is quite strong on this board. He cannot just keep hammering us because although we will keep widdling into our weakest hands with folds every time he raises, he is also dumping us chips all those times we actually have a hand (or make one).


Let's check out the range:

73 hands are very strong on this board. We want to raise 70% of hands though, so let's start by saying we raise all of these.

90% of all hands we could hold is 178 holdings.
70% of all hands we could hold is 138 holdings.

We need to find 65 (!) more combinations to raise, and then another 40 combinations to call.


ok, here goes.

Raise (65 more needed):

88, 66 12
A7s (middle pair) 3
A4s (bottom pair) 3 (A4c counted already)
AK 15 (AKc counted already)
AQ not suited in clubs 15
AJ not suited in clubs 15

that's about enough (technically, still 2 short)

Call with (40 combinations needed):

AT 15
Axs not in clubs and x below T and no pair 15
KQ not suited in clubs 2/3 of the time 10 (choose the ones with 1 club)


Now, you might object that we haven't talked about the turn yet. Well, we have narrowed our hand range and can do a similar analysis for handling the turn should he decide to bet again, and for handling what to do with the 3-bets and... it's pretty complicated, but then, we're just diving into this for the first time. So far, I'm encouraged.

Overall, this gives us a strategy of {10, 20, 70}, which intuitively seems reasonable to me given the way the big blind should run us over if we play much tighter than this. Any tighter strategy is defeated by a big blind who constantly bet/folds every time (unless of course he actually has a hand).

ok, some other quick thoughts. Our calling range is completely weak hands. In order to better deceive our opponents, it might be better to move some of the strongest hands from the raising to the calling, and move some of the strongest weaker hands into the raising category. Let's move the strongest 15% of hands into the call category, and a similar number of weak hands into the raise category, ie

raise {AA-TT, 88, 66, 99, 77, AK-AJ, any 2 clubs, JT}
call {AT, Axs not clubs, KQ with 1 club}

will become:

raise {KK-TT, 88, 66, AK-AJ, any 2 clubs, JT, KQ 2/3 of the time}
call {99, 77, AA, AT, Axs not club}


Now you might notice that we always slowplay sets. Maybe you're comfortable with this distribution. If not, you may want to mix it up a little more. Let's handle the sets by raising half the time and calling half the time. We'll have to move some KQ hands back into the call category to keep our overall strategy balanced, ie:

raise {99 50%, 77 50%, KK-TT, 88, 66, AK-AJ, any 2 clubs, JT, KQ with 1 club (6 combos)}
call {99 50%, 77 50%, AT, Axs not club, KQ, no club, with Kd (3 combos)}

Now we're really getting at a complete answer. Of course, we could do more math to try to figure the ideal distribution of slowplays for those super-strong hands. We might also want to move some of the pure draws into the call category so we don't always raise flush draws. I think I've shown the mechanics of this, maybe someone else can try to continue this a little more?


The poster wanted to know what to do with KQ. I've tried to show how we can use game theory to get at the answer, a very complete, basic-strategy-like answer which would be great against unknowns and against good players who can adjust to our play. I suggest:

FCR {38, 19, 43}

specifically,

raise: KcQc, KcQh, Kc Qd, Kc Ks, Kh Qc, Kd Qc, ks Qc
call: Kd Qh, Kd Qs, Kd Qd
fold: Kh Qd, Kh Qh, Kh Qs, Ks Qd, Ks Qh, Ks Qs

Which one did he have again? Let me check...

Kh Qd. Fold. Now let me ask you, if I just said "fold", how much would you have learned?


good luck.
eric

[flawless_victory]

[ QUOTE ]
I think you should do 33, 33, 33 for every decision ever from now on.

[/ QUOTE ]
HAAhaha.
i think im finally beginning to grasp game theory!

Hi Eric,

[ QUOTE ]
Axs: 12*4 - 3 = 45 (48 preflop, but 3 are not possible given the flop)
AK-AT 64
A9 12


[/ QUOTE ]

You double count A9s, ATs, AJs, AQs, and AKs.
You also forget to include Q9s, and QTs.
Then, you add your numbers wrong. YOUR total should be 240, not 198. But the correct total is what I had: 228.

In the next section, you double count AKc-ATc and you forget to add Q9s and QTc. Fortunately, the first 4 cancel out the second 4 and your 73 is correct. (I did mess this part up in my math, for some reason I put 108 instead of 54 for one sum. I also forgot that the 3 nonclub JTs give you a good hand on the flop.)

-v

[Pog0]

I like the way you're applying this. Rather than giving our specific hand a range of options, we give our range of hands an option and then see where our specific hand falls. This approach probably makes progressively less sense as we approach the river, but since both players know very little about each others' hands, it's nice to think about this taking villian's limited knowledge into consideration. Once the range is complete, we have to make sure that villian cannot exploit our new strategy if he knows it.

Poker Stove with our range vs a random hand:

Board: 4d 7c 9c
Dead:

equity (%) win (%) tie (%)
Hand 1: 57.5042 % 57.00% 00.50% { 66+, A2s+, K9s+, Q9s+, JTs, A9o+, KTo+, QJo }
Hand 2: 42.4958 % 42.00% 00.50% { random }

Poker stove with our range vs opponent's estimated range:

Board: 4d 7c 9c
Dead:

equity (%) win (%) tie (%)
Hand 1: 50.9383 % 49.86% 01.08% { 66+, A2s+, K9s+, Q9s+, JTs, A9o+, KTo+, QJo }
Hand 2: 49.0617 % 47.99% 01.08% { KK+, 99-22, A2s+, KJs-K8s, Q9s+, J9s+, T8s+, 97s+, 87s, 76s, 65s, 54s, ATo-A8o, KJo-KTo, Q9o+, JTo, T9o }


The above was for our original range. Below is for our premium hand range:

vs random:

Board: 4d 7c 9c
Dead:

equity (%) win (%) tie (%)
Hand 1: 70.5569 % 70.26% 00.30% { 99+, 77, AcKc, AcQc, AcJc, AcTc, Ac9c, Ac8c, Ac7c, Ac6c, Ac5c, Ac4c, Ac3c, Ac2c, KcQc, KcJc, KcTc, K9s, QcJc, QcTc, Q9s, JTs, A9o, QJo }
Hand 2: 29.4431 % 29.15% 00.30% { random }


vs range:

Board: 4d 7c 9c
Dead:

equity (%) win (%) tie (%)
Hand 1: 64.2960 % 63.47% 00.82% { 99+, 77, AcKc, AcQc, AcJc, AcTc, Ac9c, Ac8c, Ac7c, Ac6c, Ac5c, Ac4c, Ac3c, Ac2c, KcQc, KcJc, KcTc, K9s, QcJc, QcTc, Q9s, JTs, A9o, QJo }
Hand 2: 35.7040 % 34.88% 00.82% { KK+, 99-22, A2s+, KJs-K8s, Q9s+, J9s+, T8s+, 97s+, 87s, 76s, 65s, 54s, ATo-A8o, KJo-KTo, Q9o+, JTo, T9o }

[HiatusOver]

This is turning into a great thread, I am going to sleep right now but am real interested to read through all this stuff tomorrow and offer my 2 cents. Good job so far guys

[Subfallen]

[ QUOTE ]
[ QUOTE ]
I think you should do 33, 33, 33 for every decision ever from now on.

[/ QUOTE ]
HAAhaha.
i think im finally beginning to grasp game theory!

[/ QUOTE ]

[img]/images/graemlins/heart.gif[/img]

Nice analysis Eric.

A couple of comments...

An application of game theory is only optimal if our opponent is playing optimally. We can do better than this by exploiting tendencies that our opponent may have. In the heat of battle, these "reads" will prove more valuable than playing in a purely game theoretic fashion. Having said that however, this game theoretic strategy should be the starting, or bias point from which we deviate, for all but the worst players. That is, we can adjust marginally from the game theoretical solution by taking our opponents tendenceies into consideration. So, these analyses prove very useful to make sure that our "starting point" is correct in various situations.

One other point. It turns out that the probability triple for the specific KQo (no clubs) that we held was {100,0,0}. So it is a pure strategy. However, for KQo in general, we adopted a mixed strategy. The way you structured your solution, when you look at each type of hand individually, there are NO mixed strategies; hands are partitioned into specific decision classes. Another way to have structured it would have been to assign probability triples to actual hand classes. So for instance, instead of calling 100% of the time with AX no clubs, we could call 75% of the time with that and make up for it by, say, calling with KQ off no clubs 75% of the time. Just another way to structure the solution, which would have you doing 2 different things with the same exact hand.

-v

[elindauer]

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