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Game theory: finding worst hands we should defend pre / post

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Game theory: finding worst hands we should defend pre / post

Hi, guys!

I'm trying to study game theory atm and am confused in few spots. I know these spots are quite abstract and it's impossible to get an exact answer, but I hope it's possible to find some pretty good approximations.

1. How wide can we defend preflop lets say in BB vs openraise from BTN/CO/MP/UTG, assuming we're offered X:1 (lets say 2<=X<=3), we will be playing HU 100BB Deep and we know exact villain's raising range? I've heard about the R concept, even watched Sean's videos about it, but still struggling with estimating it.

2. Assume we defended a certain range pre. Now we're on the flop and villain bets Z% of the pot. Previously, I thought we had to defend 1/(1+Z) of our range, but I'm not sure about this now, especially when we had very good odds pre: our preflop calling range is a lot weaker than villain's raising range.

 

My guess is that there could be 3 possible scenarios:

a. The most common one: raiser's range is more polarized and has more equity.

b. Neither range has equity advantage, but raiser's range is more polarized.

c. Caller's range is more polarized and has more equity (probably very rare).

 

So the question is how to find the worst hand we should defend on the flop in each of these spots.

 
Any input is appreciated!

-- Sergey


17 Comments

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BigFiszh 11 years, 3 months ago

No solution, but some thoughts:

1) Preflop:

Not really sure what to add, once you saw Sean Lefort´s video, I think he explained everything in the most possible details (http://www.runitonce.com/pro-training/videos/lefort8/). Summarized you can defend any hand that has enough equity with the odds you´re given. The "R" factor is used to consider playability - you´re oop, you have no initiative, so it´s unlikely that you realize your entire equity, so R is the measure to take that into account. Like if you take an R of 65% you can defend any hand against a 3x raise that has at least 47% equity against the range:

Odds: 4.5 : 2 => equity needed = 2 / (2 + 4.5) = 31%

Equity needed after taking R into account: 31% / 60% = 47%

Against a linear 25% range the weakest hands would be: A8o, KJo, A5s, KTs and that pretty much is it. If we wanted to make it an exact science, we had to take different R´s for different hands into account, i.e. should QJs (which has too little equity given the numbers hence is not defendable) probably have a higher R than A8o. QJs has 45% equity against the given 25%-range, so the R we needed would be 31% / 45% = 69% which is still in the realistic area. So, if you wanted to guesstimate a total defense range you had to take any single hand, assign every single hand a realistic R-approximation and then construct your range, like in this example excel sheet:

2) Postflop:

Postflop is difficult. As preflop there are two different points to take into account:

a) odds we´re getting - how much can we defend?

b) odds Villain is offering himself - how much do we have to defend?

The first question is "easier" to answer and follows the same logic as described pre (equity against the betting-range, playability on later streets etc.).

The second question is more difficult, as our defense-frequency has to be calculated over ALL possible boards, it makes no sense to say "I have to defend 67% of my range against the half-pot-bet of Villain", if the board smashes his range and completely missed our range. So it´s almost impossible to calculate proper defense ranges on certain boards.


eq.fest 11 years, 1 month ago

I feel like there is something very important that you have overlooked it, and that is how well all of the hands in our range cover various board textures. I feel R is largely connected with this, but how exactly I'm not sure. Say we only call 65ss out of the BB vs the BTTN open and folded everything else. 65ss would become a losing call despite getting the odds preflop. If our opponent knew our strategy of only calling 65ss and folding everything else he would play so aggressively with his entire range on so many boards that didn't connect with our hand that eventually 65ss would become a losing call pre flop as the amount of equity we realise post flop would decrease.

BigFiszh 11 years, 1 month ago
Your point is not valid, imho. I know what you mean, but this is included in playability. Everytime your opponent knows exactly what you have, you´re always in danger, so there´s no real point to be made.

The playability takes board coverage into account ... that´s why we change the underlying equity. That´s why 87s plays (!) better than A7o, it doesn´t mean that 87s has the better equity, but it plays better because we can continue on more boards and take down the pot more often against the perceived range - and pay him off less with 2nd best hands. All that goes into "R". Obviously that lives from our opponent not knowing our exact holding. Agree?
eq.fest 11 years, 1 month ago

"Everytime your opponent knows exactly what you have, you´re always in danger, so there´s no real point to be made." When we are building optimal strategies post flop, against theoretically optimal opponents, do we not build strategies in such a way that even if they knew our strategy in a given spot they couldn't unilaterally change their strategy to exploit us? I'm not sure why this should be different pre flop.

eq.fest 11 years, 1 month ago

Maybe I'm not doing a good job of getting my point across. I agree that if our optimal opponent could could see our exact hand he could play better than if he couldn't. Perhaps when I said "and that is how well all of the hands in our range cover various board textures", I should have said " and that is how the presence of some hands in our range can effect the ev of a call with other hands in that range". I think that some hands do better when calling preflop because of the presence of other hands in that range, and we should try and account for that. It might be that this is accounted for if we rather estimated an R for each particular holding, instead of trying to apply an average R to all hands which is what I have done in the past, and I think lefort did in his videos.

Daz 11 years, 3 months ago

You could begin with an R of X% then adjust it as you see your results Defending BB, ive used 60%. I've created a table of defending ranges given villain's opening size using PPT Odds Oracle. 

im also interested in learning how to calculate 2)

Sergey Nikiforov 11 years, 3 months ago

Hi Daz,

why do you think we should start with 60%?

I'm not sure the suggested method is good since it requires a significant sample size to make any reasonable conclusions.

Daz 11 years, 3 months ago

Its Sean LeFort's baseline R. 

It makes for some easy in-session range calculations eg facing a minraise in the BB you defend twice as wide as the opening range with a upper bound of top 80% ie you can still fold the bottom 20% of hands (17% to be exact for a 70% open). so if CO minraise opens 20% i defend 40%, CO minraise open 25% i defend 50% etc 

Defending versus 3x you can actually fold plenty of hands versus a tight open. 

Well i guess that depends on your volume? surely as a pro, 5k hands per day for 20 days, means you can get a 300k+ sample in 3 months. 

Tiramisu914 11 years, 3 months ago

So it´s almost impossible to calculate proper defense ranges on certain boards.

I am just thinking out loud and I am quite possibly totally wrong but wouldn't this be the % of bluff combos in his range? So lets say the villain had 46 value combos and 20 bluff combos so would the defending frequency be 30%? This number could vary widely obviously. On a dry board this could be way above 50% so your almost forced to call with a huge part of your range regardless of equity I would think.

BigFiszh 11 years, 3 months ago
This could only be true if Villain was bluffing 100%, which is not true - and no good base for optimal defense strategies.


Adreno 11 years, 3 months ago

Typicly the PF raiser hits the high and low cards better while the PF caller hits the medium cards (6-T) better. Same applies for both SRP and 3bet pots. I ran some numbers with Pokerstove back when it used to work and in SRP pots the raiser was typicly 52%-55% ahead the caller, but could be as low as 48% on some flops. In 3bet pots the numbers vary much more if 3bettors range is polarized and doesn't have good board coverage.

HOTSANDWICH 11 years, 2 months ago

You know what man, once again, I have not read the math geeks posts.... Lets sum it up.  THIS IS THE POINT OF POKER.  TO GIVE YOUR OPPONENT A PRICE ON HIS RANGE TO MAKE A MISTAKE AGAINST YOURS.   Ive see alot of videos on here where guys are making very expliotable defends.  Looser than their cutoff range!!!  This is wrong guys.   Your defending range should be at a gto range that can defend all sorts of run outs.  Most of you and asssuming alot since overbetting is my favorite, just doesnt cover it.  My suggestion, take a few days off number one, decide to yourself, And get back at with something that makes sense and for the love of god if you find yourself against me just default fold.  I doubt you are going to defend AA and AK so you are exploitable.  (Unlike KGB)

ImMIke 11 years, 1 month ago

hey guys, i know that is a long shot that any of u play nl200 :) but still...
i would apriciate a lot if some1 can help me with estimating how different should be my BB defend range at nl200 compare to higher stakes because of the reak effect. is there any reasonable way to estimate this?

what ut thoughts guys?

Teddy 11 years, 1 month ago

I don't think you can simplify it that much. I'd start by running cr ev sims with limited options for postflop play that favors IP and see how wide you can defend then and go from there.

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