i.e.

HU NL
100bb stacks


1.) Button raises 3bb, BB makes it 9bb
2.) Button raises 3bb, BB makes it 13bb

*Assume button raises ~80% of all hands.
*Assume BB 3bets ~20% of hands.

How often do we need to defend under these 2 scenarios to prevent Button from auto profiting with his 3bet? Looking for the theoretical -- defending ranges and how the math affects it based on both 1) size of 3bet 2) how often facing 3bet.


Few more Questions for those in the know -- How could I go about looking deeper into this math? Is there any software out there that is attempting to solve preflop somewhat. Will solvers be able to map out preflop one day? whats stopping them from doing it?

1) BB risks 8bbs to win 4.5bbs. It needs to succeed 64% of the time. You need to defend 36% to prevent him from auto profiting with air. You should probably 4-bet some too so he doesn't see the flop with weak hands.

2) BB risks 12bb to win 4.5bbs. The ammount to defend this is 27%.

Basically the math is: 1 - [Their risk]/[Their risk + Pot size] to get the amount you need to defend.

I think making an 0% equity bluff indifferent is a mistake here.

That is true. When you call you allow for your opponent to see flops and hit strong hands. So you'll want to defend a bit wider than minimum defense frequency. I don't know what would be a good amount to add though. 5% maybe? I guess it partially depends on how often you 4-bet.

the fact that villain can get lucky and hit flops when he is bluffing gives you incentive to defend MORE and to 4 bet some
the fact that YOU can get lucky and hit a flop when he is actually value betting means you can call more often than you could if you lost every time you called incorrectly.

The fact that equity is not 100 and 0 percent in pre-flop poker makes these spots incredibly complex but my deep intuition is that you'd wanna add a whole lot more than 5% to the required defense if you were dead against his value and he wasn't live with his bluffs.

We just have more incentive to play flops as the caller when we are beat AND as 4 bettor to shut his hands out than what the normal defense percents assume.

I'm guessing the min defense frequencies are off by a lot and it obv depends tremendously on the stack depths. If no more money goes in post flop then it's likely closer to the math in the first answers.

I might defend with top 15% vs a pot 3-bet here, but in practice up to top 25%. Other than that, my number is top 50% vs a pot 3-bet. 2x pot 3-bet would make it top 33%, or the total amount of money he puts in there vs what is in the pot. The pot odds might add more or less implied hands to these. If I 4-bet, there are some weak hands that I don't fold then.

there is no air preflop, everything got their equity %

*Assume button raises ~80% of all hands.
*Assume BB 3bets ~20% of hands.

This doesnt really matter for your question.

(1) Cant fold more than 66% or he auto profits
(2) Cant fold more than 75% or he auto profits.

agreed

Where did we get this formula (an honest question. I'm trying to understand how MDFs work)

Am I out of line for believing that tournament play, especially late stages, would add even more considerations to this?

Thanks guys.

The formula comes from the bettor's profitability with a 0 equity hand.

For an example say the pot is $10 and you bluff $5 hoping your opponent folds. Let us pretend that this scenario happens 3 times. The first 2 times the opponent calls and you lose $5. The last time he folds and you win $10. He folded twice so you lost $10 total on the times he called. But on the time he folded you won $10. If we combine the value of the first two scenarios with the last one we get $0 total. This is the break even point. Hopefully this helps you better visualize the structure of the following equation.

$5 is the risk amount. $10 is the amount in the pot.
risk/(risk + pot)
5/(5+10) = 5/15 = 1/3.

If he folds 1/3 of the time you break even. If he folds more than 1/3 you profit. If he folds less than 1/3 you lose. If he is folding 1/3 of the time then that means the other 2/3 of the time he is calling (defending). That is why 1 subtracts from the equation afterward when we are looking at it from the caller's perspective.


There isn't anything magical about MDF and it shouldn't be followed religiously. However, it is a good way to realize when there is a spot your opponent could be betting any hand and make a profit with you folding.

OP was looking for the making sure opponent doesn't auto-profit-line which was why I originally went on the safe side. Also, for some reason my mind shifted over to fullring instead of heads up which does make a difference.

For what it is worth, in HU games I sometimes defend as much as 75% of my range to a minimum open. MDF would suggest 50%. So that is 25% more than what MDF suggests. Also for 3x 3-bet pots I defend about 50% of my opening range. Which is more than MDF as well.

However, if you are looking to play tight without getting ripped apart then MDF is a good place to start at least. I've seen some players defend a 2x open in the BB with 55% of their range and still do fairly okay. I certainly sit these people expecting to have an edge, but it isn't like I am ripping them apart. It is more of a slow process of withering their stack away.

To determine how often to defend while accounting for Villain's equity you need to do an EV calc.

Determine the equity of the worst hand in Villain's range.

Set
EV = O
x = %Call

And you will then see how much fold equity villain needs to break even with his bluff.