im trying to solve optimal 3bet/4b/5b defense ranges , but, i began wondering something and i cant find the answer:
what is the math behind WHY we open as often as we do?

ie, why is 70% open hu from the button, or 10% UTG etc, what is this rate derived from? im assuming at its most basic level it's based upon the blinds and the rate the blinds hit you?
im thinking in terms of cash games, but im sure its applicable to both.


is it just a minimum defense frequency calculated vs the billnds/the frequency the blinds hit ?

can't wait to hear any insight, tyvm

This doesn’t answer your question exactly but simplifies the situation somewhat to provide a relatively easy to understand approach. Consider a heads up situation where the small blind is thinking about a shove with his short stack of 10bb. Pot is 1.5. Small Blind knows villain will always call. At this point SB has a “good” read on villain, namely any two cards other than the ones he has.

If I did the math correctly, you can show SB needs equity of 47.6% or greater to be +EV with that size bet and villain's range. Using an equity calculator, you can then figure out the hands that give you the required equity, which will then define your betting frequency for that size shove. In this case, the betting range is the following:

22+,A2s+,K2s+,Q2s+,J3s+,T6s+,97s+,87s,A2o+,K2o+,Q3 o+,J6o+,T7o+,98o,

which represent some 58% of all hands.

Note that the criterion I used is for assuring +EV. There are other criteria/metrics such as GTO factors (e.g. indifference), MDF and variance factors. The approach can also be used when an estimate of villain’s folding probability is included. Then of course, things can get complicated because fold equity will often depend on the bet size.

Hope this helps.

interesting, ill continue pondering that ty for your response.
i guess another way to rephrase what im asking: how would i go about solving for the MINIMUM range i'd have to open ? is there a way to do that ?

That’s essentially what I did. First you set up the EV equation – in its simplest form it is

EV =eq*(Pot +Bet) - (1-Eq)*Bet,

where in my posting Bet is the effective stack. Then you set EV to 0 and solve for eq, knowing the Pot and Bet values. This is the minimum equity needed for +EV. Then using an equity calculator you find the betting range that provides that equity against villain’s calling range, which has to be estimated. For a small blind open, BB holds any two cards basically but may not open all. If so,fold equity comes into play

All starting hands are played based on their profitability. Any hand that is profitable is played and the ones that aren't are folded.

There isn't any point where it is obvious. The EV is gradual. The better the hand, the more of a mistake it is to fold it.

The answer is based on the average of all future possible outcomes. In some scenarios you will win with a given hand. In other scenarios you will lose. If on average you win more than you lose you play the hand.

There isn't a quick easy calculation you can do to solve it. Unless of course your doing a very simplistic push/fold scenario as statmanhal has shown. In a regular cash game there are many flops, turns, and rivers which are very complex. We do however have a general idea of what is good and bad based on statistics after millions of hands.

You dont have to open anything,so minimum is just dont open at all xD

If you want to know what are optimal opening ranges.Its easy.You just solve entire game of poker.

I think you can find answered to your questions(at least to some degree) in preflop section of Mattew Janda book "Applications of no-limit holdem"

"You dont have to open anything,so minimum is just dont open at all xD"
haha, i get your point, but given im trying to profit, technically i do have to open something in order to not go broke due to the blinds.

isnt that the first step of figuring out a minimum opening range - compensating for the blinds?

it's pretty simple actually

there is a range of cards that are favored over random hands. for example, in a heads up game, Q7s is the first hand favored over the random hand, so we can raise for value anything better

but because our opponent cannot see our cards and we can't see theirs, in order to win we need a mixed strategy, so we have to add hands that are not necessarily favored but have equity vs the calling range, they will be breakeven or even slightly -ev to raise but we have to play them in order to have a balanced range, like say, 56s

this concept can be extended to 3 person game, 4 person game, etc

so we can separate the hands into 3 groups, those that are favored, those that are equal, and those that are behind but can be played because they have required equity vs a 3bet range

add up the total number of hands, divide by the number of hands possible and voila

of course there is a way to do this. solvers would not exist if there wasn't

i suggest starting with what i call the "unexploitable" range. what hands are favored over the random hands behind you?

Did you mean Q5o? That hand has 50% equity vs ATC. (Q7s is much better, with about 54%).

Q7 often is known as the computer hand because it is the median hand but is above 50% equity i think around 51.4

didn't know that, just assumed Q7o as the "computer hand" would have exactly 50% equity

yeah, Q5s would be the first "raisable hand", because we are invested 1/2 blind we would have to extend our raising range to hands that have equity when re-raised like 9To

i don't play HU too much but already we are talking about a raising range of 70%, i'm sure the HU GTO from the button is much higher. seems to be solved exactly actually from what i read

the concept can be extended to any n-player game, but the GTO raising range will always be a bit looser than our standard raising range

for example only about 11% of hands are favored over the random hands behind it from UTG but a GTO raising range may contain 13-14% of hands