#PokerNugget (cool name btw)  ; ) #PreFlop

How often should we defend our blinds against the button?

On level zero we all know what good cards look like, but we also know we should be playing mostly "In Position", so how do we know when we should be 3betting (from the blinds) preflop and how often?

Lets say a guy on our right is opening 2.5x  the majority of the time he's on the button(which he knows he should be), we are trying to make the button indifferent to opening or folding the worst hands in his theoretically correct opening range. We thus only care that he loses money or breaks even with the worst hand in his range, which will of course fold to a 3bet. So how often do the blinds need to defend combined to prevent the button from opening ATC(any two cards)?

When the blinds fold the button wins 1.5bbs. When the blinds 3bet, he loses 2.5bbs

Therefore the maths is relatively simple  1.5(1-x)-2.5(x)=0 where x is going to be our optimal 3betting frequency.

So solving for X we end up with

1.5-1.5x+ -2.5x=0

1.5+ -4x=0

-4x = -1.5

x = 0.375 or 37.5%

For me, this little nugget made me look at my optimal 3betting ranges from the blinds and tweak them into another dimension, we all 3bet the obvious hands and a few extra bluffs i'm sure, but do we defend often enough?...apparently not...

So i will now reveal my own personal 3betting ranges from the blinds, they dont add up to exactly 37.5% but it's close enough when you take into account most people open between 2 and 3.5x from the button...(online at least anyway)

3BETTING RANGE SB vs BTN 18.4%

Before i learnt this stuff i definately didnt 3bet enough even though i knew i was supposed to, i always tried to maintain a polarised 3betting range so my stats looked like there was a few bluffs thrown in but not so many that they could 4bet me light too often.  Here is my Big Blind 3betting range against button opens...

3BETTING BB VS BUTTON 17.5%

These ranges are my own personal ranges so it does NOT matter what cards you chose to 3bet just so long as your percentages add up to roughly 37%, what i love about maths is that it doesn't lie...

So when you see me 3betting 4♠3♠ from the big blind, you know i've done the maths(however donk-ish it looks lol) 

I'm not great at maths but i forced myself  to learn "frequencies" so i understood the "bigger picture"...there are obvious shortcuts in basic situations, for example, most people tend to use RISK/RISK+REWARD= BREAKEVEN FOLDING FREQUENCY as the equation for solving pot odds or folding frequencies and this works totally fine, but to delve further into the other frequencies for further streets and more complex situations i find algebraic equations makes sure we dont make any mistakes and helps to train us to solve much tougher problems later.

So applying a bit of 3bet maths here shows just how silly it would actually be if the button  actually opened 100% of his range against two players (ie;he would have to defend against 3bets ridiculously wide)

I hope i kept this short enough, it's just a tiny little nugget but it helped me once upon a time and i hope it helps someone else somewhere one day, hopefully we all know this stuff and learn it, if we dont...then we're not doing enough to improve imo... i know i've got a million miles to go still, so the more gold we find the richer we'll be...hopefully ; ) 

Remember...."Sharing is Caring"  lol 

EDIT: btw as any game theorist will tell you, we can never get exact solutions, but can  only create parameters like these to show us when something is NOT correct.Rarely will the parameters show us the right answer anyway, but these restrictions are  indeed very useful for guiding us on the right path to becoming very theoretically sound players. ; )