Hi I'm trying to work out what the worst hand I can value bet is in position on the river given certain parameters.

In my example hand I have:

Pot size = bet size = 24bb

Effective stacks = 88bb

BB fold to bet frequency 50%

Equity of chosen sb betting hand vs top 50% of bb's range = 53.87%

Clearly, in a game with no check raise bluffing, our hand is a value bet.  Using the formula from Will Tipton 1 (p282) we can plug in our values:

Ev of bet fold= (S-B)+(B+P)(BB fold to bet frequency)+(2B+P)(BBcalls and we win frequency)

=(88-24)+(24+24)*0.5+(48+24)*(0.5*0.5387)=107.3932

EV of checking back=S+P*(equity of hand vs BB’s checking range)

=88+24*(0.5*0.5387)+24*0.5=106.4644

My working breaks down how our hand does vs the c/f range and vs the calling range but I am fairly sure it is correct up until this point.

Anyway, 107.3932>106.4644 so betting is better

However, I want to add in check raising to the BB’s strategy and this is where I have problems.

I want to have the BB check raising the top 5% of his previous defending range - therefore 2.5% of his total river range.  We will make his raise to pot and to balance this 2.5% have 1.25% of his total river range be check raise bluffs.

How SB’s hand does vs the value check raising range shouldn’t change - we are bet-folding and were losing to those hands anyway.

Tipton’s next formula on p282 accounts for check raise bluffing:

Evbet-fold-Ev(check)=(B)(chance BB calls with worse)-(B+P)(chance BB bluff

raises)-(B)(chance BB has a better hand)

Term 1

Chance BB calls with worse=defending frequency*equity our hand has when he calls

If we keep his overall defending frequency the same (50%) but have him no longer calling with the bottom of his range (instead using it to check raise bluff) then our equity vs his calling range surely decreases.  As far as I can tell, if we just remove 2.5% of his previous calling range then our equity vs that calling range will just decrease by exactly that amount since those were hands that we were always ahead of.  

Thus, our equity vs the calling range=0.5387-0.025=0.5137

Term 2

Chance BB bluff raises= we have defined this as 1.25%

Term 3

Chance BB has a better hand=??

Logically, he has a better hand the same amount of the time as he did before.  That is 0.5*0.4613=0.23065. However, it doesn’t feel right to be using the equity from the previous calling range when in the same equation we have changed that range and therefore it’s equity.  Can’t get my head around this.

Anyway if we use 0.4613, we have

(24)(0.5*5137)-(24+24)(0.0125)-(24)(0.5*0.4613)=0.0288

Conclusion - bet-folding is superior to checking back given our parameters.

I’m really unsure if I’ve done this right and would be grateful if people could point out my mistakes.