Quote:
Originally Posted by Bob148
Correct.
This is exactly what I was trying to get at with this thread:
https://forumserver.twoplustwo.com/1...69/index4.html
although in the excitement I forgot we were talking about turn strategy. Correction:
Deviation by hero changes the two street comaxexploitive response by the big blind, but not the 4 street strategy that guarantees a minimum ev of zero.
In this sense, the Nash equilibrium strategy is the 4 street strategy that guarantees a minimum ev of zero. The two street comaxexploitive response by the big blind is the gto strategy that a solver would produce. The maxexploitive strategy is something else which has potential for adjustments on all streets considering the mistake that the button has made.
Of course hero can make a "mistake" in the other direction where he increases the betting frequency of a bluff with equity > the equity of his bluffing range. In this case villain needs to adjust by increasing his calling frequency otherwise there would be a shift in ev in hero's favour.
I derived the below example which (hopefully) demonstrates this. This is an extension of some of the concepts covered by Janda and Tipton. I think this is correct.
EXAMPLE
At equilibrium we know the bluffing indifference holds: Ev(Check) = Ev(Bet)
Let’s apply the indifference OTT where hero’s bluffs have equity. Hero is in position.
Ev(Check) = Ev(Bet)
Pe = Pf +(1- f)(P + B)e - (1- f)B(1- e)
Pe = Pf + (1- f)(Pe + Be - B + Be)
Pe = Pf + (1 – f)(Pe + 2Be – B)
Eq. 1
Pe - Pe - 2Be + B = f(P - Pe - 2Be +B)
f = (B - 2Be)/(P + B - e(P+2B))
Eq. 2
Where:
P = Pot
B = Bet
e = equity of bluffs
f = villain’s fold frequency to maintain indifference
For zero equity bluffs we have f = B/(P + B) and when the bet is the size of the pot we have f = 1/2 = 50%. This makes sense and is what we expect to see on the river when bluffs have no equity.
For bluffs having 10% equity OTT and bet is the size of the pot we have
f = (1-0.2)/(2-0.3) = 0.8/1.7 ~ 47%.
So villain needs to call 53% of the time to maintain indifference.
Now let's say hero adds a small amount of equity to his overall bluffing range by increasing the betting frequency of hands that have > 10% equity. Let's say he increases the betting frequency of some of his flush draws ( ~ 18 % equity). Say this causes the overall equity of his bluffing range to increase from 10% to 12%. We now have f = (1-0.24)/(2-0.36) = 0.76/1.64 ~ 46%. So villain now needs to increase his calling frequency to 54%.
What happens if villain is unaware of hero’s change in strategy and continues to call at 53% ?
We can use
Eq. 1 to see if there is a change in hero’s EV.
Pe = Pf + (1 – f)(Pe + 2Be – B) Eq. 1
EV(Pot size bet,10% equity, 53% call rate) = 0.47 + 0.53(0.30 – 1) ~ 0.10 = EV(check) = 0.10
EV(Pot size bet,12% equity, 53% call rate) = 0.47 + 0.53(0.36 – 1) ~ 0.13 > EV(check) = 0.12
This tells me villain is over-folding and hero gains EV by betting instead of checking. Villain can restore equilibrium by increasing his calling frequency to 54%.