Hero doesn't gain ev here. If hero gained ev, then villain was not at equilibrium to start.

There is no immediate gain or loss vs nash equilibrium villain; there is only liability vs adjusting opponent.

This is the point where the villain deviates from self imposed indifference. He attempts to earn exploitive profits, which in effect causes a change in the guaranteed minimum ev of zero.

I need to think a bit about this. When hero increases the betting frequency of a bluff that has equity (e.g. T8s) this will affect the overall equity of his bluffing range. The equity of hero's bluffing range could increase or decrease or remain unchanged. I'm thinking this would drive villains optimal response, but I'm not sure.

Deviation by hero changes the three street flop comaxexploitive response by the big blind, but not the 4 street strategy that guarantees a minimum ev of zero.

How does this change relative to the 4 street solution? We need to look at the margins that divide the action regions for the big blind:

(check fold flop/check call flop)

(check fold flop/check raise flop)

(check call/check raise)

The "/" above is the point of profitability that separates one action from the other.

The margin that separates check fold/check call shifts because more turn calls are profitable.

The margin that separates check fold/check raise shifts because draw strength improves due to stronger pair outs.

The margin that separates check call/check raise might shift either way depending on the river strategy, or it might not shift at all.

I see all this being true provided the T8s mistake lowered the overall equity of hero's bluffs. Which will always be the case if we are talking about a marginal bluffing hand.

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%.

Yes the profitability of the bluffing range would go up in that case, but what about the profitability of the checking range? I would think this profitability will decrease and thus there is still no gain vs the Nash equilibrium strategy.