80%-$4B13

He is definitely not the toughest player so let's make some assumptions:
1. He has no r/f range
2. He has a wide floating range for that cbet size and pretty narrow r/gii range .
Maybe if he's a little bit smarter player he wants to protect his calling range on blank turns by

calling (22,44) some % of the time. But actually i found that (22,44) makes up such a small portion

of his range that we can ignore how he plays it.
So:
R/gii: (WR+FD, NFD+GD>, TP+NFD, 2PR+FD, T2P>)
Call: (WR, PR+GD>, FD, MP+BFD, TP>)-(WR+FD, NFD+GD>, TP+NFD, 2PR+FD, T2P>)

What are our options:
1. Pot/call
2. Bet around 13/fold
3. Bet around 13/call
4. Check/call turn play river
5. Check/raise all-in
6. Check/fold (assuming pot sized bet from him) 0 EV

PART 1. Pot/call

In the 1st part im only dealing with potting turn.

Let's look at his possible turn strategy as if he knows we hold KKxx. Lets divide his turn range into

3 parts:
1. SET> (33,A5,56) 20.3%, eq. 67.4
2. (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>) 26%, eq. 25
3. *-(2PR+FD, PR+FD+GD>, NFD+GD>, SET>) 54.7%, eq. 13.5


JAM SET>:
ev = 0.674*$65.5 - 0.326*$45 = $44.1 - $14.6 = $29.5 = 59bb

CALL SET>:
River pot = $60.5 Eff. stacks = $25
ev = P(we c/f)*$40.5 + P(we c/c)*(his eq.)*$65.5 + P(we c/c)*(our eq.)*-$45 + P(we c,he c)*(his check back eq.)*$40.5 + P(we c,he c)*(our eq.)*(-$20) + P(we bet, he folds)*-$20 + P(we bet,he calls)*(his eq.)*$65.5 + P(we bet, he calls)*(our eq.)*-$45

Pairing rivers (10/46):

Because his eq. is around 0% (up to 0.05 if u include slowplayed {22,44}30) our best strategy is to check 100% and his response is to check back 100%

ev = 1*0*($40.5) + 1*1*(-$20) = -$20

Flushing, non pairing rivers (18/46):

average on diamonds:

his range:
strong value R1 = (FL,ST) 94%, eq. 100
dead hand R2 = SET= 6% eq. 0

it is clear that checking dominates betting for us. let's suppose we decide to c/call 70% of the time and c/fold 30% (based on alpha = 0.41/1+0.41 = 29%). since he knows we hold KK he will bet all better hands R1 and all R2 (essentialy it's a clairvoyant game for villain where his distribution is close to 100% nuts and 0% air so he should bet 100%)

ev = 0.3*($40.5) + 0.7*0.94*($65.5) + 0.7*0.06*(-$45) = $12.15 + $43 - $1.9 = $53.25 = 106.5bb

what if we fold 100%?

ev = 1*$40.5 = 81 bb

Conclusion 1: Against 1st part of his range (SET>) on diamond rivers we should c/f 100% and he should bet 100%.

average on spades:

his range:
strong value R1 = (FL,ST) 97%, eq. 95
dead hand R2 = SET= 3% eq. 0

It's clear that ev will be the same as for diamonds

Conclusion 2: Against 1st part of his range (SET>) on spade rivers we should c/f 100% and he should bet 100%.

Blank rivers (18/46):

Turns out that even on a blank card his range is too close to (100% nuts, 0% air) that we c/f 100% and he bets 100%

ev = 81bb

Conclusion 3: Against 1st part of his range (SET>) on blank rivers we should c/f 100% and he should bet 100%.

Overall ev of calling with (SET>) if he knows he's up against KKxx:

EV = (10/46)*(-$20)+(18/46)*($40.5)+(18/46)*($40.5) = -$4.3 + $15.8 + $15.8 = $27.3 = 54.5bb

Conclusion 4: EV of jamming (SET>) on the turn is better than EV of calling and playing clairvoyant game on rivers by 4.5bb

JAM (2PR+FD, PR+FD+GD>, NFD+GD>):
ev = 0.25*($65.5)+0.75*(-$45) = 16.3 - 33.7 = -17.4 = -35bb

CALL (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>):
River pot = $60.5 Eff. stacks = $25
ev = P(we c/f)*$40.5 + P(we c/c)*(his eq.)*$65.5 + P(we c/c)*(our eq.)*-$45 + P(we c,he c)*(his check back eq.)*$40.5 + P(we c,he c)*(our eq.)*(-$20) + P(we bet, he folds)*-$20 + P(we bet,he calls)*(his eq.)*$65.5 + P(we bet, he calls)*(our eq.)*-$45

Pairing rivers (10/46):
Because his eq. is 0% our best strategy is to check 100% and his response is to check back 100%

ev = 1*0*($40.5) + 1*1*(-$20) = -$20 = -40bb

Flushing, non pairing rivers (18/46):

average on diamonds excluding Ad and 6d (7/46):

his range:
strong value R1 = (FL) 36%, eq. 86
dead hand R2 = *-(FL) 64% eq. 0

if we decide to bet and not check:

ev = 0.64*(-$20)+0.36*0.86*($65.5)+0.36*0.14*(-$45) = -$12.8 + $20.2 - $2.2 = $5.2 = 10.5bb

If we check he will bet R1 and some % of R2 and check back some % or R2. Let's suppose we decide to c/call 70% of the time and c/fold 30% (based on alpha = 0.41/1+0.41 = 29%). Since he knows we hold KK he will bet all better hands R1 and alpha*(size of R1) part of R2. [R1] = 5226 combos; aplha*[R1] = 1515 combos. [R2] = 9187 combos. 1515/9187 = 16%. So he will bet all R1 and 16% of the time with R2.Overall he bets 46% and checks 54%. when he bets he has range of (FL, {*-(FL)}16) and has eq. 62.

ev = 1*0.46*0.3*($40.5) + 1*0.46*0.7*0.62*($65.5) + 1*0.46*0.7*0.38*(-$45) + 1*0.54*1*(-$20) = 5.6 + 13 - 5.5 - 10.8 = $2.3 = 4.5bb

what if we always fold?

ev = 1*0.46*1*($40.5) + 1*0.54*1*(-$20) = $18.6 - $10.8 = $7.8 = 15.5bb

what if we always call?

ev = 1*0.46*1*0.62*($65.5) + 1*0.46*1*0.38*(-$45) + 1*0.54*1*(-$20) = $18.6 - $7.8 - $10.8 = 0

then he never bluffs and bets 36% and checks 64% :

ev = 1*0.36*0 + 1*0.36*1*0.85*($65.5) + 1*0.36*1*0.15*(-$45) + 1*0.64*1*(-$20) = $20 - $2.4 - $12.8 =

$4.8 = 9.5bb

So we tried to lower his ev by always folding but the ev increased by 11bb. Then we tried to lower his ev by always calling and it went down to 0. But then he can choose to bluff 0% and his ev goes up to 9.5bb and the difference is +5bb for him because of this adjustment. Let's assume then the first strategy optimal for both players.

Conclusion 5: c/calling is better than leading for us.
Conclusion 6: we should c/c 70% and c/f 30%

Special case. River Ad,6d (2/46):

his range:
strong value R1 = (FL,ST=) 56%, eq. 77
dead hand R2 = *-(FL, ST=) 44%, eq. 0

Again similar case with different numbers:

if we decide to bet and not check:

ev = 0.44*(-$20)+0.56*0.77*($65.5)+0.36*0.27*(-$45) = $8.8 + $28.2 - $4.3 = $32.7 = 65.5bb

if we check then once again it will probably be optimal for us c/c 70% and c/f 30%. Villain will bet all his R1 and alpha*[R1] part of R2. [R1] = 8000 combos. alpha*[R1] = 2320. [R2] = 6183 combos. 2320/6183 = 38%. So he will bet all R1 and 38% of the time with R2.Overall he bets 73% and checks 27%. when he bets he has range of ((FL, ST=), {*-(FL, ST=)}38) and has eq. 57.

ev = 1*0.73*0.3*($40.5)+1*0.73*0.7*0.57*($65.5)+1*0.73*0.7*0.43*(-$45)+1*0.27*1*(-$20) = $8.8 + $19 - $9.8 - $5.4 = $12.6 = 25bb

Conclusion 7: On Ad,6d checking dominates leading and we should c/c 70% and c/f 30%.

Spade rivers (9/46):
his range:
strong value R1 = (FL) 62%, eq. 98
dead hand R2 = *-(FL) 38% eq. 0

if we decide to bet and not check:

ev = 0.38*(-$20)+0.62*0.98*($65.5)+0.62*0.02*(-$45) = -$7.6 + $39.8 - $0.5 = $31.7 = 63.4bb

if we c/c 70% and c/f 30%. [R1] = 8634 combos. alpha*[R1] = 2503. [R2] = 5263 combos. 2503/5263 = 48%. So he will bet all R1 and 48% of the time with R2. Overall he bets 80% and checks 20%. when he bets he has range of (FL, {*-FL}48) and has eq. 80.

ev = 1*0.8*0.3*($40.5)+1*0.8*0.7*0.8*($65.5)+1*0.8*0.7*0.2*(-$45)+1*0.2*1*(-$20) = $9.7 + $29.3 - $5

- $4 = $30 = 60bb

Conclusion 8: On spade rivers leading and checking are very close in ev so we can go either way.

Blank rivers (18/46):

Q,J,T,9,8,7 river (12/46):

When he had a strong part of his range on blank rivers we played a strategy of c/f 100%. Here we have the opposite case.

his range:
dead hand R1 = (SET>) 1%, eq. 0
dead hand R2 = *-(SET>) 99% eq. 0

We should check 100% and call 100% if he bets but he should check back 100%.

ev = 1*0+1*1*(-$20) = -$20 = -40bb


5 river (2/46):
his range:
strong value R1 = (SET>) 59%, eq. 92
dead hand R2 = *-(SET>) 41% eq. 0

checking obviously dominates leading. so we check. Villain will bet all his R1 and alpha*[R1] part of R2. [R1] = 8698 combos. alpha*[R1] = 2522. [R2] = 6091 combos. 2522/6091 = 41%. So he will bet all R1 and 41% of the time with R2.Overall he bets 75% and checks 25%. when he bets he has range of (SET>, {*-(SET>)}41) and has eq. 74.

ev = 1*0.75*0.3*($40.5)+1*0.75*0.7*0.74*($65.5)+1*0.75*0.7*0.26*(-$45)+1*0.25*1*(-$20) = $9.1 + $25.4 - $6.1 - $5 = $23.4 = 47bb

6 river (2/46):
his range:
strong value R1 = (SET>) 35%, eq. 92
dead hand R2 = *-(SET>) 65% eq. 0

[R1] = 5116 combos. alpha*[R1] = 1483. [R2] = 9517 combos. 1483/9517 = 15%. So he will bet all R1 and 15% of the time with R2.Overall he bets 45% and checks 55%. when he bets he has range of (SET>, {*-(SET>)}15) and has eq. 72.

ev = 1*0.45*0.3*($40.5)+1*0.45*0.7*0.72*($65.5)+1*0.45*0.7*0.28*(-$45)+1*0.55*1*(-$20) = $5.4 + $14.8

- $3.9 - $11 = $5.3 = 10.5bb

A river (2/46):
his range:
strong value R1 = (SET>) 29%, eq. 96
dead hand R2 = *-(SET>) 71% eq. 0

[R1] = 4345 combos. alpha*[R1] = 1260. [R2] = 10622 combos. 1260/10622 = 12%. So he will bet all R1 and 12% of the time with R2.Overall he bets 38% and checks 62%. when he bets he has range of (SET>, {*-(SET>)}12) and has eq. 76.

ev = 1*0.38*0.3*($40.5)+1*0.38*0.7*0.76*($65.5)+1*0.38*0.7*0.24*(-$45)+1*0.62*1*(-$20) = $4.6 + $13.2

- $2.8 - $12.4 = $2.6 = 5bb

average on all blank rivers:

EV = (6/46)*((1/3)*$23.4+(1/3)*$5.3+(1/3)*$2.6) = (6/46)*(7.8+1.8+0.9) = (6/46)*$10.5 = (6/46)*(21bb)

Total:

EV = (10/46)*(-40bb) + (7/46)*(4.5bb) + (2/46)*(25bb) + (9/46)*(60bb) + (12/46)*(-40bb) + (6/46)*

(21bb) = -8.7 + 0.7 + 1 + 11.7 - 10.4 + 2.7 = -3bb

Conclusion 9: EV of calling turn and playing rivers clairvoyantly with range (2PR+FD, PR+FD+GD>, NFD

+GD>)-(SET>) is better than jamming turn by 32bb but still is -3bb in absolute sense.

I think it is clear that since the second range shows negative ev by calling turn then the weakest range will also show negative ev by calling let alone jamming therefore it will play better by folding.

So we have 2 ranges. 1st is playing better by jamming and 2nd is playing better by calling or folding. Therefore we've got a first approximation of strategy for villain.

Villain's strategy v pot bet:

JAM: SET> (33,A5,56) 20.3%, eq. 67.4
CALL: (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>) 26%, eq. 25
FOLD: *-(2PR+FD, PR+FD+GD>, NFD+GD>, SET>) 54.7%, eq. 13.5

OR

JAM: SET> (33,A5,56) 20.3%, eq. 67.4
FOLD: (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>) 26%, eq. 25
FOLD: *-(2PR+FD, PR+FD+GD>, NFD+GD>, SET>) 54.7%, eq. 13.5

ev1 = 0.2*$29.5 + 0.26*(-$1.5) + $0.54*0 = $5.9 - $0.4 = $5.5 = 11bb
ev2 = $5.9 = 12bb

Conclusion 10: when playing clairvoyantly our villain can play a +12bb strategy when we bet pot on the turn with KKxx.

Let's now consider our range:

preflop: 12%
flop:
let's divide our range into b/gii (R1) and b/f (R2) . the rest if c/f (R3). i dont wanna have a c/r or c/c range v this villain. once we bet that amount on the flop we should gii everything that has eq. 41 and better. let's assume villain's range for r/gii V1 = T2P>, WR+FD, TP+(NFD, FD+OE>), NFD+OE>

R1 = TP>+NFD, T2P>, NFD+OE> 35% eq. 62
R2 = (TP>, OP, TP+FD, PR+WR, BDW+BFD)-(TP>+NFD, T2P>, NFD+OE>) 65% eq. 23
R3 = *-R1-R2

When he calls we arrive to the turn with that range:
TP>, OP, TP+FD, PR+WR, BDW+BFD

Since we're looking at a pot sized bet what could we have? We could have a range that has eq. 41 when all-in. We know that he's jamming (SET>) so we get first approximation of our range for potting turn:

T1 = SET>, T2P+NFD, NFD+OE> 34% eq. 44.6

what's our ev?

ev = P(he folds)*($20.5) + P(he jams)*our eq.*($65.5) + P(he jams)*his eq.*(-$45)

ev = 0.8*($20.5)+0.2*0.446*($65.5)+0.2*0.554*(-$45) = $16.4 + $6 - $5 = $17.4 = 35bb

Conclusion 11: When potting turn with tange T1 we expect to win 35bb

Could we maximize ev of potting turn by choosing different range? Pretty much no. Our eq. will not get higher than 44.6.

Let's get back to villain then. Assuming he knows our potting range what could he do about it?

Again divide his range in 3 parts:

1. SET> (33,A5,56) 20.3%, eq. 65 (2.4 decrease as opposed to v KKxx)
2. (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>) 26%, eq. 25 (same as v KKxx)
3. *-(2PR+FD, PR+FD+GD>, NFD+GD>, SET>) 54.7%, eq. 13.5

Conclusion 12: Because his equities are the same his strategy will not change from the clairvoyant game v KKxx.

PART 1 CONCLUSION: Potting turn is definitely +ev for us (compared to c/f). Villain also has a +ev strategy to respond (compared to folding).