raise 1st 57%. 4bet 6%

i assume some slowplays from KK but raise/gii from 44,88,K8,765:

(PR+GD>, MPP+GD>, TP+BFD, BDW+BFD)-(44, 88, T2P, 765)

lets say he is only semibluffing turn with equity and then bets with nutted hands:

R1: 2PR+OE>, SET>, NFD, FD+OE> --- 28% of his flop calling range and has 66% equity v my hand.

ev = 0.33*($61) - 0.66*($39.4) = $20 - $26 = -$6 = -12bb

lets add some more bluffing hands into his range (any fd or gd>):

R2: 2PR+OE>, SET>, NFD, FD+OE>, FD, GD> ---- 73% of his flop calling range

Assuming he bet/calls with the R1 and bet/folds with R2-R1

ev = 0.62*$34.6 + 0.38*0.33*($61) - 0.38*0.66*($39.4) = $21.4 + $7.6 - $9.8 = $19.2 = 38.5 bb

Now lets compare this to the potential ev of c/c turn c/c river. Let's look at 3 different river types: pairing,flush and blank.

Pairing:

Our hand looks more like a weak made hand+fd so it makes sense for us to check pairing river representing weaker part of our range to let him continue his bluff and vbet straight.

Let's say he bets SET> for value and MPP<< as a bluff.

on a K river: ev = 0.7*($61)+0.12*($61)+0.18*($34.6) = 42.7 + 7.3 + 6.2 = $56.2 (2/40)

on a 8h river: ev = 0.36*0.9*($61)+0.36*0.1(-$39.45)+0.10*($61)+0.54*($34,6) = 19.7 - 1.42 + 6.1 + 18.6 = $43 (1/40)

on a 8s river: he has (FH, 2NF>) 25% of the time which he will bet, (SET+!FH, ST, 2NF<) 38% of the time which he will check back and we win 100%, (TP<<) 10% of the time which lets say he will bluff 50% of the time because other 50% he gets scared that this card hit our range.

ev = 0.25*($61) + 0.7*($34.6) + 0.05*($61) = $15.25 + $21 + $3.05 = $39.5 (1/40)

on a 5 river: (FH, ST) value 57%, (TP3K<<) bluff 9.6%. with a bluff he risks 26.4 to win 47.6. alpha = 0.55/1+0.55 = 0.36. we should defend 64% of our range to stay unexploitable. i think by the river we don't have pure draws because we'd rather bet/fold it OTT so our range is pretty well-defined here as a bluff-catcher and occasional slow-played KK.

our range: K8+FD, KK, AA+(OE>, FD),, K+FD+GD>

since KK make up quite a large portion of our river range (62%) we don't need to defend any other hand.

ev = 0.57*($-13) + 0.33*0.91*($34.6) +0.33*0.09*(-$13) + 0.096*(-$13) = 7.4 + 10.4 - 0.4 + 1.2 = $18.6 (3/40)

on 4 river: (FH, ST) value 45%, (TP3K<<) bluff 25%. again no need to defend K8 v a bluff

ev = 0.45*($-13) + 0.3*($34.6) + 0.25*(-$13) = -5.85 + 10.38 - 3.2 = $1.3 (3/40).

Conclusion 1: on pairing rivers we have a good ev.

EV = 1/40*[2*$56.2 + $43 + $39.5 + 3*$18.6 + 3*$1.3] = 1/40($112.4+$43+$39.5+$55.8+$3.9) = 0.025*254.6 = $6.4 = 13bb

Flush rivers.

2s: (FL) 25% (eq. 39%), (TP<<) 3%
3s: (FL) 25% (eq. 40%), (TP<<) 3%
6s: (FL) 22.4% (eq. 48%), (2PR_14<<) 17%
7s: same as 6s
Ts: (FL) 24.2% (eq. 43%), (2PR_14<<) 37%
Qs: (FL) 23.8% (eq. 23%), (2PR_14<<) 36%
As: (FL) 24.2% (eq. 24%), (2PR_14<<) 36%

ev1 = 0.25*0.61*($61) + 0.25*0.39*(-$39.45) + 0.72*($34.6) +0.03*($61) = 9.3 - 3.8 + 24.9 + 1.8 = 32.2
ev2 = 0.25*0.6*($61) + 0.25*0.4*(-$39.45) + 0.72*($34.6) +0.03*($61) = 9.2 - 3.9 + 24.9 + 1.8 = 32
ev3 = 0.22*0.52*($61) + 0.22*0.48*(-$39.45) + 0.61*($34.6) +0.17*($61) = 7 - 4.2 + 21.1 + 10.4 = 34.3
ev4 = ev3
ev5 = 0.24*0.57*($61) + 0.24*0.43*(-$39.45) + 0.39*($34.6) +0.37*($61) = 8.3 - 4 + 13.5 + 22.5 = 40.3
ev6 = 0.24*0.77*($61) + 0.24*0.23*(-$39.45) + 0.4*($34.6) +0.36*($61) = 11.2 - 2.1 + 13.8 + 22 = 44.9
ev7 = 0.24*0.76*($61) + 0.24*0.24*(-$39.45) + 0.4*($34.6) +0.36*($61) = 11.1 - 2.2 + 13.8 + 22 = 44.7

Interesting to note how ev increases the higher the the flush card falls on the river. As expected the higher the card the more flushes we beat.

Conclusion 2: On flush rivers we have +ev c/c spot.
EV = 1/40*(32.2+32+34.3+34.3+40.3+44.9+44.7) = $6.6 = 13bb

Let's now think about leading flush rivers.

Again if we had a bluff we would risk 26.4 to win 47.6. alpha = 0.55/1+0.55 = 0.36. So villain would have a incentive to defend 64% of hands that beat a bluff. The range (FL, ST, SET) makes up from 50 to 70% of his hands on a different flushing cards so he would have an incentive to defend that range.
That range has from 9% to 14% equity v our hand. So approximately

ev = 0.6*0.88*($61)+0.6*0.12*(-$39.45)+0.4*($34.6) = $32.2 - $2.8 + $13.8 = $43.2

Average ev of leading flushing rivers is bigger than the ev of c/c almost any flush river except for Qs and As.

For a lead

EV = 7/40*(43.2) = $7.5 = 15bb

Conclusion 3: On flushing rivers Lead > C/C by 2bb.

Blank rivers. (24)

2: (SET> 41%) for value (TP1K<<) 13% as a bluff, check back eq. 0% (3)
3: (SET> 40%) for value (TP1K<<) 13% as a bluff, check back eq. 0% (3)
6: (SET> 57%) for value (TP1K<<) 5% as a bluff, check back eq. 0% (2)
7: (SET> 58%) for value (TP1K<<) 6% as a bluff, check back eq. 0% (2)
9: (SET> 31%) for value (TP1K<<) 10% as a bluff, check back eq. 4% (3)
T: (SET> 30%) for value (TP1K<<) 12% as a bluff, check back eq. 12% (3)
J: (SET> 30%) for value (TP1K<<) 12% as a bluff, check back eq. 3% (3)
Q: (SET> 30%) for value (TP1K<<) 12% as a bluff check back eq. 12% (2)
A: (SET> 33%) for value, (TP1K<<) 17% as a bluff, check back eq. 32% (3)

if we c/f 100% of the time:

ev1 = 0.41*(-13)+0.46*($34.6)+0.13*(-13) = -5.3 + 15.9 - 1.7 = $8.9
ev2 = ev1
ev3 = 0.57*(-13)+0.38*($34.6)+0.05*(-13) = -7.4 + 13.1 - 0.6 = $5.1
ev4 = ev3
ev5 = 0.3*(-13)+0.6*0.96*($34.6)+0.33*0.04*(-13)+0.1*(-13) = -3.9 + 20 - 0.2 -1.3 = $14.6
ev6 = 0.3*(-13)+0.6*0.88*($34.6)+0.33*0.12*(-13)+0.1*(-13) = -3.9 + 18.2 - 0.5 - 1.3 = $12.5
ev7 = ev5
ev8 = ev6
ev9 = 0.33*(-13)+0.5*0.68*($34.6)+0.5*0.32*(-13)+0.17*(-13) = -4.3 + 11.7 - 2 + 2.2 = $7.6

EV = 24/40*((6/24)*8.9+(4/24)*5.1+(6/24)*14.6+(5/24)*12.5) = 24/40*(2.2+0.8+3.6+2.6) = (24/40)*$9.2 = $5.5 = 11bb

Now obviously villain can decrease that ev by bluffing more than just (TP1K<<). I won't get into it now but later i will look at how much he can decrease our ev.

Basically our total ev on all river cards is EV = 13bb (pairing) +13bb (flush) +11bb (blank) = 37bb. If you remember the ev of a check/shove was 38.5 bb so check-shove is slightly better under those conditions. Especially if you consider that villain can lower our ev significantly by choosing better bluffing frequency c/jam on the turn is preferable.