PART 1: http://www.runitonce.com/plo/50-hu-plo-top-set-on-the-turn-in-3b-pot/

PART 2.Bet around 13/fold and Bet around 13/call

When evaluating ev of this play i want to restrict my strategy to 3 options: bet 13/fold, bet 13/call and check/fold. Then if i know the ev of bet 13/fold and bet 13/call i can find the ranges for maximum ev and compare it to the strategy in PART 1. Again i will assume at first that villain is clairvoyant then find the best strategy for him. Then find the best ranges v his strategy. Now because my bet $13 range is gonna be much wider than bet pot range i need to make villain clairvoyant not to KKxx but to my bet $13 range. So at first i need to set a base range for betting $13 into $20.5:

(SET>, T2P, NFD, TP>+FD) 68%, eq. 32

best part of this range i will bet/call:
SET>, T2P+NFD, NFD+OE> 49%, eq. 44.5

the rest i will bet/fold:
*-(SET>, T2P+NFD, NFD+OE>) 51%, eq. 22

In other words i will bet 68% of the time on the turn with the size 13 into the pot of 20.5 and will call a shove 49% and will fold to a shove 51%

Lets divide his turn range into 3 parts:
1. SET> (33,A5,56) 20.3%, eq. 71.3 (v betting range) eq. 65 (v bet/calling range)
2. (2PR+FD, PR+FD+GD>, NFD+GD>)-(SET>) 26%, eq. 33.3 (v betting range) eq. 24 (v bet/calling range)
3. *-(2PR+FD, PR+FD+GD>, NFD+GD>, SET>) 54.7%, eq. 22.6 (v betting range) eq. 13.2 (v bet/calling range)


JAM SET>:
ev = P(we fold)*($33.5)+P(we call)*his eq.*($65.5) + P(we call)*our eq.*(-$45) = 0.51*($33.5)+0.49*0.65*($65.5)+0.49*0.35*(-$45) = $17+$20.8-$7.7 = $30.1 = 60bb

which is only 1bb bigger than ev of JAM (SET>) v pot bet and only v KKxx. i can explain it such that although he wins more when we fold to a jam he wins less when he has eq. advantage because we call less.

CALL SET>:
River pot = $46.5 Eff. stacks = $32
ev = P(we c/f)*$33.5 + P(we c/c)*(his eq.)*$65.5 + P(we c/c)*(our eq.)*-$45 + P(we c,he c)*(his check back eq.)*$33.5 + P(we c,he c)*(our eq.)*(-$13) + P(we bet, he folds)*-$13 + P(we bet,he calls)*(his eq.)*$65.5 + P(we bet, he calls)*(our eq.)*-$45

Pairing rivers (12/48):

2c river:

our range:
nut hand X1 = fh 31.6%, eq. 97
medium hand X2 = TP1K>+!fh 51.4% eq.11
air X3 = TP1K<< 16.9% eq.0

Let's define our 1st strategy as:

S1: bet X1 100%, check X2 100% and bet X3 b% and check X3 (1-b)%. When we bet X3 b% we're risking $32 to win $46.5. aplha=0.41. So b=(1/[X3])*(aplha*[X1])=(1/1694)*0.41*3166=0.76. So in our 1st strategy we bet X3 76% and check X3 24%. overall i bet 44.5% and have a range (FH, {TP1K<<}76)

when i bet
his range:
bluffcatcher 100%, eq. 0 v my value part and 100 v my bluffs

let's say he calls 100%. 71% i have a nut hand 29% i have a bluff

ev = 1*0.71*-45+1*0.29*65.5 = -31.9 + 19 = -$13 = -26bb

let's say he folds 100%.

ev = 1*-13 = -$13

So he is indifferent between calling and folding.

In the same way to make me indifferent between bluffing or checking he must fold aplha hands that beat a bluff. Since all of his hands can beat a bluff he must call 59% and fold 41%.

ev = 0.41*-13+0.59*0.71*-45+0.59*0.29*65.5 = -5.3 - $18.8 + $11.2 = -$13

As you can see he is truly indifferent.

Conclusion 1: When i bet a 2c river with range (FH, {TP1K<<}76) his ev = -26bb

when i check

his range:
nut hand 100%, eq. 82

since he doesnt have bluffs my best response is to c/f 100% and he should bet 100%

So when i play strategy S1 ev for villain:

ev = 0.555*1*1*($33.5)+0.555*1*0+0.445*0.41*(-$13)+0.445*0.59*0.29*($65.5)+0.445*0.59*0.71*(-$45) = $18.5 - $2.3 + $5.1 - $8.4 = $12.9 = 26bb

Conclusion 2: When i employ a strategy S1 on a 2 river villain's ev = 26bb

Now of course i can choose a mixed strategy for my nut hand X1 of betting x% and checking (1-x)% but because v my nut hand his range has 0 equity he will just check back 100% and still win v my X2 and X3 and not lose anything v X1. So betting X1 100% looks dominating to a mixed strategy.

3s river:

nut hand X1 = fh,fl 57%, eq. 83
dead hand X2 = *-(FH, FL) 43% eq.03

Again because i don't have intermediate part of my range that would have a reasonable equity and would need protection from bluffs there is no point to check any x% with X1. My strategy would be then:

S2: bet X1 100%, bet X2 b% and check/fold X2 (1-b)%. b=(1/[X2])*(aplha*[X1])=(1/4266)*0.41*5606=0.54. So we bet X2 54% and check X2 46%. overall i bet 80% and have a range ((FH, FL), {*-(FH, FL)}54)

Villain's response: call 59%, fold 41%. his equity when he calls is 29

ev = 0.2*1*1*($33.5)+0.2*1*0+0.8*0.41*(-$13)+0.8*0.59*0.29*($65.5)+0.8*0.59*0.71(-$45) = 6.7 - 4.2 + 8.9 - 15 = -$3.6 = -7bb

Conclusion 3: On a 3s river villain's ev with range (SET>) is -3bb.

2c river creates the same game as 2h,3c,3h,4c,4h,kh,kc
3s creates the same game as 2d,4s,kd

Conclusion 4: On 8/48 rivers his ev = 26bb and on 4/48 his ev = -3bb

On all pairing rivers:

EV = (12/48)&((8/12)*26bb+(4/12)*-3bb) = 12/46*(17bb - 1bb) = (12/46)*16bb

Flushing, non pairing rivers (18/48):

diamonds (9/48):

our range:
nut hand X1 = fl 42.7%, eq. 97
medium hand X2 = SET>+!FL 36.6% eq.24
air X3 = SET<< 20.6% eq.0

Because we have pretty clear 3 groups of hands we will probably need a mixed strategy
S2: bet X1 x%, check/call X1 (1-x)%, check/call X2 c%, check/fold X2 (1-c)%, bluff X3 b%, check/fold X3 (1-b)%.

his range:
strong value 100% eq.(v X1) = 0, eq.(v X2) = 76, eq.(v X3) = 100

his strategy: call cy%, fold (1-cy)%, bet by%, check (1-by)%

this game is pretty similar to the AKQ game, only difference is we have A,K or Q and villain has a special card K*: K*<<A + K*>K

basically his equity v A: 0; v K: 80, v Q: 100. you might think of it as when we hold a king he holds an ace (nuts) 80% and queen (air) 20%. Now it's important to see why we should not have a calling frequency with a king versus that type of card. When he holds an ace 33% and a queen 33% he will bet all his aces and appropriate frequency with queens to make us indifferent between calling or folding a king. this time however when we hold a king he always bets in 4-to-1 value:bluff ratio. since alpha = 0.41 he needs to bluff 29% times he bets to make us indifferent. here he bluffs 20% times he bets so we have incentive to fold 100%.

therefore our strategy will change:
S2: bet X1 x%, check/call X1 (1-x)%, check/fold X2 100%, bluff X3 b%, check/fold X3 (1-b)%.

because he doesn't get any value when he bets with his K* there is no point to bet for him to bet when we check. therefore there is no point for us to check/call our ace.

S2: bet X1 100%, check/fold X2 100%, bluff X3 b%, check/fold X3 (1-b)%.

his strategy: call c%, fold (1-c)%, bet 0%, check 100%

when we bluff b% of X3 we should make him indifferent between calling or folding so (0.21*b/0.43)=alpha (since we bet X1 100%). b = 2*alpha = 0.82

when he calls we have a bluff 29% and nut hand 71% so his eq. = 29.

ev = P(we c/f)*$33.5 + P(we c/c)*(his eq.)*$65.5 + P(we c/c)*(our eq.)*-$45 + P(we c,he c)*(his check back eq.)*$33.5 + P(we c,he c)*(our eq.)*(-$13) + P(we bet, he folds)*-$13 + P(we bet,he calls)*(his eq.)*$65.5 + P(we bet, he calls)*(our eq.)*-$45

ev = (0.37*1+0.2*(1-b))*1*(0.43*0+0.37*0.8+0.2*1)*($33.5) + (0.37*1+0.2*(1-b))*1*(0.43*1+0.37*0.2+0.2*0)*(-$13) + (0.43+0.21*b)*(1-c)*(-$13) + (0.43+0.21*b)*c*0.29*65.5 + (0.43+0.21*b)*c*0.71*(-$45)

b = 0.82, 1-b = 0.18, c = 0.59, 1-c = 0.41

ev = (0.37+0.036)*(0.5)*($33.5) + (0.37+0.036)*(0.5)(-$13) + (0.6)*0.41*(-$13) + 0.6*0.59*0.29*65.5+ 0.6*0*59*0.71*-45 = 6.8 - 2.6 - 3.1 + 6.7 - 11.3 = -$3.5 = -7bb

Conclusion 5: On diamond rivers (9/48) his ev with range (SET>) is -7bb.

Spade rivers (9/48):

our range:
nut hand X1 = fl 25%, eq. 97
medium hand X2 = SET>+!FL 45% eq.16
air X3 = SET<< 30% eq.0

again our strategy will look like:

S2: bet X1 100%, check/fold X2 100%, bluff X3 b%, check/fold X3 (1-b)%.

his strategy: call c%, fold (1-c)%, bet 0%, check 100%

(0.3*b/0.25)=alpha; b = 0.34

b = 0.34, 1-b = 0.66, c = 0.59, 1-c = 0.41

ev = (0.45*1+0.3*(1-b))*1*(0.25*0+0.45*0.84+0.3*1)*($33.5) + (0.45*1+0.3*(1-b))*1*(0.25*1+0.45*0.16+0.3*0)*(-$13) + (0.25+0.3*b)*(1-c)*(-$13) + (0.25+0.3*b)*c*0.29*65.5 + (0.25+0.3*b)*c*0.71*(-$45) =

ev = (0.65)*1*(0.68)*$33.5+(0.65)*1*(0.32)*(-$13) + 0.35*0.41*-13 + 0.35*0.59*0.29*65.5 + 0.35*0.59*0.71*-45 = 14.8 - 2.9 - 1.8 + 3.9 - 6.6 = $7.4 = 15bb

Conclusion 6: On spade rivers (9/46) his ev with range (SET>) is 15bb.

Blank rivers (18/48):

Q,J,T,9,8,7 river (12/48):

my range:

x1 = SET> 48%, eq. 27
X2 = *-SET> 52%, eq. 0

since we have no nut hands in our range but only hands in between and air there's no point to bet for me so i check 100% of my range.

his range:
strong value R1 = (SET>) 100%, eq. 73

since he has eq. 73 v X1 he will bet 100% and i will fold 100%

ev = 1*1*1*$33.5 = $33.5 = 67bb

i've checked that if i try to mix in some non-zero frequency of calls with X1 i only increase the value to villain.

5 river:

x1 = SET> 61%, eq. 28
X2 = *-SET> 39%, eq. 0

ev = 1*1*1*33.5 = 33.5 = 67bb

Conclusion 7: On blank rivers (18/48) his ev with range (SET>) is 67bb

Overall ev of calling with (SET>) when he's up v a range (SET>, T2P, NFD, TP>+FD):

EV = (12/48)*16bb + (9/48)*(-7bb)+ (9/48)*15bb + (18/48)*67bb = 4bb - 1.3bb + 2.8bb + 25bb = 30.5bb

Conclusion 8: Average on all rivers the ev of calling turn w/ range (SET>) for villain is 30.5bb. Since the ev of JAM (SET>) = 60bb and ev of CALL (SET>) = 30.5bb JAM is preferable.

Moving on to the second part of his range:

I think it is fair to assume that just as in PART 1 the second part of his range will do better by calling than jamming although the exact EV needs to be calculated. I will skip this part for now and maybe will do it later. And now i want to point out some conclusions that we've come so far:

1. In Part 1 i looked at EV of pot/calling turn. It turned out that villain would jam his strongest hands and fold the rest. In this case i would set up a range for potting to maximize my ev. Pot/calling the exact hand that i have KKxx is +ev.

2. In Part 2 i looked at betting $13 and set up bet/folding and bet/calling range for us and then found the optimal strategy for villain with his strongest hands (SET>). Also i found that EV of calling (SET>) is much lower than in the case when i have stronger range (for pot/calling) which means that im able to steal the pot more often with my weaker hands.

3. Probably betting smaller amount will net me more profit than pot/calling both with my range and with KKxx. This comes from the fact that if i was able to outplay stronger part of his range on the river then i would at least not lose to the weaker part of his range compared to the case when i pot and pot is bigger on the river.