[ QUOTE ]
Just curious, what kind of ranges did you give to villain? Does the 67% rule apply here? It feels like when he c/rs the river he's bluffing more often than he has a hand.
[/ QUOTE ]
i can't be sure of course but i assumed it roughly does. you say that he raised and c/r'ed the river in the past as a bluff. now we have to decide whether he will bluff-raise you more or less than optimal here. on one hand it seems that he will bluff raise too often since he's shown himself capable of it. but on the other hand, thinking players usually stop bluffing for a while after they have been caught because they know nobody trusts them anymore. so i guess it depends on additional details about the circumstances in which he bluff-raised in the past - how long ago was it, against whom, etc. ultimately it is up to you to evaluate this dynamics.
but if we assume the two factors i mentioned roughly balance, we can probably say he bluff-raises close to optimal on average, so we can apply the 67% rule.
but note:
1. 2/3 equity is a simplification which doesn't include his correct bluff check-raises. a better formula, assuming he always raises hands that beat you plus a correct number of bluffs and he never folds a better hand is:
1-1/(3-2/(pot+3))= equity vs. his non-folding and non-bluffing hands.
in our case pot=6 so we get 64% instead of 67%.
2. in reality there may be some better hands that he's not raising, like 88. then you can have slightly less equity vs. his range, or slightly more if we exclude those hands. the correct formula now becomes:
1-1/(3-2/(pot+3)-F)= equity vs. his non-folding and non-bluffing hands,
where F stands for the fraction of better hands that he just calls with instead of raising.
i agree with oink's estimate for his range. we should discount 88 and 99 of course. let's discount them by 50%, then the better calling hands are:
50% 88, 50% 99, 100% 66
or equivalently
100% 88, 100% 66
his complete range of non-folding non-bluffing hands is 88,66,44-22,A9s-A5s,K7s,K5s,Q5s,J7s,J5s,97s,87s,65s,54s,A9o-A5o,K7o,K5o,J7o,97o,87o
(for now let's assume he'll fold the weakest A highs)
F in this case is 0.23 if you count the combos. plugging into the formula, we get the necessary equity of 60.8%.
plugging into pokerstove, we get:
<font class="small">Code:</font><hr /><pre>
Board: 5d Th 7h Qc 7s
Dead:
equity win tie pots won pots tied
Hand 0: 56.349% 53.97% 02.38% 68 3.00 { Ac5c }
Hand 1: 43.651% 41.27% 02.38% 52 3.00 { 88, 66, 44-22, A9s-A5s, K7s, K5s, Q5s, J7s, J5s, 97s, 87s, 65s, 54s, A9o-A5o, K7o, K5o, J7o, 97o, 87o }
</pre><hr />
so it appears we should check under these assumptions. if, however, we assume that he calls with any ace high, we get 66% equity and a value bet is justified.