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 Originally Posted by spoonitnow
Optimal Exploitative Value Betting
Let Villain always call a value bet with n hands such that the worst hand he calls with is h_n, and that he never folds a hand that beats h_n. Then the EV of betting h_k such that 1 <= k <= n is:
...
(1+n)/2 >= k
So we should bet all hands h_k such that k >= (1+n)/2.
There is a problem here where you invert your inequality, it should be k <= (1+n)/2
Villain has n hands in his range and h_n is the worst, so h_1 is the best, and in your example, we should bet h_k with k >= 3 so QKA (not k <= 3)
So on the same basis, EV of bluffing with h_k and n < k <= T:
P(T-n-1)/(T-1) - n/(T-1)
EV of checking is the same as before:
P(T-k)/(T-1)
So bluffing is better or equivalent to checking when:
P(T-n-1)/(T-1) - n/(T-1) >= P(T-k)/(T-1)
PT-Pn-P-n >= PT - Pk
k >= n+1+n/P
So in your example, with a deck of 13 cards and villain calls with T+ (n=5), if the pot size is:
P=1: k >= 11. We can bluff profitably with h_11, h_12 and h_13 or 4, 3 and 2 (although betting 4 is equivalent to checking it EV-wise)
P=2: k >= 8.5. We can bluff profitably with {h_9-h_13} or 6-2.
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