Optimal Exploitative Value Betting
Optimal exploitative ________ is a game theory term. It essentially means that we are making every play, in this case a value bet, that is not -EV. We're going to use a slightly different game that uses T cards instead of 13 so we can generalize the solution. This is nice because it doesn't change the proof at all.
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)(k-1)/(T-1) + (1+P)(n-k)/(T-1) + (P)(T-n)/(T-1)
And the EV of checking h_k is:
(P)(T-k)/(T-1)
The the EV of betting h_k is greater than or equal to the EV of checking h_k when:
(-1)(k-1)/(T-1) + (1+P)(n-k)/(T-1) + (P)(T-n)/(T-1) >= (P)(T-k)/(T-1)
Now we multiply through by (T-1) to get:
(-1)(k-1) + (1+P)(n-k) + (P)(T-n) >= (P)(T-k)
-k + 1 + n - k + nP - kP + PT - nP >= PT - kP
-2k + 1 + n >= 0
1 + n >= 2k
(1+n)/2 >= k
So we should bet all hands h_k such that k <= (1+n)/2.
This is a fancy-looking way to say that we should bet when we have 50% equity or more against our opponent's calling range. For example, what if our opponent calls with a T or better:
A K Q T J
1 2 3 4 5
Then n = 5, and we should bet all hands such that k <= (1+5)/2 = 3. So we should bet all hands h_k such that k <= 3, which means we're betting an Ace, King, or Queen.
Conceptually
When we're value betting the optimal exploitative frequency, then by definition we open ourselves up to adjustments. The basic adjustment our opponent could make to lower the EV of our value bets is to call less. Our two basic adjustments to protect ourselves from adjustments are either to value bet fewer hands, or to value bet more hands. Each is -EV for our range in a vacuum, and are -EV long term if our opponent never adjusts correctly.
By value betting so much, we have to bluff some non-zero amount to make life difficult on our opponent. You could do the same type of analysis above to determine the optimal exploitative bluffing frequencies, but I'm going to leave that to the reader to work out.
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