Let's say for the sake of the example that we are heads up on the turn, in position. The pot size is P and villain checks to us. We have an overpair to the board. From his actions, we put villain 50% of the time on a set, and 50% of the time on a flush draw. To simplify, let's say if he has a set, our equity in the hand is 10%. If he has a flush draw, our equity in the hand is 80%. How much should we bet?

If he has a set, obviously we should bet nothing. If he has a flush draw, we should bet as much as we think he will call unprofitably with.

How to find x, our ideal bet size? Again to simplify, let's say he always calls our bet and never raises, and that no more money goes in on the river no matter what.

From Sklansky's No limit theory and practice, "when your opponent could have a range of draws from weak to strong, you are going to have to size your bet allowing him to draw profitably with his strongest draws."

My question is basically: does this apply in this case, where he does not have a range of draws, but rather he has either a made hand or a draw? If it does, then I calculate the EV of betting x as follows:

EV = 0.5 * ( 0.1*(P+x) + 0.9*(-x) ) [case he has a set]
+0.5 * ( 0.8*(P+x) + 0.2*(-x) ) [case he has a flush draw]

and I can calculate that dEV/dx = -0.1. So when x increases, my EV decreases. So to maximize my EV, I should bet 0.

Intuitively, this is correct, because we are ahead 50% of the time and we are behind 50% of the time, but when we are behind, we are behind more (90% equity for villain) than we are ahead when we are ahead (only 80% equity for us).

So the math here seems to tell me I should not bet, because that maximizes my EV.

Still, this disturbs me. Should we really not bet in such a case and give him a free card when he has a draw? This feels wrong somehow.