Sizing bets facing what is either a draw or a made hand

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.

Use ranges, not exact polarized hands to decide what to do. Remember, if he has a set he should be afraid of the flush draw.. so bet/folding like 66% pot is usually a pretty safe play in these spots.

I kinda did it on purpose to make this a typical "way ahead or way behind situation", as these happen quite regularly.

It's true that villain should be afraid of a flush draw and he makes a mistake by slow playing his set if that is what he has. But we see that a lot in practice don't we? Lots of villains even slowplay overpairs at the micros.

I don't know about betting 60% pot. If we don't give implied odds (if a flush card comes on the river, we are 100% beat here), then 35% pot is enough to price a flush draw. Why would we bet more than that in any case?

I would like to rephrase this question, to see if it attracts more interest fomulated like this:

Heads up in position on the turn, your opponent checks to you. You put him on a range against which you have less than 50% showdown equity, and no fold equity (he is a calling station and does not fold draws).

Is it generally correct to bet just because his range includes a significant amount of draws, even though you are an underdog to the range as a whole?

Sorry to insist with this, but this is a spot in which I find myself regularly. If this has been answered before, I would love a link to the thread or article in question.

Think about which ranges respond to your bet, his range beforehand is not relevant. Always consider the entire range for a given path of action. That's the whole point of using ranges rather than distinct hands

You're putting money in when you're behind your opponent's range. How is this confusing?

Also note that there is a lot more to the EV calculation than what you have put into it because you haven't included river play at all.

To be fair, I included this in the original assumptions:

let's say he always calls our bet and never raises, and that no more money goes in on the river no matter what.

Maybe simplistic, but not totally unrealistic. Many opps do not fold draws at the micros and a passive opponent may not raise a set or may keep slowplaying it by just calling.

As for the river play, it is not totally unrealistic either to assume that no more money goes in no matter what: if a card completes his draw, I am certainly not putting more money in because my equity just became close to null. If the river card does not complete his draw, he may bet with a made hand. I am still behind his range so I won't call. He might also bluff with a busted draw, but let's exclude this possibility for the sake of simplicity. Then in case we hit our set on the river, we might take his whole stack, but we likely do so whether we bet the turn or not, so these implied odds do not change dEV/dx, so that does not give us an extra reason to bet the turn.

So Spoon, are you saying no bet because we are behind the range then? So in the JJ hand I posted yesterday, would you not bet the turn? (http://www.flopturnriver.com/phpBB2/...jj-t92111.html)

Sorry if there are mistakes or dumb things in what I said. Trying to learn here.

The point is you can't value bet profitably against a range that beats you. In the JJ hand you linked to, you have to put your opponent on a range and decide your equity against that range etc.

Thanks for your reply Spoon. I did put my opponent on a range in the JJ hand analysis. While playing, I tended to weigh his range more towards draws and maybe a weak Q that a second barrel could possibly entice to fold.

After rethinking his ranges in the analysis and stoving them, I concluded that my equity was not as much as I first thought and wondered whether I should have bet the turn in the end. Other more experienced players seemed to say I played it fine, but I would value your opinion on that particular hand, since the topic of the present thread applies relatively well to that case.

So in clear, for that JJ hand:
1) if the range I put him on on the turn in my post analysis was reasonable, according to what you said above, I should not have bet the turn, right?
2) do you think the range I put him on on the turn is reasonable?

I appreciate your patience with my noob questions.

Btw the original question is not a way ahead way behind situation. It's a slighty ahead, way behind situation and therein lies all the difference. So if you have 10% equity 50% of the time and 65% equity 50% of the time you have a combined total of 37.5% equity - i.e. you are behind his range so checking is correct (as your EV calc showed)

And yeah, ranges and EV for sub ranges combined into a neat EV number for lines you could choose to play against his range anticipating specific reactions to specific betting lines. All you can do is decide on a line, decide what is reasonable that the opponent does and start crunching numbers.

Originally Posted by Erpel

Btw the original question is not a way ahead way behind situation. It's a slighty ahead, way behind situation and therein lies all the difference. So if you have 10% equity 50% of the time and 65% equity 50% of the time you have a combined total of 37.5% equity - i.e. you are behind his range so checking is correct (as your EV calc showed)

I have 10% equity 50% of the time and 80% equity (not 65%) 50 % of the time. So my combined equity is 45% and I either have a big equity or a small one, does that not make it a way ahead / way behind situation? Or put another way, am I not up against a polarized range?

We're on the turn, not the flop, so with an overpair against a flush draw (1 card to come), 80% is about right. I made a mistake in the 10% equity against the set though: it should really be closer to 5%.

I understand the bit where checking is correct when I am behind the range altogether, even if that implies giving free cards to draws. Thanks for clarifying that, that was the whole point of my original question.

Please take a look at the JJ hand I linked to above, I think it is a practical situation similar to the theoretical one in this thread. Would like to hear your opinion on that one.