I have a question about expected value in situations where you are ahead, but you give your opponent the price to draw on you. For example, lets assume i have AK on an Ac 5c 7h board with a pot of $100. For simplicity lets assume I know my opponents hole cards are exactly 10c Jc. If my opponent has $30 it is impossible to price him out of his flush draw because he's going to be getting 4-1 if i put him all in. If he is always correct to call my all in and is making a positive ev play does that mean my bet is -ev? Obviously my chances of winning the hand are much better and i'm happy to get my money in here but how is it that we can both make a +ev play? If he has more more and all other future betting rounds are checked is betting any amount that gives him the price to call a -ev play on my part?

This is a pretty common misconception.

Ok, in the example you gave:

You have ~66% equity in a $100 pot. If all betting stops here, you average a win of $66.

Now, you bet $30 and he calls allin.

You have 66% equity in a $160 pot = $106.
$30 of that is what you have put in yourself though, so subtract that. You make $76 overall.

You make $10 more by getting the money in than just by checking it down.

EV is not a zero sum game.


There are many times where both players are +EV with a bet and a call.

It's very rare to see a bet, when you're ahead, be a -EV move...it's typically when you are against multiple opponents, and all their combined outs take up most, or all, the remaining cards in the deck.

ok now lets assume i check the flop and he always checks behind. If i put him all in on the turn he no longer has the price to call. By checking the flop i can price him out on the turn, so does that mean he should always go all in on the flop? If he doesn't go all in on the flop is it more +EV to check the flop and then push the turn? I lose the extra $10 of equity from the flop but i gain a lot of equity on a blank turn. In a situation where i know his exact cards it seems like it would be correct to check one street here hoping to give him a worse price on the turn and calling if he shoves the flop. If i don't know his cards i think i have to call the turn if a club comes so its a little different but could i take a similiar line there.

He's still right to call a push on the turn in the example. He'd be getting 130/30 = 4.33 : 1 from the pot with a 4 : 1 chance of improving.

Should we maybe give him $50, so he's correct to call on the flop but wrong to call on a brick turn?

Usually in this situation, when the villain has $30 left and the pot is $100 dollars HU, you've often made him make a -EV play preflop anyway. Now, like Simo said, you can now extract more profit by putting him in and getting him to call with a smaller chance to win than you, even if his call is +EV.

I think if we play the cards face up, then it's a little better to bet the flop with $50 behind.

If we bet and he calls the flop, then we have 2/3 equity in $200 or $133, minus the $50 investment, for a net of $83.

If we wait for a brick turn, then we bet and he folds correctly. 4/5 of the time we win $100 or $80 on average. The other 1/5 of the time we fold when the flush completes.

If there's an elegant generalization, I'm not sure what it is.

No.

They're different choices. If the villain could decide for you not to bet, that would be +ev for him.

if this is the case then wouldn't you not be ahead? unless you mean its rare for a bet to be -ev when you currently have the best made hand?

I think by ahead he means currently has the best hand, but may still be an underdog to have the best hand at the river.

Yeah, but the price is right to proceed with the all in.
There's no way to be ahead, yet drawing so slim to dead cards for the villain that the price is not right here.
I don't know how else to say that.