Pot Odds Paradox in Short Stack

Question

Most if not all poker books/tutorials suggest that a good play is one that causes your opponent to do mistakes regardless of whether you end up winning the hand. For example, after the turn, you have the best hand, you know your opponent is on a flush draw (let's assume you're positive about that), the book says that you should bet your opponent a certain amount to force them make a bad call. I agree with all that. But, let's say you're short stacked in a way that your all-in after the turn would be one fourth the pot, which is a good price for your opponent's flush draw. So you think of the situation breaking it down as the following:

"What do I do? will I go all in? well I know my opponent is a good player, he knows it's a good price for his flush draw, he'd snap call. At the long run, that is a bad move, I'll end up losing more than winning in similar situations, so all-in is not the right move here, it's the sucker's move, and I don't wanna be the sucker. OK, then I'm gonna check. But wait! checking gives him a great price to beat me, the best price indeed. If I bet, there can be a chance that he folds (although it's not likely since he's a good player), so checking is extremely wrong in every aspect. On top of that, I've got the edge, we're both pot committed, so the situation I'm in right now can be considered a separate game, therefore, the right move is definitely not to check, checking is the sucker's move, I should bet, well, I'll go all in."

OK, after all that boring chain of thoughts, I got no answer. I assume that one answer has to be better than the other since after all, they're not identical (especially from the mathematical standpoint). So which one if the right move here?

Answer 1

Your "paradox" arises from the fact that aside from your bet, the pot contains enough expected value already for each player that neither could improve their expected ending stack by folding. With too small of a stack, you can't bet enough so that the opponent loses money. However, with your bet you can still reduce the expected overall gain from his point of view.

To illustrate, say the pot is 100 chips, you have 20 chips and your opponent has 100 chips remaining after the turn. You know he has a 20% chance of beating you on the river.

By checking, your EV from the pot is 80 chips; for your opponent it's 20. Your expected ending stack is 100 and his is 120. Everybody's happy, right?

Now what if you bet your 20 chips and he calls? Your EV from the new pot of 140 is 112, which will be your ending stack--an improvement! His expected ending stack will be (.20 * 140) + (80) = 108. It was beneficial to you and detrimental to him to have that bet compared to a check. However, his stack of 108 is still better than the 100 chips he's sitting with, so it's +EV for him to call--but not AS positive as it would have been to not face a bet. As a short stack, you can't force him to make a negative overall EV decision, but you can reduce the magnitude of his positive EV. The pot is big enough to essentially give him some "free" added value.

Ok, now let's say you had 40 chips instead, you bet it and he calls. Your expected ending stack is .8 * 180 = 144 compared to 40 that you started with --an improvement of 104 chips...hmmm, that's more than was in the pot to begin with. His expected ending stack is now (.20 * 180) + 60 = 96. Now he's losing money by calling. And that money is going directly to you, hence the improvement of 104 chips which was more than the pot.

To sum it up, your paradox comes from comparing an apple to an orange. The EV of the bet and call in and of itself (not considering the whole pot) can't be compared to the overall EV of the hand that does consider the whole pot. As a short stack, you can improve your EV and you can decrease the EV of your opponent compared to to EV of not betting (or betting less)--but you can't put him to an overall negative EV decision. With more chips, you can get over that break-even point (from his perspective) to end up benefitting yourself even more.

Answer 2

if your bet leaves you with the stack less than the bet itself, you should have gone all in on the flop. In general, if your bet takes the third of your stack you have to go all-in.

Answer 3

We need a bit more information. Starting stacks, bets pre-flop etc. From what it sounds like so far you should have pushed all in pre-flop or after the flop.

One thing I disagree with however is when you said that in the long run you would lose money to a flush draw. If you are positive you have him beat and the only thing that will save him is if he hits a card making his flush you will win more times than you will lose. He has at most 9 cards that will hit his draw and only one chance to get it, that is about 18%. So 82 % of the time you are winning.

I would make that push/shove all day long if you're positive they are on a flush draw.

Answer 4

When you are short stacked you unfortunately don't have the chips to force a bad decision.

Accept the opponent is not going to fold. 1/4 pot bet is not going to get them off a flush draw.

If you are short stacked then you need to look at it as you are getting 4:1 and you are not going to get a better chance to get your money in.

If you held back and drew AA the next hand you are not even getting 4:1 against two random hands.

If there is no fold equity then you don't need to go into EV

What does it take to price someone out of a flush draw on the turn?
4 on the board and 2 in the opponents hand. 52 - 4 - 2 = 46 cards out
9 cards left to make a flush
So the odd to make the flush is 9 : 37 = 1 : 4.11
What is interesting here is you know there are no flush in your hand so the actual odds are 9 : 35 = 1 : 3.89
What bet gives your opponent exactly 0 reason to call
-bet + 9/37(bet + pot) = 0
9/37 pot = (1 - 9 /37) bet
9/37 pot = 28/37 bet
bet / pot = 9 / 28 = 32%
You have to bet right at 1/3 of the pot for call to be a wash
Your opponent should call anything up to 1/3 pot bet by you
Your bet is in the pot
(4/3) / (1/3) = 4 : 1

Anything you cannot bet up to 1/3 the pot is lost opportunity. That is money you could get 4:1 on from any rational player (on a flush draw).

What about your odds based on you know you don't have a flush card
bet / pot = 9 / 26 = 35%
You actually want to bet a little more than 1/3 the pot to get them to decide to come off the pot
They are (typically) better off than they think they are

What is your EV on sizing your bet? What if you bet 1/2 the pot and they will call 1/2 the time? You have to subtract the sure call. What is the EV of the over bet. 1/2 the time they fold and you get 0 from the bet.
.5 * pot * 0 + .5 * pot * (26/35) - 9/28 * pot * 26/35 = (5 / 28)(26 / 35) pot = 13% the pot
Many people will get emotionally attached to a hand call a slight over-raise
Against a player married to their flush draw I will bet 1/2 the pot on the turn (and flop)

If you hold one of the flush cards then their odds are down to 8:36. Bet closer to that 1/3 pot to not chase them as you are making money on a 1/3 pot bet. But if you are holding a flush then less chance they are on the flush draw. Your premise is they are on a flush draw. Putting a player on A hand is very dangerous. What if they are on trips and you bet 1/2 the pot and you get a turn call? What do you do when they push on the river?

Answer 5

To my opinion you shouldn't have let this happen that your all-in is only 1/4 of pot. You should have been already all-in on flop or even preflop. If you are short-stack you should either push or fold, as you cannot afford yourself to just see the flop and bet normally.