This is something I see often enough in threads or hear at the table and the logic strikes me as wrong.

Let's assume the pot contains some tiny, insignificant amount. Player X bets $1 and Player Y raises to $3. Player X then states: "I had 2-to-1 pot odds so I called", implying 33% equity is enough to make the call.

But what about that $1 he's already put in the pot? Let's assume re-raising is not an option. Don't his choices come down to 1) fold and forfeit $1 or 2) call and risk $3 to win $3? It would seem to me he really needs 50% equity (including any implied odds he feels he might have) to call, not 33%.

Thanks in advance.

You're implying the $1 is still yours. If that is the case, you could take it back. You can't.

Thanks, Kelvis. Maybe these are not related concepts, but what triggered this post is something from the Mathematics of Poker. On page 83, they discuss this situation:

Player X has AKo. Player Y has QQ. Stacks are $800. Player Y posts a $100 blind. Player X raises to $300, and then player Y reveals his QQ. Player X knows he has 43% equity. The book then states: "... if [player Y] raises to $800, player X will be forced to call... Raising to some lesser amount will not be better, as X can simply reraise to $800 to guarantee his equity there."

That last part seems to me to imply that the betting over the street has to be taken as a whole, not in parts. In fact, that's how they calculate expected value (at least, so far).

A practical reason for the question is deciding when I have the implied odds to set mine in a 3bet pot. If stacks are $16, I open-raise to .50 and get 3bet to 1.70, I've always considered the investment to be $1.70, not the difference between my open and the 3bet ($1.20). Is that incorrect?

Yes that is incorrect. The money you put in is no longer yours. Else you could have taken it back.

Thanks. I think what's going on here is I'm mixing up equity and expected value.

Okay, I think I see where I went wrong. In that passage from the book, player X knows he has 43% equity and $500 behind. The pot contains $400 and player Y has $700 behind. By maximizing equity, the authors mean here that player X wants all the money in the middle to maximize the value he'll get if he hits his A or K. He would risk $800 to win $900. His sweet spot would be around $1060. After that, he need fold equity, too.

So I was indeed mixing up EV with equity (and therefore pot odds).

Thanks for helping me clear this up. It's been quite an adjustment since I got Equilab and started thinking of hands in terms of equity. I feel like I am relearning what I already knew before in some ways lol.

One way to define expected value is that it is the change in stack size resulting from your decision. The stack size can be your stack at the time of the decision or the stack size at the beginning of the hand, two common choices. The important point is that you are comparing EV for various decisions, such as call or fold, and choose that which maximizes EV.

So, if the baseline stack is at the decision point, then for a bet or call, any money you bet earlier is not considered as a loss if you lose but is considered as a gain if you win. If you fold, your EV is 0 for your stack size does not change.

If the baseline stack is that at the beginning of the hand, the money bet earlier is considered a loss if you bet and lose, but is also considered a loss if you fold. So the EV difference between betting and folding is the same for both cases.

Thanks for the reply. One of the aspects of EV that interests me is that it can cover multiple decision points. Also, all the examples I've seen so far consider all the possibilities until the end of a hand (only one person is left). Seems that I could even calculate the EV of open-raising AA preflop.

So if I want to know the EV of betting OOP on the turn in a particular situation, I would have to consider and add up all the outcomes: 1) I bet, you fold, 2) I bet, you re-raise, I fold, and then for all the scenarios on the turn ending in a call I'd have to go and consider the outcomes on the river based on my equity on the turn. Not sure how you'd quantify the chances of an opponent folding other than using fold equity, which is a fuzzy value at best.

I'm curious if good players actually calculate EV off table when analysing previous sessions.

To set mine vs a single villain is rarely a good idea.
There can be exceptions such as he has a huge stack and you have reason to believe he is likely to get a big part of that stack in the middle.
You are only going to hit you set about 1 in 8 times so do the math.

If V is aggro and will stack off with an overpair, I like to consider it. I play in the micro's and live 1/2 where such players are not uncommon. For set mining I've read 15x IO is a good general rule and for more speculative hands 25x. Even with the RIO involved in set mining in a 3bet pot I think sometimes it is worth it.

I used to precise EV calcs, but only so I got the general idea over which plays are profitable or not. It didn't take too long to learn which sort of spots were very profitable or very costly, and which are close (unfortunately most poker decisions are close, but as a beginner you just need to make the "obvious" folds and "obvious" calls to beat the fish). I no longer calculate precise EVs (except for answering forum questions for specific spots like calling a pre-flop shove with a specific hand vs a particular range), as all I care about is whether a play makes money or not. I don't need to know how much money a play makes. I just need to know whether or not it makes money. If it's profitable ("+EV") I do it. If it's -EV (loses money), I don't.

To use a quick and extreme example that might help to explain your earlier questions too...

Imagine you're in a tourney with 1,500 chips and the blinds are 25/50. Suppose that instead of minraising to 100, you misclicked and made it 1000 with 76s for example. That huge raise size would be a mistake and you'd wish you hadn't made it and could take back the chips and try again (it was a misclick after all). It would be a -EV open (otherwise everyone would be opening to 2/3 of their stacks with 7-high). If villain then shoves on you (for 1500 total), you should "forget" about your earlier misclick and do a rough EV calc of the new situation. You only need to call another 500 into a final pot that will be over 3000. You don't need to know your precise equity or EV. You just need to know that based on pot odds alone, you only need to win 500/3000 = one sixth of the time to break even. Since even 72o will beat AA often enough to break even at that price, you'd know it was +EV to call off with 76s. To emphasise, you don't need to know precisely how many chips you'll win/lose (i.e. the expected value) if you call with 76s vs aces (or villain's shoving range). You just need to know whether it's a good call or a losing one.

The same sort of rule applies for post-flop. You might start a hand with 1000 chips, but due to it being multiway there might be 4000 in the middle by the river, and villain might be going all in for 20% of the pot or less. It doesn't matter how the money got into the pot, or who put it there. You just have to work out whether your call will make money or not. With "sick pot odds", you should often call very light, because you don't have to win very often to make money.

Thanks, Arty. Great feedback, as usual. Online I find these spots come seldom, but live much more often.