[Buzz]

Chris - Don’t know as it matters very much, but I mis-read your post when I originally wrote “It's not clear to me exactly how much is already in the pot.” Nor do I know how I came to mis-read your very clearly written post. My apologies.

There were 6 large bets in the pot after the first betting round, then 10 large bets in the pot after the 2nd betting round with three opponents.

"Flop 7s6s6h
No high for me. I'm playing for 1/2 or 1/4 of the pot. EP bets, same MP raises. 4 call, including me. 10 big bets. Should I have considered letting this hand go?"

I earlier wrote, “Odds are thus about 11 to 10 you'll make the nut low, close to even. But because of the strong possibility of getting quartered you should fold here. However continuing is only a small mistake, IMHO.

That still seems correct to me, except that a small mistake on an early betting round tends to be compounded on later betting rounds. If you make a small mistake by calling on the river, that’s the end of your small mistake. However, if you make a small mistake on an early betting round, you may be compounding your error even when (because of the size of the pot) you correctly call on subsequent betting rounds.

Since I had originally mis-read your post, my numbers and consequently my thinking has also changed a bit regarding the third betting round.

On the third betting round, your three opponents were putting in a sub-total of 6 more big bets making a total of 16 large bets in the pot. Your cost (assuming no re-raises), would be 2 large bets to see the fifth common card. With no reasonable chance for high, You were getting 3.5 to 1 pot odds to call.

Let's look ahead and tentatively assume that if you make low on the river, there will be one bet on the river and that two of your opponents will call. In that case there will be 21 big bets in the pot at the showdown, of which you will have contributed 3 big bets on the last two betting rounds. In that case, (1) when you are the only winner for low, you will win 7.5 big bets, (2) when you are quartered for low you will win 2.25 big bets, (3) when you are sixthed for low you will win 0.5 big bets, and when you miss making the nut low on the river you will fold and lose 2.0 big bets.

To make the nut low on the river, you need 3, 4, 5, or 8 on the river. The probability of catching a 3, 4, 5, or 8 on the river is 16/44 = 0.364. You should expect to make the nut low 36.4% of the time and miss 63.6% of the time.

Against random opponents, your A2 should expect to be the only nut low 57%, get quartered 38%, and get sixthed 5%. The last time I did the calculation for your hand/flop, since you strongly suspected one of your opponents might have A2, I used 50% for you being the only nut low, 50% for your getting quartered and ignored the sixthing. There’s not a big difference between using 57%/38%/5% and using 50%/50%. No big deal. This time I’ll use 57%/38%/5%.

If you had nine opponents to start, you should expect to win:
7.5 big bets 57% of 36.4% of the time,
2.25 big bets 38% of 36.4% of the time and
0.5 big bets 5% of 36.4% of the time.

You should expect to lose
2 big bets 63.6% of the time.

Putting it all together, your e.v. is:

7.5*0.57*0.364 + 2.25*0.38*0.364 + 0.5*0.05*0.364 - 2*0.636 = +0.6 big bets.

As I already wrote, (odds-wise) it's close - and the revision doesn’t change much. I agree with Chaos that since there is a chance for a re-raise on the third betting round, it’s probably best to discontinue at that point.

As an aside, I do not think anyone can mentally do the above calculations at the table with a few possible exceptions. When I was growing up there was a radio program called the whiz kids. One of the regulars was a kid named Joel Kupperman, who performed amazing feats of mathematics in his head. Maybe a Joel Kupperman could do the math while seated at the table.

The rest of us can only prepare ourselves by making doing some sample math and making some generalizations ahead of time.

If we back it up to the third betting round where you were getting 3.5 to 1 pot odds to call, and recalculate as though you were only getting 2.5 to 1 pot odds to call, there will be 17 big bets in the pot at the showdown, of which you will have contributed 3 big bets on the last two betting rounds. In that case, (1) when you are the only winner for low, you will win 5.5 big bets, (2) when you are quartered for low you will win 1.25 big bets, (3) when you are sixthed for low you will break even, and (4) when you miss making the nut low on the river you will fold and lose 2.0 big bets.

Putting it all together, your e.v. on the third betting round would be:

5.5*0.57*0.364 + 1.25*0.38*0.364 + 0*0.05*0.364 - 2*0.636 = +0.0 big bets (if you were only getting 2.5 to 1 pot odds to call on the third betting round).

Thus the break even point with three opponents on the turn seems to be odds of 2.5 to 1 (against making your hand), the point where e.v. = 0. Looks like we maybe can call on the third betting round when we are getting 2.5 to 1 or better pot odds.

Facing a single bet on the third betting round, drawing for low with a bare ace deuce, and with no chance to win for high, then there have to be six big bets already in the pot for us to get 2.5 to 1 pot odds and call. Facing a double bet on the third betting round, drawing for low with a bare ace deuce, and with no chance to win for high, then there have to be a minimum of twelve big bets already in the pot for us to call.

When there is a pre-flop raise seen by at least six players, when there is a single bet on the flop seen by at least four players, and when facing a double bet on the turn (you must have at least two opponents to be facing a double bet), then the odds will be roughly break-even for calling the double bet. Anything over that amount already in the pot is gravy. However, if the pot is further raised after you call the double bet, although you will be getting favorable odds to call the additional raises, you were not getting proper odds to originally call the double bet.

This is a dreadful way to get stuck in a hand. I think we have to carefully avoid putting ourselves in harms way. This happens when you face a raise after calling a single bet either (1) without quite having proper odds, or (2) having barely favorable odds. Although you can play the hand perfectly from that point on, your error may be compounded on later betting rounds - and there is not really anything you can do about it! You’re stuck in the hand because of the size of the pot.

Thus when it looks like the betting will be jammed on the turn, even though you might be (barely) getting proper odds to call a double bet, I think you fold your bare ace-deuce nut-low draw, as Chaos has already suggested.

Without the raises on the first and second betting rounds there wouldn’t be enough in the pot to justify seriously considering calling a double bet on the third betting round.

Just my opinion upon re-thinking my earlier post.

Buzz