2-7 EV question.

[InWithTheBest]

Had a spot come up today thats pretty standard but made me think quite a bit. I suck too bad at math to figure this out for myself.
Assume a 3 handed game, I raise from the SB, BB defends we both draw two. I improve and bet he calls, we both draw 1 on the 2nd draw i make 86532 and bet he calls and stands pat behind me. Obviously I win here everytime.
Hes a fairly aggresive opponent and I was really surprised to see he froze me with 97632.
Now obviosuly as far as variance goes, him standing pat with a loser is a great thing because im always winning an $800 dollar pot here ($80-$160) and he is always calling a river bet.
As far as long term expected value I was wondering if you math guys could tell me how much better is it for him to raise me, assuming Ill always 3 bet and he will always break and be drawing live in a big pot. With the additional bets going in the pot is now going to be $1450 or so before the river. As far as the river goes, assume Ill always bet/call and that he will call my bet if he catches an 8 or another 9.
Let me know if I need to clairfy my question. Thanks!

[tablecop]

With 7 outs to a win and 7 outs to a crying/losing river call and no knowledge of the discards (42 card stub), his EV is exactly the same under both scenarios as described (assuming no increased rake from a 7 bet pot to a 11 bet pot). we can make some assumptions about the stub (he probably didn't discard any 4s or 5s etc) but we've started with some unrealistic assumptions anyway so i don't know that pursuing this any further will be any more illuminating.

[InWithTheBest]

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but we've started with some unrealistic assumptions anyway so i don't know that pursuing this any further will be any more illuminating.

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Which assumptions are unrealistic? And my question was more geared towards having 100% equity in an $800 dollar pot vs X amount of equity in a $1450 dollar pot.

[MarkGritter]

It's most appropriate to think of this problem in terms of hand ranges rather than the particular hands you guys held... let's try to describe a simple set of competing strategies:

Say that you are drawing to 8532x. With a 4, 6, or 7 you bet/3-bet, pat, and bet the river. With a 9 or a T you bet/break and bet the river if you hit, check/call if you are pat. Let's say that you will also snow with a K about 25% of the time, playing it like a 4. There are 3 bets already in the pot when you bet.

What then is your opponent's best strategy with a 97632? Since you bet you have one of 18 cards (4444666777 999TTTT K)

10/18 of the time he is behind and ought to draw
7/18 of the time he is ahead and his raise will induce a break
1/18 of the time he is ahead and his raise will induce a 3-bet to which he should not break

It looks to me that freezing is the best strategy, and it's not even close. 10/18 of the time he loses 2 BB and the rest of the time he makes 5BB (3 in pot + 2 additional bets if he bets when you check a 9 or T.) This averages out to +1.11BB.

Raising nets him -1.7BB to -1.5BB (not -2) when he is behind but letting you break your 9 or T costs him, so that he only earn about 3.1-3.2 BB in those cases. The average works out to +0.36BB.

For comparison, just breaking all the time earns the BB only +0.27BB.

(Of course, this is a very simple model, and SB may have a significantly better strategy...)

Fiddling with the parameters a bit:

If the stub has 38 cards instead of 42 then just breaking is superior to raise/breaking (assuming the # of outs has not decreased) but it is still much worse than freezing.

If you never snow with a K, freezing is only worth +0.88, but it's still much better than the raise.

Freezing is still better if you check/fold your 9's and T's.

[InWithTheBest]

ty mark this makes a lot of sense.. one more question how much equity does a 7632x have on the last draw against my hand?

[DeathDonkey]

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ty mark this makes a lot of sense.. one more question how much equity does a 7632x have on the last draw against my hand?

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This one's easy. He has 7 outs. The 4 fours and the 3 fives left in the deck. If I remember the OP right 16 cards have been seen that aren't 4's or 5's (except the one in your hand) so he is 29:7 against beating you which is just worse than 4:1 so probably 19% equity.

-DeathDonkey