[jbrennen]

If we can confidently put Seat 2 on starting with a three-card 7, there are 118 possible combinations of down cards he could have when the action comes to us on 5th street. We're ahead of 36 of those combinations.

So very roughly, we're ahead 30% of the time here. Figure that whichever of us is ahead is about a 2:1 favorite. Also let's give Seat 6 a charitable 10% chance to win a 3-way if he stays in.

Also assume that Seat 2 only reraises if he's ahead, and that we only cap if Seat 6 comes along.

Here are the EV deltas for raising vs. calling on this street alone, not counting implied odds:

We're ahead, Seat 6 calls:
By raising, we put an extra 1 bet into the pot. Our extra equity is 60% of the extra 3 bets, so 1.8 bets equity. Raising is +0.80 bets EV.

We're ahead, Seat 6 folds:
By raising, we put an extra 1 bet into the pot. Our extra equity is 66.7% of the extra 2 bets, plus 6.67% of the existing pot representing the times that we only win because seat 6 folded. If we had called, the pot would have been about 8 BB, so we pick up about 0.53 big bets of equity that way in addition to our 1.33 big bets of showdown equity, giving a total of +1.87 bets equity, and +0.87 bets EV.

We're behind, Seat 6 folds:
By raising and calling the reraise, we put an extra 2 bets into the pot. Our extra equity is 33.3% of the extra 4 bets, plus 3.33% of the existing pot representing the equity from Seat 6 folding. We pick up about 0.27 bets from Seat 6 folding, and we have an extra 1.33 bets equity from the bigger pot, giving a total of +1.60 bets equity, and -0.40 bets EV.

We're behind, Seat 6 calls and everyone caps:
By capping, we put an extra 3 bets into the pot. Our extra equity is 30% of the extra 9 bets, so 2.7 bets equity. Raising is -0.30 bets EV.


So to summarize, if we raise and Seat 6 calls, our EV is +0.80 EV when we're ahead, and -0.30 EV when we're behind. Given that we're ahead about 30% of the time, we get +0.24 EV when we're ahead and we get -0.21 EV when we're behind, for a grand total of +0.03 bets EV.

If we raise and Seat 6 folds, our EV is +0.87 when we're ahead, and -0.40 when we're behind. Again, figuring we're ahead 30% of the time, we get +0.26 EV when we're ahead and -0.28 EV when we're behind, for a grand total of -0.02 bets EV.


So under all of these assumptions -- Seat 2 had a three card 7 to start with, that he plays 5th perfectly (only raising when ahead), and that Seat 6 has a 10% chance to win if he hangs around -- it basically doesn't matter whether we raise or call here, the EV difference is either +0.03 bets if Seat 6 comes along or -0.02 bets if Seat 6 folds.


Personally, I think that's an argument for raising here. First of all, Seat 2 may not play 5th "perfectly" -- he may call behind when he's ahead in some misguided deception or if he's trying to actually entice Seat 6 to put more money in the pot. He may raise when he's behind in an attempt to represent that he is ahead or to drive Seat 6 out of the hand. Secondly, Seat 2 may not have started with a 4-card 7 in the first place (admittedly not a very likely chance, but we only need about 0.05 extra +EV to swing it in our direction).