TL;DR -- depending on how you range your opponents, bet seems super profitable to me.
To add some rigor here:
Quote:
Originally Posted by BlindingLaser
Sure, we could be, but against six holdings from fish, like if you say they can have basically any two suited, any connected or one gapper down to like 53, any ace, any pocket pair...your hand is way ahead of the field on average on this flop...
After card removal on the flop, there's 1,081 combinations (47*46/2) of starting hands your opponents could have. I boiled down the "fishy" strategy above to be about 611 of those combinations -- could maybe take out some of the worst suited crap, but that also takes out the 94s combos that win, so I just left it all in. TBH, I feel like ~57% is about right for a fishy lineup that sees seven way flops regularly.
Of those 611 combinations, 37 beat us and 6 tie us. So call the ties worth half, so:
Code:
((611-40)/611)^6 = 66.6% we're ahead
Like I said, though, if we're ahead against a wide field, we probably only have around 32-33% of the equity. We can increase that drastically by limiting ourselves to 1-2 opponents, and when we put money in the pot with an equity advantage, we gain.
Also, in terms of reducing combinations -- if our opponents would frequently/always 3-bet AQ+ and JJ+, it removes half the winning combinations for them while only taking out a little bit from the denominator:
Code:
((579-23)/579)^6 = 78.4% we're ahead
I dunno, it's tough to boil the final results down to math, because I don't necessarily think we lose the rest of our stack beyond this flop bet when we're called -- live players are passive, we can probably safely c/f to a lot of them. Let's say that for whatever reason, we can simplify the potential results from here as follows:
- If we check, we'll realize 25% of the pot (accounting for times we get bluffed out, are behind, or are drawn out on as a result of checking).
25% * $126 = $31.5.
- If we bet $63 and we're ahead (say 70% of the time as a hedge between the two numbers above), our bet will be called by two players with a combined 37% equity 1/3 of the time and by one player with 22% equity 2/3 of the time (there's better and worse outcomes for us being called when ahead, but I'd say this is fair, I tried 98+53, 98+54, 98+A7 and this is conservative regarding our edge). So the bet is worth:
((63*3*.63-63)*1/3)+((63*2*.78-63)*2/3) = $42.21, and our equity in the main pot goes from $31.5 to $91.98 (basically a blended equity of 73% out of $126).
So 70% of the time we gain about $100.
- If we bet $63 and we're behind, our opponent who is ahead goes all-in and we fold. So 30% of the time we lose $63.
So combining the bullets together -- .7*100-.3*63 = $51.10 value in betting.
For completeness, two closing thoughts:
1. I tightened up the opponents' ranges to suited Kx, suited two gappers, unsuited one gappers, unsuited Ax, any broadway, and J9/Q9/K9. They're still playing 499/1081 combos, the field has now found two pair or better 40% of the time against us.
2. The breakeven point between checking and betting is with the field having two pair or better around 60% of the time assuming my simplified game is correct.