A few posts ago I put a little equation up for pot sized all-in bluffs with equity. Well I went back to the original EV equations and rearranged for a more general result. When risking r to win a reward w (the pot)and estimating your equity against his calling range to be q, then shoving is a profitable play if opponent folds x% of the time such that
x > (r*(1-2*q) - q*w)/(r*(1-2*q) + w*(1-q))
My maths isn't that great, so hopefully I haven't made errors in rearranging the sums but I've put some results in (including the pot sized bet numbers I had) and it seems to work so far.
What usually happens after I work things out and see how little villain's need to fold to make bluffs with equity work is that I play a session and try it out. And often until I get a proper feel for these things I make mistakes, and I think I made one yesterday.
Firstly villain was 18/17 and had raised 3/4 times from the small blind so far. He didn't seem completely retarded, but was over the top aggressive post flop imo.
http://www.holdemmanager.net
NL Holdem $0.25(BB) Replayer Game#67004399592
joshil3 ($23.70)
Hero ($29.10)
hfly0313 ($10)
Ragham87 ($52.97)
capa_audaz ($25.65)
CrisDark19 ($25)
RCAMDESSUS ($25)
joshil3 posts (SB) $0.10
Hero posts (BB) $0.25
Dealt to Hero Ac 2c
fold, fold, fold, fold, fold,
joshil3 raises to $0.75
Hero calls $0.50
FLOP ($1.50) 7h 3d 5c
joshil3 bets $1
Hero raises to $3.75
joshil3 raises to $8
Firstly I like the flop raise initially, I have a gut shot and back door flush draw plus in all probability any ace for outs giving me probably around 28% equity versus his calling range. When he re raises I just thought he'd be bluffing some of the time. The question is how often. Putting the numbers into my little equation (we risk $19 to win $13 with equity of ~28% against a calling range) and it comes out that he needs to fold a quarter of the time to make a shove profitable. At the time I estimated that I needed less folds, maybe 15-20% so I think I was being optimistic. I certainly believe he's bluffing some of the time here intending to fold, but perhaps not as often as 26%.
Hero raises to $28.35 (AI)
joshil3 calls $14.95 (AI)
TURN ($47.40) 7h 3d 5c 9h
RIVER ($47.40) 7h 3d 5c 9h 4s
joshil3 shows Qs Qh
(Pre 68%, Flop 68.6%, Turn 84.1%)
Hero shows Ac 2c
(Pre 32%, Flop 31.4%, Turn 15.9%)
Hero wins $45.40
I get lucky and run above EV.
The important thing is that I'm still improving, and every time I start to try out new things it takes me a while (and a few mistakes) to get it right. But this work is part of a general strategy to become a little less passive and a little more aggressive so that when I reach the higher limits I'm a more difficult opponent.
What this also goes to show is that in general, if you're only betting/raising or barrelling post flop with hands that have some sort of equity against a calling range it's never likely to be a big mistake. Doing so without equity is a lot more difficult to get right. Consider the hand from last time where opponent barrels off with JJ on an AQ3Q. He had two outs on that turn so needed me to fold nearly half of the time to make it work. Perhaps this is true, but it's generally a big ask on that sort of board at this limit.
EDIT: Just ran a simulation of this hand and the fold % requirement seems a little higher at about 30%, perhaps there's an error in the equation. It's not far out though, I will re work the next time I have some time off (next weekend) and check it.
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