[Fonkey123]

[ QUOTE ]
Good post skier.

The general conclusions you have drawn are correct, even though your method is slightly off as others have pointed out.

Although most people will only care about the overall point you are making, I'll explain how I would do things for those who want to delve into the math a bit more.

I'll assume, as does the OP, that we have the following situation:

20 BB effectives stacks, shorty OTB, Hero in the BB.

PREFLOP: 1.5BB
3 folds, Shorty raises 3 BB, Hero shoves 19 BB more, Shorty... calls or folds.

So there's 1.5 + 3 + 19 = 23.5 in the pot and it's 17 more for shorty to call (the rest of his stack). We're offering him about 1.38:1.

Let:
p = probability villain folds
E = equity vs villain's call range
Mo = money in pot before we shove
Mh = money we risk by shoving
Mv = money shorty must call when we shove

EV(shove) = pMo + (1-p)E(Mo+Mv) - (1-p)(1-E)Mh

Setting EV = 0 and solving for E yields:

E = [(1-p)Mh - pMo] / [(1-p)(Mo+Mv+Mh)]

This is the minimum equity we need versus villain's calling range to make shoving neutral EV. Plugging in the numbers for the situation as described above we have:

E = [(1-p)(19) - p(4.5)] / (1-p)(40.5)

Note that if we set E = 0 so that we have no pot equity, p = .809. What this means is that, if shorty folds about 80% (or more) of the time we shove, we can profitably do it with any two! So in the first example skier gave where we have a shorty opening over 30% OTB, but folding all but {99+, AQ+} to a shove, its +EV to shove any 2!

IMO this call range is unrealistic, so I'll go ahead and use the looser one he provided: {44+, A5s+, KTs+, A7o+, KQo}. To find p, we also need to know what hands he's opening with. Let's use {22+, A2s+, K6s+, Q9s+, J9s+, T9s, 98s, 87s, 76s, 65s, A2o+, KTo+, QTo+, JTo}, which is 30.9% of hands.

So given these ranges we can find that he'll fold only 48.8% of the time. Thus, the minimum equity we need is:

E = [(1-.488)(19) - (.488)(4.5)] / (1-.488)(40.5) = .363

So any hand that has more than 36.3% equity versus the range {44+, A5s+, KTs+, A7o+, KQo} will be +EV to shove. Note that we don't want to find a range of hands that have this equity on average, we want a list of hands that each have at least this minimum equity. In this case, the hands are:

{22+, A2s+, A8o+, K9s+, KTo+, QTo+, Q9s+, J8s+, JTo, T8s+, 98s, 87s}

which makes up 24.3% of all hands. So even when the shorty calls with more than half of his opening range, we still profit by shoving hands like J8s.

Happy shoving! [img]/images/graemlins/cool.gif[/img]

[/ QUOTE ]

i came