Analysis of hands going all in from button against two bigger stacks

[Enrique]

This post was inspired by a previous post on going all in with 37s. I will use 37s as an example, but will put a formula at the end to change the hand accordingly. The objective is to see how big your folding equity needs to be to make an all in +EV and then consider different levels of "tightness" on opponents to see whether the play is +EV or not. Also consider how big your stack (or how small) it needs to be, to be profitable.

We have the following data:
Button has k chips
LB and BB cover him.
LB is 1, BB is 2, ante is 1/9 and the table is nine-handed (you can change k to kx and 1 to x, but the mathematics wil lnot change).
Hence pot is 4.
Say p is the probability both blinds fold.
Your EV is:
4p + (1-p)[(0.34)*(k+2) - 0.66k]
If we want + EV we should put the expression >= 0.
By doing simple algebraic manipulations we get:
p >= (0.32k - 0.68)/ (3.32 + 0.32k)

So let's do a table:

k-----------------p

2-----------------0
3-----------------0.065 = 6.5%
4-----------------0.13 = 13%
5-----------------0.186 = 19%
10----------------0.386 = 39%
15----------------0.507 = 51%
20----------------0.588 = 59%
30----------------0.69 = 69%
50----------------0.79 = 79%

Notice how the bigger your stack, the most folding you need. Also, notice how if your stack is twice the BB you have to go all in, since anytime they fold you gain EV. Hence, looking at this table, blinds should also note how their calling percentage should vary according to the size of the all in raiser.

Next table illustrates, how often both players fold.
Assume both players have same chance of calling (c), then both fold (1-c)^2.

Table:

Chance of calling-------Chance both players fold

5%------------------------90.25%
10%-----------------------81%
15%-----------------------72%
20%-----------------------64%
25%-----------------------56%
30%-----------------------49%

As long as the percentage of folding is greater, the play is more profitable. Notice that if k < 10, the play is correct even if the callers are as loose as calling 30% of the hands (this should be evidence enough to make you call all ins from the BB with AX, KX suited, Broadway from all in players with less than 10x, where x is the LB).
Notice tha that if k = 15 then as long as the players are not super loose (less than 30% callers), then the play is +EV.
Now if k > 25 the play becomes a mistake for the all in player, because it is not good to assume they won't call you 15% of the time.

The formula for other hands happens when you change 0.34 in the original expression to the percentage that hand wins.

The conclusion is that going all in with any two hands from the button is a good strategy if you have anywhere between 10 and 25 little blinds. Maybe not a good strategy, but at least is +EV. Now if you have less than 10, the blinds should call with almost anything, but still the play is +EV., but this computations show, how the blinds have to call with almost anything.
If you have more than 25 little blinds than the play is dumb. Although this is not news, it is intuitive to think that the lest chips you have the more hands are worth going all in, and the more chips you have the less hands you should go all in (obviously less hands to go all in with stacks that cover you, not all ins vs micro stacks).