Two Plus Two Older Archives - Math of Even Stack Bubble: How blinds, pos, calling ranges matter

I decided to examine the situation when there are 4 players left with about even stacks (all math done for 800 chip tourneys)

before the hand, your ev is .25...if u fold, it'll be roughly .24 depending on exact stacks and if u were sb or not, etc if nobody calls a bet/raise

so, then we want
ev(F)*P(F)+ev(W)*P(W) >.24
ev(F)=ev if they fold to your push
P(F)=probability that they fold
ev(W)=ev if you double up
P(W)=probability you are called times probability that you then win the hand

From SB:

ev(F)=.265 for 100/200 blinds
P(F)=.8 (opp calls with top 20% of hands only)
ev(W)=.3833

P(W)=P(c)*P(W|C)
=[1-P(F)] * P(W|C)
=.2 * P(W|C)

.8*.265 + .3833*.2*P(W|C)>.24
.212 + .07666*P(W|C)>.24
P(W|C)>.365

so you can push with any hand that is >36.5% against top 20% of hands (i can't find eastbay's page of % against diff pushing hands)

Now, let's see what affects the formula the most by playing with P(F) and ev(F), opp looseness and blind sizes...

a) if blinds are 200/400, then ev(F) becomes .2807, which is just .0157 more...however, it affects your hand winning % a lot...

.8*.2807 + .3833*.2*P(W|C)>.24
.22456 + .07666*P(W|C)>.24
P(W|C)>.201

so with 200/400 blinds, u can push anything that is 20% while in 100/200 blinds u can only push > 36.5%
so doubling the blinds makes 16.5% difference there

b) let's tighten opp to top 15% of hands

.85*.265 + .3833*.15*P(W|C)>.24
.22525 .057495
P(W|C)>.2565

so that lowered it 11% because u steal more (however, are more hands 25% against top 15 or 36% against top 20?)

c) let's loosen opp to top 30% of hands

.7*.265 + .3833*.3*P(W|C)>.24
.1855 .11499
P(W|C)>.47395

so that raised it 11%, but more hands do better against top30 than top20...again, the chart is needed to publish actual hand ranges

Position

now, let's see about if we aren't sb, but rather in the button @ 100/200, 20% chance of each calling (36% of one of em calling)

.2731 is now our ev(F) since we steal both blinds
.3875 is in middle of if we win from sb calling or win from bb calling

.64*.2731 + .3875*.36*P(W|C)>.24
P(W|C)>.4675

a little lower than if u are sb and bb has 30% calling range, but u have less hands that will be 46% against a top 20 than 47% against a top 30 hand

so to steal from button with even stacks, you need a coinflip hand like pp, broadways, a6+

UTG:
.512*.2731 + .3875*.488*P(W|C)

from utg, it goes up to 53%

the link i have to eastbay's % against different pushing hands is not working, so i can't give a range for any of these

i'll look at it more, and hopefully you guys will as well, and we can do more analysis, but preliminarily it looks like position then blinds then opp calling range have the most affect on what hands you can push with when even stacked on bubble