3way allin equity calculation with non even stack distribution

A complicated question disguised as a simple question.

There are three outcomes:

A) Win main + side pot: +1000 (SB's new stack is 1800)
B) Win side pot only: +400 (SB's new stack is 1200)
C) Lose: -800 (SB's new stack is 0)

Here's how they happen:

A) Must beat BTN and BB
B) Must beat BTN and lose to BB
C) Any other outcome

Writing this as an EV equation. Pot odds represents a breakeven call. That happens when:

0 = 1000 * P(beat BTN and beat BB) + 400 * P(beat BTN and lose to BB) - 800 * P(any other outcome)

• Pot odds for the main pot = 1000 : 800 = 44.44%
  
• Pot odds for the side pot = 400 : 800 = 66.66%
  

Unfortunately that's as simple as it gets. You have two sets of pot odds and there are an infinite number of solutions to this equation. For example, if BB flips over the nuts, then you need to beat BTN 66.66% of the time to breakeven on a call. However, that's only one solution. If you have 20% equity in the main pot, and 36.67% equity in the side pot, that would also satisfy this equation.

The sucky part about these spots is that most (all?) equity calculators can't tell you these kind of conditional probabilities.

Here's a spreadsheet calc for this scenario.

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Notes:
1) I've omitted chops for simplicity
2) I've defined value relative to starting stack: Value = stack after outcome - starting stack

Good answer.

Is there a way of quick aproximation in game of this method? I am looking to how to simplify this proccess because in my games this scenario is most common where somebody isolates shorttie with whole rfi range and so i would like to have some kind of way for quick calc with 1-2% error which i accept.

Not really because there are two pots and two pot odds.

I guess you could assume you have the same equity in both pots, (weird assumption), then it simplifies to (1000+400):800 =1400:800 ≈ 36.4%.

Well, actually after you've already put in 50 it's 1500 : 750 ≈ 33.3%.

Explanation:

Cool, you are the man tombo!

I've also had a few thoughts on this.

The best practical thing I could come up with is this:

EV(Call) = P(Win Main) x (Main + Side) + P(Lose Main | Win Side) x P(Win Side) x Side - Invest,

where
P(Win Main) is EQ in 3way pot
P(Win Side) is EQ in 2way pot
and
P(Lose Main | Win Side) is the probability that you lose the main pot given that you won the side pot.

P(Lose Main | Win Side) must be estimated. I usually look at the worst-case P(Lose Main | Win Side) = 0, best-case P(Lose Main | Win Side) = 1 and something in between P(Lose Main | Win Side) = 0.5. Another option is to assume independence and simply set P(Lose Main | Win Side) = P(Lose Main) = 1 - P(Win Main).

For the interested people: