[Aisthesis]

Now I'll try to just explain where I'm at on the 3 player game. A is UTG, B is button, and C is BB.

Ok, if A folds, the game simply reduces to the 2-player scenario, so that's already solved.

Now, if we set a as A's threshold for moving in, and b as B's threshold for calling, we now have 2 (presumably different) thresholds for C:

1) If A moves in and B folds, then C will call on [c,1]

2) If A moves in and B calls, then C will call on [d,1]

Clearly d > c, as C is going to have to be very tight about calling all-ins from both opponents.

So, I tried this by indifference equations and indeed get 4 equations with 4 variables, but they're just horribly messy since they're non-linear.

One thing I haven't yet explored is trying the same problem with stack-sizes that might make it easier (like $4 or $3), but I did try this simplification:

Let's just assume that A always moves in (call it a new game where A moves in blind).

The solution for c here is easy, but I won't even bother doing that. The real question is how it sets up the relationships between b and d.

In this case, for d, I get a value of the square root of 19/31 = 0.7958

and a value for b of 0.5238 (I won't even try to write this out combining fractions and square roots).

So, if A is always moving in, then:

B calls on [.5238,1], and
C calls B's all-in on [.7958,1]

Also haven't figured out the value of this game for B and C (presumably B will get the biggest share). But the main thing of interest is just the relative proportions for the call on the part of B and C.

Given a value of a, I don't think the equation is going to be linear, unfortunately, but this should at least allow some kind of linear approximation.

In any case, A is clearly going to be much tighter here in moving in than he was in the 2-player game, and B will call the all-in roughly in the top half of A's all-in range, and C will call both players somewhere in the top 40% of B's calling range. (if B folds, C's calling criterion will also be roughly in the top half of A's range, since he's risking $9 to win $11).