[Aisthesis]

Ok, I eventually just gave up on an exact solution in the 3-player game (any suggestions on how to do this are much appreciated), but I did come up with a way of getting an approximation that should be pretty accurate--and applicable to the larger problem as well.

Basically, I just assumed (inaccurately, but fairly close) that B would call A's all-in on the top half of A's raising range. Similarly for C if B folds. But I also assumed that C would call BOTH all-ins on the top half of B's raising range (which also seemed close enough given the results when A moves in on any hand at all).

Here's what I come up with:

A moves in on [.9535,1]
B calls on [.9767,1]
C calls if B folds on [.9767,1]
C calls if both players are all-in on [.9884,1]

The value for A (UTG) here is $.0325
The value for B (button) is $.0849

It's worth noting that this is a very slight improvement for the button over the 2-player game, which had a value of $.0826

It's also worth noting that A is already extremely tight in the 3-player game in terms of moving in, and he'll just have to get that much tighter with more players involved.

In terms of applications to a poker context, I think one problem is the stack-size assumption. While 10xBB seems rather plausible as a dividing line for moving in, this is generally happening in practice (for me, anyway) when the blinds are big and antes have been introduced.

So, let's say you're UTG with a stack-size of "x" with blinds at 100/200 and antes at, say 10. The pot before betting at a 9-player table is thus 390, or call it 400 as another approximation. The means that if your stack is 2000, the all-in is gaining you 400 rather than just 200.

As a model for this, I'm really thinking it would probably be better to set stack-sizes actually at just $5 here and keep the single blind structure (otherwise, it's going to get hopeless complicated). This will give BB much better odds to call than is the case in practice.

So, better yet, why not set up the game so that there's simply $1 dead money in the pot (BB gets it if everyone folds) and BB only puts up $1 blind. Stack-sizes now back to $10, so that you're risking $10 to gain $2.

That seems to me like at least a reasonable approximation of the kind of tournament situation we're talking about here. This should definitely loosen up all players in terms of taking a stab at the pot.

Any thoughts?

Anyhow, since this is pretty complicated even with my linear approximation method, I'd like to have the problem set up in such a way that it will hopefully shed some light on the plays to make in certain practical contexts.