Nibbana and I have been talking recently about whether ICM really captures all of the advantages of having a big stack, particularly on the bubble, the point being that with a bigger stack you can often bully the table by profitably pushing a wider range of hands than with a small stack and (this is the key point) you are therefore more likely to pick up a profitably pushable hand.
As an example, consider the bubble in a 6max SnG. There are three players left, and the payout structure is 0.65:0.35:0. My experience is that, if there are three roughly equal stacks, this is a very unstable equilibrium, and the higher the blinds, the more unstable it is. It only takes one player to successfully steal the blinds a couple of times and he can start to push even wider than before. It seems that a small chip advantage is more advantageous than ICM would indicate.
Now, assuming equal skill levels, ICM is a reasonable way of calculating the equity of the remaining players. But it is just a model (the clue's in the name!) for the distribution of equity that, for chip stacks s1, s2 and s3, assigns equities E1, E2 and E3 to each of the players as functions of s1, s2 and s3, not the actual equity. We don't know what the actual equity is. Now, apart from the constraint that if the stacks are equal, all players have equal equity, and that the equities must sum to 1, we could try to use different functions E1, E2 and E3 that give greater weight to larger stacks.
(i) What could we use instead?
(ii) How would we decide whether this new model (ICM+) was better than ICM?
Here's a suggestion for (i). Using E1 as an example, and normalizing the stacks so that s1+s2+s3=1, if E1(s1,s2) is the equity according to ICM, let e1(s1,s2) = min(1, (s1+2/3)^k*(4/3-s2)^k*E1(s1,s2)). In other words, we multiply the ICM equity by a function that is fairly flat and equal to 1 at s1=s2=1/3, where the stacks are equal, but which does not exceed the total equity available.
Here's E1
And here's e1 with k = 0.2, which is just a little bit different
We can do something similar with e2 and e3, and may have to monkey around a bit to keep things sensible, but hopefully this illustrates the idea.
Actually, it doesn't matter if the sum of the functions exceed 1. It's just a model for assigning value to a stack of chips.
Now, question (ii). Once you have the functions for ICM+, you can work out the Nash equilibrium solution for any given stack sizes. This will be different to the ICM Nash equilibrium. Which is better? Well, it just needs some simulation. We simulate lots of three handed bubbles with some players using ICM and some using ICM+ and see who has the best ROI. Simple!
Now I could go away and write some code to do this, but it seems like a real pain in the arse (is there a flush, is there a straight, is there a full house, ach! tedious) when I'm sure some of you clever computer scientists must have the appropriate software just lying around. I also know that there are other alternatives to ICM. Anyone want to work on this with me?
