So I finally have something new and interesting to write about!

Let's recap what I'm doing here. I'm looking at a Hold'em-like game, where relative hand strength is determined by a simple function that gives some qualitative similarities with Hold'em. In all the simulations below, the number of possible 'hands' is 25, ranked in order of decreasing strength from 1 to 25. Each 'hand' has an equal probability of being dealt to each player, and it is possible for players to be dealt the same 'hand'. In Hold'em, there are 169 possible starting hands, they are not all equally likely to be dealt, and their relative strengths do not decrease monotonically. Despite this, I strongly suspect that it's the dynamics of stack size and position that are most important on the bubble, not the intricacies of relative hand strength, and that this simple model captures this quite well.

I have written some code in MATLAB that lets me calculate the Nash equilibrium ranges for each player. I can also simulate HU play and a three handed bubble. I've done some Monte Carlo simulations to see how well ICM predicts the EV of each player. For the bubble simulations there are 30 big blinds in total, and for all simulations the blind level does not increase (I could easily build in rising blinds, but let's keep things simple for now.). My reasoning here is that in a super turbo on Full Tilt the bubble often arrives when the blinds are 30/60 and the average stack is 600 chips.

In the HU simulations there's good agreement between the proportion of the chips each player has initially and their EV over 10000 tourneys. This is not really surprising. Each player is using the same strategy, and there's no asymmetry to the problem, since it's just the size of the smallest stack that matters. Because of this, when I do my bubble simulations, I stop once one player is eliminated and then divide up the prize pool according to stack size in order to speed the calculation up a little. The plot below is for HU with initial stacks of 12.5 and 7.5 BB.

The graph below is for a typical three-handed bubble with a 0.65:0.35:0 payout structure and initial stacks of 15, 10 and 5 big blinds. Note that I take care to rotate the initial position of the button at the start of each tourney, and that I simulate 10000 tourneys as I expect the error to scale with 1/sqrt(N), so I should get about 2 decimal places of accuracy. This takes about 4 hours on my PC. Before you ask, I'm not sure what sort of PC I have, but it is a couple of years old. I've never been really interested in hardware, and I just picture lots of little calculating elves running around a magic hamster wheel in the PC. I know that whenever I get a new grant or my computer officer is feeling generous, a new machine appears, all my code runs faster, and presumably the little elves have been given some sort of performance enhancing drug. I also know my latest machine has two hamster wheels in it. Anyway, as you can see there is a small, but I think significant difference between each player's EV and what ICM predicts. In particular, as nibbana and I originally thought, the big stack does better than ICM predicts, and the other stacks worse.

No more graphs, but here are the numbers for some other situations. The first number is the initial stack size, and the second is the difference from the ICM prediction.

0.65:0.35 (super turbo bubble)

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15 +0.017, 10 -0.009, 5 -0.008

5 -0.010, 10 -0.004, 15 +0.014

12 +0.006, 12 -0.002, 6 -0.004

13.5 +0.008, 13.5 -0.003, 3 -0.005

15 +0.016, 7.5 -0.014, 7.5 -0.002

24 +0.016, 3 -0.012, 3 -0.004

1:0 (freezeout)

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15 -0.007, 10 +0.001, 5 +0.006

5 -0.001, 10 -0.002, 15 +0.003

0.5:0.5 (satellite)

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15 +0.028, 10 -0.008, 5 -0.0020

5 -0.032, 10 +0.014, 15 +0.018

Notice that not all equal stacks are equal! If there are two equal small stacks, the stack to act first is at a disadvantage, and vice versa for two big stacks. ICM gives equal stacks equal equity and can never capture this. Also, stacks of 5, 10 and 15 gives a different answer to stacks of 15, 10 and 5 as their relative positions are different. Not entirely sure whether or not this is just noise in the simulation though. All these effects are amplified in a more extreme (satellite) payout structure, and (I suspect) disappear in a freezeout (winner takes all) structure.

Just for a laugh (I really know how to have a good time!

) I also ran a simulation where the medium stack is a fish. As anyone who has played SnGs should expect, the fish sucks EV out of the big stack and gives it to the small stack along with some of his own EV. Sigh!

Fish range (push 30%, call 15%, overcall 5%, any stack size, any position)

0.65:0.35 (super turbo bubble)

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15 -0.002, 10(fish) -0.028, 5 +0.030

So how can ICM be improved upon? I now think that the correct sense in which an ICM+ would be 'better' than ICM is that if we use ICM+ to calculate the Nash equilibrium, the results of simulations like this should give an actual EV that agrees with the EV predicted by ICM+. I would expect that an ICM+ Nash player would do better than an ICM Nash player, not by exploiting him, but because his model would be a more accurate representation of reality, so that he would therefore be genuinely unexploitable, and that the ICM player would be doing slightly worse because he miscalculates the position of the Nash equilibrium.

Some more thought required now I think. What shall I try for ICM+? Some of the equity needs to be transferred from the small stacks to the big stack. Hmmmmm. I think I'll go back and read pzhon's and muebarek's posts more carefully in my quest for inspiration.