I know the theory behind the push or fold in heads up (Is well explained in The mathematics of poker of Chen & Ankenman). I know that exists software (e.g. ICM trainer) that computes the push or fold range, even in ICM scenario, in a multiway hand in push or fold regime. I am wondering which is the algorithm. In other words which is the fictitious play algorithm that this software are using? Well I am reading Kill Everyone of Lee Nelson, and in this is book is presented a table where you can know if push or fold the Hole cards based in in your stack and the M factor and I would like to reproduce this table. I googled a little bit but I had not found any reference.
The exact algorithm, for any particular piece of poker software, I don't know. However, I think I have some thoughts that may perhaps answer the question at hand. No human is going to run an ICM calculation in their head. Perhaps, if one practices a lot, one could calculate somewhat accurately their ICM for 6/9-Max SNGs. In a MTT there are usually hundreds of players, and we could be near the end, or near the top of the MTT.
ICM will help us to discover the theoretical $$'s to Chips ratio. If we are in a 9-Max SNG, standard payout, and there are only 3 players left, PLR1 with 3k, PLR2 and PLR3 with 1K, then we can say PLR1 has a greater %$'s lead than %Chips lead. They are much more likely to finish first because they have both players more than covered. Also, PLR2 or PLR3 could knock out the other, in which case you still have a 1.5:1 chip lead and are now heads up. The reason then, one cannot calculate ICM in their head, especially for MTT's, is that one cannot just know their % chance to win with their stack size. This is also problematic for writing an algorithm one may use to represent a theoretical ICM (Not that the software isn't trusty, it's the same as an equity calculator, accurate for the most part but shy on some opponent tells and whatnot...we shouldn't call when we know where beat, even if the odds are correct to do so. The equity calculator may say to call sometimes when we know better).
Now, to the beef of the question. Push/Fold! If one is playing push/fold poker, they almost never need to calculate their ICM. Our stack is just too small to worry about our $'s to Chips ratio, and in any case it's very low. Hence, Push or Fold. Push/fold tables can be replicated with any decent equity calculator. Plugh in a hand you would consider pushing with, plug in an opponent's range, and you will yield the % chances that each wins the hand. Since we are pushing preflop, we don't need to rely on postflop skills to win, just a call from villain! Isn't that great! The smaller ones chip stack, the wider ones push range. This again deals with how likely it is one will win the tournament, or cash in it. Assuming we are not past the bubble, If a player has 2 bb's, how likely are they to win anything? Very unlikely indeed. So any time we get say 30% chance to double up, far more than the chance we cash at all, we push.
There are some intricate details to ICM that I won't pretend to understand. However, I hope I succeeded in my explanation of how one may go about crafting a push/fold chart.
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