ICM Quantified - First Set of Results

[Shillx]

Lower stacks will find that hands that were previously easy folds now become easy pushes and conversely larger stacks have even more value than ICM gives them credit for and therefor many of the marginal and not so marginal pushes need to becomes folds.

This post is good for a variety of reasons, but this statement doesn't jive. When you go into detail and look at ICM and how it works, it always tells you to push more when you have more chips. After looking at some numbers, the whole S-curve phenomenon should be able to be explained by ICM. Here is an example.

Assume that the stacks are 4000/2000/2000/2000 and the blinds are 100/200. Everyone plays perfectly or very close to it. You have the 4k stack and are on the button. How much do you think a push is worth in this spot?

It turns out that pushing is worth .45% of the pool. You start the hand with 33% of the PP and end up with 33.45% after you shove your random hand. So eventhough it appears as though I only have 33% of the pool, I actually have a lot more since I can push every hand. The next hand when I'm in the CO and push my random hand, I'll collect .32% of the PP. Since they are playing perfectly, I'll get to push from the button 84% of the time and 63% of the time from the SB. I'll also get a walk about 32% of the time. This is where that extra equity comes from. Eventhough people don't play perfectly in the real world, this is too big of an edge for all the spite callers to overcome. Obviously you will get burned if people call you everytime, but most people play well enough that shoving everytime will make significant money. You are pulling in over 1% of the pool every lap with 20 bb behind. Even if the real world returns 30% of this, you are still in an awesome spot.

Compare this to when you are small stacked in the CO. Say the stacks are 2400/2400/2400/(1200 - you) with blinds of 100/200. The EV of pushing aginst perfect players is just .18% of the pool. With 6 BB behind, you can push roughly 25% of time from the CO and ~30% from the button. Even if everyone folded to you every time, you would only be able to get about 1.15 pushes in per lap. So when you are short but not desperate, you can't even push enough to keep up with blinds. With 10 BB behind, you can only make about .9 pushes per lap if everyone folds to you everytime. The fact that people will oftentimes be shoving in front makes this a bad spot for the hero. It would be like playing HU with 50 BB behind and just the ability to push/fold. You would get blinded down to some point and then you would begin to push enough to stabilize. The same exact thing happens to the short stack on the bubble.

So in a perfect world, I would expect to see a LARGE S-curve. The short players would get ground down to ~4 BB at which point they would start picking up enough pots to slow the decline (or get called and double up or bust). The big stack would shove every hand and eventually grind everyone down. The reason why it is an S to a lesser extent in reality is because the poor players hurt the big stack by making bad calls. The poor players also help the small stacks by getting into clashes with bigger stacks with weak hands. Just think about how many times you have made the money with no chips when someone calls all-in with A7. This would never happen against perfect players, and the short stack would have a much tougher time making it into the money. Likewise the big stack at the start of the bubble would win every hand and get ITM with a monster chip lead.

So I have to disagree that the S-curve favors the small stacked player. IMO, it is just an effect of ICM (basically, chips gravitate to chips) and it is directly related to the quality of the players. The better they are, the bigger the S is. This is still an interesting concept and I'm glad I read your OP. Very nice.

Brad