The Expectation Value of a "Re-Steal"
I commented the other day on Hoy's post about MTT resteals, that I thought what he was describing was -EV. Hoy was specifically talking about later stage MTT play, and I really can't argue with the success he has had in numerous large MTT fields. He is an expert at this stuff, and I am clearly not. What I will try to do is explain my reasoning, as to why I would make such a comment, and let others decide if I am right or wrong here.
Lets just briefly look at the "re-steal" preflop from a cash game perspective. I will stick with the idea that this can be done properly with ATC in the late stages of a tournament. Also, we are in a cash game with deep stacks, so this will not be a resteal push as in the Hoy example. So a cash game resteal from late or a blind with ATC would just be 3x times the initial raise, and not for nearly your whole stack. Well guess what? This pretty much never happens in cash games. People who know what they are doing do not "re-steal" preflop with ATC in cash games. Why do they not do this? In cash games people have plenty of time to figure out what types of plays make money and what types do not. They realize that this is a -EV move in a cash game, and don't bother with it. Re-steals preflop in cash games are done with more than ATC (are not really "re-steals") in what must be over 99% of the cases (retarded maniacs will do anything I guess). If you strip out the differences in what position you will cash in (tournament winnings expectation), and just look at the chips involved (chip expectation value), I think there is a pretty good case right here that restealing preflop with ATC is -EV.
So now for Hoy's specific example.
But now look at how great a resteal is for your same average 100k stack. Now you're in the big blind, and the action folds to the cutoff, who raises it up to 24k on a steal attempt. Sitting on the button, you sense weakness in the cutoff's move, and so you move allin over the top for your entire 100k stack, the blinds fold and the cutoff folds his naked steal. Now look what you've just won. The same 4k and 8k in blinds, plus the same 7k in antes, but now you throw in an additional 24k representing the steal-bet made by the preflop stealer. Now that is 43k added to your stack, and when you're only sitting on 100k in total tournament chips.
There is a little confusion in this as Hoy says the restealer is in both the BB and on the button. I will assume the BB, because the resteal push from the button should have a lower EV as you need to also get through both blinds. From the BB you only need to worry about the "stealer" calling.
Blinds 4000/8000, 1000 ante, 7-handed. Cut-off raises to 24000, and you push 100k from the BB on a resteal.
After your push, (for 92k more) there is 135k in the pot, and the initial "stealer" needs to call 78k more, which will make the pot 213k. The stealer is getting close to 2-1 odds here on a call. That means that if the BB does not have a big pair (bigger than both of the stealer's card), then the stealer is priced in. The push from the BB looks very weak here IMO. There is no way I put him on a big pair pushing from the BB like that. It looks like he does not have a hand he is willing to play post flop with. A couple big cards or a small pair come to mind, or even ATC. Also lets keep in mind the "stealer" is stealing from the cutoff and not the button. It takes a better hand to steal from the cutoff than the button, because you need to get through the button, and will be out of position if the button calls. The stealer does not have ATC here. The stealer's range is better than your ATC resteal range.
So now I will put some numbers in, and will probably screw this up, but bear with me a bit.
1) I assume that the stealer's range is ahead of the restealer's range. You don't resteal with a big hand, that would be a reraise value bet, and not a "resteal". So we can eliminate all of the big hands from the restealer, but none of the big hands from the stealer, as he could legitimately have a big hand. The stealer here has at least a top 50% of hands range.
2) I will also assume that the stealer has the restealer covered. If he does not have the restealer covered, the call of the push starts to become automatic, and the expectation value of the resteal goes down dramatically as a result. Everything will be decided preflop for this hand.
So the BB pushes, and the stealer is getting 2-1 odds, and will not be eliminated on this hand. People sophisticated enough to steal from the cutoff with weak holdings, will also be sophisticated enough to know they are priced in, even with their weak holdings. So I come up with these odds.
Push gets the stealer to fold preflop.
Stealer has a top 20% hand - 0% chance of folding (is not stealing, but has a hand)
Stealer has a top 50-20% hand - 30% chance of folding (is stealing, but feels priced in)
Chance of a fold is (.4)*0 + (.6)*.3 = 18%
Chance of a call = 82%
Who wins when it is called
Stealer has a top 20% hand. (40% of the time, and will be a 2-1 favorite)
Stealer calls with a 50-20% hand that is ahead (40% of the time 60/40 favorite)
Stealer calls with a 50-20% hand that is behind (20% of the time 40/60 dog)
Restealer bets 92000 and wins 43000 18% of the time
EV is (.18)*43,000 = 7,740
Stealer has a top 20% hand
EV = (.82)(.4)(.33)*213,000 = 23,055
Stealer calls and is ahead
EV = (.82)(.4)(.4)*213,000 = 27,945
Stealer calls and is behind
EV = (.82)(.2)(.6)*213,000 = 20,959
Total EV of resteal = 79,700 - 92,000 = -12,300
There are a ton of assumptions here, so you can argue some of the fine points and may come to a different conclusion. The key points from Hoy's example are that the stealer's range is ahead of the restealer's range, and the stealer is very priced in to the resteal and will know his price because the restealer is all-in preflop.
Several things could push this resteal up into positive EV territory, but I do not see it in the example. I would rather resteal a button's steal from the BB, as the button stealer's range is much wider than a cutoff stealer's range. I would rather resteal with a deeper stack so the stealer is faced with a more difficult decision, and can expect more money going in post flop. I would rather resteal with a top 50% hand vs. ATC as you can see how that swings the expectation value in your favor. Hoy, I am sure can add a ton more fine points to this that I am not aware of, and you did a pretty good job of that in the balance of your post. My main point here is don't fall in love with a resteal push over a cutoff "steal" with ATC in the late stages of an MTT. IMO, you are much better off going for a steal vs. a resteal at this stage of the MTT and with these stack sizes. I did not bother with the math on a steal, but I will just state that it can be +EV even with ATC. You may win more chips with the resteal, but you are taking a huge risk pushing ATC. Restealing works much better if you your stack is much deeper, and your cards are decent. Restealing with ATC may look cool, but I don't believe there is a mathematical foundation to justify it. Lastly, the correct way to do all of this is to do it from tournament winnings expection value. To do this you would need to know all of the stack sizes of all the players left, and all of the payout positions. Since this was not provided in the example, I am sticking with the normal expectation value in relation to chips.
Edit: I am going to do an iteration of the resteal calculation to make it more symetrical with the steal calculation in the following post. This new value should be more fair for apples to apples comparision.
New Assumptions. Everyone has 100k. Restealer has a 0-80% hand. Cutoff raiser has a 50-100% hand, and will call a push from the BB with a 70-100% hand.
Chance of getting stealer to fold = 40%
EV = (.4)(34000) = 13,600
When the stealer calls we have a 0-80% hand vs. a 70-100% hand. I will guess here, but a poker calculator could give you an exact number. For the overlapping range it is a push, which leaves a 0-80% hand vs a 80-100% hand (67/33?).
Chance of push is (1/8)(1/3) = 4.2% (and you win 1/2 of dead money)
EV of push = (.6)(.042)(4500) = 113
(.6)(.958)(.33)(109000) = 20676
(.6)(.958)(.67)(-100000) = -38512
Total EV of resteal = -4123