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Assume you are playing a NL Holdem tournament heads up. What is the best strategy against these all-in scenarios, and what is the expected win percentage when implementing the best strategy against these scenarios?

  1. Opponent goes all in every hand.
  2. Opponent plays Sklansky's System.

I'm guessing there is some software out there that has already calculated the answers?

Update: Bogdan mentions in his answer below that in the case of Sklansky's system, if the opponent moves all in on your raise, you should fold since you know he has AA, KK, or AKs. (I assume he meant fold unless you have AA.) He all but said the perfect strategy would then be to make a small raise every hand you are first to act, and therefore you will either win, or end up folding to AA, KK, or AKs. In light of this, I think we would have to revise the system a bit, or else you could simply make a small raise every hand you act first (yielding nearly 99%ish win rate of those hands), and just letting the opponent fold when he doesn't have a push hand when he's first to act, so you can blind out the opponent with this strategy. Since we're talking about heads up, I think it would be better to omit the part of the system that says what to do if the opponent raises. So for #2 above maybe we should just assume the system will use the same all-in hands regardless of first to act or not.

Note, what I'm actually after is the expected win percentage in this scenario. I'm expecting the answer to #1 to be something like: call the all in with the sklansky system hands and you'll win this tourney 5X% of the time. Or call with pocket pairs and you'll win 6X% of the time. Or call with Ax, 88+ and you'll win 7X% of the time, etc. I have no idea how to figure out how to calculate what those percentages are for this though, since you get blinded off every time you don't play a hand.

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Of course that you can beat both systems by calling with KK+, but we are here to win and exploit our opponents, so we won't play only the nuts. You can build your own calling ranges to get the preflop equity that you want, but remember that here there is a trade-off between equity and how many times you actually win the HU battle. If you play only AA, against a random hand you're an 85% favorite... but how often do we get AA?

Against shoving every hand, my best bet is to call with 22+ and AJ+. This might seem a little nitty, but you will win, in the long run, in about 68% of the time, with the added bonus that your opponent will never know if you'll call or not and, until he figures out your strategy, he'll get stacked off 7 times out of 10. You can add some more hands in the mix, if you like to gamble, but I'm more of a pragmatic type and I want to win also. So, I'll stick with the range stated earlier.

Against Sklansky's method, it's even easier. You know his ranges, you know his play, so it won't be a big deal building a counter-exploitive strategy.

If there is a raise in front of you, then you go all-in if you have any of the following hands: A pair of aces A pair of kings Ace king suited

Counter-strategy: fold every time.

If no one has raised in front of you, then you play a little bit looser. You're going to go all in with any pair, any suited connector bigger than 34 suited, and ace - little card suited, and any ace king.

Here, we can talk a little bit. The range recommended by Sklansky, 22+, 45s+, A2s+, AKo is 13.12% of the total number of combinations. To be at least flip against this range, you need... well... the same range. To win more than 50% of the time, we go back to range building, by trial and error. My rule of thumb here, when building ranges against a defined range, is to begin with a range that is half the given one. So, we'll take 13.12/2 = 6.56% of the entire range, which is:

88+, A-10s+, KJs+, AQo+

Against his range, you have 58.61% equity.

But I am not satisfied. 58% is too low to take against a mechanical, known strategy. So, we'll refine the initial range a little more. Let's get rid of the Kx hands, because we can get into a domination situation, with us on the wrong side. Our refined range #1 will be:

88+, A-10s+, AQo+

Which is 6.18% and which gives us exactly 60% equity.

How much you want to refine this range is up to you. In the end, it will be an optimum between the pre-flop equity and the actual number of times that you get your cards and he shoves.

I hope it helped.

Good luck!

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I believe your first sentence is not true. I don't think you'll win very often if you only play KK and AA. Of course it depends on the blinds to stack size, but let's say it's 100 to 1. ($1000 buyin starting with $5-10 blinds.) Against scenario 1, you'll be out $15 every two hands you don't play, or $7.50 on average per hand. So you can last 133 hands before you bust out. If you don't see KK or AA in 133 hands, you'll bust out without ever seeing a hand. Furthermore, if you finally play a hand say after blinding off half your stack, even if you win, you're still only back to where you started. – TTT Apr 10 '13 at 23:10
(Comment length is limited so I'm using multiple comments.) I think your on to something with 22+, AJ+. I think the numbers you are quoting though, are the precentages for winning the hands you play, but doesn't necessarily correlate to your chances of winning that particular tournament, since you're also losing all of the hands you don't play at all, and your stack is dwindling in the process. Your comment of "until he figures out your strategy" doesn't apply since he doesn't care- he never even looks at his cards. – TTT Apr 10 '13 at 23:15
Ahah! You're definitely on to something with the counter strategy to sklansky's system with AA, KK, AKs. You didn't actually say it but I'm going to put words in your mouth and revise the question. Great point! – TTT Apr 10 '13 at 23:18
@TTT You have a point: I didn't take into consideration the effective stack sizes and the blinds/antes of the tournament. If you have 5 BBs, then this entire theory goes out on the window and you call ATC, hoping you'll win. I took into account the fact that you have enough BBs left for you to be able to pull out some maneuvering. The 22+, AJ+ range is a variation of a principle against guys open-shoving every time, stated by Dan Harrington: "wait for a decent hand and then call". Again, we must correlate the range with how many BBs do we have. In the respect of BBs left, you're right. – Bogdan Apr 11 '13 at 5:37

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