### Saturday, August 19, 2006

## Maths Can Only Take You So Far

I already mentioned that I'm not too keen on Harrington Vol. 3. Most of the book, as I said before, is talking about very high-level thinking and I found a lot of it quite frustrating, in that for example he'll say you should bet the river because "a good player [meaning your opponent] won't bluff", without explaining why he won't bluff.

Towards the tail end though, there are a few online Sit and Go examples which delve into the world of equity calculations. As some of you know, I wrote a program to make these calculations quite a while back, and I'm very familiar with how these work. While the idea is basically sound, like most poker calculations, the results are only as valid as your initial assumptions.

Take this example from HoH 3, problem 39. Standard 50/30/20 sit and go. 4-handed, blinds 100-200, Player A (5000 chips) moves in UTG. B (1500 chips) and C (5000 chips) pass. You're in the big blind with 2000 chips. What's your minimum calling hand ? The book assigns a hand range to A and calculates the equity of all possible outcomes, concluding that the minimum calling hand here is JJ/AKs. That is, you should pass TT or AK off.

The first (and lesser) problem is that this is a completely irrational move for Player A to make. How often do you see someone actually do that ? However, the calculation would be much the same if he had made it, say, 1600, so we'll let that one slide. The much bigger problem is that the standard equity calculation takes no account of how people actually play. It starts at the top assuming that each player's chance of winning is directly proportional to his stack, and works out 2nd and 3rd probabilities in the same way. No account is taken of the fact that, in practice, Players A and C are extremely unlikely to move all their chips in together. People will play cautiously, check down hands that they might have bet in a winner-take-all sat, and so on.

Let's look at another scenario. 3 players have 3000 chips each and you have 1000. By the standard calculation, your chance of making the money is 42%. My gut feel is that, in practice, it won't be as high as that. It might be significantly lower, depending on how well your opponents play. We all know by now that if two players clash, they lose some equity between them to the two players sitting the hand out. When considering one hand "in a vacuum", this implies that tight play is required. In practice though, if you're passing and passing and passing because you don't want to give up this equity to folders, I feel (notice I'm very carefully using the word feel rather than think) that this isn't the best way to play.

I could go on, but the bottom line is that a strict equity calculation does not take into account how play changes in a proportional pay-out sit and go. It doesn't take into account the big advantage you have when you have your opponents covered, particularly those opponents on your left. If the book mentioned this, that would make a big difference. But it doesn't. It presents these equity calculations as mathematical fact, and that's wrong in my book. For sure you have to play cautiously in many bubble situations but passing TT or AK in the scenario described is too much. I stress again that the results of any calculation are only as trustworthy as your initial assumptions (remember Paul Samuel ?), and I really think that the author(s) here haven't played a lot of sit and goes. If they had, their experience might lead them to question the maths here.

Towards the tail end though, there are a few online Sit and Go examples which delve into the world of equity calculations. As some of you know, I wrote a program to make these calculations quite a while back, and I'm very familiar with how these work. While the idea is basically sound, like most poker calculations, the results are only as valid as your initial assumptions.

Take this example from HoH 3, problem 39. Standard 50/30/20 sit and go. 4-handed, blinds 100-200, Player A (5000 chips) moves in UTG. B (1500 chips) and C (5000 chips) pass. You're in the big blind with 2000 chips. What's your minimum calling hand ? The book assigns a hand range to A and calculates the equity of all possible outcomes, concluding that the minimum calling hand here is JJ/AKs. That is, you should pass TT or AK off.

The first (and lesser) problem is that this is a completely irrational move for Player A to make. How often do you see someone actually do that ? However, the calculation would be much the same if he had made it, say, 1600, so we'll let that one slide. The much bigger problem is that the standard equity calculation takes no account of how people actually play. It starts at the top assuming that each player's chance of winning is directly proportional to his stack, and works out 2nd and 3rd probabilities in the same way. No account is taken of the fact that, in practice, Players A and C are extremely unlikely to move all their chips in together. People will play cautiously, check down hands that they might have bet in a winner-take-all sat, and so on.

Let's look at another scenario. 3 players have 3000 chips each and you have 1000. By the standard calculation, your chance of making the money is 42%. My gut feel is that, in practice, it won't be as high as that. It might be significantly lower, depending on how well your opponents play. We all know by now that if two players clash, they lose some equity between them to the two players sitting the hand out. When considering one hand "in a vacuum", this implies that tight play is required. In practice though, if you're passing and passing and passing because you don't want to give up this equity to folders, I feel (notice I'm very carefully using the word feel rather than think) that this isn't the best way to play.

I could go on, but the bottom line is that a strict equity calculation does not take into account how play changes in a proportional pay-out sit and go. It doesn't take into account the big advantage you have when you have your opponents covered, particularly those opponents on your left. If the book mentioned this, that would make a big difference. But it doesn't. It presents these equity calculations as mathematical fact, and that's wrong in my book. For sure you have to play cautiously in many bubble situations but passing TT or AK in the scenario described is too much. I stress again that the results of any calculation are only as trustworthy as your initial assumptions (remember Paul Samuel ?), and I really think that the author(s) here haven't played a lot of sit and goes. If they had, their experience might lead them to question the maths here.

Comments:

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I think you've pretty much hit the N on the H - some of what you say I've sort of figured out to some extent, the rest makes good sense. I hadn't been attracted to HOH3 because, even though in 1 & 2 he attempts to give some background to the hand under discussion, I think that good old "it depends" comes into play often enough for a different answer to be "right". To take to 3K-3K-3K-1K example, sometimes it's right to sit tight because you have a good reason to believe two of the others will go to war, and in the same situation with different players it may rapidly become apparent that they're happy to steer clear of each other and blind you off, in which case you might as well get stuck in.

SNGs are a form all to themselves, or a very special form of tournament at best, and I'm increasingly coming to think that throwing them into a tournament book (usually as an afterthought) is going to mislead the punter. Which is no bad thing. Not only that, but there are distinctly different styles to be employed between say a LHE job and a turbo PLO8 one (that latter being my current favourite).

Thanks for provoking some extra thoughts.

Mike

SNGs are a form all to themselves, or a very special form of tournament at best, and I'm increasingly coming to think that throwing them into a tournament book (usually as an afterthought) is going to mislead the punter. Which is no bad thing. Not only that, but there are distinctly different styles to be employed between say a LHE job and a turbo PLO8 one (that latter being my current favourite).

Thanks for provoking some extra thoughts.

Mike

An interesting article.

I've read the 1st and 2nd edition, and am just starting the 3rd edition now.

But maybe through reading his books is thought provoking, which will improve you as a player.

It is not an exact science and every style can win...!

Cheers.

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I've read the 1st and 2nd edition, and am just starting the 3rd edition now.

But maybe through reading his books is thought provoking, which will improve you as a player.

It is not an exact science and every style can win...!

Cheers.

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