Tuesday, May 15, 2007

good ICM article

good article from zee justin on his site. i will copy the best parts of the article here, i suggest going to his site to read the whole article, and browse his other articles.

Dissecting a Hand #2 [02.19.2006]

Imagine you are playing a $200 sit’n’go on PartyPoker. Blinds are 200/400 and there are four players left. Here’s the kicker, not only does everyone have exactly 2500 chips, but everyone plays optimally. You are in the small blind and it folds to you. Which hands will you go all-in with?

Before answering this, remember that everyone plays optimally. This means that the big blind will know exactly what range of hands you are going all-in with. Despite this, the correct answer is that you should be going all-in with every hand. This will probably seem ridiculous. If the big blind knows you are moving all-in with any two cards, he will be calling with a wide range of hands, right? Wrong.

This is not a cash game we are talking about. It is a Sit’n’go. There is $2,000 in play, but it does not all go to the winner. $1,000 goes to first, $600 to second, and $400 to third. That means that if the big blind calls and loses this hand, he gets nothing.

You have just moved all-in in the small blind without looking at your cards. The big blind has two options. He can fold, in which case he will be left with 2100 chips, resulting in an equity of $446 according to ICM. It is then correct for him to call if and only if the call will result in equity of $446 or more.

Since the big blind has perfect information (in other words he knows you play perfectly and will move in with any two cards), we can calculate his equity exactly. Let’s do some math. When the big blind calls, there are two possible results. He can win or lose (technically the pot will occasionally be split, but that will complicate the calculations significantly while having virtually no effect on the conclusion, so we will ignore that possibility). He will lose X% of the time, and win Y% of the time. When he loses, he will be eliminated and will receive no prize money. His equity therefore is 0. When he wins, he will have 5000 chips. According to ICM, this will give him an equity of $766.60.

His equity when he calls is therefore 0X + 766.6Y. His equity when he folds is $446, so calling is only correct if 766.6Y > $446. This simplifies to Y > .582. In other words, the big blind will only call our all-in if he has a hand that will win against a random hand at least 58.2% of the time.

According to Poker Stove, the following hands will win at least 58.2% of the time against a random hand: AA – 55, AKs – A4s, AKo – A7o, KQs – K8s, KQo – KTo, and QJs – QTs. There are no other hands that will win more than 58.2% of the time against a random hand. Note that these hands account for 18.4% of all hands.

Now that we know how the big blind will react to our strategy, we can calculate our equity. Before the hand started, our equity was $500. If you were to fold your small blind, your equity according to ICM would be $473.6. However, instead of folding, you are going all-in with 100% of your hands. 81.6% of the time your opponent will fold and you will have a resulting stack of 2900 chips which according to ICM has equity of $549.6.

18.4% of the time, your opponent will call. Against the range of hands that he calls with, your random hand will win 35.375% of the time according to Poker Stove. This means that 64.625% of the time when you are called you will lose all your chips and have an equity of $0. The other 35.375% of the time you will win, resulting in a 5000 chip stack which has an equity of $766.6.

Your final equity therefore is:
(81.6% x $549.6) + (18.4% x 64.625% x 0) + (18.4% x 35.375% x 766.6) = $448.47 + 0 + $49.90 = $498.37. Note that this is $24.77 greater than your equity would be if you just folded all your hands. $24.77 may not seem like much in relation to a $2,000 prize pool, but remember, this is a difference made in just one hand. If you gave up $24.77 in equity on every single hand you ever played, you would go broke fast.

So far all I have proven is that pushing any two cards in this spot is better than folding every hand. Clearly no one is going to be folding pocket aces in this spot, so that really doesn’t prove much. In order to prove that this is profitable with any two cards, I need to show that it is profitable with 72o.

If you are going all-in with 72o, you will never be folding any better hands, so just like before, your opponent will know you are pushing any two cards in this spot; therefore, his calling range will remain the same. Against your opponents calling range, 72o will win 27.055% of the time (note that it performs slightly worse than 32o against that range). We can adjust our formula from earlier resulting in the following:

(81.6% x $549.6) + (18.4% x 72.945% x 0) + (18.4% x 27.055% x 766.6) = $448.47 + 0 + $38.16 = $486.63. That means with 72o, your equity from moving all in is $13.03 greater than your equity from folding. Again, I want to stress how significant this $13 is. An expert $200 sit’n’go player will often hope to make around $30 per sit’n’go entered. This means that almost half of his profits for an entire sit’n’go will be relinquished if he makes the mistake of folding 72o in this one hand.

In the scenario we are discussing, the gap increases greatly due to the bubble. The big blind is aware that if he calls and loses he will go home empty handed. Despite the fact that you will be raising more hands than normal from the small blind, the big blind will be calling with fewer hands than normal, making your equity go through the roof.

It is important to remember that in the hand we are discussing, you are against opponents who play optimally. In reality, you will never encounter opponents that play optimally. Most opponents will have no idea you are making this play with any two cards, and they will call less often as a result. At the same time, they will often misjudge the importance of the bubble. Some players will play for first and will call too often, while others will play to make the money and fold too often. For this reason, it is important to know your opponent’s style of play. Against tight opponents, you can move all-in on their blind quite often, but if your opponent is willing to call with any hand that he thinks might be best, you will have to tighten up considerably.

There is also another added benefit to moving all-in on this hand. If your opponent folds (which he will do often), you will now have the chip lead. This means that everyone else at the table will have to worry about the possibility of going broke on every future hand, while that will never be possible for you in a single hand. You will still have almost as much to gain as your opponents on every hand, but you will no longer have as much to lose. Furthermore, this advantage increases with each additional hand you win. If you can steal the blinds on the next two hands as well, you will all the sudden have a monstrous stack of 4100 chips, with the next biggest stack being only 2100.

Another concept that you should understand is that unlike in cash games, chips do not have a constant value in sit’n’gos. At the start of the tournament, you have 1,000 chips worth a total of $200. That means that each chip is worth twenty cents. At the end of the tournament, one player will have all 10,000 chips and he will be rewarded $1,000. By the time that happens, each of his chips will have dropped in value to just ten cents per chip. Over the course of a sit’n’go, a chips value will gradually drop in value until it reaches half its initial value.

If you only take away one thing from this article, it should be the importance of moving all-in in sit’n’gos when the blinds are high, especially when you are on the bubble. The most likely result will be increasing your stack by the blinds, which is a significant gain when they are large. Sure, you will occasionally be called by a good hand and lose your whole stack, but if your opponents are at risk of elimination, then they won’t be able to call as often as they would like. Edges like this may seem tiny or too risky, but it is important that you take advantage of every edge possible if you want to become a great poker player.


Keith Jackson said...

Yep nice article. I used to visit Zee Justins site a lot some time ago and his articles are great. Even better now I have some semblance of understanding them.

Good stuff and great site of yours.

GeneM said...

I only wish the guys at the $5 level would play a little closer to "optimal" and call less often when I'm jamming with 27o. LOL

Good article.

Mark said...

Thanks for the link SPR, my respect for ZeeJustin just went up. The important point for me is the link between the $ev of your error and prize expectations... been trying to explain that one for the lower-limit guys for a while but ZJ has done a great job.

Anway, enjoying your blog as always.

Cheers, Mark

Chad Carpenter (Yahoo IM: carchd) said...

I love poker and all, but I think you over analyze the GAME. I could never have fun under these circumstances :D