Posted by Junelli 3:40 PM
This is a very interesting article written by ZeeJustin.

His rationale seems to closely follow Harrington's End Game analysis from Volume II.

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Dissecting a Hand #2 [02.19.2006]
The following is more of a situation than a hand, but I think it is paramount to a proper understanding of sit’n’go play. To become a winning sit’n’go player, it is crucial to understand the following concepts.

For this article, I will be using two pieces of software. The first is Poker Stove which I discussed in my last article. The second is called the ICM or Independent Chip Model. If you plug in the appropriate stack sizes, it will give you the equity of each stack size. It does not factor in position (or skill for that matter), but that is beyond the scope of this article. You can find the ICM at http://www.bol.ucla.edu/~sharnett/ICM/ICM.html.

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?

(Note: Party has recently changed its sit’n’go structure. Under the new structure, the math will still be completely the same if you change the stack sizes to 5000 chips and the blinds to 400/800)

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.

What type of thinking should the big blind be using? There are two common answers to this question. Some people think that the big blind should play for first and ignore the bubble. Other players think that making the money is so important that it should be his primary concern. Think about these two lines of thinking for a minute and decide what line of thinking you think is correct.

In reality, both lines of thinking happen to be incorrect. A sit’n’go player should be thinking about maximizing his equity, and that requires balancing both of the above concepts.

Let’s go back to our hand. 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.

This article so far has been heavy in math. I feel that after a discussion like this, it’s important to step away from the math and make a general conclusion. There are several interesting concepts at play here.

The first is the gap concept. David Sklansky first wrote about the gap concept in Tournament Poker for Advanced Players (a book that I highly recommend). On page 27 he writes, “There is a very important general principle understood by all good poker players. That is, you need a better hand to play against someone who has already opened the betting than you would need to open yourself. For instance… in limit hold’em you would most certainly raise in middle position with [KQo] especially if no one else was in but you would rarely play against an early position raiser with that same hand.” This is mostly because the early position raiser will have a hand much better than KQo on average.

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.

This is why it is important to be aggressive later on in sit’n’gos. Once you show an aggressive action, your opponent is aware he is putting his tournament life at stake and will have to tighten up as a result. If instead you were to just call, your opponent could move all-in putting you in the tough spot where you would often need to fold the best hand.

The bubble scenario is very similar to a game of chicken. Clearly it’s wrong for both players to decide they will never swerve, but if one player tells the other, “I don’t care what you do. I’m not going to swerve.” the second player has no choice but to back out. Once you go all-in, there’s no way you can back down, so that is the poker way of letting your opponent know you won’t swerve.

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.

Obviously calculating a play’s exact equity as we have done is too complicated to do in the middle of a hand. However, it is important to have a good feel for all the concepts that have been discussed in this article. Each decision you make in poker can be considered a judgment call, and if you have a better feel for these concepts, your judgment will be a lot better.

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.

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