Okay, here are some answers to Father Luke's questions. It's easier to make a post out of them.
1. ICM
That's one way to understand ICM. Basically, it's like this. If the winner took the whole prizepool, your chips would have a linear value: in a $10 tourney, 1500 chips would be worth $10 and 13500 would be worth $900. You could evaluate actions simply by asking whether they increased your chips on average. But we don't play winner-take-all tourneys very often; we play tourneys that pay three prizes. The key point is that the top prize is only half the prizepool, so the guy who gets all the chips, he only gets half the money. So the relationship between chips and money is not linear, and we need a model of our chips' value.
What the ICM does is figure out what the stack you currently have, in relation to the other stacks, is worth. So what it does is assume that everyone has the same skill and will be affected equally by the vagaries of luck (which has the effect of making the tourney a lottery, in which bigger stacks have better chances of winning and smaller stacks worse, simply because on average the luck will even out, so having more chips will directly mean winning more). For each stack, the ICM works out how often it will win, and then how often it will be second in the case of each other player winning, and then how often it will be third in the case of each other combination of first and second. (This is why you need a computer program to figure it out: that's a hella lot of calculations.) Actually, it's calculated by taking out the winning stack and then calculating your chances of winning what is in effect a new tourney for a new prize.
Each hand affects your stack's value. It will change even if everyone folds, because the BB will gain the small blind's halfbet, making him slightly more likely to win, and changing your equity because every change in others' stacks affects how often you place. Even a change of one chip will affect you in a tiny way!
To help you understand, use this site: http://www.chillin411.com/icmcalc.php
a. Give Player 1 through Player 9 1500 chips.
b. Hit calculate ICM. You can see that everyone's equity is 0.1111. This is 1/9, and if you look under each player, you can see why: everyone has an equal chance of filling every spot (this isn't true in a real tourney of course, because you don't have equal skill).
c. Give Player 9 0 chips and increase Player 1 to 3000 chips. This is what happens when Player 1 doubles up through Player 9.
d. Hit calculate ICM. Now Player 1's chances of winning are double. (Look underneath his name and it says 0.222 for first, which is 2/9. He takes Player 2's chance of winning. But look at his ICM equity. It is not .222. Why? Because he is not twice as likely to be second! There are now eight players, and he is 2/9 to finish first. When he finishes first, someone else must come second. Do you see? The other players share Player 1's surplus minor places. In a winner-take-all tourney, Player 1 doubles the value of his stack, because there are no other prize places to be filled. But in a normal STT, he cannot double his chances of winning all three places, because someone else has to come second and third. He has taken
all Player 9's chances of winning (his chances of first go from .111 to .222, but he can't take all his chances of coming second.
2. Push/fold at <10BB
The BB is the big blind, the forced bet that one player makes each round. You begin at t20, which means that the player in the big blind pays 20 chips. Your stack is 1350, so you have 67.5BB.
As the game progresses, the blinds rise. With luck, your stack gets bigger, but even if does, you will have fewer BBs each time the blinds rise. If it's t50, and you still have 1350 chips, you have 27BB. If it's t100, and you have 1350 chips, you have 13.5BB. You have the same chips, and if everyone was still in, and still had the same stack, you'd have the same dollar equity too. But the rising blinds change everything quickly. Each orbit (round of everyone paying the blinds) costs 1.5BB (more when the antes are introduced). That's not too painful when you have 67.5BB, making only tiny changes in your equity, but it hurts when you have few BB, because the amount you are losing not only costs you a lot of equity but brings you closer to being "blinded out"--reaching the point at which paying the blinds leaves you with no chips at all.
You'll become adept at working out how many BB you have (it's important to us), and it's easy enough. If it's t100, you can take the last two figures off your stack and that's basically how many BB you have: 1400 chips is 14BB. If it's t200, you can do the same and halve the result. The same stack would be 7BB.
There are two technical reasons for saying your only play is shove or fold under 10BB. The first is that you want to increase your "fold equity". If you raise, everyone will fold some of the time, and you will win the blinds. Which is cool. We're always happy to increase our stack with no risk of being busted. If you shove, there is more chance everyone folds. Your fold equity is (increase in equity if everyone folds) x (chance everyone folds). It's usually increased if you shove because there's more chance of everyone's folding. Sometimes, you have to do the "wrong" thing because when it works it will preserve your fold equity. (The kind of thing I mean is to take a shove when it's -EV because if you don't, you will in the next couple of hands have to pay the blinds, diminishing your chances of folding everyone out next time you shove).
The other reason is that if you make a standard raise, and someone shoves, you will be priced in to call. Say you have 10BB in the cutoff. You raise to 3BB and the BB puts you all in. You are now asked to call a bet of 7BB to win 13.5BB (the 3BB you've already put in plus the 10BB the BB has put in plus the 0.5BB the SB paid). That's nearly 2 to 1. It's pretty unlikely you have a hand that isn't better than 2 to 1 against the BB's range, so folding would be a mathematical mistake. We try to avoid them. But we don't want to call for all our chips if we can help it. In an STT, we want to avoid confrontation.
3. When to call shoves
No. If the EV is
more than the EV for folding, you should call. But it's the EV in dollars, not chips. You should be calling shoves very rarely because of your strong awareness that the chips you win are not worth the same as the chips you lose.
4. Ranges and equity
Say your opponent shoves. What does he have? Well, you know he has a particular hand but you cannot know which one. So what can you do? Well, you can work out which hands he
might have. (Figuring this out is one of the hardest skills to acquire in STTs, and it's very valuable.) The hands someone might have given any particular action are his "range". Because we are maximising our wins over the long run, we try to make the correct play in each spot, because that will give us the greatest win when we average out how we did over the long run. So this time he as 87s, and we have a certain amount of equity against that, but next time he might have ATo, and we have a different amount of equity against that. But we make the play that makes most money against
both. It might make more against 87s and less against ATo and another play might make more against ATo and less against 87s, but we don't know which he has. So we play against the whole range.
What do I mean by "equity"? It's just a fancy word for your chances. We analyse poker hands with "point in time" calculations, so we give hands a value at the point of decision. The guy shoves. You have a certain hand. If you call, you're going to either win or lose. Say he covers you. You will either double up or be busted. Your probabilities for this hand are of course 1 and 0 (1 = you win, 0 = you lose). It depends what you both have and how the board runs out. But you can't read the future, so you don't know which probability you have. What you can do is work out, if you ran it a billion times, what your chances are. A billion is probably enough to work out your "real" chances (as the number of trials increases, you get ever closer, so that at infinity trials, you'd know exactly what your chances are). You aren't going to be in this spot a billion times, but you play as though you were. (If you've read boots' comments, you'll perhaps see that this is what he disparages in my method: he focuses on the truth that I am making a decision that has only two outcomes as though it had a billion. I say that without knowledge which of the two outcomes will occur, I must make my decision according to which is most likely, and risk the bad outcome when the
likelihood is that I increase the value of my stack, even though with any given decision, I will not increase it by the average amount I consider, but by either of the two extremes. Such is probability though.)
So when I say "You have to put your opponent on a range of hands (. . .) calculate your equity against (that range)", I mean:
a. Decide which hands your opponent is likely to have pushed. These generally form a range, and mostly for convenience we count them from the top down in percentiles. We say "he's pushing the top 10%" to mean he pushes the strong hands that on average do the best against a "random hand".
b. Work out how you fare against that whole range, rather than against any particular hand in it, over enough trials (running out of the board) to approximate your "real" chances.
We do this using software, because calculating it is very difficult. You can use lookup tables to figure out your chances against percentage ranges. (I have one that a friend gave me, which saved me the work I was doing on figuring it out with PokerStove.)
PokerStove is useful for calculating your chances against ranges. SNG Wiz calculates how your dollar equity will change against ranges.
The bottom line is, you do it against ranges because poker is a game of incomplete information: you don't know what cards your villain has. But you can use your skill at assembling information to figure out what cards he might have. It's the same at any point in a game. You interpret bets and calls according to the player. Each is conveying information, which you hope to be able to use.
5. Shoving vs calling
This is how a "pushbot" strategy works. Grasp this and you are on the road to becoming good at STTs.
When you push, three things can happen. Either everyone folds, or someone calls and you win, or someone calls and you lose.
When you call, only two things can happen. You either win or you lose.
So when you shove, so long as all the players have a range that is smaller than any two cards, sometimes you will win the blinds without contest. That increases your equity without risk. When called, you will then put your hand up against his hand, and you will have some chance of doubling up and increasing your equity that way.
Let's say we are in the small blind and we are shoving 10BB into the big blind. He also has about 10BB. He will call with the top 20% of hands (for our purposes here, we're assuming we know exactly what his calling range is--20% is on the loose side and most players will call a bit tighter, but this guy has told us that he will call with the top 20%). If we ran it a hundred times, 80 of those times, we would simply win the blinds. The other 20, we would be up against a top 20% hand. But whatever we hold, we have some chance of winning. Even 72o, the worst hand, has equity against AA, the best, simply because there are boards like K7743 and 8725Q.
As it happens, you should shove 72 in that spot, because the 80 times he folds increase your equity enough to make up for the 20 times you are called and mostly lose. If you had 20BB, you would have to fold, because you have more at stake and do not win enough when he folds.
But if you call a shove, you gain no risk-free chips. So you must beat his range sufficiently to make the EV of calling greater than your EV would have been if you had just folded. (Again, your actual equity will either increase by up to 1.8x or be reduced to 0, but you don't know which will happen. In effect, you work this out by solving this equation: (chances of winning x equity when you win) - (equity if you just fold). If it's greater than 0, you should call. But you need to be a pretty big favourite, usually, for it to be greater than 0.)
So don't think of it as "gambling" on shoves, because it's completely the opposite. When you shove you have fold equity: remember, the increase in your equity times the chance that everyone folds, as well as hand equity if called. Getting others to fold doesn't require gambling. It's risk-free gain the times they fold. If you call a shove, you cannot gain fold equity.
