Information is power is money

Information is power. More commonly, people say that knowledge is power, but information is more readily quanitifiable. Often, if I know how much information advantage I have over you, I know how much power I have over you. I think this is a key concept for a realist such as me. Another, related, concept is that no information is hidden. This is a different thing from saying no information is obscured or even no information is unknown. It says no information is privileged. Basically, information is in principle discoverable by anybody and is never esoteric. This doesn't mean you will be able to uncover all information, because you may not have the tools to understand or use it, but it means that you could in principle acquire it without having to know magic words, do rituals or sacrifice goats.

This is going to be a poker post, I warn those who hate posts about poker, but only because it serves as a means of talking about a concept. And there aren't too many discussions of ICM online, so maybe people will stumble on this and it will help. Maybe not.

Two things boots said in comments lead me to make this post:

I'm not sure what "ICM" is but I think it might be a mistake to equate any specific school of thought with capability.

and

It puzzles me how you can expect to grind out $50/hour playing poker when you seem to think it is a matter of maths and mindreading.

Knowing the answers to the two questions implicit in these comments is the key to winning SNGs, and I know both answers. If you bear with me, you will too.

I will begin with the second. In a hand of holdem, the players are given two cards that they can see and others cannot. The hands are dealt quasirandomly.

It's not important that random number generators are not truly random, so long as the distribution of outcomes from them resembles the distribution of random outcomes sufficiently closely. For the purposes of poker, it does. You might think that a computer could create more random outcomes than a human dealer, but you would be wrong. A well-shuffled deck (which is rather less shuffled than you might think) will give a truly random outcome.

Here is a key understanding that boots lacks. The distribution of outcomes in a poker game is normal and the outcomes will converge on expected values over enough trials. These are important things to know, because they underpin the mathematical understanding of poker. If you're not clear what I mean, I'll explain. Say you flip a coin with me. You probably know that your chances are 50/50. But you could flip a coin a hundred times and get 60 heads, 40 tails. This does not mean your coin is not fair. Chance converges on expected values over many trials. Flip the coin a million times and you'll be close to 50/50. I won't explain why (mostly because I have an intuitive grasp of why and can't explain the statistics adequately, but much of statistics depends on its being true).

Now it's true that in a normal distribution, not every outcome sits neatly around the mean. You do get outliers, and it's perfectly possible to see a long run of outlying values. So you can be "lucky" in this sense. But working on the assumption that values are close to their expected values will generally be correct. What does this mean? Two things. First, the distribution of cards dealt will tend to be "sane". You won't see many hands in which two guys have aces, and two guys have kings. You might see that hand. It's possible, and every possible deal has equal likelihood (an important thing to remember in considering random outcomes: in a lottery 1 2 3 4 5 6 truly is equally likely as 1 23 32 37 42 45; however, a mistake people make is to think that you are as likely to have the consecutive numbers as spread ones -- you aren't: there are far more outcomes with spread numbers, so they are much more likely). Second, outcomes on the flop, turn and river will tend to their expected values. Say you have four to the flush on the flop. Your chances of hitting by the river are a bit less than 2 to 1, on some (slightly dodgy but necessary) assumptions (we always assume that all unseen cards are equally likely, but of course ones held by your opponents are not). So you would expect to hit one in three times. But you can hit three, four, a dozen flushes in a row.

What can a player do about that? Try his luck and hope he gets the flush when he doesn't have the odds? No. He plays to maximise his value over the long run. What he tries to do is lay his distribution of actions over the distribution of outcomes, so that his profit over the long run, when outcomes converge on expected values, is at the maximum.

This is the correct way to play poker. Whatever you think, boots, however much you sneer at playing by maths, this is the best method to increase profit over a lifetime. Those three words are important. Remember, you can flip 60 heads from 100. Over 100 trials, you might or might not be maximising your profit by playing the odds. Over ten million, you can count on it.

I'll come back to the mindreading.

I'm not sure what "ICM" is but I think it might be a mistake to equate any specific school of thought with capability.

I'll explain ICM. It's a reasonably simple concept, but essential to SNGs, and yes, it does equate with capability.

Two concepts need to be understood. First, at each point in a tournament, all remaining players have a "share" in the pool of winnings. (Even if someone has been paid out, there is still a remaining pool that you share in.) This is called your equity. It's somewhat like equity in a company. It has a value that is not realisable on the spot but is quite real. If you have a stock, you have a share in a company that can go up and down. And your equity in a poker tourney goes up and down. Second, SNGs are not generally winner take all. In this discussion, they have a distribution of prizes of 50/30/20. In a $5 tourney, the winner takes $25, second place $15, third place $10.

When I begin an SNG, I have 1500 chips. So does everyone else. The prize pool is $50. My equity is $5. This is because I have 1500/15000 = 1/10 of the chips, so I have 1/10 of the prize pool. But this is because I have 1/10 of first ($2.50), 1/10 of second ($1.50) and 1/10 of third ($1).

Say I double up. I now have 3000 chips and one guy has gone. So I have $10 equity, right? Wrong. The guy has surrendered his entire chance at the prize pool, and you can only win half of it at most! You take most of his chances of winning but you cannot take all. Why? Because you cannot finish first, second and third. You can only fill one spot and it is not winner takes all. Everyone else has also improved their chance of a share in the prize pool. They retain the same chance of coming first (yours has doubled), but they improve their chance of coming second (because the extra time you win, you cannot also come second! Someone else must fill that spot, and now there are only nine players to share it, and each has an equal chance). There it is, the key to ICM. When we all had the same amount of chips, I had one chance of winning the tourney. When I double up, I have two chances. But I do not have two chances of coming second and third, because when I come first the extra time, I cannot also come second. My chances of coming second and third do improve dramatically, but they do not double, because of that one extra time I win.

Well, why does that matter? Remember what I said. A poker player tries to lay the distribution of his outcomes over the normal distribution of outcomes to make the most profit. We call this "expected value". Say I have four to the nut flush and I am facing an allin. The pot holds $300 and I must pay $100 to call. This is an easy call. Over a lifetime, I can expect to win the pot one in 2.86 times (I am 1.8 to 1). The pot pays three for my one. My expected value, or EV, is huge: 3/1.8. Whichever action has the highest EV is the one you should take. (If this isn't obvious, comment, and I will explain, but it should be.) This doesn't mean I will make money on this particular flush, or on any particular flush. It means that over my poker lifetime, given this spot, I will make that money. (This is a simplification, because of course my opponent can pair the board sometimes and beat me with his set, but let's say that our flush will always win to make it easy to understand.)

In a cash game, your equity in chips exactly equals your equity in dollars. In the example I give, $100 in chips is worth $100 in cash. So if I make the call, I make my EV in dollars.

In an SNG, my equity in chips corresponds to a dollar value, but not in the same way. At the start of the tourney, 1500=$5, but as my stack grows, the relationship between the two changes. As we discussed, if I improve my chances of winning, I cannot improve my chances of coming second by the same degree, because it is not winner takes all, and I cannot be second at the same time I am first. Every time you win, someone else comes second; every time you improve your chance of coming first, whether you double it, increased it by a third, or whatever proportion, you are not in the race for second that same proportion.

This is the ICM -- the independent chip model. It is the understanding that because if you have all the chips, you will win 60% of the prize, not all of it, there is a scale of value between 1500=$5 and 15000=$25. 10x the chips does not equal 10x the money! But we are playing to win money, not chips. I can't go to the bank with my virtual chips. My bank insists on hard cash.

So here's the thing. Let's say I'm playing a cash game and I have QQ. My opponent shows me that he has AK and goes all in. I am last to act and no one else has called. I should call. Not calling in this spot is horrible because you are 57/43 to win. You may not win this time (43% is quite high!) but over a lifetime you will win 57% of the time. You should also call this early in a tournament, when the value of chips and the value of money are closely correlated. This is often called a "coinflip" in poker, but it should be clear that this is not a coinflip at all. QQ is heavily favoured. The numbers look close but think about this. If I offer you a series of a million coinflips at $1 a pop, you will win 500,000 and lose 500,000 and net nothing. If I offered you 57/43 odds on heads, you can pick heads every time and win 570,000 and lose 430,000. $140,000 is a lot of money! Make the right choice on a "coinflip" in poker a million times and you will make a ton of money.

But let's say I'm playing in an SNG and we're at the bubble. The bubble is the point at which you get paid. So it's when four players remain. Whoever comes fourth gets nothing. So let's say the guy has you covered, shows you he has AJ and goes all in. You have 66. You are 55/45. So you call, right? Wrong. In a cash game, you call. In an SNG, you fold. Call the value of my hand $10. If I call and lose, the value of my hand becomes $0. You get nothing for coming fourth. If I call and win, I double up in chips, but my equity does not double, as we discussed. Nothing like it. How much it increases depends on how many chips everyone else has. But because one guy loses a ton of his equity (all of mine if I lose, most of his if he does), everyone else gains some (because their chances of coming first remain the same, but their chance of coming third has just shot through the roof! If I am knocked out, they are certain of at least third).

If I folded this hand, my equity will not change. I will have the same number of chips and the same chances of winning, placing second and placing third. If I win, my chances will improve to the value that double my chips has. But the risk I should be willing to take should not exceed new cash value of chips/old cash value of chips. In a cash game, it's simple: the cash value of my chips is their face value. But in an SNG, I need to know the ICM to know what the cash value of my chips is.

Knowing ICM is crucial to making money in SNGs. If I make calls that decrease my cash equity in the prize pool, I am losing money. It's on paper, if you like, because no one has been paid yet, but it's like having a share: 100 shares at $5 are worth $500, and if they fall to $3, you really have lost $200. Imagine that you held those shares but had to cash them out on 31 December. Whatever they're worth then, that's what you get. That fall of $200 is money that you've really lost. You are going to need to gain it back before the cashout, or your wallet takes a hit. An SNG's cashout date is the point at which you bust out! Whenever I gamble in an SNG, I'm gambling my equity. Sometimes, of course, I will bust out and my investment will be worth nothing. On a 55/45, that will happen 45% of the time. So I must ensure that the 55% of the times I get paid compensate for all those times I win nothing. In a cash game, they will (keep thinking of the coinflips if you struggle to understand why: this flip you lose your dollar, next you win: 45% of the time you lose the dollar, so out of a million, you lose 450,000 times and $450,000 dollars, but you win the dollar 550,000 times to make up for it). In an SNG, they won't.

Where does the mindreading come into it? Well, players do not show you their hands. It's tragic that they don't, but that's the cross you've got to bear. Remember what I said. Information is power. In poker, knowing what someone has is very powerful information.

Say I'm playing cash. I have 66. Some guy pushes all in. If I knew he had AJ, I have an easy call, as we've seen. But I can't know that. And as I also noted, information is not hidden. It's not unknowable that he has AJ. He knows! There's no secret to it either. If he turned his cards face up, they would be revealed as AJ. They don't magically become AJ in the act of being turned over. The information was always there. It was not created de novo.

But I do not know that the guy has AJ. What I know is that I've seen him play a few hands and he's pushed a few times. Because the cards are received at random, they have, over the long term, a predictable distribution. So you can assume that he has had that distribution. He may have had a heater, and have been dealt aces five or six times. But you cannot assume that your sample diverges from the true population of hands, even though it's perfectly possible that it does. (If they never did diverge, poker would be a lot easier!) You have to deal in models because the actual distribution of his hands is, and will remain, unknown to you. The model is an approximation and can be wrong, but it's your best guess.

So the guy has pushed a few times and you think he's doing it a bit light. He can't have been doing it that many times. So you give him a range. These are the cards you think he might have. It is not an exact science! You just do your best. The ranges you put people on get closer to what they actually have depending on how many hands they've shown down, how tricky you think they are and how much you think they balance their play (by mixing in hands that do not fit so obviously into their range -- a player might raise AA/KK/QQ UTG but also raise 76s so that he gains some value from your uncertainty over whether he does have the big pair).

You compare your 66's chances against that range. You do not know which hand he has, but you do know your chances against his range of possible hands, so far as you know them. You consider your equity vs the range your opponent has. This is how you calculate ICM. A guy pushes, you have to decide whether to call. You cannot know his cards, but you can have an idea what percentage of cards he will play here. So you calculate your chances against that percentage. He might be pushing the top end of it, and your chances are worse than you think. He might be pushing the bottom end, and they're better. But your aim, remember, is to lay all your outcomes over the distribution of outcomes, not just this one outcome. So you choose the correct action in the long run. You are not having just this one flip of the coin. There will be many many flips.

Experience helps you pick ranges that fit players. And knowledge of ICM helps you make the correct choices given those ranges. At first, you have to work it out (or use software that helps) but with training, you have a good feel for it (you might already have a good feel for it, and the training just hones your intuition).

Information is power is money in poker. If I have information about your hand (or the range of hands you might hold when you do an action), my actions will be better. I will be empowered to make the correct choices. And if I have learned ICM, I will make the choices that make me money, while you, lacking the information I have, will make the choices that lose you money. Yes, you will stumble on the right choice a lot of the time, but you will make the wrong ones sometimes, and each wrong choice will cost you just as not choosing to sell a share the day before it falls in value costs you.

posted by Dr Zen @ 9:20 am