So you want to be a proficient tournament poker player? One of the most important things to learn is how poker tournaments are different from cash games. The game of poker is the same no matter where you're playing it, but the fixed payout structure of tournaments change the mathematics behind correct decisions.
cEV and $EV in Tournaments
The first concept we have to address in a lecture about poker tournament math is that of Expected Value. In cash game poker, your EV is the amount of money that you make or lose on any action. For example if you make a really awful call, you could lose $50 in EV, whether you end up winning or losing the hand (If you don't follow me so far, go back to our Strategy lecture on Expected Value). In tournament poker, however, you're working with two kinds of EV: Chip EV (cEV) and Dollar EV ($EV). cEV represents the amount of chips you stand to make or lose on a particular action, while $EV represents the amount of money you stand to make or lose on a particular action.
The important thing to note is that these are different. Why is this important? In cash games, these two things are identical. At any point, you can stand up with your entire chip stack and turn it into dollars. Therefore, if you win $1 in chips, you also win $1 in dollars. In tournaments, you don't have the luxury of cashing in your chips. Having more chips makes it more likely that you are going to win money in the tournament, but these two variables are not aligned in a 1-to-1 ratio like they are in a cash game. Instead they begin to diverge from the first hand.
When you first sit down at a poker table, you have exactly the same number of chips as anyone else at the table and, therefore, exactly the same chance of anyone of winning money in the tournament (skill aside). This means that your cEV is equal to your $EV. After you play your first hand, if you win 100 chips (a very small amount), you have gained a cEV of +100 and slightly increased your $EV. The two values have not yet diverged significantly.
The bubble is defined as the last position in the tournament that doesn't pay out. As tournament play progresses closer to the bubble, though, significant changes happen. Imagine that there are seven players remaining and the top six places pay out, and you are dealt a hand like JJ. A player (who has you covered) goes all in. If you call, you are probably about 65% to double your chips and make at least the minimum payout, but you are also 35% to go home with 0 chips and $0. So your cEV is positive: in a cash game, if you were 65% to win, you'd want to take this gamble. But in a tournament, it's not quite so simple. Doubling up will NOT actually double the amount of DOLLARS that you have, only the amount of CHIPS. While your DOLLAR gain will be equal to the 6th place prize and then a portion of the remaining prize pool, this has much less value than the same double-up in a cash game.
It's a complicated concept, but in theory, the thing to remember is this: every chip you gain is worth slightly less than the last chip you gained. Going from 1,000 chips to 1,300 chips on the first hand of a tournament is a solid win. Going from 100,000 chips to 100,300 chips late in the tournament is less valuable in terms of dollars taken home at the end of the night. In practice, this means you should be hesitant to flip coins for very small cEV gains when close to the button, as what appears to be a profitable bet is actually (in $EV) a losing play. Winning tournament poker players only care about the $ they take home at the end of the night, not the quantity of the chips they have.
The cardinal rule of bubble play is: "get in the money first, then get to 1st." You should be doing whatever it takes to beat the bubble, whether that's playing fewer hands, stalling your table (within reason) in order to let the blinds go up just as they pass you, or folding when a player puts you to a hard decision. At this point in a tournament, bubble dynamics trump any existing poker strategy you may have.
Due to the non-linear value of tournament chips, the chips that you risk will be of a lesser value than those you stand to gain.
Independent Chip Model (ICM)
The concept that "every chip you gain is worth less than the last chip you gained" is called the Independent Chip Model. It is a mathematical way to convert the number of chips you have into your tournament "equity" -- roughly, the amount of dollars your stack is "worth."
ICM is relatively self-explanatory if you understand the differences between cEV and $EV, and we really only use ICM at the end of a tournament when players are talking about splitting the final results.
After the bubble bursts
Play for first. After the bubble bursts, all considerations surrounding the bubble become much less important. Now, the cEV and $EV disparity, which has increased throughout the tournament to this point, have disappeared. Now, your chip EV and your $ EV are identical again. The 1st place position will take home the most money, and the only way to get 1st in a tournament is to gather every chip on the table, so begin playing according to cEV (cash game) strategies once again.