This article includes excerpts from the latest Dimat release, The Math of Hold’em, by Collin Moshman and Douglas Zare.
Your equity in a poker tournament is your “rightful share,” in dollars, of the prize pool. If the game were played out a million times from that point onward, your equity is how much you would win on average.
Since chips change in value, your equity is not a linear function of how many chips you have. We will discuss a method of calculating equity as a function of chips stacks, the Independent Chip Model, in the second half of this article.
For now, just note that there are two common ways of expressing tournament equity. The first is dollars. When you purchase entry into the tournament, you are exchanging cash for chips. Cash is therefore a logical measure of equity. If everyone is equal-skilled at the beginning of a $10 tournament, then everyone has $10 in equity, regardless of whether the starting chip stack is 100 chips or 50,000 chips.
The problem with this equity measure is that it’s difficult to generalize when speaking in terms of dollars. For instance, suppose you double your starting stack of 1,000 chips in the first hand of a 10-man, 3-places-paid sit-and-go with a $10 buy-in. We will learn below how to calculate that your equity has just increased from $10 to $18.44, for a gain of $8.44.
The more general result, however, is not that you increase your equity by $8.44 with a double-up; it is that you increase your equity by 84.4% of a buy-in, or equivalently 8.44% of the total prize pool.
In this regard, another common measure of equity is percent of prize pool. Multiply the buy-in (before the rake) by the total number of entrants to get the prize pool, and divide the dollar equity under consideration by this number to get percentage of prize pool.
Now let’s look at the details of converting chips into equity.
Independent Chip Model (ICM)
The Independent Chip Model (ICM) is the most commonly used method for converting chips to equity in single table tournaments, where the difference between plays which gain chips and plays which gain money is the greatest. The ICM takes in the stack size for each player, and predicts a probability of finishing in each place for each player. When you combine this with the prize structure you get a predicted equity for each player.
The fundamental assumption behind ICM is the P(1st) Assumption: The probability that a given player wins the tournament is equal to his stack divided by the total number of chips in play. In other words, if you have 75% of the total chips, then your chance to win the tournament is 75%, regardless of how many opponents you have or how their chips are distributed.
To compute your chance to place second, the ICM chooses a winner and removes his chips from play. Then your chance to finish second given that winner is assumed to be your proportion of the remaining chips. This is summed over all possible winners.
Similarly, to estimate your chance to finish third, the ICM chooses a winner in proportion to the stacks, then a second place finisher in proportion to the remaining stacks, then says you place third with probability equal to your stack’s proportion of the remaining chips. Let’s look at an example.
There are 3 players left with 3,000, 2,000, and 1,000 chips.
Question: What are the probabilities that the player with 3,000 chips places first, second, or third?
Answer: According to the ICM, the chance to place first is equal to the player’s share of the 6,000 chips in play, 3,000/6,000 = 50%. The chip leader places second to the short stack with probability 1,000/6,000 x 3,000/5,000 = 1/10. The chip leader places second to the medium stack with probability 2,000/6,000 x 3,000/4,000 = 1/4. The total chance to place second is 10% + 25% = 35%. The chip leader places third 15% of the time.
What is independent about this? You can imagine that each chip is labeled as in bingo, and a random chip is called. The owner of that chip is the winner. Then another random chip is called, and the owner of that chip places second, and so on. Alternately, you can imagine that the chips are randomly removed from play, one by one, independently of their position. When your last chip is removed, you are eliminated.
The equity of a given player is then calculated through a standard EV equation:
Player A’s Equity = Prob (A wins 1st) x First prize
+ Prob (A places 2nd) x Second Prize
+ Prob (A finishes Nth) x Nth Prize
where N = Number of places paid.
You can apply this ICM process to convert chip stacks into dollar amounts. However, the calculation is too complicated to perform at the poker table. Instead of performing the calculation, we recommend that you leave that to one of the many ICM calculators like the one at www.ICMCalculator.com. Learn the types of decisions recommended by the ICM, and then apply those at the table, not the ICM calculations themselves.
The most common application of ICM is deciding on pre-flop all-in decisions. While The Math of Hold’em discusses manual decision-making and real-time analysis, software such as SNG Wizard is also a very useful tool for ICM calculations.
Even though the ICM is a very helpful tool, and it is used widely, there are some limitations to its utility:
- The ICM only applies to tournaments.
- The ICM says nothing special about heads-up play. When you are heads up, the ICM agrees with the count of chips.
- The ICM ignores your location relative to the blinds. When you are about to post the big blind, you have a disadvantage not considered by the ICM. When you have several free hands before you post the big blind, you have an advantage.
- The ICM does not see possible advantages from being able to use a large stack to bully your opponents. Particularly when you have a run-away chip lead so that players are very risk-averse against you, and you are not risk-averse against them, you expect to do even better than the ICM predicts.
- The ICM does not take into account any skill differences.
So, the ICM is a good starting point, but not the last word on tournament equity.
Understanding tournament equity and how it impacts your decisions during a tournament is critical to becoming a good tournament player. There are many situations in tournaments where you have a situation which is positive EV from a chip perspective, but negative EV from a tournament equity perspective.
The Independent Chip Model is discussed further in The Math of Hold’em. In next month’s article, we will look at some specific bubble situations to show how ICM can impact your decision.