In my last column, I gave an introduction on how to assess your value at the final table of any poker tournament. As long as you know the prize structure, the stack sizes of your opponents, and your own stack size, you've got all the information you need - assuming, of course, that you neglect any skill differences between the players. To keep the math simpler, and to avoid overrating your own abilities (something poker players are guilty of all the time), I recommend treating all final-table players as equally skilled when calculating your own value.
So, how do we go about calculating value? I mentioned in my last column that several poker-math people have already taken a pretty good crack at this question, and I will summarize their work here. A similar, more detailed summary can be found in the fantastic book
The Mathematics of Poker, by Bill Chen and Jerrod Ankenman.
The simplest and (therefore?) most commonly applied method of valuation is the Chip-Count Method. To use this method, first figure out what percentage you have of the total chips in play. By the Chip-Count Method, you are said to be worth that same percentage of the remaining prize pool (remember to remove the money that has already been locked up when calculating the remaining prize pool). So, if you've got 10 percent of the chips and there's $100,000 remaining, you're worth $10,000. The biggest advantage to this method is that it's relatively easy to calculate at the table. There are some enormous disadvantages, however, at least if you're interested in accuracy. Neither the prize structure nor the various stack sizes are taken into account, and ignoring those two factors is a big problem. To highlight the problem, let's say there are four players left, and the prizes are: first - $100,000, second - $50,000, third - $25,000, fourth - $0; and let's say you have 60 percent of the chips. There is $175,000 in the prize pool. The Chip-Count Method says you're worth .6 x $175,000
= $105,000. By chip count, you are worth more than first-place money! Since it doesn't take the prize structure into account at all, the Chip-Count Method highly overvalues big stacks at the expense of small stacks. If you're a big stack at a final table, and one of your opponents proposes a deal based on chip count, you probably should take it immediately. Conversely, never agree to a chip-count deal when you're a short stack.
Poker-math guys Steve Landrum and Jazbo Burns decided that they didn't like the Chip-Count Method, and wanted to propose a formula for final-table valuation that actually took the prize structure into account. They came up with FP + (1-F) x (T-P)/N, where F is your stack size as represented by a fraction of the total chips in play, P is the first-place prize, T is the total prize pool, and N is the number of opponents you have. In the above example, you would be worth .6 x $100,000 + (1-.6) x ($175,000-$100,000)/3 = $70,000. That's a much more reasonable estimate. The Burns-Landrum formula gives you your proper first-place equity (with 60 percent of the chips, an average-skilled player has a 60 percent chance of winning), and then assumes an equal chance of finishing in any of the other three places. Burns-Landrum is better than the Chip-Count Method, but it's far from perfect. Let's say that instead of having 60 percent of the chips, you have 5 percent of the chips in the above situation. Now, Burns-Landrum says you are worth .05 x $100,000 + (1-.05) x ($175,000-$100,000)/3 = $28,750. You have only 5 percent of the chips, and are very likely to finish fourth and win nothing, and yet Burns-Landrum says you are worth more than third-place money! This is because Burns-Landrum gives you more than a 30 percent chance of finishing second, which is not realistic with such a small stack size. The Burns-Landrum formula overvalues short stacks at the expense of the big stacks. If you're a short stack and can get a Burns-Landrum deal, I suggest that you take it quickly.
Is there any method that's both accurate and easy to figure on the spot? Not really, in my experience. But there is a more accurate method, and it's commonly referred to as the Independent Chip Model (ICM). ICM takes into account the entire prize structure (not just the first-place amount), and all of your opponents' individual stack sizes. Neither of the previous formulas does this. In the above example, let's say Player A has 60 percent of the chips, Player B has 20 percent of the chips, Player C has 15 percent of the chips, and Player D has 5 percent of the chips. ICM will first calculate a player's equity from finishing first (in the same manner of Burns-Landrum), but then it figures the chances of finishing second, third, and fourth, based on the other stack sizes. In other words, it attempts to tackle the question, "What happens when the player doesn't win?" According to ICM, to determine who finishes second, you look at the situation as a three-way tournament between the remaining players, with the winner's chips simply removed. Therefore, during the 20 percent of the time that Player B wins, Player A finishes second .6/(.6+.15+.05) = 75 percent of that time. Similar calculations are made for when Player C and Player D win to determine Player A's overall chances of finishing in second place. ICM compares the stack size of Player A to all of the remaining non-winners for every scenario, and then comes up with a valuation. This valuation is extremely difficult to calculate on the fly when you're playing, unfortunately. But surfing the Internet will find you some good ICM calculators that can be very useful to mess around with. Here are the ICM valuations for the players in this example.
Player A - $76,827
Player B - $45,536
Player C - $38,214
Player D - $14,423
As you can see, Player A is valued much lower than he was by the Chip-Count Method, but higher than he was by Burns-Landrum. Player D, meanwhile, is valued significantly lower than he would've been by Burns-Landrum, but actually higher than he would've been according to the Chip-Count Method ($175,000 x .05 = $8,750).
If most of this material was new to you, take some time to familiarize yourself with it. You can't possibly be overprepared for when that big day comes and you find yourself competing for boatloads of money against tough competition. You'll want every advantage you can get at the final table, and knowing your value is a big part of the battle.
Matt Matros is the author of The Making of a Poker Player, which is available online at www.CardPlayer.com.
