Most tournament poker players agree that after some point the game becomes entirely mathematical. The debate about the relevance of mathematics in poker is primarily focused on where this point is and to what degree the tournaments are mathematical before this point. Although this topic delves deeply into complex issues such as game theory, the ICM Calculator and the ICM Applicator can help any player to see and understand where and when math becomes important.

ICMs are a crucial part of any tournament poker calculation. This is because in order to convert your chipstack into money (which is trivial for a cash game) becomes a tricky problem in tournaments because you need to incorporate every player's chipstack as well as the payout structure. The ICM Calculator takes this information and does the algebraic conversion for you. The ICM Applicator takes it one step further in that it uses certain ICM calculations and opponent hand ranges to tell you, based on that information, exactly how much you make from pushing each hand. Since it can check every hand and figure out how much you make if you fold versus how much you make if you play, it can tell exactly which hands to push in any situation where 2 players have gone all in!

There are many ways to go about applying ICM calculations to model some situation. One technique that has been growing in popularity is the interpolation technique. Interpolation means that they store some data points (in this case, equities) and when they need a value somewhere inbetween this set of data points, they interpolate to get that value using the information provided by the surrounding points. Because they are often estimating equities by interpolation, they are always battling against error. Relatively speaking, increasing the number of points (in an intelligent fashion) will decrease the 'distance' of interpolation and hence, reduce error. Another way to reduce error would be to make a better interpolation function. For example if I give you a graph with just 2 points of data, (5,10) and (10,20), a linear interpolation would say, between our points we have a x distance of 5 and a y distance of 10 and rise/run provides us with a slope of 10/5 or 2. Then say we wanted to find the value at x=7. We start at our closest point and progress to x=7 using the slope that we calculated and eventually get y=14. This is the simplest kind of interpolation and likely not the function used by softwares employing this technique. To see how ICM applicators use interpolation, bring up your free copy of PokerStove. Now for Player 1 choose some fixed range and never change it in the future. Now for Player 2, choose 1% and then calculate. Okay, now repeat for 2%, 3%, 4%, 5%... and you should start to notice a trend. These sites include SnGWiz, SnGEGT, PokerHound and more.

The other technique available is an exhaustive, iterative technique. In order to exhaust all possibilites, we need to have every single point of data. The exhaustive technique guarantees 100% accurate solutions because it never has to approximate anything. This technique is seen less frequently as it requires a significant amount of extra data and computation than an interpolation technique would require. This is because for a situation where just 2 players have moved all-in we need 169*169=28,561 different preflop equity values. For a situation where 3 people have moved all-in we have 169*169*169=4,826,809 different values, thus making it impossible to do an exhaustive technique with more than 2 players on modern machines. This technique was first seen in the popular software Sit n Go Power Tools.

The ICM Poker Online Applicator also uses an exhaustive technique. If a calculation took just a couple seconds running a server with a few tens of users the server could easily enter a state of no return. Our exhaustive technique is able to compute the same values as other exhaustive techniques, but in just milliseconds. It is this secret that lets us provide our partners with a modulized and skinnable version of our tools to implement inside their own website.