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Analyzing Fifty50 poker tournaments with ICM Calculator

by Q.

The Fifty50 tournament type is quite unique, and it stands out from the long list of typical tournaments. The overwhelming majority of poker tournaments have a fixed payout for a certain finish position, which is known at the start of the tournament. Fifty50 tournaments don’t work like that.

Fifty50 poker tournaments are somewhat similar to satellite tournaments, which have a number of equal payouts and where the tournament stops once the number of remaining players is equal to the number of satellite tickets. Fifty50 tournaments stop when half of the players are eliminated. The typical number of players in Fifty50 tournaments is 10 – when only 5 players remain, the tournament is complete.

Fifty50 tournaments are also quite similar to another non-standard but rather popular tournament type – DON (double or nothing). In DONs, half of the participants of the tournament get paid, and each winner gets 2x the buy-in of the tournament. In Fifty50 tournaments, half of the tournament participants get paid. If the stacks are close when a Fifty50 tournament finishes, the payout will be quite similar to that of a DON tournament.

What is different though is that once a Fifty50 tournament is complete, the remaining players get paid according to a much more complex payout formula. Below is the quote from the official page on PokerStars regarding this tournament's payout structure.

The payout formula looks really geeky. We can tell by examining it, that a finisher with a bigger stack will get paid more than someone with a very short stack. We can also see that there is a minimum payout which is equal to the buy-in (excluding rake). So even if you have only 1 chip you will still get you buy-in back, but the product of multiplication in the brackets will be very close to $0 for someone with 1 chip.

One of the most popular questions from players who play Fifty50 tournaments is the following:

Can an ICM calculator like ICMIZER help me with analysis of hands in Fifty50 tournaments?

And the short answer is: “yes”!

ICMIZER has a special Fifty50 payout structure in the list of default payouts, which looks like the following:

Fifty50 payout structure in ICMIZER:

1st: 60%, 2nd: 10%, 3rd: 10%, 4th: 10%, 5th: 10%

Often when Fifty50 players first see this payout structure, they get confused. After all it doesn’t really look like the formula above, so it raises some questions: Is that an accurate reflection of the Fifty50 payout structure? Is it precise, or it is an approximation of some sort?

The answer to those questions is usually surprising: it is the ideal and exact ICM (independent chip model) payout structure for Fifty50 tournaments, which will result in precisely the same payouts as the formula above will produce.

In fact, what is even more striking is that Fifty50 tournaments are paid out directly according to ICM formula for nonzero chips stacks. This is different from the typical tournaments like SNGs or MTTs where only players with 0 chips (except for first place finisher) get paid with a fixed payout corresponding to their finish position.

One of the doubts about ICM and its applicability to tournament poker is that in typical tournaments like SNGs, ICM-based dollar evaluations of nonzero stacks are always performed with a certain error of an unknown magnitude. This is due to the theoretical nature of ICM-based formulas, which is used to convert a chip stack into its dollar value equivalent.

We accept that there may be imperfections with ICM in normal tournaments, but at this time (as of 2014) we simply haven’t found a better alternative to analyze poker tournaments. Since ICM has been around for years now and it has been heavily field-tested, it has gained trust and is recognized as the best model for chip / dollar EV stack conversions.

On the contrary, Fifty50 tournaments do not raise any concerns of this kind. The "magical" 60%/10%/10%/10%/10% payout just works.

Why is 60/10/10/10/10 the right payout structure for Fifty50 tournaments analysis?

This is a rather difficult question. I will provide a mathematical proof below, but right now let's simply check whether or not results are the same if we run an ICM calculation using this formula as compared to the results on the screenshot above.

http://www.icmpoker.com/icmcalculator/#WTZo

And voila – the numbers match completely. This however doesn’t explain why the formula works. Let's see why it does now.

Mathematical proof of why the [60, 10, 10, 10, 10] payout structure works for Fifty50 tournament ICM analysis

First, let's look at the standard ICM formula, which allows us to convert chip stacks into their dollar equivalent.

So for the jth stack, ICM EV is equal to the sum of the products of the probabilities to finish in a certain place and a payout for finishing at that same place.

Now let's say that the tournament has two distinct payout structures and that payouts are made in order with both of them. We'll call them X payout and Y payout structures, which can have different numbers of places paid, etc.

We need to know that we can simply add those X and Y payouts line by line and get a Z payout, and that ICM EV for stacks according to two X and Y payouts will be the same as ICM EV according to Z payout, which is sum of X and Y payouts.

We can see that:

Now we will break our payout for a Fifty50 [60, 10, 10, 10, 10] (which will play the role of Z or the sum of the two payout structures in above formula) into sum of two payouts X and Y, where X = [10, 10, 10, 10, 10] and Y = [50, 0, 0, 0, 0].

Now let's have a good look at PokerStars' formula for Fifty50 tournaments:

We can see that an actual payout consists of two parts: the left part, which is equal to the buy-in, and the right part, which is a product of multiplication. The tournament ends when half of the players are eliminated, or when 5 out of the original 10 players remain.

We can see that the initial buy-in (which is 10% of tournament prize pool) is the left part of formula and is what we now call the X payout. According to X, each of the top 5 place finishers gets 10% of prize pool or buy-in.

The second part of the formula is a multiplication product where half of the prize pool is distributed (50%). We can see that the fraction which is applied to half of the prize pool is equal to the formula which is used by ICM to calculate the probability to finish the tournament in first place.

Since our second payout Y only has a single first place payout, we can write a default ICM EV formula for Y in simplified form:

The payout for the first place is equal to half of the prize pool, and we can see that ICM EV Y is equal to the right part of the Fifty50 payout formula.

We can now see that the Fifty50 payout structure consists of the sum of ICM EV for payout X and ICM EV for payout Y. Further, we can be confident in saying that ICM EV can be calculated according to Z payout which is the sum of X and Y and is equal to [10,10,10,10,10] + [50,0,0,0,0] = [60,10,10,10,10].

We can now say that the PokerStars Fifty50 formula is an equivalent of an ICM calculation according to a [60,10,10,10,10] payout structure, and that's why we can use that payout structure for analysis of Fifty50 tournaments.

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