# Tournament Poker Variance and Backing

August 9, 2017

Poker tournaments are big business today, and for some players are fun to play and even make for a televised spectator sport. These tournaments also happen to have some interesting math associated with them, and I happen to love to write about interesting math. So, if you liked reading my past blogs about how much money Han Solo owed to Jabba the Hutt, or how to service a loan on a planet where the Roman Empire never fell, you may enjoy this.

Tournament Poker - as opposed "Cash Games" - is where the fun is. In Tournament Poker, you can buy in for $100 and first prize is $3,000, or buy in to the main event at the World Series of Poker for $10,000 and first prize is $7.5 MILLION. This is what gets the headlines and the TV coverage. It is a game where a skilled player can make money over the long run, but it has inherently high variance.

Let's start by defining and analyzing the variance. Take a hypothetical player who believes himself to be the best tournament player in the world and is probably right. We will call him, "Phil." Let's say that on average, for every 10 tournaments Phil enters, he cashes (in tournament poker, "cash" is also a verb) in one. And let's say that when Phil cashes in a tournament (10% of the time he plays), he averages a win of 50 times the buy in. This is enough information to calculate Phil's expected rate of return per tournament. So, let's say that Phil is playing in tournaments that cost $1,000 to play. Going with just the averages, it will cost Phil $10,000 to play in 10 of these tournaments, and he will cash for $50,000 one time, for a net profit of $40,000 over 10 tournaments, or an average of $4,000 per tournament. That's totally awesome. Remember that Phil is the best tournament player in the world.

But these were just averages. Now let's factor in Standard Deviation. Phil averages a cash in one out of 10 tournaments, but sometimes he will cash in two in a row, and sometimes he may go 20 or 30 tournaments between cashes. Statisticians define the amount of variance from the average by what is called standard deviation. I am not going to go into the specific math of standard deviation, but I will oversimplify the concept. If Phil is averaging one cash in ten tries than he is going, on average, nine tournaments with no cash between every time he cashes. If our standard deviation is about 10%, then we can assume that he is likely to go often only eight between cashes and often 10 or 11 between cashes. That means that over 10 events, even though he will average cashing one time, he is also likely to cash twice or cash zero times. Now this becomes important if Phil is playing in $1,000 buy-in events but only has a stake of $10,000 to play with. He has an expected win range of between two cashes and zero cashes over 10 events for a total expected profit between +$90,000 and -$10,000 (notice that the median amount is the same $40,000 we had in the previous paragraph) and an expected value per tournament of +$9,000 to -$1,000. He is still averaging +$4,000 per tournament, but now we have a high variance. If Phil plays in 10 tournaments and fails to cash once, he is broke. He can't play anymore. Phil needs to find a way to lower his variance.

Mathematically, what Phil needs to do is spread his $10,000 over 100 tournaments instead of 10. This will lower the variance but will be a whole lot more work. If Phil can now afford to have his win/loss ratio applied over 100 tournaments instead of 10, and we say that the 10% standard deviation is dictating that on average there are likely to be between nine and 11 losses between each win, then the likely outcome for Phil after 100 tournaments is now cashing somewhere between nine and 11 times. The smaller buy in means that the amount won when he does cash is now only $5,000, for a total win range of $45,000 - $55,000 on a cost of $10,000 for a net profit of $35,000 - $45,000 (still straddling our original $40,000 average; his win rate has not changed) over 100 tournaments, so we have a per tournament expected value range of +$350 to +$450. We have basically insured that for Phil to bust out is outside the realm of standard deviations, but we have also created a situation where Phil needs to work three months to grind out what he would have made in a week.

So, what is another way that we can spread Phil's risk while still allowing him to participate in the high buy in ($1,000) tournaments? Phil is going to sell shares of his action. There are a few ways to do this. Phil can get a backer to stake him with a deal that the backer pays all the buy ins and then when Phil cashes, he will pay his backer back for all of the previous losses, and then they split the remaining profit. The danger here is that a string of losses could put Phil in a hole that would become difficult to climb out of. This is called a makeup stake, and Phil wants no part of it. He is too good a player to need a deal like that.

With Phil's +$4,000/tournament expected value, there are a lot of investors who would like to take a piece of Phil's action without any expectation of getting their money back on any tournament in which Phil does not cash. Phil is going to sell stakes of his action at a 2:1 ratio. This means that for $100, you can buy a 5% stake in Phil's prize money. If Phil cashes and gets $50,000, you have won $2,500 for your $100 investment. Over 10 tournaments, Phil has shown that he will average doing this one time, so you will (if you invest in Phil ten times and he comes through the one time that is expected) expect to see +$2,500 on $1,000 in investments for a total expected profit of $1,500 or $150 per tournament. Not bad.

The deal is even better for Phil. If Phil sells 10 of these shares, he is free-rolling. So, let's say he tracks his average over 10 tournaments. He had 10 investors at $100 each for $1,000 per tournament, and his buy in was the same. He is playing for free with no risk. He is just doing all the work. He is expected to cash for $50,000 one time. That time, Phil pays out $2,500 to each of his 10 investors and turns a profit himself of $25,000 or +$2,500 per tournament. He has sacrificed +$1,500 of his expected value for the benefit of having ZERO POSSIBLE LOSS. This is called playing with other people's money. It is a tried and true practice for some tournament poker players and most stock-brokers. Now Phil still has variance but only on the positive side of his ledger. The variance on the negative side of the ledger has been reduced to precisely zero. Phil is not a gambler. He is an athlete in a mental sport getting paid to do his job. But to have such a sweet deal, Phil must be the best in the world at what he does. Once that is no longer true, Phil will lose his investors. He may have no downside, but to keep that situation alive, he has to keep winning.

What does this have to do with loan servicing? Well, Phil could track what he owes to his investors on loan servicing software. It's a long shot, but then so is being a consistent winner. Phil is clearly exceptional.