The relationship of ROI and EV in poker tournaments
One famous online poker player once turned down a (live) promotion request that this would cause him to lose 4000 dollars worth of EV. It would have been on a Sunday, which certainly is an online tournament player's most important day, but what could he have thought about and how did he come to this number?

The expected value of tournaments

Let us assume that someone proposes the following game. You pay 3 dollars, get a die, and the number you manage to roll is the number of dollars you get. Is the game worth it, and how much winnings can you expect each roll? It's gambling, so you certainly can't know the results, because those are decided by the die rolls. However, you can say that these conditions are good for you and that every roll will net us 0.5 dollar, which is what the EV is going to be with every roll.

Every decision and investment's value can be calculated if we know the chances. Knowing the cards during a poker party it's easy to calculate the expected value of the winnings in chips or dollars. But what about tournaments? If you sign up for an \$ 11 tournament, how could you tell beforehand what place you'll finish in? Is it even worth it to participate in the tournament? And what ramifications will our finishing place have? If you finished in the money, does that mean it's worth to try the same event another time? If you lost the money, should you choose another tournament the next time?

Defining ROI

We talked about this a couple times but it doesn't hurt to repeat. Tournament players' success over a long term is shown by ROI (return on investment). This shows the ratio of investment and the money earnt, and you usually use its percentage form. Calculating it is the following: ROI%=investment earnings-investment costs/investment costs*100%. Tournament players usually look at overall ROI%, which means the summarized ratio of deposits and winnings.

The relationship of ROI and EV

How to calculate the expected value at a tournament? You'll get entirely different games than yesterday or last week, and obviously, the opponents are going to be different as well. A tournament is so complicated that this seems to be impossible. You're right, you can't make calculations like this. The best you can do is estimating, and even that is only possible with a big enough pool of games. For the estimation, there is only one objective information that can help, which is your previous ROI% under similar circumstances.

The thought process of the player mentioned early on in the article was the following: during my large, 50.000 games pool I managed to accomplish 40 ROI%. You can expect that to be similar in the future, so after every 1% I invest I'll get 1,40% back in the long term. During my Sunday session, I'll probably invest about 10000 dollars worth of buy-ins, so my EV for that day is going to be +4000. If I can't play, I'll lose those 4000 dollars.

In theory, no statistics are proof that it's going to be the same in the future. It might seem a bit odd, but this explanation is acceptable, especially when someone has such a large pool of games to prove it with. (In reality, the opponents are getting better, so with the same knowledge the ROI% will decrease over time).

A few exceptions

There are tournaments which are famously featuring weaker than average players, such as the main events of the bigger online tournament series, the Sunday Million, or the Sunday Storm. We can also say this about the WSOP main event in Las Vegas. What's common in these events is that with so many opportunities to get in through satellites many amateur players participate in them. If the majority of the field is worse than you, you can count with a higher than usual (possibly even twice as much) EV. (In theory you should only participate in tournaments like this, but unfortunately, there's a problem. Where there are many amateurs, beginner players, the field is always large, so you have to deal with variance).

If there are weaker than usual tournaments, there are obviously stronger than usual ones. These are the bigger prestige or the bigger buy-in ones, such as the Sunday 500 or the Super Tuesday. We can also say that the 180 man SNGs are stronger than usual, which do have low buy-ins but feature many professional players. It's very possible that for a player with a positive ROI these tournaments will eventually result in a negative expected value.

Do you have any questions about the relationship between ROI and EV? Post them in the comment section below.

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