How to Break Even in Poker Tournaments
You probably won't but hear me out. Let's say you only play in tournaments with a total buy-in amount of $100. Casino takes a conservative 10% rake so $90 goes directly into the prize pool. If there are 50 entries, this means that the total prize pool is now $4500 and if we follow the WPT payout structure, then the top 5 will get paid accordingly:
- 1st Place: $1620 | 36%
- 2nd Place: $1125 | 25%
- 3rd Place: $720 | 16%
- 4th Place: $585 | 13%
- 5th Place: $450| 10%
Given that they only pay the top 5 out of 50 and assuming that everyone has the same skill level, you will cash 10% of the time or 1 out of 10 tournaments you will enter. At face value, your expected value (EV) is the following:
\[ EV = \$1620*\frac{1}{50}+\$1125*\frac{1}{50}+\$720*\frac{1}{50}+\$585*\frac{1}{50}+\$450*\frac{1}{50}-\$90*\frac{45}{50}-\$10 \]
\[ EV = -\$1 \]
Yikes, not off to a great start. Over the long run, you'll be losing an average of $1 per tournament. What if instead you were a beginner? We'll adjust the probability of cashing down to 5% and increase the chance of busting to 95%. How does this change our EV?
\[ EV = \$4500*\frac{1}{100}-\$90*\frac{95}{100}-\$10 \]
\[ EV = -\$50.5 \]
Oof, now it's even worse. Every time you enter this tournament, you're expected to lose half your buy-in. And over the long run, that will definitely add up. So the question you probably have is what is the mathematical edge we need to have in order to at the very least break even? In other words, what should our probability of cashing be? Let's set EV to 0 and e be the edge
\[ 0 = \$4500*e-\$90*(1-5e)-\$10 \]
So when we solve for e, we obtain:
\[ e = 2/99 \approx 0.0202 \]
Therefore, to break even in this tournament structure, the probability of cashing has to be at least 10.1% or the probability of not cashing has to 89.9%. This tiny difference of 0.1% doesn't seem like a lot, but because we are always thinking about the long run, you need to be aware of the fact that if you're always playing tournaments where you are at the bottom end of the skill ladder, you will never profit. You'll definitely get the occasional sun-run, but whatever profit you earned in that round will get swallowed up in future tournaments if you're being outplayed.
Let's now generalize this. If \(p_{o}\) is the payout percentage, \(p_{r}\) is the rake percentage, \(n_{b}\) is the number of bullets, and \(a\) is the buy-in amount, we can express our poker tournament EV as:
\[EV = a(1-p_{r})-n_{b}a(1-p_{r})(1-p_{o})-n_{b}ap_{r} \]
\[= a(1-p_{r})-an_{b}(1-p_{o}-p_{o}p_{r}) \]
We can use this equation to answer the question: how often do we need to cash in order to break even? In other words, what is our \(p_{o}\)? Set EV to 0 and solve for \(p_{o}\):
\[0 = a(1-p_{r})-an_{b}(1-p_{o}-p_{o}p_{r}) \\ n_{b}(1-p_{o}-p_{o}p_{r})= (1-p_{r}) \\ p_{o}-p_{o}p_{r} = 1-\frac{1-p_{r}}{n_{b}} \\ p_{o} = \frac{1-\frac{1-p_{r}}{n_{b}}}{1-p_{r}}\]
\[p_{o} = \frac{1}{1-p_{r}}-\frac{1}{n_{b}}\]
Notice that the number of entries and buy-in amount do not play a factor if we are strictly looking at the break-even payout percentage. What matters the most is the average tournament rake and number of bullets you fire. Let's look at rake first. For most tournaments you'll be entering, it'll range in between 10-15%. If we assume that you'll only fire 1 bullet, a 15% tournament rake means that you'll need to cash at least 18% of the time to break even. You can for most cases treat the rake as a fixed number so this is a good benchmark to shoot for.
However, let's now look at bullets. If on average you just fire 1 bullet, you don't have to do that much better than random. But the moment you fire 2 or more bullets on average, you've got quite a predicament. Setting the rake percentage at 10%, if you fire 2 bullets, then your \(p_{o}\) shoots up to 61%. You'll need to cash at least 6 out of 10 tournaments to break even if you're firing another bullet. When we start bumping that up to 3, 4, or even 5, you should start getting nervous:
| Avg # of Bullets | Avg p0 |
|---|---|
| 1 | 11% |
| 2 | 61% |
| 3 | 78% |
| 4 | 86% |
| 5 | 91% |
The results are...pretty grim. Knowing that most players are firing more than 1 bullet in a tournament, most people are not make a profit in the long run. Something we haven't even factored in to all of this is the fact that even if you make it in the money, moving towards the top is incredibly difficult, especially when facing increasing blinds. It also helps to be extremely lucky. While skill edge is still important, there will be a point in the tournament where you need the cards to fall in your favor most of the time. If 10% of the time you're just min-cashing, then your EV will drop even further down. You have to play to win.
From a theoretical standpoint, you can break even playing poker tournaments, but the odds are severely against you. Best of luck. And see you at the final table.