Our previous article only considered tournaments with 2 possible outcomes: win or lose. However, **most poker tournaments have multiple payouts and pay only 10-20% of the players**. Lower chances of winning more money substantially increases variance in potential winnings (and losses), which results in a larger bankroll required to avoid going broke.

Let’s kick this article off with a slightly more complex example than our previous article: 10-man sit & go’s with a $10+1 buy-in with the following payouts:

1st place | $50 |

2nd place | $30 |

3rd place | $20 |

12.5% x $50 = $6.25

12.5% x $30 = $3.75

12.5% x $20 = $2.50

- $11.00 entry**= $1.50 (13.6% ROI)**

We will assume that you are a talented player who will finish in the money more often than not. For this example we will give you a 12.5% chance to finish in each of the money positions, leaving a 62.5% chance of finishing out of the money in each Sit & Go. Therefore, your expected value (or EV) for each Sit & Go is $1.50 (**13.6% ROI**). As with our previous article, we have placed our more analytical sections in yellow, so that you can easily skip the math if you wish.

With a 62.5% chance of finishing out of the money, you can expect a run of 10 losses once every stretch of 119 tournaments (1 / 0.625^{10} + 9). That sure is a lot more frequent than flipping tails 10 times in a row! In fact, over a period of 1,000 Sit & Go’s, you should expect a stretch of 14 - 15 poker tournaments without reaching the money.

We simulated the results of 1,000 Sit & Go’s 200 times, with a 12.5% chance of finishing in each of the money positions in order to calculate poker variance. The results are below:

Range of Largest Downswing |
14 – 98 buy-ins |

Avg. Max Downswing |
29.8 buy-ins |

Std. Dev. of Max Downswing |
11.9 |

Even though our expected ROI is significantly higher in this case (13.6% vs 7.9% in the heads-up example), our minimum poker bankroll jumps from 34 buy-ins to 54 buy-ins (and that’s just to give us a 95% chance to avoid going broke over each stretch of 1,000 SNGs). For a 99.7% chance to avoid going broke, **we need a poker bankroll of 66 buy-ins**.

More entries and higher variation in prize money results in bigger swings in your poker bankroll!

There are several other factors to consider that introduce even more poker variance:

- Each Sit & Go will probably be played against
**different opponents**, so some stretches you will face tougher competition than others (meaning that your expected EV will fluctuate). In many SNGs, you may even have a negative EV, making a long stretch of losses that much more likely. - You are not a machine, so
**the quality of your play will fluctuate**. During stretches where you are not playing your best poker, the likelihood of long stretches of bad results will increase. - In the coin flip example, each “win” paid the same. With Sit & Go’s, you could reach the money 4 times out of 10 (an impressive 40%) and still lose (four 3
^{rd}place finishes would pay $80, but ten entries would cost $110, for a net loss of $30).

## Variance in Multi-Table Poker Tournaments

Multi-player tournaments typically pay a much smaller percentage of the field (typically 10-20%), and the difference between finishing first and just barely making the money is immense. In fact, for many tournaments, a guaranteed bottom of the money finish in 50% of the tournaments played would be a losing proposition!

Let’s look at a $1 buy-in 45-player Sit & Go from PokerStars, which pays:

1st place | $12.73 |

2nd place | $8.80 |

3rd place | $6.75 |

4th place | $5.11 |

5th place | $3.68 |

6th place | $2.45 |

7th place | $1.43 |

Only 15.6% of the field will finish “in the money” as opposed to 30% in the 10-player sit & go example, and just barely making the money will not recoup the costs of a single loss. Let’s assume that your skill increases your chances of finishing in the money to 21% (3% chance of finishing in each of the top 7 spots). Your EV per tournament would be **$0.21** (a very respectable **21% ROI**).

3% x $12.73 = $0.38

3% x $8.80 = $0.26

3% x $6.75 = $0.20

3% x $5.11 = $0.15

3% x $3.68 = $0.11

3% x $2.45 = $0.07

3% x $1.43 = $0.04

- $1.00**= $0.21 (21% ROI)**

Even though your ROI is significantly better in this scenario (21% vs 13.6%), **your swings will be much worse**. Although you are reaching the money more often than the average player, you can still expect to finish out of the money 10 times in a row once every 20 tournaments! (1/0.79^{10} + 9). Over a stretch of 1,000 tournaments, **you can expect 29 - 30 losses in a row at some point**!

Once more we simulated the results of 1,000 Sit & Go’s 200 times, with a 3% chance of finishing in each of the money positions and calculated the poker variance. The results are below:

Range of Largest Downswing |
27 – 108 buy-ins |

Avg. Max Downswing |
50.4 buy-ins |

Std. Dev. of Max Downswing |
16.1 |

To have a 95% chance of not going broke during a run of 1,000 sit & go’s, you would **need a poker bankroll of 83 buy-ins**. Again, even though your ROI is higher, the increased number of players (and thus more possible outcomes in each tournament) dramatically increases your poker variance! To have a 99.7% chance to avoid going broke over a period of 1,000 tournaments, you would need a bankroll of 99 buy-ins.

Any statistician who is reading this article is no doubt cringing since our sample sizes are most definitely not big enough. However, this is intentional on our part. If you play 10 sit & go’s each day, 5 days a week, it will take 20 weeks to play 1,000 tournaments. Although a larger sample size would give a more accurate minimum bankroll calculation, we are more interested in demonstrating the huge differences in the maximum downswing over a period of 1,000 tournaments.

The maximum downward swing in our 200 samples of 1,000 tournaments ranged from 27 to 108 buy-ins. All too often, a player who experiences a maximum downward swing on the low end of that range attributes this to skill instead of normal statistical variation. As a result, they begin to risk a larger percentage of their poker bankroll each tournament. Then, during the next run of 1,000 tournaments, they don’t get as lucky and quickly find themselves broke!

## Massive Tournaments = Massive Poker Variance!

For our final tournament example, let’s consider an event with a very large field of players: the Sunday Million at PokerStars. This is a $200+$15 weekly event and on 23 Mar 2014, there were 7,789 entries, with a **top prize of $233,674.32**. This tournament paid the top 1,170 players (15.0%).

For our example, we’ll assume that you are able to reach the money 20% of the time (with equal probability to finish in each of the paying positions). This will result in an EV of $51.29 per tournament (**23.9% ROI**). Over a period of 1,000 tournaments, you will need a bankroll of $101,628 (473 buy-ins) to have a 95% chance to avoid going broke. If you want a 99.7% chance to avoid going broke, you need a bankroll of $121,660 (566 buy-ins).

Once more we simulated the results of 1,000 tournaments 200 times, this time with a 20% chance to finish in the money (equal chance for each position). The data for the maximum downward swings in each grouping are shown in the table below:

Range of Largest Downswing |
83 – 485 buy-ins |

Avg. Max Downswing |
286.3 buy-ins |

Std. Dev. of Max Downswing |
93.2 |

To play the Sunday Million 1,000 times, you will need to play it **every week for over 19 years**! Even over such a long period, with a 20% chance to finish in the money and an expected ROI of 24%, 91 of our 200 samples resulted in an overall loss! One of our 200 samples of winning players was **$115,784 in the red** after 1,000 tournaments played! The average loss incurred by the 91 losing winners was $43,057!

While it’s not uncommon for a winning tournament player to have an ROI of 20% or more, most top players achieve this ROI, not by finishing in the money significantly more, but rather by finishing higher in the money significantly more. This actually introduces even more variation in results since wins will be even further apart.

It requires an incredible amount of discipline to have a bankroll of $120,000 and only risk $215 (<0.2%) per tournament. Sadly, if you exclusively play tournaments with thousands of entries, buying in for more than 0.2% of your bankroll means you will need to get very lucky to avoid going broke. In other words, you’re no longer playing poker and are just another gambler!

Large field tournaments are not a profitable investment because the variance in poker results is so extreme that even winning players may never show a profit.

If you’re only risking 0.2% of your poker bankroll per tournament, it will be almost impossible to either earn enough money to justify sustaining such a large bankroll or to grow your bankroll so you can start playing higher buy-in tournaments.

The more entries there are in each tournament, the more our minimum bankroll size increases, until eventually we end up with a bankroll requirement that is unrealistic. Since we still want to avoid going broke, the next article in the series will discuss ways to mitigate risk and reduce bankroll requirements.

Did you find our bankroll calculations for large field tournaments as shocking as we did? We'd love to hear your feedback in the comments below!