^Mod edit, new link
In addition to this
thread i wrote a spreadsheet program that simulates the variance STT results based on your input. You may then review some typical runs and your profit distribution.
For example, here are some results from my 9-player 20+2 STT simulation, 5000 tournaments, ROI=9%:
Furthermore, by performing simulations I obtained some dimensionless graphs that might show variance in a graphical way and is useful for most buyin levels.
1. Profit distribution
The profit distribution is shown as probability density function, which in this case follows a normal distribution (as expected):
Notice variance decrease considerably with increasing ROI and tournaments.
Furthermore, average and standard deviation is calculated which provide quantative statements about confidence intervals. For example, over 5000 tourneys with a 10% ROI the standard deviation is about 0.218*Expected value (which is just the expected profit based on ROI). You may then conclude that:
- There is about 68% chance your profit is between 0.782*EV and 1.218 EV
- There is 2.2% chance your profit is smaller than 0.564*EV
- There is a 0.1% chance your profit is larger than 1.654*EV
Use your own parameters for simulations using the excelsheet to create insight in possible profit distributions. You may want to check wikipedia
on normal distribution and confidence intervals.
2. Chance of going broke
The neccessary bankroll for playing SNG's highly depends on your true ROI. The chance of going broke converges with increasing number of tournaments, as is also shown in the following graph. Note that in this case there are no withdrawals.
I'm currently working on graphing the chance of a downswing of a certain amount of buyins after N tournaments (based on ROI=x%). However, the algorithm of identifying downswings takes an enormous amount of calculation time for lare number of simulations and tourneys.
To perform simulations for your own parameters, you can download the spreadsheet. As VB code is used, you may need to enable macros in your Excel security settings.
Note 1: Nondimensional profit distribution is based on 3 sets of 32000 simulations for each roi and number of tournaments.
Note 2: Nondimensional results are based on a 10% rake situation. Results may change with other rake levels.
Note 3: For the nondimensional simulations the following finish distribution is applied: 40% first, 30% second, 30% third (percentages of ITM part). This is quite a realistic finish distribution for a good player but can be changed in the sheet for your own simulations.