Let's look at how a rating system might be developed that you can use to evaluate your own play.
First look at a hypothetical heads up Holdem game, where you have twice the skill of your opponent. Since, this is a single trial situation and we'll progress to more opponents later, it will be best if we start by using the negative, "chance you will lose," instead of "chance you will win." There are two possible out comes. You will win, or you will lose. For every time you lose, your opponent should lose two times. The odds that you will be eliminated first are 1:2 or 1/3. (Of course that means you win 2/3.)
Now we'll continue and do three players and still a skill level twice that of our opponents. In this game, for every time you are eliminated first, each of your opponents will be eliminated first twice. So the odds of you being eliminated first is 1:(2+2), 1:4 or 1/5. So you will be eliminated first 20% of the time. Then the problem reduces to a heads up match for second place and you will be eliminated 1/3 or 33% of the time. The chance that you will be eliminated in 2nd or 3rd equals .33*(.80)+.20 = .266.+.20 = 46.6% of the time. You should win about 53% of the time.
You should have an idea how to calculate your expected win rate. Let's move out to a 10 seat Sit & Go and calculate the chance that you will be eliminated at each successive level. (To skip the math go to the **)
10th place: 1:(2+2+2+2+2+2+2+2+2)=1:18=1/19=.053=5.3%
9th place: 1:(2+2+2+2+2+2+2+2)=1:16=1/17=.059=5.9%
8th place: 1:(2+2+2+2+2+2+2)=1:14=1/15=.067=6.7%
7th place: 1:(2+2+2+2+2+2)=1:12=1/13=.077=7.7%
6th place: 1:(2+2+2+2+2)=1:10=1/11=.091=9.1%
5th place: 1:(2+2+2+2)=1:8=1/9=.111=11%
4th place: 1:(2+2+2)=1:6=1/7=.143=14%
3rd place: 1:(2+2)=1:4=1/5=.200=20%
2nd place: 1:(2)=1:2=1/3=.333=33%
Then we can calculate the chance that we will still be in at a particular level.
9th level: 1-.053=.947=94.7%
8th level: 1-(059*(.947)+.053)=1-.109=89.1%
7th level: 1-(067*(.891)+.109)=1-.169=83.1%
6th level: 1-(077*(.831)+.169)=1-.233=76.7%
5th level: 1-(091*(.767)+.233)=1-.303=69.8%
4th level: 1-(.111 *(.698)+.303)=1-.380=62%
**
3rd level: 1-(.143 *(.62)+.38)=1-.469=53%
A player with twice the skill of the other 9 opponents should make the money in a 10 seat tournament 53% of the time. On Full Tilt Poker, where the singles are 9 seat, you should make the money 56% of the time. Most of you probably find this number rather surprising
2nd level: 1-(.200 *(.531)+.469)=1-.575=42.5%
Winner: 1-(.33 *(.425)+.575)=1-.715=28.5%
*Major ASSUMPTIONS: For these calculations we assumed that you were at exactly twice the skill level of every opponent at each level. In truth, as the lesser skilled players are eliminated first, the average skill level increases and your skill advantage decreases. I'm working on the formulas to incorporate this increasing average into the numbers, but I can forecast the results. In large tournaments with a limited number of skilled entries, the impact will be small at the early levels, and become significantly more pronounced as you progress to the end of the tournament. The most significant change occurs when the average skill level approaches your own. If this convergence occurs prior to the bubble, your positive expectation will be skewed significantly. Also the adoption of a survival tactics by weaker players could affect your results. (Most of my math skills are rather rusty, and I would appreciate any assistance by more skilled mathematicians. If you can help with this project please, post it in the forums or PM jbharshaw.)
This methodology can be used to calculate the expected win rate in tournaments with any number of entries, and even for large Multi-Table Tournaments the numbers are just as surprising. A player with twice the average skill level should make the money against 45 opponents 44% of the time, against 500, paying 60 places, 55% of the time *.
What conclusions can we draw from these numbers? The differences in skill levels of competent poker players are significantly smaller than we might think. We probably aren't three times better than our opponents, but Phil Ivey and Daniel Negreanu aren't three times better than us, either. Based on the tournament results, The relationship between the best in the world, and the average poker player is probably a factor of only four maybe five, or possibly as low as three.