EPT[Note: All of the numbers have been rounded for display purposes only. The math was done using 4 decimal places accuracy.]
Using simple probability, if two evenly skilled players play in a best of 3, then the odds that the winner of game 1 will win is 75%. [Math: If game 1 is a given, then the opponent needs to win 2 games in a row. At 50% chance each, this is only like 25% of the time]. Now, obviously not all matches have perfectly evenly matched players. I checked all of the GSL season 1,2,3,4 games and found the following interesting piece of information. In all tournaments at all levels, the winner of the best of 3 won the first match 81% of the time. What does this mean? It means that on average, there was a difference in skill between the players such that it resulted in 81%. By doing a little math and probability, this occurs if one player, on average, was the favorite to win 74.5% of the time (rounded to 75% for good ease). With a 75% chance of winning a game, the players chance of winning the series is 84%. Playing in a best of 3 instead of a best of 5 means that the better player has a 9% increased chance of winning the series. Playing in a best of 5 increases your chances to 90% or another 6% improvement. Playing in a best of 7 increases your odds to 93%.
Now, this is an average difference. If seeded correctly we can assume that the first round of games that the winner had a more that 75% chance of winning and that the final game had roughly a 50% chance. (which is mostly confirmed from gsl results, still need a little more information). Then here are some interesting facts from GSL.
Assuming a 32 player tournament, the average skill difference will be 85%-15% of an ELO difference of 300. The round of 16 has an ELO difference of 170. Round of 8 a difference of 130, round of 4 a difference of 90 and little or no difference for the top 2. (Note this is an average distribution, at the very highest level I doubt that the distribution is perfectly normal, but if they did 10,000 GSLs, this is what it would look like).
So with that information let's look at the possibilities.
For round of 32. With a 85% chance of winning for the average better player, what is the difference between a best of 1 and best of 3. A best of 3 improves the odds of winning to 93% for the better player. A best of 5 improves the odds to 97%.
Is it worth it?
Let's see. If you play a best of 1, the odds that all of the top players go through to the round of 16 is 7% (mean of 12/16 correct players). If you change it to a best of 3, the odds increase to 31% (mean of 14/16 correct players advancing) if you do best of 5 is 61% (mean of 15/16 correct players advancing)!
Is 12/16 correct players advancing good (4 upsets)? Probably not. By doubling the # of games, on average 2 better players will advance (only 2 upsets). If we do a best of 5, there will be on average only 1 upset.
After best of 5s the aggregate value decreases drastically.
Therefore, I would recommend the ro32 be a bo3.
Ok ro16. Best of 1's will result in an average of 3.5 upsets. Best of 3's will result in an average of 2.3 upsets. Best of 5's will result in an average of 1.5 upsets. For best of 7 (1.2 upsets). I would say that for this level, a best of 5 is preferred, but best of 3 is good enough.
Now ro8, Best of 1's will result in a little more than 1 upset , best of 3 will have around 1 upset, and best of 5 will have around 1/2 an upset.
At this level. bo5's are probably needed.
After that, it's really a crap shoot. The information I have gathered on the ro4 is so small that it's too hard to make any meaningful data. At the highest level, the difference between the #1 and #2 player could be 0 ELO or it could be 100 ELO. An interesting stat though is that for GSL finals/semi-finals, in all but 1 case, the winner of the best of 7 would have won the best of 3, but that the winner of the best of 7 only wins game 1 63% of the time. Best of 3 are probably good enough for finals.
The next part I want to talk about is non-GSL tournaments. Dreamhack had their best of 1 format for a lot of the tournament. Using the normal distribution information, if you are trying to run a tournament that is cost effective, best of 1 should be good enough until the round of 16 at which point you can just do a best of 3 until the finals and even a best of 3 for the finals and still be very accurate.
tldr; best of 5's are not really worth it before the round of 8 and are only marginally better than a best of 3. Best of 7 is pretty unnecessary. Best of 1's are good for open tournaments until the round of 16. Best of 3's for round of 16 on is good enough for most tournaments (except maybe world championships). The difference in skill between the top 16 and bottom 16 in code S is around 300 ELO on average (which is huge). Difference between top 4 is probably around 80 (which is not huge).
