Inspired by ez4u's previous thread (http://www.lifein19x19.com/forum/viewtopic.php?f=10&t=9217), where he calculated the probability of a player repeating as champion under various tournament formats, I sat down to look at the round-robin tournament format. I've been thinking about such things for a while, but Dave's work prompted me to sit down and actually do it.
This work considers round-robin tournaments having 6, 8, 10 or 12 players. Each player's playing strength is modelled with an Elo rating, with each player being set a fixed number of Elo points stronger ("the player rating difference") than the next stronger competitor. This gives a ladder of ratings, rather than a single player dominating a group of roughly comparable competitors. For reference, a 100 Elo-point difference corresponds to a 64% chance of the stronger player winning and a difference 200 Elo points gives a 76% winning probability. For each combination of tournament size and player rating difference I simulated 1,000,000 fictitious tournaments and calculated the final standings of the players. Standings were calculated only using the number of wins; no tiebreaks were used.
Initially we are interested in two situations: the top player wins the tournament outright or is tied for first place. We start by considering the probability of winning the tournament outright, shown in the first graph below. There appears to be three regimes of interest. In the first, for a player rating difference larger than ~125 Elo points, the size of the tournament has very little impact on the winning chance of the strongest player. Presumably this is the because the players at the bottom of the field are so weak that their chances of staging an upset against a top-ranked player is minimal--adding a bunch of kyu players to the Meijin tournament is not going to affect how Cho U does.
A second regime, below ~25 Elo points, has the strongest player doing worse the larger the tournament. This is because the players are so close in strength and upsets occur across the entire field. The winner of the tournament is approximately chosen at random, and the chance of being selected improves with a smaller field. Finally, in the transition regime between ~25 and ~125 Elo points we see the effects of both player strength and tournament size asserting themselves, although there are diminishing returns to larger tournaments.
The probability that the top player is involved in a tie for first place is shown below. As before, in the regime above ~125 Elo points the effect of tournament size is negligible. The regime below ~25 Elo points, however, completely vanishes, and the probability of being involved in a tie actually reaches a maximum between 75-100 Elo points. This came as a surprise given that that top player is noticeably stronger than the rest of the field.
Taken together, the probability that the top player is in contention (outright win or tied) is shown below. For rating difference on the order of 50 Elo points, which corresponds a winning percent of ~57% for the stronger player, the top player is in contention roughly half the time. In go terms, most tournaments are going to involve players within a stone of each other, roughly less than 100 Elo points. The top player is not necessarily going to find it easy going, which helps keep the go scene interesting for the fans.
, similar to calculating the probability is the median, only for the maximum