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Example?
Posted By: Roland Scheicher
Date: Monday, 29 June 2015, at 12:40 p.m.
In Response To: Example? (Chuck Bower)
In my opinion there are two principles, which should be adhered to, whenever there are seeded players and byes.
a) Every player – whether seeded or not – should have the same chance of winning a bye.
b) The byes should be distributed as equally as possible within the tournament tree.
However, these two principles can be accomplished in the following cases:
We want to assume, that the number of seeded players is a power of 2 (which we can always achieve by simply seeding additional players if necessary)
1. there are no seeded players, but possibly a few byes
2. there are no byes, but possibly a few seeded players
3. the number of byes is a power of 2 and less than or equal to the number of seeded players
4. the number of byes is a multiple of the number of seeded players.
Note that in the example given, principle a) is not adhered to.
Since there are 13 byes and 51 entrants, each player should have a probability of 13/51 = 25,49% to win a bye.
Let’s have a look at the tournament tree: there will be 8 subbrackets with 8 slots each. 5 subbrackets contain 2 byes, 1 seeded player and 5 nonseeded players, the remaining 3 subbrackets contain 1 bye, 1 seeded player and 6 nonseeded players.
A player in a subbracket with 2 byes has a chance of 2/6 = 1/3 to receive a bye, a player in one of the other subbrackets has a 1/7chance of winning a bye.
Now, let’s look at a seeded player. A seeded player’s chance to win a bye is (5/8) * (1/3) + (3/8) * (1/7) = … = 11/42 = 26,19%, which is about 0,6 points of percentage higher than 25,49%.
On the other hand a nonseeded player’s probability to win a bye equals to (25/43) * (1/3) + (18/43) * (1/7) = … = 229/903 = 25,36 %, which is slightly less than 25,49%
Note: At tennis tournaments, the byes are assigned to the seeded players in the order of seeding. If we had the above example at a tennis tournament, all of the eight seeded players would be assigned byes and three lucky non seeded players.
However, this procedure may be common practice at tennis, I do not like this idea at backgammon. I do not object seeding, since I would like the idea of having two or four very good players as guests, who will fight against each other in the final rounds. This might be quite a thrilling attraction to a club tournament. But since seeding in general favours the seeded players, assigning byes to these players means going much too far in my mind.

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