Tournament Seed Optimizer

A knockout bracket is governed by two pieces of arithmetic and one probability model. Since every game eliminates exactly one competitor, a single-elimination event with a bracket of size B always takes B minus 1 games and log base 2 of B rounds, and any entry count that is not a power of two is padded up to the next one, with the surplus slots handed to the highest seeds as byes. Double elimination roughly doubles that to 2B minus 1. The seeding order comes from a recursion that pairs seed s with seed n plus 1 minus s within each half, producing 1 against 16, 8 against 9, 5 against 12, 4 against 13 and so on, which guarantees that if every favourite wins, seeds 1 and 2 cannot meet before the final and the top four cannot meet before the semi-finals. The probability layer is the Elo logistic curve, in which a rating gap of 100 points implies about a 64 percent win chance, 200 points about 76 percent, and 400 points about 91 percent by construction, since 400 is defined as a tenfold odds ratio. Ratings between the top and bottom seed are interpolated linearly here, a strong assumption given that real fields are usually top-heavy, with a long thin tail rather than an even ladder. The championship path multiplies the favourite's win probability against the nominal seed expected in each round, which understates the top seed's true odds because earlier upsets usually deliver a weaker opponent than the seeding implies. Elo also assumes transitivity and ignores venue, rest, injury, and stylistic matchups; for reference, in NCAA men's basketball a number one seed has lost to a number sixteen only twice, in 2018 and 2023.

The word seeding comes from lawn tennis, where organisers in the 1920s began deliberately planting the strongest entrants at separate points in the draw so that a random pairing could not eliminate two favourites in the opening round. Wimbledon adopted the practice during that decade, initially as a way of spreading players from different nations before evolving into ranking-based placement. American collegiate basketball took much longer: the NCAA tournament began in 1939 with eight teams and did not seed its entire field until 1979, the year of the Magic Johnson and Larry Bird final, before expanding to 64 teams in 1985 and 68 in 2011. The rating half of the problem was solved by Arpad Elo, a Hungarian-born physics professor at Marquette University and a strong amateur player, who devised a statistically grounded replacement for the older Harkness system used by the United States Chess Federation. His method was adopted by the USCF in 1960 and by FIDE, the international chess federation, in 1970, and he set out the reasoning in The Rating of Chessplayers, Past and Present in 1978. Elo originally assumed normally distributed performance; the logistic form used almost everywhere today was substituted later because it fitted results at the lower end of the scale better. The framework spread far beyond chess: Mark Glickman added a rating-reliability term with Glicko in 1995, Microsoft Research introduced TrueSkill in 2005 to match Xbox Live players and published it the following year, and sports outlets from Jeff Sagarin's newspaper ratings to FiveThirtyEight's NFL and NBA models applied the same curve to team sports.