A faster and open license multi-team, multiplayer rating system comparable to TrueSkill.
This is a port of the amazing openskill.js package and some of it’s models are based on this wonderful paper.
>>> from openskill import Rating, rate >>> a1 = Rating() >>> a1 Rating(mu=25.0, sigma=8.333333333333334) >>> a2 = Rating(mu=32.444, sigma=5.123) >>> a2 Rating(mu=32.444, sigma=5.123) >>> b1 = Rating(43.381, 2.421) >>> b1 Rating(mu=43.381, sigma=2.421) >>> b2 = Rating(mu=25.188, sigma=6.211) >>> b2 Rating(mu=25.188, sigma=6.211)
a2 are on a team, and wins against a team of
b2, send this into rate:
>>> [[x1, x2], [y1, y2]] = rate([[a1, a2], [b1, b2]]) >>> x1, x2, y1, y2 (Rating(mu=28.669648436582808, sigma=8.071520788025197), Rating(mu=33.83086971107981, sigma=5.062772998705765), Rating(mu=43.071274808241974, sigma=2.4166900452721256), Rating(mu=23.149503312339064, sigma=6.1378606973362135))
You can also create
Rating objects by importing
>>> from openskill import create_rating >>> x1 = [28.669648436582808, 8.071520788025197] >>> x1 = create_rating(x1) >>> x1 Rating(mu=28.669648436582808, sigma=8.071520788025197)
When displaying a rating, or sorting a list of ratings, you can use
>>> from openskill import ordinal >>> ordinal([43.07, 2.42]) 35.81
By default, this returns
mu - 3 * sigma, showing a rating for which
there’s a 99.7% likelihood the player’s true rating is higher, so
with early games, a player’s ordinal rating will usually go up and could
go up even if that player loses.
If your teams are listed in one order but your ranking is in a different order, for convenience you can specify a ranks option, such as:
>>> a1 = b1 = c1 = d1 = Rating() >>> result = [[a2], [b2], [c2], [d2]] = rate([[a1], [b1], [c1], [d1]], rank=[4, 1, 3, 2]) >>> result [[Rating(mu=20.96265504062538, sigma=8.083731307186588)], [Rating(mu=27.795084971874736, sigma=8.263160757613477)], [Rating(mu=24.68943500312503, sigma=8.083731307186588)], [Rating(mu=26.552824984374855, sigma=8.179213704945203)]]
It’s assumed that the lower ranks are better (wins), while higher ranks are worse (losses). You can provide a score instead, where lower is worse and higher is better. These can just be raw scores from the game, if you want.
Ties should have either equivalent rank or score.
>>> a1 = b1 = c1 = d1 = Rating() >>> result = [[a2], [b2], [c2], [d2]] = rate([[a1], [b1], [c1], [d1]], score=[37, 19, 37, 42]) >>> result [[Rating(mu=24.68943500312503, sigma=8.179213704945203)], [Rating(mu=22.826045021875203, sigma=8.179213704945203)], [Rating(mu=24.68943500312503, sigma=8.179213704945203)], [Rating(mu=27.795084971874736, sigma=8.263160757613477)]]
You can compare two or more teams to get the probabilities of each team winning.
>>> from openskill import predict_win >>> a1 = Rating() >>> a2 = Rating(mu=33.564, sigma=1.123) >>> predictions = predict_win(teams=[[a1], [a2]]) >>> predictions [0.45110901512761536, 0.5488909848723846] >>> sum(predictions) 1.0
You can compare two or more teams to get the probabilities of the match drawing.
>>> from openskill import predict_draw >>> a1 = Rating() >>> a2 = Rating(mu=33.564, sigma=1.123) >>> prediction = predict_draw(teams=[[a1], [a2]]) >>> prediction 0.09025541153402594
Sometimes you want to know what the likelihood is someone will place at a particular rank. You can use this library to predict those odds.
>>> from openskill import predict_rank, predict_draw >>> a1 = a2 = a3 = Rating(mu=34, sigma=0.25) >>> b1 = b2 = b3 = Rating(mu=32, sigma=0.5) >>> c1 = c2 = c3 = Rating(mu=30, sigma=1) >>> team_1, team_2, team_3 = [a1, a2, a3], [b1, b2, b3], [c1, c2, c3] >>> draw_probability = predict_draw(teams=[team_1, team_2, team_3]) >>> draw_probability 0.329538507466658 >>> rank_probability = predict_rank(teams=[team_1, team_2, team_3]) >>> rank_probability [(1, 0.4450361350569973), (2, 0.19655022513040035), (3, 0.02887513234594437)] >>> sum([y for x, y in rank_probability]) + draw_probability 1.0
The default model is
PlackettLuce. You can import alternate models
openskill.models like so:
>>> from openskill.models import BradleyTerryFull >>> a1 = b1 = c1 = d1 = Rating() >>> rate([[a1], [b1], [c1], [d1]], rank=[4, 1, 3, 2], model=BradleyTerryFull) [[Rating(mu=17.09430584957905, sigma=7.5012190693964005)], [Rating(mu=32.90569415042095, sigma=7.5012190693964005)], [Rating(mu=22.36476861652635, sigma=7.5012190693964005)], [Rating(mu=27.63523138347365, sigma=7.5012190693964005)]]
BradleyTerryFull: Full Pairing for Bradley-Terry
BradleyTerryPart: Partial Pairing for Bradely-Terry
PlackettLuce: Generalized Bradley-Terry
ThurstoneMostellerFull: Full Pairing for Thurstone-Mosteller
ThurstoneMostellerPart: Partial Pairing for Thurstone-Mosteller
Which Model Do I Want?#
Bradley-Terry rating models follow a logistic distribution over a player’s skill, similar to Glicko.
Thurstone-Mosteller rating models follow a gaussian distribution, similar to TrueSkill. Gaussian CDF/PDF functions differ in implementation from system to system (they’re all just chebyshev approximations anyway). The accuracy of this model isn’t usually as great either, but tuning this with an alternative gamma function can improve the accuracy if you really want to get into it.
Full pairing should have more accurate ratings over partial pairing, however in high k games (like a 100+ person marathon race), Bradley-Terry and Thurstone-Mosteller models need to do a calculation of joint probability which involves is a k-1 dimensional integration, which is computationally expensive. Use partial pairing in this case, where players only change based on their neighbors.
Plackett-Luce (default) is a generalized Bradley-Terry model for k ≥ 3 teams. It scales best.