TrueSkill¶
the video game rating system
What’s TrueSkill?¶
TrueSkill is a rating system among game players. It was developed by Microsoft Research and has been used on Xbox LIVE for ranking and matchmaking service. This system quantifies players’ TRUE skill points by the Bayesian inference algorithm. It also works well with any type of match rule including N:N team game or freeforall.
This project is a Python package which implements the TrueSkill rating system:
from trueskill import Rating, quality_1vs1, rate_1vs1
alice, bob = Rating(25), Rating(30) # assign Alice and Bob's ratings
if quality_1vs1(alice, bob) < 0.50:
print('This match seems to be not so fair')
alice, bob = rate_1vs1(alice, bob) # update the ratings after the match
Learning¶
Rating, the model for skill¶
In TrueSkill, rating is a Gaussian distribution which starts from \(\mathcal{ N }( 25, \frac{ 25 }{ 3 }^2 )\). \(\mu\) is an average skill of player, and \(\sigma\) is a confidence of the guessed rating. A real skill of player is between \(\mu \pm 2\sigma\) with 95% confidence.
>>> from trueskill import Rating
>>> Rating() # use the default mu and sigma
trueskill.Rating(mu=25.000, sigma=8.333)
If some player’s rating is higher \(\beta\) than another player’s, the player may have about a 76% (specifically \(\Phi(\frac {1}{\sqrt{2}})\)) chance to beat the other player. The default value of \(\beta\) is \(\frac{ 25 }{ 6 }\).
Ratings will approach real skills through few times of the TrueSkill’s Bayesian inference algorithm. How many matches TrueSkill needs to estimate real skills? It depends on the game rule. See the below table:
Rule  Matches 

16P freeforall  3 
8P freeforall  3 
4P freeforall  5 
2P freeforall  12 
2:2:2:2  10 
4:4:4:4  20 
4:4  46 
8:8  91 
Headtohead (1 vs. 1) match rule¶
Most competition games follows 1:1 match rule. If your game does, just use
_1vs1
shortcuts containing rate_1vs1()
and quality_1vs1()
.
These are very easy to use.
First of all, we need 2 Rating
objects:
>>> r1 = Rating() # 1P's skill
>>> r2 = Rating() # 2P's skill
Then we can guess match quality which is equivalent with draw probability of
this match using quality_1vs1()
:
>>> print('{:.1%} chance to draw'.format(quality_1vs1(r1, r2)))
44.7% chance to draw
After the game, TrueSkill recalculates their ratings by the game result. For example, if 1P beat 2P:
>>> new_r1, new_r2 = rate_1vs1(r1, r2)
>>> print(new_r1)
trueskill.Rating(mu=29.396, sigma=7.171)
>>> print(new_r2)
trueskill.Rating(mu=20.604, sigma=7.171)
Mu value follows player’s win/draw/lose records. Higher value means higher game skill. And sigma value follows the number of games. Lower value means many game plays and higher rating confidence.
So 1P, a winner’s skill grew up from 25 to 29.396 but 2P, a loser’s skill shrank to 20.604. And both sigma values became narrow about same magnitude.
Of course, you can also handle a tie game with drawn=True
:
>>> new_r1, new_r2 = rate_1vs1(r1, r2, drawn=True)
>>> print(new_r1)
trueskill.Rating(mu=25.000, sigma=6.458)
>>> print(new_r2)
trueskill.Rating(mu=25.000, sigma=6.458)
Other match rules¶
There are many other match rules such as N:N team match, N:N:N multiple team match, N:M unbalanced match, freeforall (Player vs. All), and so on. Mostly other rating systems cannot work with them but TrueSkill does. TrueSkill accepts any types of matches.
We should arrange ratings into a group by their team:
>>> r1 = Rating() # 1P's skill
>>> r2 = Rating() # 2P's skill
>>> r3 = Rating() # 3P's skill
>>> t1 = [r1] # Team A contains just 1P
>>> t2 = [r2, r3] # Team B contains 2P and 3P
Then we can calculate the match quality and rate them:
>>> print('{:.1%} chance to draw'.format(quality([t1, t2])))
13.5% chance to draw
>>> (new_r1,), (new_r2, new_r3) = rate([t1, t2], ranks=[0, 1])
>>> print(new_r1)
trueskill.Rating(mu=33.731, sigma=7.317)
>>> print(new_r2)
trueskill.Rating(mu=16.269, sigma=7.317)
>>> print(new_r3)
trueskill.Rating(mu=16.269, sigma=7.317)
If you want to describe other game results, set the ranks
argument like the
below examples:
 A drawn game –
ranks=[0, 0]
 Team B won not team A –
ranks=[1, 0]
(Lower rank is better)
Additionally, here are varied patterns of rating groups. All variables which
start with r
are Rating
objects:
 N:N team match –
[(r1, r2, r3), (r4, r5, r6)]
 N:N:N multiple team match –
[(r1, r2), (r3, r4), (r5, r6)]
 N:M unbalanced match –
[(r1,), (r2, r3, r4)]
 Freeforall –
[(r1,), (r2,), (r3,), (r4,)]
Partial play¶
Let’s assume that there are 2 teams which each has 2 players. The game was for a hour but the one of players on the first team entered the game at 30 minutes later.
If some player wasn’t present for the entire duration of the game, use the
concept of “partial play” by weights
parameter. The above situation can be
described by the following weights:


As a code with a 2dimensional list:
# set each weights to 1, 0.5, 1, 1.
rate([(r1, r2), (r3, r4)], weights=[(1, 0.5), (1, 1)])
quality([(r1, r2), (r3, r4)], weights=[(1, 0.5), (1, 1)])
Or with a dictionary. Each keys are a tuple of
(team_index, index_or_key_of_rating)
:
# set a weight of 2nd player in 1st team to 0.5, otherwise leave as 1.
rate([(r1, r2), (r3, r4)], weights={(0, 1): 0.5})
# set a weight of Carol in 2nd team to 0.5, otherwise leave as 1.
rate([{'alice': r1, 'bob': r2}, {'carol': r3}], weights={(1, 'carol'): 0.5})
Backends¶
The TrueSkill algorithm uses \(\Phi\), the cumulative distribution function; \(\phi\), the probability density function; and \(\Phi^{1}\), the inverse cumulative distribution function. But standard mathematics library doesn’t provide the functions. Therefore this package implements them.
Meanwhile, there are thirdparty libraries which implement the functions. You
may want to use another implementation because that’s more expert. Then set
backend
option of TrueSkill
to the backend you chose:
>>> TrueSkill().cdf # internal implementation
<function cdf at ...>
>>> TrueSkill(backend='mpmath').cdf # mpmath.ncdf
<bound method MPContext.f_wrapped of <mpmath.ctx_mp.MPContext object at ...>>
Here’s the list of the available backends:
None
– the internal implementation. (Default) “mpmath” – requires mpmath installed.
 “scipy” – requires scipy installed.
Note
When winners have too lower rating than losers, TrueSkill.rate()
will
raise FloatingPointError
. In this case, you need higher
floatingpoint precision. The mpmath library offers flexible floatingpoint
precision. You can solve the problem with mpmath as a backend and higher
precision setting.
Win probability¶
TrueSkill provides a function (quality()
) to calculate a draw probability
between arbitrary ratings. But there’s no function for a win probability.
Anyway, if you need to calculate a win probability between only 2 teams, this code snippet will help you:
import itertools
import math
def win_probability(team1, team2):
delta_mu = sum(r.mu for r in team1)  sum(r.mu for r in team2)
sum_sigma = sum(r.sigma ** 2 for r in itertools.chain(team1, team2))
size = len(team1) + len(team2)
denom = math.sqrt(size * (BETA * BETA) + sum_sigma)
ts = trueskill.global_env()
return ts.cdf(delta_mu / denom)
This snippet is written by Juho Snellman in issue #1.
API¶
TrueSkill objects¶

class
trueskill.
Rating
(mu=None, sigma=None)¶ Represents a player’s skill as Gaussian distrubution.
The default mu and sigma value follows the global environment’s settings. If you don’t want to use the global, use
TrueSkill.create_rating()
to create the rating object.Parameters:  mu – the mean.
 sigma – the standard deviation.

mu
¶ A property which returns the mean.

sigma
¶ A property which returns the the square root of the variance.

class
trueskill.
TrueSkill
(mu=25.0, sigma=8.333333333333334, beta=4.166666666666667, tau=0.08333333333333334, draw_probability=0.1, backend=None)¶ Implements a TrueSkill environment. An environment could have customized constants. Every games have not same design and may need to customize TrueSkill constants.
For example, 60% of matches in your game have finished as draw then you should set
draw_probability
to 0.60:env = TrueSkill(draw_probability=0.60)
For more details of the constants, see The Math Behind TrueSkill by Jeff Moser.
Parameters:  mu – the initial mean of ratings.
 sigma – the initial standard deviation of ratings. The recommended
value is a third of
mu
.  beta – the distance which guarantees about 76% chance of winning.
The recommended value is a half of
sigma
.  tau – the dynamic factor which restrains a fixation of rating. The
recommended value is
sigma
per cent.  draw_probability – the draw probability between two teams. It can be
a
float
or function which returns afloat
by the given two rating (team performance) arguments and the beta value. If it is afloat
, the game has fixed draw probability. Otherwise, the draw probability will be decided dynamically per each match.  backend – the name of a backend which implements cdf, pdf, ppf. See
trueskill.backends
for more details. Defaults toNone
.

create_rating
(mu=None, sigma=None)¶ Initializes new
Rating
object, but it fixes default mu and sigma to the environment’s.>>> env = TrueSkill(mu=0, sigma=1) >>> env.create_rating() trueskill.Rating(mu=0.000, sigma=1.000)

expose
(rating)¶ Returns the value of the rating exposure. It starts from 0 and converges to the mean. Use this as a sort key in a leaderboard:
leaderboard = sorted(ratings, key=env.expose, reverse=True)
New in version 0.4.

make_as_global
()¶ Registers the environment as the global environment.
>>> env = TrueSkill(mu=50) >>> Rating() trueskill.Rating(mu=25.000, sigma=8.333) >>> env.make_as_global() trueskill.TrueSkill(mu=50.000, ...) >>> Rating() trueskill.Rating(mu=50.000, sigma=8.333)
But if you need just one environment,
setup()
is better to use.

quality
(rating_groups, weights=None)¶ Calculates the match quality of the given rating groups. A result is the draw probability in the association:
env = TrueSkill() if env.quality([team1, team2, team3]) < 0.50: print('This match seems to be not so fair')
Parameters:  rating_groups – a list of tuples or dictionaries containing
Rating
objects.  weights – weights of each players for “partial play”.
New in version 0.2.
 rating_groups – a list of tuples or dictionaries containing

rate
(rating_groups, ranks=None, weights=None, min_delta=0.0001)¶ Recalculates ratings by the ranking table:
env = TrueSkill() # uses default settings # create ratings r1 = env.create_rating(42.222) r2 = env.create_rating(89.999) # calculate new ratings rating_groups = [(r1,), (r2,)] rated_rating_groups = env.rate(rating_groups, ranks=[0, 1]) # save new ratings (r1,), (r2,) = rated_rating_groups
rating_groups
is a list of rating tuples or dictionaries that represents each team of the match. You will get a result as same structure as this argument. Rating dictionaries for this may be useful to choose specific player’s new rating:# load players from the database p1 = load_player_from_database('Arpad Emrick Elo') p2 = load_player_from_database('Mark Glickman') p3 = load_player_from_database('Heungsub Lee') # calculate new ratings rating_groups = [{p1: p1.rating, p2: p2.rating}, {p3: p3.rating}] rated_rating_groups = env.rate(rating_groups, ranks=[0, 1]) # save new ratings for player in [p1, p2, p3]: player.rating = rated_rating_groups[player.team][player]
Parameters:  rating_groups – a list of tuples or dictionaries containing
Rating
objects.  ranks – a ranking table. By default, it is same as the order of
the
rating_groups
.  weights – weights of each players for “partial play”.
 min_delta – each loop checks a delta of changes and the loop will stop if the delta is less then this argument.
Returns: recalculated ratings same structure as
rating_groups
.Raises: FloatingPointError
occurs when winners have too lower rating than losers. higher floatingpoint precision couls solve this error. set the backend to “mpmath”.New in version 0.2.
 rating_groups – a list of tuples or dictionaries containing
Default values¶

trueskill.
MU
= 25.0¶ Default initial mean of ratings.

trueskill.
SIGMA
= 8.333333333333334¶ Default initial standard deviation of ratings.

trueskill.
BETA
= 4.166666666666667¶ Default distance that guarantees about 76% chance of winning.

trueskill.
TAU
= 0.08333333333333334¶ Default dynamic factor.

trueskill.
DRAW_PROBABILITY
= 0.1¶ Default draw probability of the game.
Headtohead shortcuts¶

trueskill.
rate_1vs1
(rating1, rating2, drawn=False, min_delta=0.0001, env=None)¶ A shortcut to rate just 2 players in a headtohead match:
alice, bob = Rating(25), Rating(30) alice, bob = rate_1vs1(alice, bob) alice, bob = rate_1vs1(alice, bob, drawn=True)
Parameters: Returns: a tuple containing recalculated 2 ratings.
New in version 0.2.

trueskill.
quality_1vs1
(rating1, rating2, env=None)¶ A shortcut to calculate the match quality between just 2 players in a headtohead match:
if quality_1vs1(alice, bob) < 0.50: print('This match seems to be not so fair')
Parameters:  rating1 – the rating.
 rating2 – the another rating.
 env – the
TrueSkill
object. Defaults to the global environment.
New in version 0.2.
Functions for the global environment¶

trueskill.
setup
(mu=25.0, sigma=8.333333333333334, beta=4.166666666666667, tau=0.08333333333333334, draw_probability=0.1, backend=None, env=None)¶ Setups the global environment.
Parameters: env – the specific TrueSkill
object to be the global environment. It is optional.>>> Rating() trueskill.Rating(mu=25.000, sigma=8.333) >>> setup(mu=50) trueskill.TrueSkill(mu=50.000, ...) >>> Rating() trueskill.Rating(mu=50.000, sigma=8.333)

trueskill.
rate
(rating_groups, ranks=None, weights=None, min_delta=0.0001)¶ A proxy function for
TrueSkill.rate()
of the global environment.New in version 0.2.

trueskill.
quality
(rating_groups, weights=None)¶ A proxy function for
TrueSkill.quality()
of the global environment.New in version 0.2.

trueskill.
expose
(rating)¶ A proxy function for
TrueSkill.expose()
of the global environment.New in version 0.4.
Draw probability helpers¶
Mathematical statistics backends¶

trueskill.backends.
choose_backend
(backend)¶ Returns a tuple containing cdf, pdf, ppf from the chosen backend.
>>> cdf, pdf, ppf = choose_backend(None) >>> cdf(10) 7.619853263532764e24 >>> cdf, pdf, ppf = choose_backend('mpmath') >>> cdf(10) mpf('7.6198530241605255e24')
New in version 0.3.

trueskill.backends.
available_backends
()¶ Detects list of available backends. All of defined backends are
None
– internal implementation, “mpmath”, “scipy”.You can check if the backend is available in the current environment with this function:
if 'mpmath' in available_backends(): # mpmath can be used in the current environment setup(backend='mpmath')
New in version 0.3.
Changelog¶
Version 0.4.4¶
Released on Dec 31 2015.
Fixed documentation error. See issue #11. Thanks to Russel Simmons.
Version 0.4.3¶
Released on Sep 4 2014.
Fixed ordering bug on weights argument as a dict. This was reported at issue #9.
Version 0.4¶
Released on Mar 25 2013.
 Added dynamic draw probability.
 Replaced
Rating.exposure()
withTrueSkiil.expose()
. Because the TrueSkill settings have to adjust a fomula to calculate an exposure.  Deprecated headtohead shortcut methods in
TrueSkill
. The toplevel shortcut functions are still alive.
Version 0.3.1¶
Released on Mar 6 2013.
Changed to raise FloatingPointError
instead of ValueError
(math
domain error) for a problem similar to issue #5 but with more extreme input.
Version 0.3¶
Released on Mar 5 2013.
TrueSkill
got a new option backend
to choose cdf, pdf, ppf
implementation.
When winners have too lower rating than losers, TrueSkill.rate()
will
raise FloatingPointError
if the backend is None
or “scipy”. But
from this version, you can avoid the problem with “mpmath” backend. This was
reported at issue #5.
Version 0.2¶
Released on Nov 30 2012.
 Added “Partial play” implementation.
 Worked well in many Python versions, 2.5, 2.6, 2.7, 3.1, 3.2, 3.3 and many interpreters, CPython, Jython, PyPy.
 Supported that using dictionaries as a
rating_group
to choose specific player’s rating simply.  Added shorcut functions for 2 players individual match, the most usage:
rate_1vs1()
andquality_1vs1()
,  Renamed
TrueSkill.transform_ratings()
toTrueSkill.rate()
.  Renamed
TrueSkill.match_quality()
toTrueSkill.quality()
.
Version 0.1.4¶
Released on Oct 5 2012.
Fixed ZeroDivisionError
issue. For more detail, see issue#3. Thanks
to Yunwon Jeong and Nikos Kokolakis.
Version 0.1.1¶
Released on Jan 12 2012.
Fixed an error in “A” matrix of the match quality algorithm.
Version 0.1¶
First public preview release.
Further more¶
There’s the list for users. To subscribe the list, just send a mail to trueskill@librelist.com.
If you want to more details of the TrueSkill algorithm, see also:
 TrueSkill: A Bayesian Skill Rating System by Herbrich, Ralf and Graepel, Thore
 TrueSkill Calculator by Microsoft Research
 Computing Your Skill by Jeff Moser
 The Math Behind TrueSkill by Jeff Moser
Licensing and Author¶
This TrueSkill package is opened under the BSD license but the TrueSkill™ brand is not. Microsoft permits only Xbox Live games or noncommercial projects to use TrueSkill™. If your project is commercial, you should find another rating system. See LICENSE for the details.
I’m Heungsub Lee, a game developer. Any regarding questions or patches are welcomed.