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A simplified version of the TrueSkill algorithm


ModeratelyAccurateSkill is a simplified variant of Microsoft's TrueSkill algorithm that uses just Bayesian nets with tabular CPDs as implemented in Programming Assignment 4 of the Coursera PGM Course.

The basic data structures

Since we are using tables, we cannot use the continuous skill variable and its variance, but must represent our knowledge of the skill of a player as a distribution over a fixed set of skill levels. So each player will be represented as a factor with one variable, and whenever we see a new player we do not yet know anything about, we give her a uniform skill distribution:

% Create a new factor for a new player we do not
% yet know anything about
function player = MakePlayer(variable, steps)
  if nargin < 2
    steps = 10;
  player = struct('var', [variable],
                  'card', [steps],
                  'val', ones(1, steps)/steps);

The next important ingredient is the probability that a player wins against another player, which we assume to depend only on the difference of skill levels. A bit of repmat-magic (yeah, it's an .m-function, so sue me) gives us a nice rectangular matrix of differences:

diff = repmat((1:5)(:), 1, 5) - repmat(1:5, 5, 1)
            diff =
               0  -1  -2  -3  -4
               1   0  -1  -2  -3
               2   1   0  -1  -2
               3   2   1   0  -1
               4   3   2   1   0

which we can transform into a nice probability using the logistic distribution, which we scale so that a difference of one skill level results in an 80% chance of winning:

            ans =
               0.5000000   0.1999831   0.0588118   0.0153798   0.0038894
               0.8000169   0.5000000   0.1999831   0.0588118   0.0153798
               0.9411882   0.8000169   0.5000000   0.1999831   0.0588118
               0.9846202   0.9411882   0.8000169   0.5000000   0.1999831
               0.9961106   0.9846202   0.9411882   0.8000169   0.5000000

The function ExpectedOutcome produces a factor that encodes this probability distribution for the expected outcome of a match between two players. The outcome variable is one if the first player wins and two if the second one wins (for sake of simplicity we assume that our game never produces a draw):

function outcome = ExpectedOutcome(outcomeVariable, player1, player2)
  steps1 = player1.card(1);
  steps2 = player2.card(1);
  diffs = repmat((1:steps1)(:), 1, steps2) - repmat(1:steps2, steps1, 1);
  probs = logistic_cdf(1.3864*(diffs))(:);
  val = [ probs, (1 - probs)]'(:)';
  outcome = struct('var', [outcomeVariable, player1.var(1), player2.var(1)],
                   'card', [2, player1.card(1), player2.card(1)],
                   'val', val);

Now we can make our first skill estimate based on match between two players where the first player wins:

p1 = MakePlayer(1);
p2 = MakePlayer(2);
ex = ExpectedOutcome(3, p1, p2);
m = ComputeExactMarginalsBP([p1, p2, ex], [0 0 1], 0);
[m(p1.var(1)).val; m(p2.var(1)).val]
                ans =

                 Columns 1 through 5:

                   0.015587   0.031588   0.050411   0.070102   0.090020
                   0.184413   0.168412   0.149589   0.129898   0.109980

                 Columns 6 through 10:

                   0.109980   0.129898   0.149589   0.168412   0.184413
                   0.090020   0.070102   0.050411   0.031588   0.015587

As we can see, our skill estimate for the first player has been moved to higher values, while the one for the second to lower value. This becomes even more visible if player one wins ten games in a row:

for i = 1:10
  m = ComputeExactMarginalsBP([p1, p2, ex], [0 0 1], 0);
  p1 = m(p1.var(1));
  p2 = m(p2.var(1));
[p1.val; p2.val]
            ans =

             Columns 1 through 4:

               7.4135e-09   6.1870e-06   3.8620e-04   5.0852e-03
               2.9838e-01   2.5766e-01   2.0248e-01   1.3723e-01

             Columns 5 through 8:

               2.6033e-02   7.2738e-02   1.3723e-01   2.0248e-01
               7.2738e-02   2.6033e-02   5.0852e-03   3.8620e-04

             Columns 9 and 10:

               2.5766e-01   2.9838e-01
               6.1870e-06   7.4135e-09

Now our estimate is quite certain that player one has a skill in the upper half of the distribution:

                ans =  0.96849

Adding some elements of chance

In TrueSkill, there is one more factor between the skill and the outcome, the actual performance of the player, which is a random variable that depends on the skill of the player and among other things models if the game is more dominated by skill or by chance. For a moderately chancy game, we can model this for example with a binomial distribution where the total number of balls in the urn is one more than the number of skill levels, and we draw one time less:

function strength = Performance (variable, player)
  steps = player.card(1);
  v = zeros(steps,steps);
  for i = 1:steps
  v(i, :) = binopdf(0:(steps -1), (steps - 1), i / (steps + 1));
  strength = struct('var', [variable, player.var(1)],
                    'card', [steps, steps],
                    'val', v(:)');

With this choice, a player at the highest skill level has a appreciable probability of actually performing at one of the two highest performance levels:

p=Performance(2, MakePlayer(1));
ObserveEvidence(p,[1 10]);
d = FactorMarginalization(ObserveEvidence(p,[1 10]), 1).val;
            ans =

             Columns 1 through 4:

               6.3520e-10   3.2522e-07   1.2503e-05   1.6651e-04

             Columns 5 through 8:

               1.2406e-03   6.4014e-03   2.5633e-02   8.5255e-02

             Columns 9 and 10:

               2.4609e-01   6.3520e-01

Now a single match involves the two player factors, the two performance factors and the expected outcome factor, which now depends on the performance factors, and the resulting change from the initial distribution is weaker than in the previous example:

p1 = MakePlayer(1);
p2 = MakePlayer(2);
pf1=Performance(3, p1);
pf2=Performance(4, p2);
ex=ExpectedOutcome(5, pf1, pf2);
m = ComputeExactMarginalsBP([p1, p2, pf1, pf2, ex], [0 0 0 0 1], 0);
[m(p1.var(1)).val; m(p2.var(1)).val]
            ans =

             Columns 1 through 5:

               0.016361   0.036312   0.056516   0.077556   0.099135
               0.117171   0.170364   0.163175   0.142543   0.120867

             Columns 6 through 10:

               0.120867   0.142543   0.163175   0.170364   0.117171
               0.099135   0.077556   0.056516   0.036312   0.016361

The same thing happens when you iterate ten times:

for i = 1:10
  m = ComputeExactMarginalsBP([p1, p2, pf1, pf2, ex], [0 0 0 0 1], 0);
  p1 = m(p1.var(1));
  p2 = m(p2.var(1));
[p1.val; p2.val]
            ans =

             Columns 1 through 4:

               8.2230e-10   2.2340e-06   1.5321e-04   2.5803e-03
               4.9292e-03   3.1480e-01   3.8291e-01   2.0059e-01

             Columns 5 through 8:

               1.8680e-02   7.5345e-02   2.0059e-01   3.8291e-01
               7.5345e-02   1.8680e-02   2.5803e-03   1.5321e-04

             Columns 9 and 10:

               3.1480e-01   4.9292e-03
               2.2340e-06   8.2230e-10

The algorithm is still quite sure that the first player is above average:

                ans =  0.97858

but it is far more reluctant to assign any player to one of the extreme values.


Now a tournament can be modelled as a connected series of matches where every player has won against the following player in the resulting list:

p1 = MakePlayer(1);
p2 = MakePlayer(2);
p3 = MakePlayer(3);
pf1=Performance(4, p1);
pf2=Performance(5, p2);
pf3=Performance(6, p3);
ex12=ExpectedOutcome(7, pf1, pf2);
ex23=ExpectedOutcome(8, pf2, pf3);
factors = [p1, p2, p3, pf1, pf2, pf3, ex12, ex23];
obs = zeros(1, 8);
obs(7) = 1;
obs(8) = 1;
m = ComputeExactMarginalsBP(factors, obs, 0);
[m(p1.var(1)).val; m(p2.var(1)).val; m(p3.var(1)).val]
                ans =

                 Columns 1 through 5:

                   0.0069047   0.0180071   0.0333445   0.0538342   0.0797502
                   0.0392870   0.0794268   0.1095116   0.1304529   0.1413217
                   0.1519890   0.2109338   0.1868481   0.1473617   0.1110269

                 Columns 6 through 10:

                   0.1110269   0.1473617   0.1868481   0.2109338   0.1519890
                   0.1413217   0.1304529   0.1095116   0.0794268   0.0392870
                   0.0797502   0.0538342   0.0333445   0.0180071   0.0069047


The final building block are teams, which TrueSkill models by just adding the player performance. In ModeratelyAccurateSkill, this can be modelled with a deterministic factor that encodes the addition, although you will probably want to add an offset so that a team of four lowest level players gets a combined performance at the lowest level, and not at level four. Implementing this is left as an exercise for the readers. (You should also check that I did not get row and column ordering wrong in ExpectedOutcome if you do so).

Adding teams will also lead to rather huge factors: A team of four players can have 37 different performance levels, and a match between two of these teams requires an ExpectedOutcome factor that has 37 * 37 * 2 elements in its val array. On the bright side, inference in the resulting net is still exact and we get around the iterated inner schedule of TrueSkill which is required as the updates on the continuous distributions used there are only approximate.

That's it

As far as I can see, this covers all the basic ideas of the TrueSkill algorithm in this simplified, discrete skill level reformulation. Whats left are some catchy simple numbers to use in leaderboards or so. First you have to calculate the expectation and deviation of a player's skill:

function [skill, sigma] = Skill(player)
  ll = 1:(player.card(1));
  skill = ll * player.val(:);
  sigma = sqrt((ll .* ll) * player.val(:) - skill*skill);

So an unknown player has an expected skill level of 5.5:

 [sk, sd] = Skill(MakePlayer(1))
                sk =  5.5000
                sd =  2.8723

while the player who has won ten games in a row has a skill level of 7.9:

 [sk, sd] = Skill(p1)
                sk =  7.9062
                sd =  1.0203

The reported skill level used for player ranking is a certain number of standard deviations below the skill estimate. TrueSkill uses a factor of three and adjusts the prior for an unknown player so that a new player enters the leaderboard from the bottom, a value of 1.75 gives us roughly the same effect for reasonably low numbers of skill levels.

function mas = ModeratelyAccurateSkill(player)
  [sk, sd] = Skill(player);
  mas = sk - 1.75 * sd;

With this choice we have for an unknown player and the winner and loser of our ten games:

                ans =  0.47351
                ans =  6.1207
                ans =  1.3082

and for the three participants after one round of our tournament:

                ans =  3.6928
                ans =  1.3722
                ans = -0.042808
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