[MikeE]

Originally Posted by Ether

The criterion for "no surrogates" is M*6/T = N, where

M is the number of qual matches
T is the number of teams
N is a whole number (the number of matches played by each team)

At CMP, T=100 and M=134, so N was not a whole number; thus there were surrogates.

If instead T=96 and M=128, N would be a whole number (namely 8) and there would be no surrogates.

I prefer to think of the surrogate issue in terms of looking at
(T*N) mod 6
i.e. how many teams are left over if everyone plays a certain number of rounds.

If there are any teams left over then we need at least one surrogate match. The scheduling software also adds the reasonable constraint that there can only be one surrogate team per alliance. Putting this all together there are between 0 and 3 matches with surrogates in qualification rounds.

Clearly if either T or N are multiples of 6 then the remainder is zero so no surrogates.

Choosing N=12 guarantees no surrogates however many teams are at the event, gives plenty of matches for each team and also has the nice property that M=2*T so it's easy to estimate the schedule impact. I'm sure the designers of FiM and MAR settled on 12 matches per event through similar reasoning.

Originally Posted by Ether

What do you mean by "maximum likelihood estimate models" in this context?

(I'll try to keep this accessible to a wider audience but we can go into further details later.)

OPR estimates a single parameter model for each team, i.e. what is the optimal solution if we model a team's contribution to each match as a constant. We can also use regression (or other optimization techniques) to build richer models. For example we can model each team with two parameters: a constant contribution per match similar to OPR, plus a term which models a team's improvement per round.

But these type of models are deterministic. In other words if we use the model to predict the outcome of a hypothetical match we will always get the same answer. That means we can't use a class of useful simulation methods to get deeper insight into how a collection of matches might play out.

Here's an alternative approach.
Instead of modeling a team's score as a constant (or polynomial function of known features), we treat each team's score as if it is generated from an underlying statistical distribution. Now the problem becomes one of estimating (or assuming) the type of distribution and also estimating the parameters of that distribution for each team.

With OPR we model team X as scoring say 15.3 points every match, so our prediction for a hypothetical match is always 15.3 points.
With a statistical model we would model team X as something like 15.3 +/- 6.3 points. To predict the score for a hypothetical match we choose randomly from the appropriate distribution, and this will obviously be different each time we "play" the hypothetical match.

So with OPR if we "play" a hypothetical match 100 times where OPR(red) > OPR(blue), the final score would be the same every time so red will always win. But if we use a statistical model then red should still win most matches but blue will also win some of the time. Now we have an estimate of the probability of victory for red, which is potentially more useful information than "red wins", and can be used in further reasoning.

MLE is just an approach for getting the parameters from match data. For simplicity I assume a Gaussian distribution, use linear regression as an initial estimate of each team's mean and linear regression on the squared residuals as an initial estimate of each team's variance.

Originally Posted by Ether

In this context, I'm assuming "the binary matrix" refers to the 2MxN design matrix [A] of the overdetermined system.

Do you then use QR factorization directly on the binary matrix to obtain the solution, or do you form the normal equations and use Cholesky?

Yes, I mean the design matrix.

I've implemented many numerical algorithms over the years and the main lesson it taught me is not to write them yourself unless absolutely necessary!
So for linear regression I solve the normal equation using Octave (similar to MATLAB). I don't see any meaningful difference between my results and other published sources on CD.