Bongle
18-04-2008, 13:55
It struck me this morning while watching RS's after one match that it should be somewhat possible to recover penalty counts for the losing side in the first n matches given every team's RS after those n matches at a regional with n teams. Since a team's RS is dependent on the pre-penalty losing alliance's score, it therefore contains information about penalties that we can extract by being clever.
Given a team that:
-won games that had reported losing scores of a, b, c, and d
-lost games with reported losing scores of e, f, g, and h
-an unknown penalty vector p, with elements pa, pb, pc, pd... representing how many points were deducted in penalties.
Then this team's RS can be written as:
RS = ((a+pa) + (b+pb) + (c+pc) + (d+pd) + e + f + g + h)/[matches played = 8 in this case]
MatchesPlayed*RS - a - b - c - d - e - f - g - h = pa+ pb+ pc+ pd
We can do this for every team, and we end up with a bunch of equations with the form
(MatchesPlayed*RS minus losing scores) = sum of unknown penalty scores
We can write these into a matrix, solve it for the p vector, and thus derive the penalty point counts. The cell in the ith column and jth row is 1 if team j participated in and won the i'th match, 0 if they did not. The constant vector's j'th row is ([number of matches the team played]*RS - sum of losing scores in their games). Matrix height is how many teams there are, matrix width is how many matches you're using.
I don't have time to implement this (as a student, I actually do have studying to do), but I thought I'd put it out there for people that might want to try it.
Limitations I can think of:
-I don't think it'd be solveable once the number of matches exceeds the number of teams, though if you grabbed every team's RS data at the half-way point you could probably use that data to extract penalty counts for the losing side in every match
-Only works for the losing side, I don't think there is any information about the winning side encoded in match scores or team rankings
-If there are matches that end with 0 points, the potential number of penalties is infinite, and will probably ruin the matrix. Fortunately, there have been no such matches on Galileo so far.
Given a team that:
-won games that had reported losing scores of a, b, c, and d
-lost games with reported losing scores of e, f, g, and h
-an unknown penalty vector p, with elements pa, pb, pc, pd... representing how many points were deducted in penalties.
Then this team's RS can be written as:
RS = ((a+pa) + (b+pb) + (c+pc) + (d+pd) + e + f + g + h)/[matches played = 8 in this case]
MatchesPlayed*RS - a - b - c - d - e - f - g - h = pa+ pb+ pc+ pd
We can do this for every team, and we end up with a bunch of equations with the form
(MatchesPlayed*RS minus losing scores) = sum of unknown penalty scores
We can write these into a matrix, solve it for the p vector, and thus derive the penalty point counts. The cell in the ith column and jth row is 1 if team j participated in and won the i'th match, 0 if they did not. The constant vector's j'th row is ([number of matches the team played]*RS - sum of losing scores in their games). Matrix height is how many teams there are, matrix width is how many matches you're using.
I don't have time to implement this (as a student, I actually do have studying to do), but I thought I'd put it out there for people that might want to try it.
Limitations I can think of:
-I don't think it'd be solveable once the number of matches exceeds the number of teams, though if you grabbed every team's RS data at the half-way point you could probably use that data to extract penalty counts for the losing side in every match
-Only works for the losing side, I don't think there is any information about the winning side encoded in match scores or team rankings
-If there are matches that end with 0 points, the potential number of penalties is infinite, and will probably ruin the matrix. Fortunately, there have been no such matches on Galileo so far.