Need some help from the math-type folks here.

[Pulsar]

Ok. There are 40 teams playing in a competition. Each team plays 10 games. The games are of a format where 3 teams play together to achieve a final score, such that:

A + B + C = Final Score, where A, B, and C is each team's contribution.

I would like to determine the contribution of each team. I know this involves matrix math, but I'll be VERY frank and it's been 20 years since I've touched it, and I don't have the foggiest recollection of how it works. (PS - this isn't homework. I graduated 14 years ago and this is for a side hobby: I'm working with high school kids and would like to walk this through with them in language they can understand).

Now, how would I go about starting to solve that? I know it has to be an interative solution, but it's been so long since I've done any math like this I'm afraid I don't remember where to even start.

What I'd like to do is once I'm clear on the math, I'd like to plug the numbers into an excel spread sheet and calculate it out.... but I'm lost with the math side of it.

 

[Crusty]

Your question isn't quite clear. Do you have the breakdown of the scores from each game?
So a list of 40 rows of data that satisfy your equation(A + B + C = FS)? Do you know which teams played in which game? How is that data formatted? What do you mean by contribution, are you looking for the teams total scores across all games? Like a breakdown of A scores, B scores, C scores etc?

 

[Pulsar]

I have a list of which 3 teams participated in each match, and the final score from each match.

So, for instance,

Match 1: Team1 + Team2 + Team3 = Final Score
Match 2: Team2 + Team4 + Team10 = Final Score 2
Match 3: Team4 + Team1 + Team7 = Final Score 3
and so on.

The objective is to iteratively solve the equations such that I get the contribution that best fits the 40 Final scores (by each team). Each team plays an equal number of times (10).

The final result will be the contributed score, or how many points you can assume a player or team is going to score.

In this manner, you don't need to keep track of individual scores of each team to see how much they normally score.

You can do this with any sport - baseball, basketball, etc. In those cases, you can see whether players on the team drive the point average of their team up. For instance let's say in basketball they have a lot of assists, that will show up in a higher final score and therefore a higher contribution. That would NOT show up if you were to take a simple average of that player's score.

 

[eLiu]

Uhhh I must be missing something still. This is not solveable. Each match generates 1 equation with 3 unknowns. So you will never have a unique solution.

In situations like this you can do something approximate (i.e., least squares), but it is unclear to me how that is appropriate to this situation.

I have a list of which 3 teams participated in each match, and the final score from each match.

So, for instance,

Match 1: Team1 + Team2 + Team3 = Final Score
Match 2: Team2 + Team4 + Team10 = Final Score 2
Match 3: Team4 + Team1 + Team7 = Final Score 3
and so on.

The objective is to iteratively solve the equations such that I get the contribution that best fits the 40 Final scores (by each team). Each team plays an equal number of times (10).

The final result will be the contributed score, or how many points you can assume a player or team is going to score.

In this manner, you don't need to keep track of individual scores of each team to see how much they normally score.

You can do this with any sport - baseball, basketball, etc. In those cases, you can see whether players on the team drive the point average of their team up. For instance let's say in basketball they have a lot of assists, that will show up in a higher final score and therefore a higher contribution. That would NOT show up if you were to take a simple average of that player's score.

 

[DrPizza]

I have a vague idea of what you're talking about, but not sure how to do it. eLiu, it's sort of a best fit type of curve.

 

[CycloWizard]

As far as I can tell, you are trying to do a statistical analysis of the data to determine which teams were most likely to play in each game... Is that correct? It sounds like you have the following data:
1. Each team plays exactly 10 games;
2. Each game has exactly 3 teams;
3. There are 40 teams total; and
4. The total score for each game is known.

If this is correct, there are a few ways to proceed. The simplest is to assume that each team scores a constant number of points for each game (though this may not be appropriate, depending on what you're after... you might assume a mean and standard deviation for each as well, but that gets more complicated). If you take this approach, then I would use the following general algorithm:
1. List the teams from 1-40;
2. Assign each team a score as an initial guess (simply go down the line with 1 having the highest, decreasing the score for each);
3. Assemble a coefficient matrix [A], limiting each column to a maximum of ten entries; then
4. Solve the linear algebra problem [A][x]=, where [x] is the list of teams {x1, x2, x3...x40}, is the score for each team, and is the score for each game.

The last step would probably best be done in a least-squares sense, unless the assumption of "constant score" is valid for your case. It's early in the morning now, so my brain isn't going to let me state the problem exactly in this framework (and there are probably some transposes in step #4 that I omitted for this reason as well), but this should get you going in the right direction.

 

[eLiu]

I have a vague idea of what you're talking about, but not sure how to do it. eLiu, it's sort of a best fit type of curve.

Yeah I know what least squares is Probably wikipedia will explain it more clearly than I could on a forum, lol. My point was more that I'm not sure if that is the best choice for this task, supposing that this is the path he wants to pursue. What I felt uncertain about is after setting up Ax=b & solving least squares, you have the x s.t. ||b-Ax||_2 is minimized. But it wasn't clear to me that the 2-norm "A-distance" is the best metric for this problem.

I know practically nothing about statistics, but it seems like some kind of statistical analysis would be useful here... if you're doing this to track like individual players' contributions to teams over the NBA season, I'd guess you get enough data to do things like estimate the average score per game per player & put down some error bars? Dunno :/

edit: from CycloWizard's post, it sounds like the statistical analysis approach IS least squares? Huh, cool.

Also cyclo, I am confused by your notation. Why do you have Axb? Shouldn't there be #teams*#matches variables? So x_{1,1} = team1, match1. x_{10,30} = team10, match30. Some variables can be removed since in general it doesn't look like you need to assume every team plays in every match. But that vector of x's are your unknowns. Then you have teh right hand side b_i, where b_i is the score for the i-th match. Solve:
Ax = b via least squares... i.e.,
A = QR (look up QR factorization)
QRx = b
Rx = Q^T *b
x = R^{-1}*Q^T * b

 

[Swivelguy2]

If you have 40 teams, and each team plays in 10 games, that's 400 total instances of a team participating in a game. Because there are 3 teams in each game, that means a total of 133 1/3 games were played. Oops?

Anyway, for this to be solvable, you're going to have to make the assumption that each team always scores the same number of points in every game they play. Or something like that.

 

[Pulsar]

I guess I'm not explaining it very well.

What your looking to solve for is basically the point contribution of a team to the final scores of the games they played in.

To do that, you've got this system of equations. The numbers I'm using are just approximations to give you the idea (hence the 1/3 of a match that someone else pointed out).

In every match, 3 teams play together to get a total score. Therefore, while there is no EXACT answer for all these equations because teams obviously will score differently in each match, there is an inexact answer for every team's score that will get you close to the final score in every case.

Yes, it is 'one' number for each team. It is their contribution (I hesitate to use the word average because that isn't correct) to the final score.

This is different from an average score. An average score is a very bad way to order teams because it doesn't take interactions into account.

For example, if you're looking at an NBA player, you don't just look at how many points they scored to determine their quality. That would be leaving out things like rebounds and assists that lets other members of their team score points.

By iteratively solving the system of equations above, you end up solving for a teams contribution. That is because if a team is great at enabling other teams to score, that will reflect in the final scores in the games they play in being higher - hence their contribution will reflect it.

In that manner, this system takes into account things like assists, rebounds, etc, without having to actually worry about them numerically.

Does that make more sense?

It's a new-ish field in sports that many big-name teams are starting to use to grade players and their efficacy.

Read a bit more here: http://hoyaprospectus.blogspot.com/2008/10/individual-offensive-defensive-ratings.html

This type of analysis can be used in just about anything - from the military to the NBA to a silly little robotics competition =).

 

[Pulsar]

As a follow up, look at this excel spreadsheet. It's doing exactly what I'm talking about, but while I can code I can't seem to wrap my head around the math they are using.

http://www.chiefdelphi.com/media/papers/download/2321

 

[CycloWizard]

I guess I'm not explaining it very well.

What your looking to solve for is basically the point contribution of a team to the final scores of the games they played in.

To do that, you've got this system of equations. The numbers I'm using are just approximations to give you the idea (hence the 1/3 of a match that someone else pointed out).

In every match, 3 teams play together to get a total score. Therefore, while there is no EXACT answer for all these equations because teams obviously will score differently in each match, there is an inexact answer for every team's score that will get you close to the final score in every case.

Yes, it is 'one' number for each team. It is their contribution (I hesitate to use the word average because that isn't correct) to the final score.

This is different from an average score. An average score is a very bad way to order teams because it doesn't take interactions into account.

For example, if you're looking at an NBA player, you don't just look at how many points they scored to determine their quality. That would be leaving out things like rebounds and assists that lets other members of their team score points.

By iteratively solving the system of equations above, you end up solving for a teams contribution. That is because if a team is great at enabling other teams to score, that will reflect in the final scores in the games they play in being higher - hence their contribution will reflect it.

In that manner, this system takes into account things like assists, rebounds, etc, without having to actually worry about them numerically.

Does that make more sense?

It's a new-ish field in sports that many big-name teams are starting to use to grade players and their efficacy.

Read a bit more here: http://hoyaprospectus.blogspot.com/2008/10/individual-offensive-defensive-ratings.html

This type of analysis can be used in just about anything - from the military to the NBA to a silly little robotics competition =).

It's just regression analysis, which is what we've been saying (in so many words). There are a million and one ways to do it, depending on how rigorous you want to be and your level of experience with statistics. Assigning a constant score rather than using a stochastic method automatically removes any possibility of interactions as you have assumed that the "effect" is constant. This is still a decent place to start, though model accuracy can of course be improved by adding non-constant effects (e.g. linear and quadratic terms and interactions).

 

[DominionSeraph]

How is what you're looking for not included in the sum of a team's final scores?

Just average them.