Hey, that doesn’t sound very nice. Not that I expect to be able to build a rigorous and elegant model…but I’m also not trying to pass myself off as a great mathematical theorist, either.
This not being my field of expertise, I fully expected that my propsed way of solving the problem was open for debate. That’s why I threw it out to the Teeming Millions. But the problem still led me to a fundamental curiosity that is mostly detached from the original issue, and that is whether there is a way to quickly and easily maximize the value of the diagonal elements of a very large matrix.
And when you couple that mental exercise with the problem at hand, it seems satisfying on the surface. After all, it’s for an ego-stroking awards ceremony meant to maximize the recognition for all the teams. If points awarded correlate to happiness, the maximum value of the diagonal would therefore maximize the happiness quotient. Good enough for government work, as they way.
Besides, I threw the eigen-reference out somewhat tongue-in-cheek, only because there are areas of math that make me nauseated, and matrix algebra is one of them. My background in this area has been of the forced-down-my-throat type. I had some graduate level applied math as part of my Master’s in Aerospace Engineering, but it’s not something I use everyday, or want to.
Well, the OP said he might want to maximize the norm of the diagonal of the matrix. Since this is the maximization solved in the assignment problem, I thought it was worth mentioning that. I indicated that I was speculating about his objective function; I thought I’d made that clear:
I agree that a maximum-scoring assignment may not be what the OP wants. It maximizes some sort of “overall satisfaction,” but doesn’t guarantee that the top-ranked team (for reasonable definitions of “rank”) is happiest. A greedy algorithm of some sort may be a better solution, though the outcomes of greedy algorithms depend strongly on the order in which the suboptimizations are performed, and hence on some (more-or-less arbitrary, and hence subject to endless bickering; cf. BCS) ranking of the teams.
I meant no offense. How shall I put this… I don’t expect anyone in America whose credentials I haven’t seen to be able to form a remotely passable mathematical model.
As for jumping on “eigen-somethings”, could you read your post without reading your mind and know it was meant in jest and not a classic example of someone trying to pull in some powerful technique they barely remember or learned properly just because it’s a powerful technique?
Basically, look at it from my side: someone with no references to indicate they have any mathematical knowledge beyond maybe a few undergrad courses God-knows-how long ago, throws out a suggested technique involving a seemingly half-remembered term which doesn’t remotely relate to the problem at hand and doesn’t throw so much as a smiley after it to signal that it’s a joke. Doesn’t this throw any of the rest of the suggested models into the debatable zone at best?
Oh, I understand. I’m sure it irritates you when people defile the sanctity of mathematics. I get the same way about dumb mechanical questions.
No offense taken. I was momentarily peeved because it just came off as a bit…well, snotty. And if there’s one thing I’ve learned in all the times I’ve tried to be impressive with my knowledge, it’s that few people appreciate condescending geekiness. Even (especially?) from us engineers.
But…no harm, no foul. 
I think I got a few ideas through this thread that ought to get me a suitable answer.
Well, if your idea of “optimizing the diagonal” is indeed the proper model, then Omphaloskeptic’s algorithm should work.
This sort of thing, though, used to come up with my old high school math team.
The first, second, and third places countywide were the best, so they were awarded on total score. Then prizes for the top person in each class (freshman, sophomore, junior, senior) with the first three winners out of play. Then prizes for the top person in each school, with the first seven winners out. For example: when I got 2nd place overall, I was also the top freshman and top from my school, so those prizes went to the runners-up in those categories.
Now, this picks award categories by restricting the field, but something like it was the inspiration for my tossed-out idea.