'Test' categories with total weights each

[Bakerbakura]

So after chatting with Herman about the weights for Stellenbosch, April, and IMO selection, it seems like we're going to go with a scheme wherein, taking the April Camp selection as an example, 50% is allocated to Stellenbosch Senior tests, 30% to monthlies and 20% to the SAMO 3rd round. So I think we should have each test/source of scores being allocated to a category, and the weights specified on a category level, with each category weight divided equally among the tests in that category to calculate the test weights which are used in the objective function.

[dlnnlsn]

I think that the normal way that we think of test weights of course doesn't make sense in the Magic Method context. Usually, we take a weighted sum (or average) of all of the test scores. If test j has weight w_j, then student i's overall score is sum_j (w_j a_{ij}). Now the whole point of the Magic Method is to interpolate scores where they are not available, so we would then replace a_{ij} with s_i d_j, making the student's overall score sum_j (w_j d_j s_i) = (sum_j w_j d_j) s_i. But (sum_j w_j d_j) is just a constant, so ranking students according to this total is then equivalent to ranking them according to the ideal score s_i produced by the Magic Method, so weights in the traditional sense are meaningless.

I think that what Herman's intuition when we say something like "50% is allocated to Stellenbosch Senior tests, 30% to monthlies" is that Senior tests are 5/3 as important as monthlies. If we actually go by 50% as a category for Senior tests that is then divided equally among the tests, then each Stellenbosch test becomes 1/3 as important as the monthlies as a whole. In this case to capture our intuition about what weights mean, I think that every Stellenbosch test should be weighted 50, and every monthly should be weighted 30, so that each Stellenbosch test is 5/3 as important as each given monthly.

On the other hand, if "50% is allocated to Stellenbosch Senior tests, 30% to monthlies" should mean that the total for the Stellenbosch tests is 5/3 as important as the total for the monthlies, then I think that the reasonable interpretation is to divide the 50% among the Stellenbosch tests and weight each one 10%. Though in this case, if we have 3 monthlies that count (approximately like usual), then this division would make each individual Stellenbosch test as important as each individual monthly.