Wholistic metrics on winner sets

It was suggested by @parker_friedland that I use some more wholistic metrics to evaluate the performance of winner sets in multimember systems.

I am already using 12 metric in the simulation but they are all pretty targeted at specific things. They are listed in the readme file.

Based on some optimal methods I have come up with the following metrics.

Harmonic Quality is just the Quality function from Harmonic voting.

Unitary Quality is the Quality function from the optimal version of Sequentially Spent Score

Monroe Quality is not really the Monroe function because voters can be in each of the V/W quotas more than once but this implementation should be reasonable and is very fast.

Ebert cost is not exactly what @Toby_Pereira suggested because it does not use the KP transform but it is hopefully reasonable. It is worth noting that this is very similar to a metric I already have called Utility Deviation. The Score Deviation and Favored Winner Deviation are also similar but care more about how voters are represented by multiple winners.

Are there others?

OK so here is what I am thinking. I have Narrowed it down to 6

  1. Harmonic Quality

  2. Unitary Quality

  3. Ebert Cost (I could easily be convinced that the Max Phragmen based one is better since that satisfies Proportional Justified Representation).

  4. Most blocking unelected candidate. This is basically the unelected candidae with the highes capture count over the whole winner set. It is a simple method for checking the stability for all S’ of size 1.

  5. Largest totally unserved group. The max count of voters who did not get any winner but who are all voting for a nonwinner. This is basically the test for the simple Justified representation

  6. Total utility gain by one extra winner. It may not be totally obvious but this is the same quantity as the prior if the score ballots are passed through the KP-Transform. Recall that Justified representation is not defined for score but approval.

I cut the monroe quality metric since it is not really well founded in theory. Also it can go up and down for good reasons so it is not useful for comparison.