So I have some voting data. There were 25 possible things to vote for, and each voter had to fill out a 10 position ballot, ie each user picked 10 from the 25 possible things for each ballot. Additionally, each spot on the ballot had a rank, 1-10. The goal was for the user to pick their top 10 from the 25 things and rank them.
The goal now is determine the 10 winners from the 25 things, and their rank in the winning 10. Some of the 25 things have the most votes for multiple ranks, ie object A has the most votes for rank 1, and the most votes for rank 2.
To determine the winning 10 and their rank, I used a weighted sum for each of the 25 things. Every vote was worth (10/rank), with a vote being a rank 1-10, with each user having 10 votes. (10 votes per ballot)
So my question is, would it be better to use a 10/rank weight? (ie, 10, 10/2, 10/3...ect) weighted system or use a (10,9,8,6....) weighted system? (linear vs log weights)? I used both methods to compute weighted sums for each of the 25 things and then I took the top 10. In both cases, the ones with the top5 weights weighted sums are the same, but the bottom 5 are different. I am not sure which method of weighting is better for this situation.
The votes are pretty evenly distributed, so my logic was to have rank 1 have the most weight relative to everything else. This is so that if a object of the 25 has a lot of votes for 1 rank, but less for 2-10, it would still rank higher.