Rating Formula Design (conceptual not coding) - php

Question

One element of my site is a rating system. I am puzzled a bit by how best to set up the formula and I hope someone with more math aptitude can help me.
Users upload pictures that are rated 1-10 by other users. The users rating then is the average of those scores.
Simple enough. However, I want to add some system which rewards users for uploading more pictures. SO that the formula would be average of ratings + some function of the number of pictures uploaded.
An example might be, Rating = AVG + .05 * Count
This formula would be somewhat fair for users who have uploaded 1-20 pictures... However, if users upload 2000 pictures they will have bipassed the entire rating system and automatically will have a 10/10.
So, my limited knowledge of post algebra math is failing. What would be some formula that would produce the desired effect? The word "log" keeps bouncing around in my head--but I honesty can't remember anything about why... :)

Answer 1

Just do something like:
avg + numofpics*scale*(.9^numofpics)
This will make it do that as they upload more pictures they get less and less. You can change .09 (the rate of decay) depending on how many pictures you expect the average user to upload.
This the equation used for half life decay

Answer 2

you could do something like:
Rating = Average + (\sum_{i=1}^numofuploads 1/i)*scalefactor
though the sum grows to infinity - the sum grows very slowly
Edit:
The idea is basically the same as with #maxhud's solution you add less points to your rating for every picture, and for simplicity say scaling is 1/3, and for now i use exact not floating point math
1 -> avg + (1/1)*.3 = avg + 1/3
2 -> avg + (1/1+1/2)/3 = avg + (3/2)/3) = avg + 1/2
3 -> avg + (3/2+1/3)/3 = avg + (10/6)/3 = avg + 10/18 = avg + 5/9 ~ avg + .55555
4 -> ...
technically the series (1+1/2+1/3+…) is going to infinity but you'd have to upload a huge amount of pictures to go over 50 - so you'd better choose your scaling factor carefully and give a bit, of thought. If you want to have a maximum of points that can be achieved via uploading this is the WRONG solution. You should rather go with something like
avg + scaling*(.9^n)
where n is the number of pictures. if you could upload infinitely many pictures you would have
avg + scaling*(1/(1-.9)) = avg + 10*scaling
for your rating: which is, as I commented, much better.
ps: I think #maxhud should leave
avg + numofpics*scale*(.9^numofpics)
^^^^^^^^^
because after uploading 10 pictures you have outweight your shrinking growth function.

Answer 3

Consider an alternative approach - you want to reward users by increasing for uploading images but perhaps the user is uploading many many images that are not rated highly. Do you want to reward them if the majority of their images are poor?? Consider stackoverflow as an example - you can answer many many questions, but if they are not considered "good" by the rest of the community your reputation will not increase, no matter how many answers you provide.
It is possible that this is not how you want to do it and want to reward quantity rather than quality, but should you decide the opposite you could try something like
UserRating = 10 * (AverageRating/10 + Scalefactor*((AverageRating/10)^2 * ImageUploadCount)))
You choose scale factor to be what you want and obviously limit the rating as maximum 10. This way, you reward mutiple image uploads, but you reward users with higher quality image uploads too. Consider someone automates uploading images with some type of web bot and all images are considered poor by users - do you want to reward it? This way you can reward multiple uploads, but better quality uploads are treated more favourably. Maybe not what you are looking for but worth considering perhaps - only you can decide....