I'm a part of a volunteer organization that organizes a bunch of events. For each event, members need to submit an application in order to attend the event. A lot of people want to go to these events and we only have a limited number of spots available, so the selection process is usually pretty competitive. The application selections (accept/reject) for each event are usually handled by a different group of people, so it can be somewhat challenging to ensure that the selection process is as "fair" as possible across the board. I happen to have access to all of the relevant data (who applied to which events, who was accepted/rejected, some stats on the applicant, etc.), and I'm trying to assess whether the current acceptance policy is "fair" or not.
In a perfect world, each applicant would be accepted the same proportion of the time. Obviously in real life this may not necessarily be the case. For example, if 5 people apply to 10 events each, and are accepted to 5,7,6,6,5 of them, the system appears to be mostly fair, since the difference between the acceptance proportions isn't that big. On the other hand, if the acceptance numbers are 2,3,9,8,3, the system is obviously unfair since a few people are being selected repeatedly at the expense of the others. What's the most meaningful way to quantify this in terms of actual statistics?
Also, I have access to some data which might influence the acceptance rate of a particular individual, for example the number of articles written by them in the past year. It makes sense that someone who has written more articles should be accepted to more events. Also, if someone has, for example, only applied to 1 or 2 events ever, it makes sense for them to have a higher acceptance rate than someone who has applied to dozens.
So in short, I'm trying to come with a way to figure out if the current distribution of acceptance proportions is correlated in terms of some of the variables to which I would expect it to be correlated, as opposed to based solely on luck, and if it does turn out to be based solely on luck, whether the proportions are roughly uniform across the population or some people are repeatedly being favored at the expense of others.
Can someone guide me in the right direction here?