how to fairly select a leader

Question

I recently came across a rather practical problem:

A large group (around 30 people) wanted to elect a new leader (someone who is not part of the group) of 4 possible candidates. Each of the candidates can be interviewed by the group once (e.g. for one hour). Each member of the group attends at least 3 of the 4 interviews.

What fair ways do you suggest to elect the leader? (And if you know, what properties does your choosing system have?)

I see the main problem that the 'best' candidate could be in the interview where the least number of group members attend. (Another problem would be whether the final decision should be made after all the interviews (so that each member could make a priority list) or rather that each member should cast his 'vote' (in any form) at each of the interviews.)

PS: In the end the system still needs to be so simple that the group of non-mathematicians will grasp it without hours of lecutres about the topic=)

PPS: I think I should tell what system I suggested: Each member should make a priority list of those candidates who's interview he attended. That means someone who attended all interviews distributes 3,2,1,0 points to the four candidates, thos who only attended in three interviews distributes 2,1,0 points to the three candidates who's interview he attended. The candidate with the most number of points wins. (Do you thinkt that is a good system? Could it be improved? What do you think are it's weaknesses?)

Answer 1

Experience tells us that this problem is difficult also when all the voters know what all the candidates stand for. There are also different practical solutions in this case, each with different mathematical properties.

For instance, voters can vote on who they think is the best candidate, and the candidate with most votes will be the winner. Or the number of candidates may be reduced by elimination in some way or other, and there will be a runoff between two candidates. If the voters have given full rankings of all the candidates, then the result of the runoff may be computed without holding a new election. Books have been written about this problem.

What seems to be the issue here is that not all the voters have full information. To focus on that aspect, let any method be given which provides a winner when all voters are able to rank all the candidates. If a voter is only able to rank the candidates $a,b,c$, then replace his or her vote by four votes, each with weight $1\over 4$, where the last candidate $d$ is inserted in each of the four possible places. (I.e., the incomplete ranking $abc$ is replaced by the four rankings $abcd$, $abdc$, $adbc$, $dabc$, and each of these is given $1\over 4$ of a vote.) Then use your given method to compute the winner.

The mathematical properties of this method depends on your solution to the case with full information. Once you have decided that, you may start to look for properties preserved under incomplete information. (Example: If the method of election is to disregard the votes and always choose the oldest candidate, then the method suggested here will give the same result also in the case of incomplete information.)

Answer 2

Have a look at Arrow's impossibility theorem. If by 'fair' you mean Arrow's fairness criteria, then your question has no answer.

Answer 3

As noted in other answers and in comments, there is not one right (or perfect) answer to questions like this--and even if all group members attended all interviews, there still wouldn't be a perfect answer.

Here are a couple of other possibilities based on 'approval voting':

(1) Absolute approval voting: After each interview, each attendee approves the candidate (as acceptable) or disapproves the candidate (as unacceptable). The winning candidate is the one with the most approval votes. Too bad for a candidate who has less people attending her/his interview.

(2) Relative approval voting: Same voting as above, but now each candidate's score is the proportion of approval votes. So for instance if a candidate had 25 interview attendees, of whom 19 approved, that candidate's score would be $0.76$. High score wins.