Fair Rank Voting When Some Members Do Not Vote

I am currently trying to work out a problem with a voting system for a local club. Competitions are held frequently and the current system works as such:

• Everbody who enters the competition gets to vote
• Everybody gets 3 votes; first second and third
• First is worth 3 points, second is 2 and third is 1
• After all votes are submitted the points are added up those with the most points win (currently top 6)
• You cannot give any vote for yourself

The problem we have is that there are sometimes situations where one or more members are unavailable to cast votes. This is gives them an unfair advantage.

E.g. Members A, B, C and D enter. D does not vote. This means that A, B and C must spread their votes out among the other 2 voters plus D which means that every voter must vote for D whereas the other 3 contestants do not have the same advantage.

I ran some simulations and for 10 contestants and 1 non-voter, the non-voter scores an average of 6 points and the others score an average of about 5.3.

Intuitively the advantage for the non-voters decreases as the total number of contestants increases.

We have considered a couple of solutions, such as taking 1 point off the total of the non-voters, or spreading the number of votes the non-voters would have had evenly across the rest of the contestants.

Neither of these seem very elegant and I am sure there is a mathematical solution to this that depends on the number of contestants and number of non-voters.

Is there a way to keep the voting fair under the above system?

• It's changing the rules, but you might want to consider not requiring voters to use all three choices. They could vote for just two candidates, giving them $3$ and $2$ points, and not use a $1$. See en.wikipedia.org/wiki/Bullet_voting Mar 23, 2018 at 16:17

1 Answer

I think that it would be more fair to spread the non-voters' hypothetical votes according the actual vote percentages, rather than evenly. So if D isn't voting, and A gets more votes as B, then A should get more of D's votes than B does.

Mathematically, I think you should multiply all the voting candidates' total by (number of voters)/(number of voters-1). Note than in the case of where there are 9 voters, this means multiplying by 9/8. Taking the numbers that you got in your simulation, (9/8)(5.3) = 5.96, which is close to the 6 that the non-voter got (presumably, rounding and margin of error is enough to explain the difference). The logic for this is that in that if there are V people who vote, then the non-voters have V people who can vote for them, but the voters have only V-1 people. So the non-voters have V/(V-1) times as many potential votes. An equivalent method is multiplying the non-voters' totals by V-1 and the voters' by V; this keeps all the numbers integers, if you like them better than fractions.

I do think that if you aren't able to find a method that everyone agrees is perfectly fair, you should lean to erring on the side of people who vote having an advantage.

PS The name for giving decreasing number of points as you go down the ballot is "Borda count".