Ballot: use range [0, MAX]
- for each candidate, the ratings received are sorted from lowest to highest.
- calculate: Q = “number of votes + 1” / (MAX+1)
- the breakdown points (position in the rating list) are Q, Q•2, …, Q•(MAX+1). If the breakdown point (position) isn’t an integer, the rating in the previous and next position is averaged.
- for each candidate, the values in breakdown points are added and the highest sum wins.
In case of a tie, the process is repeated (only for the candidates in a tie), changing the value of Q:
Q = “number of votes + 1” / (MAX+2)
In case of a tie, it will repeat, using (MAX+3) in Q, and so on.
Why multiple breakdown points?
The classic median considers only 50% as breakdown points, and this can lead to bad results. Example:
A: 0,0,0,5,5 --> Median 0
B: 0,0,1,1,1 --> Median 1
B beats A even though A is clearly better than B.
Assuming we have range [0,3], we observe the following case:
A: 0,0,0,0,0,0,0,0,0,0,0,3,3,3,3 --> Sum breakdown: 3
B: 0,0,0,1,1,1,1,1,1,1,1,1,1,1,1 --> Sum breakdown: 3
The sum of the ratings of A and B (Score Voting) would be equal, so tie makes sense.
Now, maximize A’s ratings, and make B slightly better:
A: 0,0,0,0,0,0,0,0,3,3,3,3,3,3,3 --> Sum breakdown: 3
B: 0,0,0,1,1,1,1,1,1,1,1,2,2,2,2 --> Sum breakdown: 4
In this system B wins while with Score Voting there would be: A B and A would win.
- This is one of the worst cases (rare).
- B isn’t very far from A (as happens using the classical median).
- The ratings (utility) in B are more distributed among the voters than in A and this, for some people, can make B consider better than A (or anyway not much worse), even if it has a lower total utility.
- The high resistance to the strategies of the median concept may be sufficient to justify this case.