Ballot: use range [0, MAX]

Counts:

- 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[21] B[16] and A would win.

Remarks:

- 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.