English: 3 voters (blue circles) and 3 candidates (red circles) in a 2-dimensional political space. The voters prefer candidates closest to themselves, leading to a circular societal preference of A > B > C > A... and no Condorcet winner.
Voter locations:
1: (1, 1)
2: (6, 3)
3: (1, 7)
Candidate locations:
A: (2, 3)
B: (5, 1)
C: (4, 6)
Distances from voters to candidates:
1: A = 2.2, B = 4.0, C = 5.8
2: B = 2.2, C = 3.6, A = 4.0
3: C = 3.2, A = 4.1, B = 7.2
Note that the distances are not equal/symmetrical. A score voting system with honest voters would select A as the winner, since A is closest to the centroid of the voters. See below.
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Score voting
Under Score voting, candidate A would win, as they are the closest to the centroid of voters. For instance, if the largest distance (7.2) is assigned a score of 0, and all voters vote proportionally to that distance on a scale of 0 to 6, the scores would be:
A
B
C
1
4
3
1
2
3
4
3
3
3
0
3
Total:
10
7
7
If instead the voters normalize their votes based on their own maximum and minimum distance, with the middle candidate scored proportionally, the results are:
A
B
C
1
6
3
0
2
0
6
1
3
5
0
6
Total:
11
9
7
Other ways of normalizing the scores also lead to A winning.