Let’s assume we have voters with the following honest interests:

G1: “I love A almost double B, I love B triple C and D, E is the worst candidate”.

G2: “I love B to the max, all the others are the worst in the same way”

I could represent these interests in an honest and simple way in the form of proportions (the values are assigned to the candidates in this way A,B,C,D,E):

G1: 5,3,1,1,0

G2: 0,1,0,0,0

In the example I want to propose you have 3 voters for G1 (75%) and 1 voter for G2 (25%).

For convenience, I represent the real interests by using 100 points for both, so that they can be compared a little better:

G1-3: 50,30,10,10,0

G2-1: 0,100,0,0,0

**Excluding any tactical vote, who is the single winner in this context?**

**SNTV - Single non-transferable vote**

The votes will take the following form:

G1-3: X,-,-,-,-

G2-1: -,X,-,-,-

**A** wins.

**IRV - Instant-runoff voting**

The votes will take the following form:

G1-3: 1st,2nd,3rd,4th,-

G2-1: -,1st,-,-,-

They are eliminated in order E,D,C, and finally **A** wins.

**Borda**

The grades take this form (I use the scores immediately, not the ranking):

G1-3: 4,3,2,1,0

G2-1: 0,4,0,0,0

**A** wins.

**DV - Distributed Voting** (definition)

Votes take this form:

G1-3: 50,30,10,10,0

G2-1: 0,100,0,0,0

They are eliminated in order E,D,C, redistributing the points, and finally **B** wins.

Note that: G1<4 wins B, G1-4 draw of A and B, G1>4 wins A.

Intuitively it makes sense that by increasing the voters of G1, A gets to win.

**AV - Approval Voting**

Votes take this form:

G1-3: X,X,-,-,-

G2-1: -,X,-,-,-

**B** wins.

Note that: G1>0 always wins B.

Observing the real interests, G1 voters prefer A more than B, therefore by increasing a lot G1, A should win sooner or later, instead in AV he continues to lose.

The incorrect representation of the interests of the AV is clearly a problem, making it wrong (compared to the DV).

**SV - Score Voting (range 0-100)**

Votes take this form:

G1-3: 100,60,20,20,0

G2-1: 0,100,0,0,0

**A** wins.

Note that: G1<3 wins B, G1≥3 wins A.

To deal with this case we need to make 3 observations:

- Same interests: G1’s interests regarding A and B, which are “A is almost double better than B” are perfectly respected (at the same identical proportion) from DV and SV.
- Same weight: in democracy it’s necessary that the voters (and the votes) have the same weight.

The fact that in DV, A wins when G1≥5, while in SV, A wins when G1≥3, means that either DV or SV are poorly representing the weight of the voters G1. - The vote of the DV, after eliminating C,D,E, takes this form:

G1: 62.5 , 37.5 (worth 100 points, exactly as G2’s points is 100 points).

The SV vote instead, excluding C,D,E, who are losers in any case, takes this form:

G1: 100, 60 (uses 160 points while G2 uses only 100 points).

The SV turns out to be wrong because it badly represents the weight of the voters (giving more weight to G1 than G2, risking to cancel democracy itself). This doesn’t happen in the DV where it’s guaranteed that the votes are always worth 100 points.

This thing in AV is masked by the bad representation of interests (which always makes B win, for G1>0).

All that has been said, applies both to the sum and the average of the points (therefore, it also applies to the Range Voting, STAR Voting and similar).

**Conclusion**

In this context, the DV is the only one that correctly wins B (if G1<4, otherwise with G1≥5 obviously wins A). The only other method that makes B win is the AV, which however would make B win with G1>0 (also with G1-1000), so it’s wrong.

**In contexts, without tactical votes and single-winner, DV can be the best.**

*I could also say that: if the DV returns the wrong candidate because of tactical votes, then the other methods will return a wrong candidate because of the same type of tactical votes, or bad representation of interests. Try to disprove this with a counterexample.*