User:Psephomancy/Three tribes

From electowiki

Short version

Imagine a society has three tribes, and each runs a candidate in an election. Every voter loves their own tribe's candidate, while hating the other two tribes' candidates.

There's also a fourth candidate, who is liked by every voter and is listed as everyone's second choice.

Who should win? A candidate hated by two-thirds of the population, or a candidate who is liked by everyone?

Longer version

Let's say there's a nation with three tribes in it (A, B, and C), who haven't been getting along. They're voting for mayor, and a candidate from each tribe runs. There's also a fourth-party candidate who is respected by all of the tribes, but no one's favorite. Their preferences are:

  • Tribe A:
    • 100 people
    • Ranks candidates: A > D > B = C
  • Tribe B:
    • 100 people
    • Ranks candidates: B > D > A = C
  • Tribe C:
    • 101 people
    • Ranks candidates: C > D > A = B

Candidate D is everyone's second favorite, but nobody's favorite. If a binary election were held between only A and D, D would win by a landslide (67% to 33%), since 2 out of 3 tribes prefer them. Likewise, D would win if an election were held against only B and likewise if D ran against only C. D is, therefore, the Condorcet winner, or "beats-all winner".

If instead of rankings, you asked each tribe how much they liked each candidate, they would give their own tribe's candidate the highest rating, candidate D a high rating, and the other tribes' candidates the lowest rating. So D would have the highest overall approval rating (say, 60-80%), since they're supported by all of the tribes, vs the 33% approval rating for each of the tribes' own candidates. It's pretty clear that D would also be the best winner if based on approval ratings.

So D is both the most-preferred candidate and the most-approved candidate.

Yet, if this nation used any of the common voting systems used in real-world elections, D would be eliminated immediately, since they got zero first-preference votes:

All of these systems would then eliminate A and B, leaving C as the winner (by 1 vote), despite 2/3 of the population listing C last on their ballots.

Is this a good result? Is this democratic? Will relations between the tribes become better or worse after C wins? Which candidate was the best representative of the nation as a whole?