Foundation

The voters of Johnnashville are having an election! All eleven voters will get to choose who the next mayor will be. With ten voters, it should be easy to determine who’s going to win right? Not exactly. It all depends on who’s on the ballot.

Because you see, Johnnashville is funny. The law actually requires that all voters vote for the candidate whose views most represent their own, and they are not allowed at all to consider electability. Failure to comply opens up a trap door leading to a pool of acid (with no saving throw), and nobody wants that.

Let’s consider one case a lot of expecting to happen. With two candidates on the ballot, candidates C and G, it’s expected that candidate G will squeak by on a 6-5 vote, snagging voters 1-6 while losing voters 7-11. Let’s write that down, the preferences of the voters:

1. G < C
2. G < C
3. G < C
4. G < C
5. G < C
6. G < C
7. C < G
8. C < G
9. C < G
10. C < G
11. C < G

However candidate L has been making noises about running. That messes everything up; in that case, candidate C wins on a 5-4-2 vote:

1. L < G < C
2. L < G < C
3. G < L < C
4. G < L < C
5. G < L < C
6. G < L < C
7. C < G < L
8. C < G < L
9. C < G < L
10. C < G < L
11. C < G < L

Looking at the preference chart, all the voters who hate candidate C shook their heads and sighed. “If only we could avoid this law where we have to vote our preferences!” they moaned, until one came up with an idea: “Let’s form a political party! That way we can beat C and L and win!”

Candidate L even upped the stakes: “Hey, that’s great! If you guys nominate one of these other guys”, pointing to candidates M and T, “then I won’t even run, and victory is assured!”

The low-digit voters of Johnnashville loved this idea, and immediately got to work planning their low digit primary. Candidates T, G, and M were on the ballot, and the winner was… G?

1. L < M < T < G < C
2. L < T < M < G < C
3. G < T < M < L < C
4. G < T < M < L < C
5. G < M < T < L < C
6. G < M < T < L < C

Realizing that they were doomed, the finger pointing began. “One and two, it’s all your fault! You people need to get right in the head and not vote L! And L, you need to not run because we all hate C!” But one and two countered with just as much anger: “It’s not our fault that’s how the system works! You guys need to get over G because he just can’t win!”

Three through six then got red in the face “You people, one and two, you need to break the law and vote tacticially in the general! The fact that you won’t shows you secretly WANT C to win anyway!” One and two countered, spittle flying, “You all need to vote tactically in the primary so we can band together in the general! If anyone wants C to win, it is you!”

Then they all thought in unison: “If only we could unite behind M or T, because together we all prefer either of them to C, and know that’s the only way to win. Curse this system that doesn’t let us vote tactically in the primary or the general election! Why on why can’t we follow the advice of our town’s namesake?”