Last night, using simple logic, Simon proved to me why a two-candidate plurality voting system is better than many others.

There’s a theorem called Arrow’s Theorem which says that any ranked voting system with 3 or more candidates cannot have all of the following three properties:

*Non-dictatorship:A dictatorship is a system where one person decides the entire election, even if everyone else is against them. This should obviously not be the case.

*Unanimity:If literally everybody ranks candidate A over candidate B, the results should also rank candidate A over candidate B.

*Independence of Irrelevant Alternatives:If some people just change the ranking of C, that shouldn’t affect A or B in the results.

The pictures are of me trying to prove this theorem.

Each of these properties are great for a voting system to have. Removing any one of these properties will lead to strategic voting (intentionally lying in the hope that that helps your favorite candidate).

Note that plurality voting is also a ranked voting system; it just only takes advantage of voters’ 1st-place candidates, not the full rankings.

So what to do if you have more than 2 candidates?

Score voting (based on giving every candidate an individual rating) is the optimal system in that case. There is also approval voting (which is actually just a simpler version of score voting), but people tend to mistake it for plurality voting, which, as I said already, is bad.

In The Netherlands (elections coming up in March!), we have sort of a rescaled version of plurality voting (as Simon has defined it), when the seats in the parliament are distributed among many parties according to their ranking across the country. So the Dutch system doesn’t withstand the criticism of Arrow’s Theorem!

I just want to be clear here. It doesn’t matter how you try to improve the voting system (for example with rounds, or like in the US, the Electoral College), as long as it’s a ranked system with more than two candidates, it’ll fall victim to Arrow’s Theorem.

Inspired by Undefined Behavior and Jordan Ellenberg’s book *How Not to Be Wrong?*