Political Analysis: Median Voter Theorem and Voting Systems

In this final, though not very conclusive, post on political analysis, we’re talking voting, voting systems, and ice cream.

The Median voter theorem helps explain the importance of the swing voter. For the MVT there are four elements:

  1. A set of n voters where n is odd (apparently an even number can work too, but we didn’t really talk about it, and, as in Minecraft, odd numbers just work better.)
  2. Unidimensional policy space (i.e. right vs. left, socialist vs. capitalist, more funding vs. less funding)
  3. Voters have quadratic or “single-peaked” preferences that can be represented by the equation U[x] = [x1 – x’]2. So they have one ideal point (x’) and whichever option falls closer to their ideal point is what they’ll vote for.
  4. The group makes decisions by majority rule (and we’ll talk about alternatives to this in a  bit.)

The theorem states that, if every voter in the group has an ideal point, the voter with the median ideal point is an indicator of how the group will vote.

To apply this, here’s an example:

The International Space Committee is voting between two bills to send colonists to Mars. There are seven seats on the committee. Three of them are hardliners who think colonization is a waster of resources, and their ideal point is 0 colonists. On the other end of the spectrum is a sci-fi fan who wants to send a hundred colonists. Then in the middle we have someone who wants to send three colonists, someone who wants to send five, and someone who wants to send twelve. So, to find the median, we list all these different voters out from the lowest to the highest ideal point, like so:

  1. 0
  2. 0
  3. 0
  4. 3
  5. 5
  6. 12
  7. 100

So, if someone wants their bill to pass, how many colonists should they call for? If you took an average of everyone’s ideal points, you’d get approximately 20—but if you pitted that bill against one calling for 0 colonists, the only people who would vote for 20 would be #6 and #7, because 20 is closer to their ideal points than 0. If you look at the mode, you see that the highest number of people want 0 colonists. But if you put that bill up against a bill for 5 colonists, that bill would fail. So really, the way to get a bill to pass is to look at the person in the middle—in this case, 3 is their ideal point, and each side should ensure that their bill is closer to that number than the other side’s bill if they want it to pass. We can look at any two bills (or a yes/no vote on any one bill) and see which one will win a majority based entirely on how the median voter will vote.

But this spatial model doesn’t always work—sometimes an issue isn’t just right/left, and it causes majority voting that appears irrational. Consider the following example, with this voting profile:

There are thirty people in the Assembly of Ice Cream Experts, and they are voting on the national ice cream for the nation of Myimagi. There are four options: Chocolate, Vanilla, Strawberry, and Mint. Every member has a ranked order for these choices, which I will list below.

Number of Voters Preference
3 Chocolate > Strawberry > Mint > Vanilla
6 Chocolate > Mint > Strawberry > Vanilla
3 Vanilla > Strawberry > Mint > Chocolate
5 Vanilla > Mint > Strawberry > Chocolate
2 Strawberry > Vanilla > Mint > Chocolate
5 Strawberry > Mint > Vanilla > Chocolate
2 Mint > Vanilla > Strawberry > Chocolate
4 Mint > Strawberry > Vanilla > Chocolate

There are a few different ways that they could conduct this vote, each of which greatly affects the outcome.

  1. Plurality. The choice with the most 1st place votes wins. A majority isn’t required. This is how the UK does parliamentary elections. It’s pretty straight forward and quick. In this case, Chocolate, with 9 votes, would win.
  2. Dual ballot. Like a run-off election. In the first round, if there is no majority, you determine the top two choices. In this case there’s no majority, so the top two choices would be Chocolate with 9 votes, and Vanilla with 8. Then there’s another ballot with just those two, and whoever wins that wins—and if you look at the guys who put Strawberry and Mint first, you can instantly see that they won’t vote for Chocolate in the run-off, because they all have that ranked last. Apparently Chocolate is pretty polarizing. Anyway, in this case Vanilla would win with 21 votes.
  3. Instant run-off. Like the dual ballot, but with multiple rounds, and instead of narrowing it down to two, only the lowest candidate is dropped each time. This is how Australia elects their representatives. In the case of the AICE, Mint (with only 6 votes in the first round) would be eliminated first. In the second election, chocolate would be eliminated, because the 9-member Chocolate caucus is the only group that actually likes it. But check this—the Chocolate caucus vehemently hates Vanilla, so with just Vanilla and Strawberry remaining, they’ll cast their nine votes in with Strawberry, giving them a majority without even needing to include the four mint guys who like Strawberry second best. So in this case, Strawberry is the winner, with 20 votes.
  4. Borda count. A point system, based off how each voter ranks their choices. Lots of awards use this system. There are many ways to assign points, but a simple way (and what I’m going to do here) is to have the lowest rank weighted at 0, second lowest at 1, and so on up until the highest ranked choice, which would be rated at n – 1. All the points are added up for each choice, and whichever one gets the highest score would win.

Borda counts can be difficult to calculate, clearly, so to do this quickly we’ll make a dominance matrix.

Chocolate Vanilla Strawberry Mint
Chocolate / 9 9 9
Vanilla 21 / 10 10
Strawberry 21 20 / 13
Mint 21 20 17 /

What this table is saying is, how many people would vote for the row item over the column item? So you can see by looking down the Chocolate row, it’s the same 9 people voting every time. And if you look at the Chocolate column, you can see that everyone hates Chocolate.

To get the Borda count, we sum each row (trust me, this is mathematically sound), and get the following:


So with our Borda count, Mint actually comes out as the winner.

So, what the heck? We have four different procedures here, each of which produces a different result. If there was a Grand Master of Ice Cream charged with determining the voting procedure for this, they could effectively decide the outcome just based on what procedure they choose.

So, what is the best procedure? Unfortunately, all four have their problems. Even the Borda count, which seems pretty democratic, ends up with a vote that only 6 people are really thrilled about. Imagine if everyone in the AICE really loves their first choice, but hates their second, third, and fourth choices to slightly varying degrees. Then the Borda count would be horrible, and you’d probably be better off just doing a plurality vote, because at least you’d have the most people happy as you possibly could. Each works better given different circumstances. The GOP probably wishes the primary could’ve been a Borda count, because Trump is sort of Chocolate.

Well, that’s it for political analysis. I may do some posts about applying political analysis to writing, but as far as this kind of stuff, I’ve run through the entirety of my notes. All due respect to my professor, Douglas Dion, who did an excellent job of teaching this material in a clear and entertaining way, which I have hopefully emulated here.

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