Scott asks why, if the Median Voter Theorem is true, American politicians aren't all middle of the road, and barely distinguishable from each other.

Elegant as this proof may be, it fails to describe the real world. Democrats and Republicans don’t have platforms exactly identical to each other and to the exact most centrist American. Instead, Democrats are often pretty far left, and Republicans pretty far right. What’s going on?

He suggests a number of reasons. And they're all probably true at the margin. But there's a much more basic reason why parties aren't clones of each other: preferences are correlated.

I'm going to give a really simple toy example, and we're going to see that correlated preferences + first pass the post is enough to blow the Median Voter Theorem out of the water.

Let's pretend there's only two issues that matter to the American voter. Immigration, and Abortion. And let's pretend that these issues are binary - they're either fully legal, or completely banned^[1].

40% of voters want abortion to be banned, and 60% want it to be legal.

60% of voters want immigration to be banned, and 40% want it to be legal.

But opinions on these two issues are not uncorrelated. Almost nobody is against abortion, but loves immigration. Pretty much everyone who wants abortion to be banned wants immigration to be banned, and everyone who want immigration to be legal wants to abortion to be legal.

So voter preferences look like this.

Now, if we wanted to satisfy the most preferences, under reasonable assumptions^[2] we should ban immigration and legalize abortion.

But if you had 4 candidates, representing each of the 4 possible policy choices, the one who advocated this policy would come in a distant third:

I think this models the real world pretty well. Republicans and Democrats don't just differ a bit on the details, they represent baskets of policies, each of which the majority of their voters want. A party that chose the most popular option for each policy would be less tempting both to democrats and to republicans than their current party.

Other Related Stuff

That's the gist of the post over, view this as an appendix.

Ranked Voting

Ranked voting schemes tend to mitigate this issue. e.g.

Let's assume that when picking a 2nd/3rd choice candidate 50% of voters care about abortion more and 50% care about immigration more. Then we get this:

If we use a simple Borda count to  give 1 point to 3rd choice, 2 for second and 3 for 1st we would end up with scores:

However even using a Condorcet method we cannot avoid this problem entirely. Consider 5 voters and 3 policies, with the following preferences:

Then a candidate who stood for Abortion, Bail and Curfew would be preferred by a majority of voters to a candidate who stood against all 3, even though a majority of voters are against each individual policy when looked at independently^[3].

Median Voter Theorem doesn't apply to First Pass The Post

The Median Voter Theorem only applies to voting mechanisms where if a candidate would be preferred to everyone else in a head to head match, they would also win in a multi-candidate match. This doesn't apply to FPTP where two similar candidates can split the vote, letting a less preferred candidate win.

Median Voter Theorem basically never applies

The Median Voter Theorem only applies to single peaked preferences. This means that all possible outcomes can be ordered from most X to least X, and everybody just picks an amount of X they want.

So for example, the Median Voter Theorem would apply if everybody was voting on how many immigrants should be let into the country, since then everybody picks a number, and the further away the number of immigrants is from this number the sadder you are.

But it wouldn't apply if everybody was picking which countries to let immigrants in from, because everybody would rank them in different orders - some people would say the only people we should let in are Canadians, others might say Japanese or Europeans or whatever.

And it definitely definitely definitely doesn't apply to Presidential elections where you're voting on an incredibly broad set of policies, each of which people will have their own complex opinions on.

Median Voter Theorem is kind of trivial

As discussed, the Median Voter only applies when:

a) If a candidate would be preferred to everyone else in a head to head match, they would also win in a multi-candidate match.

b) All possible outcomes can be ordered from most X to least X, and everybody just picks an amount of X they want.

At which point it's obvious the Median candidate will win, because they are preferred to every other candidate in a head to head match (by definition of Median).

Parliamentary Democracies

The Median Voter Theorem only applies when there is a single winner of the election. But most countries don't actually have a single winner - they have a parliament, and unless a single party is preferred by a majority of the electorate over all other parties, the country will usually be ruled by a coalition.

Under these circumstances parties have an incentive to cater to a particular group of people. Since they can almost exactly match what the group wants, they are almost guaranteed their votes. Someone who tries to take the middle of the road approach won't get any votes. If proportional representation is used, you would naively expect there to be as many parties as there are seats, each getting exactly one seat. In practice people vote strategically + vote for the most salient parties, so parties tend to be significantly bigger than this.
 

1. ^{^}

   Whilst this obviously isn't true in real life, in practice the real question tends to be whether to move the current policy a bit more to the left, or a bit more to the right, which ends up as effectively a binary choice for most people, unless you happen to want a position exactly between where it currently is and where it will probably end up if your party wins.

2. ^{^}

   E.g. 
   - Those who want to ban abortion don't feel much more strongly about the issue than those who want to make it legal.
   - People's opinion on about one issue doesn't depend on what policy is chosen for the other, so if you make abortion legal, nobody will as a result change their mind about immigration.

3. ^{^}

   This is an example of Ostrogorski's paradox.