Utopia. Class struggle. Liberty. Tyranny. Property. Natural law. Human rights. Rousseau, Locke, Paine, Plato, Spinoza, The Federalist Papers, Marx, Rawles, and the rest. What is a “good” society and how can “we” make our society better?

For me there was a time (age 18) when these things seemed very important. I’m a socially minded guy, and political problems seem to always be f**king things up for people who don’t need their lives f**ked with. If you fancy yourself compassionate and intelligent, it’s natural to be drawn to political problems. For me it was an ego draw — the appeal of “doing good” with my mind.

After a while, though, I started to feel like I was going in circles, endless debates that seemed to dance around — but never solve — certain fundamental problems (and meanwhile Idi Amin killing his countrymen, Bosnians and Serbians tearing each other apart, etc). Schools of thought seemed to coalesce around personalities (not facts) and I felt this pursuit was going nowhere.

I wanted a way out…

The Median Voter Theorem

Imagine 10,000 people were voting on which of 2 congressional candidates to elect. Each candidate is represented merely by a real number* which indicates liberal -vs- conservative. Once elected, the candidate implements policies robotically. “I am 23% liberal and 77% conservative, therefore I will do exactly what a 23% liberal would do.”

If voters have single-peaked, symmetrical preferences over the same real number* spectrum and everyone knows the formulas and figures involved, then there is only one Nash equilibrium strategy: run to the middle. Campaigning on a policy that pleases the median voter is the only Nash equilibrium and therefore what rational, winning politicians would do.

* or element of any measurable space, like a sig-algebra

Interpretation

This result is niche famous. People whose friend took a game theory class in college might have heard a version of this as a “proof” that the 2-party system is better or more centrist than multi-party systems.

I’m telling the more mathematical version because, when “popular accounts” try to relate an important result like the median voter theorem while taking the math out, they end up making no sense or accidentally lying.

I ain’t gonna talk down to you. You’re smart enough to look up what a Nash equilibrium is. And you can decide for yourself what it means if a mathematical model sounds somewhat like human reality but isn’t exactly like it.

The median voter theorem doesn’t say that 2-party systems are better, it suggests something — or maybe it doesn’t. It’s just a piece of math that may be relevant to real life, may serve as a mental model, may serve as a basis for intuition, or … may be misleading.

“Political Philosophy” from a different perspective

The Median Voter Theorem naturally brings up questions — questions that you wouldn’t think of if you framed your thinking in response to Rousseau, Locke, and other famous writers.

• Are these preferences symmetric?
• Are these preferences 1-dimensional?
• Are these preferences stable over time?
• Do these preferences map onto the real numbers? (or something isomorphic to R)
• Are these preferences single-peaked?
• Are these preferences symmetrical?
• Can you represent a congressional representative’s behaviour in office by a single number?
• What about after they’re elected? Won’t they deviate from what they said they would do?
• What about party politics? Won’t the party whips keep them in line?
• Back to the voters; what if the politicians don’t know what they want?

  (This last point turns out to be very important in Persson & Tabellini’s theory, which explains that George W. Bush could be re-elected not because the hateful simpletons who voted for him were numerous, but because they are predictable.)

More sophisticated would be to ask: “to what degree or in what ways / cases are the above things true or false?” And those questions tend to be more answerable.

Other kinds of questions you might want to add to the mathematical framework begun here:

• Is this in just one district? How do inter-district politics factor in?
• What about redistricting? What about gerry-mandering?
• What about fact X about Country A’s constitution vis-a-vis Country B’s constitution?
• What about cultural fact Y? How could we take account of that mathematically?

I could go on and on, and in fact many people have. I think this is how fields of research get started. This one is called “spatial voting theory”.

But this is ridiculous. People are not one-dimensional.

One surprising result, due to Rosenthal & Poole, about the unidimensionality question, is that — yes! — politicians are pretty well represented as just a number on a one-dimensional scale — like maybe 85% of their votes can be characterized this way.

Also surprising. Judges are even more unidimensional than politicians. However, voters are decidedly not uni-dimensional.

Not what I would have assumed, although I can make up a story to “explain” these findings post hoc. Actually I could make up lots of different stories and am just left with more questions. But. At least I’ve moved outside of the narrative of political philosophy handed down to me in college.

Wrap Up

Things I like about this book:

• politics = relevant  +  math = logical
• application of math to something more interesting than bridge engineering
• sorry engineers, but all the engineering in the world isn’t going to solve global poverty — that’s a political problem
• and it would be sweet if logical thinking could lead to an optimal constitution (if such a thing exists).

Things I don’t like about this book:

• didn’t know enough math at the time I read it to think deeply or broadly about what they were saying
• as far as I know, no practical applications (yet)

Too long, didn’t read: Variations on a theme, the theme being the Median Voter Theorem. Game theory leads to a framework for political analysis called “spatial voting theory” which is alternative to the “pure-humanities” approach from my college political philosophy courses.