Race to the centre

Seth is running against Olivia for Class President. It's not as exciting as it sounds, because the Class President only really gets to decide one thing: how long lunch time is.

It's a pretty divisive issue. Some of their classmates enjoy playing outside and would love for lunch to be longer. Others are total bookworms and want shorter lunch times so that they can get back to reading The Magic School Bus.

Still a better love story...

You might even say that the whole situation is a bit like a political spectrum:

I had some trouble deciding which way around to make the diagram. On the one hand, longer lunch times are definitely right-wing, because they represent greater social autonomy. On the other hand, longer lunch times are definitely left-wing, because they represent greater social autonomy.

Seth and Olivia have their own personal opinions on the idea lunch time length, which we can mark out on the spectrum like so:

When the election occurs, students will vote for whichever candidate's position is closest to theirs. This means that the outcome of the election is actually very predictable: whoever's policy is closer to the median is going to win.

Let's get technical. Let \(v_1, v_2 \in [0,1]\) be the candidates' actual preferences on the spectrum (where the spectrum has the obvious distance function), and let \(v_m\) be the median. Then:

Lemma 1: In the election scenario described above, the candidate who is strictly closer to the median will win the election.

Proof: Assume w.l.o.g. that the closest candidate position is \(v_1\); assume w.l.o.g. that \(v_2 < v_m\). Then candidate #1 wins the vote of the median voter plus everyone with a higher value on the spectrum. Hence, a majority.

Of course, the candidates can use this knowledge to help their cause. Both Seth and Olivia can commit to an election platform (an arbitrary point on the spectrum), such that if they get voted in, they will set the lunch time length to exactly what they proposed in their platform^*. So by making a small compromise towards the median, they might be able to get their own preferred policy.

Call the candidates' platforms \(x_1, x_2 \in [0,1]\). Let \(w: [0,1]^2 \rightarrow [0,1]\) denote the winning platform, i.e.: \[ w(\hat{x}) = \left\{ \begin{array}{ll} x_1, & |x_1 - v_m| < |x_2 - v_m| \\ x_2, & |x_1 - v_m| > |x_2 - v_m| \\ \dfrac{x_1+x_2}{2}, & |x_1 - v_m| = |x_2 - v_m| \end{array} \right. \]

(The third branch above is chosen as the average to disincentivise ties; it's the most sensible choice I could see for a scenario where the deciding voter is 'torn'.)

We can now define the system as a two-player game, with the payoff for candidate \(i\) defined as: \[ f_i(\hat{x}) = |x_i - w(\hat{x})| \]

Lemma 2 (political centrism): Assume the candidates are on opposite sides of the spectrum, i.e. \(v_1 \leq v_m \leq v_2\) or \(v_2 \leq v_m \leq v_1\). Then the unique (weak) Nash equilibrium is \(x_1 = x_2 = v_m\).

Proof: (Centrism is a weak Nash equilibrium:) \(w(\hat{x})\) doesn't move if either of the candidates change their platform, hence neither do the payoffs. (Nothing else is a Nash equilibrium:) [Omitted, but essentially, push further in if you're losing; scale back if you're winning; one of the two will always apply unless everyone is at the median.]

So far, not surprising. In this simplified voting system, the median voter ends up happiest.

What happens if both Olivia and Seth have magicschoolbus-wing political views?

Lemma 3: Assume the candidates are on the same side of the spectrum, i.e. \(v_1, v_2 \leq v_m\) or \(v_1, v_2 \geq v_m\). Then the unique Nash equilibria are when the candidates both propose the same point between \(v_m\) and the closer of the two \(x_i\), inclusive.

(E.g. if \(v_1 \leq v_2 \leq v_m\), then for any \(x^* \in [v_2, v_m]\), \(x_1 = x_2 = x^*\) is a Nash equilibrium.)

Proof: [Omitted. Similar to the previous, but too many words for a blog post.]

Some follow-up questions:

Q: So then why are there distinct political parties at all?

A: Limitations of the model. Firstly, not everyone trusts politicians to hold to their promises, and offering a suggestions that departs dramatically from your known values may actually undermine your chances. Secondly, the median is hard to precisely calculate (and hell, if it was easy, you could just run a country by plebiscite!). Thirdly, once you get close enough to the centre, politics gets complicated enough that there isn't a well-defined median on a spectrum.

Single-issue elections don't seem to happen in practice. (Corollary: If anyone tells you that a country's election results show how the people have spoken on their pet issue, like humanitarian foreign policy, workplace relations, gender equality or gay rights, they are probably feeding you bullshit.)

Also, some countries, like the USA, choose party candidates by (approx.) public vote, which means it's no longer sensible to model a unified party trying to optimise its outcome.

Q: So in this model, do the elections maximise net utility?

Um, what the fuck is utility in this context?

And even if we defined sensible utility functions, what the fuck is net utility? Utility functions aren't additive. They're weird abstract objects that are invariant under affine transforms; there's no such thing as zero utility; how the hell would you add things that don't have an additive identity?

I am confused.

Q: Did that girl turn into a plant?

Creepy, no?

Q: Why does this model privilege two random people to be the candidates? Wouldn't it make more sense to let everybody run if they chose to?

Modulo barriers to entry, yeah. Which actually makes me wonder now whether we could derive political parties by producing a model where a group of people collude to buy a candidate slot. Food for thought!

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* I am of course assuming that lunch time lengths are a core promise.