A Game Theory of Primaries

The usual setup is, suppose political philosophies are entirely one-dimensional, so that each voter can be identified by a real number \(x\) (say, less than zero for left-wing and more than zero for right-wing, with opinion on all issues always in perfect correlation), and let the cumulative density function of voters' opinions to be \(\mu : \mathbb{R} \to [0, 1]\).¹ [¹ I'm implicitly assuming that the set of voters is static—\(\mu\) is independent of politicians' positions. So turnout effects are implicitly abesnt; maybe I'm modeling a mandatory voting system.]

Then suppose there are two candidates, and suppose a voter just votes for whichever candidate is closer to her opinion. The candidates, in this game, are allowed to pick their politics; what will they choose?

A candidate will choose an opinion \(x\), but since she wants to win, she must ensure that \(\mu\left(\frac{x + y}2\right) \ge \frac12\), where \(y\) is the opinion chosen by the other candidate, if \(x < y\). Likewise if \(x > y\) then \(\mu\left(\frac{x + y}2\right) \le \frac12\). What is the equilibrium? Given the other candidate's opinion \(y\), there are three cases: \(\mu(y) < \frac12\), \(\mu(y) > \frac12\), and \(\mu(y) = \frac12\). In all three cases, choosing \(x = \mu^{-1}(\frac12)\) is a best strategy, so symmetrically \(y = \mu^{-1}(\frac12)\) and thus \(x = y\).

In other words, in this setup, both candidates choose to emulate the opinion of the median voter, \(\mu^{-1}(\frac12)\).² [² Note that the number of voters with that median opinion, \((\partial \mu)(\mu^{-1}(\frac12))\), is irrelevant: the result does not rely on centrism among the voters.]

However, the model above doesn't match most countries' political systems, since it leaves off primaries. And primaries change the game significantly, since they provide an incentive to move toward the median of the party, which will not be the median of the overall voter base. So let's set up a game with primaries.

Let's again have number line of voters, with the same \(\mu\) function. However, voters \(x < 0\) belong to one party (the Left party, say), while voters \(x > 0\) belong to the other (the Right).³ [³ I'm imagining that the parties are well sorted by ideology, but not that the dividing line is the "center", though that assumption will reappear later. I'm also imagining that a voter's party is independent of politician opinions that cycle. In other words, in this model voters don't change party to vote in a different primary, and every voter votes in the primary.] There are four candidates: two running for the Left candidacy, and two for the Right candidacy. Let's focus on the Left, where the two candidates choose \(x\) and \(x'\) as their opinions.

Remember that only voters in the Left party vote for the Left candidate; there are \(\mu(0)\) Left voters total. How do those voters act? Let's imagine (as we implicitly did before) that all candidates choose their positions, and the voters vote in both the primary and election afterwards.⁴ [⁴ So in this game candidates may not drift to the center after the primaries.] Voters want a candidate most like them to win the election. In the election, that means voting for the most similar candidate. But in the primary, that results in complex voting behavior. A far-Left voter prefers both Left candidates to the Right candidate, so will vote for the further-left candidate only if that candidate can win. A center-Left voter may prefer the Right candidate to either Left candidate⁵ [⁵ I guess they're in the wrong party.] and would sandbag the Left by voting for a further-left candidate.⁶ [⁶ I've never heard of this behavior in the real world, likely because in the real world you don't need to vote in primaries, which this model doesn't include. Patches welcome!]

So to win the Left primary, the further-left candidate \(x\) must remain able to win the election, thus \(\mu\left(\frac{x + y}2\right) \ge \frac12\),⁷ [⁷ Where \(y\) is the right candidate.] and accumulates votes \(\mu\left(\frac{x+x'}2\right)\) from organic voters plus \(\mu(0) - \mu\left(\frac{x' + y}2\right)\) from sandbaggers. \(x\) wants the total to surpass \(\frac12\mu(0)\), while \(x'\) wants to avoid this.

Well, the second term in the \(x\) vote share is independent of \(x\), so \(x\) only benefits from moving right toward \(x'\). So we must have \(x = x'\). But \(x'\) will move to ensure that the Left primary vote is exactly 50/50. And of course the Right will choose a candidate to ensure that the election vote is 50/50. Thus we have: \[ \mu(x) + \frac12\mu(0) = \mu\left(\frac{x + y}2\right) = \frac12 = (1 - \mu(y)) + \frac12(1 - \mu(0)) \]

What's interesting about this is that it is an overdetermined system! We have:

\begin{align*} x &= \mu^{-1}\left(\frac12 - \frac12\mu(0)\right) \\ y &= \mu^{-1}\left(1 - \frac12\mu(0)\right) \\ \frac12 &= \mu\left(\frac{x + y}2\right) \end{align*}

For all three to be true, we must have: \[ \mu^{-1}\left(\frac12 - \frac12\mu(0)\right) + \mu^{-1}\left(1 - \frac1\mu(0)\right) = 2 \mu^{-1}\left(\frac12\right) \]

I'm not sure how to characterize all solutions \(\mu\) to this equation,⁸ [⁸ Among continuous, monotonically increasing \(\mu\). Comments welcome!] but solutions always exist if \(\mu(0) = \frac12\) and \(\mu^{-1}(\frac14) = - \mu^{-1}(\frac34)\). This can be called the balanced parties assumption: the median voter is also precisely at the boundary of the two parties, whose medians are equally far apart from this median.

In this case, the two candidates chosen are:

\begin{align*} x &= \mu^{-1}(\frac14) \\ y &= \mu^{-1}(\frac34) \end{align*}

Instead of seeing a run toward the center, both candidates end up being the median of their own party, and are thus quite ideologically distinct.

With this setup, neither candidate has an incentive to change opinion. If the Left candidate were to move left, they'd lose the election; if they were to move right, they'd lose the primary.

If that system of three equations is not simul-satisfiable, you end up with distortions in the electoral system. It can become impossible for the center-Left to win the primary because almost all Left voters are to the left of the boundary of electability. In more extreme situations, the bizarre sandbagging effect is decisive, with the Left being unable to win because it includes too many voters who prefer the Right candidate. Turnout effects would likely solve this.

I do wonder if "distortions" like these could point to some notion of electoral dynamics, where the split between parties \(0\) is variable and some plausible mechanism allows these distortions to move it. That could produce some explanation of how changes in \(\mu\) cause changes in the structure of parties.

This model didn't have voters choosing their party; which is partly responsible for the bizarre sandbagging behavior which I doubt occurs in reality. Adding party choice would make the analysis a lot more difficult, however, since it would allow candidate behavior to change the voting base in the primaries.

But the most important lack in the model is the lack of any turnout effects. I'm not sure how to model turnout,⁹ [⁹ Does vote chance depend on opinion difference? Is it probabilistic or strategic?] and I don't know how it differs for primaries and for the election, but this is the most important change worth exploring.