Median Voter Theorem in other Voter Models?

The Economist's Why a left-wing nominee would hurt Democrats explicitly invoked an implicitly assumed proposition in political discourse: the median voter theorem "holds" in US elections.

Briefly put, the median voter theorem states that, if we have an odd number1The assumption of an "odd number" of voters is technically not needed, if we have some way to break the tie, or if (for an even number of voters) we have made sure the 2 median voters agree on how they'll vote. The assumption on the odd number of voters is not strictly necessary, it's just a helpful assumption. of voters who are rational and the vote is about an issue describable by a one-dimensional issue space, then polling the median voter will tell us the outcome of the vote. I won't digress on the assumptions of this theorem or the conditions when it holds too long, the take-away message is the median voter's vote coincides with the result of the vote, so a rational candidate would run as close to the median as possible.

But I draw your attention to the fact that the median voter theorem assumes the voters are rational, in the game theoretic sense of the term.

Puzzle: Does the median voter theorem hold for other voter models?

We can be generous and weaken the median voter theorem statement to be something like There exists a voter, the Median Voter, and the Median Voter's vote "correlates strongly" with the outcome in a first-past-the-post vote with two candidates.

I don't have an answer to this puzzle, nor have I searched the literature hard enough to be satisfied. (There may be some obscure article on this very puzzle, I am unaware of it though.) The answer is famously "no" for ranked-choice voting with multiple candidates, but for incomplete information or alternative voter models...these have not been adequately explored in the literature. I just have accumulated some notes that may be germane to this puzzle.

Aside 1: Half-life of information. It has been noted that the half-life of information, for the average voter, is less than half a week2The conclusion to be drawn from these analyses is clear: memory for campaign messages not only fades but fades quickly, the half-life of the message typically being less than a week., page 315 (bottom of the right column) of Milton Lodge, Marco R. Steenbergen, Shawn Brau, The Responsive Voter: Campaign Information and the Dynamics of Candidate Evaluation. The American Political Science Review, Vol. 89, No. 2. (Jun., 1995), pp. 309-326.. Lets use an exponentially decaying moving average with the half-life set to half a week. (So W = 3.5 ln(2) days.) Specifically, integrating new information Y_n (which occurs at time t_n) into a "stockpile" or "accumulator" S_n obeys the formula: \[ \begin{split} S_n &= \alpha(t_n - t_{n-1}) Y_n + \left[1 - \alpha(t_n - t_{n-1})\right] S_{n-1} \\ &= \left[1 - \exp\left({-\frac{t_n - t_{n-1}}{W}} \right)\right] Y_{n} + S_{n-1}\exp\left({-\frac{t_n - t_{n-1}}{W}} \right) \end{split} \]

For very frequent events, the exponential function becomes approximately linear (since \(t_{n}-t_{n-1}\) is "small").

Remark. We need some way to translate an event, a speech, news coverage, etc., into values for Y_n. Presumably it follows a log-normal distribution. For a one-dimensional issue-space, we could arbitrarily pick positive Y to benefit one candidate, and negative Y to benefit the other. We'd need another random variable, Bernoulli distributed, which controls the sign of Y.

References

• Milton Lodge, Marco R. Steenbergen, Shawn Brau, The Responsive Voter: Campaign Information and the Dynamics of Candidate Evaluation. The American Political Science Review, Vol. 89, No. 2. (Jun., 1995), pp. 309-326.