Note: “the MVT is a good empirical first approximation” is not the same as “the MVT is a good predictor of politician behaviour”.
This is because of two things: first, the MVT does not necessarily hold when issues are multidimensional. Plott (1967)’s AER article demonstrates that when voter preferences are multidimensional, then the requirements for a stable majority vote to exist at all are quite stringent and unlikely to obtain in reality. The usual voting problem issues crop up. The winner is ultimately the agenda-setter, who can control the final vote and therefore the outcome.
It is however true that most contemporary issues are observed to align along a single axis. But this is the second issue: the more true the MVT is as an empirical first approximation, the more similar politicians will be, and the more they will need some way to distinguish themselves from their competitors! All that the MVT tells you is that over the vast majority of the policy possibility space, politicians will have similar views. And so they do. We could fund programs to search for the Lost City of R’lyeh, but we don’t, etc.
Because politicians are so similar over the vast majority of the space, the gains from introducing additional dimensions for voters to puzzle over are potentially enormous. All they need to do is introduce a single issue which breaks the pairwise symmetry of the existing median-voter-favoured equilibrium—some issue where the degree to which people care is deeply asymmetric. If it ‘sticks’, then they are assured of a victory. If it doesn’t, then all you’ve lost is some advertising budget.
So that’s what politicians do: they try to find issues which, at least for a while, don’t align cleanly along the predominant axis. Most issues won’t stick, because the possibility space is enormous.
Hmm. The article is technically correct but irrelevant. The case where necessity fails relies on three conditions: (1) the number of voters is even (2) the number of voters is small (3) at least one voter has their optimal preferences exactly identical to the proposed equilibrium; not merely ‘very close’ but exactly. All three (plus some additional, complicated conditions) must hold for Plott’s conditions to be sufficient but not necessary.
(2) is obviously not a concern here, for nation-state electorates. (3) is implausible: just introduce a suitably fine-grained continuum of possible policies. If you still have an ideal voter at the equilibrium, it’s not fine-grained enough.
On (3), in particular: in general, mainstream economics ignores degenerate cases in utilitarian analysis. That’s why the additional conditions are not mentioned: it requires that of a (finite) number of voter ideal points, at least one of them must fall on the equilibrium. But in a multidimensional phase space, the set of equilibrium points is a set of measure zero! Why would you care about that case?