How is voting (not) like buying a hot dog?

Imagine a beach with two hot dog vendors and a crowd of hungry vacationers. Each vacationer will buy a hot dog from the vendor located closest to their spot on the beach. Where should each vendor set up shop if they want to maximize their sales?

It may be intuitive to think that the vendors give each other space. Perhaps they set up equidistant from each other and the end of the beach, dividing it into thirds. But if they start there, Vendor A will quickly notice that they can attract more customers by moving toward Vendor B – inching into their territory in the middle third. The vacationers at the end of the beach may have a longer distance to walk, but they’ll make the trek since Vendor A is still the closest one to them. Vendor B, in an attempt to win their customers back, follows suit. Before long, the two vendors are bumping shoulders, set up right next to each other.

This thought exercise is attributed Harold Hotelling, an economist who was chiefly interested in explaining differentiation (or lack thereof) in firms. It explains why, for example, gas stations and fast food chains tend to cluster in similar locations. In 1957, Anthony Downs adapted it for electoral politics in An Economic Theory of Democracy. Per Downs’s Median Voter Theorem, when voters’ preferences and candidates’ positions can be located along a single dimension, then elections held under majority rule will be decided by the voter with the median preference. Even if you haven’t read An Economic Theory of Democracy, if you pay even passing attention to political discourse, you’re familiar with this part of Downs’s work. Any time you hear a political commentator or strategist arguing that one party or the other (usually the Democrats) should move to the center because they’ve gotten too extreme, the Median Voter Theorem is lurking in the background.

The Median Voter Theorem is an intuitive model with many attractive qualities. But its central prediction – that a two-party contest will produce identical platforms as both parties attempt to win the median voter, as both hot dog vendors attempt to capture the largest market share by setting up in the same spot on the beach – is clearly wrong. Our two major parties have arguably never been farther apart. This point was not lost on Downs, who spent the latter sections of the book thinking through the assumptions necessary to make this model of voting behavior work, and why those assumptions are unlikely to hold in practice, to explain why we do not observe convergence to identical party platforms.

Given the renewed debate over whether and how the Democratic Party should move toward the center and redouble its efforts to win the median voter, it is worth revisiting those assumptions. Voting is clearly not like buying a hot dog, but how is it not like buying a hot dog, and why might this make it difficult for a party to simply “move to the center” in order to win? Here is a (likely non-exhaustive) list of assumptions necessary for the theorem to work:

One dimension. The hot dog vendors are competing on a beach, where they can only move left or right. They are not in a shopping mall where they can move left, right, up, or down. Ideal candidate positioning becomes much less obvious the moment additional dimensions are introduced, especially since different voters place different weights on different dimensions. The one-dimension assumption also implies that candidates are offering the same goods; to keep with the analogy, they’re both selling hot dogs and neither is selling ice cream. But in politics, setting the agenda to issues more favorable to your candidate – selling ice cream when your opponent is selling hot dogs – can be a winning strategy. If you can make the election about issues where more voters already agree with you, you don’t need to change your positions.

Exogenous preferences. This is a fancy way of saying that voters’ preferences are taken as a given, and do not change in response to the positions the candidates take. The hungry beachgoers will keep their beach umbrellas planted in the same spot on the beach regardless of where the hot dog vendors set up; they won’t move them so that they don’t have to walk as far to get a hot dog. But in politics, we know that politicians are often able to persuade voters, meaning that the voters’ positions in left/right space are to some degree dependent on the positions the parties and candidates take. A party may be willing to take an extreme position now in hopes of moving voters toward that position over time.

Full participation / no disaffection. Every hungry beachgoer buys a hot dog, regardless of how far they have to walk. This is clearly not the case in politics; in the most recent presidential election, 64 percent of eligible voters cast a ballot. Even if we adopted mandatory voting, voters at the extremes of the left/right spectrum would still be able to cast protest votes for minor party candidates if they felt that the major party candidate closest to them was still too far away to support. This can pull candidates away from the center, as they cannot take their base supporters for granted.

Complete information. The hungry beachgoers can see exactly how far they have to walk to get a hot dog from each vendor. In politics, this would imply that voters know where both they and the respective candidates are located on the left/right dimension. This assumption would be nearly impossible for voters to satisfy even if candidates didn’t intentionally make their locations difficult to pin down. Most people, most of the time, have better things to do with their lives than to maintain a full accounting of every position each candidate has taken and precisely summarize those positions to a point on the left-right spectrum. This can give parties and candidates some flexibility to take non-centrist positions without the proverbial median voter noticing.

No uncertainty. The vendor isn’t promising how far you’ll need to walk for a hot dog tomorrow, they’re showing you how far you need to walk for a hot dog today. The opposite is true in elections, where candidates make claims about what they will try to do if they are elected. But in politics, talk is, famously, cheap. Candidates cannot simply announce their ideological position and leave it at that – they must persuade voters that this is the position they really hold and would pursue in government. When a candidate says they have taken the median voter’s position, have they really, or is that just the position the Median Voter Theorem says they should take? When a candidate takes a more extreme position, it is likely a more credible signal of the position they really hold.

Freedom of movement. The vendor can move their hot dog stand wherever they want, whenever they want, to maximize their sales. In politics, positions are sticky for a few reasons. First, voters’ perceptions of parties are sticky – even if a candidate or party really has moved, their reputations precede them. Second, movement generates uncertainty – does the candidate really hold their new position, or are they just saying that because it’s convenient? Third, candidates and parties do not have full control over how voters see them. Others, such as the media or their political opponents, also make claims about how moderate or extreme their positions are.

Unitary actor. The vendor, and the vendor alone, decides where to locate their hot dog stand. If the consumers on the edge of the beach had an ownership stake in the business and, by extension, a say in where the vendor sets up shop, they might be willing to sacrifice some market share to shorten their own walk to get a hot dog. This essentially describes the relationship between political parties – whose members care about more than winning for winning’s sake – and the candidates they nominate for office.

This isn’t to say that the Median Voter Theorem is useless. This is to say that it’s a model that works under a particular set of assumptions that are never fully realized. The closer these assumptions are to being met, the better the model works. If and when voters’ preferences are well-described by a single dimension, and candidates can persuade the median voter that they are closer to their preference without disaffecting their base, then it is in their interests to do so. But pulling this off is harder than it might seem at first glance.