When I began dipping my toes into game theory and rational choice theory, like many others, I learned about the Median Voter Theorem (MVT). This theory is essentially the Hotelling’s Law of voting, in which two competing politicians, on any given issue, will adopt views similar to the median on a spectrum of views of that issue, in order to maximize the number of votes they receive. Any movement toward either extreme, so the theory goes, would allow the opponent to gain the votes of centrists by moving in the same direction, but not as far, effectively gaining all voters on the other extreme AND the centrists. According to MVT, the most successful politicians should, if rational choice theory can be said to apply to elections, represent (if not hold) the views closest to those of the median voter, who should be relatively “centrist” even if extremist voters outnumber centrists.

This is, rather dramatically, not the case. History and current events offer a plethora of examples: a brief look at the makeup of the US government implies that centrist voices (and especially centrist voters) are outnumbered and drowned out. If MVT has any effect at all, why is increasing political polarization such a hot topic?

What are the possible explanations for this? Is MVT fundamentally wrong in its core idea, and is voting in fact not possible to model in rational choice theory? I do not think so, so here are several ideas, some old and one novel, about why the application of MVT does not lead to centrist politicians winning most elections in practice:

Voter preferences are polarized . If voter opinions are not only not normally distributed, but are in fact gathered at two poles with no centrist voters, MVT may actually function, but it would predict that centrists would lose a lot (because the median voter would be at one pole or the other). The idea that voters resemble a barbell graph more than a normal distribution may be very salient, because many issues are dominated by extreme views. However, the voting population is demonstrably more centrist on some issues, so this cannot fully explain the difference between reality and MVT.

. If voter opinions are not only not normally distributed, but are in fact gathered at two poles with no centrist voters, MVT may actually function, but it would predict that centrists would lose a lot (because the median voter would be at one pole or the other). The idea that voters resemble a barbell graph more than a normal distribution may be very salient, because many issues are dominated by extreme views. However, the voting population is demonstrably more centrist on some issues, so this cannot fully explain the difference between reality and MVT. Third parties spoil things . This hearkens back not only to Arrow’s Impossibility Theorem, but also to the influence of Ross Perot and Ralph Nader on presidential elections. Third party participation does not spoil MVT, because it still fundamentally follows the idea of rational choice theory about elections, but it complicates two-candidate models. Further note: multi-party systems are vulnerable to extremists . The possibility of invasion by extreme views is higher in multi-party systems, because if multiple centrist parties compete for median voters, extreme groups gain power disproportionate to their constituency.

. This hearkens back not only to Arrow’s Impossibility Theorem, but also to the influence of Ross Perot and Ralph Nader on presidential elections. Third party participation does not spoil MVT, because it still fundamentally follows the idea of rational choice theory about elections, but it complicates two-candidate models. Primaries spoil things . This is an old idea in US politics: in primaries, you have to run to the extremes, and then in the general election, run to the center. The very logic behind this idea, and any empirical evidence of it, proves the viability of MVT, but it does introduce complications to any model of it that fails to account for the two different elections many politicians must win. Also, we must remember to factor in the fact that politicians have a negative utility in abandoning past positions (especially recent decisions) of seeming inauthentic or losing their base.

. This is an old idea in US politics: in primaries, you have to run to the extremes, and then in the general election, run to the center. The very logic behind this idea, and any empirical evidence of it, proves the viability of MVT, but it does introduce complications to any model of it that fails to account for the two different elections many politicians must win. Also, we must remember to factor in the fact that politicians have a negative utility in abandoning past positions (especially recent decisions) of seeming inauthentic or losing their base. The Electoral College spoils things . Outside of US presidential elections, “swing states” may not have the same power, but in the US, candidates would be “rational” to pursue votes in swing states much more assiduously than they do in states that are nearly sure bets. This is only narrowly applicable, but the campaign focus on swing states indicates that candidates do at least campaign rationally (see academic studies or news reports), in that they seek votes strategically and not indiscriminately.

. Outside of US presidential elections, “swing states” may not have the same power, but in the US, candidates would be “rational” to pursue votes in swing states much more assiduously than they do in states that are nearly sure bets. This is only narrowly applicable, but the campaign focus on swing states indicates that candidates do at least campaign rationally (see academic studies or news reports), in that they seek votes strategically and not indiscriminately. Voter turnout skews MVT . In elections without mandatory voting, those who are less committed to the issues at stake—who are more likely to be centrists—tend to vote less than those who care greatly one way or another. Therefore, voting may be more about mobilizing one’s base than appealing to centrists. (Note: this is not an endorsement of mandatory voting, which can allow more zero-sum games in politics and the enforcement of which seems worse than the problem).

. In elections without mandatory voting, those who are less committed to the issues at stake—who are more likely to be centrists—tend to vote less than those who care greatly one way or another. Therefore, voting may be more about mobilizing one’s base than appealing to centrists. (Note: this is not an endorsement of mandatory voting, which can allow more zero-sum games in politics and the enforcement of which seems worse than the problem). People do not vote based on rational weighing of stances . This would be a troubling conclusion, and suggests that people may align with parties and candidates based on motives other than matching candidate views to their own. While this is certainly true to some extent, MVT can still predict the overall “centrism” of politicians. Also, just because many people have an irrational method of weighing stances does not mean that the aggregate result does not mean that the utility of favoring centrist positions would not be positive.

. This would be a troubling conclusion, and suggests that people may align with parties and candidates based on motives other than matching candidate views to their own. While this is certainly true to some extent, MVT can still predict the overall “centrism” of politicians. Also, just because many people have an irrational method of weighing stances does not mean that the aggregate result does not mean that the utility of favoring centrist positions would not be positive. Voting is multidimensional . Since single candidates represent dozens if not hundreds of salient issues in a single election, voters are forced to compromise on some issues in order to win others. This multidimensionality is not a counterargument to MVT, but an extreme complication, because “median” politicians could lose based on voter preference not within an issue, but between issues. That is, a politician can gain voters by recognizing weighting each voter’s stance by how likely that issue is to change their vote. This can be observed in the fact that referenda tend to follow MVT in a more straightforward fashion than elections. It is also a possible explanation for the rise of certain coalitions: no matter if constituents have deep disagreements on many issues, they end up aligned behind the same candidate based on inter-issue preferences. This may have been a major motivation for federalization and separation of power: different decisions are constrained to certain elections, allowing voters to communicate on each individual issue more clearly because of the reduced multidimensionality. The game theory on multidimensional voting is well developed, but still has some complications that have not been specifically argued: Complication: Special interest compared to general interest . Multidimensional voting allows politicians to promise concentrated special interest to voters on certain issues in order to gain votes rather than appeal to general interest across many issues. The implications of this have been shown in game form, especially in special interest influence on enforcement. In this, public choice theory proves the idea that governments tend to favor dispersed costs and concentrated gains. Complication: Singe issue voters . This is a subset of the above, or at least related to it. In multidimensional elections, voters obviously have to weigh issue preferencing. However, if a voter decides that they would choose a candidate so long as he agrees with the voter on a single issue, then on that issue, the single-issue voter has a hugely disproportionate influence on the candidate’s opinion on that issue. The reason: if the candidate runs to the center on that single issue, he risks his opponent capturing the vote of all single-issue voters on that side by running slightly more to the extreme. The Nash Equilibrium of such a situation would depend on how many single-issue voters there are at either extreme (I assume here that single-issue voters are not centrists, based on examples shown below, but I am open to argument) and how much of the non-single-issue constituency is ceded by focusing on single-issue voters, but it is distinctly possible that issue preferencing, especially the power of the ultimatum implied in single-issue voting, makes it rational for politicians to run to the extremes. Interesting examples of this phenomenon include background checks for guns, defense spending, and possibly marijuana legalization. (Please note that this does not constitute an endorsement of these ideas—whether the majority is correct is a different question from whether the majority idea is enacted by politicians). The single-issue voter idea continues to fascinate me . Most of all, it fascinates me how little (apart from some basic preferencing models in the literature) I can find that either theorizes or empirically examines the specific influence of single-issue voters and voter preferencing. I hope it is out there, and I just can’t find it (so send it to me if you have found one!). But if not, is anyone out there a public choice theorist who wants to help me figure this out?

. Since single candidates represent dozens if not hundreds of salient issues in a single election, voters are forced to compromise on some issues in order to win others. This multidimensionality is not a counterargument to MVT, but an extreme complication, because “median” politicians could lose based on voter preference not an issue, but issues. That is, a politician can gain voters by recognizing weighting each voter’s stance by how likely that issue is to change their vote. This can be observed in the fact that referenda tend to follow MVT in a more straightforward fashion than elections. It is also a possible explanation for the rise of certain coalitions: no matter if constituents have deep disagreements on many issues, they end up aligned behind the same candidate based on inter-issue preferences. This may have been a major motivation for federalization and separation of power: different decisions are constrained to certain elections, allowing voters to communicate on each individual issue more clearly because of the reduced multidimensionality. The game theory on multidimensional voting is well developed, but still has some complications that have not been specifically argued:

Thanks for sticking with a long read, and please give me feedback if you have any examples of this phenomenon or another angle on MVT!