Can Median Voter Theorem explain political polarization?

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.
  • 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.
  • 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.
  • 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.
  • 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).
  • 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.
  • 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?

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!

Rules of Warfare in Pre-Modern Societies

As my first foray into NOL blogging, I figured I would bring up a recent debate I had liberty, war, and peace that lingered in my mind: how have rules of war been maintained throughout history without a central enforcing agency? This question is fundamental to the understanding of the nation-state in IR theory, and is also an astonishing example of spontaneous order in an anarchic and chaotic scenario.

The quandary exists because even the laudable negative rights of life, liberty, and property ownership, as Eric Mack discusses in his essay on Just War Theory, require a positive enforcement by others. Similarly, “rules of war”–such as refraining from attacking non-regulars, not attacking neutral parties, abiding by the terms of treaties, treating prisoners of war with respect, etc.–are, theoretically, difficult to establish and dependent on positive enforcement. This is because if Party A respects these rules, they provide a perverse incentive to Party B to take advantage of Party A’s restraint, and if doing so gives Party B the upper hand, they can enjoy the benefits of betraying the rules of war with impunity. This is a classic Prisoner’s Dilemma, and if it generalized across many nations, the theory of rational choice would lead us to expect a coordination problem, in which those using the strategy of Party B would dominate the Party A’s.

I am certainly not the first to identify this, and the literature on overcoming coordination problems through iteration of the Prisoner’s Dilemma, regime collaboration, and international organizations and treaties is incredibly thorough (just for a taste, you can see James Morrow’s book, F.V. Kratochwil’s book, and articles by Duncan Snidal, Arthur A. Stein, and even James Buchanan and Victor Vanberg). However, I thought it would be interesting to examine the historical evidence of effective rules of war, particularly from the premodern period. Because global communication technology and networks, international courts, treaties, and organizations, and deterrence based on the terrifying weapons of modern war were lacking in antiquity and on through roughly the 18th century (open to argument on that one), premodern societies seem to be the best test of the effectiveness of rules of war and their mechanisms. I won’t discuss any in detail, and I am skipping many rules of war for which their effectiveness is not discernable (such as the Mahabarata, Deuteronomy, and the Quran), but here is a list of interesting examples for discussion:

  • The archaic Greek poleis:
    • As Victor Davis Hanson argues in his influential book, the Western Way of War, the incentive to focus on agricultural production and the fact that citizen-warriors were personally responsible for military service made the costs of long-term campaigns, especially given the lack of siege technologies and the difficulty in laying waste to wheat fields and olive trees, higher than the potential benefits. However, there were still disputes to be resolved, and raiding was still harmful to the agriculture of polis that was raided. In order to limit costs to both invader and defender, the poleis developed the hoplite warfare strategy, in which citizen-soldiers met for decisive conflicts in traditional, if not previously agreed, locations, in which limited territorial gains were afforded to the victor. While this does not describe every aspect of 7th-5th century warfare in Greece, this strategy pervaded the Greek mainland and allowed disputes to be resolved with minimal collateral damage and investment.
  • Thucydides’ Athens:
    • Though Thucydides’ History of the Peloponnesian War is seen as the invention of realism based on its “the strong did what they could, and the weak suffered what they must” representation of self-interest in foreign policy, his narrative as a whole shows an important constraint in war: if a military power makes war with the expressed intent of empire-building without casus belli, they will entrench their enemies, alienate neutral states, and cause divisiveness on the home front because they have lost the high ground. Thucydides notes that the majority of Greeks opposed Athens on the grounds of their selfish empire-building, and because of their inability to convince Sparta of their just motives, brutality to neutral states, internal dissension during the Sicilian expedition, and many other misfortunes of war (plague, death of Pericles, Persian intervention), Athenian power was broken. The lesson: Party B (from above) must consider the international reaction to Party A, and at least make a public showing that the war is just. Also, if Hitler had only read his Thucydides, he might have known that marching through Belgium may be tactically sound, but he was risking the same reaction that the Athenians risked in the Melian massacre.
  • POW’s and ransoming in antiquity
    • Several rules of warfare were maintained through the mutual benefits to combatants, the most notable being the conventions concerning ransoming. From at least 5th century Greece (in the Sphacteria incident) to Caesar, citizens could be ransomed following a battle—and there were even conventional levels of payment for these POW’s. This was a benefit specifically afforded to “civilized” foes, and Roman practice increasingly became enslavement rather than ransom, but this convention was widespread for centuries, possibly showing that ransoming enemies is an Iterated Prisoner’s Dilemma.
  • Ancus Marcius and Just War Theory:
    • Along the same lines as the Thucydides example, the Romans engaged in the ritual of the fetiales, including the enumeration of the just cases for war, before invading an enemy. This limited war to official disagreements with neighboring states, and other religious conventions were maintained that limited certain tactics in war (a noteworthy passage of the Aeneid shows that putting on the armor of your enemies for stealth purposes would be doubly punished by the gods). These conventions included looking down on poison as woman’s weapon and on taking some religious statuary as booty, and though Roman generals still poisoned wells or robbed cities of their gods, they received negative reactions by their contemporaries.
  • Hostage policies throughout antiquity:
    • Another problem with the rules of war is the enforcement of treaties, which have credible commitment problems. Both Greeks and Romans made imperial gains by breaking treaties, but it was common practice to overcome the credible commitment problems of both alliances and treaties to end wars that hostages, usually the children of influential citizens or nobles, were exchanged. Whether they were exchanged both ways (more common in alliances) or passed only one way (usually from the defeated to the victorious), hostages were used at least 250 times by Rome and countless times by other ancient civilizations to ensure the enforcement of treaties.
  • Carthage’s “Truceless War”:
    • While we often think of ancient war as anarchic and based on the whims of generals, wars that completely lacked conventions or limitations were rare. In fact, following truces that allowed for collection of the dead, ransoming of both the living and the dead, and supplication for one’s own life go back at least as far as the Iliad, and wars that lacked such conventions were shocking to ancient historians. Such wars occurred when one side broke a general convention, usually the convention of allowing enemies to surrender alive and be ransomed. Because of this betrayal, their opponents would also stop following any rules of war, and such wars became not about achieving strategic goals but annihilating the opponent entirely. Carthage, following their loss in the First Punic War, fought a truceless war with their former mercenaries due to lack of payment that featured escalations in mutilation and crucifixion until the mercenaries were wiped out, at great cost in men and money to Carthage.
  • Roman 3rd party arbitration or intervention:
    • The Romans, after they gained international prominence but before they ruled the whole Mediterranean, took an interest in wars between their neighbors. While this sometimes included imperialism, in several instances they served as a 3rd party arbitrator of peace, and even as an enforcer of peace in Antiochus IV’s invasion of Egypt.
  • Blood feuds:
    • While mentioning blood feuds brings up images of Hatfields, McCoys, and senseless brutality over generations, blood feuds were actually a mechanism for limiting violence through threat of reprisal. While the effectiveness of this mechanism may be debatable, its intention as a limitation of violence is notable in several pre-modern societies, especially the Scots and Slavs.
  • Chivalric codes:
    • We should be careful of romanticizing this example, but from the 12th to 14th centuries, chivalry established rules of conduct for how knights should treat knights on and off the battlefield. Much of the conception of chivalry comes from poetic fictions about historical figures that were vicious or corrupt in many ways. However, it was actually the battlefield codes, such as ransoming rather than killing noble foes, that were actually practiced the most often, a trend that saw a brutal reversal in the War of the Roses. One might point out that neither the chivalric codes nor the earlier Roman codes of war included avoidance of harming civilians. This shows that, while rules of war were effective in practice at many points in history, they did not always have the same conceptions of what these rules were made to protect.
  • The Roman Catholic Church:
    • Catholicism influenced the rules of war in two ways: like the fetiales of the Romans, it established the grounds on which war was justifiable (and was influential on the ideals of chivalry), and the pope himself, through the power of excommunication, could limit the warring impulses of kings and lords. While many popes used their power to cause conflict, the church still had both moral influence and bargaining power, and was a powerful international institution for centuries that forced treaties on Christian rulers, provided a court of arbitration, and, several times, that tried to unite these leaders in war against non-Christians. The influence of Catholic peacekeeping measures waxed and waned from Charlemagne onward, but the Peace and Truce of God was one of the earliest attempts to protect non-combatants in wartime

This very incomplete list represents a lot of the more conventional examples of this phenomenon (sorry, but I am very conventionally educated). I would love if those who have other examples, especially from outside of Greece, Rome, and the Western World, would bring them up in the comments so I can expand my knowledge of the history of the rules of war!

The many iterations of rules of war in pre-modern societies shows the effectiveness of spontaneous order in creating systems that promote liberty and peace. These rules did not eliminate violence, cruelty, or imperialism, but they forced self-interested parties to check their selfish impulses. This is not an argument that international organizations with the goal of limiting war are unnecessary (and the Geneva Conventions are a laudable example of voluntary self-enforcement), but rather a demonstration of the wide reach of both Smith’s invisible hand and Hayek’s spontaneous order: even in the most anarchic of trades, long-term individual self-interest can support general interest, and a certain level of order is imposed on the chaos of war through the unplanned conventions of societies.


ἐν μὲν γὰρ τῇ οἱ παῖδες τοὺς πατέρας θάπτουσι, ἐν δὲ τῷ οἱ πατέρες τοὺς παῖδας

In [peace], sons bury their fathers, but in [war], fathers bury their sons.

–Herodotus, The Histories, 1.87.4.