Why doesn't Hotelling's law seem to apply to the stances taken by political parties?

Question

Hotelling's law is an observation that opposing entities in the same market derive benefit from being more similar, or physically closer, to their competitors. Leading to situations where competing shops are right next to each other. The idea being, that if initially both shops are equidistant from each other, either shop can increase the number of customers for which they are the closest shop by moving towards their competitor.

It seems like this should apply to the core ideologies of political parties as well. In a 2 party system, if either party shifts its ideology towards its opponent, then it would increase the number of people who will most closely identify with the party. In practice though it seems that in 2 party systems, each party has quite different ideologies.

Why don't political parties core ideologies in 2 party systems shift closer to the mean of both parties as Hotelling's law would predict?

Answer 1

"Hotelling's law" or the "Median voter theorem" says that where customers/voters care only about one variable (e.g. distance or when in a pregnancy abortions should no longer be allowed)—i.e. we're in a one-dimensional context—AND customers/voters have single-peaked preferences, then the producers/politicians will tend to move to the center/median.

But this doesn't generalize to higher dimensions. As soon as customers/voters care about more than one variable (as is almost always the case), there usually isn't a unique center/median (the exceptions are special and contrived situations).

As usual, there is a gigantic literature on this. You can start with e.g. https://en.wikipedia.org/wiki/Median_voter_theorem#Extensions_to_higher_dimensions

Answer 2

Why don't political parties core ideologies in 2 party systems shift closer to the mean of both parties as Hotelling's law would predict?

Personal ideologies are more stable than political party time horizons

Because the ideologies of members of the general public are largely pre-existing, and are stable on multi-generational time scales.

Political parties have short time horizons - typically four to six years or so for a full cycle of elections, which is much shorter than the time frame in which ideologies of members of the public change.

So, it is more fruitful to shuffle the deck if ideologies to assemble them into coalitions in advance of an election, in an effort to secure a minimal winning coalition, that can be coherent enough to last for the next election cycle.

In the short run of single election cycle, if a party changes its ideology to be more like the other party in a two party system, it loses the supporters who valued the old plank of its platform (possibly to not voting at all, or voting for a doomed third party or independent candidate), while gaining probably less than half of the people in the other party who value the new plank of the platform since political inertia based upon party labels is non-zero.

Issues constantly shift as consensus is reached on old issues

The other factor is that once there is a two party consensus on an issue in a two party system, it ceases to be a political issue, almost by definition.

For example, for almost the entire first century of U.S. history, slavery was the #1 political issue. But when slavery was abolished and the abolition of slavery received bipartisan support, as it eventually did, it ceased to be a political issue. Instead, people found other things to disagree on, like segregation and then affirmative actions.

Basically, the political community only has the bandwidth to think about so many political issues at a time. When old political issues ceases to be relevant, new ones fill the slots of the public consciousness that the old ones vacated.

As long as you have two parties, the top ten or twenty (or however many) political differences between the two parties will always exist, even if the particular issues on that list shift over time.

One could argue that list is growing less and less significant over time, consistent with Hotelling's law, but that is merely a subjective evaluation. intuitively, it doesn't feel like that is the case. Today's issues don't seem intrinsically more minor than those of fifty years ago, for example.

Hotelling's law isn't very strong

Indeed calling it a law is something of an overstatement. As explained at the link:

The opposing phenomenon is product differentiation, which is usually considered to be a business advantage if executed properly. . . .

When people along the street, or along the range of possible different product positions, consume more than a minimum number of goods (i.e. have discretionary income), companies can position their products to sections where consumers exist to maximize profit; this will often mean that companies will position themselves in different sections of the street, occupying niche markets. When prices are not fixed, companies can modify their prices to compete for customers; in those cases it is in the company's best interest to differentiate themselves as far away from each other as possible so they face less competition from each other.<2>

Political science

In a democracy, and especially in the American two-party system, political parties want to maximize the vote share allocated to their candidate. In theory, this means that political parties will adjust their platform to comply with the median voters' preferences. The Comparative Midpoints Model represents this idea best: Both political parties will get as close as possible to the competing party's platform while preserving its own identity.<3> However, party primaries can complicate this dynamic and make the stable points harder to find.<4>

The citations above are:

<2> D'Aspremont, C.; Gabszewicz, J. Jaskold; Thisse, J.-F. (1979). "On Hotelling's "Stability in Competition"". Econometrica. 47 (5): 1145. doi:10.2307/1911955. JSTOR 1911955.

<3> Westley, Christopher; Calcagno, Peter T.; Ault, Richard (2004). "Primary Election Systems and Candidate Deviation". Eastern Economic Journal. 30 (3): 365–376. JSTOR 40326400.

<4> Harrenstein, P; Lisowski, G; Sridharan, R; Turrini, P (2021). "A Hotelling-Downs framework for party nominees". Association for Computing Machinery. ISSN 2523-5699. Retrieved 2023-06-05.

There is no obvious way to analogize Hotelling's law to politics.

Finally, since Hotelling's law is a law of economics and not political science, and is conditioned as a core assumption upon prices being fixed as a core assumption (with the opposite conclusion sometimes applying when this and other unspecified conditions are not met). Therefore, one needs to discern what a "price" means in the context of political activity to determine if this law applies.

The fact that determining if the preconditions for this law to apply are non-obvious in a political context, strongly suggests that this law from the field of economics may not have a close parallel in political science.

Answer 3

Ah, the spatial model! Literature on this topic is quite extensive (you could write a whole thesis on it), so I fear it might not be feasible to cover it all in a single answer without major omissions. However, on the same note, I do believe there is still a lot to add to what the existing answers have gone over. As mentioned in user182601's linked Wikipedia article, work started on it's application to politics from Black's 1948 coining of the median voter theorem and Downs's (1957) An Economic Theory of Democracy (which was hugely influential), so while ohwilleke is right that such applications are not necessarily obvious, we have a lot to say on what factors might cause divergence instead of convergence.

A good starting point is an essay by Grofman (2004), which goes over 15 or so assumptions required for models to predict complete convergence. Grofman indicates the violation of any one of these assumptions is likely to lead to some divergence:

1. There are only two political parties.
2. There is a single-round election for any office.
3. The election chooses a single candidate.
4. Elections take place within a single constituency.
5. The election is decided by a plurality vote.
6. Policies can be located along a single (left-right) dimension.
7. Candidate policy positions are well defined.
8. Candidate policy positions are accurately estimated by each voter.
9. Voters look no further than the next election.
10. Eligible voters go to the polls if the expected benefits of their vote’s contribution to the election of the candidate for whom they would vote exceed the "costs" of voting.
11. a) Voters care only about which candidate/party will enact policies closest to their preferences. They vote for the candidate closest to their own policy location. b) If there are no policy differences among the candidates/parties, then voters will be equally likely to support each of the candidates/parties.
12. Parties/candidates care only about winning.
13. Parties/candidates look no further than the next election.
14. Candidates/parties accurately estimate the policy preferences of voters, or at minimum, they can identify the location of the median voter overall and the median voter in each party.
15. Candidates are part of a unified party team.

The US, for example, satisfies at most a couple of these assumptions. Now, going over exactly how much each individual factor affects things would probably span several pages, so instead I will yoink this one sentence from the conclusion, taking into account all of the factors: "we would expect that, under plurality, candidates will in general be much closer to the median voter in their own party than to the overall median voter, but will be shifted somewhat toward the views of potential swing voters." Their book with Adam et al. (2005) also goes into various factors in much more detail, I've dropped a link in § Further Reading below. Nonetheless, I will quickly go over a few possible factors:

The primary system

Conventional wisdom indicates that there may be some centrifugal effect caused by a two round, primary/general election system. This is, for example, modeled by Owen and Grofman (2006), though later work by Serra (2015) notes that the empirical evidence is mixed, and introduces modelling demonstrating that it's possible for primaries alone to not result in divergence. However, we don't necessarily need to go to that level of complexity/accuracy.

The mechanism for primaries to increase polarization of candidates can be boiled down to something quite simple: If each party holds its own mini-election to determine who the candidate would be, the there the obvious possibility for any centripetal tendency to cause primary candidates to converge to the median of the party itself instead of the whole electorate. Said candidates, once confirmed to be the nominee, can then veer towards the center of the electorate as a whole, of course, and the members of a party might be willing to strategically vote for more centrist candidates in the hope of an advantage in the general election, but the degree this might be feasible would vary.

Convex utility functions

Kamada and Kojima (2014) point out that some voter preferences may be naturally convex, and these are the issues that are probably the most visible, hot button topics. Things like gay marriage, abortion, the death penalty are all, less formally, very unlikely to satisfy voter preferences in some middle ground. Essentially, as in the comment to the question, "an individual wouldn't vote for the candidate that was more to the right, because both are too left for them." Or rather, voters, for convex issues, can be strongly distinguishing between candidates close to their bliss points, and almost indifferent and unable to distinguish candidates further from their bliss points.

There are, of course, as Starship points out, some such issues that have settled solidly into one answer as an equilibrium, and are unlikely to move in the other direction. These issues might even be country or region specific. But with any issue where there is a polarised electorate and convex utility functions (note, convex utility functions means concave preferences) the equilibrium positions of political parties are likely to be polarised, and strong polarisation on a few such issues would look like strong polarisation as a whole.

Uh, I think I spent a bit too much time on this, if anyone wants me to add more citations let me know and I'll do it later.

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Bibliography

Grofman, Bernard (2004-05-17). "Downs and two-party convergence". Annual Review of Political Science. 7 (1): 25–46. https://doi.org/10.1146/annurev.polisci.7.012003.104711. ISSN 1094-2939.

Owen, Guillermo; Grofman, Bernard (2006-05-11). "Two-stage electoral competition in two-party contests: persistent divergence of party positions". Social Choice and Welfare. 26 (3): 547–569. https://doi.org/10.1007/s00355-006-0087-1. ISSN 0176-1714.

Serra, Gilles (2015). "No Polarization in Spite of Primaries: A Median Voter Theorem with Competitive Nominations". In Schofield, Norman; Caballero, Gonzalo (eds.). The Political Economy of Governance. Cham: Springer International Publishing. pp. 211–229. https://doi.org/10.1007/978-3-319-15551-7_11. ISBN 978-3-319-15550-0.

Kamada, Yuichiro; Kojima, Fuhito (2014-11-01). "Voter Preferences, Polarization, and Electoral Policies". American Economic Journal: Microeconomics. 6 (4): 203–236. https://doi.org/10.1257/mic.6.4.203. ISSN 1945-7669.

Further reading

Adams, James F.; Merrill III, Samuel; Grofman, Bernard (2005). A Unified Theory of Party Competition: A Cross-National Analysis Integrating Spatial and Behavioral Factors. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511614453. ISBN 978-0-521-83644-9. OCLC 61227822.

Downs, Anthony (1957). An economic theory of democracy. New York: Harper. ISBN 978-0-06-041750-5. OCLC 1247693190.

Jones, Matthew I.; Sirianni, Antonio D.; Fu, Feng (2022-02-02). "Polarization, abstention, and the median voter theorem". Humanities and Social Sciences Communications. 9 (1). https://doi.org/10.1057/s41599-022-01056-0. ISSN 2662-9992.

Answer 4

The other answers already make a fairly good point, though I think there's value seeing these effects in combination and not just standalone.

So first of all @Starship makes a really good point with a frame challenge, in the sense that the question focuses on the differences in the stance of political parties, ignoring that the parties themselves are already pretty similar. Like the political spectrum ranges from direct democracies or anarchism to rigid caste systems and autocracies. Or from participatory volatile systems to exclusionary rigid systems. Also the power that economic wealth has on the political spectrum ranges from independent to plutocracy. In general economic ideals range from collective to private ownership, between a cooperative economy or a competitive one. The role of the state can be interpreted vastly different, from the ruler and owner of everything to a collectively owned service provider.

So while from an insiders perspectives parties may look like they are far apart, from the perspective of an outsider or with anticipation of a full left-right-political spectrum, the overtone window of acceptable ideas is/was already quite narrow and the two mainstream parties still largely tried to cover the center in that narrow window.

Like historically the 2 parties have formed out of just 1 party and have swapped positions at one point and even if you look at their healthcare proposals where they apparently so drastically differ: https://www.diffen.com/difference/Obamacare_vs_Romneycare

So it's more or less a matter of details rather than a "radical" difference.

So TL;DR: Frame challenge: Does Hotelling's law really not already apply to the stances taken by political parties?

Furthermore @StigHemmer and @user182601 make good points that the topology of the spectra might actually be a more complicated than one where you have to pick the spatial center between two competitors.

So rather than having a 1 dimensional and 1 center of attention, you could have a hyperplane of several dimensions and several local extreme points.

So less of a shape like this:

and more of a shape like this:

where if you'd place the parties on the left and right local maximum, they both would occupy a position where movement in any direction would mean a decrease in voters, yet where neither of them actually sits on the actually optimal spot.

And in the worst case it might not just look like that but even something like this:

Where peaks and valleys are even more difficult to find. And those lines don't have to be continuous but could also have sudden jumps so that iteratively approaching a point could have vastly different results depending on the direction and speed in which you approach.

Also as @StigHemmer has mentioned the electorate itself might not be continuous and stable, but voter might join or leave the electorate or deliberately "spoil" their vote, by voting third party in places that are "safe" in order to send a message of not approval despite even weaker approval of the other candidate.

Though contrary to what they claim, that does not necessarily mean that Hotteling's law doesn't apply (for example the democratic campaign apparently shifted their focus away from unique selling points of social justice and moved towards a seeming middle ground of tougher border control, making the two parties even more indistinguishable), but actually finding that center position is a lot less easier in the first place.

Another good point mentioned by @ohwilleke is about ignoring the problem of times scale and the fact that this is a dynamic equilibrium.

So for example a definitive measure of a parties performance is given on elections, however that isn't just a performance of the politics they claim to stand for but also a measure on how effective the electorate thinks that they were on these issues. So a poor result could refer to poor goals or poor performance on good goals.

Likewise, while short term polling exists (accuracy may vary due to how close the parties and thus the races are), parties can't nonetheless directly react to those polls. Because on the one hand the people that gave them a mandate expect them to be stable in their positions as that is the reason they voted for them in the first place, while on the other hand if you commit to a losing strategy, you might not lose voters, but you also don't win elections either.

So while an equilibrium seeking algorithm, that is behind Hotelling's law, would incentivize fast reactions to changing circumstances to always stay right in the sweet spot. In reality parties need to strike the right balance between consistency and adjustments to changing situations. So despite theoretically being able to change the platform in hours, it often takes years to build the credibility for such a change of platform to be accepted and for the politicians to implement it to be given the level of assumed expertise.

Also Hotelling's law is just one part of the equations. With Hotelling's law you make sure that you can access to an equal share of the population of potential buyers if you are close to the competitor. But at the end of the day you also need to make sure that they buy at your place and not the place of the competitor. So while you try for overall similarity, you also like to make sure that you appear just that little bit better, so that all people coming to the shared big tent will flock to your stand and not that of the competitions.

So while essentially selling the same you might want to make a huge deal out of the differences even if they are actually fairly minor. So idk Obamacare vs Romneycare seems to be less of a fundamental battle between different systems and more of a minor disagreement in terms of nuances to essentially very similar ideas. So less about content and more about packaging.

Like apparently the Democrats still follow the strategy of Hotelling's law and move closer to the Republican platform, while the Republican's try to distance themselves from the Democrats and like to pretend that they are so unique, but when push comes to shove usually deliver nothing but hot air (Obamacare replacement) or more of the same with different packages or if they actually do something unique, it's not even all that popular or working as intended.

So it's kinda interesting that the Democrats from a positions of having won the last election, moved towards closing that gap between them and the Republicans rather than making Republicans try to close that gap by moving more to the left. So while Republicans ramp up a left-scare tactic of people moving further and further to the left, it's actually rather that Republicans take a far right turn and Democrats seem to follow.

And last but not least Hotelling's law kinda assumes a somewhat stable competition in which the the competitors compete for a popular vote.

Which is apparently not how the U.S. elects its government. Like election results of the popular vote of:

year	popular vote (R - D)
2024	50% - 48.3%
2020	46.8% - 51.3%
2016	46.1% - 48.2%
2012	47.2% - 51.1%
2008	45.7% - 52.9%
2004	50.7% -48.3%
2000	47.9% - 48.4%

Even "landslide" victories such as

year	popular vote (R - D)
1988	53.4% - 45.6%
1984	58.8% - 40.6%
1980	50.7% - 41.0%

Are by and large the result of the electoral college and not the popular vote. I mean you have to go back to 1820 and before to find an election for government with a margin wider than a 30% and that's because that guy ran unopposed... Winning by little more than 20% is a HUGE decisive victory, which still usually means 40%-60% (in case you lead by 20% but got less than 50% just in case you wonder about that math) of the population didn't vote for that government.

So coming back to Starship's observations:

Does Hotelling's law really not already apply to the stances taken by political parties?

The thing is even if it does it doesn't have the same desired effect as a close second isn't awarded a near 50% share of the costumers and is able to built a comfortable position to adjust from there. But a close second has lost the election and gets 0 representation in government. So despite being near indistinguishable and scoring almost equal in almost all elections, the result is that one wins and one loses.

Even worse so when there are apparently goals outside of democracy where making sure the electoral college votes reflect the popular votes even less or when government officials and judges that are appointed FOR LIFE and which do not underly the necessity of a democratic mandate are ways in which parties do politics. So that winning one election might ensure you win the next regardless of your political proposals or your performance in office.

So TL;DR, the law does actually seems to apply. Though applying it is more complicated given that it might not be clear where the center is, it might not be as easy to shift positions even if one knows one would need to and ultimately finding the center doesn't ensure you actually win the election as there's a whole different game being played with regards to that point.

Answer 5

I think the biggest reason for the persistent distance between opposing party's positions is that people care enough about the issues in question that, if the distance between people's own preferences and those of the ideologically-nearest party is too large, they will resort to the unspoken third option of creating their own new party in order to have a party that has ideals within an acceptable distance of their own.

Any main party that gets too close in its political positions to the other party, and thus too far from the political preferences of its voter base, will find itself replaced by a new party that holds more closely to the core preferences of its voters.

Answer 6

Why doesn't Hotelling's law seem to apply to the stances taken by political parties?

TL;DR It does. Mainstream parties agree with each other on almost everything and only debate semantics.

It might not seem that way, though. That's because focusing a campaign on an issue your opponents agrees with you on a pretty bad idea, because even if your opponents win, why does it matter? In order for people to have a reason to vote for you over your opponent, you being elected needs to make some difference over your opponent being elected. That's why the semantics are highlighted. But, these are semantics. Here's some examples of things both the Democrats and the Republicans agree on:

1. Slavery should be illegal
2. America should have laws
3. America should have a monetary system
4. Leaving America should generally be legal
5. America should have a government
6. Terrorism is bad
7. Murder should be illegal
8. A path should exist to immigrate to America
9. All people should have human rights
10. America should exist

All of these probably seem pretty obvious. But, at one point or another, there have been nations/societies where these things have not been the case. The semantics of all of these things are also debated (ex. is abortion murder?, what should the immigration process be?, how should America prevent terrorism?) but the basic idea is agreed upon.

There are plenty more examples and they vastly outnumber the controversial ideas of each party. But the controversial ideas are what matters. If you are a voter, you know that no matter what America will continue to have laws. But the result of an election could potentially determine the semantics/controversial issues. When you vote for a candidate, you are voting for them because you think, overall, they are more likely than the other candidate to resolve the controversial issues in the way you like.

Let's also do a thought experiment. Let's say both parties agree on all the top controversial issues. What happens now? Will there be no more differences and controversy? Will there be no debate between the parties?

No. Say that the issues resolved were the top 25. Excluding any new issues that come up, what would happen is that issues 25-35 or 45 would become the new focus of political campaigns and the main party differences. Parties would still seem to be quite different, but in reality they are much more similar than they were before.

Answer 7

There are many good points about how the parties are similar in most ways. There's one thing that helped me understand the behavior of the parties and party voters.

Think of this like sports teams and fans. People are often diehard fans of their chosen team. When the ref makes a call against their team, the ref is biased and it's not that bad. When the ref makes a call for their team, it's deserved.

Parties shift positions just to oppose each other often. Vaccine doubters were very cross-party, until the position because associated with (in the US) the Republicans. That caused many on the left who would have been skeptical to decide vaccines were ok, and some on the right who would have gotten them to reconsider.

Also, as an example of the parties moving towards one another, once again in the US there's a good bit of noise about trans rights pro and con right now. However, both parties have pretty much agreed now on gay marriage and such, so you don't hear much. That affected a much greater section of the populace, which has now become pretty much a settled matter. Trans is going through the same thing now, but it's a much smaller section of the populace (admittedly, that doesn't help anyone who is trans now, but it offers glimpse of the future, and a positive picture of human rights over all).

Answer 8

I don't know if the idea originates with him, but William Spaniel presents the interesting idea that poor polling can lead to violations of Hetelling's Law. As I understand it, the argument is that if the parties are unsure about what the position of the median voter is, and they have a utility function that values extreme positions, then the expected utility of the taking the position of the expected median voter can be lower than that of other positions.

For instance, suppose a party has modelled the electorate as having three possibilities for the median voter, A, B, and C, and has assigned probabilities 40%, 20%, and 40%, and utilities 100, 10, and 0, respectively, to them. If they guess that the actual median is B, they will minimize their "error" for some metrics, but their utility is only 6 (they could have as much as a 60% chance of winning, but if they win, they only get 10 utility). On the other hand, if they go for A, then their expected utility goes up to 40; their chances of winning go down, but their reward for winning goes up enough to compensate.

Answer 9

(Like any political theory, the following is very simplified)

In a two-party system, there are (at least) four groups. Using the US as an example:

1. Democratic non-voters
2. Democratic voters
3. Republican voters
4. Republican non-voters

The non-voters on each side think that both parties are wrong, and they see no point in voting for anyone. They still have a preferred party if pressed.

Now, if a Democrat politician tries to move their stance to capture Republican voters, they will also lose voters to become non-voters.

They can easily lose more voters than they win, so this is a losing strategy.

In the brilliance of hindsight, this is what happened in the last election.

Trump gained a lot of support among former non-voters. He made them actually go and vote. He also lost some traditional Republican voters, but less than he gained.

Harris thought that Trump had abandoned the traditional values and tried to grab voters in the center by flagging traditional values.

She may have won some of those, but she also lost a lot of more radical voters.

And the rest is history.