Talk:Median voter theorem

Confusing image

I think the image is confusing. What does y-axis mean (what kind of value)? Must point M exactly coincide with the peak? Is it important that the graph is so symmetrical?
What does "already won" mean? (I think some "gray" voters will vote for A or B too) --Y2y 11:12, 12 March 2007 (UTC) Y2y 11:59, 12 March 2007 (UTC)

The main issue with the image is that there are no "hard" values for the graph. The 'M' stands for median and by its very nature must be at the peak of the curve, which by its very nature must be symmetrical. The Y-axis is somewhat arbitrary and the X-axis represents increments of people, and goes from 0% to 50% towards the M.

"Already won" signifies the voters who have made up their minds, or decided to vote for a certain candidate who endorsed their opinion on the issue. The idea is that there are some people who will be extremists and others who are more middle of the road - say the issue at hand is the eradication of all apple trees. Some people will right away be for it (those maybe allergic to apples?) while others will right away be against it. Those are the starting points, the edges of the skirt of the graph. Others might need more convincing to make a decision. In such an idealized representation, one would think that the people with the most extreme views are very small in number, and that most people are indifferent or wavering on the topic. In this schematic Party A only has to approach the median - or water down their policy on the matter - to a much smaller degree than Party B.

Sorry, I can't get into too many details about the exactness of this type of graph, but hopefully this explanation will allow you to see it in a different light and reflect on its data. JesseRafe 02:01, 7 April 2007 (UTC)

I think this image is very useful. It explained the basic thrust of the article to me in a single glance! It might benefit from a couple of more labels for people unaccustomed to interpreting such a graph, but overall it was very helpful to me as a reader. I must take issue with Jesse's comment above, though. The median is not the center of the X axis, it's the position along the X axis where the size of the left and right slices cover an equal area. Also, not all opinion graphs need be bell-shaped. A highly polarized opinion graph would have two humps, and it's easy to think of an example that would have many humps.

Here are some suggested changes to the graph:

• Strike the arrow at the right of the X axis. There is nothing further to the right than 100% support for choice B. (or add an arrow to the left)
• Drop the entire Y axis line.
• Spell out the L, M, and R labels. Try adding the words 'left' and 'right' floating in the whitespace above the curve. The median might be represented by a pale vertical line and the word median above it.
• Consider making the curve asymetric. This will allow the median to be distinguished from the mean, by having the median also positioned asymetrically. Maybe the hump could be shifted left, and the red area shrunk a bit. Then the median line could be visibly off center. On the other hand, such a change may introduce additional confusion for some readers by adding another avenue for misinterpretation.
• Change the caption to something like this: "A conceptual graph of a one-dimensional policy space, with potential voters represented as two-dimensional space bonded by a curve. Candidates A and B currently have the support of a shaded area, and attempt to attract more uncommitted voters by migrating their posotions toward the moderate center.

By making the graph look a bit less graph-like, you clue the reader that it's not meant to be properly quantitative, but conceptual. Anyway, I like the graph, even in its current form. I'm sure it can be adjusted or just left as-is. --Loqi T. 03:32, 15 May 2007 (UTC)

What about this: make a series of 3 graphs, with the first as an initial condition (neither at the median), the second like the first but with the formerly losing candidate moving to the median and winning, and the third at equilibrium.

CRGreathouse (t | c) 20:24, 3 July 2007 (UTC)

Normal Distribution

I don't think it is necessary that the graph forms a normal distribution. Claiming the /median/ voter would always give a majority, regardless of the shape of the curve, no?

Exactly. What if the issue is "how tough should the law be against pedophilic murder"? The median voter would probably be far away from the median possible position (say, giving only a few years prison, or 50% of all people arrested for pedophilic murder automatically go free). Sagittarian Milky Way 00:06, 21 September 2007 (UTC)

Is this a good way to write a sentence?

This article now begins as follows:

The median voter theory, also known as the median voter theorem and the median voter model, is a famous voting model positing that in a majority election, if voter policy preferences can be represented as a points along a single dimension, if all voters vote deterministically for the politician that commits to a policy position closest to their own preference, and if there are only two politicians, then if the politicians want to maximize their number of votes they should both commit to the policy position preferred by the median voter.

So it says "if blah blah, if blah blah blah, and if blah blah, then if blah blah, ...." Michael Hardy (talk) 02:53, 17 April 2008 (UTC)

Majority cycle trap

Searching for the "majority cycle trap" on google gives exactly one hit - this article. What is it? 212.130.46.190 (talk) 14:26, 14 November 2008 (UTC)

See Condorcet paradox.radek (talk) 15:48, 17 April 2009 (UTC)

Rename

Shouldn't we move this to "Median Voter Theorem" which is a phrase much more common in literature (at least that part of it with which I'm more familiar)?radek (talk) 15:47, 17 April 2009 (UTC)

Good point. I moved it. CRGreathouse (t | c) 23:27, 6 February 2010 (UTC)

Need to clarify things up a bit

With regard to when the theorem holds and when it doesn't and the issue of single peaked preferences and single dimensional choice space: Basically, single peakness gets you everything - it is a sufficient condition for the theorem. Of course, the converse is not necessarily true; the result does not necessarily fail if preferences are not single peaked, though it might. The thing about single-dimensionality arises because if the choice is over just a single variable, then single peakness follows from concavity of the utility function (which is a standard assumption). So essentially what we have is (single dimension)+(concave utility)-->(single peakness)-->theorem. Now, again, the converse is not necessarily true. You could have multi dimensional choice space (and concave utility) and you could still get the result, though you might not. The difficulty arises because it's hard to define a single valued "median" in two+ dimensions.radek (talk) 19:17, 17 April 2009 (UTC)

Information considerations

Should the ``Shortcomings" section be broadened to include other problems with the theorem? There are two related to information:

• On the voters' side, the theorem relies on the implicit assumption that all voters are perfectly informed about the candidates' platforms. This makes the existence of campaign contributions a paradox. There are models in the literature that resolve the paradox by allowing for uninformed voters (e.g., McKelvey, Richard, and Peter Ordeshook, 1985, ``Elections with Limited Information: A Fulfilled Expectations Model Using Contemporaneous Poll and Endogenous Data as Information Sources," Journal of Economic Theory 36:55-85; or Baron, David P. , 1994, ``Electoral Competition with Informed and Uniformed Voters", The American Political Science Review, Vol. 88, No. 1, pp. 33-47).
• On the candidates' side, the theorem depends on there being no uncertainty regarding the location of the median voter. In the presence of uncertainty about the location of the median and positional preferences on part of the candidates, policy platforms diverge.

Is there a systematic way of incorporating this into the article, however? Certainly there are many other refinements in the literature, these are just the ones I'm familiar with. Is there perhaps a good review article on the subject that could be referenced? Tpudlik (talk) 00:13, 7 February 2010 (UTC)

The first assumption isn't implicit: "if all voters vote deterministically for the politician who commits to a policy position closest to their own preference". But it is certainly a strong limitation regardless!

CRGreathouse (t | c) 02:00, 7 February 2010 (UTC)

Dr. Winer's comment on this article

Dr. Winer has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

The writing is not elegant. It is not incorrect though. It would be useful to compare the median to a Condorcet winner (an outcome that beats any other in a pairwise sequence of majority rule votes. See R. Congleton, The Median Voter Theorem, Encyclopedia of Public Choice.

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Winer has published scholarly research which seems to be relevant to this Wikipedia article:

• Reference : Ferris, J. Stephen & Park, Soo-Bin & Winer, Stanley L., 2008. "Studying the role of political competition in the evolution of government size over long horizons," POLIS Working Papers 111, Institute of Public Policy and Public Choice - POLIS.

ExpertIdeasBot (talk) 15:44, 24 June 2016 (UTC)

Dr. Caleiro's comment on this article

Dr. Caleiro has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"The median voter theorem implies that voters have an incentive to vote for their true preferences." As is known, the median voter theorem has numerous applications, namely in the determination of the amount of public good to be provided. As is also well known, there is a problem in the revelation of preferences by individuals about the desirable amount of public good but, on the other hand, in the median voter theorem it seems to be assumed that this problem does not exist since it is considered that politicians have the full knowledge of the preferences of voters. This raises the question: what if the voters take a strategic vote, distorting their electoral preferences so that the final outcome of the votes is closer to their true peak in preferences? In this context, would still be valid the median voter theorem?

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Caleiro has expertise on the topic of this article, since he has published relevant scholarly research:

• Reference : Caleiro, Antonio, 2008. "How Can Voters Classify an Incumbent under Output Persistence," Economics Discussion Papers 2008-16, Kiel Institute for the World Economy.

ExpertIdeasBot (talk) 19:06, 30 August 2016 (UTC)

Relevance of Harold Hotelling

Harold Hotelling does indeed reason that "political candidates' platforms seem to converge during majoritarian elections." However, that doesn't seem relevant to Black's work. — Preceding unsigned comment added by 85.171.133.169 (talk) 18:04, 18 August 2017 (UTC)

Makes no sense

I find this article unintelligible. Way down, as assumption (3), it says that ‘voters are only choosing between two options’, but the lead para says that ‘voters have one alternative that they favor more than any other’ (my italics) which seems to imply more than two.

A theorem is a ‘statement which has been proven to be true’, but the ‘median voter theorem’ does not seem to be a theorem at all (and no proof is offered); some people evidently call it a model. Either there should be a proof (or a reference to a proof) or a disclaimer that it is a true theorem. Colin.champion (talk) 17:12, 6 January 2021 (UTC)

Another confusion in my mind derives from the use of the term ‘voter’ in the lead para. In the UK this is often applied to electors and never to their representatives, although it is the latter who make laws (and the latter alone who make binary decisions). Thus, taken at face value, the lead para seems to claim a relation between decisions taken and the ‘median elector’. I assume this is not its intention. I think it would be much clearer if it said in black and white that the ‘theorem’ applies to direct votes on binary decisions (eg. ‘should the rate of income tax be >30%?’). Colin.champion (talk) 19:08, 6 January 2021 (UTC)

I’ve replaced the lead para. I have no knowledge of my own to contribute; I’ve simply tried to write something that makes sense based on what was there already. I do not intend to make changes elsewhere in the article. If anyone with more knowledge of the subject than me wants to change the substance, they will have my blessing, but what they write needs to be intelligible.

Another confusion as the article originally stood is that there are two quite distinct claims associated with the ‘median voter theorem’, and it was impossible to tell at most places whether it was referring to one or the other or confusing them together.

After looking at the literature I found a form of the theorem which is better than any I’d detected in the original article, namely the one saying that Condorcet-compliant ranked preference systems produce winners close to the median voter. I’ve given it some space in the article. Maybe I should have said less about the application to binary decisions. Colin.champion (talk) 17:48, 11 January 2021 (UTC)

Having read Black’s paper I can see where some of the confusion comes from. He is interested in n-way votes with n>2, but he analyses them by looking at the n(n-1)/2 pairwise preferences which he presents as if they were separate binary votes (although they are not suitable for voting because they would present a false dilemma). This presentational expedient has been misuderstood as containing his substantial content.

I’ve had another go at the article. In my view all the material from ‘Accuracy’ onwards could profitably be deleted. It’s essay-like and worries too much about the literal truth of Hotelling’s claim, which is only a wishy-washy generalisation in the first place. [Edited Colin.champion (talk) 16:30, 16 January 2021 (UTC).]

But I don’t like deleting material which other people have worked hard to contribute, particularly off my own bat, so I’m letting it be. Colin.champion (talk) 12:21, 13 January 2021 (UTC)

Well, in the end I deleted the essay-like sections at the end of the article, meanwhile adding a few words on the extension to more than one dimension. If anyone wants to reinstate my deletions, they need to go the article as it stood at 16 Feb 2021. Colin.champion (talk) 08:21, 16 February 2021 (UTC)

Ending of proofs

There are two section of this article which end in "∎", which I interpret as Q.E.D. However, Wikipedia:WikiProject Mathematics/Proofs#Style says to just end the section at the end of the proof, or to collapse proofs so they can be skipped. I'm not sure what section header to give to the material that follows "∎" in these cases, but perhaps someone with more expertise can rearrange things to comply with the Manual of Style? -- Beland (talk) 05:42, 16 March 2021 (UTC)

It’s helpful to have a visual clue indicating the end of a proof. Presumably the tombstone caught on to avoid the need to artificially break the text with a new heading whenever a proof comes to its end. Why the MoS deprecates it I have no idea. It’s true that some readers may not have encountered it, but if you eschew everything that readers may not have encountered you end up with babyspeak. Colin.champion (talk) 08:10, 16 March 2021 (UTC)

What is a "median voter"?

The term of art "median voter" is not defined anywhere in the article. What exactly is the magical property Marlene has? You can't state a theorem without definitions. And the "theorem" should be extended to cover even numbers of voters and multiple medians. It would be even nicer to extend the theorem to show that pairwise median voting satisfies Arrow's axioms (IIA in particular) on the domain of single-peaked preferences.

I hadn’t thought that “median voter” was a technical term at all: I’d thought it was a regular construct derived from the familiar term “median”. Colin.champion (talk) 06:35, 29 April 2021 (UTC)

However I have a lot of sympathy with your other two points, though I don’t feel I’m in a position to address them.

• My impression is that the median voter theorem is generally seen as giving enlightment concerning the behaviour of elections with large electorates, and that one voter more or less is considered to be lost in the noise. This may be a sloppy approach, but Wikipedia can’t steer a different course than the literature it relies on. For instance a thorough treatment might require the Condorcet Criterion to be extended to require that if a subset of m candidates tie with each other and beat all the rest of the field, then the final winner is restricted to this set.
• I haven’t seen a succinct and satisfying interpretation in a trustworthy source of the relationship between the median voter theorem and Arrow’s impossibility theorem, but if anyone knows of one, it might well improve the article. Colin.champion (talk) 07:13, 5 May 2021 (UTC)

A further reply. I am not an expert on this topic, and I don’t find the style of reasoning at all intuitive. I made extensive edits to this page because I thought I could improve it even from readily available sources; I feel a little awkward about responding to proposals as if I was a resident expert.

I find a lot of the writing on the topic rather vague (the Congleton article cited by Dr Winer being a case in point). I have never seen an adequate statement of the median voter theorem which makes proper allowance for even numbers of voters. Black adopts the artifice of assuming a chairman with a casting vote, which essentially makes the number of voters odd in all cases. I believe that a true statement would be that the Copeland winner (that is, a candidate chosen in any way from those maximising the Copeland score) will belong to the median set of candidates. This set is defined as follows:

The left and right median voters are defined in the natural way. The left and right median candidates are those preferred by the left and right median voters. The median set comprises all candidates not to the left of the left median candidate and not to the right of the right median.

More specifically, I think that every candidate within the median set has a higher Copeland score than every candidate outside it. I have not seen this, or anything analogous, written down anywhere. If anyone knows of a reliably sourced statement applicable to even numbers of voters, I’d be very pleased if they were to add it. Colin.champion (talk) 09:44, 11 May 2021 (UTC)