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# Bleg: Median Voter Theorem

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In my Cost/Benefit Analysis course, I teach one segment on Public Choice. One of the issues I get into is the median voter theorem. I point out that it applies more directly to direct democracy, i.e., voting on initiatives and referenda, than to representative democracy, i.e., voting on candidates. The reason is that on the latter, people typically vote for a candidate who takes positions on more than one issue, usually many more.

But I have a problem. I'll quote from the text I'm using, Harvey S. Rosen's and Ted Gayer's Public Finance. The book gives the example of Donald, Daisy, Huey, Dewey, and Louie voting on how much money to spend on a party. Their preferred expenditures are: \$5, \$100, \$150, \$160, and \$700. So the median voter is Huey and so if the outcome reflects the preference of the median voter, it will be \$150. Here's what the text says:

A movement from zero party expenditure to \$5 would be preferred to no money by all voters. A movement from \$5 to \$100 would be approved by Daisy, Huey, Dewey, and Louie, and from \$100 to \$150 by Huey, Dewey, and Louie. Any increase beyond \$150, however, would be blocked by at least three voters: Donald, Daisy, and Huey. Hence the majority votes for \$150.

HERE'S WHAT I DON'T GET:

Who is deciding what gets voted on? If someone decides that we're voting between spending \$5 and spending \$160, then \$150 CAN'T emerge as the outcome. Indeed, what is the actual experiment that Rosen and Gayer are running? Are they going pairwise, from \$0 to \$5, and then \$5 to \$100, and then \$100 to \$150, etc.? It sounds like it. But have you ever heard of a vote like that?

I can see that if representatives are running and there's only one issue they'll decide, there's a tendency for them to choose to run at or close to the median voter's preference. But, typically, representatives will vote on many issues--that's why they're representatives. So if the median voter theorem means anything, it should apply to referenda and initiatives.

### Comments and Sharing

CATEGORIES: Public Choice Theory

COMMENTS (18 to date)
writes:

I use this idea in a class that's about political philosophy, not econ or public choice, and that may make a difference. Anyhow, I present it as the outcome of a method that, intuitively, seems to be the best for discovering what "the will of the majority" is: a) the proposals are all made by the voters themselves and b) we keep voting until we discover the one proposal that cannot be defeated. If the issue has the relevant cost-benefit structure, one person will win every time. Ladies and gentlemen, meet your ruler: l'homme moyen sensuel!

Alex Tabarrok writes:

I would put it this way, the median voter model is really a model about equilibrium and not a dynamic model. The MVM tells us that there is only one Nash equilibrium, the ideal point of the median voter. Any other point cannot be an equilibrium thus the implicit argument is that through some unspecified dynamics we will get to the equilibrium (just as demand and supply doesn't specify price dynamics). One possibility, for example, would be for the two candidates/parties to think through the model and jump straight to the equilibrium (not so unrealistic given polls, focus groups etc.) Other possibilities are for the candidates/parties to explore the space with one candidate winning until the second candidate comes up with another proposal and eventually this settles down.

Having said all that there are interesting cases where, for example, a bureaucracy such as a school board puts say a bond referendum to the public. In these cases by carefully choosing the two choices the agenda setter can get what they want which is far from that of the ideal point of the median voter.

writes:

I see Rossen and Gayer's example as a bureaucrat process, not a public vote. Think of the Ducks as members of a school board or city council who want to institute a new program. By definition, the \$0 option isn't on the table. The Ducks' task therefore, is to decide how much to spend on the program or to ask voters to approve, if it's to funded by a bond issue. The amount proposed by Duck reflects that Duck's assessment of what he/she is willing to commit to the program, given other programs that he/she wants in the budget, or given that Duck's assessment of the maximum amount that voters are likely to approve.

david writes:

The median voter theorem indeed only applies to issues where all voters have unimodal preferences over a scalar. If you have a vector of issues, or if preferences are complicated, then you can have a perverse situation where the ordering of the sequence of pairwise votes arbitrarily decides the outcome, and at the end there may be individuals who wish to stage yet another pairwise vote. The relevant theorem here becomes Arrow's theorem, instead with its alarmingly negative result.

Real-life governments that fall into this trap rapidly tear themselves apart in disputes over legitimate procedure, so it might not be terribly relevant. The tripartite French Fourth Republic had a rather short life. The only stable democracies that one might observe, tend not to be democracies where vote cycling is a concern.

david writes:

For more discussion of the problem, I should recommend chapter 7 "Spatial Theories of Electoral Competition" in Green & Shapiro's Pathologies of Rational Choice Theory. They discuss the multidimensionality problem in detail, and survey some of the empirical research on candidates and whether candidates solve the multiple-issue problem in voter cycling.

David R. Henderson writes:

@Alex Tabarrok,
Thanks so much, Alex.
@david,
You're right but you're talking about the more complicated situation. I'm talking here about the simple situation where preferences are single-peaked.

John writes:

I always took the MVT to include the campaigning of candidates and the attempts to discover where the majority votes would exist -- with the assumption that this information is perfectly revealed/discovered through political competition in the campaign.

I would suggest not worrying too much about MVT. It's a fairly uninteresting theorem that has limited application in political economic inquiry. Exactly how it might be important or irrelevant in a Cost-Benefit Analysis course I'm not sure -- who is the target audience?

If it's for business/MBAs or Public Policy students I think cycling and agenda setting insights from Public Choice would be more valuable, than MVT. This might also be true if the audience in more engineering types.

John Goodman writes:

The median voter theorem is wrong. It is hard to believe that economists (who are supposed to believe in the principle of marginalism) ever accepted it.

Think about what it implies: The only opinion that matters is the median voter's opinion. If he changes his preference, the political equilibrium changes. But if everyone else makes (small) changes in their preferences, the equilibrium doesn't change at all!

That is, everybody in the whole country (accept for the median voter) can increase his desired spending on a project (the political demand curve shifts to the right), but the equilibrium does not change unless the median voter changes his mind.

Hogwash. Everything I have done in public choice is a complete rejection of this idea.

John Goodman writes:

Followup comment:

My approach to public choice is to introduce the principle of marginalism into politics the same way it is encorporated in economics.

That means that any change in the prefernces or actions of any individual -- no matter how small -- can affect the equlibrium platform.

Jim Glass writes:

This is why the person who controls the agenda before a vote to determine what is voted upon has so much power in all voting situations -- from local clubs to corporate boards to the US Congress. Even if those with such control have no official votes themselves.

there are interesting cases where, for example, a bureaucracy such as a school board puts say a bond referendum to the public. In these cases by carefully choosing the two choices the agenda setter can get what they want

You are being kind, "interesting" is not the word. Where I am in the suburbs north of NYC if the budget the school board wants doesn't pass it produces an alternate budget that has zero rational cost reductions but which guts the programs of highest personal value to parents and students. They are so brazen about it, it's impressive. But what can one do? Nobody else has any say in producing alternative budget proposals.

The words "extortion" and "blackmail" come to mind, not sure which fits best. But it sure works! The final result always comes from a free and fair election.

The battle over controlling the agenda is where a great many elections are won and lost, rather than on election day.

David R. Henderson writes:

@John Goodman,
The only opinion that matters is the median voter's opinion. If he changes his preference, the political equilibrium changes. But if everyone else makes (small) changes in their preferences, the equilibrium doesn't change at all!
If everyone else makes (small) changes in their preferences, the odds are that the previously median voter is no longer the median voter.

I think you're asking about political/public entrepreneurship. See Buchanan & Congleton's *Politics by Principle, Not Interest* chapter 5 (it's specifically about the political agenda), Oakerson and Parks's 1988 paper "Citizen voice and public entrepreneurship: The organizational dynamic of a complex metropolitan county", and Boettke & Coyne's book *Context Matters: Institutions and entrepreneurship*.

Christopher Kam writes:

I believe Duncan Black's original paper (1948, I think) stressed that the median outcome hinged on an open agenda. If the agenda is open it can be shown that all agenda trees lead to an outcome = median voter's position. Roemer & Rosenthal (1978, again, I think) relaxed this assumption, and considered the outcome given an actor with agenda-setting power. The result is a function of the median voter's position, the status quo's location, and the agenda setter's ideal point.
ck

writes:

I see it as the Hotelling Theorem, which describes the equilibrium outcome of a two party election with candidates both of whom want to win.

writes:

As a regular chair in large democratic general assemblies, I can say what way I would choose to vote over such an issue (which, by the way, is utterly regular: it is basically every contested issue when voting on budgets).

I would arrange the motions in a regular order, from most to least expensive in this case. The most radical choice would be to spend most - spending nothing could also be a very radical choice but usually less radical than spending the most.

Then you vote for the most expensive versus all the other possible amounts. If that gets a majority, people want to spend a lot. But people who prefer 150 to 160 would vote against. People who prefer 160 would vote for, obviously.

So: if 160 falls, we vote for 150. People preferring 150 to 160 vote for, but almost in every case would people preferring 160 also prefer 150 to 100, so they would also vote for.

After that, we just keep going down the list until one amount gets a majority, picking up the votes of the losing motions on the way.

Carl C writes:

The kind of voting you describe is actually used - even at the national level. It's called an 'elimination runoff' system, and there are examples of systems like it in Roberts Rules of Order.

An automated version, called IRV (instant runoff voting) has been used in Australia since the early 1900s. I believe it's also used in a few other countries as well.

writes:

The MVT is not so much a rigorous conclusion that precisely flows from a particular set of democratic procedures as it is a descriptive rule of thumb for explaining why the prediction of the MVT is often a good first order approximation of what happens in actual democratic decision making processes for all manner of context specific reasons.

The key insights of the MVT are that electoral decision-making processes whether they are direct or indirect, are largely indifferent to the preferences of non-voters, and that voters with extreme minority views in either direction have only moderate impact on questions of distributive justice where the outcome must be some scalar value. These insights have considerable predictive power even if they aren't terribly precise. They demonstrate how regulating the franchise has powerful policy outcomes and this result is a strong counterargument to the argument that all that matters is that the smartest people be allowed to vote. MVT is principle that demonstrates in a simple and intuitive way how power can be as important as rational choice in political decision-making.

It is also worth noting that while relative values on issues like how much to spend on public schools per student, for example, are often quite partisan, it is frequently the case that neither party's ideology provides much of a method for determining from first principles how high is high and how low is low, and that values of zero are typically very extreme on the kinds of issues to which the MVT is applied. MVT is a useful way of estimating where the left-right divide will be drawn in a quantitative way.

There are all sorts of processes that can produce results close to the MVT, just as sums of all sorts of variables tend to be approximately Gaussian (i.e. normally distributed) without much regard to the underlying distribution of the variables. For example, in a representative system, there is a natural tendency for parties in the minority to moderate towards 50-50 divisions of power and for partisan biases relative to voter preference of elected officials of one party in a two party system to be balanced by partisan biases of elected officials of the other party in a two party system.

Also, while in principle, representatives make lots of decisions on lots of issues, in practice, a very high share of the distributive justice issues that officials consider that are suitable for MVT analysis are highly correlated with each other, and the orthogonal and independent views of representatives typically involve non-MVT issues.

Moreover, some MVT issues are not all that salient to voters and in those cases you get non-partisan votes where the random sampling of views found in the legislature on the non-salient issue will tend to be similar to those of a random sample of voters with a median voter prevailing.

In general, the assumptions necessary to get approximately MVT results are quite weak.

david writes:

@David R. Henderson

It's a complicated situation that is assumed to be simple (through the focus on the Nash outcome as Tabarrok explains, rather than how the outcome is reached to begin with), because the complications allow highly perverse mathematical outcomes that are not relevant to real experience.

In real life there are strong forces that pressure people who lose votes regularly to nonetheless respect the vote and process. These forces are undermined if perceived manipulation becomes systemic, so that the legitimacy of procedure is regularly assailed. If it is not possible to diverge from the Nash equilibrium profitably, then the manipulation doesn't occur.

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