Prof. Bryan Caplan

Econ 410


Week 3: The Median Voter Model

I.             Rational, Instrumental Voting

A.           Let us begin with two standard assumptions about voters.  We will think about relaxing these in the second part of the course, but for now we will stick with them.

B.           Assumption #1: Rationality.  Specifically, while they make mistakes, their mistakes are random, as opposed to systematically biased.

1.            In more technical terms, voters have rational expectations.

C.           Assumption #2: Instrumental goals.  Voters care about nothing except the policies they get.  They aren't interested in personalities, entertainment, impressing their friends with their social conscience, etc.

D.           Neither of these require that voters be selfish.  They might be rational, instrumental voters who care only about the liberalism/conservatism of policy, for example.

II.            Single-Peaked Preferences

A.           Next, let us assume that voters' preferences are "single-peaked."  This means that voters have an "ideal point" (aka "bliss point"), and that the further policy moves from their ideal point, the unhappier they get.

B.           For example, one voter's ideal point might be a world where people are allowed to own any weapon up to and including a machine gun.  This voter would be less happy in both:

1.            A world where fewer weapons were legal (e.g. where the semi-automatic gun is the most dangerous legal weapon).

2.            A world where more weapons are legal (e.g. artillery, tanks, nuclear bombs).

C.           Or a voter's ideal point could be a world with a 5% tax rate.  This voter would be less happy in both:

1.            A world where taxes were higher (more services, but they aren't worth the taxes).

2.            A world where taxes were lower (more after-tax income, but the service cuts aren't worth it).

D.           Aren't all preferences "single-peaked"?  Probably not.  Consider for example a wealthy parent.  If spending on education is high, she sends her kids to public school.  But otherwise she sends them to private school, and gets no benefit from education spending.  So her preferences would look like this:

1.            #1 pick: high spending

2.            #2 pick: low spending

3.            #3 pick: medium spending

III.          Two-Party, Winner-Take-All Elections

A.           Now let's take these voters and put them in a hypothetical situation.  Suppose we have a two-party (or two-candidate) election.  Voters care about and are perfectly informed about party positions on exactly one issue: liberalism versus conservatism. 

B.           The electoral rule is "winner-takes-all" - whoever gets more votes wins.

1.            Picky point - ties.  When in doubt, assume ties are resolving by flipping a coin.

C.           Assumption about party/candidate motivation: They want to win, and care more about that than everything else put together.

D.           The two parties compete in exactly one way: By taking a stand on the issue.

E.           Imagine graphing the distribution of voter ideal points.  There are many ways that it could look, but it is easiest - and pretty realistic - to draw it as a "bell curve."

F.            Now suppose for a moment that each party's position is randomly assigned.  Who votes for which party?

G.           The electorate may be divided into three groups: those who definitely vote for the more liberal party, those who definitely vote for the more conservative party, and the people in the middle, who pick whichever party is closer to them.

IV.          Political Competition and Platform Convergence, I

A.           Now put yourself in the shoes of the party that will lose the election if it stays where it is. 

B.           Question:  What will it do to get more votes? 

C.           Answer:  Move to the center.  They don't lose any of the extreme votes, and get more of the "swing" votes.

1.            In fact, you get the maximum possible number of votes by moving as close as possible to the rival party without overlapping.

D.           But then put yourself in the shoes of the rival party.  It won't take this lying down.  It will move towards the center as well.  If necessary, it will "leap frog" over you.

E.           Standard economist's question thus emerges: What is the equilibrium?

F.            You can't have an equilibrium where the parties' platforms are different, because both parties gain votes by moving closer to each other.

G.           You can't have an equilibrium where one party gets more than 50% of the votes.  Why?  Because you can always win 50% by simply offering exactly the same platform as your competitor.

H.           Thus, equilibrium platforms "converge" - both parties offer the same policy.

V.           Political Competition and Platform Convergence, II

A.           But what specific policy do they converge to?  To answer this, recall that the median of a distribution is the point with half of the distribution's density above it and half below it.

1.            Ex: What is the median in this series: {100, 7, 3, 2, 2} ?

B.           Could the equilibrium platform ever be one where both parties are above the median of the distribution of voter preferences?  No.  Why?  Because one party would get more than 50% of the votes by moving a little closer to the median.

C.           Could the equilibrium platform ever be one where both parties are below the median of the distribution of voter preferences?  No, for the same reason.

D.           Could the equilibrium platform be the median of the distribution?  Yes!  If both parties are at the median, then staying there gets you 50% of the votes, but moving a little to the left or right gets you fewer than 50%.

E.           Thus, we arrive at the famous Median Voter Theorem:  Given the preceding assumptions, both parties move to the exact median of the distribution of voter preferences.  In other words, they both offer platforms identical to the bliss point of the median voter.

F.            Note: These results hold up as long as parties prefer winning to losing, all else equal.  We don't need to assume that parties are unprincipled; it's just that political competition forces them to either compromise their principles or lose.

VI.          Voter Participation and Franchise Restrictions

A.           Competing parties thus find themselves under intense pressure to cater to the median voter.

B.           But the identity of the median voter typically changes when the electorate changes.

1.            In a conservative state, for example, the median voter will be more conservative, and both parties will have to offer conservative policies to win.

2.            Similarly, if poorer voters are less likely to vote, parties will move to the median of the distribution of voters, not the distribution of citizens.

C.           There are many factors that affect participation: age, education, what's on the ballot... even the weather.

D.           If proportional amounts of all political persuasions don't vote, the median stays the same, and so does the electoral outcome.

E.           But if participation changes in a disproportionate way, this changes the median, and thereby changes the nature of the winning platform.

F.            There are also legal restrictions on voting. 

1.            Non-citizens normally can't vote at all. 

2.            Citizens have to register in advance to vote. 

3.            Non-residents in a state can't vote in that state. 

4.            Convicted felons and children can't vote.

G.           Similarly, you have to be a Supreme Court justice to vote in the Supreme Court, a U.S. Senator to vote in the U.S. Senate, etc.  As you would expect, measures that come before these bodies are usually tailored to the median of those who vote on these measures.

H.           In the past, there were often stronger restrictions on voting, also known as "restrictions of the franchise."

1.            Non-property-holders

2.            Non-whites

3.            Women

4.            18-21 year-olds

I.             Like them or hate them, the key idea of franchise restrictions is to change the median voter.  If kids could vote, for example, school might become a lot more fun.

J.            Corporations typically have voting, but it is voting proportional to your number of shares.  (Turnout of small share-holders is also typically very low).  Thus, the median corporate voter is usually a large shareholder with a big stake in the company's financial success.

K.           In the past, some countries (like Sweden) also had "plural voting," with extra votes for the aristocracy.

VII.        The Effect of Fringe Parties

A.           In many cases, we see people with extreme preferences deciding not to vote because "their" candidate is an unprincipled "sell-out."

B.           This is probably a major force that keeps real-world parties from completely converging.  They have to trade-off extra moderate votes for foregone extremist votes.

C.           Fringe, "extremist" parties do much the same thing.  For example, if a far-left Green Party exists, then the Democrats have to worry about two things:

1.            Extremists stay home

2.            Extremists vote Green

D.           How do fringe parties affect the outcome?  They tend to push the median in the opposite of the direction they favor!  If the 5% of most-left-wing Democrats vote Green, the median of the remaining voters shifts to the right.

1.            So what are fringe parties doing?  In many cases, they say they are working for long-run attitudinal change. 

E.           In the 2000 presidential election, fringe parties wound up making a huge difference in spite of their small vote shares.

VIII.       Multi-Peaked Preferences and Intransitivity, I

A.           We have already considered an example of multi-peaked preferences:  A wealthy parent's first choice may be high educational spending, even though medium spending is her third choice.

B.           With multi-peaked preferences, the analysis of elections becomes far more complicated because electoral outcomes may cease to be transitive.

C.           What does "transitive" mean?  For an individual, preferences are transitive if the facts that (i) you prefer A to B and (ii) B to C imply that (iii) you prefer A to C.

D.           For example, I prefer watching Dark City to Excalibur, and Excalibur to Desperado, so I prefer Dark City to Desperado.

E.           This seems like a trivial assumption, and for the most part it is.  (Though there are many experiments that "trick" people into making intransitive choices).

F.            Moreover, if someone has intransitive preferences, it is unclear what they would choose.  If for example I preferred Desperado to Dark City, and someone told me to choose one movie, what would I do?

G.           Think Rock, Scissors, Paper.

H.           If they repeatedly offered me new choices, and charged me a penny each time, they could use me as a "money pump"!

IX.           Multi-Peaked Preferences and Intransitivity, II

A.           Key conclusion: With multi-peaked preferences, electoral outcomes can be intransitive, even though no individual voter has intransitive preferences!

B.           Proof by example.  Going back to the school case, imagine we've got 3 voters.

C.           Voter #1's preference ordering: {high, low, medium}

D.           Voter #2's preference ordering: {medium, high, low}

E.           Voter #3's preference ordering: {low, medium, high}

F.            Imagine giving this 3-person electorate two choices at a time.

1.            High versus low: 2 for, 1 against

2.            Low versus medium: 2 for, 1 against

3.            Medium versus high: 2 for, 1 against

G.           Notice: High beats low, low beats medium, and medium beats high!

H.           For many, this example shows that the "will of the people" may be a meaningless concept.  What level of education spending does "the people" "will" in this example?

X.           Cycling and Condorcet Winners

A.           When some voters have multi-peaked preferences, there may be no stable electoral outcome.  If you keep voting, you keep "cycling" through the possibilities.

B.           So what happens?  It all depends on which vote happens last.  If you can manipulate the sequence of votes, you can pick the answer.

C.           The ability to manipulate the answer depends, however, on the range of the transitivity.  An electorate may be intransitive over 6 out of 7 options, but still unambiguously prefer the 7th to all of the other 6.

D.           If there is one option that can beat all other options in pair wise elections (option A beats option B, option A beats option C, etc.), it is called a "Condorcet winner."

E.           If there is a Condorcet winner, manipulating the sequence of votes doesn't matter much even if there are intransitivities.  Without a Condorcet winner, manipulating the sequence can matter a lot.

XI.          Multiple Voting Dimensions

A.           The Median Voter Theorem only strictly holds if there is a single issue.

B.           If there are two or more issues that parties take stands on, but only one election, there is no guarantee that the median voter's preference will prefer on any issue.

C.           Moreover, even with single-peaked preferences, multiple voting dimensions make it possible for voting cycles to arise.

D.           At this point, you might say: "But all real-world elections have multiple issues.  So the Median Voter Theorem is useless."

E.           Possibly so, but matters are more complicated than that.  In particular, we will see that to a large extent, platforms empirically boil down to a single dimension - in the U.S., position on the liberal-conservative spectrum.