Median Voter Theorem: Preferences & Equilibrium in 1D Spatial Models, Exercises of Logic

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THE MEDIAN VOTER THEOREM

(ONE DIMENSION)

Single Peaked Preferences

With single peaked preferences, utility is a decreasing function of the distance between the alternative and the ideal point.

• Non-single peaked preferences
• Single peaked preferences

utility

utility

Symmetric Preferences

• Symmetric preferences
• Asymmetric preferences

With symmetric preferences, individuals prefer alternatives closer to their ideal point more than alternatives farther away.

utility

utility

Example

Suppose policy is one-dimensional and that a legislator has single-peaked and symmetric preferences with an ideal point at

2. If the status quo policy is located at 7, what is the set of policies that the legislator prefers to the status quo?

Suppose policy is one-dimensional and that a legislator has single-peaked and symmetric preferences with an ideal point at

9. If the status quo policy is located at 7, what is the set of policies that the legislator prefers to the status quo?

Equilibrium Concept

• Core : Alternative x is an element of the core of an f -voting rule game if there does not exist another alternative y that f individuals prefer to x. - Ex: x is an element of the majority rule core if there does not exist another alternative y that a majority of individuals prefer to x. - The core is an equilibrium concept for spatial voting games.

Proof of The Median Voter Theorem ( n odd)

Notation

tm = median’s ideal point q = the status quo. L = (n-1)/2 number of ideal points to the left of tm R = (n-1)/2 number of ideal points to the right of tm

Assume q = tm. First show that q is in the core.

Consider an arbitrary x such that x < tm. Note that R ∪ {t (^) m } individuals prefer q to x; thus, a majority do not not prefer x to q. Consider an arbitrary y such that y > tm. Note that L ∪ {t (^) m } individuals prefer q to x; thus, a majority cannot not prefer y to q. The proof follows by showing that any z ≠ t (^) m is not in the core, which follows because q will attain either R ∪ {t (^) m } votes or L ∪ {t (^) m } votes, and defeat z.

Proof of The Median Voter Theorem ( n even)

Prove the MVT for n even.

• Order the voters ideal points from smallest to largest and note that the median pivots are in position M1 = n/2 and M2 = (n+2)/2. The total number of voters to the right (larger) than M 2 are n − (n+2)/2 = (n − n/2) − 1 = n/2 − 1. This means that there is less than a majority of the members to the right of M 2 (larger than M2). Hence, any alternative to the right of M (^2) (larger) cannot receive majority of votes in favor of it.
• Similar reasoning shows that there is not a majority of individuals to the left (smaller) than M 1.
• Hence, for n even, the core is [M 1 , M2].
  • EX: on board.

Sketch of proof of the Corollary

x y

x y

l m r

l m r

x y

l m r

Note: the alternative closer to the median gets the median’s vote and half the voters to one side. That’s why the closer alternative always wins.

Because of the corollary, alternatives will

be drawn toward the median.

x y

l m r

y

Other Results

Given:

1. n > 2 (and n is odd),
2. pairwise majority rule voting,
3. alternatives are on a single dimension,
4. preferences are single peaked,
5. and preference are symmetric.

2. The social preference ordering formed by majority rule is the same as the median voter’s preference ordering.
3. Social preferences created by majority rule are transitive (as are the median voter’s preferences).

Example

x y z

l m r

What is m’s preference order for x, y, z?

What wins under pairwise majority rule: x vs. y, y vs. z, x vs. z?

Social preferences are:

Application 1: Location of the Capital

On July 28, 1788 Congress began to vote on the location of the capital. Can you guess what location they agreed upon? D.C. was not an option. NH MA CT RI NY NJ PA DE MD VA NC SC GA Status Quo: New York City

Application 2: Two Candidate Election

Voters: A B C D E

Candidates:

Obama: Romney: A, B, C D, E Obama wins, because he gets C’s vote and half of the others.

Who is at the median?

Who is closer to the median?

What should Romney do?