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The Median Voter Theorem is a fundamental result in political science and economics that explains the equilibrium concept in spatial voting games. the theorem in the context of single-dimensional spatial models, single-peaked preferences, symmetric preferences, and pairwise majority rule. It also includes proofs for odd and even numbers of voters and corollaries.
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Typology: Exercises
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With single peaked preferences, utility is a decreasing function of the distance between the alternative and the ideal point.
utility
utility
With symmetric preferences, individuals prefer alternatives closer to their ideal point more than alternatives farther away.
utility
utility
Suppose policy is one-dimensional and that a legislator has single-peaked and symmetric preferences with an ideal point at
Suppose policy is one-dimensional and that a legislator has single-peaked and symmetric preferences with an ideal point at
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.
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.
x y
l m r
y
Given:
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:
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
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?