**The Median Voter Theorem**states that in one-dimensional elections, the candidate closest to the median voter will be elected.

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**In one-dimensional elections, the Condorcet winner is the candidate closest to the median voter.** - 🔑
**Condorcet methods will elect the candidate preferred by the median voter.** - 🔑
**Hotelling's law suggests that politicians gravitate toward the position favored by the median voter.** - 🔑
**The theorem is valuable for understanding the optimality of voting systems and their outcomes.**

- The theorem applies to one-dimensional elections where candidates and voters are distributed along a spectrum.
- The Condorcet winner is the candidate closest to the median voter.
- Proof sketch involves the closest candidate receiving the first preference vote of the median voter.

- The theorem can be extended to even numbers of voters and more complex spatial models.
- Spatial models with valence included can still apply the median voter theorem.

- Hotelling's law suggests that politicians move toward the median voter's position.
- This behavior is akin to businesses in the same area maximizing market share.

- The theorem provides insights into the optimality of voting systems.
- It has been applied to understand relationships between voting populations and policy outcomes in various areas like income inequality and taxation policies.

- "In general elections, a Condorcet winner might not exist."
- "Systems that fail the median voter criterion encourage extremism rather than moderation."
- "Hotelling's law states that politicians gravitate toward the position favored by the median voter."
- "The median voter theorem proved extremely popular in the Political Economy literature."
- "The main reason is that it can be adopted to derive testable implications about the relationship between some characteristics of the voting population and the policy outcome."

This summary contains AI-generated information and may have important inaccuracies or omissions.