Median voter theorem

In majority-rule voting on a single dimension, if preferences are single-peaked, the median voter's most-preferred outcome defeats every alternative in pairwise votes.

Single-peaked preferences: Preferences over an ordered set of outcomes that rise toward one most-preferred point and fall beyond it. This restriction makes voters on either side of the median prefer the median's choice to alternatives farther away.
Majority rule: A decision rule under which an option wins when it receives more than half the votes cast. The theorem concerns outcomes selected by majority comparisons.
Downsian model: A model in which vote-seeking candidates choose policy positions to maximize electoral support. With two candidates and standard assumptions, it predicts convergence toward the median voter's position.
Arrow's impossibility theorem: A theorem showing that no ranked-choice social welfare rule satisfies a set of fairness conditions for three or more alternatives. It highlights broader limits on aggregating preferences beyond the theorem's restricted setting.
Duncan Black: A Scottish economist and political scientist who developed formal theories of committee decision-making and majority voting. Black established the theorem's central result for single-peaked preferences.
Median voter: The voter whose position divides an ordered electorate into equal-sized groups on either side. The theorem's winning outcome is this voter's most-preferred point.
Median: The middle value of an ordered set, with at least half the values no greater and half no less. The median voter's position divides the electorate into two sides.
Median voter model: A political competition model in which candidates choose policies to win a majority by appealing to the median voter. It applies the theorem to strategic candidate positioning.
Condorcet paradox: A cycle in which majority preferences rank A over B, B over C, and C over A. Such cycles can arise without the theorem's single-peaked, one-dimensional structure.
The Theory of Committees and Elections: Duncan Black's 1958 book presenting a systematic analysis of voting, committees, and majority decision-making. The book gave the theorem a foundational treatment.

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