Line voters up from left to right and let two candidates compete for their votes; each voter picks whoever’s NEARER. The winning move is always to move toward the MIDDLE — and both candidates end up crowding the median voter, which is why rivals so often sound alike near an election. Majority rule hands the decision to the exact middle. Slide a candidate and watch the median decide.
The Median Voter Theorem (Black 1948; Downs 1957): if preferences are single-peaked along one dimension and the rule is majority vote, the outcome is the position of the MEDIAN voter, and it is a Condorcet winner — it beats every other position pairwise. In two-candidate competition each candidate’s best response is to move toward the median (voters choose the nearer one), so both converge there. It explains platform convergence, the pivotal centrist, and why fringe positions lose. Here the median of five voters sits at the centre and defeats any challenger. A fail-loud self-check throws unless the median position beats positions on either side in a majority vote. ◆ real political-economy theory, node-verified.
A one-dimensional, single-peaked, two-candidate model (the exact median result); real politics is multi-dimensional (where a median may not exist), has turnout and primaries pulling to the wings — the median-wins-under-its-assumptions theorem is exact.