This simple change makes the rule strategyproof: no voter can gain by reporting a false peak.
The median voting rule holds in any setting in which the agents have single peaked preferences.
Several other works allow rules that handle only a subset of single-peaked preferences: Berga and Serizawa[3]: Sec.4 seek rules that are both strategyproof and satisfy a condition they call "no vetoer": no individual should be able to avoid any alternative to be the outcome by declaring some preference.
Barbera, Gul and Stacchetti[6] also generalize the notions of single-peaked preferences and median voting rules to multidimensional settings.
Barbera and Jackson[7] characterized strategyproof rules for weakly-single-peaked preferences, in which the maximal set may contain two alternatives.
Moulin characterized strategyproof rules on single-plateau preferences - a generalization of single-peaked in which each agent is allowed to have an entire interval of ideal points.
They consider three classes of preferences: In quadratic non-separable domains, the only strategyproof mecanisms are dictatorial.
Annually, each member company's role was weighted by its fixed share of the total output.
The theorem says that, if the agents' preferences are single-peaked, then every Condorcet method always selects the candidate preferred by the median voter (the candidate closest to the voter whose peak is the median of all peaks).
This may be a reasonable model of expressive voting, but the rule will not be strategyproof in situations where voters have single-peaked preferences over the outcome (winner) of the election.
Dummet and Farquharson present a sufficient condition for stability in voting games.