File:Majority of consensus histograms.gif

Summary
A more natural notion of majority can be defined in terms of the spatial model of voters.

We consider the smallest region around the consensus which contains a majority of voters within it. This is a true majority, a property of the voters that is independent of candidates and whatever factions they create.

In this diagram, this majority of consensus is denoted as a red circle around the consensus.

Now, here's the very important part: in this diagram, the candidate closest to the consensus is "magically" picked as the "winner", coloring the interior of this circle. There is no "voting" taking place! It is a completely geometric property being depicted.

At the bottom, we have a distribution of distances from voters to both candidates. This is what voters would be intuitively measuring during an election. The vertical line is the mean of the distributions (not the median, as one would expect), that is, the mean distance.

This is analogous to voters voting in a continuous scale from 0 to infinity, mapping distance perfectly to this scale.

Observe that the mean distances exactly match the coloring of the majority of consensus circle: if mean distance to the yellow candidate is lower than that of the purple candidate (the "voting"), the yellow dot is geometrically closer to the consensus ("magically" selected from the geometry of the problem).

Under an actual cardinal voting scheme, the mapping of distances to the ballot scale are bounded, confined to discrete steps, and may not be linear.