Visual explanation of the Condorcet method

From electowiki

This animation illustrates how Condorcet methods find the "faction which is closest to the consensus".

Under a spatial model of voters, a Condorcet method partitions the ideological space into two halves for each pair of candidates. If a Condorcet winner exists, they are on the majority side of each partition.

In this animation voters are small dots, candidates are big black or blue dots, and the Condorcet winner is a big red dot. The big blue dot is the current candidate being pairwise compared to the Condorcet winner. The thin line joining the two is a visual aid to highlight the comparison being made.

Each pairwise comparison defines a cut (thick blue line across the image) and creates a "dominant region" (in light blue background) on the majority side. The Condorcet winner is always within this dominant region, by definition.

Over all possible partitions, the Condorcet winner will "carve" itself a "Condorcet region" of the ideological space (yellow background), and every voter within that region (red dots) is the "Condorcet winner's faction", as defined by this current distribution of candidates and voters. (Note that a different distribution of candidates, even with the same voters, would create different factions.)

This "Condorcet faction" is the group of voters that are more closely represented by the Condorcet winner, and thus the election results.

The consensus of all voters within this faction is shown as the small black cross, and the population's overall consensus is the origin over the entire space, as denoted by the axes. (Note that this is different than claiming the consensus is a "centrist" or "moderate" position. This ideological consensus could very well be in an extreme position, and we just shifted it to the center of the image for simplicity).

The yellow "Condorcet region" always contains the overall consensus (the origin of the space), and the Condorcet winner is the closest candidate to the population consensus under all of these examples. But this is not in general true.

In real life elections, we have no information about this abstract ideological space. All we have are the ballots and election results. Nevertheless, such illustrations are helpful to understand what sort of representation is achieved by different voting methods under different scenarios.

As we can see, the Condorcet winner is a good proxy for the ideological consensus of a population, even though it may favor a majority faction too much under polarization (as shown in the middle-right example).