Spoilers in Clustering Methods

In voting schemes, when there are more than two cadidates, there is the possibility of a 'spoiler'. That is, if a third candidate is introduced, votes might be taken away only from the formerly winning candidate, 'spoiling' a chance of victory, letting a candidate not preferred by the majority to win because the majority is split between two similar candidates.

This is similar to a clustering algorithm, where having set three clusters, the agglomeration is set to distinguish between two smaller clusters such that, if only two clusters were desired, together would be bigger than the third.

(from Pier Luca Lanzi)

In the example figure, the parameter to the system for k-means clustering is 3. The upper right set is split into two separate clusters. But if the parameter were 2, then those two clusters might combine to make a cluster larger than the lower left.

Of course, that doesn't mean the lower left cluster would be preserved, some elements may move back or forth. This shows that clustering can have anomalies like voting schemes, even though clustering doesn't account for all possible orderings (permutations) of 'candidates' and the correspondence of cluster label with candidate is not perfect.