  In this paper, we propose a natural notion of individual preference (IP)
stability for clustering, which asks that every data point, on average, is
closer to the points in its own cluster than to the points in any other
cluster. Our notion can be motivated from several perspectives, including game
theory and algorithmic fairness. We study several questions related to our
proposed notion. We first show that deciding whether a given data set allows
for an IP-stable clustering in general is NP-hard. As a result, we explore the
design of efficient algorithms for finding IP-stable clusterings in some
restricted metric spaces. We present a polytime algorithm to find a clustering
satisfying exact IP-stability on the real line, and an efficient algorithm to
find an IP-stable 2-clustering for a tree metric. We also consider relaxing the
stability constraint, i.e., every data point should not be too far from its own
cluster compared to any other cluster. For this case, we provide polytime
algorithms with different guarantees. We evaluate some of our algorithms and
several standard clustering approaches on real data sets.
