A Strongly Polynomial-Time Combinatorial Algorithm for the Nucleolus in Convex Games

Abstract: The nucleolus is a fundamental solution concept in cooperative game theory, yet computing it is NP-hard in general. In convex games-where players' marginal contributions grow with coalition size-the only existing polynomial-time algorithm relies on the ellipsoid method. We re-examine a reduced game approach, refuting a previously claimed polynomial-time implementation and clarifying why it fails. By developing new algorithmic ideas and exploiting the structure of least core polyhedra, we show that reduced games can in fact be used effectively. This yields the first combinatorial and strongly polynomial algorithm for computing the nucleolus in convex games.