Computing the nucleolus of weighted voting games

TL;DR

This paper presents a pseudopolynomial time algorithm for computing the nucleolus in weighted voting games, solving an open problem and providing a general framework applicable to broader classes of coalitional games.

Contribution

It introduces a pseudopolynomial time algorithm for the nucleolus in WVGs and generalizes the approach to k-vector weighted voting games, advancing computational methods in cooperative game theory.

Findings

Nucleolus can be computed in pseudopolynomial time for WVGs.

The framework extends to k-vector weighted voting games.

Provides an efficient algorithm solving an open problem.

Abstract

Weighted voting games (WVG) are coalitional games in which an agent's contribution to a coalition is given by his it weight, and a coalition wins if its total weight meets or exceeds a given quota. These games model decision-making in political bodies as well as collaboration and surplus division in multiagent domains. The computational complexity of various solution concepts for weighted voting games received a lot of attention in recent years. In particular, Elkind et al.(2007) studied the complexity of stability-related solution concepts in WVGs, namely, of the core, the least core, and the nucleolus. While they have completely characterized the algorithmic complexity of the core and the least core, for the nucleolus they have only provided an NP-hardness result. In this paper, we solve an open problem posed by Elkind et al. by showing that the nucleolus of WVGs, and, more generally, k-vector weighted voting games with fixed k, can be computed in pseudopolynomial time, i.e., there exists an algorithm that correctly computes the nucleolus and runs in time polynomial in the number of players and the maximum weight. In doing so, we propose a general framework for computing the nucleolus, which may be applicable to a wider of class of games.