On the Complexity of the Decisive Problem in Simple, Regular and Weighted Games

TL;DR

This paper investigates the computational complexity of decisiveness in simple, regular, and weighted games, showing polynomial-time algorithms for certain classes and exploring structural implications of regularity.

Contribution

It introduces a hypergraph-theoretic approach to decisiveness and establishes complexity bounds, highlighting the impact of regularity and properness on computational difficulty.

Findings

Decisiveness can be decided in quasi-polynomial time for simple games.

Decisiveness can be decided in polynomial time for regular and weighted games.

Regularity influences the complexity and representation of problem instances.

Abstract

We study the computational complexity of an important property of simple, regular and weighted games, which is decisiveness. We show that this concept can naturally be represented in the context of hypergraph theory, and that decisiveness can be decided for simple games in quasi-polynomial time, and for regular and weighted games in polynomial time. The strongness condition poses the main difficulties, while properness reduces the complexity of the problem, especially if it is amplified by regularity. On the other hand, regularity also allows to specify the problem instances much more economically, implying a reconsideration of the corresponding complexity measure that, as we prove, has important structural as well as algorithmic consequences.