Bounds for the Nakamura number

TL;DR

This paper surveys bounds on the Nakamura number, an invariant of simple games, exploring known results, formulas for special cases, and connections with other game theory concepts.

Contribution

It compiles and discusses existing bounds and characterizations of the Nakamura number across various subclasses of simple games.

Findings

Nakamura number for symmetric quota games can be calculated with a simple formula.

Characterizations of Nakamura number exist for certain subclasses of simple games.

The paper highlights gaps and connections with other game-theoretic concepts.

Abstract

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.