Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with Application to False-name Manipulation

TL;DR

This paper establishes worst-case bounds on the ratio of power to votes in weighted voting games, revealing bounded ratios for some indices and unbounded for others, with implications for false-name manipulation strategies.

Contribution

It introduces a novel modeling of big and small players in weighted voting games and analyzes the bounds of their power ratios, including applications to false-name manipulation.

Findings

Bounded worst-case ratios for Shapley-Shubik and Deegan-Packel indices.

Unbounded ratio for the Banzhaf index.

Analysis of false-name splitting strategies in voting games.

Abstract

Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs.~small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t.~their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together, our results provide foundations for the implications of players' size, modeled as their ability to split, on their relative power.