On minimum integer representations of weighted games

TL;DR

This paper investigates the existence and properties of minimum integer representations in weighted voting games, establishing their uniqueness for two voter types and exploring conditions for more types.

Contribution

It proves the existence and uniqueness of minimum integer representations for weighted games with two voter types and provides examples and characterizations for more than two types.

Findings

Minimum integer representations exist and are unique for two voter types.

Examples show non-existence of minimum representations for more than two types.

Characterization of weights in minimum representations for different voter types.

Abstract

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for $t\ge 4$ types of voters without a minimum integer representation preserving types, i.e., where we additionally require that the weights are equal within equivalence classes of voters.