The Complexity of Testing Properties of Simple Games

TL;DR

This paper analyzes the computational complexity of various problems related to simple voting games, including representation transformations, property testing, and classification of game types, providing polynomial and exponential time results.

Contribution

It offers a comprehensive complexity classification for transforming and analyzing simple and weighted games across multiple representations and properties.

Findings

Some representation transformations are polynomial-time computable.

Testing whether a game is simple or weighted can be done in polynomial time.

Complexity results for testing properties like strongness, properness, decisiveness, and homogeneity.

Abstract

Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of ``yea'' votes yield passage of the issue at hand. A collection of ``yea'' voters forms a winning coalition. We are interested on performing a complexity analysis of problems on such games depending on the game representation. We consider four natural explicit representations, winning, loosing, minimal winning, and maximal loosing. We first analyze the computational complexity of obtaining a particular representation of a simple game from a different one. We show that some cases this transformation can be done in polynomial time while the others require exponential time. The second question is classifying the complexity for testing whether a game is simple or weighted. We show that for the four types of representation both problem can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. In this way, we analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game.