Thesis introduction

...few structures arise in more contexts and lend themselves to more di- verse interpretations than do simple games

- Taylor and Zwicker [202]

1.2.1 Overview

This thesis examines the computational and algorithmic aspects of simple voting games or cooperative simple games which are not only an important class of co- operative games but also a widely used voting model. The mathematical model of simple games is generic enough to model various scenarios. The research fo- cusses on algorithms and the complexity of analysing the influence of players in game theoretic situations. This study of influence is significant in fields as diverse as percolation theory, reliability theory, political science and game theory [116]. The thesis also examines susceptibility of simple games, especially weighted vot- ing games, to various kinds of manipulations. Manipulation is an urgent issue in multiagent systems and it has been observed that not only do coalitional voting games model various multiagent scenarios well, but computational complexity is seen as a useful barrier against manipulation.

1.2 Thesis introduction 15

A comprehensive investigation of the influence of players in simple games promises to be a useful contribution to the literature considering that the notion has not been explored much in the works [213] and [202]. Taylor and Zwicker note in the preface of [202] that the cardinal notions of power have not been men- tioned in their book. Interestingly, such notions of power are now being explored much more in communities as diverse as reliability theory, political science and multi-agent systems. Voting power is also used in joint stock companies where each shareholder gets votes in proportion to the ownership of a stock [94]. An al- gorithms and complexity study of the influence of players is particularly relevant with the increase of large scale multi-agent systems. Moreover, in the manuscript on ‘Challenges for Theoretical Computer Science’ by Johnson [115], the fol- lowing challenges are highlighted: preventing strategic voting, computing power indices, continuing exploring the impact of bounded rationality and developing a theory of algorithmic mechanism design.

1.2.2 Simple games

...we will arrive at an extensive class of games, to be called simple. It will be seen that a study of this class yields a body of information which is of value for a deep understanding of the general theory...

- von Neumann and O. Morgenstern [213]

Simple games (which are yes/no decision games) were introduced in the clas- sical work of von Neumann and Morgenstern [213]. Von Neumann and Mor- genstern point out that a study of simple games makes it possible to get an un- derstanding of more general but harder to study zero-sumn-person games. Sim- ple games have a rich mathematical history with contributions from game theo- rists, computer scientists, electrical engineers and combinatorialists. The history of simple games could even be stretched back to the famous Dedekind prob- lem[120]. In 1897, Dedekind asked for the numberd(n) of free distributive lat- tices onnelements. This problem is equivalent to the number of simple games on

nplayers. The Dedekind problem has been well studied. The functiond(n) grows rapidly and d(n) is only known for very small n. Various algorithms have been

16 1 Introduction

proposed for efficient computation ofd(n) [85]. Simple games also have connec- tions with Sperner theory[71]. As in the case of Taylor and Zwicker [202], we will discuss simple games in a voting-theoretic context. This is convenient both from an intuition and notation point of view.

Simple games and weighted voting games (which are a sub-class of simple games) are known in different literatures and communities by different names. There is considerable work on these models in threshold logic [216, 109, 154] and also in game theory (see [202] for a detailed literature references).

Weighted voting games (WVGs) are mathematical models which are used to analyze voting bodies in which the voters have different number of votes. In WVGs, each voter is assigned a non-negative weight and makes a vote in favour of or against a decision. The decision is made if and only if the total weight of those voting in favour of the decision is greater than or equal to some fixed quota. Since the weights of the players do not always exactly reflect how critical a player is in decision making, voting power attempts to measure the ability of a player in a WVG to determine the outcome of the vote. WVGs are also encoun- tered in threshold logic, reliability theory, neuroscience and logical computing devices ([202], [208]). Parhami [171] points out that voting has a long history in reliability systems dating back to von Neumann [212]. For reliability systems, the weights of a WVG can represent the significance of the components whereas the quota can represent the threshold for the overall system to fail. Systems of this type are used in various areas such as target and pattern recognition, safety monitoring and human organization systems. WVGs have been applied in various political and economic organizations ([1]).

1.2.3 Approach of the thesis

...every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution

1.4 Thesis outline 17

The approach of the thesis is algorithmic. For many problems in cooperative game theory and social choice theory, there are mathematical results such as the existence or non-existence of properties. However, there is a need for an algorith- mic study of these topics so that efficient constructive methods can be devised to test different properties of games. For various computational problems asso- ciated with simple coalitional games, polynomial time exact algorithms, pseudo- polynomial algorithms, approximation algorithms and parameterized algorithms are presented. In other cases, a proof is provided that the problem is, for instance, NP-hard or #P-complete.