Power measures in large weighted voting games [Elektronische Ressource] : asymptotic properties and numerical methods / vorgelegt von Ines Lindner

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• Mathematics

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• Publié par: universitat_hamburg
• Publié le: 01 janvier 2004
• Nombre de lectures: 28
• Langue: English

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Power Measures
in
Large Weighted Voting Games
Asymptotic Properties and Numerical Methods
Dissertation
zur Erlangung des Doktorgrades
des Fachbereichs Mathematik
der Universit¨at Hamburg
vorgelegt von
Ines Lindner
aus Tubingen¨
Hamburg 2004Als Dissertation angenommen vom Fachbereich
Mathematik der Universit¨at Hamburg
auf Grund der Gutachten von
Prof. Dr. Claus-Peter Ortlieb, Universit¨at Hamburg
und
Prof. Dr. Mosh´e Machover, London School of Economics
Hamburg, den 21.10.2003
Prof. Dr. Alexander Kreuzer
Dekan des Fachbereichs MathematikAcknowledgments
IwanttoexpressmygratitudetoeveryonewhosupportedmesothatIcould
complete this work. Firstly, I am grateful to the people that educated me,
and to my parents in particular.
I am especially grateful to Mosh´e Machover and Claus Peter Ortlieb where
the latter already acted as a supervisor to my diploma thesis. Without
their support, it wouldn’t have been possible to realize my PhD in applied
mathematics as an external candidate. In that respect I also wish to thank
the Department of Mathematics at Hamburg University, whose cooperative
andencouragingattitudetowardsstudentshasalwaysbeenmostimpressing.
Some of the thesis has been written during my time as a research assistant
at the Institute of SocioEconomics at Hamburg University. I thank Manfred
Holler who supervised me on part of this research and for giving me the
invaluable opportunity to frequently represent my research on conferences. I
also profited a lot from the cooperation with Mathew Braham.
Since his profile is very close to mine, my coauthor Mosh´e Machover repre-
sented the strongest influence on this research on voting power. I am also
very grateful to Abraham Neyman and Guillermo Owen for their kind help
on Part I of this document.
I thank Maurice Koster for wonderful kinds of reasons.
Asignificantcontributionwasbyallthosepeoplethatkeptmefromworking.
Many thanks in that respect go to my dear old soulmates Volker Schirp
and Susanne Vorreiter. I thank them for contributing to make my life less
ordinary and I am especially grateful for their support in gloomy periods.
Lucas Erbsman has always been a very special and important person - I
thank him for his precious time. I am grateful to Isabel Guill´en who always
filled my office with laughter and energy and to whom I owe many wise
decisions. Moreover, my PhD time was particularly enlightened by my dearfriends Olaf Casimir, Barbara Dziersk, Melanie and Tillmann Esser, Daniel
Friedrich, Christian Hahne, Ilona Isforth, Stefan Kock, Holger Strulik and
ArnoldvanMeteren. IamalsogratefulforthestimulatingsupportofJochen
Bigus. With greatest pleasure I look back to our female circle of friends at
Hamburg University (’die Kreischziegen’) which has given academic life a
most cheerful aspect. In this respect I thank Silke Bender, Nicola Brandt,
Heide Coenen, Daniela Felsch, Sandra Greiner, Isabel Guill´en, Ria Steiger
for their company. Hamburg, March 2004.Contents
1 Preliminaries 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 5
I L.S. Penrose’s Limit Theorem 7
2 Introduction to Part I 9
3 PLT for Binary WVGs 15
3.1 PLT for Replicative q-Chains and the S-S index . . . . . . . . 15
3.2 PLT for 1/2-Chains and the Banzhaf Index . . . . . . . . . . . 17
4 PLT for Ternary WVGs 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Probabilistic Interpretation. . . . . . . . . . . . . . . . . . . . 25
4.3 Nature of Abstention . . . . . . . . . . . . . . . . . . . . . . . 28
4.4 PLT in WVGs with Abstentions . . . . . . . . . . . . . . . . . 31
5 Discussion of Part I 37II Global Asymptotic Properties 39
6 Introduction to Part II 41
6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 General Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7 Complaisance of WVGs 49
7.1 Complaisance in Binary WVGs . . . . . . . . . . . . . . . . . 49
7.2 Complaisance in Ternary WVGs . . . . . . . . . . . . . . . . . 52
7.3 Convergence Characteristics . . . . . . . . . . . . . . . . . . . 57
7.4 The EU Council of Ministers . . . . . . . . . . . . . . . . . . . 61
8 In Search of the Truth 65
8.1 Introduction to Condorcet’s Jury Theorem . . . . . . . . . . . 65
8.2 Generalization of Condorcet’s Jury Theorem . . . . . . . . . . 66
9 Discussion of Part II 73
III Numerical Methods for Large WVGs 75
10 Introduction 77
11 Exact Evaluation of WVGs 79
11.1 Common Structure of Classical Measures . . . . . . . . . . . . 79
11.2 Complaisance and the Banzhaf Measure . . . . . . . . . . . . 80
11.3 Computation of Jury Competence and the S-S Index . . . . . 83
11.4 Evaluating TWVGs . . . . . . . . . . . . . . . . . . . . . . . . 86
11.5 Storage Schemes for Sparse Matrices . . . . . . . . . . . . . . 90
12 Approximation Methods 93IV Appendix 95
13 Tables 97
14 Miscellaneous 103
15 Source Code 105
16 Basic Notations and Abbreviations 111
17References113
18 Abstract 117
19 Zusammenfassung - German Abstract 119
20 Curriculum Vitae 121Chapter 1
Preliminaries
1.1 Introduction
This thesis deals with the asymptotic properties of simple weighted voting
games when there are many ‘small’ voters. Institutions with a large number
of participants are common in political and economic life; examples are mar-
kets, stock companies and executive boards, for example the EU Council of
Ministers. The analysis of structural properties of collective decision-making
rules has a well established history in game theory and social choice theory.
Measurements of voting power have been proven to be useful as instruments
to analyze collective decision-making rules which can be modelled as a sim-
ple (voting) game. This relates to any collective body that makes yes-or-no
decisions by vote.
This thesis deals almost exclusively with a priori voting analysis. Contrary
to actual (a posteriori) analysis, it models the voting system as an ‘abstract
shell’, without taking into consideration voters’ preferences, the range of
issues over which a decision is taken or the degree of affinity between the
voters. This abstraction seems to be necessary to focus on the legislature
itself in a pure sense. Roth (1988, p. 9) puts it this way:
‘Analyzing voting rules that are modelled as simple games abstracts away
from the particular personalities and political interests present in particular
votingenvironments,butthisabstractioniswhatmakestheanalysisfocuson
therulesthemselvesratherthanonotheraspectsofthepoliticalenvironment.
This kind of analysis seems to be just what is needed to analyze the voting
rules in a new constitution, for example, long before the specific issues to be2 Preliminaries
voted on arise or the specific factions and personalities that will be involved
can be identified.’
The theoretical and empirical literature on the field of voting power can
roughly be divided into two fields: one studying individual and the other fo-
cussingonglobalmeasures. Individualpowermeasures–suchastheclassical
power measure proposed by Shapley & Shubik (1954) and Banzhaf (1965) –
focus on the question to what ‘extent’ a given member is able to control the
outcome of a collective decision. Global measures deal with global charac-
teristics of decision rules, for example the ease with which the decision rule
responds to fluctuations in the voters’ wishes or the propensity to approve
bills (which was introduced by Coleman (1971) as the power of a collectivity
to act).
Power measures represent a useful instrument to shed light on the differ-
ent aspects of voting scenarios, both in political as well as financial fields.
However, the extent of common acceptance and applicability to real-life vot-
ing design is still modest. A major limiting factor is presumably that the
computation of power measures is not straightforward – especially when the
number of voters is large – such that specific software has to be written or
installedinordertobeabletoevaluatevotingsystemsatall. Inthisrespect,
the limit theorems developed for weighted voting games in this thesis clearly
serve as a convenient approximation for large weighted voting games.
Weighted voting games play a central role, not only because they are very
common in economic and political organizations but also because many vot-
1ing systems can be equivalently represented as such. In a weighted voting
gameeachboardmemberisassignedtoanon-negativenumberasweight,and
acertainpositivenumberisfixedasquota. Manyorganizationshavesystems
of governance by weighted voting, examples for economic organizations are
the International Monetary Fund, the World Bank, stock companies, etc. In
federalpoliticalbodiestheweightsareusuallydesignedtoreflectthenumber
of inhabitants of each represented state; examples are the EU’s Council of
Ministers and the US Presidential Electoral College.
Awidespreadfallacyisthatunderaweightedvotingdecisionrulethepowers
of the voters are proportional to their respective weights. A simple coun-
terexample is a game of three voters, one being endowed with 2% of the total
weight sum and the other 98% evenly split up among the other two voters.
If