Evaluating Voting Systems with Probability Models: Essays by and in Honor of William Gehrlein and Dominique Lepelley

412 Pages·2021·9.841 MB·English

by  Mostapha Diss, Vincent Merlin

Studies in Choice and Welfare
Mostapha Diss
Vincent Merlin   Editors
Evaluating
Voting Systems
with Probability
Models
Essays by and in Honor of William
Gehrlein and Dominique Lepelley
Studies in Choice and Welfare
Editors-in-Chief
Marc Fleurbaey, Paris School of Economics, Paris, France
Maurice Salles, University of Caen, Caen, France
Series Editors
Bhaskar Dutta, Department of Economics, University of Warwick, Coventry, UK
Wulf Gaertner, FB Wirtschaftswissenschaften, Universität Osnabrück,
Osnabrück, Niedersachsen, Germany
Carmen Herrero Blanco, Faculty Economics and Business, University of Alicante,
Alicante, Spain
Bettina Klaus, Faculty of Business & Economics, University of Lausanne,
Lausanne, Switzerland
Prasanta K. Pattanaik, University of California, Riverside, CA, USA
William Thomson, Department of Economics, University of Rochester,
Rochester, NY, USA
More information about this series at http://www.springer.com/series/6869
Mostapha Diss Vincent Merlin
(cid:129)
Editors
Evaluating Voting Systems
with Probability Models
Essays by and in Honor of William Gehrlein
and Dominique Lepelley
123
Editors
Mostapha Diss Vincent Merlin
Department ofEconomics Faculty of Economics, Management, and
CRESE EA3190, UniversitéBourgogne Geography
Franche-Comté CNRSand UniversitédeCaenNormandie
Besançon,France Caen,France
ISSN 1614-0311 ISSN 2197-8530 (electronic)
Studies in ChoiceandWelfare
ISBN978-3-030-48597-9 ISBN978-3-030-48598-6 (eBook)
https://doi.org/10.1007/978-3-030-48598-6
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Foreword
The history of social choice theory is mainly a history of negative results. The
rebirth of this theory in modern times is essentially due to Kenneth J. Arrow. In
1948, he demonstrated the inconsistency of several properties of procedures of
aggregation of individual preferences into a ‘social’ preference. Some of these
properties are generally considered as basic elements of democratic procedures,
suchastheabsenceofadictatorandtherespectofunanimity(thispropertyisbeing
associated with Pareto and the so-called Pareto optimality, a major concept of
microeconomictheory).Otherpropertieshavebeencalledintoquestionsuchasthe
so-called independence of irrelevant alternatives and the transitivity of the social
preference. Arrow’s independence condition amounts to limit the information that
can be used in the aggregation procedure to pairwise preference considerations.
A second negative result, due to Amartya Sen, concerns an inconsistency
betweencollectiverationality(forinstance,somekindoftransitivitypropertyofthe
social preference), unanimity, and a degree offreedom conferred to individuals.
AthirdmajornegativeresultduetoAlanGibbard,PrasantaPattanaik,andMark
Satterthwaite concerns a property of strategyproofness: Aggregation procedures
shouldbeimmunetoastrategicbehaviorofindividuals(inthecaseofvoters,this,
in consequence, concerns ‘useful or tactical’ voting—it can advantageous for a
votertomisrepresentherpreferencesothatthevotingrulegeneratesanoutcome—
for instance, a candidate—that this voter prefers to the outcome that would have
prevailed if she had not misrepresented her preference).
Longbeforetheseresultswereobtained(in1948andinthe1970s),Condorcetin
1785 had shown that majority rule, a procedure that obviously satisfies Arrow’s
properties, could generate a cycle of the social preference. A very simple example
(a lot simpler than Condorcet’s example!) could be the following. Consider three
persons who want to have dinner together and have a choice between three
restaurantsa,b,andc.Person1prefersatobandbtoc,andsincesheisarational
person she prefers a to c. Persons 2 and 3 are also rational persons, and person 2
prefersbtocandctoawhileperson3prefersctoaandatob.Theyagreethatthe
v
vi Foreword
choice of the restaurant will be made by using majority rule: If the number of
personswhoprefersayatobisgreaterthanthenumberofpersonswhopreferbto
a,thenawillbe‘socially’preferred tob.Butgiventheindividualpreferences just
mentioned,onecanseethataissociallypreferredtobwhichissociallypreferredto
c which is socially toa. So there isno restaurant which issocially preferred tothe
other two, and the choice is problematic.
However, this outcome is based on a specific configuration of individual pref-
erences. A natural question arises regarding the probability that such situations
occur.Forinstance,giventhreeoptions(thethreerestaurants),therearesixpossible
rational preference orderings (excluding ties). If each person has one of these six
preferenceswithprobability1/6,whatistheprobabilitytoobtainacyclicoutcome
or a situation where there is no option defeating the other two. The French math-
ematician G. Th. Guilbaud indicated in 1952 that for our example a cycle will be
obtained in less than 6% of the situations. In a rather enigmatic (enigmatic at the
time of its publication) footnote, Guilbaud gave a limit for a ‘large’ number of
individuals, limit being less than 9%. Although Guilbaud’s work was largely
ignored in the English-speaking world, this kind of analysis took off at the end
of the 1960s as indicated by Sen in his book of 1970 and by Peter Fishburn in his
1973 treatise on social choice theory.
The works of Fishburn and William Gehrlein in the 1970s establish a new
sub-domain of the theory of social choice where various paradoxical situations
generated by various voting rules under various combinatorial/probabilistic
assumptions were studied. William Gehrlein has been the most prolific and most
influential author in this sub-domain over the last decades, and he published a
wonderful book in 2006 on Condorcet’s paradox.
At the University of Caen, under the leadership of Dominique Lepelley, this
sub-domain was eagerly developed (in particular by several of the contributors to
this volume). I must outline that Dominique Lepelley and Boniface Mbih were, to
the best of my knowledge, the pioneers regarding exact calculations related to the
manipulationofvotingrules(previousworkswerebasedonsimulationsandMonte
Carlo techniques).
Whatshouldhappendidhappen:AcollaborationbetweenGehrleinandLepelley
began (a collaboration which was extended to a few others). The result of this
collaborationisthepublicationofmanyjointpapersandoftwoexceptionalbooks.
Another outcome of this collaboration is the present volume partially based on a
conference which took place at the University of Caen-Normandy in 2018.
Foreword vii
I am very proud to have been a very minor element in the success of this
scientific accomplishment.
Maurice Salles
Normandy University, UNICAEN, CNRS,
UMR 6211, CREM
Université de Caen Normandie
Caen, France
e-mail: [email protected]
CPNSS, London School of Economics
London, UK
Murat Sertel Center for Advanced
Economic Studies
Bilgi University
Istanbul, Turkey
Acknowledgements
The papers in this volume have been cross-reviewed by the contributors and by
Daniela Bubboloni, Conal Duddy, Annick Laruelle, Ashley Piggins, Maria
Polukarov,JérômeSerais,FatyMbayeTop,andFabriceValognes.Wethankallthe
refereesfortheirthoughtfulcommentsandeffortstowardimprovingthechaptersof
this volume. We also thank Martina Bihn, Marc Fleurbaey, Johannes Glaeser, and
Maurice Salles for their contribution to the project.
ix
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Mostapha Diss and Vincent Merlin
The Condorcet Efficiency of Voting Rules and Related Paradoxes
Analyzing the Probability of Election Outcomes with Abstentions . . . . . 15
William V. Gehrlein and Dominique Lepelley
Condorcet Efficiency of General Weighted Scoring Rules Under IAC:
Indifference and Abstention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Mostapha Diss, Eric Kamwa, Issofa Moyouwou, and Hatem Smaoui
The Effect of Closeness on the Election of a Pairwise Majority
Rule Winner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Mostapha Diss, Patrizia Pérez-Asurmendi, and Abdelmonaim Tlidi
Analyzing the Practical Relevance of the Condorcet Loser Paradox
and the Agenda Contraction Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Felix Brandt, Christian Geist, and Martin Strobel
Other Voting Paradoxes
On the Probability of the Ostrogorski Paradox . . . . . . . . . . . . . . . . . . . 119
William V. Gehrlein and Vincent Merlin
Violations of Reversal Symmetry Under Simple and Runoff
Scoring Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Raouia Belayadi and Boniface Mbih
Binary Voting in Federations
Majority Efficient Representation of the Citizens in a Federal Union. . . 163
Marc Feix, Dominique Lepelley, Vincent Merlin, Jean-Louis Rouet,
and Laurent Vidu
xi