MODELS:
a critical perspective
EVALUATION AND DECISION MODELS:
a critical perspective
Denis Bouyssou ESSEC
Thierry Marchant Ghent University
Marc Pirlot SMRO, Facult´ e Polytechnique de Mons
Patrice Perny LIP6, Universit´ e Paris VI
Alexis Tsouki` as LAMSADE - CNRS, Universit´ e Paris Dauphine
Philippe Vincke SMG - ISRO, Universit´ e Libre de Bruxelles
KLUWER ACADEMIC PUBLISHERS
Boston/London/Dordrecht
1 Introduction 1
1.1 Motivations . . . 1
1.2 Audience . . . 2
1.3 Structure . . . 3
1.4 Outline . . . 3
1.5 Who are the authors ? . . . 5
1.6 Conventions . . . 5
1.7 Acknowledgements . . . 6
2 Choosing on the basis of several opinions 7 2.1 Analysis of some voting systems . . . 8
2.1.1 Uninominal election . . . 9
2.1.2 Election by rankings . . . 13
2.1.3 Some theoretical results . . . 16
2.2 Modelling the preferences of a voter . . . 18
2.2.1 Rankings . . . 18
2.2.2 Fuzzy relations . . . 21
2.2.3 Other models . . . 23
2.3 The voting process . . . 23
2.3.1 Definition of the set of candidates . . . 23
2.3.2 Definition of the set of the voters . . . 24
2.3.3 Choice of the aggregation method . . . 24
2.4 Social choice and multiple criteria decision support . . . 25
2.4.1 Analogies . . . 25
2.5 Conclusions . . . 26
3 Building and aggregating evaluations 29 3.1 Introduction . . . 29
3.1.1 Motivation . . . 29
3.1.2 Evaluating students in Universities . . . 30
3.2 Grading students in a given course . . . 31
3.2.1 What is a grade? . . . 31
3.2.2 The grading process . . . 31
3.2.3 Interpreting grades . . . 37
3.2.4 Why use grades? . . . 40
3.3 Aggregating grades . . . 41
3.3.1 Rules for aggregating grades . . . 41 v
3.3.2 Aggregating grades using a weighted average . . . 42
3.4 Conclusions . . . 51
4 Constructing measures 53 4.1 The human development index . . . 54
4.1.1 Scale Normalisation . . . 56
4.1.2 Compensation . . . 57
4.1.3 Dimension independence . . . 58
4.1.4 Scale construction . . . 59
4.1.5 Statistical aspects . . . 59
4.2 Air quality index . . . 61
4.2.1 Monotonicity . . . 61
4.2.2 Non compensation . . . 62
4.2.3 Meaningfulness . . . 62
4.3 The decathlon score . . . 63
4.3.1 Role of the decathlon score . . . 65
4.4 Indicators and multiple criteria decision support . . . 66
4.5 Conclusions . . . 69
5 Assessing competing projects 71 5.1 Introduction . . . 71
5.2 The principles of CBA . . . 73
5.2.1 Choosing between investment projects in private firms . . . 73
5.2.2 From Corporate Finance to CBA . . . 75
5.2.3 Theoretical foundations . . . 76
5.3 Some examples in transportation studies . . . 79
5.3.1 Prevision of traffic . . . 80
5.3.2 Time gains . . . 80
5.3.3 Security gains . . . 81
5.3.4 Other effects and remarks . . . 82
5.4 Conclusions . . . 83
6 Comparing on several attributes 87 6.1 Thierry’s choice . . . 87
6.1.1 Description of the case . . . 88
6.1.2 Reasoning with preferences . . . 91
6.2 The weighted sum . . . 97
6.2.1 Transforming the evaluations . . . 98
6.2.2 Using the weighted sum on the case . . . 99
6.2.3 Is the resulting ranking reliable? . . . 99
6.2.4 The difficulties of a proper usage of the weighted sum . . . 101
6.2.5 Conclusion . . . 105
6.3 The additive value model . . . 106
6.3.1 Direct methods for determining single-attribute value functions . . . 107
6.3.2 AHP and Saaty’s eigenvalue method . . . 111
6.3.3 An indirect method for assessing single-attribute value func- tions and trade-offs . . . 117
6.3.4 Conclusion . . . 124
6.4 Outranking methods . . . 124
6.4.1 Condorcet-like procedures in decision analysis . . . 124
6.4.2 A simple outranking method . . . 129
6.4.3 Using ELECTRE I on the case . . . 131
6.4.4 Main features and problems of elementary outranking ap- proaches . . . 139
6.4.5 Advanced outranking methods: from thresholding towards valued relations . . . 141
6.5 General conclusion . . . 144
7 Deciding automatically 147 7.1 Introduction . . . 147
7.2 A System with Explicit Decision Rules . . . 149
7.2.1 Designing a decision system for automatic watering . . . 150
7.2.2 Linking symbolic and numerical representations . . . 150
7.2.3 Interpreting input labels as scalars . . . 153
7.2.4 Interpreting input labels as intervals . . . 156
7.2.5 Interpreting input labels as fuzzy intervals . . . 161
7.2.6 Interpreting output labels as (fuzzy) intervals . . . 164
7.3 A System with Implicit Decision Rules . . . 170
7.3.1 Controlling the quality of biscuits during baking . . . 170
7.3.2 Automatising human decisions by learning from examples . 171 7.4 An hybrid approach for automatic decision-making . . . 174
7.5 Conclusion . . . 176
8 Dealing with uncertainty 179 8.1 Introduction . . . 179
8.2 The context . . . 179
8.3 The model . . . 180
8.3.1 The set of actions . . . 180
8.3.2 The set of criteria . . . 181
8.3.3 Uncertainties and scenarios . . . 182
8.3.4 The temporal dimension . . . 184
8.3.5 Summary of the model . . . 186
8.4 A didactic example . . . 186
8.4.1 The expected value approach . . . 187
8.4.2 Some comments on the previous approach . . . 187
8.4.3 The expected utility approach . . . 189
8.4.4 Some comments on the expected utility approach . . . 191
8.4.5 The approach applied in this case: first step . . . 193
8.4.6 Comment on the first step . . . 196
8.4.7 The approach applied in this case: second step . . . 198
8.5 Conclusions . . . 200
9 Supporting decisions 205 9.1 Preliminaries . . . 206
9.2 The Decision Process . . . 207
9.3 Decision Support . . . 210
9.3.1 Problem Formulation . . . 211
9.3.2 The Evaluation Model . . . 213
9.3.3 The final recommendation . . . 219
9.4 Conclusions . . . 226
Appendix A . . . 228
Appendix B . . . 231
10 Conclusion 237 10.1 Formal methods are all around us . . . 237
10.2 What have we learned? . . . 239
10.3 What can be expected? . . . 243
Bibliography 247
Index 262
1
INTRODUCTION
1.1 Motivations
Deciding is a very complex and difficult task. Some people even argue that our abil- ity to make decisions in complex situations is the main feature that distinguishes us from animals (it is also common to say that laughing is the main difference).
Nevertheless, when the task is too complex or the interests at stake are too impor- tant, it quite often happens that we do not know or we are not sure what to decide and, in many instances, we resort to a decision support technique: an informal one–we toss a coin, we ask an oracle, we visit an astrologer, we consult an expert, we think–or a formal one. Although informal decision support techniques can be of interest, in this book, we will focus on formal ones. Among the latter, we find some well-known decision support techniques: cost-benefit analysis, multiple crite- ria decision analysis, decision trees, . . . But there are many other ones, sometimes not presented as decision support techniques, that help making decisions. Let us cite but a few examples.
• When the director of a school must decide whether a given student will pass or fail, he usually asks each teacher to assess the merits of the student by means of a grade. The director then sums the grades and compares the result to a threshold.
• When a bank must decide whether a given client will obtain a credit or not, a technique, called credit scoring, is often used.
• When the mayor of a city decides to temporarily forbid car traffic in a city because of air pollution, he probably takes the value of some indicators, e.g.
the air quality index, into account.
• Groups or committees must also make decisions. In order to do so, they often use voting procedures.
All these formal techniques are what we call (formal)decision and evaluation models, i.e. a set of explicit and well-defined rules to collect, assess and process information in order to be able to make recommendations in decision and/or eval- uation processes. They are so widespread that almost no one can pretend he is
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not using or suffering the consequences of one of them. These models–probably because of their formal character–inspire respect and trust: they look scientific.
But are they really well founded ? Do they perform as well as we want ? Can we safely rely on them when we have to make important decisions ?
That is why we try to look at formal decision and evaluation models with a critical eye in this book. You guessed it: this book is more than 200 pages long.
So, there is probably a lot of criticism. You are right.
None of the evaluation and decision models that we examined are perfect or the best. They all suffer limitations. For each one, we can find situations in which it will perform very poorly. This is not really new: most decision models have had contenders for a long time. Do we want to contend all models at the same time ? Definitely not ! Our conviction is that there cannot be a best decision or evaluation model–this has been proved in some contexts (e.g. in voting) and seems empirically correct in other contexts–but we are convinced as well that formal evaluation and decision models are useful in many circumstances and here is why:
• Formal models provide explicit and, to a large extent, unambiguous represen- tations of a given problem; they offer a common language for communicating about the problem. They are therefore particularly well suited for facilitating communication among the actors of a decision or evaluation process.
• Formal models require that the decision maker makes a substantial effort to structure his perception or representation of the problem. This effort can only be beneficial as it forces the decision maker to think harder and deeper about his problem.
• Once a formal model has been established, a battery of formal techniques (often implemented on a computer) become available for drawing any kind of conclusion that can be drawn from the model. For example, hundreds of what-if questions can be answered in a flash. This can be of great help if we want to devise robust recommendations.
For all these reasons (complexity, usefulness, importance of the interests at stake, popularity) plus the fact that formal models lend themselves easily to criti- cism, we think that it is important to deepen our understanding of evaluation and decision models and encourage their users to think more thoroughly about them.
Our aim with this book is to foster reflection and critical thinking among all individuals utilising decision and evaluation models, whether it be for research or applications.
1.2 Audience
Most of us are confronted with formal evaluation and decision models. Very often, we use them without even thinking about it. This book is intended for the aware or enlightened practitioner, for anyone who uses decision or evaluation models–for research or for applications–and is willing to question his practice, to have a deeper understanding of what he does. We have tried to keep mathematics and formalism
at a very low level so that, hopefully, most of the material will be accessible to the not mathematically-inclined readers. A rich bibliography will allow the interested reader to locate the more technical literature easily.
1.3 Structure
There are so many decision and evaluation models that it would be impossible to deal with all of them within a single book. As will become apparent later, most of them rely on similar kinds of principles. We decided to present seven examples of such models. These examples, chosen in a wide variety of domains, will hopefully allow the reader to grasp these principles. Each example is presented in a chapter (Chapters 2 to 8), almost independent of the other chapters. Each of these seven chapters ends with a conclusion, placing what has been discussed in a broader context and indicating links with other chapters. Chapter 9 is somewhat different from the seven previous ones: it does not focus on a decision model but presents a real world application. The aim of this chapter is to emphasise the importance of the decision aiding process (the context of the problem, the position of the actors and their interactions, the role of the analyst, . . . ), to show that many difficulties arise there as well and that a coherence between the decision aiding process and the formal model is necessary.
Some examples have been chosen because they correspond to decision models that everyone has experienced and can understand easily (student grades and voting). We chose some models because they are not often perceived as decision or evaluation models (student grades, indicators and rule based control). The other examples (cost-benefit analysis, multiple criteria decision support and choice under uncertainty) correspond to well identified and popular evaluation and decision models.
1.4 Outline
Chapter 2 is devoted to the problem of voting. After showing the analogy between voting and multiple criteria decision support, we present a sequence of twelve short examples, each one illustrating a problem that arises with a particular voting method. We begin with simple methods based on pairwise comparisons and we end up with the Borda method. Although the goal of this book is not to overwhelm the reader with theory, we informally present two theorems (Arrow and Gibbard- Satterthwaite) that in one way or another explain why we encountered so many difficulties in our twelve examples.
Then we turn to the way voters’ preferences are modelled. We present many different models, each one trying to outdo the previous one but suffering its own weaknesses. Finally, we explore some issues that are often neglected: who is going to vote? Who are the candidates? These questions are difficult and we show that they are important. The construction of the set of voters and the set of candidates, as well as the choice of a voting method must be considered as part of the voting process.
After examining voting, we turn in Chapter 3 to another very familiar topic for the reader: students’ marks or grades. Marks are used for different purposes (e.g.
ranking the students, deciding whether a student is allowed to begin the next level of study, deciding whether a student gets a degree, . . . ). Students are assessed in a huge variety of ways in different countries and schools. This seems to indicate that assessing students might not be trivial. We use this familiar topic to discuss operations such as evaluating a performance and aggregating evaluations.
In Chapter 4, three particular indicators are considered: the Human Devel- opment Index (used by the United Nations), the ATMO index (an air pollution indicator used by the French government) and the decathlon score. We present a few examples illustrating some problems occurring with indicators. We assert that some difficulties are the consequences of the fact that the role of an indicator is often manifold and not well defined. An indicator is a measure but, often, it is also a tool for controlling or managing (in a broad sense).
Cost-benefit analysis (CBA) is a decision aiding method that is extremely popular among economists. Following the CBA approach, a project should only be undertaken when its benefits outweigh its costs. First we present the principles of CBA and its theoretical foundations. Then, using an example in transportation studies, we illustrate some difficulties encountered with CBA. Finally, we clarify some of the hypotheses at the heart of CBA and criticise the relevance of these hypotheses in some decision aiding processes.
In Chapter 6, using a well documented example, we present some difficulties that arise when one wants to choose from or rank a set of alternatives considered from different viewpoints. We examine several aggregation methods that lead to a value function on the set of alternatives, namely the weighted sum, the sum of utilities (direct and indirect assessment) and AHP (the Analytic Hierarchy Pro- cess). Then we turn to the so called outranking methods. Some of these methods can be used even when the data are not very rich or precise. The price we pay for this is that results provided by these methods are not rich either, in the sense that conclusions that can be drawn regarding a decision are not clear-cut.
Chapter 7 is dedicated to the study of automatic decision systems. These systems concern the execution of repetitive decision tasks and the great majority of them are based on more or less explicit decision rules aimed towards reflecting the usual decision policy of humans. The goal of this section is to show the interest of some formal tools (e.g. fuzzy sets) to model decision rules but also to clarify some problems arising when simulating the rules. Three examples are presented:
the first one concerns the control of an automatic watering system while the others are about the control of a food process. The first two examples describe decision systems based on explicit decision rules; the third one addresses the case of implicit decision rules.
The goal of Chapter 8 is to raise some questions about the modelling of un- certainty. We present a real-life problem concerning the planning of electricity production. This problem is characterised by many different uncertainties: for example, the price of oil or the electricity demand in 20 years time. This prob- lem is classically described by using a decision tree and solved with an expected utility approach. After recalling some well known criticisms directed against this
approach, we present the approach that has been used by the team that “solved”
this problem. Some of the drawbacks of this approach are discussed as well. The relevance of probabilities is criticised and other modelling tools, such as belief functions, fuzzy set theory and possibility theory, are briefly mentioned.
Convinced that there is more to decision aiding than just number crunching, we devote the last chapter to the description of a real world decision aiding process that took place in a large Italian company a few years ago. It concerns the eval- uation of offers following a call for tenders for a GIS (Geographical Information System) acquisition. Some important elements such as the participating actors, the problem formulation, the construction of the criteria, etc. deserve greater con- sideration. One should ideally never consider these elements separately from the aggregation process because they can impact the whole decision process and even the way the aggregation procedure behaves.
1.5 Who are the authors ?
The authors of this book are European academics working in six different universi- ties, in France and in Belgium. They teach in engineering, business, mathematics, computer science and psychology schools. Their background is quite varied as well: mathematics, economics, engineering, law and geology but they are all ac- tive in decision support and more particularly in multiple criteria decision support.
Among their special interests are preference modelling, fuzzy logic, aggregation techniques, social choice theory, artificial intelligence, problem structuring, mea- surement theory, operations research, . . . Besides their interest in multiple criteria decision support, they share a common view on this field. Five of the six authors of the present volume presented their thoughts on the past and the objectives of future research in multiple criteria decision support in the Manifesto of the new MCDA era (Bouyssou, Perny, Pirlot, Tsouki`as and Vincke 1993).
The authors are very active in theoretical research on the foundations of de- cision aiding, mainly from an axiomatic point of view, but have been involved in a variety of applications ranging from software evaluation to location of a nu- clear repository, through the rehabilitation of a sewer network or the location of high-voltage lines.
In spite of the large number of co-authors, this book is not a collection of papers. It is a joint work.
1.6 Conventions
To refer to a decision maker, a voter or an individual whose sex is not determined, we decided not to use the politically correct “he/she” but just “he” in order to make the text easy to read. The fact that all of the authors are male has nothing to do with this choice. The same applies for “his/her”.
None of the authors is a native English speaker. Therefore, even if we did our best to write in correct English, the reader should not be surprised to find
some mistakes or inelegant expressions. We beg the reader’s leniency for any incorrectness that might remain.
The adopted spelling is the British and not the American one.
1.7 Acknowledgements
We are ggreatly indebted to our /////////collEague friend Philippe Fortemps \cite{Fortemps99}
.
Without him and his knowledge of Late-
x, this book would look like this paragraph.%\newline The authors also wish to thank J.-L. Ottinger, who contributed to Chapter 8, H. M´elot, who laid out the complex diagrams of that chapter, and Stefano Abruzzini, who gave us a number of references concerning indicators. Chapter 6 is based on a report by S´ebastien Cl´ement written to fulfil the requirements of a course on multiple criteria decision support. Large part of chapter 9 uses material already published in (Paschetta and Tsouki`as 1999).
A special thank goes to Marjorie and Diane Gassner who had the patience to read and correct our continental approximation of the English language and to Fran¸cois Glineur who helped in solving a great number of latex problems.
We thank Gary Folven from Kluwer Academic Publisher for his constant sup- port during the preparation of this manuscript.
2
CHOOSING ON THE BASIS OF SEVERAL OPINIONS: THE EXAMPLE OF VOTING
Voting is easy! You’ve voted hundreds of times in committees, in presidential elections, for the senate, . . . Is there much to say about voting ? Well, just think about the way heads of state or members of parliament are elected in Australia, France, the UK, . . .
United Kingdom’s members of parliament The territory of the UK is di- vided into about 650 constituencies. One representative is elected in each constituency. Each voter chooses one of the candidates in his constituency.
The winner is the candidate that is chosen by more voters than any other one. Note that the winner does not have to win an overall majority of votes.
France’s members of parliament As in the UK, the French territory is divided into single-seat constituencies. In a constituency, each voter chooses one of the candidates. If one candidate receives more than 50 % of the votes, he is elected. Otherwise a second stage is organised. During the second stage, all candidates that were chosen by more than 12.5 % of the registered voters may compete. Once more, each voter chooses one of the candidates. The winner is the candidate that received the most votes.
France’s president Each voter chooses one of the candidates. If one candidate has been chosen by more than 50 % of the voters, he is elected. Otherwise a second stage is organised. During the second stage, only two candidates remain: those with the highest scores. Once again, each voter chooses one of the candidates. The winner is the candidate that has been chosen by more voters than the other one.
Australia’s members of parliament The territory is divided into single-seat constituencies called divisions. In a division, each voter is asked to rank all candidates: he puts a 1 next to his preferred candidate, a 2 next to his second preferred candidate, then a 3, and so on until his least preferred candidate.
Then the ballot papers are sorted according to the first preference votes. If a candidate has more than 50 % of the ballot papers, he is elected. Otherwise, the candidate that received fewer papers than any other is eliminated and the corresponding ballot papers are transferred to the candidates that got
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a 2 on these papers. Once more, if a candidate has more than 50 % of the ballot papers, he is elected. Otherwise, the candidate that received fewer papers than any other is eliminated and the corresponding ballot papers are transferred to the candidates that got a 3 on these papers, etc. In the worst case, this process ends when all but two candidates are eliminated, because, unless they are tied, one of the candidates necessarily has more than 50 % of the papers. Note that, as far as we know, it seems that the case of a tie is seldom considered in electoral laws.
Canada’s members of parliament and prime minister Every five years, the Canadian parliament is elected as follows. The territory is divided into about 270 constituencies called counties. In each county, each party can present one candidate. Each voter chooses one candidate. The winner in a county is the candidate that is chosen by more voters than any other one. He is thus the county’s representative in the parliament. The leader of the party that has the most representatives becomes prime minister.
Those interested in voting methods and the way they are applied in various countries will find valuable information in Farrell (1997) and Nurmi (1987). The diversity of the methods applied in practice probably reflects some underlying complexity and, in fact, if you take a closer look at voting, you will be amazed by the incredible complexity of the subject. In spite of its apparent simplicity, thousands of papers have been devoted to the problem of voting (Kelly 1991) and our guess is that many more are to come.
Our aim in this chapter is, on the one hand, to show that many difficult and interesting problems arise in voting and, on the other hand, to convince the reader that a formal study of voting might be enlightening. This chapter is organised as follows. In Section 1, we make the following basic assumption: each voter’s preferences can accurately be represented by a ranking of all candidates from best to worse, without ties. Then we show some problems occurring when aggregating the rankings, using classical voting systems such as those applied in France or the United Kingdom. We do this through the use of small and classical examples. In Section 2, we consider other preference models than the linear ranking of Section 1. Some models are poorer in information but more realistic. Some are richer and less realistic. In most cases, the aggregation remains a difficult task. In Section 3, we change the focus and try to examine voting in a much broader context.
Voting is not instantaneous. It is not just counting the votes and performing some mathematical operation to find the winner. It is a process that begins when somebody decides that a vote should occur (or even earlier) and ends when the winner begins his mandate (or even later). In Section 4, we discuss the analogy with multiple criteria decision support. The chapter ends with a conclusion.
2.1 Analysis of some voting systems
From now on, we will distinguish between the election—the process by which the voters express their preferences about a set of candidates—and the aggregation
method—the process used to extract the best candidate or a ranking of the can- didates from the result of the election. In many cases, the election is uninominal, i.e. each voter votes for one candidate only
2.1.1 Uninominal election
Let us recall the assumption that we mentioned earlier and that will hold through- out Section 1. Each voter, consciously or not, ranks all candidates from best to worse, without ties and, when voting, each voter sincerely (or naively) reports his preferences. Thus, in a uninominal election, we shall assume that each voter votes for the candidate that he ranks in first position. For example, suppose that a voter prefers candidateatobandb toc(in shortaP bP c). He votes fora. We are now ready to present a first example that illustrates a difficulty in voting.
Example 1. Dictature of majority
Let{a, b, c, . . . , y, z} be a set of 26 candidates for a 100 voters election. Suppose that
51 voters have preferences aP bP cP . . . P yP z and 49 voters have preferences zP bP cP . . . P yP a.
It is clear that 51 voters will vote for a while 49 vote for z. Thus a has an absolute majority and, in all uninominal systems we are aware of, a wins. But is a really a good candidate ? Almost half of the voters perceivea as the worst one. And candidate b seems to be a good candidate for everyone. Candidate b could be a good compromise. As shown by this example, a uninominal election combined with the majority rule allows a dictatorship of majority and doesn’t favour a compromise. A possible way to avoid this problem might be to ask the voters to provide their whole ranking instead of their preferred candidate. This will be discussed later. Let us continue with some strange problems arising when using a uninominal election.
Example 2. Respect of majority in the British system
The voting system in the United Kingdom is plurality voting, i.e. the election is uninominal and the aggregation method is simple majority. Let {a, b, c} be the set of candidates for a 21 voters election. Suppose that
10 voters have preferences aP bP c, 6 voters have preferences bP cP a and 5 voters have preferences cP bP a.
Then a (resp. b and c) obtains 10 votes (resp. 6 and 5). Thus a is chosen.
Nevertheless, this might be different from what a majority of voters wanted. In- deed, an absolute majority of voters prefers any other candidate to a(11 out of 21 voters preferbandc toa).
Let us see, using the same example, if such a problem would be avoided by the two-stage French system. After the first stage, as no candidate has an absolute majority, a second stage is run between candidatesaandb. We suppose that the voters keep the same preferences on{a, b, c}. Thus a obtains 10 votes andb, 11 votes so that candidateb is elected. This time, none of the beaten candidates (a andc) are preferred tobby a majority of voters. Nonetheless we cannot conclude that the two-stage French system is superior to the British system from this point of view, as shown by the following example.
Example 3. Respect of majority in the two-stage French system Let{a, b, c, d}be the set of candidates for a 21 voters election. Suppose that
10 voters have preferences bP aP cP d, 6 voters have preferences cP aP dP b and 5 voters have preferences aP dP bP c.
After the first stage, as no candidate has absolute majority, a second stage is run between candidatesb andc. Candidatebeasily wins with 15 out of 21 votes though an absolute majority (11/21) of voters prefer a and d to b. Because it is not necessary to be a mathematician to figure out such problems, some voters might be tempted not to sincerely report their preferences as shown in the next example.
Example 4. Manipulation in the two-stage French system
Let us continue with the example used above. Suppose that the six voters having preferences cP aP dP bdecide not to be sincere and vote fora instead of c. Then candidate a wins after the first stage because there is an absolute majority for him (11/21). If they had been sincere (as in the previous example),bwould have been elected. Thus, casting a non sincere vote is useful for those 6 voters as they prefer a to b. Such a system, that may encourage voters to falsely report their preferences, is called manipulable. This is not the only weakness of the French system as attested by the three following examples.
Example 5. Monotonicity in the two-stage French system
Let {a, b, c} be the set of candidates for a 17 voters election. A few days before the election, the results of a survey are as follows:
6 voters have preferences aP bP c, 5 voters have preferences cP aP b, 4 voters have preferences bP cP a and 2 voters have preferences bP aP c.
With the French system, a second stage would be run, between aand b and a would be chosen obtaining 11 out of 17 votes. Suppose that candidate a, in order to increase his lead overband to lessen the likelihood of a defeat, decides to strengthen his electoral campaign againstb. Suppose that the survey did exactly
reveal the preferences of the voters and that the campaign has the right effect on the last two voters. Hence we observe the following preferences.
8 voters have preferences aP bP c, 5 voters have preferences cP aP b and 4 voters have preferences bP cP a.
After the first stage, b is eliminated, due to the campaign ofa. The second stage opposes a to c and c wins, obtaining 9 votes. Candidate a thought that his campaign would be beneficial. He was wrong. Such a method is called non monotonic because an improvement of a candidate’s position in some of the voter’s preferences can lead to a deterioration of his position after the aggregation. It is clear with such a system that it is not always interesting or efficient to sincerely re- port one’s preferences. You will note in the next example that some manipulations can be very simple.
Example 6. Participation in the two-stage French system
Let{a, b, c} be the set of candidates for a 11 voters election. Suppose that 4 voters have preferences aP bP c,
4 voters have preferences cP bP a and 3 voters have preferences bP cP a.
Using the French system, a second stage should opposeatocandcshould win the election obtaining 7 out of 11 votes. Suppose that 2 of the 4 first voters (with preferences aP bP c) decide not to vote because c, the worst candidate according to them, is going to win anyway. What will happen ? There will be only 9 voters.
2 voters have preferences aP bP c, 4 voters have preferences cP bP a and 3 voters have preferences bP cP a.
Contrary to all expectations, candidatec will loose whileb will win, obtaining 5 out of 9 votes. Our two lazy voters can be proud of their abstention since they preferbtoc. Clearly such a method does not encourage participation.
Example 7. Separability in the two-stage French system
Let{a, b, c}be the set of candidates for a 26 voters election. The voters are located in two different areas: countryside and town. Suppose that the 13 voters located in the town have the following preferences.
4 voters have preferences aP bP c, 3 voters have preferences bP aP c, 3 voters have preferences cP aP b and 3 voters have preferences cP bP a.
Suppose that the 13 voters located in the countryside have the following pref- erences.
4 voters have preferences aP bP c, 3 voters have preferences cP aP b, 3 voters have preferences bP cP a and 3 voters have preferences bP aP c.
Suppose now that an election is organised in the town, with 13 voters. Candi- datesaandc will go to the second stage andawill be chosen, obtaining 7 votes.
If an election is organised in the countryside, awill defeat bin the second stage, obtaining 7 votes. Thus ais the winner in both areas. Naturally we expectato be the winner in a global election. But it is easy to observe that in the global election (26 voters)a is defeated during the first stage. Such a method is called non separable.
The previous examples showed that, when there are more than 2 candidates, it is not an easy task to imagine a system that would behave as expected. Note that, in the presence of 2 candidates, the British system (uninominal and one-stage) is equivalent to all other systems and it suffers none of the above mentioned problems (May 1952). Thus we might be tempted by a generalisation of the British system (restricted to 2 candidates). If there are two candidates, we use the British system;
if there are more than two candidates, we arbitrarily choose two of them and we use the British system to select one. The winner is opposed (using the British system) to a new arbitrarily chosen candidate. And so on until no more candidates remain.
This would requiren−1 votes between 2 candidates. Unfortunately, this method suffers severe drawbacks.
Example 8. Influence of the agenda in sequential voting
Let{a, b, c} be the set of candidates for a 3 voters election. Suppose that 1 voter has preferences aP bP c,
1 voter has preferences bP cP a and 1 voter has preferences cP aP b.
The 3 candidates will be considered two by two in the following order or agenda:
a and b first, then c. During the first vote, a is opposed to b and a wins with absolute majority (2 votes against 1). Thenais opposed tocandcdefeatsawith absolute majority. Thuscis elected.
If the agenda is a and c first, it is easy to see that c defeats a and is then opposed tob. Hence,bwins againstcand is elected.
If the agenda isb andcfirst, it is easy to see that, finally,ais elected. Conse- quently, in this example, any candidate can be elected and the outcome depends completely on the agenda, i.e. on an arbitrary decision. Let us note that sequential voting is very common in different parliaments. The different amendments to a bill are considered one by one in a predefined sequence. The first one is opposed to the status quo, using the British system; the second one is opposed to the winner , and so on. Clearly, such a method lacks neutrality. It doesn’t treat all candidates in a symmetric way. Candidates (or amendments) appearing at the end of the agenda are more likely to be elected than those at the beginning.
Example 9. Violation of unanimity in sequential voting
Let{a, b, c, d}be the set of candidates for a 3 voters election. Suppose that 1 voter has preferences bP aP dP c,
1 voter has preferences cP bP aP d and 1 voter has preferences aP dP cP b.
Consider the following agenda: a and b first, then c and finally d. Candidate a is defeated byb during the first vote. Candidatec wins the second vote anddis finally elected though all voters unanimously prefera to d. Let us remark that this cannot happen with the French and British systems.
Up to now, we have assumed that the voters are able to rank all candidates from best to worse without ties but the only information that we collected was the best candidate. Why not try to palliate the many encountered problems by asking voters to explicitly rank the candidates ? This idea, though interesting, will lead us to many other pitfalls that we discuss just below.
2.1.2 Election by rankings
In this kind of election, each voter provides a ranking without ties of the candidates.
Hence the task of the aggregation method is to extract from all these rankings the best candidate or a ranking of the candidates reflecting the preferences of the voters as much as possible.
At the end of the 18th century, two aggregation methods for election by rank- ings appeared in France. One was proposed by Borda, the other by Condorcet.
Although other methods have been proposed, their methods are still at the heart of many scientists’ concerns. In fact, many methods are variants of the Borda and Condorcet methods.
The Condorcet method
Condorcet (1785) suggests to compare all candidates pairwise in the following way.
A candidateais preferred tobif and only if the number of voters rankingabefore bis larger than the number of voters rankingbbeforea. In case of tie, candidates a and b are indifferent. A candidate that is preferred to all other candidates is called a (Condorcet) winner. In other words, a winner is a candidate that, opposed to each of the n−1 other candidates, wins by a majority. It can be shown that there is never more than one Condorcet winner.
Note that both the British as well as the two-stage French methods are different from the Condorcet method. In example 2, candidateais elected by the British method butb is the Condorcet winner. In example 3, ais the Condorcet winner althoughb is chosen by the French method.
Although the principle underlying the Condorcet method—the candidate that beats all other candidates in a pairwise contest is the winner—seems very natural, close to the concept of democracy and hence very appealing, it is worth noting that, in some instances, this principle might be questioned: in example 1,ais the
Condorcet winner, although almost half of the voters consider him to be the worse candidate. Consider also example 10 taken from Fishburn (1977).
Example 10. Critique of the majority principle
Let{a, b, c, d, e, f, g, x, y}be a set of 9 candidates for a 101 voters election. Suppose that
19 voters have preferences yP aP bP cP dP eP f P gP x, 21 voters have preferences eP f P gP xP yP aP bP cP d, 10 voters have preferences eP xP yP aP bP cP dP f P g, 10 voters have preferences f P xP yP aP bP cP dP eP g, 10 voters have preferences gP xP yP aP bP cP dP eP f and 31 voters have preferences yP aP bP cP dP xP eP f P g.
Candidate x wins against every other candidate with a majority of 51 votes.
Thus x is the Condorcet winner. But let us focus on the candidates x and y.
Let us summarise their results in Table 2.1. In view of Table 2.1, it seems thaty should be elected.
k
1 2 3 4 5 6 7 8 9
x 0 30 0 21 0 31 0 0 19
y 50 0 30 0 21 0 0 0 0
Table 2.1: Number of voters who rank the candidate ink-th place in their prefer- ences
Furthermore, there are cases (called Condorcet paradoxes) where there is no Condorcet winner. Consider example 8: a is preferred to b, b is preferred to c and c is preferred toa. No candidate is preferred to all others. In such a case, the Condorcet method fails to elect a candidate. One might think that example 8 is very bizarre and very unlikely to happen. Unfortunately it isn’t. If you consider an election with 25 voters and 11 candidates, the probability of such a paradox is significantly high as it is approximately 1/2 (Gehrlein 1983) and the more candidates or voters, the higher the probability of such a paradox. Note that, in order to obtain this result, all rankings are supposed to have the same probability. Such an hypothesis is clearly questionable (Gehrlein 1983).
Many methods have been designed that elect the Condorcet winner, if he exists, and choose a candidate in any case (Fishburn 1977, Nurmi 1987).
The Borda method
Borda (1781) proposed to use the following aggregation method. In each voter’s preference, each candidate has a rank: 1 for the first candidate in the ranking, 2 for the second, . . . andnfor the last. Compute the Borda score of each candidate, i.e. the sum for all voters of that candidate’s rank. Then choose the candidate with lowest Borda score.
Note that there can be several such candidates. In these cases, the Borda method does not tell us which one to choose. They are considered as equivalent.
But the likelihood of indifference is rather small and decreases as the number of candidates or voters increases. For example, for 3 candidates and 2 voters, the probability of all candidates being tied is 1/3; for 3 candidates and 50 voters, it is less than 1 %. Note that once again, we supposed that all rankings have the same probability.
Note that the Borda method not only allows to choose one candidate but to rank them (by increasing Borda scores). If two candidates have the same Borda score, then they are indifferent.
Example 11. Comparison of the Borda and Condorcet methods Let{a, b, c, d}be the set of candidates for a 3 voters election. Suppose that
2 voters have preferences bP aP cP d and 1 voter has preferences aP cP dP b.
The Borda score ofais 5 = 2×2+1×1. Forb, it is 6 = 2×1+1×4. Candidates canddreceive 8 and 11. Thusais the winner. Using the Condorcet method, the conclusion is different: bis the Condorcet winner. Thus, when a Condorcet winner exists, it is not always chosen by the Borda method. Nevertheless, it can be shown that the Borda method never chooses a Condorcet looser, i.e. a candidate that is beaten by all other candidates by an absolute majority (contrary to the British system, see Example 2).
Suppose now that candidates c and d decide not to compete because they are almost sure to lose. With the Borda method, the new winner is b. Thus b now defeats a just because c and d dropped out. Thus the fact that a defeats or is defeated byb depends upon the presence of other candidates. This can be a problem as the set of the candidates is not always fixed. It can vary because candidates withdraw, because feasible solutions become infeasible or the converse, because new solutions emerge during discussions, . . .
With the Condorcet method, b remains the winner and it can be shown that this is always the case: if a candidate is a Condorcet winner, then he is still a Condorcet winner after the elimination of some candidates.
Example 12. Borda and the independence of irrelevant alternatives Let{a, b, c} be the set of candidates for a 2 voters election. Suppose that
1 voter has preferences aP cP b and 1 voter has preferences bP aP c.
The alternative with the lowest Borda score isa. Now consider a new election where the alternatives and voters are identical but they changed their preferences aboutc. Suppose that
1 voter has preferences aP bP c and 1 voter has preferences bP cP a.
It turns out that b has the lowest Borda score. However, none of the two voters changed their opinion about the pair{a, b}. The first (resp. second) voter prefers a (resp. b) in both cases. Only the relative position of c changed and this was enough to turn b into a winner and a into a looser. This can be seen as a shortcoming of the Borda method. One says that the Borda method does not satisfy the independence of irrelevant alternatives. It can be shown that the Condorcet method satisfies this property.
2.1.3 Some theoretical results
We could go on and on with examples showing, that any method you can think of suffers severe problems. But we think it is time to stop for at least two reasons.
First, it is not very constructive and, second, each example is related to a particular method; hence this approach lacks generality. A more general (and thus theoretic) approach is needed. We should find a way to answer questions like
• Do non manipulable methods exist ?
• Is it possible for a non separable method to satisfy unanimity ?
• . . .
In another book, in preparation, we will follow such a general approach but, in the present volume, we try to present various problems arising in evaluation and decision models in an informal way and to show the need for formal methods.
Nevertheless, we cannot resist to the desire to present now, in an informal way, some of the most famous results of social choice theory.
Arrow’s theorem
Arrow (1963) was interested by the aggregation of rankings with ties into a ranking, possibly with ties. We will call this ranking the overall ranking. He examined the methods verifying the following properties.
Universal domain. This property implies that the aggregation method must be applicable to all cases. Whatever the rankings provided by the voters, the method must yield an overall ranking of the candidates. This property rules out methods that would impose some restrictions on the preferences of the voters.
Transitivity. The result of the aggregation must always be a ranking, possibly with ties. This implies that, if aP b and bP c in the overall ranking, then aP c in the overall ranking. Example 8 showed that the Condorcet method doesn’t verify transitivity: a is preferred to b, b is preferred to c and c is preferred toa.
Unanimity. If all voters are unanimous about a pair of candidates, e.g. if all voters rank a before b, then a must be ranked before b in the overall preference.
This seems quite reasonable but example 9 showed that some commonly used
aggregation methods fail to respect unanimity. This property is often called Pareto condition.
Independence. The relative position of two candidates in the overall ranking de- pends only on their relative positions in the individual’s preferences. There- fore other alternatives are considered as irrelevant with respect to that pair.
Note that we observed in example 12 that the Borda method violates the independence property. This property is often called Independence of irrel- evant alternatives.
Non-dictatorship. None of the voters can systematically impose his preferences on the other ones. This rules out aggregation methods such that the overall ranking is always identical to the preference ranking of a given voter. This may be seen as a minimal requirement for ademocratic method.
These five conditions allow to state Arrow’s celebrated theorem.
Theorem 2.1 (Arrow) When the number of candidates is at least 3, there ex- ists no aggregation method satisfying simultaneously the properties of universal domain, transitivity, unanimity, independence and non-dictatorship.
To a large extent, this theorem explains why we encountered so many diffi- culties when trying to find a satisfying aggregation method. For example, let us observe that the Borda method satisfies the universal domain, transitivity, una- nimity and non-dictatorship properties. Therefore, as a consequence of theorem 2.1, we can deduce that it cannot satisfy the independence condition. What about the Condorcet method ? It satisfies the universal domain, unanimity, independence and non-dictatorship properties. Hence it cannot verify transitivity (see example 8). Note that Arrow’s theorem uses only five conditions that, in addition, are quite weak (at least at first glance). Yet, the result is powerful. If, in addition to these five conditions, we wish to find a method satisfying neutrality, separability, monotonicity, non-manipulability, . . . we face an even more puzzling problem.
Gibbard-Satterthwaite’s theorem
Gibbard (Gibbard 1973) and Satterthwaite (Satterthwaite 1975) were very inter- ested by the (non-)manipulability of aggregation methods, especially those leading to the election of a unique candidate. Informally, a method is non-manipulable if, in no case, a voter can improve the result of the election by not reporting his true preferences. They proved the following result.
Theorem 2.2 (Gibbard-Satterthwaite) When the number of candidates is larger than two, there exists no aggregation method satisfying simultaneously the proper- ties of universal domain, non-manipulability and non-dictatorship.
Example 4 concerning the two-stage French system can be revisited bearing in mind theorem 2.2. The French system satisfies universal domain and non- dictatorship. Therefore, it is not surprising that it is manipulable.
Many other impossibility results can be found in the literature. But this is not the place to review them. Besides impossibility results, many characterisations are available. A characterisation of a given aggregation method is a set of properties simultaneously satisfied by only that method. These results help to understand the fundamental principles of a method and to compare different methods.
At the beginning of this chapter, we decided to focus on elections of a unique candidate. Some voting systems lead to the election of several candidates and aim towards achieving a kind of proportional representation. One might think that those systems are the solution to our problems. In fact, they are not. Those systems raise as many questions (perhaps more) as the ones we considered (Balinski and Young 1982). Furthermore, suppose that a parliament has been elected, using proportional representation. This parliament will have to vote on many different issues and, very often, only one candidate or law or project will have to be chosen.
2.2 Modelling the preferences of a voter
Let us consider the assumption that we made in Section 1: the preferences of each voter can accurately be represented by a ranking of all candidates from best to worse, without ties. We all know that this is not always realistic. For example, in some instances, there are several candidates that a voter cannot rank, just because he considers them as equivalent. Those candidates are tied. There are many other reasons to question our assumption. In some cases, a voter is not able to rank the candidates; in others, he is able to rank them but another kind of modeling of his preferences would be more accurate. In this section, we list different cases in which our initial assumption is not valid.
2.2.1 Rankings
To model the preferences of a voter, we can use a ranking without ties. This model corresponds to the assumption of Section 1. This implies that when you present a pair of candidates (a, b) to a voter, he is always able to tell if he prefersato bor the converse. Furthermore, if he prefersatobandb toc, he necessarily prefersa toc (transitivity of preference).
Indifference: rankings with ties
In some cases, a voter is unable to state if he prefersatobor the converse because he thinks that both candidates are of equal value. He is indifferent betweenaand b. Thus, we need to model his preferences by a ranking with ties. For each pair of candidates (a, b), we have “ais preferred tob”, the converse or “ais indifferent to b” (which is equivalent to “bis indifferent to a”). Preference still is transitive.
Suppose that a voter prefersatob,candd, he is indifferent betweenbandcand, finally, he prefersa,bandctod. We can model his preferences by a ranking with ties. A graphic representation of this model is given in Fig. 2.1 where an arrow between two candidates (e.g. a and b) means that a is preferred to b and a line between them means thatais indifferent to b. Note that, in a ranking with ties,
a
b c
d
Figure 2.1: A complete pre-order. Arrows implied by transitivity are not repre- sented
indifference also is transitive. If a voter is indifferent betweenaandband between bandc, he is also indifferent betweenaandc.
Incomparability: partial rankings
It can also occur that a voter is unable to rank the candidates, not because he thinks that some of them are equivalent but because he cannot compare some of them. There can be several reasons for this.
Poor information Suppose that a voter must compare two candidatesaand b about which he knows almost nothing, except that their names are aandb and that they are candidates. Such a voter cannot declare that he prefersa tobnor the converse. If he is forced to express his preferences by means of a ranking with ties, he will probably rankaandbtied rather than ranking one above the other. But this would not really reflect his preferences because he has no reasons to consider that they are equivalent. It is very likely that one is better than the other but, as he doesn’t know which one, he is better off not stating any preferences about them.
Conflicting information Suppose that a voter has to compare two candidatesa andbabout which he knows a lot. He might be embarrassed when asked to tell which candidate he prefers because, in some respects,ais far better than b but, in other respects,b is far better thana. And he does not know how to balance the pros and cons or he does not want to do so for the moment.
Confidential information Suppose that your mother invited you and your wife for dinner. At the end of the meal, your mother says “I have never eaten such a good pie! Does NameOfYourWife prepare it as well as I do ?” No matter what your preference is, you would probably be very embarrassed to answer. And your answer is very likely to be “Well, it is difficult to say.
In fact they are different. I like both but I cannot compare them.” Such situations are very common in real life where people do not tell the truth, all the truth and nothing but the truth about their preferences.
Of course, this list is not exhaustive. We therefore need to introduce a new model in which voters are allowed to express incomparabilities. Hence, when comparing two candidatesaandb, four situations can arise:
1. ais preferred tob,
2. b is preferred toa, 3. ais indifferent tobor 4. aandb are incomparable.
If we keep the transitivity of preference (and indifference), the structure we obtain is called a partial ranking.
Example 13. Transitivity and coffee: semiorders
Consider a voter who is indifferent between a andb as well as between b and c.
If we use a ranking with ties to model his preferences, he is necessarily indifferent betweenaandc, because of the transitivity of indifference. Is this what we want ? We are going to borrow a small example from Luce (1956) to show that transitivity of indifference should be dropped, at least in some cases. Let us suppose that I present two cups of coffee to a voter: one cup without sugar, the other one with one grain of sugar. Let us also suppose that he likes his coffee with sugar. If I ask him which cup he prefers, he will tell me that he is indifferent (because he is not able to detect one grain of sugar). He equally dislikes both. I will then present him a cup with one grain and another with two. He will still be indifferent. Next, two grains and three grains, and so on until nine hundred ninety nine and one thousand grains. The voter will always be indifferent between the two cups that I present to him because they differ by just one grain of sugar. Because of the transitivity of indifference, he must also be indifferent between a cup without sugar and a cup with one thousand grains (2 full spoons). But of course, if I ask him which one he prefers, he will choose the cup with one thousand grains. Thus transitivity of indifference is violated. A possible objection to this is that the voter will be tired before he reaches the cup with one thousand grains. Furthermore–this is more serious–the coffee will be cold and he hates that.
There is a structure that keeps transitivity of preference and drops it for in- difference. Consequently, it can model the preferences of our coffee drinker. It is called semiorder. For details about semiorders, see Pirlot and Vincke (1997).
Example 14. Transitivity and poneys: more semiorders
Do we need semiorders only when a voter cannot distinguish between two very similar objects ? The following example, adapted from (Armstrong 1939) will give the answer. Suppose that you ask your child to choose between two presents for his birthday: a poney and a blue bicycle. As he likes both of them equally, he will say he is indifferent. Suppose now that you present him a third candidate: a red bicycle with a small bell. He will probably tell you that he prefers the red one to the blue one. “So, you prefer the red bicycle to the poney, is that right ?” you would say if you consider a transitive indifference. However, it is obvious that the child can still be indifferent between the poney and the red bicycle.
poney
blue bike red bike
Figure 2.2: The poney vs bicycles semiorder Other binary relations
Rankings with or without ties, partial rankings and semiorders are all binary relations. Many other families of binary relations have been considered in the literature in order to formally model the preferences of individuals as faithfully as possible (e.g. Roubens and Vincke 1985, Abbas, Pirlot and Vincke 1996). Note that even the transitivity of strict preference can be questioned due to empirical observations (e.g. Fishburn 1988, Fishburn 1991, Tversky 1969, Sen 1997). Let us now focus on another kind of mathematical structure used to model the preferences of a voter.
2.2.2 Fuzzy relations
Fuzzy relations can be used to model preferences in at least two very different situations.
Fuzzy relations and uncertainty
When a voter is asked to express his preferences by means of a binary relation, he has to examine each pair and choose “ais preferred to b”, “b is preferred to a”,
“a is indifferent to b” or “aand b are incomparable” (if indifference and incom- parability are allowed). In fact, reality is more subtle. When facing a question like “do you preferatob”, a voter might hesitate. It is easy to imagine situations where a voter would like to say “perhaps”. And it is just a step further to imag- ine different situations where a voter would hesitate but with various degrees of confidence: almost yes but not completely sure, perhaps but more on the side of yes, perhaps, perhaps but more on the side of no, . . . There can be many reasons for his hesitations.
• He does not have full knowledge about the candidates. For example, in a legislative election, a voter does not necessarily know what the position of all candidates is regarding a particular issue.
• He does have full knowledge about the candidates but not about some events that might occur in the future and affect the way he compares the candi- dates. For example, again in a legislative election, a voter might ideally know everything about all candidates. But he does not know if, during the forth- coming mandate, the representatives will have to vote on a particular issue.
If such a vote is to occur, a voter might prefer candidate ato candidate b.
In the other case, he might preferbtoabecause there is just one thing that he disapproves of the policy ofb: his position about that particular issue.
• He does not fully know his preferences. Suppose that the community in which you live has decided to build a new recreational facility. There are two options: a tennis court or a playground. You have to vote. You perfectly know the two options (budget, time to completion, plan, . . . ). You like tennis and your children would love that playground. You will have access to both facilities under the same conditions. Can you tell which one you will choose ? What will you enjoy more ? To play tennis or to let your children play in the playground ?
These three cases can be seen as three facets of a single problem. The voter is uncertain about the final consequences of his choice.
Fuzzy relations can be used to model such preferences. The voter must still answer the above mentioned question (do you prefer a to b?), but by numbers, no longer by yes or no. If he feels that “ais preferred tob” is definitely true, he answers 1. If he feels that “ais preferred tob” is definitely false, he answers 0. For intermediate situations, he chooses intermediate numbers. For example, perhaps could be 0.5 and almost yes, 0.9. A typical fuzzy relation on three candidates is illustrated by Fig. 2.3 where a number on the arrow between two candidates (e.g.
aandb) is the answer of the voter to the question “isapreferred tob”.
a
c b
0.6 0.4
0.8 0.3
1.0 0.0
Figure 2.3: A fuzzy relation
Note that, in some cases, a probability distribution on the possible conse- quences is assumed to exist. In such cases, the problem faced by the voter is no longer uncertainty but risk. In these cases, probabilities of preference might be assigned to each pair.
Fuzzy relations and preference intensity
In some cases, when a voter is asked to tell if he prefers a to b, he will tend to express faint differences in his judgement, not because he is uncertain about his judgement, but because the concept of preference is vague and not well defined.
For example, a voter might say “I definitely prefer a to b but not as much as I preferc to d”. This is due to the fact that preference is not a clear-cut concept.
We might then model his preferences by a fuzzy relation and choose 0.5 for (a, b) and 0.8 for (c, d). A value of 0 would correspond to no preference.
Note that in many cases, uncertainty and vagueness are probably simultane- ously present. For a thorough review of fuzzy preference modelling, see (Perny and Roubens 1998).
2.2.3 Other models
Many other models can be conceived or have been described in the literature. An important one is the utilitarian one: a voter assigns to each candidate a number (the utility of the candidate). The position of a candidate with respect to any other candidate is a function only of the utilities of the two candidates. If the utilities of a and b are respectively are 50 and 40, the implication is that a is preferred tob. In addition, if the utilities ofc and dare respectively 30 and 10, it implies that the preference betweenc and dis twice as large as the preference betweenaandb.
Another important model is used in approval voting (Brams and Fishburn 1982). In this voting system, every voter votes for as many candidates as he wants or approves. Consequently, the preferences of a voter are modelled by a partition of the set of candidates into two subsets: a subset of approved candidates and a subset of disapproved candidates. Approval voting received a lot of attention during the last twenty years and has been adopted by a number of committees.
We will not continue our list of preference models any further. Our aim was just to give a small overview of the many problems that can arise when trying to model the preferences of a voter. But there is an important issue that we still must address. We encountered many problems in Section 2.1. In this section, we were using complete orders to model voters’ the preferences. We then examined alternative models. Is it easier to aggregate individual preferences modelled by means of complete pre-orders, semiorders, fuzzy relations, . . . ? Unfortunately, the answer is no. Many examples, similar to those in Section 1, can be built to demonstrate this (Sen 1986, Salles, Barrett and Pattanaik 1992).
2.3 The voting process
Until now, we considered only modelling the preferences of a voter and aggregating the preferences of several voters. But voting is much more than that. Here are a few points that are included in the voting process, even if they are often left aside in the literature.
2.3.1 Definition of the set of candidates
Who is going to define the candidates or alternatives that will be submitted to a vote ? All the voters, some of them or one of them ? In some cases, e.g. presidential elections, the candidates are voters that become candidates on a voluntary basis.
Nevertheless, there are often some rules: not everyone can be a candidate. Who should fix these rules and how ? There is an even more fundamental question:
who should decide that voting should occur, on what issue, according to which
rules ? All these questions received different answers in different countries and committees. This may indicate that they are far from trivial.
Let us now be more pragmatic. The board of directors of a company asks the executive committee to prepare a report on the future investment strategies. A vote on the proposed strategies will be held during the next board of directors meeting. How should the executive committee prepare its report ? Should they include all strategies, even infeasible ones ? If infeasible ones are to be avoided, who should decide that they are infeasible. To find all feasible strategies might be prohibitively resource and time consuming. And one can never be sure that all feasible strategies have been explored. There is no systematic way, no formal method to do that. Creativity and imagination are needed during this process.
Finally, suppose that the executive committee decides to explore only some strategies. A more or less arbitrary selection needs to be made. Even if they do make this selection in a perfectly honest way, it can have far reaching consequences on the outcome of the process. Remember example 11 in which we showed that, for some aggregation methods, the relative ranking of two candidates depends on the presence (or absence) of some other candidates. Furthermore, some studies show that an individual can prefer ato bor b toa depending on the presence or absence of some other candidate (Sen 1997).
2.3.2 Definition of the set of the voters
Who is going to vote ? As in the previous subsection, let us look at different democracies, past or present. Citizens, rich people, noble people, men, men and women, everyone, white men, experts who have some knowledge about the dis- cussed problem, one representative for each faction, a number of representatives proportional to the size of that faction, . . . There is no universal answer.
2.3.3 Choice of the aggregation method
Even the choice of the aggregation method can be considered as part of the voting process for, in some cases, the aggregation method is at least as important as the result of the vote. Consider two countries, A and B: A is ruled by a dictator, B is a democracy. Suppose that each time a policy is chosen by voting in B, the dictator ofA applies the same policy in his country, without voting. Hence, all governmental decisions are the same inA and B. The only difference is that the people in A do not vote; their benevolent dictator decides alone. In what country would you prefer to live ? I guess you would choose B, unless you are the dictator. And you would probably choose B even if the decisions taken in B were a little bit worse than the decisions taken in A. What we value in B is freedom of choice. Some references or more details on this topic can be found in (Sen 1997, Suzumura 1999).
2.4 Social choice and multiple criteria decision support
2.4.1 Analogies
There is an interesting analogy between voting and multiple criteria decision sup- port. Replace criteria byvoters, alternatives by candidates and you get it. Let us be more explicit. In multiple criteria decision support, most papers consider an entity, called decision-maker, that wants to choose an alternative from a set of available alternatives. The decision-maker is often assumed to be an individual, a person. To make his choice, the decision maker takes several viewpoints called criteria into account. These criteria are often conflicting, i.e. according to a cri- terion, a given alternative is the best one while, according to another criterion, other alternatives are better.
In a large part of the literature on voting, there is an entity called group or society that has to choose a candidate from a set of candidates. This entity consists of individuals and, for some reasons, that can vary largely in different groups, the choice made by this entity must reflect in some way the opinion of the individuals. And, of course, the individuals often have conflicting views about the candidates. In other words, the preferences of an individual play the same role, in social choice, as the preferences along a single viewpoint or criterion in multiple criteria decision support. The collective or social preferences, in social choice theory, and the global or multiple criteria preferences, in multiple criteria decision support, can be compared in the same way.
The main interest of this analogy lies in the fact that voting has been studied for a long time. The seminal works by Borda (1781), Condorcet (1785), and Arrow (1963) have led to an important stream of research in the 20th century. Hence we have a huge amount of results on voting at our disposal for use in multiple criteria decision support. Besides, this similarity has widely been used (see e.g. Arrow and Raynaud 1986, Vansnick 1986).
In this chapter, we only discussed elections in which only one candidate must be chosen (single-seat constituencies, prime ministers or presidents). However, it is often the case that several candidates must be chosen. For example, in Belgium and Germany, in each constituency, several representatives are elected so as to achieve a proportional representation. A committee that must select projects from a list often selects several ones, according to the available resources. In multiple criteria decision support, such cases are common. An investor usually invests in a portfolio of stocks. A human resources manager chooses amongst the candidates those that will form an efficient team, etc.
In fact, the comparison can be extended to the processes of voting and decision- making. In multiple criteria decision support, the decision process is much broader than just the extraction, by some aggregation method, of the best alternative from a performance tableau.
The very beginning of the process, the problem definition, is a crucial step.
When a decision maker enters a decision process, he has no clearly defined problem.
He just feels unsatisfied with his current situation. He then tries to structure his
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