Mostapha Diss
email: mostapha.diss@univ-fcomte.fr.

Eric Kamwa
email: eric.kamwa@univ-antilles.fr.

Simulations in Models of Preference Aggregation

Keywords: Social choice, voting, preference aggregation, simulations, models JEL Codes: C15, C18, D71, D72 Choix social, vote, agrégation des préférences, simulations, modèles JEL Codes: C15, C18, C63, D71, D72

Introduction

Over the past few decades, social choice theory has been one of the areas in economics that has seen a boom in work using models based on the behavior of individuals involved in collective decision-making. These models have helped in designing renowned and robust results in the eld of preference aggregation. The frameworks developed on the basis of these models made it possible, by theoretical and/or computer simulations, to validate or invalidate several analytical results established in the literature. The aim of this paper is to oer, to the uninitiated, a methodological presentation of these dierent models, as well as the associated techniques of theoretical calculations and simulations, and then to report on recent developments concerning new models and advances in calculation techniques and simulations.

Computer simulations emerged in social choice theory almost at the same time as in political science. From the late sixties onwards, more and more researchers turned to computer simulations to study the behavior of agents involved in collective decision-making processes.

This method of analysis diers from those that have traditionally prevailed. This new approach mainly relies on the use of statistical tools to model behaviours, as well as the use of empirical analyses on real databases (results of elections, referenda or public consultations, etc.) or data collected through opinion polls or surveys.

We could actually argue that a large literature using computer simulations in social choice theory is devoted to the evaluation of voting systems according to dierent normative criteria, in order to obtain a hierarchy of the voting rules under investigation. Basically, this literature can be divided into two main categories:

1. Research that aims to compare various voting rules on the basis of their ability to lead to some desirable voting outcomes. The selection of the Condorcet winner, when he/she exists, is one of these desirable voting outcomes. Indeed, there is a large literature in social choice theory devoted exclusively to the Condorcet eciency 1 of voting rules.

The question of the concordance between voting rules 2 has also been widely studied in this literature. Naturally, there are a variety of other considerations that go into this category.

2. Much of the research in social choice theory is concerned with whether a paradox can occur for a given voting rule or not. We dene a voting paradox as an undesirable outcome to which a ballot can lead under a given voting rule and which may be regarded as surprising or counterintuitive. Many voting paradoxes has been the focus of numerous investigations in the literature: Condorcet's paradox, Borda's paradoxes, referendum paradox, monotonicity paradoxes, susceptibility to strategic manipulation, etc. This list of examples is of course a partial one, but they all clearly demonstrate that the notion of the probability of voting paradoxes is a central one. 1 The Condorcet eciency of a voting rule is dened as the conditional probability that the procedure select the Condorcet winner, given that a Condorcet winner exists. 2 The analysis is of interest since it allows to know how the choice of the voting rule is susceptible to impact the determination of the winner.

All these questions and many others have been investigated via simulations. For a detailed survey of early research on the two items, the reader can refer to [START_REF] Felsenthal | Review of Paradoxes Aicting Procedures for Electing a Single Candidate[END_REF], [START_REF] Felsenthal | Voting Procedures for Electing a Single Candidate[END_REF], Gehrlein andLepelley (2011, 2017), [START_REF] Nurmi | Voting Paradoxes and How to Deal with Them[END_REF][START_REF] Nurmi | Comparing Voting Systems[END_REF], and [START_REF] Saari | Basic Geometry of Voting[END_REF], among others.

Throughout this paper, the only illustration considered is the likelihood of Condorcet's paradox which is one of the themes that has most mobilized researchers from both the traditional line and the simulations approach. This choice can be understood, given the historical and theoretical importance of this paradox in the social choice literature which has largely been dominated by studies that are associated with this paradox. According to this paradox, it is possible, by aggregating the preferences of a group of individuals who are asked to rank three propositions (let us say A, B, and C) in order of preference, that a majority of voters prefers A to B, another majority prefers B to C and another prefers C to A. In such a case, we get a majority cycle which is the main drawback of the voting rule suggested by [START_REF] Condorcet | Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix[END_REF] as an alternative to that proposed by [START_REF] De Borda | Mémoire sur les élections au scrutin[END_REF]. Indeed, at the end of the 18th century, Borda and Condorcet, both members of the Paris Royal Academy of Sciences, proposed alternative voting rules to the one that was in use in the academy. The Borda rule picks as the winner the candidate with the highest Borda's score.

3 Condorcet criticized this rule in that it allows the existence of a candidate that is preferred by more than half of the electorate to the Borda winner; he proposed a rule based on pairwise comparisons. According to this rule, a candidate should be declared the winner if he/she beats all the other candidates in pairwise majority; such a candidate is called the Condorcet winner. In the end, the members of the academy leaned in favor of the Borda rule to the detriment of the Condorcet rule.

Although the Borda-Condorcet debate concerned only two rules for collective decisionmaking, it helped lay the groundwork for what can be described as the quest for the best rule for collective decision: a quest that is built around the comparison of the merits of dierent voting rules against each other. Decision rules can therefore be compared either on the basis of normative properties which they meet or not (the axiomatic approach), or on the basis of the frequency with which they satisfy or fail a given criterion (the probabilistic approach). These two ways of proceeding dene the two principal approaches in the theoretical study of social choice theory. We present these two approaches in Section 2 where we show that they complement each other.

As mentioned before, the second approach experienced a boom in the late fties, which saw much work on the occurrence of Condorcet's paradox. One of the paths taken to evaluate the Condorcet paradox is empirical studies based on data collected during real decisionmaking processes, elections, surveys or polls. For an overview of the work that falls within this framework, the reader may refer to [START_REF] Chamberlin | Social choice observed: Five presidential elections of the American Psychological Association[END_REF], [START_REF] Dobra | An approach to empirical measures of voting paradoxes : an update and extension[END_REF], [START_REF] Dobra | An approach to empirical measures of voting paradoxes[END_REF], [START_REF] Kurrild-Klitgaard | An empirical example of the Condorcet paradox of voting in a large electorate[END_REF][START_REF] Kurrild-Klitgaard | Voting paradoxes under proportional representation: Evidence from eight Danish elections[END_REF], [START_REF] Niemi | The occurrence of the paradox of voting in university elections[END_REF], Regenwetter et al. (2002a,b), [START_REF] Riker | The paradox of voting and congressional rules for voting on amendments[END_REF][START_REF] Riker | Arrow's theorem and some examples of the paradox of voting[END_REF], [START_REF] Taylor | A glimpse of impossibility: Kenneth Arrow's impossibility theory and voting[END_REF] and [START_REF] Tideman | Collective decision and voting: Cycles[END_REF]. Notice that the results of these various empirical studies are summarized in [START_REF] Gehrlein | Condorcet's paradox[END_REF] and Gehrlein andLepelley (2011, 2017). However, empirical analyses are not always possible, because the data for 3 With m candidates, the Borda rule awards m -j points to a candidate each time he/she is ranked j-th in a voter's ranking. The total number of points received by a candidate denes his/her Borda's score.

such studies are rarely available, accessible or even reliable; this may limit the scope of the empirical approach. The solution to such a limit is the use of probabilistic models describing the behavior of individuals and then try to nd the theoretical probabilities of voting events according to these models.

The rst use of probabilistic models in social choice theory dates back to the paper by [START_REF] May | Probability of Certain Election Results[END_REF] who calculated the overall likelihood of the referendum paradox which is the one that occurred, for instance, when Donald Trump was elected in 2016.

4 It deserves to be mentioned that, some years later, [START_REF] Guilbaud | Les théories de l'intérêt général et le problème logique de l'aggrégation[END_REF] also published an important paper on the probability of the existence of a Condorcet winner when three options are in the contest. This practice spread in the late 1960s with the work of [START_REF] Campbell | A measure of the importance of cyclical majorities[END_REF], [START_REF] Garman | The paradox of voting: Probability calculations[END_REF] and [START_REF] Niemi | A mathematical solution for the probability of the paradox of voting[END_REF] on the probability of a cyclical majority.

For the state of the art on the Condorcet paradox using probabilistic models, the reader can refer to the books of [START_REF] Gehrlein | Condorcet's paradox[END_REF] and Gehrlein andLepelley (2011, 2017). The use of probabilistic models requires us to make a priori assumptions on the distribution of preferences in order to build a model describing the behavior of the individuals. We discuss the main assumptions and models in Section 3. The probabilistic model approach faces some criticisms, chiey: i) even with a fairly small number of candidates in the running, probabilistic models can very quickly become intractable; ii) the realism of the assumptions underlying most models is questionable; and iii) the results obtained depend strongly on the hypothesis underlying the model and can thus vary from one hypothesis to another. We give a detailed discussion of each of these limitations later in the paper.

As mentioned before, the use of simulations has emerged as a means of transcending the main limitations of the two traditional approaches of preference aggregation analysis. In a general sense, simulation is the systematization and formulation of a model for determining the main characteristics of a system, a transaction or a process. In the framework of the aggregation of preferences, simulations mean the construction of a model which tries to simulate to the best degree possible the behaviours (i.e., preferences) of the individuals (i.e., voters). The use of simulations in social choice theory is not as recent as one might think; it was a result of the pioneering work of [START_REF] Arrow | Social Choice and Individual Values[END_REF][START_REF] Arrow | Social Choice and Individual Values[END_REF] that the rst results based on simulations appeared (see for instance, [START_REF] Klahr | A computer simulation of the paradox of voting[END_REF][START_REF] Weisberg | Probability calculations for cyclical majorities in Congressional voting[END_REF][START_REF] Weisberg | A pairwise probability approach to the likelihood of the paradox of voting[END_REF], mainly around the probability of the Condorcet paradox, in which analytical calculations were no longer possible due to mathematical limits when the number of candidates or voters increases. We come back to these dierent studies later in the paper, where we take the opportunity to provide a brief history of simulations in social choice theory and to present the dierent approaches adopted as well as review their scope.

The rest of the paper is organized as follows: First of all, we must familiarize the reader with the object of social choice theory, namely the aggregation of preferences. Section 2 is therefore dedicated to this end. In Section 3, we present the main models or hypotheses on which the theoretical works are based. Section 4 is devoted to the methods of simulations that have been developped in contrast to that of the theoretical approach. Section 5 concludes.

Aggregation of individual preferences

Preference aggregation: a brief history

The aggregation of preferences is at the heart of social choice theory, the essential purpose of which is to study ways of coherently aggregating individual preferences into a collective choice or a collective ranking of candidates. Given a group of individuals, who have to choose between at least two options (alternatives or candidates), a collective decision procedure, also called an aggregation rule, associates with each state of nature a collective ranking of options or a subset of winners, ideally a singleton. This theory leans on fundamental microeconomic principles and aims to understand the decision-making of rational individuals as regards to economic phenomena or beyond. According to [START_REF] List | Social Choice Theory[END_REF], social choice theory is not a single theory but a cluster of models and results concerning the aggregation of individual inputs.

Indeed, it covers, by its vast eld of applications, a multitude of contexts where the problem is formally similar: a group of individuals (e.g., experts, judges, jury, voters, etc.) who face a set of options (e.g., resource allocations, economic projects, candidates in a competition or an election, etc.), must reach a collective decision based on the opinions and interests of the dierent members. In other words, the scope and stakes of this theory are, in fact, potentially far-reaching, and may interest, in addition to economics and politics, areas as diverse as management, psychology, computer science and philosophy. This theory is now a recognized branch of modern microeconomics.

Historically, since the seminal works of [START_REF] Arrow | Social Choice and Individual Values[END_REF][START_REF] Arrow | Social Choice and Individual Values[END_REF], the interest of economists in the question of collective choice has had its source in the new economy of well-being developed in the 1940s, thanks in particular to the works of [START_REF] Pigou | The Economics of Welfare[END_REF], [START_REF] Bergson | A reformulation of certain aspects of welfare economics[END_REF],

and [START_REF] Samuelson | Foundations of Economic Analysis[END_REF]. The traditional economy of well-being was for a long time dominated by an almost total adherence to the utilitarian approach of [START_REF] Bentham | An Introduction to the Principles of Morals and Legislation[END_REF], and settled down, with the work of [START_REF] Edgeworth | Mathematical psychics: An Essay on the Application of Mathematics to the Moral Sciences[END_REF], [START_REF] Marshall | Principles of Economics[END_REF][START_REF] Pigou | The Economics of Welfare[END_REF], in a very dierent framework from that of social choice theory. It was only from the late 1940s, with the work of [START_REF] Arrow | Social Choice and Individual Values[END_REF][START_REF] Arrow | Social Choice and Individual Values[END_REF], [START_REF] Black | Justication of the Borda count[END_REF][START_REF] Black | The Theory of Committees and Elections[END_REF][START_REF] Black | On the rationale of group decision making[END_REF], and [START_REF] May | A set of independent necessary and sucient conditions for simple majority decisions[END_REF][START_REF] May | Some mathematical remarks on the paradox of voting[END_REF], that the context of the collective decision has been conjoined with that of the welfare economy, leading to the birth of the social choice theory in its modern form.

More precisely, in utilitarian calculations, the preferences of individuals are represented by numerical utility functions dened on all social states, and judgments on social interest are obtained by maximizing the sum of individual utilities. This implies that one can measure satisfaction, or happiness, in the form of utilities and that these utilities are comparable between individuals. This cardinal approach was called into question in the 1930s and nally abandoned in favor of the new welfare economy, which banned any possibility of interpersonal comparison of individual utilities and which gave an important place to the criterion of Pareto eciency. This criterion tells us that one allocation is preferable to another if it allows an increase in the level of satisfaction of one or more individuals without reducing the utility of others. This principle proved to be insucient, however, since many possible allocations can be Pareto optima. Hence, an economic theory of collective choice has become indispensable. The notion of a collective aggregation function introduced by [START_REF] Arrow | Social Choice and Individual Values[END_REF][START_REF] Arrow | Social Choice and Individual Values[END_REF] ts into this framework insofar as it allows aggregating individual preferences into a collective preference, so that society can choose between the dierent optima in the sense of Pareto. [START_REF] Arrow | Social Choice and Individual Values[END_REF][START_REF] Arrow | Social Choice and Individual Values[END_REF] showed that inconsistencies related to collective choice are not a surprise as they aect a very wide class of aggregation procedures. The method adopted in Arrow's theorem consists of choosing a certain number of properties which it seems reasonable to impose on the aggregation procedures and then demonstrating that the satisfaction of these properties taken together leads to an inconsistency, in the sense that such an aggregation procedure does not exist. In other words, there is no aggregation procedure that veries all the desirable conditions taken into account.

5

2.2

The axiomatic approach and the probabilistic approach By accepting the idea of discarding or weakening some properties and adding others, Arrow's theorem has given rise to countless contributions using what we call the axiomatic approach, which is the more common approach in social choice theory. Moreover, the majority of the convincing results in this eld have been obtained in the form of impossibility theorems, which highlights the diculty of designing a method allowing a reasonable aggregation of the opinions expressed by individuals in a decision procedure. However, the most important limitation of the axiomatic approach is that it does not oer information on the frequency of situations where aggregation procedures violate the desirable properties taken into account.

The probabilistic approach has been developed in the framework of social choice theory in order to deal with this limit of the axiomatic approach. This approach starts by using models describing the behavior of individuals involved in the aggregation process and then to quantify the probability of occurrence of certain types of collective outcomes for a given aggregation rule under the assumption xed on the distribution of individual preferences. Most of the models that are widely used in the literature are based on two pioneer assumptions:

the Impartial Culture (IC) and the Impartial and Anonymous Culture (IAC) assumptions.

These models will formally be dened and discussed later. Most of the models used are special cases of the multinomial law, and one of their limitations lies in the fact that even for a limited number of alternatives and individuals, the multinomial law becomes dicult to manage except, in this case, by resort to numerical or computer simulations. Indeed, simulations have made it possible to validate or invalidate several results established in the literature. 5 The desirable conditions are among others: i)completeness and transitivity according to which the collective ranking is a complete and transitive binary relation; ii)the condition of independence of irrelevant alternatives requires that the relative rankings of two alternatives depends only on the relative rankings of these candidates in the voters' rankings; the Pareto principle according to which when all the voters have the same strict preference over a pair of candidates, the social ranking is the same as the voters' unanimous preference; and iv) the non-dictatorship imposes that a given individual cannot impose his choice to the whole society.

Individual preferences

Let N be the set of n ≥ 2 individuals who have to decide on the set A of m ≥ 3 alternatives. 6

In order to decide, each voter must make a judgment of his/her own on the candidates in the running. This judgment is part of the process of preference formation which in social choice theory is not the subject of a study. The preferences are then assumed to be exogenous.

In addition, it is generally assumed that each voter votes sincerely and acts according to his/her true preferences.

The binary relation

R over A is a subset of the cartesian product A × A. For a, b ∈ A, if (a, b) ∈ R,
we write aRb to say that a is at least good as b. ¬aRb is the negation of aRb. If we have aRb and ¬bRa, we will say that a is better or strictly preferred to b. In this case, we write aP b with P the asymmetric component of R. If we assume that voters only have strict preferences also called linear orders, 7 the preference prole π = (P 1 , P 2 , ..., P i , ..., P n ) gives all the linear orders of the n voters on A. The set of all the preference proles of size n on A is denoted by P(A) n . A voting situation ñ = (n 1 , n 2 , ..., n t , ..., n m! ) indicates the number of voters for each linear order such that m! t=1 n t = n.

For m = 3 and linear orders assumed, Table 1 describes the possible strict rankings on A = {a, b, c}. In this table, it is indicated that n 1 voters have the ranking abc; this means that they rank candidate a at the top followed by candidate b and candidate c is the least preferred. et al. (1995) pointed out that this is not without impact on the results obtained; they also 6 As mentioned before, in order to show the increasing interest for simulations in social choice theory, our illustrative example is the Condorcet's paradox which requires that there are at least three alternatives. However, notice that an entire component of the literature on probability calculations and simulations in social choice theory is ignored in this paper: the one that considers the two-alternative case. First, some voting paradoxes can occur with only two alternatives (e.g., the referendum paradox, see [START_REF] Miller | Election Inversions by the U.S. Electoral College[END_REF]; second (and most importantly) a large number of studies deal with the question of voting power, which can be measured as the probability of being pivotal for a voter, in a two-candidate (voting "yes" or "no") framework. The two most famous power indices, the Banzhaf index and the Shapley-Shubik index, are respectively based on IC and IAC. On this topic, see e.g., [START_REF] Stran | The Shapley-Shubik and Banzhaf power indices as probabilities[END_REF] and [START_REF] Breton | Correlation, partitioning and the probability of casting a decisive vote under the majority rule[END_REF].

7 A linear order is a binary relation that is transitive, complete and antisymmetric. The binary relation R on A is transitive if for a, b, c ∈ A, if aRb and bRc then aRc. R is antisymmetric if we have aRb and bRa, then a = b. R is complete if and only if for all a, b ∈ A, we have aRb or bRa.

emphasize that admitting the possibility of having weak orders rather than strict orders would make preferences more realistic and would greatly reduce the probability of certain events such as the Condorcet paradox. Analytically, [START_REF] Fishburn | The paradox of voting: Eects of individual indierence and intransitivity[END_REF] reached the same conclusion. In a recent book, as part of their work on Behavioral Social Choice, Regenwetter et al. (2002b) have developed the tools to overcome the tradition of a priori preferences. In addition to highlighting the dierent limits of the traditional approach of social choice theory, they have developed methodologies to (re)construct preference distributions from incomplete data and a statistical sampling and Bayesian inference framework for the theoretical and empirical analysis of preference aggregation in samples drawn from practically any distribution over any family of binary relations. We say more on this in Section 4.

We are now equipped to introduce the main theoretical models used in social choice theory when dealing with voting events.

3

Simulations based on theoretical models on agents' behavior

The main purpose of using theoretical assumptions to model the behavior of a group of individuals is to derive a representation of the probability of a given event. The starting point in the dierent models is to assume an a priori distribution or assumption under which the samples of the individual preferences are drawn. In this paper, we will only focus our attention on the most popular and widespread models.

The main theoretical models

The impartial culture model

The impartial culture (IC) model was introduced for the rst time in the social choice literature by [START_REF] Guilbaud | Les théories de l'intérêt général et le problème logique de l'aggrégation[END_REF], who was interested in calculating the probability of the Condorcet paradox. As mentioned before, this model is among the most used in the literature, as shown by the large body of work on calculating the probabilities of electoral events produced since the seventies.

Under this model, it is assumed that all voting proles have the same probability of appearing. This means that each individual randomly and independently chooses his/her preference on the basis of a uniform probability distribution across all linear (or weak) orders.

It follows that, with linear orders, where each of m! linear orders has the same probability

1 m!
of being chosen by an individual, and the probability of attaining a voting situation ñ is given by:

Prob(ñ) = n! m! j=1 n j ! (m!) -n (1)
It is worth noting that the IC model is only a special case of the multinomial law. [START_REF] David | The variance of spearman's rho in normal samples[END_REF], Gehrlein and Fishburn (1978a,b) and [START_REF] Plackett | A reduction formula for normal multivariate integrals[END_REF] showed that for an innite number of individuals, an application of the central limit theorem allows an approximation of the multinomial law by a multivariate normal law. Recall that a multivariate normal distribution is a vector of multiple normally distributed variables, such that any linear combination of the variables is also normally distributed.

8 The central limit theorem states that averages calculated from independent, identically distributed random variables have approximately normal distributions, regardless of the type of distribution from which the variables are sampled, provided it has nite variance. However, the use of the multinomial law, as well as its approximation, can quickly become intractable even with a relatively small number of individuals. The various probability calculation techniques suggested under the IC model (e.g., Gehrlein andFishburn, 1978a,b, Saari and[START_REF] Saari | The likelihood of dubious election outcomes[END_REF] have the disadvantage of leading to dierent formulas which are often not compact and are dicult to handle. [START_REF] Fishburn | The paradox of voting: Eects of individual indierence and intransitivity[END_REF] was the rst to introduce an extension of IC in order to take into account the possible indierence in the agents' preferences: the impartial weak ordering culture (IWOC). More recently, [START_REF] Diss | On the Condorcet eciency of approval voting and extended scoring rules for three alternatives[END_REF] have provided another extension of IC that allows the possibility for voters to have dichotomous preferences with complete indierence between two or more candidates: the extended impartial culture (EIC). Note that for selected values of the parameters in the EIC and IWOC models, the IC model is easily found. Also, we can easily nd the IWOC model from EIC. As it can be seen, these extensions are only renements of IC that tend to take into account the remarks of [START_REF] Bjurulf | A probabilistic analysis of voting blocs and the occurrence of the Paradox of Voting[END_REF] and [START_REF] Jones | Condorcet winners and the paradox of voting: Probability calculations for weak preference orders[END_REF] according to which admitting the possibility of having weak orders rather than strict orders would make preferences more realistic and would greatly reduce the probability of certain events such as the Condorcet paradox.

The dual culture model

The dual culture (DC) model rst introduced by [START_REF] Gehrlein | Condorcet winners in dual cultures[END_REF] operates as the IC model; it was dened only for strict rankings. Under DC, the probability that a given individual chooses his/her preference is the same as that of a voter with the inverted (dual) ranking.

Let us illustrate this with the preferences of Table 1. In this table, the rankings abc and cba, acb and bca, bac and cab are dual; so, the distribution is as follows:

P rob(abc

) = P rob(cba) = p 1 P rob(acb) = P rob(bca) = p 2 P rob(bac) = P rob(cab) = 1 2 -p 1 -p 2
One may notice in this case that we recover the IC model when p 1 = p 2 = 1 6 .

8 Recall that the normal distribution is the most common type of distribution assumed in probabilistic analyses. The standard normal distribution has two parameters (the mean and the standard deviation) and has the main following properties: i) The mean, mode and median are all equal. ii) The curve is symmetric at the center (i.e., around the mean). iii) Exactly half of the values are to the left of center and exactly half the values are to the right. iv) The total area under the curve is 1.

The impartial and anonymous culture model

The impartial and anonymous culture (IAC) model was introduced for the rst time in the literature of social choice theory by [START_REF] Gehrlein | Condorcet's paradox and anonymous preference proles[END_REF] and [START_REF] Kuga | Voter antagonism and the paradox of voting[END_REF]. Under this model it is assumed that all voting situations are equally likely to be observed, then the probability of a given event is calculated according to the ratio between the number of voting situations in which the event occurs and the total number of possible voting situations. The possibility of computing the probability of an event as a ratio is not specic to IAC: with IC, the probability is the ratio between the number of preference proles in which the event occurs and the total number of possible preference proles. Both models are based on a notion of equiprobability, but the elementary events are preference proles under IC and voting situations under IAC. Notice that the IAC model allows to obtain closed form representations and this is one of its main advantages compared to the IC model. The probability of getting a given voting situation ñ with n voters and m candidates is given as follows:

Prob(ñ) = n!(m! -1)! (n + m! -1)! (2)
Under the IAC model, the number of voting situations associated with a given event can also be reduced to the solutions of a nite system of linear constraints with rational coecients.

For instances, using the labels of Table 1, the number of voting situations associated with the Condorcet paradox (of the type a is majority preferred to b, b is majority preferred to c, and c is majority preferred to a) is reduced to the solutions of the following system: 9

           n 1 + n 2 + n 5 > n 3 + n 4 + n 6 (a is majority preferred to b) n 1 + n 3 + n 4 > n 2 + n 5 + n 6 (b is majority preferred to c) n 4 + n 5 + n 6 > n 1 + n 2 + n 3 (c is majority preferred to a) n i ≥ 0 for i = 1, 2, . . . , 6 6 i=1 n i = n (3)
Dierent techniques and algorithms for nding solutions for such systems have been proposed in the literature; the reader may refer to the works of [START_REF] Cervone | Which scoring rule maximizes Condorcet eciency under IAC?[END_REF], El [START_REF] Ouafdi | IAC-probability calculations in voting theory: Progress report[END_REF], [START_REF] Gehrlein | Voting paradoxes and group coherence[END_REF], [START_REF] Lepelley | On Ehrhart polynomials and probability calculations in voting theory[END_REF] and [START_REF] Wilson | Probability calculations under the IAC hypothesis[END_REF].

The maximal culture model

The maximal culture (MC) model is due to [START_REF] Fishburn | An Analysis of Voting Procedures with Nonranked Voting[END_REF]. MC is quite similar to IAC with the exception that there is no need to x the number of voters in the random voting situation. It xes an integer L (L > 0) and each ranking is drawn from a uniformly random distribution over [0, L]. According to this, for three-candidate elections, the number of possible equally likely voting situations is equal to (L + 1) 6 and the expected number of voters in a voting situation is 3L. 9 The second possible cycle is dened in the same way using the symmetry of the IAC assumption with respect to candidates.

The urn model: The Pólya-Eggenberger model According to Berg (1985a,b) and [START_REF] Berg | A note on the paradox of voting : anonymous preference proles and May's formula[END_REF], the IC and IAC models are in fact only two special cases of the more general Pólya-Eggenberger model; this is not the case for the MC model. This model was introduced into the social choice literature by Berg (1985a). 10 In this model, everything happens as if from an urn containing B balls including B j balls of color j (j = 1, 2, . . . , m!), where each individual involved in the collective decision process chooses his/her preference by means of a random draw of a ball in the urn, and at each draw, α balls of the same color as the one drawn by the individual are added into the urn. The quantities B and α are assumed to be positive real numbers. So, the probability of getting a given voting situation ñ under the Pólya-Eggenberger model is:

Prob(ñ) = n! B [n,α] m! j=1 B [n,α] j n j ! (4)
where B [n,α] = n-1 i=0 (B + iα) is a generalized ascending factorial and the B j are positive numbers associated with each order such that B = m! j=1 B j .

According to [START_REF] Berg | A note on the paradox of voting : anonymous preference proles and May's formula[END_REF], α is a parameter measuring the level of social cohesion: the larger it is, the more the preferences of the individuals tend to be homogeneous. [START_REF] Berg | A note on the paradox of voting : anonymous preference proles and May's formula[END_REF] showed that if we x B j = 1 for all j = 1, 2, . . . , m!, the Pólya-Eggenberger model leads to the IC model for α = 0 and to the IAC model for α = 1.

Compared to the IC and IAC models, the Pólya-Eggenberger model therefore has the advantage of taking into account all possible degrees of interdependence in the preferences that individuals adopt.

Table 2 reports the limit probability, i.e., when the number of voters tends to innity, of the Condorcet paradox in three-candidate elections obtained under each of the above theoretical models. All these results are drawn from Gehrlein andLepelley (2011, 2017).

-3 2π arcsin( 1 3 ) ≈ 0.0877 DC 1 4 -1 2π arcsin(1 -4p 1 ) + arcsin(1 -4p 2 ) + arcsin(4p 1 + 4p 2 -1) IAC 1 16 MC 11 120 + 99L 4 +341L 3 +474L 2 +305L+109 120(L+1) 5
The dierent models that we have just presented have governed most of the theoretical analyses and have been used to develop the probability representations of electoral events in 10 The reader may refer to [START_REF] Johnson | Urns Models and their Application[END_REF] for an overview of this model.

particular for three-candidate elections. Despite the fact that for the same event, the models can lead to dierent probabilities, [START_REF] Gehrlein | Probability calculations in voting theory: An overview of recent results[END_REF] put forward a certain number of arguments to justify the use of such models. Let us summarize these arguments:

• It can be useful to nd out if the relative probabilities of paradoxical outcomes on various voting mechanisms behave in a consistent fashion over a number of dierent assumptions about the likelihood that voting situations or voter preference proles are observed.

• With real elections, large amounts of empirical data are not available; the use of theoretical models is thus found to be very useful.

• Despite the fact that they are generally believed to represent situations that exaggerate the probability that paradoxical events will occur, the theoretical models can show that some paradoxical events are very unlikely to be observed in reality.

• Theoretical models can show the relative impact that paradoxical events can have on dierent types of voting situations.

• By using probability models to obtain closed form representations, it is easy to observe the impact of varying dierent parameters (e.g., parameters of specic measures of social homogeneity or group coherence) of voting situations or voter preference proles;

this is somewhat more dicult to do with simulation studies.

• The obtained probability representations are directly reproducible and veriable with mathematical analysis, which is not as simple to do with simulation analysis.

The last two arguments express the main advantages of theoretical models on approaches based on simulations. However, we must also admit that very early on (and continue to do so even today, despite the increase in computer processing power) the analytical approach has shown its limits when trying to explore situations with more than three candidates.

Thus, alongside the analytical approach, many works have been developed on the basis of simulations based on the a priori theoretical hypotheses that we have just presented.

Theoretical-based simulations

Initially, the studies of voting situations involved only two or three candidates and were limited to a nite or a very small number of agents. This is due to the fact that the analytical calculations, which were done by hand, rapidly became complicated and indeed unmanageable or untractable. One way to overcome the constraints and limits of the analytical approach would be to operate on real data; however, these are dicult to access, or rarely available. Even if they are available, the reliability of expressed preferences may be questioned, and there is no guarantee that the voters interviewed will all be able to really express their preferences when the number of candidates is large. To circumvent this obstacle, several authors quickly opted for the assistance of computer science through simulations. Over time, simulations have come to be no longer conned to the subelds of economics, and are spreading to almost all social sciences (see for instance, [START_REF] Axelrod | Advancing the art of simulation in the social sciences[END_REF][START_REF] Fontana | Simulation in economics: evidence on diusion and communication[END_REF]. The principle of simulation, in the common sense of the term, is to use a model, that is to say an abstract representation of a system or a problem, and to study the evolution of this model without operating the actual system.

In their rst usage in social choice theory, the applications of the simulations focused for the most part on the evaluation of the probability of the Condorcet paradox. Among these applications, without being exhaustive, are the works of [START_REF] Campbell | A measure of the importance of cyclical majorities[END_REF], [START_REF] Demeyer | The probability of a cyclical majority[END_REF], [START_REF] Gehrlein | Condorcet's paradox and anonymous preference proles[END_REF], [START_REF] Klahr | A computer simulation of the paradox of voting[END_REF] and Weisberg andNiemi (1978, 1973); most of these works involve only strict orders for agent preferences.

According to [START_REF] Jones | Condorcet winners and the paradox of voting: Probability calculations for weak preference orders[END_REF] this could be justied by the performance of computers at that time. Taking advantage of the computer advances of the 90s, [START_REF] Jones | Condorcet winners and the paradox of voting: Probability calculations for weak preference orders[END_REF] conducted an analysis of the simulated probability of the Condorcet paradox when weak preferences are allowed.

The simulations are made assuming a certain distribution a priori on the preferences of individuals, i.e., recourse to one of the theoretical models presented earlier. Once the distribution is chosen, the preferences are generated using the Monte Carlo simulation method, which is a method of estimating a numerical quantity that uses random numbers. Note that this method was introduced by Von Neumann and Ulam ( 1945), referring to games of chance in casinos, during the Manhattan project.

11 This method has the advantage of being easy to use. It is now applied to a very wide range of problems. Let us briey present the methodology of Monte Carlo simulations under IC and IAC models as they are carried out as part of the aggregation of preferences for generating samples of preferences. For our presentation, we will focus on cases where only strict preferences are allowed and we will use the notation of Section 2.3.

Simulations under IC:

The goal is to generate, equiprobably, a prole of total orders with n voters and m candidates. So each of the m! possible total orders is chosen equiprobably; to choose a total order is therefore equivalent to choose an integer between 1 and m!. An integer is chosen over this interval for each of the voters, one after the other and independently.

The chance of occurrence of each prole in this process is actually 1 m! n . At the end of the process, which is anonymous, we count the number of voters assigned to each strict order; we then obtain a m!-uple of integers whose sum is equal to n. Concretely, the routine used is the following:

• We start from the prole (0, 0, . . . , 0) which is a null vector with m! components.

• From step 1 (voter 1) to step n (voter n), an integer of 1 to m! is equiprobably selected. If the result is j, add 1 to the component j of the prole. • At the n th stage, the prole is indeed a m!-uple of integers whose sum is equal to n. 11 Project Manhattan is the code name for the research project that produced the rst atomic bomb during the Second World War.

To generate a sample of size T (number of repetitions), we run the previous routine T times while keeping the result; this gives a T -tuple of proles with total orders.

Simulations under IAC:

Given the m! possible strict orders, the objective is to generate the voting situations (anonymous proles) equiprobably. As a reminder, a voting situation is an m!-uple (n 1 , n 2 , . . . , n m! ) of natural numbers whose sum is equal to n for which it will be necessary to randomly generate each of the components. To do this, m! -1 numbers are generated equiprobably in [0; 1] which we rank in increasing order, say x 1 , x 2 , . . . , x m!-1 ; this series will be completed by 0 the smallest possible value and 1 the largest possible value such that 0 = x 0 , x 1 , x 2 , . . . , x m!-1 , x m! = 1. Then, the value n(x j -x j-1 ) is assigned to n j (j = 1, 2, . . . , m!) that is to say n(x 1 -x 0 ) is assigned to n 1 , n(x 2 -x 1 ) to n 2 and so on until n(x m! -x m!-1 ) which is assigned to n m! . Note that the values obtained may not be integers; they are then rounded to the lower unit. After rounding, if the sum of the numbers thus assigned is less than n, the dierence observed will be added randomly and equiprobably to one of the components. By this process, voting situations have the same chances of being observed. To generate a sample of size T , we run the previous routine T times.

It is worth nothing that [START_REF] Feix | L'espace des phases électoral et les statistiques quantiques[END_REF] showed that there is a connection between the probability models (IC and IAC) and probabilistic models or distributions that are widely used in physics, particularly in quantum mechanics and statistical physics: the IC model is linked to Maxwell-Boltzmann distribution 12 and the IAC model is related to the Bose- Einstein statistic; 13 quantum statistics would thus be another gateway for calculating the probabilities of voting events. [START_REF] Feix | L'espace des phases électoral et les statistiques quantiques[END_REF] complete their analysis by simulating the probabilities of existence of the Condorcet winner under IC and IAC with a number of candidates ranging from 3 to 8 and electorates of innite size. Their calculations show a certain convergence between IC and IAC when the number of candidates increases. Table 3 reports likelihood of the Condorcet paradox when preferences are simulated according to the IC and IAC models for voting situations with three to eight candidates.

14 The values in this table are derived from those of Table 4 and 5 by [START_REF] Feix | L'espace des phases électoral et les statistiques quantiques[END_REF].

12 The Maxwell-Boltzmann statistic is a probability law or distribution used in statistical physics (thermal equilibrium) to determine the distribution of particles between dierent energy states.

13 It describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic equilibrium.

14 These gures are consistent with those obtained under the IC model by [START_REF] Gehrlein | The Condorcet criterion and committee selection[END_REF], [START_REF] Klahr | A computer simulation of the paradox of voting[END_REF], [START_REF] Niemi | A mathematical solution for the probability of the paradox of voting[END_REF] and [START_REF] Weisberg | Probability calculations for cyclical majorities in Congressional voting[END_REF]. Other simulation results on the likelihood of the Condorcet paradox with a given number of candidates and a given number of voters, are available in the literature not only the IC and IAC models but also for many other assumptions; for an overview, the reader may refer to the papers of [START_REF] Fishburn | Majority eciencies for simple voting procedures: Summary and interpretation[END_REF], [START_REF] Gehrlein | Condorcet's paradox and the Condorcet eciency of voting rules[END_REF], Jones et al. (1995), Pomeranz and[START_REF] Pomeranz | The cyclical majority problem[END_REF] and [START_REF] Weisberg | Probability calculations for cyclical majorities in Congressional voting[END_REF]. It comes from all these results that the probability of the Condorcet paradox tends to increase with the number of candidates and the number of voters. The literature is now full of numerous probabilities of various electoral events, obtained by simulations of the IC and IAC models. See for instance, the works of It should be noted that in the days of the rst simulation work in the theory of social choice, computer workstations were almost non-existent or at least expensive; access to mainframes was even more so. Thus, the simulations, which for the most part were conned to the probability of the Condorcet paradox, were limited to voting situations with three candidates and a very small number of voters. After a certain number of voters, the calculations were time consuming and the results were based on samples generated from a low number of repeats; this therefore casts doubt on the accuracy of the results. With the development of mathematical, statistical and computer techniques, over time, many (more or less complex)

programming languages have been developed, as well as software and toolkits that meet the particular needs of the simulation, particularly for generating samples of preferences.

Today, easily accessible Microsoft Excel spreadsheets oer many possibilities for simulations using simple macros and VBA language. We can also turn to more advanced tools such 15 Among others, Rousseauist cultures, impartial culture, distributives cultures and spatial Euclidian cultures. Rousseauist cultures are adapted from Rousseau's ideal of a general will. Distributive cultures describe societies of complete antagonism with a context comparable to that which governs the problems where a unit of a divisible good has to be shared between individuals. Spatial Euclidian cultures are consistent with what we present in Section 4.1.

as Maple, MATLAB or Mathematica based on sharp programming languages more or less comprehensible only for insiders. This development of methods and techniques today makes available to researchers turnkey kits to eectively conduct their simulations that today can be done on personal computers or on dedicated servers, or even on supercomputers [START_REF] Macal | Tutorial on agent-based modelling and simulation[END_REF]. The saving of time is remarkable and the accuracy of the obtained results is indisputable. The advances made in current computer simulation techniques have made it possible to correct or rene several theoretical results obtained in the past.

Like the analytical approach based on theoretical models, the theoretical simulations approach is strongly criticized. According to [START_REF] Tideman | Developing the aggregate empirical side of computational social choice[END_REF], under the theoretical-based models, the analysis and consequently the results assume frequency distributions chosen just because of the convenient mathematical properties, while these distributions are far from reecting what is happening in real elections. In fact, there is no evidence that voters' choices obey any probabilistic distribution, let alone a uniform distribution. No work has ever supported or even established that the above theoretical models reect the reality in a particular situation. On the basis of their criticisms of theoretical models, several authors have argued for simulations based on more realistic distributions and assumptions.

4

Other approaches of agent-based modeling and simulation models

Besides the models just discussed, two main other approaches emerge: spatial voting models and models inspired by psychology. The modeling under these approaches seeks not to assume a certain behavior of voters but to determine a distribution of preferences that is closest to reality. These approaches have the common feature of analyzing and generating preferences so as to reect or to come close to real elections' data samples.

4.1

The spatial voting models Spatial voting models were rst applied specically to elections by [START_REF] Downs | An Economic Theory of Political Action in a Democracy[END_REF] to study the relative positioning of political parties and voters using a spatial approach built on the pioneering work of [START_REF] Black | On the rationale of group decision making[END_REF], [START_REF] Hotelling | Stability in Competition[END_REF], [START_REF] Lerner | Some Notes on Duopoly and Spatial Competition[END_REF], [START_REF] Smithies | Optimum location in spatial competition[END_REF] and [START_REF] Greenhut | Plant Location in Theory and Practice, The Economics of Space[END_REF], who addressed the problem of location between two competing rms in order to optimally choose their setting in a market of undierentiated goods. Under a spatial model, it is assumed that both candidates and voters are placed in a unidimensional or multidimensional space according to the position they take or prefer on certain issues, each of which corresponds to a dimension. In such a setting, a voter tends to choose the candidate who is closest to his/her position while a candidate will tend to choose a position that maximizes the number of electoral votes.

Let us notice that the most basic spatial model inspired by [START_REF] Downs | An Economic Theory of Political Action in a Democracy[END_REF], involves an election based on a single dimension under which candidates can be ordered on a left-right axis, such that for each voter, his/her utility is decreasing with the distance to his/her preferred alternatives along this axis. Given his/her location (i.e., ideal or bliss point) and knowing the locations of the candidates on the spectrum, each voter casts a vote for the candidate who is closest to his/her location. The locations of the voters along the line follow a specic distribution and the Euclidean distance serves as a tool for measuring the electorcandidate proximity. For a given voter i and party or candidate j, if we denote by v i the voter's position and by p ij the party's position as perceived by voter i, the Downsian utility (U ij ) of voter i is given by:

U ij = -α × (v i -p ij ) 2 (5)
In Eq. 5, the overall policy importance is captured through the parameter α.

Besides the Downsian-inspired model, we note the existence of the so-called directional models. Under the directional model developed by [START_REF] Rabinowitz | A Directional Theory of Issue Voting[END_REF], 16 it is assumed that utilities are determined by both the intensity and communality of direction of voters' and candidates' positions. So, voters have a diuse preference for certain direction on an issue but vary in the intensity with which they hold that preference. Under this model, the voter's utility is a product of the policy positions of the voter i and the party j:

U ij = α × v i × p ij (6)
We owe to [START_REF] Rabinowitz | A Directional Theory of Issue Voting[END_REF] the introduction of a mixed model that combines the directional and the Downsian logic: a voter's choice is determined both by a proxy of proximity and by a directional component. A voter's utility under this model is dened as follows:

U ij = α × -β(v i -p ij ) 2 + (1 -β)v i p ij (7)
where β ∈ [0, 1] is a relative weight of the two components of voter utility. As one can see, when parameter β is equal to 1, we get the Downsian model; when β = 0, we get the directional model. More recently, [START_REF] Kedar | When Moderate Voters Prefer Extreme Parties: Policy Balancing in Parliamentary Elections[END_REF] introduced a model combining the Downsian approach with a compensatory component which captures the outcome orientation of the voters. According to [START_REF] Kedar | When Moderate Voters Prefer Extreme Parties: Policy Balancing in Parliamentary Elections[END_REF], when a voter is outcome-oriented, it is assumed that he/she compares the expected policy outcome P if all parties are elected and a counterfactual policy outcome P -p j where one party p j is excluded from the policy process P ; then, he/she will choose the party where the distance between the two scenarios is greatest, providing the party shifts the expected policy outcome in the desired direction. Under the compensational model, a voter's utility is dened as follows:

U ij = α -β(v i -p ij ) 2 -(1 -β)[(v i -P ) 2 -(v i -P -p j ) 2 ] + δ j z i (8)
where, given p j the position of party and s j ∈ [0, 1] the relative impact of party j such that j s j = 1; P = j s j p j ; δ j is a vector indicating the eect of background variables z i on voter utility for party j.

Each of the above models has been described in the one-dimensional framework; they are easily extensible and adaptable to the multidimensional framework. However, in the 16 Please refer to [START_REF] Merrill | A unied theory of Voting: Directional and Proximity Spatial Models[END_REF] for an overview of all the so-called directional models. multidimensional framework, resorting to Euclidean distances requires assumptions about agent preferences. These hypotheses are, for the most part, improbable and empirically unrealistic; worse, they can complicate the theoretical analyses [START_REF] Enelow | Advances in the Spatial Theory of Voting[END_REF].

Hence the use of models based on simulations. So, utilizing data from real elections, many variations of spatial models have been used to test theoretical results and also to show how institutional context aects voter behavior. See for instance, the work of [START_REF] Merrill | A comparison of eciency of multicandidate electoral systems[END_REF], [START_REF] Plassmann | How to predict the frequency of voting events in actual elections[END_REF], [START_REF] Tideman | Collective decision and voting: Cycles[END_REF] and [START_REF] Tideman | Developing the aggregate empirical side of computational social choice[END_REF].

4.2

The behavioral social choice approach

Popularized by [START_REF] Regenwetter | Behavioral social choice[END_REF], behavioral social choice is the counterpart of the traditional social choice theory which integrates into the analysis some realistic psychological factors (limited rationality, cognitive biases, etc.) that can inuence individuals's choices in a real world. A behavioral approach usually tries to confront what should be (the normative aspect) with what is (the empirical aspect). So, behavioral social choice compares how supposedly rational individuals should make their decisions with how real decision makers behave empirically. It provides a framework for crafting more realistic models of social choice by embedding social choice analysis into a psychological representation of preferences and choice behavior, alongside a statistical evaluation of these models against empirical data (see for instance, [START_REF] Regenwetter | Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering[END_REF], Regenwetter et al., 2002a,b, Tsetlin and Regenwetter, 2003); it also develops methodologies to (re)construct preference distribution from incomplete data [START_REF] Regenwetter | Behavioral social choice[END_REF].

Behavioral social choice challenges the analyses carried out in social choice theory based on a priori assumptions on the distribution of agents' preferences. The group of authors behind behavioral social choice support the idea that the results obtained from the theoretical models are highly dependent on the a priori assumptions considered in generating elections scenarios; these hypotheses, by restricting the behavior of individuals to probabilistic distributions (normal law), are themselves very far from reecting the behavior of individuals in the real world. Thus, the results of the theoretical models based on a priori assumptions tend to promote views that are too pessimistic regarding the probability of many voting events such as the Condorcet paradox. According to [START_REF] Popova | A behavioral perspective on social choice[END_REF], these results may magnify gloomy predictions found in the axiomatic literature on the inability of an electorate to make a group decision. [START_REF] Efron | Bootstrap Methods: Another Look at the Jackknife[END_REF].

Generally speaking, the main idea behind a bootstrap is to make, on sample data, inferences about an estimate of sample statistics (sample mean, standard deviation, etc.) for a population statistical parameters (its mean, its standard deviation, etc.). Concretely, going from a data sample of size N with complete or incomplete voters' preference from real elections, it proceeds by carrying out sampling with replacement: a sample of size N is independently drawn from the original sample with replacement and replicated T times. For each of the T bootstrap samples, the estimates of the population parameters are evaluated;

then a sampling distribution is built with all these estimates and used for the statistical inference. Ideally, it would be nice if T were large enough to ensure meaningful statistics;

this is generally possible when using Monte Carlo simulations on fairly powerful computers by generating random samples.

In a behavioral social choice context, a bootstrap appears as a computer-based method for statistical inference that without relying on too many assumptions, is a way of simulating possible sources of uncertainty 17 in the results; it assesses how the results would be aected by small disturbances in the distribution of votes (preferences) and helps infer condence levels about at which point estimates of model parameters would not be aected by such disturbances. Using this inference approach of preference aggregation, [START_REF] Regenwetter | Behavioral social choice[END_REF] and related papers have established the robustness of the empirical absence of majority cycles for a wide range of realistic modeling assumptions; they also came to the conclusion that the theoretical assumption quite often used in the literature give a pessimistic view, assigning high probabilities to the existence of electoral paradoxes, and indeed considering them as virtually certain when in fact in the real world this is not the case.

Concluding remarks

Over time, simulation models have emerged as an indispensable tool in many disciplines and elds of study. They oer a way to overcome the limits or constraints of theoretical modeling. In social choice theory as well as in political science or psychology, simulations quickly found their place as a way of dealing with the complexity of the topic and the challenges of the modeling of human behavior in a decision-making framework. They appear as a springboard allowing us to complete the analyses carried out in theoretical approach, or at least to question them. The models developed in theoretical work have shown some limits when it comes to modeling the behavior of individuals involved in a process of collective decision: models can become intractable; and indeed, given certain parameter values (number of agents, number of alternatives, etc.) some analyses are almost impossible. Moreover, the results obtained depend strongly on the assumptions on the behavior of the agents that support the models. It is also true that these assumptions are deemed relevant to a universe that is actually very far from reality.

Since social choice theory has been one of the areas in economics that has seen a boom in work using models based on the behavior of individuals involved in collective decisionmaking, the purpose of this paper has been to oer to the uninitiated in the social choice theory, a methodological presentation of some well-known models and the techniques of theoretical calculations and simulations, and then to report on recent developments of new models and advances in calculation techniques and simulations.

After briey presenting the general framework of the aggregation of preferences, we presented the most widespread theoretical models and their extensions, and then discussed their strengths and weaknesses. We have particularly emphasized the two models that are most prevalent in the literature: the model of impartial culture (IC) and that of impartial and anonymous culture (IAC). The model IC, introduced by [START_REF] Guilbaud | Les théories de l'intérêt général et le problème logique de l'aggrégation[END_REF], is based on the idea that all preference proles are equiprobable and that each individual chooses his/her preference in a uniform probability distribution. For instance, when the individual preferences are expressed as linear orders on a set of alternatives, the IC assumption indicates that the preference relation of each voter is drawn uniformly at random from the set of all possible linear orders. On the other hand, the IAC model, introduced by Gehrlein and [START_REF] Gehrlein | Condorcet's paradox and anonymous preference proles[END_REF] and [START_REF] Kuga | Voter antagonism and the paradox of voting[END_REF], assumes the equiprobability of voting situations. Most theoretical results are based on these two models. Since these models are special cases of the multinomial law, one of their limitations lies in the fact that even for a limited number of alternatives and individuals, the multinomial law becomes dicult to manage.

Indeed, we have shown how simulation models (notably with the Monte Carlo method) developed under these models may be helpful in analyzing complex problems in social choice theory; they have made it possible to validate or invalidate several results established in the literature. In short, the simulations implemented under these assumptions have helped to produce well-known and robust results in the eld of preference aggregation.

The theoretical modeling has been strongly criticized for being based on distributions that do not reect what happens in real elections; in fact, there is no evidence that voters' choices obey any probabilistic distribution, and no work has ever supported or even established that the theoretical models reect the reality in a particular situation. These criticisms gave rise to the emergence of modelling that is not built on a priori assumptions on the preferences of agents. In this paper, we have presented two approaches that fall within this framework: spatial voting models and behavioral social choice. Under spatial voting models, inspired by [START_REF] Downs | An Economic Theory of Political Action in a Democracy[END_REF], it is assumed that both candidates and voters are placed in a unidimensional or multidimensional space according to the position they take or prefer on certain issues, each of which corresponds to a dimension. In such a setting, a voter tends to choose the candidate who is closest to his/her position, while a candidate will tend to choose a position that maximizes the number of electoral votes. According to [START_REF] Merrill | A comparison of eciency of multicandidate electoral systems[END_REF] Remarkable advances in computer science and mathematical and statistical calculation techniques are giving more and more prominence to simulations. This suggests that new opportunities are opening to theorists to rene the results found in the literature, but also to revisit certain problems whose resolution was previously impossible.

  [START_REF] Aleskerov | On the manipulability of voting rules: The case of 4 and 5 alternatives[END_REF],[START_REF] Brandt | Analyzing the Practical Relevance of Voting Paradoxes via Ehrhart Theory, Computer Simulations, and Empirical Data[END_REF],[START_REF] Diss | Multi-winner scoring election methods: Condorcet consistency and paradoxes[END_REF],[START_REF] Kelly | Almost all social choice rules are highly manipulable, but a few aren't[END_REF],[START_REF] Lepelley | Computer simulations of voting systems[END_REF] and others. Notice that based on a number of probabilistic models, 15 Laslier (2010) simulated the frequency of the existence of a Condorcet winner, for several prole sizes and also the likelihood of the election of the Condorcet winner, when he/she exists, for several voting rules. It comes from the simulation results that the way we should judge voting rules depends also on the context (political election, aggregation of judgments, jury, etc.) and the right model could depend on the type of collective decision problem under consideration.

  [START_REF] Plassmann | How to predict the frequency of voting events in actual elections[END_REF] and[START_REF] Tideman | Developing the aggregate empirical side of computational social choice[END_REF], when generating candidates and voters by means of simulations based on a spatial model, outcomes come astonishingly close to describing the distribution of actual outcomes, and ranking data simulated with the spatial model are very similar to observed ranking data. The spatial-model results thus tend to be more realistic. Behavioral social choice, popularized by the book of[START_REF] Regenwetter | Behavioral social choice[END_REF], provides a framework for crafting more realistic models of social choice by embedding social choice analysis into a psychological representation of preferences and choice behavior, and a statistical evaluation of these models against empirical data; it also develops methodologies to (re)construct preference distributions from incomplete data. Contrary to the theoretical models, these two approaches describe a modeling in which one confronts what must be with what is, the goal being to get as close as possible to what happens in real situations of collective decision. Practice has shown that the developed models perform well in this task.

Table 1 :

 1 Possible strict rankings on A = {a, b, c} n

1 : abc n 2 : acb n 3 : bac n 4 : bca n 5 : cab n 6 : cba It should be noted that, in most social choice literature, it is explicitly admitted that the preferences of individuals are linear orders. This implies that agents cannot be indierent between two or more alternatives. Using Monte Carlo simulations, Bjurulf (1972) and Jones

Table 2 :

 2 Limiting probability of the Condorcet paradox in three-candidate elections

	Model	Limiting probability of the Condorcet paradox
	IC	1 4

Table 3 :

 3 Limiting probabilities of the Condorcet paradox obtained by simulations under IC

	and IAC
			Number of candidates
	Models	3	4	5	6	7	8
	IC	0.0877 0.1756 0.2513 0.3152 0.3694 0.4140
	IAC	0.0624 0.1616 0.2477 0.3143 0.3691 0.4170

More precisely, Donald Trump won enough states to secure the majority in the Electoral College, while Hillary Clinton received 2.87 million more votes than Trump.

According toRegenwetter et al. (2002b[START_REF] Regenwetter | Behavioral social choice[END_REF] uncertainty can come from various factors, such as voters' uncertainty about their preference, unreliability of voter turnout, counting of ballots, etc.