1Over 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 field 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 offer, to the uninitiated, a methodological presentation of these different 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.
2Computer 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 differs 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.
3We could actually argue that a large literature using computer simulations in social choice theory is devoted to the evaluation of voting systems according to different normative criteria, in order to obtain a hierarchy of the voting rules under investigation. Basically, this literature can be divided into two main categories:
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- 1 The Condorcet efficiency of a voting rule is defined as the conditional probability that the proced (...)
- 2 The analysis is of interest since it allows to know how the choice of the voting rule is susceptibl (...)
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 efficiency1 of voting rules. The question of the concordance between voting rules2 has also been widely studied in this literature. Naturally, there are a variety of other considerations that go into this category.
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Much of the research in social choice theory is concerned with whether a paradox can occur for a given voting rule or not. We define 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.
4All 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 Felsenthal (2012); Felsenthal and Nurmi (2018); Gehrlein and Lepelley (2011, 2017); Nurmi (1989, 1999), and Saari (1995), among others.
- 3 With m candidates, the Borda rule awards m − j points to a candidate each time he/she is ranked j-t (...)
5Throughout 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 Condorcet (1785) as an alternative to that proposed by Borda (1781). 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.
6Although the Borda-Condorcet debate concerned only two rules for collective decision-making, 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 different 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 define the two principal approaches in the studies dealing with social choice theory. We present the two approaches in Section 1 where we will show that they complement each other.
7The probabilistic approach experienced a boom in the late fifties, 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 decision-making processes, elections, surveys or polls. For an overview of the work that falls within this framework, the reader may refer to Chamberlin et al. (1984); Dobra (1983); Dobra and Tullock (1981); Kurrild-Klitgaard (2008, 2001); Niemi (1970); Regenwetter et al. (2002a,b); Riker (1958, 1965); Taylor (1997) and Tideman (1992). Notice that the results of these empirical studies are summarized in Gehrlein (2006) and Gehrlein and Lepelley (2011, 2017). However, empirical analyses are not always possible, because the data for 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 find the theoretical probabilities of voting events according to these models.
- 4 More precisely, Donald Trump won enough states to secure the majority in the Electoral College, whi (...)
8The first use of probabilistic models in social choice theory dates back to the paper by May (1948) 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, Guilbaud (1952) gives in his important paper the probability of Condorcet’s paradox when three options are in contest. This practice spread in the late 1960s with the work of Campbell and Tullock (1965); Garman and Kamien (1968) and Niemi and Weisberg (1968) 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 Gehrlein (2006) and Gehrlein and Lepelley (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 2. The probabilistic model approach faces some criticisms, chiefly: 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.
9We give a detailed discussion of each of these limitations later in the paper.
10As 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 Arrow (1951, 1963) that the first results based on simulations appeared (see for instance, Klahr, 1966; Weisberg and Niemi, 1978, 1973), 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 different 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 different approaches adopted as well as review their scope.
11The 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 1 is therefore dedicated to this end. In Section 2, we present the main models or hypotheses on which the theoretical works are based. Section 3 is devoted to the methods of simulations that have been developped in contrast to that of the theoretical approach. A conclusion follows.
12The 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. This theory leans on fundamental microeconomic principles and aims to understand the decisionmaking of rational individuals as regards to economic phenomena or beyond. According to List (2013), 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 field 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 different 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.
13Historically, since the seminal works of Arrow (1951, 1963), 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 Pigou (1920), Bergson (1938), and Samuelson (1947). The traditional economy of well-being was for a long time dominated by an almost total adherence to the utilitarian approach of Bentham (1789), and settled down, with the work of Edgeworth (1881), Marshall (1960), and Pigou (1920), in a very different framework from that of social choice theory. It was only from the late 1940s, with the work of Arrow (1951, 1963), Black (1948, 1958, 1976), and May (1952, 1971), 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.
14More precisely, in utilitarian calculations, the preferences of individuals are represented by numerical utility functions defined 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 finally 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 efficiency. 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 insufficient, 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 Arrow (1951, 1963) fits into this framework insofar as it allows aggregating individual preferences into a collective preference, so that society can choose between the different Pareto optima.
15Arrow (1951, 1963) showed that inconsistencies related to collective choice are not a surprise as they affect a very wide class of aggregation procedures.
16The method adopted in Arrow’s theorem is called the axiomatic approach. It consists of choosing a certain number of properties which seem too reasonable to impose on the aggregation procedures and then demonstrating that the satisfaction of these properties all 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 allows to verify all the desirable conditions taken into account at the same time. More precisely, the desirable conditions stated by Arrow are: i) Universal domain: all individually rational preference orderings are allowed as inputs into the aggregation method; ii) Completeness and Transitivity: the derived social preference ordering should be complete and transitive; iii) Unanimity condition (also called the Pareto principle): 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. In other words, if every individual prefers X to Y , then the aggregation method should similarly rank X and Y ; iv) Independent of Irrelevant Alternatives: the social ranking of X and Y should depend only on how individuals rank X and Y and not on how they rank some “irrelevant” alternative W relative to X and Y ; v) Non-imposition: an outcome is not to be imposed. vi) Non-dictatorship: the aggregation method cannot be based solely on one individual’s preferences.
17By accepting the idea of discarding or weakening some of the Arrow’s axioms and/or adding others, Arrow’s theorem has given rise to countless contributions using his axiomatic method which still is the most common approach in social choice theory. However, it is important to mention that the majority of the convincing results using the axiomatic method have been obtained in the form of impossibility theorems, which highlights the difficulty of designing a method allowing a reasonable aggregation of the opinions expressed by individuals in a decision procedure. Moreover, the most important limitation of the axiomatic approach is that it does not offer information on the frequency of situations where a given aggregation procedure violates a desirable property/axiom.
18The 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 fixed on the distribution of individual preferences. The most significant part of the research on this topic make use of sophisticated analytical techniques in order to obtain exact results describing the theoretical probabilities of the studied voting events. 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). These models will formally be defined and discussed later. The IC and IAC assumptions as well as most of the other used models 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 difficult to manage, except by resorting to numerical or computer simulations. Indeed, simulations have made it possible to find and (in)validate several results established in the social choice literature.
A voting profile consists of a list of each of the individual voter’s identities along with their respective preference. In other words, a profile is defined as a collection of individual preferences expressed on the set of candidates A. Moreover, a voting situation is an anonymous profile. In other words, a voting situation is a vector denoting any particular combination of natural integers that sum to ; each component of this vector denotes the number of voters associated with the corresponding ranking. For and linear/strict orders assumed, Table 1 describes the possible strict rankings on . In this table, it is indicated that voters have the ranking ; this means that they rank candidate at the top followed by candidate and candidate is the least preferred. In this setting, a profile is a sequence of linear orders – one for every individual – over the three candidates and a voting situation is defined by the vector , where in parentheses, we refer the number of voters endowed with each of the six orders such that
Table 1: Possible strict rankings on A = {a,b,c}
19As it is the case in Table 1, 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 indifferent between two or more alternatives. Using Monte Carlo simulations, Bjurulf (1972) and Jones et al. (1995) pointed out that this is not without impact on the results obtained; they also 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, Fishburn and Gehrlein (1980) reached the same conclusion. In a recent book, as part of their work on “Behavioral Social Choice”, Regenwetter et al. (2006) have developed the tools to overcome the tradition of a priori preferences. In addition to highlighting the different 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 3.
20We are now equipped to introduce the main theoretical models used in social choice theory when dealing with voting events.
21The main purpose of using theoretical assumptions to model the behavior of a group of individuals is to derive an exact theoretical representation of the probability of a given event. The starting point in the different 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 impartial culture (IC) model was introduced for the first time in the social choice literature by Guilbaud (1952), 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 profiles 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 linear orders has the same probability of being chosen by an individual, and the probability of attaining a voting situation is given by:
- 6 Recall that the normal distribution is the most common type of distribution assumed in probabilisti (...)
22As mentioned before, it is worth noting that the IC model is only a special case of the multinomial law. David and Mallows (1961); Gehrlein and Fishburn (1978a,b) and Plackett (1954) showed that for an infinite 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.6 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 finite 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 and Fishburn, 1978a,b; Saari and Tataru, 1999) have the disadvantage of leading to different formulas which are often not compact and are difficult to handle.
23Fishburn and Gehrlein (1980) was the first to introduce an extension of IC in order to take into account the possible indifference in the agents’ preferences: the impartial weak ordering culture (IWOC). More recently, Diss et al. (2010) have provided another extension of IC that allows the possibility for voters to have dichotomous preferences with complete indifference 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 find the IWOC model from EIC. As it can be seen, these extensions are only refinements of IC that tend to take into account the remarks of Bjurulf (1972) and Jones et al. (1995) 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.
24The dual culture (DC) model was first introduced by Gehrlein (1978) in the literature of social choice theory and operates as the IC model but it was only defined for the framework of 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 defined as follows:
One may notice in this case that we recover the IC model when .
The impartial and anonymous culture (IAC) model was introduced for the first time in the literature of social choice theory by Gehrlein and Fishburn (1976) and Kuga and Nagatani (1974). 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 specific to IAC: with IC, the probability is the ratio between the number of preference profiles in which the event occurs and the total number of possible preference profiles. Both models are based on a notion of equiprobability, but the elementary events are preference profiles 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 with voters and candidates is given as follows:
25Different techniques and algorithms for finding exact theoretical solutions for such systems have been proposed in the literature; the reader may refer to the works of Cervone et al. (2005); El Ouafdi et al. (2020); Gehrlein and Lepelley (2011); Lepelley et al. (2008) and Wilson and Pritchard (2007).
The maximal culture (MC) model is due to Fishburn and Gehrlein (1977). MC is quite similar to IAC with the exception that there is no need to fix the number of voters in the random voting situation. It fixes an integer and each ranking is drawn from a uniformly random distribution over . According to this, for three-candidate elections, the number of possible equally likely voting situations is equal to and the expected number of voters in a voting situation is given by 3L.
where is a generalized ascending factorial and the are positive numbers associated with each order such that .
According to Berg and Bjurulf (1983), is a parameter measuring the level of social cohesion: the larger it is, the more the preferences of the individuals tend to be homogeneous. Berg and Bjurulf (1983) showed that if we fix for all , the Pólya-Eggenberger model leads to the IC model for and to the IAC model for . 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.
26Table 2 reports the limit probability, i.e., when the number of voters tends to infinity, of the Condorcet paradox in three-candidate elections obtained under each of the above theoretical models. All these results are drawn from Gehrlein and Lepelley (2011, 2017).
Table 2: Limiting probability of the Condorcet paradox in three-candidate elections
27The different 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 particular for three-candidate elections. Despite the fact that for the same event, the models can lead to different probabilities, Gehrlein and Lepelley (2004) put forward a certain number of arguments to justify the use of such models. Let us summarize these arguments:
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It can be useful to find out if the relative probabilities of paradoxical outcomes on various voting mechanisms behave in a consistent fashion over a number of different assumptions about the likelihood that voting situations or voter preference profiles are observed.
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With real elections, large amounts of empirical data are not available; the use of theoretical models is thus found to be very useful.
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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.
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Theoretical models can show the relative impact that paradoxical events can have on different types of voting situations.
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By using probability models to obtain closed form representations, it is easy to observe the impact of varying different parameters (e.g., parameters of specific measures of social homogeneity or group coherence) of voting situations or voter preference profiles; this is somewhat more difficult to do with simulation studies.
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The obtained probability representations are directly reproducible and verifiable with mathematical analysis, which is not as simple to do with simulation analysis.
28The 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.
29Initially, the studies of voting situations involved only two or three candidates and were limited to a finite 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. As mentioned before, one way to overcome the constraints and limits of the analytical approach would be to operate on real data; however, these are difficult 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 confined to the subfields of economics, and are spreading to almost all social sciences (see, for instance, Axelrod, 1997; Fontana, 2006). 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.
30In their first 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 Campbell and Tullock (1965); DeMeyer and Plott (1970); Gehrlein and Fishburn (1976); Klahr (1966) and Weisberg and Niemi (1978, 1973); most of these works involve only strict orders for agent preferences. According to Jones et al. (1995) this could be justified by the performance of computers at that time. Taking advantage of the computer advances of the 90s, Jones et al. (1995) conducted an analysis of the simulated probability of the Condorcet paradox when weak preferences are allowed.
- 9 Project Manhattan is the code name for the research project that produced the first atomic bomb dur (...)
31The 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 Ulam and Von Neumann (1945), referring to games of chance in casinos, during the Manhattan project.9 This method has the advantage of being easy to use. It is now applied to a very wide range of problems. Let us briefly 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 (linear) preferences. For our presentation, we will focus on cases where only strict preferences are allowed and we will use the notation of Section 1.3.
The goal is to generate, equiprobably, a profile of total orders with voters and candidates. So each of the possible total orders is chosen equiprobably; to choose a total order is therefore equivalent to choose an integer between 1 and . An integer is chosen over this interval for each of the voters, one after the other and independently. The chance of occurrence of each profile in this process is actually . At the end of the process, which is anonymous, we count the number of voters assigned to each strict order; we then obtain a -uple of integers whose sum is equal to . Concretely, the routine used is the following:
To generate a sample of size (number of repetitions), we run the previous routine times while keeping the result; this gives a -tuple of profiles with total orders.
Given the possible strict orders, the objective is to generate the voting situations (anonymous profiles) equiprobably. As a reminder, a voting situation is an -uple of natural numbers whose sum is equal to for which it will be necessary to randomly generate each of the components. To do this, numbers are generated equiprobably in [0;1] which we rank in increasing order, say ; this series will be completed by 0 the smallest possible value and 1 the largest possible value such that . Then, the value is assigned to that is to say is assigned to to and so on until which is assigned to . 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 , the difference 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 , we run the previous routine times.
- 10 The Maxwell-Boltzmann statistic is a probability law or distribution used in statistical physics (t (...)
- 11 It describes one of two possible ways in which a collection of non-interacting, indistinguishable p (...)
- 12 These figures are consistent with those obtained under the IC model by Gehrlein (1985); Klahr (1966 (...)
32It is worth noting that Feix and Rouet (2005) 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 distribution10 and the IAC model is related to the BoseEinstein statistic;11 quantum statistics would thus be another gateway for calculating the probabilities of voting events. Feix and Rouet (2005) 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 infinite size. Their calculations show a certain convergence between IC and IAC when the number of candidates increases. Table 3 reports the likelihood of the Condorcet paradox when preferences are simulated according to the IC and IAC models for voting situations with three to eight candidates.12 The values in this table are derived from those of Table 4 and 5 by Feix and Rouet (2005).
Table 3: Limiting probabilities of the Condorcet paradox obtained by simulations under IC and IAC
- 13 Among others, Rousseauist cultures, impartial culture, distributives cultures and spatial Euclidian (...)
33Other 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 under the IC and IAC models but also for many other assumptions; for an overview, the reader may refer to the papers of Fishburn and Gehrlein (1982); Gehrlein (1997); Jones et al. (1995); Pomeranz and Weil (1970) and Weisberg and Niemi (1978). 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 Aleskerov et al. (2012); Brandt et al. (2016); Diss and Doghmi (2016); Kelly (1993), Lepelley et al. (2000) and others. Notice that, based on a number of probabilistic models,13 Laslier (2010) simulated the frequency of the existence of a Condorcet winner, for several profile 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.
34It should be noted that in the days of the first 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 confined to the probability of the Condorcet paradox, were limited to voting situations with three candidates and a very small number of voters. With more than 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 offer many possibilities for simulations using simple macros and VBA language. We can also turn to more advanced tools such 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 effectively conduct their simulations that today can be done on personal computers or on dedicated servers, or even on supercomputers (Macal and North, 2010). 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 refine several theoretical results obtained in the past.
35As it is the case for the analytical approach which aims to obtain exact results describing the theoretical probabilities of the studied voting events, the simulation approach is strongly criticized. According to Tideman and Plassmann (2013), the analysis and consequently the results assume frequency distributions chosen just because of the convenient mathematical properties, while these distributions are far from reflecting what is happening in real elections. In fact, there is no evidence that voters’ choices obey any probabilistic distribution, let alone a uniform distribution. On the basis of their criticisms of theoretical models, several authors have argued for simulations based on more realistic distributions and assumptions.
36Besides the models that have just been discussed, two 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. More exactly, these approaches have the common feature of analyzing and generating preferences so as to reflect or to come close to real elections’ data samples.
37Spatial voting models were first applied specifically to elections by Downs (1957) to study the relative positioning of political parties and voters using a spatial approach built on the pioneering work of Black (1948); Hotelling (1929); Lerner and Singer (1937); Smithies (1941) and Greenhut (1956), who addressed the problem of location between two competing firms in order to optimally choose their setting in a market of undifferentiated goods. Under a spatial model, it is assumed that both candidates and voters are placed in an 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 Downs (1957), 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 specific distribution and the Euclidean distance serves as a tool for measuring the elector-candidate proximity. For a given voter and party or candidate , if we denote by the voter’s position and by the party’s position as perceived by voter , the Downsian utility of voter is given by:
In Eq. 5, the overall policy importance is captured through the parameter .
- 14 Please refer to Merrill and Grofman (1999) for an overview of all the so-called directional models.
Besides the Downsian-inspired model, we note the existence of the socalled directional models. Under the directional model developed by Rabinowitz and Macdonald (1989),14 it is assumed that utilities are determined by both the intensity and communality of direction of voters’ and candidates’ positions. So, voters have a diffuse 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 and the party :
38We owe to Rabinowitz and Macdonald (1989) 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 defined as follows:
where is a relative weight of the two components of voter utility. As one can see, when parameter is equal to 1, we get the Downsian model; when , we get the directional model. More recently, Kedar (2005) introduced a model combining the Downsian approach with a compensatory component which captures the outcome orientation of the voters. According to Kedar (2005), when a voter is outcome-oriented, it is assumed that he/she compares the expected policy outcome if all parties are elected and a counterfactual policy outcome where one party is excluded from the policy process ; 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 defined as follows:
where, given the position of party and the relative impact of party such that ; is a vector indicating the effect of background variables on voter utility for party .
39Each 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 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 (Enelow and Hinich, 1990). 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 affects voter behavior. The reader may refer, for instance, the work of Merrill (1984); Plassmann and Tideman (2011); Tideman (1992) and Tideman and Plassmann (2013).
40Popularized by Regenwetter et al. (2006), 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 influence 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, Regenwetter and Grofman, 1998; Regenwetter et al., 2002a,b; Tsetlin and Regenwetter, 2003); it also develops methodologies to (re)construct preference distribution from incomplete data (Regenwetter et al., 2006).
41Behavioral 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 reflecting 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 Popova et al. (2013), these results may magnify gloomy predictions found in the axiomatic literature on the inability of an electorate to make a group decision.
42Behavioral social choice aims to empirically analyze the rules or methods of preference aggregation by abstracting useless and/or unsubstantiated assumptions about human behavior. It turns out, therefore, that for any analysis, one has to state, very explicitly, tested and validated hypotheses about human behavior. Behavioral social choice considers empirical data on social choice from an inferential statistical point of view. If the empirical data are considered as imperfect and incomplete reflections of the voters’ preference, one must evaluate the replicability of social choice outcomes and assess to what extent one can be confident about the search for correct collective outcomes. Thus, under each behavioral model, the maximum likelihood estimate is used to calculate the probabilities of the voting events, and statistical confidence levels are generated through a nonparametric bootstrap (Efron, 1979).
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 with complete or incomplete voters’ preference from real elections, it proceeds by carrying out sampling with replacement: a sample of size is independently drawn from the original sample with replacement and replicated times. For each of the 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 were large enough to ensure meaningful statistics; this is generally possible when using Monte Carlo simulations on fairly powerful computers by generating random samples.
- 15 According to Regenwetter et al. (2002b, 2006) uncertainty can come from various factors, such as vo (...)
43In a behavioral social choice context, a bootstrap appears as a computerbased method for statistical inference that, without relying on too many assumptions, is a way of simulating possible sources of uncertainty15 in the results. It assesses how the results would be affected by small disturbances in the distribution of votes (preferences) and helps infer confidence levels about at which point estimates of model parameters would not be affected by such disturbances. Using this inference approach of preference aggregation, Regenwetter et al. (2006) 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.
44Over time, simulation models have emerged as an indispensable tool in many disciplines and fields of study. They offer a way to overcome the limits or constraints of theoretical modeling. In social choice theory, 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.
45Since 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, the purpose of this paper has been to offer 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.
46After briefly 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 IC model, introduced by Guilbaud (1952), is based on the idea that all preference profiles 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 Fishburn (1976) and Kuga and Nagatani (1974), 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 difficult 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 field of preference aggregation.
47The theoretical modeling has been strongly criticized for being based on distributions that do not reflect 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 reflect 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 Downs (1957), 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 Merrill (1984); Plassmann and Tideman (2011) and Tideman and Plassmann (2013), 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 Regenwetter et al. (2006), 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.
48Remarkable 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 refine the results found in the literature, but also to revisit certain problems whose resolution was previously impossible.
The authors would like to thank the Editor and two anonymous reviewers for their comments and suggestions. The first author would like to acknowledge financial supports from Université de Lyon (project INDEPTH Scientific Breakthrough Program of IDEX Lyon) within the program Investissement d’Avenir (ANR-16-IDEX-0005) and from Université de FrancheComté within the program Chrysalide-2020.