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Understanding Political Agreements and Disagreements: Evidence from the 2022 French Presidential Election

Since the seminal works of Condorcet and Borda, social choice theory has explored how to aggregate individual preferences into collective decisions. Yet, social choice theory has focused primarily on identifying winners in elections involving few candidates, leaving questions about direct participation on multiple issues relatively unexplored. Here we analyze data collected in a direct participation experiment where people built their own government programs using 120 proposals from the candidates of the 2022 French presidential. We find that in this setting it is useful to introduce a measure of"divisiveness,"which can be constructed for any voting rule, is orthogonal to them, and helps identify polarizing proposals. We show that divisiveness captures fragmentation across multiple dimensions (sex, age, political orientation, and urban-rural divide) and explore some of its axiomatic properties. These results suggest divisiveness is a relevant aggregate in direct forms of participation.


Introduction
Social choice theory studies how to aggregate individual preferences into collective decisions 1 3 . To achieve this goal, social choice theory builds on axioms that define which systems are normatively desirable. A seminal example here is Arrow theorem 4,5 , which states that when choosing among three or more options, no ranked electoral system can create a complete and transitive ranking satisfying four key axioms: independence of irrelevant alternatives (IIA) (changing the outcome of an election by removing a nonwinning candidate), weak Pareto efficiency (an election that cannot be made better off for every individual), non-dictatorship (no single voter can impose its order of preferences), and unrestricted domain (all possible orderings of preferences are allowed).
But social choice theory involves multiple criteria, such as efforts to avoid the tyranny of the majority, which prevents disregarding the preferences of minorities, or Condorcet efficiency, which estimates the percentage of times that the winner of an election also wins every possible head-to-head contest. These and other axioms have been explored extensively in electoral contests focused on identifying a winner. But while this goal is reasonable for elections involving a few political representatives, it may be too narrow when analyzing electoral systems involving direct forms of participation for multiple proposals.
Here we use data collected in an organic direct democracy experiment conducted during the 2022 French presidential election to argue that in elections involving dozens or hundreds of issues, social choice theory could benefit from methods that can distinguish among similarly ranked proposals. For instance, two proposals ranked 50 and 51 in a 100-issue election, could achieve that ranking for very different reasons. On the one hand, individuals could simply not have a strong preference on that proposal. On the other hand, is could be that half of the population strongly prefers the proposal while the other half strongly rejects it. This difference has important implications, since in the former case the proposal is more-or-less irrelevant, while in the latter, it points to a major source of disagreement. To separate among these reasons, we need aggregates that can rank proposals not based on of agreement, but instead, on their level of disagreement.
In this paper, we define simple measures of divisiveness, which can be defined for any voting rules (measures of agreement), and that help identify controversial proposals. We show that divisiveness provides new information, since it is uncorrelated with its corresponding voting rule, and explore its axiomatic properties. To conclude, we use matrix factorization techniques to show that divisiveness is related to the second eigenvector of the matrix of pairwise preferences (while voting rules are related to the first one), suggesting that divisiveness represents a natural extension of voting rules for high dimensional data.
These findings should be of the interest of both, social choice theorists and practitioners working to design more direct forms of civic participation.

Digital Platforms for Civic Participation
Online participation systems involving dozens or hundreds of alternatives have gained prominence in recent years thanks to planned activities, such as participatory budgeting exercises 6 9 , and emergency response scenarios, where platforms were deployed in response to mobilizations, such as the networked social movements 10 , in Taiwan 11 , Chile 12 , and Lebanon 13 . These participation efforts aim to provide a more nuanced preferences, and in the case of mobilizations, a mean to mitigate instances of tactical freeze 14 16 22 . The platform asked users to select items from pairs to derive a partial ordering for the items. It is related to voting rules based on the pairwise majority matrix 23 ( 1 ).
Today, paired comparisons are a popular research tool in psychology, computer science, and statistics. 16 22 But they have also been used to understand political preferences during mobilizations 12,24 hey provide a method to quantify and aggregate subjective judgments. Unlike traditional survey methods, which focus on stated preferences, paired comparisons have a more behavioral component, since they can be used to infer agreements and disagreements from population level behaviors. For instance, we can learn that a policy proposal (e.g. France leaving the Schengen agreement ) is a source of disagreement, not because participants tell us this issue is polarizing, but because the proposal is popular among some participants and unpopular among others.
But paired comparisons also have limitations. For instance, an electoral contest involving 120 proposals involves 7,140 possible pairs. This means that it would be impractical for a user to evaluate all possible pairs. To mitigate that issue, we designed a multiple comparison platform, where we present users a set of 4, 5 or 6 proposals (randomly assigning participants to each of these conditions but finding no differences in their level of participation).
Participation then occurred in two iterative steps. First, we asked users to select which proposals they would like to include in their own government program, and then, to rank sort the proposals they had selected (

Results
First, we study proposals associated with high levels of agreement and disagreement among the participants. Figure 3 shows the ranking of proposals sorted by their win percentage (W): the fraction of times a proposal was selected over other proposals divided by the number of times it was presented to a user. This voting rule is our main measure of agreement and is related to the Borda count (a voting rule that awards points to a proposal based on the number of other proposals ranked lower than them). The top proposals include the use of 100% renewable energy by 2050, the increase of the personnel in public hospitals, and the increase of the minimum wage.
Top priorities, however, tell us little about population level frictions among the participants, since they involve issues agreed upon within this population. This motivates us to explore a second form of aggregation focused on disagreements. Our intuition is that divisive proposals, which are strongly supported by one group but opposed by another, can represent opportunities for political trading or bargaining when groups may consider accepting proposals that are low priority for them but important for the opposite group. Divisive proposals can also point to items that need to be deliberated upon or discussed, since they involve issues that the population disagrees on (and hence, may hold different views on their potential causes and consequences). So next, we explore measures of divisiveness, first by  We define the divisiveness di of a proposal with respect to a split in the population (A and not A ( )) as the difference in the score Si that the proposal gets using any voting rule (e.g. Borda, Copeland, etc.) when evaluated among participants from these two populations. For instance, if a proposal has a win percentage of 60% when evaluated among participants above the median age, and a win percentage of 40% when evaluated among participants below the median age, its divisiveness with respect to median age would be d(Age)=20%. Thus, we define the divisiveness d of proposal i with respect to population A and the score of a voting rule S as: Where Si(A) is the score of proposal i when evaluated in population A and is the score of proposal i when evaluated in population not A. a proposal that ranks higher among participants who self-identified with the political left, but relatively low among those who self-identify from the right (rank 6, , We observe less divisiveness when we split the participants by their self-reported sex, geography (capital versus regions, urban versus rural), age (older and younger people), and education, suggesting that divisiveness is a more political than demographic phenomenon.
Still, we identify some divisive proposals that satisfy common sense. For instance, a proposal to (rank 41 overall) is more popular among older participants than among younger participants. Also, we find that Restoration of border control by France leaving the Schengen agreements among people without tertiary studies. This suggests that the divisiveness of a proposal is nevertheless a multidimensional phenomenon, and that specific proposals may be seen as divisive depending on how we split the population.
But is there a way to understand divisiveness in absence of self-reported demographic and political orientation data?
Next, we estimate divisiveness by introducing a method that is agnostic about any self-   This is an early indicator that divisive issues can be relevant in terms of public discussion. Next, we explore the convergence of measures of divisiveness and agreement by estimating the fraction of the sample needed to obtain the observed final ranking (with the full sample).
Interestingly, we find that the ranking of divisiveness converges more slowly than that of agreements. Figures 6 a and b show the correlation among rankings estimated using nonoverlapping set of preferences. To achieve a Kendall-Tau correlation of 0.9 among priorities we need only 40,000 preferences, but for the ranking of divisiveness to converge to the same level we need around 300,000 preferences (7.5 times more). This is likely because divisiveness is more of an ensemble property, that unlike agreements, would be non-sensical in a two-candidate election. This suggest divisiveness as a measure that could be more difficult to manipulate by a strategic voter.

The empirical boundaries of social choice theory
Finally, we explore our data using the axiomatic properties of social choice theory. We look at Condorcet efficiency, the robustness to irrelevant alternatives, and robustness to scenarios of the tyranny of the majority.
We test the Condorcet efficiency of the rankings of agreements by counting the number of times in which a pairwise contest contradicted the order of the ranking. This is an extended   proposals that are ranked higher tend to beat proposals ranked lower. This difference in win percentage increases with differences in ranking. Condorcet violations are values larger than 50% in the lower triangle or lower than 50% in the upper triangle. Aggregating across the whole matrix we find a Condorcet efficiency of 97.6%, meaning that although the ranking is not perfect with respect to Condorcet efficiency, it is close to it.
We can create tighter bounds by progressively softening this definition to be more forgiving of smaller violations. That is, we do not want to count equally a violation where the lower ranked proposal won by 50.1% (0.2% difference) or 60% (a 20% difference). Figures 8 c and d show the Condorcet efficiency of these matrices comparing three voting rules or aggregation methods (Win Percentage, Copeland, and Elo). As one would expect, removing small violations increases the Condorcet efficiency of these rankings. In fact, once we remove all violations smaller than 20% we get rankings that are 100% Condorcet efficient for both Copeland and Win Percentage (Elo is not as robust). We also explore the Condorcet efficiency of the full matrix as a function of sample size (Figures 7 c and d inset) finding that the rankings achieve about 90% Condorcet efficient for samples involving around 100,000 preferences.
Next, we study the robustness of the rankings of agreements (win percentage) and disagreements (divisiveness) to the removal of single proposals (independence of irrelevant alternatives or IIA) he robustness of the full ranking (regardless of how high or low a proposal ranked). Figure 8 c and d present matrices showing the differences in ranking experienced by proposals after removing the proposal represented by the column (sorted by the win percentage). We only count changes in ranking of more than 2 positions (for the full range of values see SM). With this definition we find over 99% robustness for the top proposal in the rankings of agreements and disagreements. To look at the overall robustness, we count the total number of ranking changes finding a robustness of 87.7% for the ranking of agreements and 90.7% for the ranking of divisiveness.
To explore the robustness of these rankings to scenarios of the tyranny of the majority, we split the data into two sets: one for users who self-reported identifying with the left or center left and one for users who self-reported identifying with the right or center-right. We estimate separate rankings for both groups to simulate a situation in which only one group would have participated. Figures 8 e and f show that the rankings of agreements and divisiveness are not very robust (respectively R 2 =33.9% and R 2 =34.5%).
These axiomatic tests, however, are limited by our sample, and thus, do not represent general conclusions about the axiomatic properties of divisiveness. They can be thought of as initial explorations of these properties given the data at hand. Moreover, since divisiveness converges more slowly than its corresponding voting rule, it is likely that divisiveness has not fully converged and that these empirical results could change for larger samples. Finally, we analyze the matrices of pairwise preferences (figure 7 a) using singular value decomposition (SVD) (Figure 8 a and b). SVD is a fundamental matrix factorization technique that generalizes the concept of eigenvectors and eigenvalues to non-square matrices. SVD can be used to approximate a matrix as a sum of other matrices that is guaranteed to provide the best possible approximation with given a number of factors (mathematically, you cannot provide a better single factor approximation to the full matrix than using the first factor of SVD) 26

Discussion
Using online platforms to understand growing phenomenon. Here, we present our findings from an online participation platform designed to understand the preferences of a group of self-selected citizens during the 2022 French presidential election.
This sample allowed us to experimentally explore an electoral contest involving over one hundred proposals, finding that efforts to understanding preferences in such a context requires complementing traditional measures of consensus with measures of divisiveness. This is interesting for a variety of reasons.
First, divisiveness provides information that cannot be surfaced through aggregation methods focused on agreements, since divisive proposals cannot climb to the top of a priority ranking. This is because for every participant that pushes a divisive proposal up, there will be other participants pushing it down. The proposal for increasing the retirement age can be clearly identified as the top proposal when we look at the ranking of divisiveness, whereas it would be harder to identify using a ranking of agreements (buried below the top 20). This is an interesting observation considering that it is based on data collected in the middle of the electoral campaign, and that this proposal was among the most discussed by the winner of the election.
There are good reasons to believe that divisive issues are politically relevant. According to some scholars, political position in front of society. 27 For instance, a presidential candidate supporting an increase in the minimum wage a popular but not very divisive proposal would fail to signal their political identity at the time of the campaign. Thus, one could argue that during campaign periods, it may be strategic for politicians to focus on divisive issues instead of areas of wide agreement.
Second, divisiveness represents a form of aggregation that has been understudied in social choice theory. This may be because social choice theory generally focuses on identifying winners in elections involving a handful of political representatives. In such cases, finding a remains key, since it would be odd to argue that the most divisive candidate should be the one elected. But since our platforms focused on 120 issues instead of a few candidates, it represented a large enough change of scope and scale to generate the need for a form of aggregation that is orthogonal to consensus aggregates and that would be irrelevant in a two-candidate election. These are aggregates focused on divisiveness, which based on the SVD decomposition, represent a fundamental expansion of the voting rules used to identify agreements. These higher-order aggregates become more relevant in contests involving dozens or even hundreds of issues, suggesting an avenue of research for social choice theory focused on the axiomatic properties of metrics of divisiveness.
Using online tools for digital participation is a growing phenomenon connected to a rising interest in online civic deliberation 7, 28 35 . Civic participation platforms have been deployed with varying levels of success in Taiwan 11,36,37 , Spain 38 , Italy 39 , and Chile 12 . Yet, there are important risks that need to be considered. Although the platform did not go viral, it suffered cyberattacks (see SM for details). Also, the experience showed that combining approval and rank voting provides an efficient alternative data collection (we obtained nearly 2 million preferences from less than 2,000 participants). However, the voting system used by this platform can be hard to understand for people used to episodic elections in plurality forms of voting (e.g. first-past-the-post ) 40 . The design and development of digital participation tools, therefore, needs to overcome important technical and social challenges before these systems can be more widely adopted. Moreover, the adoption of this platform varied enormously by location. The project enjoyed the widest adoption in Haute-Garonne (home of the city of Toulouse, where the platform was created and launched), and less adoption in Île-de-France (Paris). This shows the key role that the media can play to engage users in digital forms of participation.
Going forward, we expect to see more work on digital participation efforts and social choice theory, due to the increasing prevalence of decentralized movements and the increased availability of communication and participation technologies. We hope these findings help inspire and inform future research.

Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code Availability
The code that supports the findings of this study is available from the corresponding author upon request on reasonable request.

Methods
All procedures performed in studies involving human participants were in accordance with the ethical standards of the TSE-IAST Review Board for Ethical Standards in Research, under the reference code 2022-02-001. Informed consent was obtained from all individual participants involved in the study.