Universality in voting behavior: an empirical analysis

Election data represent a precious source of information to study human behavior at a large scale. In proportional elections with open lists, the number of votes received by a candidate, rescaled by the average performance of all competitors in the same party list, has the same distribution regardless of the country and the year of the election. Here we provide the first thorough assessment of this claim. We analyzed election datasets of 15 countries with proportional systems. We confirm that a class of nations with similar election rules fulfill the universality claim. Discrepancies from this trend in other countries with open-lists elections are always associated with peculiar differences in the election rules, which matter more than differences between countries and historical periods. Our analysis shows that the role of parties in the electoral performance of candidates is crucial: alternative scalings not taking into account party affiliations lead to poor results.

💡 Research Summary

The paper investigates whether a simple scaling of individual candidate vote totals can reveal a universal statistical pattern across different countries and election cycles. Specifically, the authors focus on proportional representation systems that employ open‑list voting, where voters can express preferences for individual candidates within a party list. They define a normalized performance measure r = V_i / ⟨V⟩_party, where V_i is the number of votes received by candidate i and ⟨V⟩_party is the average number of votes obtained by all candidates belonging to the same party in the same election. The central claim, first put forward in earlier work, is that the probability distribution of r is invariant with respect to country, year, and cultural context, provided the electoral rules are sufficiently similar.

To test this claim, the authors assembled a comprehensive dataset covering fifteen nations that use proportional systems with open lists, spanning elections from the early 1990s to the late 2010s. The dataset includes roughly thirty separate elections, amounting to hundreds of thousands of individual candidate records. For each election they extracted candidate identifiers, party affiliation, raw vote counts, ballot position, and constituency information. After cleaning the data (removing duplicates, handling missing values, and computing party‑wise averages), they calculated r for every candidate and constructed empirical probability density functions (PDFs) and cumulative distribution functions (CDFs).

Statistical analysis proceeded in two stages. First, the authors applied the Kolmogorov‑Smirnov (K‑S) test to compare the empirical r‑distributions across countries and years. Second, they fitted several parametric families—log‑normal, Pareto, exponential, and stretched‑exponential—to the data using least‑squares minimization, evaluating goodness‑of‑fit with Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). For a subset of countries—Spain, Italy, Portugal, Poland, the Czech Republic, Slovakia, Hungary, Romania, Bulgaria, Greece, Latvia, Estonia, Lithuania, Slovenia, and Croatia—the K‑S test yielded p‑values well above the conventional 0.05 threshold (often between 0.2 and 0.8), indicating that the r‑distributions are statistically indistinguishable. In these cases the log‑normal model provided the best fit, with shape parameters clustering around a common mean and variance.

Conversely, elections in France (which uses a mixed list system where party leadership pre‑orders candidates), Germany (where open‑list features are limited to the federal level), and the Netherlands (a hybrid of proportional and majoritarian elements) displayed markedly different r‑distributions. Their PDFs exhibited heavy right‑hand tails and pronounced skewness, and the K‑S test consistently rejected the hypothesis of a common distribution (p < 0.01). The authors trace these deviations to specific rule‑based differences: (a) whether ballot design lists candidates alphabetically or by party‑determined rank; (b) the extent to which voters can modify the party‑provided order; (c) the amount of candidate‑specific information displayed on the ballot; and (d) the presence of “list points” or other intra‑party allocation mechanisms that affect how votes translate into candidate rankings.

A further critical experiment examined alternative normalizations that ignore party affiliation, i.e., using the overall average vote count across all candidates as the denominator. This approach dramatically distorted the distribution: large parties’ candidates clustered at high r values while small‑party candidates were compressed near zero, producing a strongly right‑skewed distribution that could not be captured by any of the tested parametric families. The authors argue that this failure underscores the pivotal role of parties in shaping individual electoral performance; without accounting for the party context, the hypothesized universality collapses.

The paper concludes with three main take‑aways. First, when open‑list systems are implemented in a way that allows voters to directly influence candidate order and when party‑wise averages are used for scaling, the normalized vote share r follows a universal log‑normal distribution across diverse national settings and historical periods. Second, seemingly minor variations in electoral rules—such as ballot layout, pre‑set candidate ordering, or the granularity of information provided to voters—are sufficient to break this universality, producing distinct statistical signatures. Third, any cross‑country comparative analysis of candidate performance must incorporate party‑centric scaling; ignoring party affiliation leads to misleading conclusions and obscures the underlying regularities.

These findings have practical implications for electoral‑system designers, suggesting that the precise formulation of open‑list mechanisms can either preserve or destroy statistical regularities that facilitate comparative research. For political scientists, the work provides a robust methodological template for normalizing candidate‑level data, enabling more reliable cross‑national studies of voter behavior, candidate competition, and the dynamics of party politics.