How voting rules impact legitimacy

The paper addresses how different voting methods shape the perceived legitimacy of collective decisions. Legitimacy is treated as a multi-dimensional construct grounded in input (representation and participation opportunities), throughput (procedural quality/fairness), and output (compliance with outcomes) perspectives, drawing on Tyler (2006), Scharpf (1999), and Schmidt (2013). The authors argue that voting mechanisms differ in their ability to elicit detailed preferences and in incentives for truthful revelation, and that avoiding tyranny of the majority is a key goal. There is limited experimental evidence on how alternative voting methods influence legitimacy perceptions, creating a knowledge gap relevant to democratic design and policymaking. The study tests whether more expressive, preferential input methods can enhance perceived legitimacy compared to majority voting, and whether this effect depends on context (neutral vs. critical/polarized decisions) and on individual differences in preference stability. The research examines context-dependence (e.g., COVID-19 issues vs. color choices) and hypothesizes that legitimacy evaluations can vary with issue criticality and the voter’s motivational landscape. A single legitimacy measure is used given high correlations among fairness, acceptance, trust, and representation in prior work.

Design and platform: A pre-registered, controlled online human-subject experiment (ETH Zurich DeSciL; pre-registration: https://doi.org/10.1257/rct.7871-1.0; ethics approval 2021-N-28) used the open-source smartphone application Votelab to administer four input methods: majority voting (mv; choose one; binary {0,1}), combined approval voting (cav; disapprove/neutral/approve {-1,0,1}), range voting (rv; 0–4 scale), and modified Borda count (mbc; ranking with points {0,1,2,3,4}; unranked options receive 0). Input method order was fixed for all participants (mv → cav → rv → mbc) to reflect increasing complexity and cognitive demand. Aggregation rules beyond the input mechanism were not the focus.

Contexts and questions: Participants voted in two contexts: a neutral, non-political color preference context and a political COVID-19 context. In the COVID-19 context, participants answered four questions (vaccine, icu, protection, lockdown), each with five options. Screenshots illustrated the interfaces for each input method and question.

Legitimacy measure: After voting with each input method in each context, participants rated the statement: “I would comply with the result and accept it as fair, reflecting my and others’ opinions.” Ratings used a 5-point Likert scale (0–4). Legitimacy ratings were collected for each method (mv, cav, rv, mbc) in both contexts (color, COVID-19).

Sample and procedure: N=120 participants (36% female; mean age 25.47; 22 nationalities; modal education Bachelor’s degree, 37%) took part in July 2021 across three sessions. The experiment had three stages: (I) introduction to COVID-19 questions and input methods; (II) voting and legitimacy ratings (color, then COVID-19; question order randomized to mitigate recency effects); (III) control questions on socio-demographics, strategic voting, and polarization.

Preference profiles and metrics: For each individual i, question q, and voting method vm, a preference profile consists of the ratings assigned to the five options. To understand variation in expressed preferences across methods and questions, two metrics were used: (1) standard deviation of ratings per individual profile (σ), summarizing the spread in preferences; and (2) divisiveness D (mean difference in ratings between voters who prefer option om over on and those with the opposite preference), computed per method and question. Higher σ and D indicate sharper preference differentiation and polarization.

Max-choice profiles: To analyze consistency across methods, the highest-rated option per method was encoded as 1 and all others as 0, yielding max-choice profiles such as 1111 (option is top-rated in all four methods). Because mv and mbc are exclusive and require ranking, profiles with leading/trailing zeros in these positions can indicate inconsistency in ranking across methods. Theoretical maxima for profile counts were derived given exclusivity and number of options.

Clustering analysis: To test whether satisfaction with topic options contaminated legitimacy ratings, the authors reduced dimensionality by averaging rescaled ratings per option across methods for each participant and question, then clustered these reduced preference profiles. Nine clustering methods and nine evaluation indices supported three clusters per question. Kruskal–Wallis tests assessed differences in legitimacy ratings across clusters for each method and question.

Additional analyses: The area under the preference curve (AUC) was computed by ordering options by rescaled scores per participant and method to capture how broadly a participant assigns high ratings. Relationships between AUC and perceived legitimacy were examined using local polynomial regression.

Ethics and consent: Approved by ETH Zurich Human Research Ethics Committee; informed consent obtained. Data are available on GitHub.

• Different input methods often yield different winners on the same question (Table 2). For example, in COVID-19 questions: vaccine winner varied (mv: o2; cav/rv/mbc: o4), icu varied (mv: o5; cav: o2; rv/mbc: o1), protection was stable (o4), lockdown varied (mv/rv/cav: o4; mbc: o1).
• Preference structure explains outcome dependence: The COVID-19 question “protection” exhibited the lowest standard deviation (σ_protection = 0.395) and second-lowest divisiveness (D_protection = 0.45), corresponding to minimal sensitivity of the outcome to input method. Across pairs, higher σ and D were associated with greater outcome sensitivity to the voting method.
• Consistency across methods was limited: 32.8% of voters did not vote consistently (i.e., max-choice profile 1111 not achieved for all). The fully consistent profile (1111) did not differ by question, suggesting consistency is an individual trait rather than question-driven.
• Legitimacy ratings by context and method: Range voting (rv) was perceived as more legitimate than majority voting (mv) in both contexts. In the color context, medians were equal (η=3) but pairwise comparisons showed mv < cav/mbc < rv. In the COVID-19 context, mv and cav were least legitimate, mbc intermediate, and rv highest; rv and mbc were rated equally legitimate in COVID-19.
• Context dependence: Majority voting was rated more legitimate in color than in COVID-19 (paired Wilcoxon signed rank, p = 7.02e-06). Range voting showed the opposite pattern, being more legitimate in COVID-19 than in color (p = 0.043). This supports the hypothesis that perceived legitimacy is context-dependent.
• Preference stability matters: Consistent voters gave higher legitimacy ratings to rv and mbc than inconsistent voters (Wilcoxon rank-sum, rv p = 0.0387; mbc p = 0.0228). Among consistent voters, median legitimacy: mv = 1; cav = 3; mbc = 3; rv = 4, with most pairwise differences significant. Among inconsistent voters, mv was lowest (η=2) and cav/rv/mbc shared η=3 with several non-significant pairwise differences, indicating simpler methods may be perceived as more legitimate by less decided individuals.
• Preference breadth and legitimacy (AUC analysis): For rv, the AUC–legitimacy relationship showed an inverse U-shape: low AUC (few options rated highly) and very high AUC (almost all options rated highly) were associated with lower legitimacy, while moderate-to-high AUC (appreciation of flexibility) corresponded to higher legitimacy.
• Topic preference clusters did not drive legitimacy: Across three preference clusters per question, Kruskal–Wallis tests detected no significant differences in legitimacy ratings for any method across vaccine, icu, protection, lockdown (all p>0.05), indicating legitimacy ratings were not confounded by specific COVID-19 topic preferences.

The study investigates whether and how the choice of voting input method affects perceived legitimacy, and under what conditions such effects are stronger. The results show that when preferences are more polarized or differentiated (higher σ and D), outcomes depend more on the input method, and voters value the expressive capacity of preferential methods. Range voting consistently achieved higher legitimacy than majority voting, particularly in the politically salient COVID-19 context, indicating that richer preference elicitation can enhance perceived procedural fairness and acceptance in complex decisions. However, the benefits of more nuanced methods are contingent on individual preference stability: participants with consistent, well-defined preferences viewed range voting and modified Borda count as more legitimate, while inconsistent voters tended to favor less complex methods or saw little difference among the more expressive methods. These findings align with the input/throughput legitimacy framework: richer input mechanisms can increase perceived representation and fairness for those able to articulate nuanced preferences, especially in critical contexts. The analysis further supports that legitimacy judgments are context-dependent, with majority voting seen as more legitimate in low-stakes, neutral choices and less legitimate in high-stakes, polarized ones. The absence of systematic effects of COVID-19 topic preference clusters on legitimacy ratings suggests the legitimacy scale captured evaluations of the method rather than spillovers from issue positions. Overall, the results underscore the importance of matching voting methods to both contextual complexity and voter preference stability to support legitimate collective decisions.

The paper demonstrates that (1) different input methods can produce different winners even for the same electorate and options; (2) these differences are amplified when preferences are more polarized; (3) perceived legitimacy is context-dependent, with preferential methods gaining legitimacy over majority voting in complex, politically salient decisions; (4) this legitimacy gain is concentrated among individuals with clear, consistent preferences, whereas undecided voters may find complex methods burdensome; and (5) voters with nuanced preferences perceive nuanced methods as more legitimate, and vice versa. Policy implications include adopting more expressive methods like range voting in high-stakes, polarized contexts to enhance legitimacy, while considering voter heterogeneity in preference stability. A practical recommendation is a phased approach: pre-select and communicate the deciding method, but engage voters through successive rounds moving from simple to more complex input methods to help undecided voters crystallize preferences without overwhelming them. Future research should examine how outcome favorability influences legitimacy, explore allowing voters to choose their preferred voting method with outcomes aggregated proportionally, assess cognitive factors related to preference stability and comprehension of complex methods, and test these dynamics across diverse political settings such as participatory budgeting, constitutional processes, and protest-related policymaking.

• The study varies only the input mechanism; effects of alternative aggregation rules (e.g., Condorcet vs. sum of scores) on perceived legitimacy are not directly tested within the main design.
• The relationship between outcome favorability (whether one’s preferred option wins) and perceived legitimacy is not explicitly addressed and is proposed for future work.
• Fixed order of input methods (mv → cav → rv → mbc) may introduce order or learning effects despite design justification; cognitive load may differentially affect methods.
• The sample (N=120, young and educated, recruited via ETH platforms) may limit generalizability to broader populations and cultural contexts.
• While question order was randomized and clustering suggests minimal contamination, potential carryover from satisfaction with options to legitimacy ratings cannot be fully excluded.
• The political context tested centers on COVID-19; results may vary for other salient or polarized issues.