The New Mathematics of Democracy

Abstract: This article surveys emerging directions in the mathematics of democracy. It uses three case studies --- voting theory, participatory budgeting, and deliberative democracy --- to highlight how contemporary challenges motivate rigorous mathematical research that incorporates real-world data, institutional constraints, and implementation feasibility. Within each case, we highlight active and promising research frontiers, evidence of real-world impact, practical applications, and opportunities for getting involved.

• The paper argues that empirical election data and realistic voter behavior should complement classical voting theory, noting that IRV and Condorcet methods agree on winners in over 99% of roughly 3,000 elections studied.
• Participatory budgeting research combines knapsack-based preference elicitation with the Method of Equal Shares, which reduced voters receiving no approved projects from 28% to 18% in one deployment while improving geographic inclusion.
• The paper shows how mathematical tools including convex optimization, discrepancy theory, and PAC learning support fair sortition and deliberative assemblies, with equality-optimizing algorithms already used in more than 1,000 assemblies during 2024–2025.

This survey by Flanigan and Volić (2608.16869) examines how contemporary democratic practice is reshaping the mathematical study of collective decision-making. Organized around three case studies—voting theory, participatory budgeting (PB), and deliberative democracy—the article argues that classical, idealized models of preference aggregation are being supplemented (and in places corrected) by approaches that incorporate real election data, institutional constraints, behavioral realism, and deployment infrastructure. The unifying thesis is that practical "messiness" is not an obstacle to formal work but a source of new, mathematically rich questions spanning probability, optimization, convex geometry, game theory, dynamical systems, discrepancy theory, and learning theory.

Voting theory: from impossibility to empirics

The voting section traces the standard arc from Borda and Condorcet through Arrow's Impossibility Theorem, which shows that non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives are mutually incompatible for social welfare functions on universal domain. The authors emphasize how restrictive this framework's assumptions are relative to actual elections: complete orderings, full turnout, universal domain, and single-peakedness are all idealizations that fail in practice. A worked example with eleven voters and four candidates illustrates that Borda, Condorcet, and instant runoff voting (IRV) can each select a different winner from the same profile.

The substantive contribution of the section is its account of the empirical turn. A database of roughly 3,000 IRV and STV elections from the U.S., Scotland, and Australia supports several strong claims: IRV and Condorcet methods agree on the winner over 99% of the time; spoiler effects and vote splitting are extremely rare for both; strategic truncation and burying are effectively non-issues for IRV but a slight weakness for Condorcet methods; plurality performs worst across measures. The authors also flag a methodological caveat that deserves emphasis: cast-vote data from IRV/STV elections may not reflect what voters would do under other rules, so conclusions about counterfactual rules rest on an assumption about behavioral invariance.

A sharper illustration of why modeling assumptions matter comes from two conflicting results on candidate moderation. One line of work using Cooperative Election Study (CES)-based synthetic electorates found that Condorcet methods elect centrists more often than IRV; subsequent work incorporating observed behavior—most notably the roughly 35% bullet-vote rate—found that this difference largely disappears under more realistic conditions of truncated ballots, abstention, and noisy voter perceptions. The paper presents this contradiction plainly as evidence that simulation choices drive substantive policy-relevant conclusions, and it identifies data-driven models of voter behavior whose robustness properties can be formally analyzed as a central open problem. Smoothed-analysis extensions of classical impossibility results (including Arrow's theorem) are noted as one promising direction, though the authors concede that current noise models remain strongly assumption-laden and not data-informed.

On multi-winner systems, the paper reports that STV with multi-member districts substantially curtails partisan gerrymandering in simulations across all 50 states, and—more strikingly—that race-blind neutral maps achieve proportional representation for racial and ethnic minorities comparable to maps optimized for minority representation. An empirical analysis of Portland, Oregon's 2024 STV city council elections found people of color elected candidates of choice in every district. The paper notes the legal context (Rucho v. Common Cause and the 2026 Louisiana v. Callais ruling) motivating interest in systems structurally resistant to manipulation, while acknowledging the 1967 Uniform Congressional District Act's ban on multi-member congressional districts as an implementation hurdle.

Three frontier directions are identified: richer voter-behavior models; preference formation and opinion dynamics (network models of polarization, sheaf-theoretic representations of context-dependent communication); and voting advice applications, where recent work using Swiss Smartvote data shows that small design choices in questionnaires, similarity metrics, and weighting can substantially change recommendations—an axiomatic analysis of these pipelines remains largely open.

Participatory budgeting: elicitation and proportionality

PB is framed as knapsack-style combinatorial optimization without an exogenous objective: $m$ projects with costs, budget $B$, and $n$ voters whose preferences must be both elicited and aggregated. The paper structures the literature around these two problems.

Elicitation via knapsack voting: Goel et al.'s format asks each voter to submit a budget-feasible bundle. In the continuous relaxation, knapsack voting is strategy-proof and welfare-maximizing; for indivisible projects the guarantees hold approximately. Deployment evidence exists—32 documented elections on the Stanford PB platform used knapsack ballots—and empirical evaluation shows voters select cheaper projects under knapsack than under $k$-approval ballots, consistent with the theory that approval formats obscure budget trade-offs. The limitation is acknowledged directly: a single feasible bundle conveys little about marginal values, substitutions, or complementarities, so richer tractable preference representations remain an open problem.

Aggregation via the Method of Equal Shares (MES): Peters, Pierczyński, and Skowron's rule divides the budget into virtual per-voter accounts of $B/n$ and iteratively funds the project with the smallest per-supporter price $\rho_j$, satisfying Extended Justified Representation (EJR). The empirical record here is concrete: in Wieliczka, Poland, MES reduced the share of voters receiving none of their approved projects from 28% to 18% relative to the greedy rule, and funded projects in a region the greedy rule shut out entirely; in Aarau, Switzerland, MES selected at least one project from every district where greedy selection favored only city-wide or central-district projects. These results carry a clear implication: proportionality axioms have measurable distributional consequences, not merely theoretical ones. Open questions include principled completion rules (MES may leave budget unspent), incentive effects (proportional rules may encourage narrow approval strategies), and welfare–proportionality trade-offs.

The frontier discussion covers two areas. First, new preference models: bundle-level voting that can surpass welfare impossibilities when voters weigh collective interests, and formal treatment of project dependencies (substitutes, complements, category limits). Second, project list design—the upstream shortlisting stage that exerts substantial power over outcomes yet is understudied—with emerging work on strategic proposers, cost manipulation, and ML/NLP tools for proposal filtering, for which no formal algorithmic guarantees (diversity preservation, robustness, end-to-end proportionality) currently exist. The existence of shared infrastructure—the Stanford platform and the Pabulib dataset library—is highlighted as lowering the barrier to testing new methods.

Deliberative democracy: sortition as a mature case study

The deliberation section treats aggregation-based mechanisms as insufficient for decisions requiring learning and preference revision, and documents the rapid institutionalization of deliberative mini-publics: over a thousand documented processes, with binding or permanent examples including Ireland's Citizens' Assembly (whose abortion recommendation was ratified by referendum), France's Citizens' Convention for Climate, Paris's permanent assembly, the Ostbelgien Model, and Mongolia's constitutional requirement of deliberative polling.

The mathematically developed core is sortition under demographic quotas. Because quotas overlap across marginal groups, feasibility itself is NP-complete, and the number of feasible panels precludes explicit enumeration. The equality-optimizing algorithms exploit Carathéodory's theorem: optimal lotteries exist over small supports, enabling column generation with duality-based certificates. Game-theoretic results later showed that equalizing individual selection probabilities also prevents manipulation via attribute misreporting, converting a normative preference into a formal requirement. When the objective shifts from first-moment equality to controlling joint selection probabilities—maximizing Shannon entropy over panels—Carathéodory sparsity no longer applies, and the solution requires convex duality combined with dynamic programming. Extensions use discrepancy theory for publicly transparent rounding of lotteries and PAC-learning bounds for selecting alternates against dropout risk.

Deployment is unusually direct: the equality-optimizing algorithm runs in StratifySelect and Panelot.org, which drew 1,071 assemblies over 294 distinct days during 2024–2025. This is among the clearest instances in the paper of theory-to-practice transfer at scale.

Frontier directions include group-division as constrained submodular maximization, interaction protocols optimizing outcome quality, and—most conceptually significant—the problem that deliberation's alternative space is generated endogenously rather than fixed. Generative social choice combines justified-representation guarantees with LLMs to produce proportionally representative statements from free text; the Habermas Machine synthesizes jointly endorsable statements; platforms like Pol.is and Remesh confront sparse participant–statement matrices where both rows and columns arise from participants, raising open questions about adaptive statement display and partial-preference aggregation. The paper is careful to note that these tools leave open their own axiomatic guarantees and implicit assumptions about opinion structure.

Limitations and open questions

The paper is candid about the field's dependence on assumptions. Empirical voting conclusions inherit the counterfactual-behavior caveat noted above; smoothed-analysis guarantees rest on noise models disconnected from real data; MES requires completion heuristics and has unresolved incentive properties; sortition's quota structure handles marginal groups better than intersectional ones; and generative deliberation tools lack established guarantees. The most consequential cross-cutting open question is whether robust, data-driven models of voter and participant behavior can be built such that conclusions about voting rules, allocation rules, and deliberation protocols remain stable under them—a question on which the IRV-versus-Condorcet moderation discrepancy shows substantive stakes.

Conclusion

The article's central claims are twofold: real democracy is simultaneously a consumer and a generator of mathematics, and normative questions about what mechanisms should do cannot be settled formally—mathematics is most valuable in partnership with political science, law, practitioners, and citizens. On AI, the authors take a specific position worth noting: because democratic input is naturally linguistic, LLMs enable participation at scale, but collapsing the pipeline into model output would outsource democratic judgment; the productive role they propose is AI as translator between human expression and formal objects governed by explicit axioms and guarantees. The concrete open problems identified across the three case studies—behaviorally grounded voting models, tractable rich preference elicitation in PB, end-to-end guarantees for agenda-setting pipelines, and axiomatic foundations for generative deliberation tools—define a research agenda that is mathematically substantive and directly coupled to deployed systems.