DistriVoting: Distributed Voting Systems

• DistriVoting is an umbrella framework encompassing voting systems that output probability distributions over alternatives or distribute aggregation across districts and networked agents.
• It integrates approaches like randomized social choice, district-based mechanisms with distortion guarantees, expressive cumulative ballots, and fully decentralized, verifiable protocols.
• Research in DistriVoting focuses on resolving trade-offs between expressiveness, efficiency, fairness, secrecy, and robustness across diverse technical implementations.

“DistriVoting” (Editor's term) is best understood as an umbrella label for voting systems in which either the social outcome is itself a distribution over alternatives, the aggregation process is distributed across districts or networked agents, or both. In the literature considered here, that label spans randomized social choice under hidden cardinal utilities, district-based distributed voting with distortion guarantees, cumulative and quadratic ballots in participatory budgeting, fine-grained delegation over cumulative ballots, fully decentralized voting/ranking protocols on communication graphs, and distributed end-to-end verifiable election infrastructures (Ebadian et al., 2022, Filos-Ratsikas et al., 2023, Salehkaleybar et al., 2017, Chondros et al., 2016). This suggests that the unifying object is not a single mechanism, but a family of architectures for distributing either outcomes, information, authority, or computation.

1. Conceptual scope

A first line of work studies voting rules that output a probability distribution over alternatives rather than a single deterministic winner. A second line studies two-stage distributed aggregation, where agents are partitioned into disjoint districts or groups, each district selects a representative alternative, and a final winner is chosen from those representatives. A third line studies fully decentralized or threshold-cryptographic protocols in which tallying, verification, or vote storage are themselves distributed across nodes, trustees, or peer-to-peer overlays (Ebadian et al., 2022, Filos-Ratsikas et al., 2019, Chondros et al., 2016).

These strands differ in what is “distributed.” In randomized social choice, the distribution is the outcome. In district-based voting, the decision process is geographically or institutionally distributed. In network protocols such as DMVR, VotingFarm, D-DEMOS, BVOT, and Kademlia-based reputation voting, the computation, storage, or verification path is distributed across communicating nodes or authorities (Salehkaleybar et al., 2017, Florio, 2016, Javani et al., 2020, Evseenko, 2013).

A persistent source of ambiguity is the distinction between distributed voting and distributional voting. The former concerns decentralization of aggregation or infrastructure; the latter concerns lotteries, support allocation, or probabilistic outcomes. The recent literature contains both, and several papers make them interact: randomization can be the social outcome, the preference-generation model, the in-district selection rule, or the over-district rule (Filos-Ratsikas et al., 2023, Goyal et al., 2024).

2. Randomized outcomes and probabilistic preference generation

In “Optimized Distortion and Proportional Fairness in Voting” (Ebadian et al., 2022), a voting rule decides on a probability distribution over a set of $m$ alternatives based on rankings, while agents are assumed to have hidden cardinal utility functions over those alternatives. The paper studies distortion, defined as a worst-case approximation ratio comparing the utilitarian social welfare of the optimum outcome to the social welfare produced by the outcome selected by the voting rule, and shows the first voting rule achieving the optimal distortion $\Theta(\sqrt{m})$ for unit-sum utilities. It also achieves optimal $\Theta(\sqrt{m})$ distortion for a larger class of utilities, including unit-range and approval utilities, and gives a voting rule that can achieve a $\Theta(\log m)$-approximation to proportional fairness, and thus also to Nash welfare and to the core, with $\Theta(\log m)$ shown to be best possible. In this strand, randomization is not tie-breaking; it is the intended social choice object.

A related but distinct direction appears in “Metric distortion Under Probabilistic Voting” (Goyal et al., 2024), which replaces deterministic metric-induced rankings with stochastic rankings generated from the metric. In the Plackett–Luce model, rankings are sampled according to

$$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$

with candidate strength taken to be inversely proportional to distance raised to power $\theta$. Under this model, the distortion landscape changes sharply: with strength inversely proportional to the square of metric distance, Copeland’s distortion is at most $2$, whereas Random Dictator has distortion $\Omega(\sqrt{m})$ in large elections; for Borda, distortion is $\Theta(m^{1-2/\theta})$ when $\Theta(\sqrt{m})$0 and $\Theta(\sqrt{m})$1 otherwise (Goyal et al., 2024). The paper’s broader claim is that distortion under probabilistic voting better corresponds with conventional intuitions regarding rules such as Plurality, Copeland, Random Dictator, and Borda.

Taken together, these papers separate two roles for randomness. One role is outcome randomization, where the rule outputs a lottery over alternatives. The other is preference stochasticity, where the latent metric generates a distribution over rankings. Both matter for DistriVoting, but they evaluate different objects: one studies what should be selected; the other studies how reported rankings arise (Ebadian et al., 2022, Goyal et al., 2024).

3. District-based distributed voting and distortion

“The distortion of distributed voting” formalized the district-based model in which voters are partitioned into disjoint districts, each district elects a local winner, and the final outcome is selected by aggregating district winners rather than by running one centralized election over all voters (Filos-Ratsikas et al., 2019). For symmetric elections, the paper proves exact distributed distortion

$\Theta(\sqrt{m})$2

for Range Voting and

$\Theta(\sqrt{m})$3

for Plurality Voting, and shows that any deterministic ordinal voting rule has distributed distortion $\Theta(\sqrt{m})$4. It also shows that districting can be harmful even in best-case design: for every $\Theta(\sqrt{m})$5, there exists an instance such that no symmetric $\Theta(\sqrt{m})$6-districting allows the welfare-optimal alternative to win when districts use Range Voting, whereas the centralized Plurality winner can always be preserved by a suitable symmetric $\Theta(\sqrt{m})$7-districting.

“Revisiting the Distortion of Distributed Voting” generalized this model to the two-stage class $\Theta(\sqrt{m})$8-of-$\Theta(\sqrt{m})$9, with deterministic and randomized in-district and over-district rules (Filos-Ratsikas et al., 2023). It establishes asymptotically tight bounds for deterministic distributed mechanisms: $\Theta(\sqrt{m})$0 in the cardinal setting, $\Theta(\sqrt{m})$1 in the ordinal setting, and $\Theta(\sqrt{m})$2 for deterministic strategyproof distributed voting. It also shows that randomizing only at the global stage does not asymptotically help when the in-district rule is deterministic and ordinal, but that fully randomized distributed implementations of point-voting schemes can induce exactly the same probability distribution as the corresponding centralized rule. This yields distributed ordinal distortion between $\Theta(\sqrt{m})$3 and $\Theta(\sqrt{m})$4, independent of $\Theta(\sqrt{m})$5, via distributed implementations of point-voting schemes such as BCHLPS (Filos-Ratsikas et al., 2023).

The line-metric case has been studied much more sharply. “Tight Distortion Bounds for Distributed Single-Winner Metric Voting on a Line” considers agents and alternatives embedded on the real line and resolves the distortion of the average-of-max and max-of-average objectives at

$\Theta(\sqrt{m})$6

via the mechanisms $\Theta(\sqrt{m})$7-Leftmost-of-Rightmost and Rightmost-of-$\Theta(\sqrt{m})$8-Leftmost, with $\Theta(\sqrt{m})$9 (Voudouris, 2023). A later paper, “Tight Bounds On the Distortion of Randomized and Deterministic Distributed Voting,” gives a near-complete characterization for four objectives—$\Theta(\log m)$0, $\Theta(\log m)$1, $\Theta(\log m)$2, and $\Theta(\log m)$3—in general metrics (Abam et al., 21 Sep 2025). It reduces the deterministic upper bound for $\Theta(\log m)$4 from $\Theta(\log m)$5 to $\Theta(\log m)$6, establishes a tight lower bound of $\Theta(\log m)$7 for $\Theta(\log m)$8, tightens the deterministic upper bound for $\Theta(\log m)$9 from $\Theta(\log m)$0 to $\Theta(\log m)$1, and proves tight randomized bounds such as $\Theta(\log m)$2 for $\Theta(\log m)$3, $\Theta(\log m)$4 for $\Theta(\log m)$5 and $\Theta(\log m)$6, and $\Theta(\log m)$7 for $\Theta(\log m)$8 when only the second stage is randomized. When both stages may randomize, the exact distortion becomes $\Theta(\log m)$9 for $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$0 and $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$1, and nearly tight around $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$2 for $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$3 and $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$4 (Abam et al., 21 Sep 2025).

A further extension replaces desirable alternatives with obnoxious ones. “Metric Distortion of Obnoxious Distributed Voting” studies the goal of maximizing total distance from all agents under a two-step distributed mechanism and proves tight full-information distortion

$$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$5

in general metrics and $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$6 on the line or with two alternatives, as well as tight ordinal distortion

$$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$7

in general metrics and $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$8 on the line or with two alternatives (Voudouris, 2024). This variant shows that the interaction between districting and limited information is not confined to welfare-from-proximity models.

Across this strand, one repeated conclusion is that districting is not a benign implementation detail. The two-stage restriction can exclude the globally best alternative from the final stage, and the performance loss depends critically on whether randomization is introduced inside districts, only after district representatives are selected, or not at all (Filos-Ratsikas et al., 2019, Filos-Ratsikas et al., 2023, Abam et al., 21 Sep 2025).

4. Expressive ballots, cumulative support, and delegation

Another strand treats DistriVoting as the allocation of divisible support rather than the selection of a single winner. In “Fair and Inclusive Participatory Budgeting: Voter Experience with Cumulative and Quadratic Voting Interfaces,” cumulative and quadratic voting are described as distributional voting methods that enable participants to distribute a number of points across multiple options based on the intensity of their preferences (Wellings et al., 2023). The paper reports an implementation on Stanford Participatory Budgeting and a modest user study with 27 respondents: cumulative voting received a mean expression score of $$\Pr(\pi \mid v) = \prod_{k=1}^{m} \frac{w_{\pi_k}(v)}{\sum_{\ell=k}^{m} w_{\pi_\ell}(v)},$$9 versus $\theta$0 for ranked voting; $\theta$1 preferred cumulative voting, $\theta$2 preferred ranked voting, and $\theta$3 were unsure; and in the interface-layout comparison, Top + Side Bar was preferred by $\theta$4, compared with $\theta$5 for Top Bar, $\theta$6 for Side Bar, and $\theta$7 for no additional interface. At the same time, the semantic differential on the “complex but accurate” versus “simple but less accurate” scale had a normalized mean of $\theta$8, and the paper reports a p-value of $\theta$9 while also claiming significance at the $2$0 level (Wellings et al., 2023). The central empirical lesson is the tension between expressiveness and simplicity.

“Fine-Grained Liquid Democracy for Cumulative Ballots” provides a formal model for cumulative ballots in which each voter has a nonnegative vector over candidates summing to $2$1, and may partition the candidate set into bundles, assign a nonnegative budget to each bundle, and delegate that bundle to another voter (Köppe et al., 2022). The paper studies transitive partial delegation through fixed-point semantics over the resolved ballot profile $2$2, with per-voter and per-bundle constraints

$2$3

It analyzes four semantics—Exact Proportionality (EP), Exact Proportionality with Thresholds (EP-T), Exact Proportionality with Thresholds, Interpolated (EP-TI), and Weighted Convex Combinations (WCC)—and proves existence for EP via Kakutani, and for EP-TI and WCC via Brouwer, while showing that EP-T may fail to admit any solution (Köppe et al., 2022). The paper also proves that WCC is not a contraction, that the pseudo-gradient of $2$4 is not pseudo-monotone, and that fixed points can be non-unique. For fixed $2$5 and $2$6, an $2$7-strong approximation can be computed in time polynomial in the encoding size and $2$8 via Renegar’s algorithm (Köppe et al., 2022).

These works move DistriVoting away from winner determination toward support allocation, interface design, and delegation semantics. They are especially relevant to participatory budgeting and liquid-democratic settings, where the central object is not a district representative or lottery winner, but a divisible budget of influence distributed across projects or candidates (Wellings et al., 2023, Köppe et al., 2022).

5. Networked, verifiable, and self-tallying protocols

A different meaning of DistriVoting appears in fully distributed protocols. “Distributed Voting/Ranking with Optimal Number of States per Node” presents DMVR, a pairwise-interaction algorithm on a connected undirected graph with $2$9 nodes and $\Omega(\sqrt{m})$0 choices (Salehkaleybar et al., 2017). Each node maintains a set-valued state updated by union and intersection operations, and the protocol determines either the plurality winner or the full ranking of all choices. It converges in finite time with probability one, uses $\Omega(\sqrt{m})$1 nodal states for voting and $\Omega(\sqrt{m})$2 nodal states for ranking, proves the ranking-state count optimal, and gives $\Omega(\sqrt{m})$3 time complexity on complete graphs for given vote percentages, with runtime inversely proportional to the minimum vote-percentage gap among choices (Salehkaleybar et al., 2017).

At the systems level, “The Voting Farm: A Distributed Class for Software Voting” implements a distributed software voting mechanism for EPX-like message-passing multithreaded environments (Florio, 2016). A local voter is attached to one user module and connected to a clique of peer voters; the user module interacts only with its local voter, while the local voters broadcast inputs, gather all $\Omega(\sqrt{m})$4 values, and execute a selected voting algorithm. The class supports exact consensus, majority voting, generalized median, plurality voting, weighted averaging, simple majority voting, and simple averaging (Florio, 2016). The target use case is $\Omega(\sqrt{m})$5-modular redundancy and restoring organs rather than political elections, but the architecture is a direct example of distributed vote computation.

Cryptographic distributed voting protocols push further. “Distributed, End-to-end Verifiable, and Privacy-Preserving Internet Voting Systems” introduces D-DEMOS, with a distributed Vote Collection subsystem, a replicated and fault-tolerant Bulletin Board, and trustees who control tally production while guaranteeing privacy and end-to-end verifiability as long as a strong majority is honest (Chondros et al., 2016). The suite contains D-DEMOS/IC, which uses Interactive Consistency and minimal timing assumptions for better performance, and D-DEMOS/Async, which uses asynchronous binary consensus and has higher message and signature overhead. The system’s voting operation is designed to be human verifiable: a voter can vote over the web even when the web client stack is potentially unsafe, without performing cryptographic operations locally, and still be assured the vote was recorded as cast (Chondros et al., 2016).

“BVOT: Self-Tallying Boardroom Voting with Oblivious Transfer” addresses the small-$\Omega(\sqrt{m})$6, boardroom setting (Javani et al., 2020). It uses multiparty threshold homomorphic encryption, masked unique primes associated with candidates, and oblivious transfer with an untrusted distributor so that each voter privately receives the masked prime for the chosen candidate. Ballots are encrypted and broadcast; after all votes are cast, multiplying all ballots, decryption shares, and the unmasking factor yields a product of primes whose exponents encode the tally. The protocol provides ballot secrecy, fairness, dispute-freeness, self-tallying, and support for multiple candidates without zero-knowledge proofs, but it is not robust and not coercion resistant (Javani et al., 2020).

At the peer-to-peer layer, “New hybrid distributed voting algorithm” proposes a decentralized vote system on structured overlay Kademlia networks (Evseenko, 2013). Vote data are stored under a hashed document ID or info-hash, replicated to the $\Omega(\sqrt{m})$7-nearest nodes, and maintained using a 24-hour ring buffer with separate positive and negative HyperLogLog counters. The protocol introduces announce_vote and get_votes, adopts a one-vote-per-IP heuristic, and aims at low local resource consumption and low traffic. The same design, however, leaves Sybil resistance, malicious-node filtering, vote aggregation across replicas, and NAT collisions only weakly addressed (Evseenko, 2013).

These protocol papers treat DistriVoting as a systems problem: how to distribute the act of tallying, storing, verifying, or protecting votes across nodes or trustees without collapsing the system back into a single point of failure (Salehkaleybar et al., 2017, Chondros et al., 2016, Javani et al., 2020, Evseenko, 2013, Florio, 2016).

6. Trade-offs, misconceptions, and open directions

One recurring misconception is that randomization in voting is merely tie-breaking. The randomized social-choice literature instead treats the lottery itself as the social outcome, with distortion and fairness evaluated on expected welfare under hidden utilities; the probabilistic-voting literature treats randomness as part of the preference-generation model and shows that rule quality can reverse relative to deterministic distortion benchmarks (Ebadian et al., 2022, Goyal et al., 2024). These are technically distinct roles for probability and should not be conflated.

A second misconception is that distributing authority automatically improves outcomes. District-based distortion results show the opposite in the worst case: partitioning voters into districts can considerably increase distortion, and randomizing only after district representatives are chosen often does not repair the information loss introduced at the local stage (Filos-Ratsikas et al., 2019, Filos-Ratsikas et al., 2023). The strongest positive results in this line come either from highly structured settings, such as the line metric, or from introducing randomization inside districts through point-voting or randomized representative selection (Voudouris, 2023, Abam et al., 21 Sep 2025).

A third recurring trade-off is expressiveness versus usability. In participatory budgeting, the Stanford deployment study reports that cumulative voting was preferred over a simpler ranked method, yet respondents also leaned toward a “simple but less accurate” method in the abstract, and the interface required explicit token tracking, dynamic feedback, and strong visual affordances such as the Top + Side Bar combination (Wellings et al., 2023). Fine-grained liquid democracy sharpens the same trade-off at the semantics layer: robust delegation rules such as WCC guarantee existence and continuity, but fixed points need not be unique and simple iterative solution procedures do not enjoy contraction guarantees (Köppe et al., 2022).

A fourth trade-off is fairness and secrecy versus robustness. BVOT obtains fairness by using an effectively $\Omega(\sqrt{m})$8-out-of-$\Omega(\sqrt{m})$9 threshold structure, but any voter can prevent completion by withholding a vote or decryption share (Javani et al., 2020). D-DEMOS removes many classical single points of failure by distributing vote collection, bulletin-board publication, and tally control, yet its setup still uses an Election Authority assumed to be destroyed after initialization, and its two variants expose a performance-versus-timing-assumptions trade-off (Chondros et al., 2016).

Finally, decentralization at the infrastructure layer is not an automatic remedy. The thesis “Design of Distributed Voting Systems” concludes that peer-to-peer networks and blockchain provide many advantages, like decentralization, which might be applicable to electronic voting systems, but also emphasizes serious problems for political elections, especially coercion or receipt-freeness and fairness; its proposed contribution in that direction is a modification of proof-of-stake intended to allow smartphones and tablets to participate in verification and ballot inclusion (Meter, 2017). On the social-choice side, open problems remain for randomized-first-stage deterministic-second-stage mechanisms, for the remaining narrow gaps in $\Theta(m^{1-2/\theta})$0 and $\Theta(m^{1-2/\theta})$1, and for extending line-metric techniques to general metrics (Abam et al., 21 Sep 2025, Voudouris, 2023).

The literature therefore presents DistriVoting not as a settled framework but as a set of intersecting research programs. Some treat distribution as a lottery over outcomes, some as a decomposition of aggregation across districts, and some as a systems architecture for distributed tallying and verification. The common theme is that voting can be distributed along multiple axes, but each axis introduces its own distortion bounds, interface burdens, cryptographic assumptions, and robustness constraints.