Multi-dimensional voting beyond peak-preferences 

Advance publication December 14, 2025

Abstract

We study voting in continuous multi-dimensional spaces beyond single-peak preference ballots. While single-peak methods allow voters to specify a singular most-preferred candidate, they do not allow voters to express indifference, different importance, or a lack of knowledge regarding particular dimensions.

We propose Nested Convex Voting, which we believe to be a novel method for continuous multi-dimensional voting. In this approach, each voter submits a finite sequence of nested convex sets of candidates with decreasing weights. The first convex set represents a set of equally ranked peak preferences, while subsequent convex sets allow voters to express a willingness to compromise within delimited regions of the candidate space.

Based on the submitted ballots, each candidate is assigned a depth based on half-spaces bounded by affine hyperplanes passing through that candidate. Candidates with maximum depth are selected as winners. In the special case where all ballots consist of a single candidate, the method coincides with the Tukey median.

We further outline Nested Convex Delegated Voting, combining Nested Convex Voting with elements of Liquid Democracy by resolving indifference in certain dimensions through depth-first delegation traversal.

A heuristic implementation based on randomly sampled hyperplanes is provided as a concrete implementation example, while an axiomatic and complexity-theoretic analysis is left for future work.

The presented approach is mainly intended as a contribution to theory-building and to the discussion of voting in continuous multi-dimensional spaces. It is not proposed as a general-purpose democratic voting rule, but might be a suitable method applicable to contexts in which voters' preferences can be reasonably assumed to be convex and in which deliberation or prior discretization of candidates is limited.

Peak-preference-based voting systems

Preferential voting systems allow voters to express a preference order (usually a weak order) on all proposed candidates on their ballot.

Single-peak-preference-based voting systems, in contrast, allow voters to express only their topmost choice when casting a ballot. This is a relevant simplification which makes multi-dimensional continuous voting possible, in which the number of candidates (i.e. possible winners) is infinite.

An example for an infinite set of candidates (potential winners) is the set of all non-negative rational triples with a sum of 1, containing, for example (1/2, 1/4, 1/4) or (1/14, 7/14, 3/7), which could reflect relative allocations of a fixed budget to different projects (also called “portioning”).

Another example for an infinite set of candidates is the Cartesian product of all applicable temperatures, pressures, pH values, and flow velocities for operating a chemical reactor.

It is possible to define single-peak-based voting systems in which each voter may freely pick their most-preferred choice from such an infinite set of candidates, and where, given a suitable tie-breaker, a unique winner is selected from that set. This class of systems includes the following voting rules, among others:

• Metric based systems, minimizing distances from winner to each voter's peak preference
  • Minimizing sum of squared Euclidean distances (simple averaging)
  • Minimizing sum of Manhattan distances [Lindner2008]
  • Minimizing sum of Mahalanobis distances [ElMahdi2023]
• Moving phantoms
  • Independent markets [Freeman2019]
  • Pareto optimal moving phantoms [Freeman2019]
• Tukey median / Condorcet ex ante [Nehring2023]

All systems, where each voter may cast a single peak preference only, suffer certain weaknesses by principle:

1. Single-peak-preference systems require voters to make a choice even if some of the available dimensions do not matter for any voter. For example, when assigning resources to projects, some of these projects (i.e. dimensions) might be clones, in the sense that it does not matter whether budget is assigned to one or the other project. Single-peak-based systems, however, require each voter to make a choice also with respect to those dimensions. This choice may have an impact on the selection of the winner.
2. Because voters only cast a singular peak preference, the voting rule cannot take into account that some voters may consider the importance of distinct dimensions in a different manner. For example, some voters might be entirely indifferent about certain dimensions (“I'm indifferent about shifting budget from A to B or vice versa.”), while other voters are indifferent about others (“I'm indifferent about shifting budget from B to C or vice versa.”). The true degree of prioritization versus indifference regarding specific dimensions might also be non-binary (e.g. “I have a clear opinion on A's and B's ideal budget, but assigning a specific budget to C is twice as much important for me.”). Single-peak voting systems, in contrast, make it entirely impossible for voters to express willingness to compromise with regard to certain dimensions.
3. Some voters might lack knowledge about what is best with regard to certain dimensions, but might have a strong opinion on other dimensions. Also here, single-peak-based systems do not allow voters to express that lack of knowledge.

We therefore assume that all single-peak multi-dimensional voting rules are only suitable for domain-specific problems, for which at least the following criteria are met:

• Voters are not indifferent regarding any dimension and can therefore pick a peak preference without having to make an arbitrary choice.
• The relative importance of dimensions may differ between different dimensions but must be (at least roughly) the same among all voters. (Note that metric-based systems additionally require the relative importance to be known or assumed beforehand, such that a suitable metric, like a specific Mahalanobis distance, can be chosen. [ElMahdi2023])
• For each dimension, a sufficient number of voters is knowledgeable.

Nested Convex Voting

With the aim of finding a voting system that may address the previously discussed constraints, we propose a system for continuous multi-dimensional voting, novel to our knowledge, which we name “Nested Convex Voting”.

Ballot design

Consider an infinite convex set describing all available candidates, for example the convex hull of (100%, 0%, 0%), (0%, 100%, 0%), (0%, 0%, 100%) in a three-dimensional space when distributing a budget to three different projects (Figure 1). Note that in this example, the resulting set can be depicted as a two-dimensional triangle because we demand that the total budget is spent, which leaves two degrees of freedom. Generally any number of dimensions is possible.

Each voter's ballot consists of a finite sequence of convex sets, each being a subset of the set of all candidates, and each having a positive numeric rational weight assigned by the voter.

Additionally, the convex sets and associated weights on a ballot are subject to the following constraints:

• the convex sets on the ballot can be described as a convex hull of a finite set of vertices,
• each convex set on the ballot, except the first one, must be a proper superset of the previous one,
• the weights associated with convex sets in the sequence must be strictly monotonically decreasing.

Ballot semantics

The first convex set listed on the ballot reflects the voter's set of equally ranked peak preferences. This allows voters to express indifference in one or more particular dimensions. For example, consider deciding on distributing a budget to three projects A, B, and C, where B and C are clones. Each dimension determines resources assigned to the respective project. If a voter wants to assign 40% of the resources to A but is indifferent about B and C, the voter may state the convex hull defined by the vertices (40%, 60%, 0%) and (40%, 0%, 60%) as their set of peak preferences. In this case, the convex hull is essentially a line segment.

While a set of peak preferences (opposed to a single peak preference) solves the issue that one or more dimensions might be entirely irrelevant for a particular voter, it doesn't allow expressing a gradual prioritization of dimensions yet. For example, assume that a voter wants to assign 40% to A, but also has an idea about how to distribute the remaining funds on B and C: He or she wants to assign 35% to B and 25% to C. Let's assume that while the voter assesses the 40% assignment to A as a high-priority, the voter has no strong opinion on favoring B over C, or vice-versa, as long as C retrieves at least 10% funding. In that case, the voter could cast the following ballot:

• Weight 10: Convex hull of a single candidate (40%, 35%, 25%), resulting in a single candidate
• Weight 5: Convex hull of two candidates (40%, 50%, 10%) and (40%, 0%, 60%), resulting in line segment of candidates

The weights reflect a relative prioritization. For example, changing the weight of the line segment from 5 to 9 (while maintaining a weight of 10 for the peak preference) would reflect that the voter is more willing to compromise on shifting budget between B and C (within the given bounds), away from their topmost preference. A weight of only 1 for the line segment, in contrast, would reflect that the voter is much less willing to compromise their topmost preference.

Weights are relative within a ballot. Multiplying or dividing all weights of a ballot by a certain number does not change their meaning, e.g. picking weights 10 for the first and 5 for the second set is the same as picking weights 2 for the first and 1 for the second set).

While the notion of weights is some sort of cardinal utility, it doesn't directly reflect the cardinal utility with respect to the candidates in each set. Instead, the weight expresses willingness to compromise or to partially abstain with regard to certain dimensions and bounds.

Tallying

Based on the voters' ballots, a winner is calculated as follows.

As a preparation, we normalize all weights by dividing them by the first weight on the same ballot.

All vertices from every ballot are taken to create a set A of potential winners. Optionally, all possible candidates or any other suitable subset of them could be taken as a set of potential winners. However, this may drastically increase implementation complexity, depending on the particular choice and size of that set.

Each candidate C ∈ A gets a depth d_C assigned as follows.

1. We let H_C be the set of all closed half-spaces whose boundary is an affine hyperplane passing through candidate C.
2. For each half-space H ∈ H_C and each ballot B, we define w_C,H,B as the maximum weight of those candidate sets on that ballot of which at least one vertex lies within that half-space (including its hyperplane boundary). If no vertex of any candidate set of a ballot B lies in the half-space H, we let w_C,H,B = 0.
3. For each half-space H ∈ H_C, we calculate w_C,H as the sum of the selected weights from step 2 over all ballots, i.e. w_C,H = Σ_B w_C,H,B
4. Finally, we determine the depth d_C of the candidate C as the minimum of w_C,H over all closed half-spaces H ∈ H_C

The candidate C with the maximum depth d_C is then the winner. If there are several candidates sharing the same maximum depth, a tie-breaker is needed.

In cases where all voters cast only a singular convex set consisting of a singular point as a ballot, this method is equivalent to the Tukey median.

Implementation

Implementing the described tallying algorithm is not trivial, because the depth d_C of each candidate C is defined using an infinite set H_C of half-spaces.

As an appendix, we provide a Python program, which implements the voting rule heuristically as a Monte Carlo-style search strategy over randomly sampled hyperplanes using independent Gaussian components scaled by per-dimension variances.

The implementation optionally creates linear combinations of the vertices of convex candidate sets on the ballots as additional candidates in the heuristic search. This allows a winner to be found which has not been explicitly stated by one of the voters as a vertex but lies within a convex set of a ballot. Note that the number of linear combinations created has an impact on the calculation speed.

No absolute termination criterion is provided by our implementation. Instead, the program indefinitely attempts to improve the upper bounds for the depths d_C of the most promising candidates and prints outs a set of winners, along with a suitable tie-breaking result if more than one candidate is in the set of winners.

Example data and result

Consider the 12 example ballots depicted in Figure 5.

In this example, we have 12 voters, of which 8 report only a single convex set with a singular candidate as their peak-preference (black points in Figure 5). The remaining 4 voters (colored in Figure 5) report two convex sets each. The first set of those 4 voters, respectively, is a singular candidate (their peak-preference), and the second set contains a 1-dimensional or 2-dimensional region with a lower weight.

Using our implementation, we feed these example ballots from Figure 5 into our tallying program:

from nested_convex_voting import *
from fractions import Fraction as Frac

nested_convex_voting([
    [(1, [(20,200)]), (Frac(9,10), [(10,500), (40,300), (10,100)])],
    [(1, [(30,300)]), (Frac(3,10), [(30,300), (40,350)])],
    [(1, [(40,400)])],
    [(1, [(60,600)]), (Frac(7,10), [(45,300), (65,700)])],
    [(1, [(70,700)]), (Frac(5,10), [(70,800), (80,800), (80,100), (50,100)])],
    [(1, [( 5,600)])],
    [(1, [(10,600)])],
    [(1, [(15,600)])],
    [(1, [(20,600)])],
    [(1, [(50,800)])],
    [(1, [(55,800)])],
    [(1, [(60,800)])],
])

The final output (intermediate results omitted) is as follows:

1 winner with depth 21/5:
(40, 400)

Here, the winner's depth is 21/5 = 4.2. We note that while there may be candidates with a greater depth, those might not be found if they are not in the set of potential winners considered by the heuristic. In the given example, such a candidate exists, e.g. (40, 600), but it is not considered as a potential winner because it is not within any ballot's convex sets.

When we remove the voters' willingness to compromise on their peak-preferences, i.e. when we remove all candidate sets but the first one on their ballots, we obtain a different result:

from nested_convex_voting import *

nested_convex_voting([
    [(1, [(20,200)])],
    [(1, [(30,300)])],
    [(1, [(40,400)])],
    [(1, [(60,600)])],
    [(1, [(70,700)])],
    [(1, [( 5,600)])],
    [(1, [(10,600)])],
    [(1, [(15,600)])],
    [(1, [(20,600)])],
    [(1, [(50,800)])],
    [(1, [(55,800)])],
    [(1, [(60,800)])],
])

The final output (intermediate results omitted) of the second example is as follows:

1 winner with depth 4:
(20, 600)

Here, a different candidate wins with a depth of 4.

For completeness, we mention that in the situation depicted in Figure 6, the winning candidate (20, 600) is not the only point with a (Tukey) depth of 4. For example, the point (30, 600), which is a bit to the right, also has a Tukey depth of 4. That point, however, is not considered as a potential winner by the heuristic search algorithm because it is not part of any voter's set and thus not included in a tie-breaking.

Comparison with preferential voting

Without analyzing specific properties of the tallying method of Nested Convex Voting, we still can compare it with preferential voting systems by considering the difference in ballot design.

Advantages

The presented system may have an advantage to preferential voting, when the number of candidates cannot be restricted to a small finite set. This is, because in multi-dimensional scenarios, the number of potential candidates grows fast with an increase of dimensions, even if each dimension is discretized.

By allowing voters to specify convex sets through a finite number of vertices, it is possible to express an opinion regarding an infinite number of candidates on one's ballot. Note, however, that the number of vertices to describe, for example, an n-dimensional hypercube still grows exponentially (2ⁿ). Here, suitable input methods would be required for a participant of such a voting system, which automatically creates the vertices, for example, based on several specified ranges. See also section “Challenge of UI design” below.

Another advantage of Nested Convex Voting is that it allows voters to partially abstain in certain dimensions. Preferential voting usually allows voters to rank two candidates equally (if it demands a weak order), but then the voter expresses complete indifference between them. Nested Convex Voting, in contrast, allows a voter to express a continuous degree of abstaining with regard to certain dimensions or bounds by giving additional convex sets on their ballot a certain weight.

Disadvantages

The most obvious potential disadvantage of Nested Convex Voting is the rather complex ballot in the sense that the concept of (nested) convex subsets of a candidate set might be difficult to grasp for humans in case of a dimensionality of 4 or higher.

While the ability to select a candidate from a potentially infinite number of candidates on one's ballot can be seen as an advantage, it also comes with a risk: Voters may cast an opinion on their ballot, which has not been considered as part of a preliminary deliberation. In a democratic context, we anticipate with concern that this might foster egoistic behavior such that minority viewpoints are not sufficiently considered.

Moreover, Nested Convex Voting presumes that voters' preferences are convex: A voter cannot express that, for example, they would be okay with assigning € 100,000 to a project or € 0 to that project, but not € 10,000. This renders the system unsuitable for non-convex problems, which might be predominant in the real-world. Considering our previous example of an industrial machine whose temperature and pressure is adjusted, we would also like to note that many physical processes are non-linear by nature, and convex sets might be unsuitable to describe safe operating regions.

Last but not least, computing the winner poses computational challenges. Potential optimizations of our prototype algorithm could be explored in future research.

Deliberation as a discretizer

Emphasizing the importance of deliberation, it is worth considering whether reducing multi-dimensional spaces to a finite number of candidates might be a better approach to high-dimensional problems. For example, the decision-making process as implemented in LiquidFeedback [PLF] already performs such a discretization in its admission, discussion, and verification phases.

This process of agreeing on a small number of candidates may facilitate an exchange of arguments and could lead to better informed decisions including the consideration of minority viewpoints. [PLF]

Use cases with limited deliberation

There may be scenarios, e.g. in a business context, where deliberation is not possible or wanted, e.g. because it is inefficient or uneconomical. Specifically scenarios in which there is no time (or not enough time) for deliberation may be a potential use case for Nested Convex Voting. An example would be a group of experts who need to agree in real-time on adjusting a specific set of parameters for a complex machine or for a company's strategy. Here, a system like the described Nested Convex Voting method might be of interest, as it allow voters to cast a ballot independently of a previous agreement on discrete voting options.

Another potential use case covers artificial intelligence, where the (partially indefinite) output of different components shall be aggregated. Using Nested Convex Voting, each AI component could specify its confidence with regard to specific dimensions and bounds.

Challenge of UI design

In case of a human electorate, creating a practical user interface for Nested Convex Voting is a major challenge. In case of budget assignments to three projects where the total budget must be spent, the convex sets and selected weights may be displayed on a two dimensional screen or marked on a paper ballot (compare Figure 1). But as soon as the number of dimensions increases, displaying such a ballot to participants on electronic media or conceiving a suitable paper form becomes difficult.

At this point, we have no feasible idea for UI design yet, but one possibility might be to present each dimension as a slider with selectable minimum and maximum value. However, constraints such as requiring the sum of two scaled values being within certain limits would need to be selectable somehow, which seems to be an interesting challenge in UI design.

Alternatively, additional constraints could be added to all ballots in advance, such that the voters are limited in expression. For example, the number of convex sets on each ballot or the number of vertices for each convex set could be limited. This of course, would lead to a different voting system that might lack certain properties in comparison.

As previously mentioned, reducing each voter's ballot to a single candidate (by limiting the number of convex sets to 1 and requiring that the set may only consist of a single vertex), we effectively obtain the Tukey median as a resulting voting rule. However, other less restrictive simplifications could be made, resulting in a trade-off between ballot complexity and voting system properties.

Combination with Liquid Democracy

When voters have a lack of knowledge regarding certain decisions, Liquid Democracy provides an interesting approach to ensure voters' representation. Liquid Democracy, among other principles, allows participants to choose whether casting a direct vote or delegating a decision to another person. [PLF]

We will now combine Nested Convex Voting with aspects of Liquid Democracy, creating another voting system, which we will call “Nested Convex Delegated Voting”.

The general idea is as follows: Recalling that Nested Convex Voting allows voters to express indifference in certain dimensions or bounds, we now allow voters to provide a delegate either before or after their own vote on the ballot. If a delegate is provided after the own vote, then the delegate can refine the vote only within the indifferent dimensions and bounds of one's own vote. If a delegate is provided before the own vote, then the delegate decides primarily, but one's own vote can make refinements regarding the delegate's vote.

A conceptually related approach has been proposed in a different context under the term “Pairwise Liquid Democracy”, in which delegations can be made for voters' indifference classes. [Brill2018]

Note, however, that Nested Convex Delegated Voting, as outlined in the following, does not allow delegating different dimensions to different voters. Instead, we propose that a voter can provide a list of delegates, which are resolved in a depth-first manner. Compare also [Brill2022].

The tallying procedure for Nested Convex Delegated Voting starts with collecting for each voter a chain of votes, using a depth-first traversal, i.e. the votes of the voter itself and all his or her delegates are flattened into a sequence. Each vote in that sequence then has the same format as the ballots in (non-delegated) Nested Convex Voting and consists itself again of a sequence of nested candidate sets with an assigned weight.

We recall that in Nested Convex Voting it is sufficient that only some vertices of a ballot's set lie within a given half-space: If not all vertices are lying outside of that half-space, this is still counted in favor of the candidate. Nested Convex Delegated Voting is a modification of this rule such that if some vertices lie inside and some outside a half-space, then the candidate set of a vote will be considered “indifferent” with regard to that half-space.

In case of such an indifference, subsequent votes in the chain of votes are considered, which may resolve the indifference by counting it in favor of the candidate (all vertices of a convex set are in the half-space) or against it (all vertices of a convex set are outside of the half-space). If, within that process, all votes are indifferent regarding that candidate, we still count it (for this half-space) in favor of the candidate, same as in the non-delegated variant of the algorithm.

The actual rules are somewhat more complex because they have to take the different weights into account. For details, we refer to the implementation in the appendix.

As an interesting side remark, we would like to point out that counting indifference (that is not resolved by delegation) in favor of a candidate resembles the notion of measuring pairwise links in the Schulze method by the strength of the winning votes, i.e. an abstention in a pairwise comparison by the Schulze method makes it more difficult for a candidate to defeat another candidate. Refer to [Schulze2017] as well as [PLF], page 99, for a discussion of strengths in the context of the Schulze method.

A potential extension to Nested Convex Delegated Voting could be to restrict different delegations to different dimensions. This might be achieved by providing a set of hyperplane normal vectors per delegation, describing a subset of “directions”. Considering the ballot complexity, however, this might be specifically interesting for otherwise simplified variants of the proposed system, e.g. when restricting voters' ballots in certain ways as discussed in the previous section on UI design.

Limitations and scope

Neither Nested Convex Voting nor Nested Convex Delegated Voting are intended as a general purpose democratic voting rule.

The method should instead be understood as an aggregation mechanism for convex preferences on candidates in a continuous multi-dimensional space in expert driven or technical contexts, in which heuristic evaluation and high ballot complexity are acceptable.

Furthermore, it is meant as a contribution to theory building in the context of (computational) social choice.

Summary

Multi-dimensional voting systems, which allow voters only to present a singular peak preference on their ballot, suffer problems related to

• voters being indifferent with regard to certain dimensions,
• voter-dependent importance of different dimensions, and
• insufficient knowledge of voters regarding specific dimensions.

We presented a system named “Nested Convex Voting” aiming to overcome these issues. However, further analysis regarding suitability and properties is required. In any case, the voting system serves as show case for an extended kind of ballot covering a continuous multi-dimensional candidate space and arguably contains more valuable information than (singular) peak preferences. We also pointed out an interesting parallel between Nested Convex Voting and the Schulze Method with regard to pairwise defeats.

A rough sketch of how to combine the proposed voting method with aspects of Liquid Democracy has been provided along with a respective prototype implementation.

Considering the complexity of multi-dimensional ballots, modified versions of “Nested Convex Voting” are thinkable, for which a different trade-off between ballot complexity and desirable properties of the system is made. Depending on these simplifications, this might also allow for extensions of the delegation model such that different dimensions could be delegated to different voters.

Given the importance of deliberation, we recall that existing methods with a preferential voting, such as the LiquidFeedback opinion formation and decision-making process [PLF], might be already well suitable for high-dimensional decisions, considering that the final voting can be accompanied by a deliberation during which discrete proposals are developed and justified. In contrast, voting systems in which participants may freely state any point in a multi-dimensional space on their ballot might bear the risk of egoism-driven behavior that disregards relevant minority viewpoints previously discussed during deliberation.

Nonetheless, the presented aggregation mechanism might be suitable in scenarios in which deliberation is infeasible or even outside of democratic contexts such as aggregating the output of different artificial intelligence subsystems or combining expert opinions or strategies in a time-efficient manner.

Updates / Errata

This document has been updated on December 15, 2025.
In the context of the example depicted in Figure 5, we clarified that the winner found by the heuristic is not (necessarily) the candidate with the maximum depth.

Acknowledgement

The authors acknowledge support by the European Union under the Horizon Europe Perycles project (Participatory Democracy that Scales).

Funded by the European Union. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Appendix: Implementation

• Python module nested_convex_voting
• Examples
  • example1.py
  • example2.py
  • example-with-delegations.py