Finding Possible Winners in Spatial Voting with Incomplete Information

Andreas Wiese; Hadas Shachnai; Rotem Shavitt

REVIEW 2 major objections 4 minor 34 references

Finding Possible Winners in Spatial Voting with Incomplete Information

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that the possible-winner problem in one-dimensional spatial voting is polynomial-time solvable for every fixed top-k scoring rule, and fixed-parameter tractable in the number of candidates in any dimension.

desk verdict The paper's headline result, polynomial-time possible winners for constant-k truncated rules in 1D, is not established: the reduction to shapes scheduling relies on Lemma 5, and Lemma 5 is false. read the letter →

arxiv 2505.12451 v1 pith:ZZNLM4OB submitted 2025-05-18 cs.GT

Hadas Shachnai , Rotem Shavitt , Andreas Wiese This is my paper · ORCID

classification cs.GT MSC 91B1291B1468Q1768Q27

keywords possiblewinnerproblemspatialvotingpositionalscoringrulesk-truncatedapprovalshapesschedulingfixed-parametertractabilityweightedvoters

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks when a designated candidate can still win an election in the spatial model when each voter's ideal point is known only by an interval in each issue. Its central result is that in one dimension, for any scoring rule that gives positive weight to only a fixed number k of top ranks, the possible-winner question can be decided in polynomial time; the previous tractable cases were much narrower. In higher dimensions, where the same question is NP-complete even for simple rules, the paper shows the problem is fixed-parameter tractable in the number of candidates for every positional scoring rule and for approval voting. It also maps the weighted one-dimensional case: two-valued rules are easy exactly when they approve at least half the candidates, and Borda becomes NP-complete already at four candidates. A reader should care because these results draw a much clearer boundary between tractable and intractable uncertainty in a natural model of elections.

What carries the argument

The load-bearing device is the reduction from voter uncertainty to shapes scheduling. A shape is a vector saying how many machines a job occupies in each of k consecutive time slots; in the voting instance, time slot t is candidate c_t, and a voter-job's shape at start time t records the scores the voter could give to candidates c_t, ..., c_{t+k-1} from some feasible position. The generated job set is P-structured: every job has the same processing length, interior times share one global set of shapes, and jobs can be ordered so that deadlines and deadline shape sets are monotone. A dynamic program, built on a known equal-length jobs scheduling recursion, then decides whether the jobs fit on M* machines with all machines busy during the slot of the designated candidate. The monotonicity of a voter's top-k window as her ideal point moves rightward is what makes the instance P-structured.

What would settle it

Compare the paper's dynamic program with exhaustive search over all finitely many position segments for small one-dimensional instances under truncated Borda; any mismatch about whether a designated candidate can win would refute the claimed polynomial-time algorithm.

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Extended reading notes

Core claim

The paper establishes that the one-dimensional possible-winner problem can be solved in $O(n^{1+k+3k^2}m^3)$ time for every k-truncated positional scoring rule with constant k. The proof converts each voter into a job in a new 'shapes scheduling' problem, where time slots are candidates and the shape a job occupies records the scores the voter would hand to a sliding window of k candidates; a feasible schedule with all M* machines busy at the target candidate's slot is exactly a preference completion in which that candidate wins. For d >= 2 dimensions, it proves that the same problem is fixed-parameter tractable when the parameter is the number m of candidates, for any positional scoring rule, by typing each voter according to which of finitely many voting vectors it can realize and solving a fixed-variable integer program. The same FPT conclusion holds for approval voting. For weighted voters in one dimension, the paper gives a dichotomy for two-valued rules and proves that Borda is NP-complete even when there are only four candidates.

Load-bearing premise

The reduction behind the main polynomial-time result assumes that, as a voter's ideal point moves rightward on the line, the set of candidates in her top k moves rightward monotonically without skipping; if a different tie-breaking rule or a different distance made the set skip around, the scheduling construction would no longer be valid.

Editorial extensions

If this is right

Any fixed-top-k truncated rule in one dimension, including k-truncated Borda and common contest scoring vectors, becomes polynomial-time solvable.
In any number of dimensions, the possible-winner problem for every positional scoring rule is fixed-parameter tractable in the number of candidates, so few-candidate elections remain solvable even in high-dimensional issue spaces.
Approval voting, NP-complete in every dimension, is also fixed-parameter tractable in the number of candidates.
For weighted voters on a line, a two-valued rule is tractable exactly when it approves at least half of the candidates in every candidate count; otherwise the problem is NP-complete.
Weighted Borda is NP-complete with only four candidates on a line, in contrast to the polynomial-time unweighted one-dimensional case.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The FPT result suggests that the practical bottleneck is the number of candidates, not the dimension; one natural test is whether the same voter-typing argument survives when distances are not Euclidean.
The weighted dichotomy implicates the weights themselves, rather than spatial geometry alone, as the source of NP-hardness; a plausible extension is that other multi-valued scoring rules become hard in the weighted setting once the rule gives positive weight to fewer than half the candidates.
The shapes-scheduling formulation could be reused directly for the necessary-winner problem, since the same schedule already encodes upper bounds on every candidate's score.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the main claim is not proven. Lemma 5, the bridge from possible winners to shapes scheduling, is false. The promised P-structured order does not exist, so the DP in Section 3.3 has no valid recursive basis.

Counterexample: take candidates at 0, 10, 20, 30 and the 2-truncated rule (2,1). Voter 1 with interval [10.1,14.9] always ranks c2,c3 as its top two, so its endpoint shape set is {(2,1)}. Voter 2 with interval [15.1,19.9] always ranks c3,c2, so its endpoint shape set is {(1,2)}. Both jobs have the same deadline, but neither shape set is a subset of the other. Therefore no total order satisfies Definition 3's inclusion condition. Lemma 5's proof only handles the case where the later interval lies to the left of the segment defining the shape; the right-side case is exactly this counterexample. Since Lemma 6's swap argument and the DP's choice of a 'last job' use the inclusion property, Theorem 1's proof collapses at this point.

Credit where it is earned: the reduction idea is natural, the shapes scheduling framework is a nice contribution, and the parameterized results are genuinely useful. Theorem 4's FPT algorithm via voter types and an ILP in fixed dimension is plausible, as is the approval-voting variant. The weighted dichotomy for two-valued rules and the Borda hardness for m=4 (Theorems 6 and 7) are independent of the broken lemma and look mostly solid, though I did not verify every case in Theorem 6. The paper builds honestly on Imber et al. and gives detailed proofs, which is why the false lemma stands out as a real error rather than a sloppy statement.

The notational issues the reader flagged (j'' vs j, j') are minor by comparison. The real problem is that the reduction does not produce P-structured instances in general. I cannot tell from this version whether a different proof strategy can save Theorem 1; the property as defined is simply not satisfied.

Who this paper is for: researchers in computational social choice working on possible winners and spatial voting. The FPT and weighted results may be worth citing on their own, but the polynomial-time result for truncated rules should not be cited as established until Lemma 5 and the DP are repaired. A serious referee should see this paper, because the problem is important and some results are valuable, but the current version is not acceptable.

Referee Report

2 major / 4 minor

Summary. The paper studies the possible-winner (PW) problem in spatial voting where each voter's ideal point is known only through a box of intervals in d-dimensional Euclidean space and each candidate's position is known exactly. The main claims are: (i) for any fixed k, PW in one dimension under any k-truncated positional scoring rule is polynomial-time solvable via a reduction to a new 'shapes scheduling' problem; (ii) for any dimension d and any scoring rule or approval voting, PW is fixed-parameter tractable in the number m of candidates; and (iii) a complexity classification for the weighted version in one dimension, including NP-completeness for Borda with four candidates and a dichotomy for two-valued rules. The paper introduces the shapes scheduling model, proves a structural lemma (P-structured jobs), and gives a dynamic program based on a technique of Baptiste.

Significance. If the results were correct, the paper would be significant: it would resolve the tractability of the possible-winner problem for one-dimensional spatial voting with interval uncertainty for the broad and practically relevant class of k-truncated scoring rules with constant k, and it would provide the first FPT algorithms parameterized by the number of candidates for multidimensional spatial voting. The reduction-to-scheduling framework is a potentially useful new idea, and the weighted-case classification addresses an open direction. However, the central positive result of Section 3 rests on a structural lemma that is false, so the main theorem is not established by the proofs given.

major comments (2)

[Section 3.2, Lemma 5] Lemma 5 is false. The proof orders jobs by nondecreasing upper bounds u_j and considers a shape f(E) that lies in F_{d_j}^{(j)} but not in F_{d_j'}^{(j')}. It treats only the case where the later interval lies to the left of the segment E, leading to u_j < e_1 and a contradiction. The opposite case, where the later interval lies to the right of E, is never ruled out. This case actually occurs. Take candidates at 0, 10, 20, 30 and the 2-truncated rule (2,1). Voter v1 with interval [10.1,14.9] has r_1=2, d_1=4 and F_2^{(1)}={(2,1)}. Voter v2 with interval [15.1,19.9] also has r_2=2, d_2=4, but F_2^{(2)}={(1,2)}. These two endpoint sets are incomparable, so no total order can satisfy Definition 3's condition that equal deadlines imply inclusion of the endpoint shape sets. Thus the jobs generated by the reduction are not necessarily P-structured. Since Lemma 6 (the swap argument) and Lemma 7 (the DP) both rely on the existence of this order and on the inclusion property to swap jobs with equal deadlines, the dynamic program in Section 3.3 cannot be applied to the instances produced by the reduction. Consequently, Theorem 1, the paper's main positive result, is not proved.
[Section 4, integer program for Theorem 4] The integer program given for the FPT algorithm does not constrain the variables x^z_tau when z is not in the type tau. The text says that a voter of type tau may vote exactly according to the vectors in tau, yet variables x^z_tau are introduced for every z in Z and no constraint forces x^z_tau=0 for z not in tau. As written, the ILP can assign voters to voting vectors that are not allowed by their type, producing solutions that do not correspond to any profile completion. The constraint sum_z x^z_tau = n_tau does not repair this. This needs an explicit restriction (for example, defining x^z_tau only for z in tau, or adding x^z_tau=0 for z not in tau) before Theorem 4 is established.

minor comments (4)

[Section 3.3, Lemma 6] The statement of Lemma 6 refers to a job j'' that is never defined in the proof; the proof uses j and j'. This makes the lemma difficult to parse and should be corrected.
[Section 3.2 and Definition 3] The notation F_{d_j}^{(j)} is used for the set of shapes at start time d_j-k, but the reader is left to infer this. The paper should define this explicitly, since the subscript d_j does not match the start-time convention used elsewhere.
[Section 3.3, Lemma 7] The recurrence in Lemma 7 uses an undefined term b(S_j', f(j')). It presumably represents the contribution to the objective when job j' is scheduled in shape f(j'), but it is never defined in the text.
[Throughout] There are several typographical and grammatical errors, including 'V oting' in the title, 'traceable' for 'tractable' in Section 5, and 'P +n + 2' in the proof of Theorem 6. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new results are derived from explicit reductions to scheduling and integer programming, not from reusing their own conclusions.

full rationale

The central claims are self-contained or use external algorithmic machinery. For Theorem 1, the chain is explicit: Lemma 1 bounds the reachable candidate window, Lemma 2 constructs shape sets from midpoint segments, Lemma 3 gives a bijection between voter positions and scheduled jobs, Lemma 4 translates possible-winner existence into an all-machines-busy schedule condition, and Lemmas 5-7 plus the DP yield the polynomial bound. The if-and-only-if statements are factual equivalences of the reduction; they do not assume the target result. For Theorem 4 and Theorem 5, the algorithms use type-based ILP formulations and fixed-dimension LP/IP solvers (Lenstra; Reis and Rothvoss; Grigor'ev and Vorobjov), which are external, and the parameter m bounds the number of variables. The weighted hardness results in Theorems 6 and 7 are reductions from PARTITION with explicit score computations. The self-citation to Imber et al. [22] supplies the model and previous baselines, but the new polynomial-time and FPT proofs do not depend on [22] for their load-bearing steps. A possible mathematical gap in Lemma 5's ordering argument would be a correctness concern, not circularity, because it would not make any output equal to an input by construction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The paper is a theory paper, so the main assumptions are mathematical modeling assumptions (Euclidean spatial voting, interval uncertainty, tie-breaking) and external algorithmic results (IP in fixed dimension, quantifier elimination). No data-fitting parameters are used. The paper is fairly explicit about its model and the external results it relies on.

free parameters (2)

k = constant parameter of the rule
The polynomial-time algorithm for k-truncated rules treats k as a fixed constant; the running time grows exponentially in k. This is not fitted to data, but the complexity result depends on k being constant.
m (number of candidates) = parameter of FPT algorithms
In the FPT results, m is the parameter; the running time is f(m) n^{O(1)}. The algorithms depend on m being small, which is an assumption about practical instances.

assumptions (4)

domain assumption Euclidean distances induce single-peaked-like monotone candidate windows in one dimension (Lemma 1).
The reduction to shapes scheduling assumes that for each voter, the set of candidates that can receive positive scores as the voter position varies over an interval is a contiguous window, and that as the position moves right, the window shifts right monotonically. This is Lemma 1, which relies on Euclidean distance in R^1 and a fixed tie-breaking rule.
standard math Integer programming with fixed number of variables can be solved in FPT time.
The FPT algorithms in Section 4 invoke Lenstra's algorithm and its improvements [26, 30] for integer linear programming with fixed dimension, which is a standard external result.
standard math Solving systems of polynomial inequalities of bounded degree in fixed dimension takes polynomial time.
In Lemma 9, the paper invokes Grigor'ev and Vorobjov [21] to check feasibility of distance inequalities for approval voting.
domain assumption The type-counting approach of Kimelfeld, Kolaitis, Tibi [24] is applicable to spatial voting.
The FPT algorithm groups voters into 2^{m!} types based on which voting vectors are feasible; this assumes that the only relevant information about a voter for the possible winner problem is the set of feasible voting vectors, which is argued in Section 4.

invented entities (1)

Shapes scheduling problem
purpose: A new scheduling problem introduced to model PW⟨1⟩, where jobs require varying machine resources over time.
This is a new algorithmic abstraction introduced in the paper. It is not independently evidenced, but it is a mathematical tool, not a physical or empirical entity; its value is demonstrated by the reduction.

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Cite this review

Pith. "Pith review of Finding Possible Winners in Spatial Voting with Incomplete Information." pith.science (2026). https://pith.science/paper/ZZNLM4OB

@misc{pith2026250512451,
  author       = {Pith},
  title        = {Pith review of: Finding Possible Winners in Spatial Voting with Incomplete Information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZNLM4OB}},
  note         = {Machine review of arXiv:2505.12451}
}

abstract

We consider a spatial voting model where both candidates and voters are positioned in the $d$-dimensional Euclidean space, and each voter ranks candidates based on their proximity to the voter's ideal point. We focus on the scenario where the given information about the locations of the voters' ideal points is incomplete; for each dimension, only an interval of possible values is known. In this context, we investigate the computational complexity of determining the possible winners under positional scoring rules. Our results show that the possible winner problem in one dimension is solvable in polynomial time for all $k$-truncated voting rules with constant $k$. Moreover, for some scoring rules for which the possible winner problem is NP-complete, such as approval voting for any dimension or $k$-approval for $d \geq 2$ dimensions, we give an FPT algorithm parameterized by the number of candidates. Finally, we classify tractable and intractable settings of the weighted possible winner problem in one dimension, and resolve the computational complexity of the weighted case for all two-valued positional scoring rules when $d=1$.

Figures

Figures reproduced from arXiv: 2505.12451 by the authors.

**Figure 2.** Illustration of a partial spatial profile and two different spatial completions. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Two schedule options for a job at time t by shape f = (2, 1) with processing time p = 2. In the shapes scheduling problem we are given a set of M identical machines for some M ∈ N, and a set of jobs J. Each job j ∈ J is associated with (i) a processing time pj ∈ N, (ii) a release time rj ∈ N0, (iii) a deadline dj ∈ N with rj + pj ≤ dj , and (iv) a set of shapes F (j) t , each with processing time pj , for any time t… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Example of iL and iR for a voter vj described by Pj . 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Two positions of a voter vj and the corresponding shapes [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Partition into segments of a set of m = 4 candidates under 3-truncated Borda. the segment E2 between m1,2 and m1,3. Then RE2 = (c2, c1, c3, c4), thus zE2 = 1. As for f(E2), czE2+0 = c1 is ranked second, therefore M f(E2) 0 = s(RE2 , czE2+0) = s(RE2 , c1+0) = 2. Also, c…

**Figure 7.** Figure 7: Schedule example of job j ′ in L shape 1 2 3 4 5 6 M1 M2 M4 M5 (a) Resource profiles of left subproblem. 4 5 6 7 8 M4 M5 M7 M8 (b) Resource profiles of right subproblem [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: Resource profiles after the partition due to Figure 7. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

34 extracted references · 33 canonical work pages

[1]

Al ´os-Ferrer and G

C. Al ´os-Ferrer and G. D. Grani´c. Political space representations with approval data. Electoral Studies, 39:56–71, Jan. 2015

work page 2015
[2]

Baptiste

P. Baptiste. Scheduling equal-length jobs on identical parallel machines. Discrete Applied Mathematics, 103(1- 3):21–32, 2000

work page 2000
[3]

J. J. Bartholdi, C. A. Tovey, and M. A. Trick. The computational difficulty of manipulating an election. Social choice and welfare, 6:227–241, 1989

work page 1989
[4]

Baumeister, P

D. Baumeister, P. Faliszewski, J. Lang, and J. Rothe. Campaigns for lazy voters: truncated ballots. In AAMAS, pages 577–584, 2012

work page 2012
[5]

Baumeister, M

D. Baumeister, M. Roos, and J. Rothe. Computational complexity of two variants of the possible winner problem. In The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 2, pages 853–860, 2011

work page 2011
[6]

Baumeister, M

D. Baumeister, M. Roos, J. Rothe, L. Schend, and L. Xia. The possible winner problem with uncertain weights. In ECAI 2012, pages 133–138. IOS Press, 2012

work page 2012
[7]

Baumeister and J

D. Baumeister and J. Rothe. Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules. Information Processing Letters, 112(5):186–190, 2012. 21

work page 2012
[8]

Betzler and B

N. Betzler and B. Dorn. Towards a dichotomy for the possible winner problem in elections based on scoring rules. Journal of Computer and System Sciences, 76(8):812–836, 2010

work page 2010

Show all 34 references

[9]

D. Black. On the rationale of group decision-making. Journal of political economy, 56(1):23–34, 1948

1948
[10]

Bogomolnaia and J.-F

A. Bogomolnaia and J.-F. Laslier. Euclidean preferences. Journal of Mathematical Economics , 43(2):87–98, 2007

2007
[11]

Brandt, M

F. Brandt, M. Brill, E. Hemaspaandra, and L. A. Hemaspaandra. Bypassing combinatorial protections: Polynomial-time algorithms for single-peaked electorates. Journal of Artificial Intelligence Research, 53:439– 496, 2015

2015
[12]

Brandt, V

F. Brandt, V . Conitzer, U. Endriss, J. Lang, and A. D. Procaccia. Handbook of computational social choice . Cambridge University Press, 2016

2016
[13]

Chakraborty and P

V . Chakraborty and P. G. Kolaitis. Classifying the complexity of the possible winner problem on partial chains. In AAMAS’21: Proceedings of the 20th International Conference on Autonomous Agents and MultiAgent Systems , 2021

2021
[14]

Conitzer and T

V . Conitzer and T. Sandholm. Complexity of manipulating elections with few candidates. In AAAI/IAAI, pages 314–319, 2002

2002
[15]

Conitzer, T

V . Conitzer, T. Sandholm, and J. Lang. When are elections with few candidates hard to manipulate? Journal of the ACM (JACM), 54(3):14–es, 2007

2007
[16]

Cygan, F

M. Cygan, F. V . Fomin, Ł. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized algorithms, volume 5. 2015

2015
[17]

Do ˘gan and A

O. Do ˘gan and A. E. Giritligil. Implementing the borda outcome via truncated scoring rules: a computational study. Public Choice, 159:83–98, 2014

2014
[18]

R. G. Downey, M. R. Fellows, et al. Fundamentals of parameterized complexity, volume 4. 2013

2013
[19]

Elkind, M

E. Elkind, M. Lackner, and D. Peters. Preference restrictions in computational social choice: A survey. arXiv preprint arXiv:2205.09092, 2022

2022 arXiv
[20]

Faliszewski, E

P. Faliszewski, E. Hemaspaandra, L. A. Hemaspaandra, and J. Rothe. The shield that never was: Societies with single-peaked preferences are more open to manipulation and control. In Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge, pages 11...

2009
[21]

D. Y . Grigor’ev and N. N. V orobjov Jr. Solving systems of polynomial inequalities in subexponential time. Journal of symbolic computation, 5(1-2):37–64, 1988

1988
[22]

Imber, J

A. Imber, J. Israel, M. Brill, H. Shachnai, and B. Kimelfeld. Spatial voting with incomplete voter information. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 9790–9797, 2024

2024
[23]

B. Kenig. The complexity of the possible winner problem with partitioned preferences. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, pages 2051–2053, 2019

2019
[24]

Kimelfeld, P

B. Kimelfeld, P. G. Kolaitis, and M. Tibi. Query evaluation in election databases. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pages 32–46, 2019

2019
[25]

Konczak and J

K. Konczak and J. Lang. V oting procedures with incomplete preferences. In Proc. IJCAI-05 Multidisciplinary Workshop on Advances in Preference Handling, volume 20, 2005

2005
[26]

H. W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematics of operations research, 8(4):538–548, 1983

1983
[27]

H. Moulin. Generalized condorcet-winners for single peaked and single-plateau preferences. Social Choice and Welfare, 1(2):127–147, 1984. 22

1984
[28]

Niedermeier

R. Niedermeier. Invitation to fixed-parameter algorithms. Habilitationschrift, University of T¨ubingen, 19, 2002

2002
[29]

M. S. Pini, F. Rossi, K. B. Venable, and T. Walsh. Incompleteness and incomparability in preference aggregation: Complexity results. Artificial Intelligence, 175(7-8):1272–1289, 2011

2011
[30]

Reis and T

V . Reis and T. Rothvoss. The subspace flatness conjecture and faster integer programming. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS 2023), pages 974–988, 2023

2023
[31]

Stockemer, A

D. Stockemer, A. Blais, F. Kostelka, and C. Chhim. V oting in the eurovision song contest. Politics, 38(4):428– 442, 2018

2018
[32]

Terzopoulou and U

Z. Terzopoulou and U. Endriss. The borda class: An axiomatic study of the borda rule on top-truncated prefer- ences. Journal of Mathematical Economics, 92:31–40, 2021

2021
[33]

T. Walsh. Uncertainty in preference elicitation and aggregation. In AAAI, volume 7, pages 3–8, 2007

2007
[34]

Y . Yang. On the complexity of borda control in single-peaked elections. In Proceedings of the 16th Conference on Autonomous Agents and MultiAgent Systems, pages 1178–1186, 2017. 23

2017

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Al ´os-Ferrer and G

C. Al ´os-Ferrer and G. D. Grani´c. Political space representations with approval data. Electoral Studies, 39:56–71, Jan. 2015

work page 2015

[2] [2]

Baptiste

P. Baptiste. Scheduling equal-length jobs on identical parallel machines. Discrete Applied Mathematics, 103(1- 3):21–32, 2000

work page 2000

[3] [3]

J. J. Bartholdi, C. A. Tovey, and M. A. Trick. The computational difficulty of manipulating an election. Social choice and welfare, 6:227–241, 1989

work page 1989

[4] [4]

Baumeister, P

D. Baumeister, P. Faliszewski, J. Lang, and J. Rothe. Campaigns for lazy voters: truncated ballots. In AAMAS, pages 577–584, 2012

work page 2012

[5] [5]

Baumeister, M

D. Baumeister, M. Roos, and J. Rothe. Computational complexity of two variants of the possible winner problem. In The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 2, pages 853–860, 2011

work page 2011

[6] [6]

Baumeister, M

D. Baumeister, M. Roos, J. Rothe, L. Schend, and L. Xia. The possible winner problem with uncertain weights. In ECAI 2012, pages 133–138. IOS Press, 2012

work page 2012

[7] [7]

Baumeister and J

D. Baumeister and J. Rothe. Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules. Information Processing Letters, 112(5):186–190, 2012. 21

work page 2012

[8] [8]

Betzler and B

N. Betzler and B. Dorn. Towards a dichotomy for the possible winner problem in elections based on scoring rules. Journal of Computer and System Sciences, 76(8):812–836, 2010

work page 2010