Pref.Tools: ABC Voting

For more information, see How to Sample Approval Elections? Stanisław Szufa, Piotr Faliszewski, Łukasz Janeczko, Martin Lackner, Arkadii Slinko, Krzysztof Sornat, Nimrod Talmon. arXiv:2207.01140. Code is based on the prefsampling package.

• Impartial Culture (IC)
  p : float in [0, 1] - probability of approving a candidate
• Impartial Culture (IC) with fixed size
  setsize : int - number of candidates to approve
• Euclidean VCR (Voter Candidate Range)
  radius : float - voters will approve candidates located within this radius
  1D Interval (uniform [0,1]) 1D Gaussian (0.5 ± 0.15) 2D Square (uniform [0,1]²) 2D Disc (uniform, radius 1) 2D Gaussian 2D Gaussian restricted to disc voter_prob_distribution : str - probability distribution of the voter points
  1D Interval (uniform [0,1]) 1D Gaussian (0.5 ± 0.15) 2D Square (uniform [0,1]²) 2D Disc (uniform, radius 1) 2D Gaussian 2D Gaussian restricted to disc candidate_prob_distribution : str - probability distribution of the candidate points
  Error: Voter and candidate distributions have different dimensions.
• Euclidean fixed size
  setsize : int - number of candidates to approve
  1D Interval (uniform [0,1]) 1D Gaussian (0.5 ± 0.15) 2D Square (uniform [0,1]²) 2D Disc (uniform, radius 1) 2D Gaussian 2D Gaussian restricted to disc voter_prob_distribution : str - probability distribution of the voter points
  1D Interval (uniform [0,1]) 1D Gaussian (0.5 ± 0.15) 2D Square (uniform [0,1]²) 2D Disc (uniform, radius 1) 2D Gaussian 2D Gaussian restricted to disc candidate_prob_distribution : str - probability distribution of the candidate points
  Error: Voter and candidate distributions have different dimensions.
• Truncated Mallows
  dispersion : float in [0, 1] - dispersion parameter of the Mallows model
  setsize : int - number of candidates that each voter approves
• Urn
  p : float in [0, 1] - If a new vote is generated, each candidate is approved with likelihood p.
  replace : float - New balls added to the urn in each iteration, relative to the original number. A value of 1.0 means that in the second iteration, there is a chance of 0.5 that the ballot of the first iteration is chosen and a chance of 0.5 that a new ballot is drawn from p-IC.
• Urn with fixed size
  setsize : int - number of candidates that each voter approves
  replace : float - New balls added to the urn in each iteration, relative to the original number. The urn starts with (num_cand choose setsize) balls, each representing a set of candidates with size setsize. This quantity is normalized to 1.0. The replace value is a float that indicates how many balls are added using this normalization. Specifically, replace * (num_cand choose setsize) are added in each iteration.
• Truncated Urn
  setsize : int - number of candidates that each voter approves
  replace : float - New balls added to the urn in each iteration, relative to the original number. The urn starts with num_cand factorial balls, each representing a ranking of candidates. This quantity is normalized to 1.0. The replace value is a float that indicates how many balls are added using this normalization. Specifically, replace * (num_cand factorial) are added in each iteration.
• Resampling
  p : float in [0, 1] - Fraction of candidates that will be approved by the central ballot. Precisely, the central ballot will approve the first floor(p * num_cand) candidates.
  phi : float in [0, 1] - Probability to resample an approval. For each generated vote, we go through all candidates. For each candidate, we copy the approval of the central ballot with probability 1 - phi. Otherwise, with probability phi, we resample the approval of the candidate (so that the generated vote approves that candidate with probability p).
• Disjoint Resampling
  num_groups : int - Corresponds to the parameter g in (p,phi,g)-Disjoint Resampling. The model randomly partitions the candidates into num_groups groups. To generate a vote, the model first randomly selects a group. Then, it samples a vote from a (p,phi)-resampling model, where the central ballot approves exactly all candidates in the selected group. (Thus, the parameter p is not used to obtain the central ballot, but is used in case of resampling.)
  phi : float in [0, 1] - Probability to resample an approval. For each generated vote, we go through all candidates. For each candidate, we use copy the approval of the central ballot with probability 1 - phi. Otherwise, with probability `phi`, we resample the approval of the candidate (so that the generated vote approves that candidate with probability p).
  p : float in [0, 1] Probability of approving a candidate in case that the model resamples. Note: num_groups * p should be at most 1.
• Random Noise
  p : float in [0, 1] - Fraction of candidates that will be approved by the central ballot. Precisely, the central ballot will approve the first floor(p * num_cand) candidates.
  phi : float in [0, 1] - Probability to resample an approval. For each generated vote, we go through all candidates. For each candidate, we copy the approval of the central ballot with probability 1 - phi. Otherwise, with probability phi, we resample the approval of the candidate (so that the generated vote approves that candidate with probability p).
  Hamming Jaccard Zelinka Bunke-Shearer distance : string - Distance metric used to measure the distance between two rankings.

• Load PAV compared to Phragmén-style rules (shown at startup) Peters and Skowron, 2020, "Proportionality and the Limits of Welfarism", Introduction
• Load Running example used by Lackner and Skowron (2023) Lackner and Skowron, 2023, "Multi-Winner Voting with Approval Preferences", Springer, Example 2.1
• Load PAV fails core Aziz et al, 2016, "Justified Representation in Approval-Based Committee Voting", Example 6
• Load Seq-PAV fails JR (minimal example, k = 6, 108 voters) Sánchez-Fernández et al, 2016, "Proportional Justified Representation", Table 2
• Load Seq-Phragmén fails EJR Brill et al, 2021, "Phragmen's Voting Methods and Justified Representation", Example 6
• Load leximax- and var-phragmen fail EJR Brill et al, 2021, "Phragmen's Voting Methods and Justified Representation", Example 5
• Load Rev-Seq-PAV fails JR Aziz, 2017, "A Note on Justified Representation Under the Reverse Sequential PAV rule", Proposition 2 for k = 5
• Load Seq-Phragmén/seq-PAV/Equal Shares fail Pareto optimality Lackner and Skowron, 2020, "Utilitarian welfare and representation guarantees of approval-based multiwinner rules", Example 2