[Feature Request] Additional Probabilistic Voting Methods: Conditional Utilitarian Rule and MaxParC

[mensch72]

It would be nice to have two more probabilistic methods:

Conditional Utilitarian Rule

• Input: approval ballots.
• Output: probability distribution.
• A non-abstaining ballot is one that approves at least one candidate.
• P(candidate x) = probability arising from the following random process:
  • score(candidate x) = number of approvals for x
  • Draw a ballot uniformly at random.
  • Let A be the set of candidates approved on that ballot.
  • Let M be the subset of A with largest score.
  • Draw a member of A uniformly at random.

Maximum Partial Consensus (MaxParC)

• Input: ratings ballots (a "UtilityProfile") with ratings between 0 and 100 (inclusive)
• Output: probability distribution.
• Procedure:
  • Let u(i,x) be the rating ("utility") voter i assigned to candidate x.
  • Let r(i,x) = 100 if u(i,x) >= u(i,y) for all candidates y, otherwise let r(i,x) = u(i,x)
    • This ensures max{ r(i,x) : x } = 100
  • Let N be the number of voters.
  • For each x, find the smallest integer m in {0,...N} so that |{ i : r(i,x)/100 > m/N }| >= N - m. Denote this value t(x).
  • Construct an approval profile as follows:
    • i approves x iff r(i,x)/100 > m/N
  • Apply the following version of the Conditional Utilitarian Rule to this approval profile:
    • score(candidate x) = number of approvals for x + (avg. u(i,x) over all voters i) / 101
    • Draw a ballot uniformly at random.
    • Let A be the set of candidates approved on that ballot.
    • Let M be the subset of A with largest score.
    • Draw a member of A uniformly at random.

Reference: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3751225 (accepted for Soc Choice Welf)