User:Matijaskala/Probabilistic Approval Voting

Probabilistic Approval Voting is a sequential proportional voting system that uses either approval or score ballots. Its winners are found by the use of probabilistic calculations.

Derivation

Given:

• ${\textstyle C}$ ... the set of all candidates
• ${\textstyle W}$ ... the set of already elected candidates
• ${\textstyle V(A)}$ ... number of voters who approve A
• ${\textstyle V(A\land B)}$ ... number of voters who approve both A and B
• ${\textstyle V(A\lor B)}$ ... number of voters who approve A or B or both

Let's say that there is a function $P:C\times C\to [0,1]$which returns the probability that two given candidates belong to the same political faction. Then for a given candidate A, the expected number of elected candidates belonging to the same faction equals ${\textstyle \sum _{X\in W}P(A,X)}$ and A's voter efficiency equals ${\textstyle {\frac {\sum _{X\in W}P(A,X)}{V(A)}}}$. If A is not already elected then we can calculate the A's voter efficiency after A's election as ${\textstyle {\frac {1+\sum _{X\in W}P(A,X)}{V(A)}}}$. Let's call ${\textstyle {\frac {V(A)}{1+\sum _{X\in W}P(A,X)}}}$ A's score and ${\textstyle {\frac {1+\sum _{X\in W}P(A,X)}{V(A)}}}$ A's inverse score. If we keep electing candidates whose inverse score is at least as low as inverse Hare quota then we can expect an outcome where each winner's voter efficiency is below a certain limit aka a proportional outcome.

In practice, electing a candidate whose inverse score is at least as low as inverse Hare quota will not always be possible. In that case we need to either add fail-safe approvals which we previously didn't consider or elect the candidate with the lowest inverse score and hope it is low enough.

A reasonable choice for P(A,B) would be ${\textstyle {\frac {V(A\land B)}{V(A\lor B)}}}$ which would give ${\frac {V(A)}{1+\sum _{X\in W}{\frac {V(A\land X)}{V(A\lor X)}}}}$ as the formula for the score of candidate A. In each step we elect the candidate with the highest score. This version of the system is 2-level precinct-summable and passes the universally liked candidate criterion.

In party list case P(A,B) equals 1 if A and B belong to the same party and 0 otherwise. Because of that the system decays into D'Hondt.

Example

29 AB

1 B

14 C

${\textstyle {\frac {V(A\land B)}{V(A\lor B)}}={\frac {29}{30}}}$

${\textstyle {\frac {V(A\land C)}{V(A\lor C)}}={\frac {0}{43}}}$

${\textstyle {\frac {V(B\land C)}{V(B\lor C)}}={\frac {0}{44}}}$

First seat:

A: ${\textstyle 29/1=29}$

B: ${\textstyle 30/1=30}$

C: ${\textstyle 14/1=14}$

B is elected

Second seat:

A: ${\textstyle 29/(1+{\frac {V(A\land B)}{V(A\lor B)}})=29/(1+{\frac {29}{30}})=14,745762711864}$

C: ${\textstyle 14/(1+{\frac {V(C\land B)}{V(C\lor B)}})=14/(1+{\frac {0}{44}})=14}$

A is elected

Score ballots

Probabilistic voting can be done with score ballots. We start by treating maximum score as approval. Once every candidate's score falls below Hare quota we progressively add lower scores.