Probabilistic Approval Voting is a sequential proportional voting system that uses either approval or score ballots. Probabilistic calculations are used as a tool to predict potential power balance of elected political factions. The method itself is deterministic.

## 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 of A
• ${\textstyle V(A \and B)}$ ... number of voters who approve of both A and B
• ${\textstyle V(A \or B)}$ ... number of voters who approve of A or B or both

Let's say that there is a function ${\displaystyle P:C \times C \to [0, 1]}$ that maps two given candidates to the probability that they 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 ${\textstyle \frac{1+\sum_{X \in W} P(A,X)}{V(A)}}$ (P(A,A) equals 1). 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 ${\textstyle \frac{V(A \and B)}{V(A \or B)}}$ which would give ${\displaystyle \frac{V(A)}{1+\sum_{X \in W} \frac{V(A \and X)}{V(A \or X)}}}$ 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

In each step we elect the candidate with the highest ${\textstyle \frac{V(A)}{1+\sum_{X \in W} \frac{V(A \and X)}{V(A \or X)}}}$.

${\textstyle \frac{V(A \and B)}{V(A \or B)} = \frac{29}{30}}$

${\textstyle \frac{V(A \and C)}{V(A \or C)} = \frac{0}{43}}$

${\textstyle \frac{V(B \and C)}{V(B \or C)} = \frac{0}{44}}$

First seat:

A: ${\textstyle V(A)/1 = 29/1 = 29}$

B: ${\textstyle V(B)/1 = 30/1 = 30}$

C: ${\textstyle V(C)/1 = 14/1 = 14}$

B is elected

Second seat:

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

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

A is elected

## Score ballots

One possible procedure to elect a candidate using 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.