User:Matijaskala/Probabilistic Approval Voting

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. The system is 2-level precinct-summable and passes the universally liked candidate criterion.

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

Let's say that a candidate A proposes a decision. The probability that a candidate supports the decision equals ${\textstyle {\frac {V(A\land B)}{V(B)}}}$. Expected number of votes that the decision gets equals ${\textstyle \sum _{B\in W}{\frac {V(A\land B)}{V(B)}}}$ or ${\textstyle {\frac {\sum _{B\in W}{\frac {V(A\land B)}{V(B)}}}{V(A)}}}$ per voter. To minimize number of votes per voter we maximize ${\displaystyle {\frac {V(A)}{1+\sum _{X\in W}{\frac {V(A\land X)}{V(X)}}}}}$ for each newly elected candidate.

Example

29 AB

1 B

14 C

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

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

${\textstyle {\frac {V(A\land C)}{V(C)}}={\frac {0}{14}}}$

${\textstyle {\frac {V(B\land C)}{V(B)}}={\frac {0}{30}}}$

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

C: ${\textstyle V(C)/(1+{\frac {V(C\land B)}{V(B)}})=14/(1+{\frac {0}{30}})=14}$

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.