An alternative to SPAV


#1

This is my idea how approval ballots could be converted into proportionally allocated seats.

The procedure:

  • Define influence of candicate x as the expected value of number of winning candidates for which a randomly chosen voter that approves the winning candidate also approves x.
  • Elect a candidate with the highest number of approvals per unit of influence, where influence is calculated as if that candidate was already elected.
  • Repeat until all the seats are filled.

The formula for the quotient between approval ratio and influence can be written as
P(Ax)/(1+∑winning y≠xP(Ax|Ay))
where P stands for probability and Ax is the event that a randomly chosen voter approves x. In each turn, the candidate with the highest quotient is elected.

This method is equivalent to SPAV when voters vote along party lines, but doesn’t disadvantage overlapping voter groups.
It’s also possible to use it with score ballots. In the score-based variant, voters would be able to state a probability with which they approve each candidate.


#2

Interesting.

Does it have a non-sequential version?

Does it pass the ULC criterion (a criterion that both SPAV and it’s non-sequential version PAV fail)?


#3

No, but it can be seen as the sequential version of min (∑winning x1/P(Ax)+∑winning {x,y}P(Ax∩Ay)/(P(Ax)P(Ay))), which is similar to Ebert’s function (his function is equivalent to min (∑winning x1/P(Ax)+2 ∑winning {x,y}P(Ax∩Ay)/(P(Ax)P(Ay)))).

Assuming that u is elected and P(Ax|Au)/P(Ax) does not depend on x (it equals 1 if u is universally liked):
P(Ax)/(1+∑winning y≠xP(Ax|Ay)) = 1/((1+∑winning y≠xP(Ax|Ay))/P(Ax))
P(Ax)/(1+∑winning y≠xP(Ax|Ay)) = P(Ax)/(1+∑winning y≠x,uP(Ax|Ay)+P(Ax|Au)) = 1/((1+∑winning y≠x,uP(Ax|Ay))/P(Ax)+P(Ax|Au)/P(Ax))
When comparing values of the quotient for different candidates, we can just compare values of the denominator of the expression on the right. Because P(Ax|Au)/P(Ax) is a constant, it doesn’t affect the inequations. Therefore, the subsequently elected candidates would be the same as if u was not present.
So it does seem to pass ULC.