[EM] Definition of proportional electoral system

[Kristofer Munsterhjelm]

On 2023-08-28 13:46, Luděk Belán wrote:
> Thank you for answer.
> I didn't express myself accurately. I wanted to know whether the 
> mentioned principle is a necessary condition of a proportional electoral 
> system, not a sufficient condition.

It might be, for quota-based proportional representation.

The Droop proportionality criterion says that if a solid coalition (in 
this case, a party) obtains more than 1/(n+1) of the vote, it should get 
at least 1/n of the seats.

So if we have two parties for two seats, and one party has more than 33% 
of the vote, then it must get at least one seat. Then the other party 
also has more than 33% and must get the other seat. So there's no room 
for nonmonotonicity in the mono-add-plump sense (i.e. additional votes 
for A won't harm A).

For things that generalize Webster and other divisor methods, it's even 
easier: the party list case is just Webster (or D'Hondt or etc.), which 
themselves are monotone functions and so pass your condition.

Things get a lot messier for ranked voting methods, though, because 
"votes for" someone don't neatly map into "support for" everybody in his 
party. I'd say the obvious generalized criterion is mono-add-plump (i.e. 
if you only use your first preference, then whoever you vote for 
shouldn't be harmed), which I'd say that most sensible methods should 
pass; but I don't think it's *necessary*.

So to sum up, I think proportional methods have to pass your condition 
in the "party list, each voter chooses one party" case.

-km