Question about Limited Voting

[Donald E. Davison]

- - - - - - - - - - - - - - - - - - - - - 10-25-00
Abaughm wrote:

I was wanting to know how to compute the probability of a candidate from
each group of  voters being elected.  I have three populations of  60%,
20%, and 20%.  There are 5 seats and a person can vote for 3 candidates.  I
actually need to know the probability of voting in 3 from the 60% and 1
each from the 20%.  Thank you.

- - - - - - - - - - - - - - - - - - - - - 10-28-00
Dear Abaughm,

     The probability of each population electing one seat per twenty
percent depends on how the people of all populations vote.  Your 60%
population has a probability of one hundred percent of voting in three
candidates. All the population needs to do is to run three candidates and
if all members of the population vote for all three then each candidate
will receive sixty votes per hundred cast in the election. And, the two
twenty factions can elect one seat each.
     But, the leader of the sixty percent population may want to try for
all five seats, it is possible.
     He could run five candidates and the members of the population would
be instructed to vote according to the last digit of their house number, as
follows:

     Last Digit    Vote for these candidates:
          0              A B C
          1              B C D
          2              C D E
          3              D E A
          4              E A B
          5              A B C
          6              B C D
          7              C D E
          8              D E A
          9              E A B

     This will result in each of his candidates receiving about thirty
votes per hundred, which is more than the 20 votes/100 a twenty percent
faction could muster.
     So you see, the probability of a twenty percent faction electing one
candidate has now changed due to a different voting plan by the sixty
faction.

     Of course, if the two twenty factions put their heads and votes
together they could come up with three candidates with forty votes per
hundred each. They, the two factions, would have to agree on three
candidates and then instruct their combined supporters to vote for the
three, but it could result in electing three seats between them.

     If it looks like this is going to happen, the sixty faction would need
to go to `Plan B' which is to drop one of the five candidates and instruct
supporters to vote as follows:

     Last Digit    Vote for these candidates:
          0              A B C
          1              D A B
          2              C D A
          3              B C D
          4              A B C
          5              D A B
          6              C D A
          7              B C D
          8              A B C
          9              D A B

     This voting will yield:
       48 votes/100 for candidate A and B
       42 votes/100 for candidate C and D
     Now the twenty percent factions will be able to elect only one seat
between them.
     Limited Voting is only a slight improvement over Plurality(FPTP). It
will not always yield Fair Representation, if that is what you are trying
to obtain. With Limited Voting, each twenty percent may not be able to
elect one of five seats.

     We should now consider three other methods - Cumulative Voting, Single
Non-Transferable Vote(SNTV), and Bottoms Up(Alternative Vote for
Multi-seat).
     In these three methods, the two twenty factions should be able to
elect one seat each, which leaves three seats for the sixty faction to
elect. These three methods will yield a higher probability of Fair
Representation than Limited Voting for your example, but this is not a
perfect world, people do not always do what we expect them to do.
     The sixty faction may have a very popular candidate and most
supporters of the faction may insist on voting for him. If the voting on
their three candidates becomes too much out of balance, then one of their
three seats will be at risk. Suppose the sixty faction votes: 38A 20B 2C
and the two twenty factions vote: 18X 16Y 6Z.
     The sixty faction needs to do something to balance up their votes in
these three methods. There is a method that will do the balancing for the
sixty faction or any faction over twenty percent. I speak of Choice
Voting(STV).
     Choice Voting will balance out the votes according to the quota of
twenty votes needed to be elected. It does this by transferring surplus
votes over the quota to other candidates according to the next choices on
the ballots. Choice Voting will give us Fair Representation.

     If your example is a partisan election, then the Party List methods
will also elect one seat per twenty percent. I suggest Open Party List, it
is more democratic than Closed Party List. Choice Voting(STV) is more
democratic than Open Party List in that voters are allowed to go across
party lines. For large partisan elections, I favor a mix of Choice Voting
and Party List, but that's another discussion.

     Thank You for writing.

Regards, Donald E. Davison