Approval Vote: reply to Bart Ingles
Craig Carey
research at ijs.co.nz
Fri Mar 3 00:09:05 PST 2000
Approval Vote has for an axiom, utility values((?)).
At 17:55 03.03.00 , Bart Ingles wrote:
>
>Craig Carey wrote:
:: :>Not sure why you mention a voter with 3 votes -- is this something you
:: :>consider for other systems?
:: :
>> > I was imposing a constraint that the counts be integral.
>> > How would you argue that such aa constraint should be imposed?.
That should have read:
"I was Not imposing a constraint that the counts be integral."
>
>Bart Ingles:
>> The only constraint with approval voting is that a voter is allowed one
>> vote per candidate.
You of course mean the Approval Vote sub-votes, but the vote itself being
a real number (or if integers are insisted upon, then rationals).
>
>I should have said "The only constraint with approval voting is that a
>voter is allowed to give either 0 or 1 vote to a given candidate."
The value 1 is selected if the voter provides a preference ranking, (an
integer between 1 and k, k = the number of sub-votes).
>
>It would be possible to have a system where the voter is allowed to give
>a fractional vote between 0 and 1 to a candidate, but it would never
>make sense to vote that way -- optimal strategy is always to give either
>0 or 1 votes. Thus the system would be equivalent to approval voting,
>except that the additional possibilities might confuse ignorant voters.
>
Until some numbers on giving the weights are given, that paragraph hasn't
a lot of clarity or weight. The words "optimal strategy", i.e. strategy
of the method designer, is perhaps a implicit reference to Ossipoff's
"utility" idea.
////////////////////////////////////////////////////////////////////////
At 17:46 03.03.00 , Bart Ingles wrote:
>
>
>Craig Carey wrote:
>[...]
>> "Is Approval Voting an 'Unmitigated Evil?' A Response to Brams,
>> Fishburn, and Merrill," by Donald G. Saari and Jill Van Newenhizen,
>> Public Choice (1988), 59:133-147.
>>
>> "A Case Against Bullet, Approval and Plurality Voting," by
>> Donald G. Saari and Jill Van Newenhizen, January 1985.
>>
>> "The problem of indeterminacy in approval, multiple, and truncated voting
>> systems, Public Choice (1988), 59:101-120, by Donald G. Saari and
>> Jill Van Newenhizen,
>
>
>The titles of the latter two apparently point out Saari's objection to
>allowing voters to truncate ballots. The version of Borda he favors
>actually penalizes truncators more than the "standard" version. To wit:
>
>Sincere Borda points with three candidates, for a single voter with
>preferences ABC:
>A = 2, B = 1, C = 0
>
>Points for same candidate who truncates (standard Borda):
>A = 2, B = 0.5, C = 0.5 (or A=1.5, B = 0, C = 0)
I.e. paper = (A . .)
>Points for truncator in Saari's version:
>A = 1, B = 0, C = 0
>
>(I got this directly from Prof. Saari, in response to a question)
>
I'll check these methods against (my) P2:
---------------------------------------------------------------------------
Consider the Borda said to be of D. G. Saari: Case 3 candidates:
A.. Za0 2t . (2, 0, 0)
AB. Zab0 -t . (2, 1, 0)
AC. Zac0 -t . (2, 0, 1)
Sum = (0,-t,-t)
That fails P2({1}) since adding (2:A, -1:AB, -1:AC) can make A change
from a loser into a winner.
---------------------------------------------------------------------------
Consider the 4 formula of the given 'Standard Borda' passes P2. I find that
it doesn't for 4 candidates. It does pass P2 for 3 or less candidates.
P2({0}): Passes if winners unaltered on adding (t,t,t,t) to
(Za0,Zb0,Zc0,Zd0):
A B C D
A... Za0 t . ( 3, 1/3, 1/3, 1/3)
B... Zb0 t . (1/3, 3, 1/3, 1/3)
C... Zc0 t . (1/3, 1/3, 3, 1/3)
D... Zd0 t . (1/3, 1/3, 1/3 3 )
P2({1}): Winners unaltered on adding (3t,-t,-t,-t) to (Za0,Zab0,Zac0,Zad0)
A... Za0 3t . (3, 2/3, 2/3, 2/3)
AB.. Zab0 -t . (3, 2, 1/2, 1/2)
AC.. Zac0 -t . (3, 1/2, 2, 1/2)
AD.. Zad0 -t . (3, 1/2, 1/2, 2 )
Sum = (0, (2-2-1/2-1/2), (2-1/2-2-1/2), (2-1/2-1/2-2))
= (0, -1, -1, -1).
So P2({1}) is Not held, since adding (3:A, -1:AB, -1:AC, -1:AD) will
sometimes change A from a loser into a winner.
P2({2}): Winners unaltered on adding (2t,-t,-t) to (Zab0,Zabc0,Zabd)
AB.. 2t . (3, 2, 1/2, 1/2)
ABC. -t . (3, 2, 1, 0 )
ABD. -t . (3, 2, 0, 1 )
Sum: (0, 0, 0, 0 ) Hence P2({2}) is held
P2({3}): There is no failure of P2({3}) since altering ABC. into ABCD
makes no difference to the weights multiplied by the vote.
So the defined Standard Borda is OK for 3 candidate elections but it is
failed by P2 in 4 candidate elections. [Did I extrapolate the definition
correctly?]
---------------------------------------------------------------------------
...
>> If the paper allows 30 Approval sub-votes, any voter can mark just
>> a single candidate as the best way to get the method to try hard.
>
>Not sure what you mean by sub-votes -- the voter is allowed to give one
>vote each to any candidate(s) he desires. Thus if there are 30
>candidates, the voter is allowed to cast anywhere from 0 to 30 votes (0
>and 30 are both forms of abstention).
...
>Are you sure you aren't confusing approval voting with cumulative
>voting? That would explain the "30 sub-vote" mention above, if you
>intended other than 30 candidates.
You have the example immediately below to show my understanding of the
Approval Vote's definition. In the first, C & D win because 11 & 12 are
greater than 9 and 9.
>
>> >> In the voting booth the voter can't easily tell if D should
>> >> be voted for. The[re] may be >7 candidates so the voter may
>> >> imagine that voting for D gives him/her more power.
>> >> In this example voting for D is to the voter's disadvantage:
>> >>
>> >> A B C D
>> >> 3 Voters: 3 3 3 3
>> >> Others: 6 7 8 9
>> >> Total: 9 9 11 12
>> >> Winners = {C,D}
>> >>
>> >> A B C D
>> >> 3 Voters: 3 3 3
>> >> Others: 6 7 8 9
>> >> Total: 9 10 11 9
>> >> Winners = {B,C}
>> >
>> >You don't support your claim, since you don't show how many votes the
>> >remaining candidates are likely to receive. Voting for D may well be
>> >the right choice if E,F, or G are also likely to receive 11 or 12 votes.
>>
>> You need to interpret "there may be >7 candidates" to
>> "there are 4 candidates". Then the proof of subsequent preferences
>> harming candidates supported by earlier preferences is done and visible.
>>
>> Of course, a single numerical example is able to allow the Approval Vote
>> to found to be failed by that rule.
>
>If A, B, C, and D are the only candidates, and D is the least preferred,
>then there would _never_ be any reason to vote for D under approval
>voting.
Voters are not constrained by "reason". If any wants to define a constraint
of reason, then let's have that in an existential logic form.
Anyway, when trying to find out if the method fails the rule, it is most
certainly reasonable to examine the method in the very regions where it
fails that rule.
When there are 4 candidates, then specifying the last preference can alter
the set of winners.
>> >This example is not really sufficient to predict voter strategy, since
>>
>> You wrote about "efficiency" in text deleted, and questioned my using
>> 3 votes for one voter, which indicates a viewpoint difference.
>
>I mentioned "Condorcet efficiency" and "social utility efficiency", but
>I haven't a clue what those have to do with using 3 votes per voter.
>
>
>> Suppose a person was ignorant and didn't use enough Approval sub-votes.
>
>He would presumably still have voted for his more highly-favored
>candidates. He may not have supported the necessary compromises, but
>you can do the same with any other system.
Suppose he/she just voted for one candidate; the method will cause a vote of
that type to lack power. If Approval Vote is implemented in a real election
then there 'should be' a TV or pamphlet publicity effort funded by the
government that educates voters on what a bad method it is, and how they
should not cast too few sub-votes, e.g. out of mistaken presumption that
the method is fair and democratic and does not give an undue unfluence to
those individuals that will vote randomnly for half of the, say 88,
candidates. Suppose one of the 88 was the current president fighting of
scores of no hopers fattened in industry, and the same president that twice
led the nation to military victories in the previous 4 years against foreign
dictators.
...
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