From: ramsay@unixg.ubc.ca (Keith Ramsay)
Newsgroups: can.general,can.politics,soc.culture.canada
Subject: Re: Voting Systems (was Re: Proportional representation)
Date: 10 Nov 1992 04:11:02 GMT
Organization: University of British Columbia, Vancouver, B.C., Canada
NNTP-Posting-Host: unixg.ubc.ca

In article <1992Nov8.074732.7435@ee.ubc.ca> jmorriso@ee.ubc.ca 
(John Paul Morrison) writes:
[Regarding the Arrow impossibility theorem:]
>I would like to see how this following system is affected. I believe
>that the theorem you mention can't apply in this case.
                              ^^^^^ 
It can apply, and I will explain again how it may be applied.  In
order to apply it properly, you have to choose appropriate definitions
of the terms involved.

In particular, you have to use the notion of "alternative" which is
relevant to the theorem. This was the key point of my last posting:
what the correct notion of "alternative" is, for the theorem, and how
to apply it. In what follows farther below, you have, in effect,
decided to impose a certain choice for the meaning of the term
"alternative", and then claimed Arrow's theorem *cannot* be applied,
because it can't be applied with your sense of "alternative"
substituted for the one to which the theorem naturally applies!

Arrow's theorem applies to voting systems which can be described as
producing one of various "outcomes" or "alternatives", depending upon
the preferences of each of a group of voters. An outcome consists of a
complete description of the decision arrived at. If A and B are
distinct outcomes, then A and B are incompatible, since they both
give complete accounts of the result of the vote, and the accounts
differ from each other.

As I wrote before:
|For this voting system the "alternatives" being chosen from, i.e., the
|possible outcomes of the vote, include all the possible ways of
|selecting a group of delegates, with weights assigned to them (which
|total up to the number of voters). It is clearly not allowed for
|voters to arrange all of these (zillions of) possibilities in priority
|order.

For instance, suppose we have 10 voters, who are also the people
eligible for office. Then there are 92378 possible arrangements of the
resulting "parliament"- one for each choice of "weights" which add to
10. Each of these is an outcome which could occur in this system, if
we were all to vote in a certain way.

A voter can have quite complex preferences among these alternatives.
Perhaps I am a voter who remembers the kind of mess we had after the
last election, when each of us voted for himself, and would like to
have a smaller number of representatives this time. It is conceivable
that *all* of us would rather have one certain arrangement, such as 
{A has 3 votes, D has 4, H has 3} (since we regard A, D, and H as
being good, balanced representatives of our various opinions, likely
to produce results akin to what we'd produce ourselves), than have the
situation where each of us packs up and goes to parliament with his or
her own vote.

One or two of the conditions mentioned in Arrow's result imply that in
such a case, the outcome will never be that each of us ends up with
just 1 vote in parliament-- since we all prefer the {A:3, D:4, H:3}
outcome. This is a condition which can easily be failed by this voting
system.

It also does not express itself as directly as a function of the
voters' preferences among the available possible outcomes. Suppose I
happen to know precisely which compositions of parliament I prefer to
which other ones. It still may not be clear who I will vote for, in
order to get the composition I want most. Even if we all have written
down the 92378 possibilities in rank order, the system does not give
us a well-defined outcome. So it fails that (fairly basic) condition.

It is sometimes implicitly assumed in discussions of politics that
"what we want" out of a legislature is to have people in it whose
opinions are as close to our own as possible. This is, however, rather
a simplification of the real situation. Take a certain acquaintance of
mine, for instance, who said that one of the things he wanted was for
there *not* to be an overwhelming majority of any one party in the
parliament, even his own favorite.

I wrote:
>There's no way to express the preference, for instance, of those who
>might mainly want some kind of coherence and consensus of purpose in
>the parliament, for efficiency, and aren't as concerned about getting
>their own first choice of representative into it.

In article <1992Nov9.022817.20512@ee.ubc.ca> jmorriso@ee.ubc.ca 
(John Paul Morrison) writes:
>You don't read very closely do you? 

I read the articles which I reply to fairly closely, and I'm
disappointed that you are so quick to take so poor a view of my
reading.

...
>What you don't understand, is that in this hypothetical system, the votes
>of each delegate are not equal

No, I did understand that. What I've been writing is true independent
of that.

I wrote:
>It also fails another condition of the theorem, in that it is possible
>for a person to cause an option A to be selected instead of an option
>B, by deciding to try to get C instead of D. Suppose there are a
>number of us who'd like to vote for Smith, but for whatever reason
>we'd rather she not have too much power. If, however, she already has
>sufficient votes, then we'd rather vote for Jones.

John Paul Morrison writes:
>This isn't the case. Your logic system requires that out of the set 
>{A,B,C,D}, exactly one item is selected. 

"Logic system"? :-)

>In the system I am suggesting
>(to demonstrate that this line of reasoning does not apply) you actually
>have a "fuzzy" set where ALL of the set {A,B,C,D} is chosen. 

I think this is because you are identifying "outcomes" or "options"
with people to be selected. It is true, in most of the cases being
discussed before in this group, that the outcome can be specified by
saying which individual won the election, but there's no reason to
restrict the applicability of the theorem to voting systems of that
kind.

I wrote:
>Let A, B, C, and D be possible compositions of the legislature, 

As you can see, I am talking about whole compositions of the
legislature, not individual people. And by "composition", perhaps I
should have pointed out, I mean to include not only who is there, but
how many votes they're given to cast. But I think that is clear from
the following:

>where
>A differs from B only by Smith getting some of the votes Jones got in
>case B, 

I.e., both Smith and Jones are chosen (as well as whoever else is
chosen), but with different numbers of votes in the two scenarios.

>and likewise for C and D, where A and B are alternatives in
>which Smith gets enormous power, and C and D are alternatives in which
>Smith gets either a modest amount or none at all. The condition
>described by the theorem, requiring that changing preferences between
>A and B will not affect whether C or D is chosen, is violated by this
>voting system. By trying to ensure Smith gets a modicum of influence,
>we could end up giving her more than we had wanted, or conversely.

John Paul Morrison writes:
>It is completely independent! You haven't proven anything.

We can suppose that there are four candidates, Smith, Jones, Cheng,
and Lu. Suppose 25 voters prefer Cheng best, and 20 voters like Lu
best. Suppose that the 55 other voters would like for *both* Smith and
Jones to be elected, for balance, with neither one having enough votes
to legislate autocratically (but, say, trust Smith enough more than
Jones that they want him to have enough more power than Jones that if
it is Smith and Lu vs.  Cheng and Jones on a given vote, that Smith-Lu
wins). What would those voters do, to make this most likely?  It is
possible that by all choosing Smith, they will give him more power
than they want for him to have. So if

     A= {Cheng:25, Lu:20, Smith:55, Jones:0}
     B= {Cheng:25, Lu:20, Smith:35, Jones:20}
     C= {Cheng:25, Lu:20, Smith:20, Jones:35}
     D= {Cheng:25, Lu:20, Smith:0, Jones:55}

then these voters prefer B to C to A to D. Preferring C to D could
wind up getting them A instead of B.

I hope it is now clear what I was trying to say.

>the Arrow theorem assumes that only one candidate is elected. The example
>posted does not follow Arrows assumptions, so it doesn't apply.

For "candidate" it is natural to read "outcome", and this allows it to
apply to a wider variety of examples.

>IF you prefer A to B to C, voting for A will always help A and never C,
>because in this system, a delegate's power depends only on the votes
>he receives.

Some of our preferences, in complex situations like this, aren't
reducible to which potential delegates we "prefer". There may not even
be a well-defined answer to "who do you prefer?", if your preferences
with regard to outcomes are complex in some way. It is in cases where
only one delegate were chosen that this simplifying assumption would
make more sense. If there are many, perhaps we prefer "diversity"
(forcing compromises), or prefer "unanimity" (perhaps for efficiency)
in the government. These are preferences which are not often
explicitly taken into account in voting systems. It seems more often
to be a state of affairs which is more-or-less "hard wired" by the
electoral system, in one direction sometimes by forms of "winner take
all", or in the opposite direction, by forms of PR. Whether or not
that is what the electorate as a whole wants.

To relate this back to Canada, :-) what do people think is the
distribution of opinion on this in Canada? Do people like better for
the government to be more-or-less dominated by one party at a time, or
would they rather it be a matter of constant compromise among various
groups? Or something else?

Keith Ramsay
ramsay@unixg.ubc.ca
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