Approval

[Bart Ingles]

Craig Carey wrote:
> >> Are people writing to this list advocating some "Approval Vote" method?.
> >> Isn't it one of the very worst methods around?.
> >
> >In whose estimation?  Prof. Saari?
> >
> Was that part of a quite public dispute?. I suppose Borda is better
>  since the Approval method is a Borda method but with the weights
>  (1, 1, 1, ...) for each preference.

I recall seeing a brief comment by Saari in either Science News or
Scientific American where he touts Borda over Approval.  I think he
co-wrote an article for an academic periodical with a title something
like "Approval Voting: Unmitigated Evil?", but I didn't have ready
access to the article.  I don't recall exactly where I saw the mention,
but you might try looking for Saari's web site.

I might have been motivated to track down the article, but for the fact
that Saari is proposing Borda as a better alternative.  Borda is so
strategy prone as to be useless for public elections; the fact that
Saari promotes it makes me think that he must be using an awfully narrow
set of criteria to evaluate systems.  In fact in the short comment I did
see, Saari uses a simple 3-voter ranked example to show why he thinks
Approval fails, but doesn't seem to realize that rankings alone are not
enough to predict voter behavior in Approval (or probably any other
voting system).


> The power a voter has will be a hump shaped curve with the
>  voters having no power if the number of votes they cast is zero
>  or if the number votes they cast is equal to the number of
>  candidates.
> 
> Power
>  +
>  +                             +     +     +
>  +                       +                       +
>  +                 +
>  +           +
>  +     +
>  +-----+-----+-----+-----+-----+-----+-----+-----+-----+---->
>  0     1     2     3     4     5     6     7  Number of Votes
> 
> Voters are expected to want to go into a voting booth and while
>  in there they want to (a) retain or (b) change the government.
>  They do not want to consider the latest, IF ANY, on hump shaped
>  curves.

It is true that under approval voting, a voter can make more
distinctions between pairs of candidates by voting for exactly half of
the candidates, but this doesn't necessarily translate to voting power. 
Brams-Fishburn (1982) deals with this inequity question in terms of
average utility gain an average voter can expect from voting for
different numbers of candidates, and shows that FPP is much more
inequitable than Approval.

And of course, with Runoff or IRV, a voter who votes for one of the top
two front-runners only has one vote counted, while supporters of losing
candidates are able to have many distinctions counted.


> It is not a method that tries particularly hard to elect their
>  first preference.

The method doesn't try -- the question of how hard to try to elect a
first choice is left up to the voter.  


> Subsequent preferences act against earlier preferences.

To the extent that this is true, I view it as an advantage, contributing
to approval's unique lack of adverse behavior in the face of voter
strategy.

Additionally, the fact that the voter must make trade-offs when
considering compromise candidates insures that any vote given to a
candidate comes with a meaningful amount of voter support.  This
accounts for approval's good performance in both Condorcet efficiency
and social utility efficiency.


> Example: Suppose that in a election using a preferential vote,
>  a voter would have voted A>B>C>D. The voter very much (or
>  infinitely) prefers A to B to C to D.

I would think that an infinite preferences, if possible, would by nature
be dichotomous, so there would be a set of infinitely preferred
candidates, and a set of non-preferred candidates.  Such a situation is
tailor-made for approval voting.

I'm not sure what it would mean to have three different infinities
between the AB, BC, and CD.  How would you translate this into von
Neumann-Morgenstern utilities?


>  Suppose the method was electing two winners.
>  Approval vote isn't a preferential method.
>  This voter is either important with 3 votes or the there are
>  3 voters.

Not sure why you mention a voter with 3 votes -- is this something you
consider for other systems?

>  In the voting booth the voter can't easily tell if D should
>  be voted for. They 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.

This example is not really sufficient to predict voter strategy, since
we don't know how strongly the 3 voters support the various candidates,
or which candidates among those shown and not shown are the
front-runners.

In the unlikely event that nothing is known about which candidates are
front-runners, the voter should vote only for those candidates whose
utility to that voter is above average.  Since the underlying tradeoffs
are simple and straightforward, most people should be able to do this
intuitively, once they know what to aim for.

In the more likely case where the top two front-runners are known, you
simply vote for your favorite front-runner along with anyone you like
better.


> Mr Brams or whomever might call it a simple method. I can't recall
>  who declared that the Approval method was simple. Perhaps it might
>  be called "a simple method for a simple people" (after carefully
>  excluding unavoidable comments from senators and congressmen who
>  probably haven't studied the method for long enough). But voters
>  may think it is more a method that demands that they be a nation
>  of Einsteins. The information they need to decide whether a voter
>  should list all of what they want or just a fraction of the
>  candidates they want elected, is completely unavailable to them
>  when they are voting, and even if accurate polls from the
>  previous day were available, it may be difficult to interpret
>  because the problems with the Approval Vote occur when candidates
>  have similar support.

I suggest reading Samuel Merrill, III "Making Elections More
Democratic", Princeton University Press (1988), for a good explanation
of voter strategy for both approval and FPP.

> It would need to be shown that voters do actually know whether
>  they prefer candidates over other candidates rather than knowing
>  that they are indifferent to a group of three (lest a fourth wins
>  and makes the 2nd lose), or is it all four? (in those cases where
>  the three would win anyway).
> 
> What are the names of the theorists that would want to replicate
>  those sort of considerations over groups containing at thirty
>  thousand people?.
> 
> Surely it is better to stay with a genuinely simple method like
>  First Past the Post, or else use a well designed preferential
>  voting method like STV?

The book listed above also shows computer simulation performance in
regard to Condorcet efficiency and social utility efficiency for various
systems, including approval, FPP, and single-winner STV.  Approval
compares well to the others, despite the fact that the comparisons
assume no information is available to identify the front-runners.  Thus
Approval at its worst compares favorably to the other methods under more
typical conditions.

What is striking about the simulations is that approval voting seems to
hold up even when there are large numbers of candidates, while FPP and
even STV decline in both Condorcet and social utility efficiency as the
number of candidates increases.  This contradicts the intuitive view
that you need to use ranking in order to deal with large numbers of
candidates.


> A problem with the Approval Vote is that it creates extremely
>  difficult stragic voting mathematical problems for individual
>  voters. They do not have the data or the time to allow them
>  to run computer simulations just in a hopeless aim of trying
>  to have their vote have no less influence than their
>  neighbours.

Only if you don't understand approval strategy.


> It is beyond imagination that the Democrats and the Republicans
>  would not be running their own simulations, etc. They have the
>  intelligence and the money: they jsut don't have the Approval
>  Voting method.
>
> Electorates may be better off with multiwinner First Past the
>  Post. At least FPTP/FPP is a true preferential method.
>  Even Borda (2 versions perhaps) is a better, but by how much
>  is unclear. I believe FPTP IS a better method than the Approval
>  Voting method.

How would you use multiwinner FPP in a single-winner election? 
Incidentally, it sounds like you are talking about SNTV, which is a
well-known semi-proportional method for multi-winner elections.

 
> How has the Approval Voting method evolved over the last decade
>  or years?.

It was apparently discovered by several people independently ca. 1970,
and has been used by several prominent math and engineering
organizations for internal elections since around 1987.  It is also
reportedly used in various places in eastern europe, and in the U.N.
Security Council (subject to veto).  Most of this information is
available on the approval voting web site:

   http://bcn.boulder.co.us/government/approvalvote/center.html