[EM] Bounded AV

[Forest Simmons]

On Sun, 28 Apr 2002 hager2002 at lsh107.siteprotect.com wrote:

> I have a question that has been bugging me, and now that I'm not a
> candidate I can indulge myself.
>
> I read a paper by Marji Lines (cited by Brams) dealing with the Venetian
> Republic's use of AV.  If I didn't misunderstand, the Venetian Republic
> used AV but required that the winner get 66%.  I later read in another
> paper about a "rule of 64%".  In essence it said that single winner would
> be assured with no cycles in something called an "n-dimensional issue
> space" as n approached infinity at just under 64%.

The 64 percent sounds like the probability of a normal random variable
landing within one standard deviation of the mean, so it must have
something to do with the Central Limit Theorem.  It makes sense that the
voters and candidates would be distributed with a multivariate normal
distribution in the limiting case if the issues were truly independent,
but that condition seems unlikely in practice.

The "cycle" part is unclear, since there are never any cycles in simple
Approval.  Could it be referring to the cycles in repeated balloting
Approval that we were discussing a few weeks ago?

Could it be saying that (in some limiting case) if you require 64 percent
quota in repeated balloting approval, once the voters compromise enough to
reach that quota, then it would be so stable that further repeated
balloting wouldn't change it, i.e. there would be no cycling?

> Is an "n-dimensional
> issue space" just a recondite way of saying n choices?

No.  It means that there are n key issues among which both the voters and
the candidates can have differing opinions.  A typical point in the space
is represented by an n-tuple of real numbers (x1,x2, ... , xn), where each
of the numbers is between zero and one, representing opposite extremes of
opinion on the respective issue, i.e. one and zero representing pro and
con, respectively.  In the limiting case it wouldn't really matter if
values between zero and one were disallowed, because the Central Limit
Theorem applies to discrete random variables as well as continuous random
variables.

In the discrete version, your position in issue space could be determined
by a questionnaire asking whether you agreed with the pro or con side of
each issue.  If your answers matched exactly with those of another voter
or candidate, then you would share a point in issue space.

There are 2 to the n such "corner points" in an n dimensional issue space.

In practice, your positions on a few key issues make it possible to
reliably predict your positions on the remaining issues, so the effective
dimension of issue space is not likely to be large.  In other words, the
vast majority of the "corner points" in a space based on a large number of
issues would be vacant or sparsely inhabited by non-conformist voters.

> And, if it does
> mean that, was the Venetian system "perfect" in the sense that it avoided
> cycles?

I don't know what it is supposed to mean.


> This suggests replacing the idea of None of the Above (NOTA) with the
> Venetian system.  NOTA is a veto such that if it gets a majority, all
> candidates are rejected and a new election with different candidates is
> held.  With the Venetian system, set it up so that a failure of any
> candidate to get at least 66% means that all candidates are rejected and a
> new election with new candidates must be held.

It seems unlikely to get a 66 percent approval for somebody the first time
around, so I suspect the method would require repeated balloting.

Forest


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