by Warren D. Smith, May 2008

**Abstract.**
Some people have argued that the Borda Count, or Condorcet methods,
are superior single-winner voting methods to range voting.
Here, we shall discuss the "Gazillion Transform" of a voting method, which is a way to add
new artificial "candidates" to any election. It turns out that the gazillion transform
*converts* Borda and
Copeland voting into range voting and median-based range voting,
respectively.
That is, in the new election, the topmost-finisher (using Borda or Copeland) among
the non-artificial candidates *is* the range-voting winner – assuming honest voters.
Range voting remains apparently superior even with strategic voters
in the sense that
strategy is exactly the same in the
gazillionized Borda and Copeland elections as it is in the range election,
*except* that the Borda and Condorcet voters then would be able
also to apply *more* strategies too. So if having strictly
more strategies (a superset) available makes (all other things
being equal) a voting method worse, then they are worse than range voting.

This "proves" that range voting is at least as good as either Borda or Copeland-Condorcet voting.
The only way I see to deny this "proof" is to believe
that adding extra candidates
hurts Borda or Condorcet *more* (as ordering methods) than it hurts range voting.
(But if so, then that itself is an argument that range is superior.)

Simply add a "gazillion" (more precisely G, in the limit G→∞)
extra artificial candidates to the election.
Candidate *p*, where *p* is an arbitrary real number with 0≤*p*≤1,
is equivalent in utility (from the point of view of voter *v*) to
a lottery in which, with probability *p* voter *v*'s favorite wins, while with
probability 1-*p* voter *v*'s most-detested choice wins.

The *p*-values of the gazillion new candidates
are assumed to be uniformly spaced throughout [0,1].

I am here describing the "normalized Gazillion transform." One could also
consider the "*un*normalized Gazillion transform" in which the *p*-lottery instead is
among these two alternatives: (A) everybody lives to be 140 years old happily all the way,
and (B) everybody is tortured to death.

We now conduct the election using any election method which outputs a rank-ordering of the
candidates (e.g. Borda voting). The winner is declared to be the topmost-finisher
among the *original* candidates.

**1.**
Note that the addition of the gazillion new candidates
of course leaves unaltered the identity of the "best"
winner among the *original* candidates.

**2.**
(Normalized) range voting is *invariant* under the (normalized) Gazillion transform;
the same winner results (with honest voters) as in the original election.

**3.**
Obviously, voting in the new gazillionized election is very difficult and expensive (since,
e.g. rank-ordering a billion candidates is difficult for both voters and talliers).
However, that would not matter if
we were interested in obtaining the best possible winner
no matter how much it cost. And this expense only
makes our arguments about the superiority of range voting
over, e.g, Borda, become more true.

**4.**
The gazillion transform will effectively convert any voting method based on rank-order ballots,
into one instead based on rating (range-voting style) ballots.

In Borda voting, each voter rank-orders all
*N* candidates.
The *k*th-ranked candidate gets score *N-k*. The candidate with
the highest score-sum is elected, and more generally, the voting system can be
regarded as outputting
an ordering of all the candidates by decreasing "Borda score."

Borda Count voting is transformed into (Normalized) Range Voting by the (Normalized) Gazillion transform, because the Borda scores are in the limit the same, up to a rescaling, as the range-voting scores.

By "Copeland" voting, we shall mean the following procedure.
Each voter rank-orders all
*N* candidates.
The "Copeland score" of candidate C is then the number of "pairwise victories" he
enjoys over his rivals.
The candidate with maximum score is the winner, and more generally the voting system can be regarded
as outputting a full
ordering of all the candidates by decreasing Copeland score.
(A serious problem with Copeland is its propensity for tied-winners, but
that will turn out generically not to matter for our purposes here.)
Copeland is a "Condorcet voting method," i.e. elects any pairwise-undefeated candidate.

Copeland voting is transformed into median-based
Range Voting (with normalized votes)
by the (normalized) gazillion transform.
To see that, argue as follows.
The artificial candidates are ordered, unanimously, by all voters by *p*-value.
Each voter may be regarded as assigning a *p*-score between 0 and 1
to each candidate based on where she places that
candidate in her overall ordering.
Among the artificial candidates, consider the one Q whose *p*-value is
the same as the greatest median score enjoyed by any
original candidate. That candidate with the greatest median score will be exactly tied
with Q (with pairwise margin=0), but will be defeated by (margin≤0)
every artificial candidate
whose *p*-value exceeds Q's, and will defeat
every artificial candidate
whose *p*-value is below Q's (ignoring pairwise ties where exactly half the voters
say A>B and half B>A; these cannot occur if the number of voters is odd).
Thus the Copeland score of each candidate will be, in the limit, proportional
to the median of the *p*-scores the voters award to that candidate.

The gazillion transform actually can be applied to *any*
voting method
which outputs a full ordering of the candidates.

Further, any method which merely
outputs a single "winner" can be *regarded* as outputting a full ordering by
the following trick: Output the winner, then remove that candidate from the set of candidates and redo
the election to yield the "second finisher," then remove it and redo to get the "third," etc.
This will be called "removal order." (It is not necessarily the same as
the "natural" order for a voting method for which a natural order exists.)

It is interesting to explore what effect gazillionizing has on different voting methods.

Assuming each voter has a fixed private utility threshhold and approves everybody above-threshhold, then approval voting is invariant under the gazillion transform.

Plurality voting is invariant. (And so is anti-plurality voting.)

By "Simpson-Kramer minmax" voting, we shall mean the following procedure.
Each voter rank-orders all
*N* candidates.
We compute the
N×N "pairwise margins table," saying, for each ordered pair (A,B) of candidates, the
number of voters who prefer A>B minus the number who prefer B>A.
The "Simpson-Kramer score" of candidate C is then the
margin of the worst pairwise-defeat C
suffered.

Gazillionizing Simpson-Kramer (with removal order) converts it into a simple but extremely bad voting method. Specifically, it is this:

- Each voter awards a rating in the real interval [0,1] to each and every candidate as her vote.
- We find the candidate-voter pair with the greatest score. (For example, if Jill Voter rated Candidate Nixon 0.99, whereas every other voter rating were below 0.99, then Nixon is the winner.)

The same silly voting method arises also as the Gazillion transform of the *Nanson-Baldwin*
voting method
(of repeatedly eliminating the Borda-score loser), using the natural elimination-ordering.

"Upper-truncated" range voting means that we elect the candidate
with the greatest average score *after* all scores greater than T are truncated down to T.
(We similarly can define "lower-truncated" range voting; and we can do
this for median-based range voting too.)
I mention these because they seem to be important ingredients when trying to understand
the gazillion transforms of
various Condorcet and Borda-related voting methods.

By "Basic Condorcet" voting, we shall mean the following procedure.
Each voter rank-orders all
*N* candidates.
We compute the
N×N "pairwise margins table," saying, for each ordered pair (A,B) of candidates, the
number of voters who prefer A>B minus the number who prefer B>A.
The "score" of candidate C is then the minimum number of "pair-vote-reversals" we need
to cause C to beat every rival pairwise. For example, if C defeats every rival pairwise
(i.e. has positive margin over) except for A and B – becasue A beats C by a 10-vote margin, and
B beats C by a 5-vote margin – then C's score would be 10+5=15.

Gazillionizing Basic Condorcet (with removal order)
*almost* converts it into median-based range voting.
The reason is that every artificial candidate whose *p*-value
exceeds the maximum median score of any candidate, is a Condorcet winner among them.
We keep removing these from the top down. Eventually we get a situation in which
the median-range winner among the original candidates beats every artificial candidate hence has
boundedly large Condorcet score.
*But*
the other original candidates still are (generically) defeated by an arbitrarily-large number
of artificial candidates, hence have arbitrarily-large Condorcet score, hence cannot win.

This would have proven the winner must be
the median-based range winner,
*except* that at this point a non-removed
artificial candidate might still be the winner.

So all that we have really proven is that, using removal ordering, at some point in the removal sequence (at which moment all the original candidates still remain) we shall reach a situation in which the median-based range winner among the original candidates, has far smaller Condorcet score than any other original candidate.

Gazillionizing Schulze Beatpaths (with removal order,
generically) also *almost*
converts it into median-based range voting (up to ties, which is an issue we ignore
rather badly).

The argument is as follows.
The median-based range winner W (with median score M) plainly has a beatpath
to everybody who has median-score below M (simply walk to an artificial candidate wth
*p*=M-ε then downward in *p*, then to the target candidate). Every defeat on the
beatpath is unanimous (100-0) except for the first and last ones which might be by tiny margins.
The first hop
(from W to an artificial candidate Q whose *p*-value lies ε below M)
always has minimally small victory margin, namely 1 or 0 votes.
There does not exist a beatpath from anybody
to any artificial candidate with *p*>M, and the only beatpaths from original candidates
to the artificial candidate with score M-ε, have minimal
strength. Therefore, no rival of W among the original candidates can be
a beatpath winner because it has greater minimum beatpath strength to Q than does W.
So we would be able to conclude W must be a beatpath winner (although
perhaps a tied co-winner) *except* that, again, some
non-removed
artificial candidate might still be the winner.

So, again, all that we have really proven is that, using removal ordering, at some point in the removal sequence (at which moment all the original candidates still remain) we shall reach a situation in which the median-based range winner among the original candidates, is the only one with a beatpath to every still-not-removed candidate.

Black's method employs rank-order ballots; it elects a Condorcet winner if one exists, otherwise it elects the Borda winner.

Black (with removal order) is converted by the Gazillion transform into this voting system: If the ordinary (averaging-based) range voting winner has greater average score than the median-based range voting winner's median score, then elect the former. Otherwise, if the median-based range voting winner is a Condorcet winner, then elect it. Otherwise something else (more mysterious) happens.

First, all the pairwise defeats of artificial candidates with lesser *p* by
those with greater *p* are "locked in" immediately since they all are 100-0 defeats.
But what happens next is more mysterious.
If the median-based range voting winner is a Condorcet winner, it will be elected.

IRV under the Gazillion transform (with removal ordering) will actually act
exactly the same as IRV (but using range-style ballots,
albeit with the usual IRV counting procedure that ignores the numerical preference *strengths*)
with only a *single* extra
candidate E added, where E is given some common score (also called E)
for all voters, and we just keep decreasing E a smidgen until this IRV
procedure stops electing E, and then the new winner
is the winner of Gazillion(IRV).

So it is possible to devise a reasonably efficient algorithm to find the Gazillion(IRV) winner, but I do not think it has any simpler description and it is sometimes the same as and sometimes not the same as, the ordinary IRV winner.

The same "shrinking E" algorithm idea can be used to determine efficiently
who wins Gazillion(Schulze)
elections.