Fishburn Rich Parties Example

[Mike Ossipoff]

An example with Fishburn//Copeland//Condorcet & the Rich Parties
problem:

3 parties: A, B, & C

5 candidates:

A party candidates: A1 & A2
1 B party candidate: B
C party candidates: C1 & C2

The percents in the left column are the percents of the voters
who vote the ranking that follows the percents in the same line:

35%: A1, A2
32%: B
33%: C1, C2, B

Though B beats everyone by abaout a 2/3 majority, A1 wins
in Fishburn//Copeland//Condorcet.

It'a a circular tie in which B beats the A candidates, the
A candidates beat the C candidates, & the C candidates beat B.

The circular tie is the result of truncation by A voters.

The winner by Copeland, A1, isn't disqualified by Fishburn;
A1 is a "Fishburn winner". That's because, though B beats
A1, B is beaten by something that doesn't beat A1; and
though A1 is beaten by something that doesn't beat the C
candidates, A1 beats the C candidates.

***

Bruce, did you say that Fishburn//Copeland//Condorcet doesn't
have the Rich Problem? Why did you say that?

***

But wait--it's better than that:

Let's consider it in a more general way:

There are 3 parties: A, B, & C
The letters A, B, & C will stand for the numbers of candidates
of the parties.

There is a circular tie in which the B candidates beat the A
candidates beat the C candidates beat the B candidates.

Within each party, there's a candidate who beats each of the
other candidates of that party.

***

What does it take for an A candidate to be the winner?

A1's Copeland score is A-1+C-B (Where A1 is the A candidate who
beats the other A candidates.

That's because A1 beats A-1 A candidates & C C candidates, & is
beaten by B B candidates.

The scores of the top candidates in the other parties are
calculated exactly the same way.

So then, what does it take for the top A candidate to beat
the top B candidate?

A-1+C-B > B-1+A-C   Subtract A-1 from each side:

C-B > B-C   Subtract -C & -B from both sides:

2C > 2B     Divide both sides by 2:

C > B

So the top A candidate will have a higher Copeland score than
the top B candidate if C can afford to run more candidates than
B can.

What does it take for the top A candidate to have a higher
Copeland score than the top C candidate?

A-1+C-B > C-1+B-A  Subtract C-1 from each side:

A-B > B-A  Subtract -B & -A from both sidses:

2A > 2B   Divide both sides by 2:

A > B

So what does it take for the top A candidae to have a higher
Copeland score than the top C candidate? That will be so if
Party A can afford to run more candidates than party B.

Since party B is the party whose candidates in the circular
tie beat the party A candidates, all it takes for party A
to steal the election by truncation (or order-reversal)
is for the victim party to not be able to afford to run
as many candidates as the other 2 parties.

***

It's no exaggeration to say that Copelands 
(and Fishburn//Copeland//Condorcet) is remarkably co-operative
with truncation or order-reversal by one of the rich parties.

It's no exaggeration to say that Copeland & its various versions,
is made to order for the rich parties problem & for successful
stealing of an election by truncation or order-reversal.

A muck-raker once said "Follow the money", and so please allow
me to point out that it's academics who tout Copeland's method,
their employers, universities, and research grants, tend to be
receive grants from foundations paid-into by rich families.

Guess what? The 2 sentences before this one have a word in
common: Rich.

***

Another thing: Bruce, is it true that you're employed by
the Institute for Defense Analysis, and that the Institute
for Defense Analysis is associated with the Defense Department?

No doubt we've all heard about the ways in which the Pentagon
has made a reputation as a defender of democracy around the
world  :-)

***

Mike









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