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### Condorcet Criterion

```
Date: 02/13/2002 at 16:13:02
From: Marge Hughes
Subject: Modern Math - Condorcet candidate

I am taking a "Modern Math" course in college. My book does not
explain clearly the "Condorcet candidate" when using various ways to
determine a winner in an election. Do you have an explanation for
this?
```

```
Date: 02/13/2002 at 16:16:26
From: Doctor Paul
Subject: Re: Modern Math - Condorcet candidate

matchups with all other candidates should be the winner.

Suppose 4 candidates A, B, C, and D run for mayor of a small town (a
very small town). There are 20 registered voters. The local newspaper
performs a post-election survey of each of the 20 registered voters.
Among other things, the survey asks the voters who they preferred in a
two-way race between candidate C (the one endorsed by the paper's
editorial staff) and each of the other candidates. Here are the
results:

11 voters preferred candidate C over candidate A
11 voters preferred candidate C over candidate B
17 voters preferred candidate C over candidate D

other candidates.

Wouldn't it seem unfair if candidate C was not declared the winner?

When the actual votes were tabulated, candidate A got 9 first place
votes, candidate B got no first place votes, candidate C got 8 first
place votes, and candidate D got 3 first place votes. If candidate C
is not declared the winner, it will be a violation of the Condorcet
Criterion.

more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/13/2002 at 16:47:47
From: Marge Hughes
Subject: Modern Math - Condorcet candidate

others, i.e. preference of A over B, A over C, and A over D,

OR

do I compare all voters against the others: A against B, C, D,  and
B against C and D, and C against D ?
```

```
Date: 02/13/2002 at 16:55:10
From: Doctor Paul
Subject: Re: Modern Math - Condorcet candidate

indicate that the first scenario is correct.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 02/13/2002 at 17:04:15
From: Marge Hughes
Subject: Modern Math - Condorcet candidate

I don't have it yet.  Here's the example I am trying to do:

# votes  27  19   8  15   2
1st    B   A   D   C   A
2nd    D   D   C   A   C
3rd    A   C   A   D   D
4th    C   B   B   B   B

B has greatest number of 1st place votes, so B is the winner. The
Condorcet Criterion should (I think) look at each candidate against A.

So
27  19   8   15  2
1st   B   A   D   C   A
2nd   D   D   C   A   C
3rd   A   C   A   D   D
4th   C   B   B   B   B

B has 27 1st place votes over any other choice
A has 21 1st place votes over C or D
D has  8 1st place votes over A or C
C has  15 1st place votes over A or D

Collectively there are 44 1st place votes that don't want B (21+8+15),
so of the 44, A has most votes. Therefore I think A is the Condorcet
Criterion.
```

```
Date: 02/13/2002 at 17:31:59
From: Doctor Paul
Subject: Re: Modern Math - Condorcet candidate

Why did you pick A? If you want to see if this election satisfies the
Condorcet criterion, compare each candidate against the declared
winner (i.e., candidate B) to see how B fares in head-to-head matchups
against the rest of the competition.

It's clear that B wins by the Majority Criterion. If you want to see
whether or not this voting scheme satisfies the Condorcet criterion,
compare the winner (i.e., candidate B) against each of his (her)
opponents:

B vs A:

27 people preferred B to A but 19 + 8 + 15 + 2 = 44 people preferred A
to B. Thus since B did not beat A in a head to head matchup, if B is
declared the winner, then this voting scheme will violate the
Condorcet Criterion.

We don't need to consider B vs. C or B vs. D because we already know
that this voting scheme violates the Condorcet criterion if B is
declared the winner.

If you are looking for a candidate who should win if this election is
to satify the Condorcet criterion, consider candidate A:

We know that A beats B (see my note above).

How does A do against C?  A is preferred over C by 27+19+2 = 48 of the
voters, while C is preferred over A by 8+15 = 23 of the voters. So A

What about A vs. D? 35 prefer D to A, but 36 prefer A to D, so A beats

Thus if this election is going to satisfy the Condorcet criterion,
then candidate A had better be declared the winner.

We know that Candidate C can't win this election if it is to satisfy
matchup.

Similarly, we know that Candidate D can't win this election because D

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Discrete Math
High School Discrete Mathematics

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