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

The Condorcet criterion says that a candidate who wins head-to-head 
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

So, in head-to-head competition, candidate C won against each of the 
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.

I hope this helps.  Please write back if you'd like to talk about this 
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

If A wins the most votes, do I then compare A head-to-head against the 
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

A "head to head" (i.e., one person against another) matchup would 
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 
beats C in a head-to-head matchup as well.

What about A vs. D? 35 prefer D to A, but 36 prefer A to D, so A beats 
D in a head-to-head matchup as well.

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 
the Condorcet criterion, because C doesn't beat A in a head-to-head 
matchup.

Similarly, we know that Candidate D can't win this election because D 
doesn't beat A in a head-to-head matchup.

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