Essai sur l'application de l'Analyse à la probabilité des décisions
rendues à la pluralité des voix (1785)
M. le Marquis de Condorcet, secrétaire perpétuel de l'Académie des Sciences

Voting

Quod si deficiant vires,
audacia certe laus erit :
in magnis et voluisse sat est.

Elegiarum [II, x, 5]

If our powers are not sufficient,
fortitude would surely be praiseworthy.
In great deeds, willingness is everything.

(2010-03-06) Condorcet's Paradox
A group of voters may prefer A to B, B to C and C to A !

When faced with a choice among three options A, B and C,
each individual may express his or her own preferences in one of six
distinct ways.
For the sake of future convenience,
we may tally the number of voters in each of those 6 categories
using the 6 quantities a,b,c,x,y,z introduced in the following table:

Notations

Order of preferences

Number of voters

C A B

a + x

B A C

a

A B C

b + y

C B A

b

B C A

c + z

A C B

c

With those notations, the outcomes of the three possible straight votes are:

When A is opposed to B

Breakdown

Total

Votes for A

[CAB] + [ABC] + [ACB]

(a+b+c) + x+y

Votes for B

[BAC] + [CBA] + [BCA]

(a+b+c) + z

When B is opposed to C

Breakdown

Total

Votes for B

[BAC] + [ABC] + [BCA]

(a+b+c) + y+z

Votes for C

[CAB] + [CBA] + [ACB]

(a+b+c) + x

When C is opposed to A

Breakdown

Total

Votes for C

[CAB] + [CBA] + [BCA]

(a+b+c) + x+z

Votes for A

[BAC] + [ABC] + [ACB]

(a+b+c) + y

Thus, the aforementioned
paradoxical result occurs if and only if
x, y and z verify the so-called
triangular inequalities
(which state that the sum of any pair of quantities is never less than the third).

In that case, those 3 triangular inequalities do imply that x,y,z are positive. For
example, we have
2x+y > x+z > y which implies x>0.

The other paradoxical case (B>A, A>C, C>B)
occurs when the three triangular inequalities are all
backward, which implies that x,y,z are negative. In all other cases,
there is no paradox (which is to say that the collective preferences of the
voters are consistent).

How frequent is the paradox ?

Sometimes, ranking three options really boils down to a simple choice between
two options (whenever the third choice is either clearly inferior or
clearly superior). The paradox will occur with vanishing probability
in such cases, since a vote between two options is never paradoxical.

To evaluate the situation when the three options offered
to the voters are a priori on the same footing,
we shall determine the probability of the above paradoxical situation
when all preferences are assumed to be equiprobable...

The number of triples of integers forming
the sides of a triangle of perimeter n is either
(n+2)(n+4)/8 (if n is even)
or (n-1)(n+1)/8 (if n is odd).

Curiously, that's also exactly
the number of ordered triples of integers forming
the sides of a triangle of nonzero area and perimeter n+3.

Number of triples (x,y,z)
forming a triangle of perimeter n = x+y+z

The paradoxical situation A>B>C>A among v voters is obtained
by choosing a positive n of the same parity as v
and three nonnegative integers
a,b,c which add up to m = (v-n)/2.

(2013-09-18) Ranking Individuals According to Multiple Qualities
On the relative size of the convex hull in many dimensions.

(2013-09-18) Encouraging or Discouraging Multiples Political Terms
About incumbents.

(2013-09-18) What's Political Clout?

The work "clout" itself has an elusive meaning.

(2013-09-18) Are Quotas a Good Thing?
The dangers of slicing the citizenry into categories.

(2013-09-18) When Should Political Decisions be Ruled Out?
The proper political domain is the boundary of a convex set.

Very often, choices have to be made between uncommensurable things.
Weighing the basic rights of every individual and

(2013-09-18) Clienteles and Coalitions?
Minorities may triumph.

(2013-09-18) The Evil of Dogmas and Agendas
Dogmatism reduces or eliminates the tossing of ideas.