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Final Answers
© 2000-2016   Gérard P. Michon, Ph.D.

Social Choice

 Nicolas de Caritat 
 1743-1794  Enjoy your own life, without
 comparing it to that of another.

  Nicolas de Caritat,
Marquis de Condorcet  (1743-1794)

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Social choice theoryx   |   Voting paradox

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

 
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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.
 
Sextus Propertius (50-16 BC) 

(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 Ba + x
B A Ca
A B Cb + y
C B Ab
B C Ac + z
A C Bc

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

When A is opposed to B BreakdownTotal
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 BreakdownTotal
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 BreakdownTotal
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
n 012 345678910 11121314151617
A008795 103163106 15282136284536 456655

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.

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 still working on this one...


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

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 still working on this one...


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

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

The work "clout" itself has an elusive meaning.

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

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(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

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

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

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