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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.

Social Choice Theory
An Introductory Survey

Many forms of Government have been tried,  and will be tried in this world of sin and woe.  No one pretends that democracy is perfect or all-wise.  Indeed, it has been said that democracy is the worst form of Government, except for all those other forms that have been tried from time to time...
  Winston L. Spencer-Churchill  (1874-1965)
(in the House of Commons,  on 1947-11-11)

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Related articles on this site:

Related Links (Outside this Site)

An Introduction to Vote-Counting Schemes  Levin  &  Nalebuff  (1995).
The Mathematics of Voting  by  Larry Bowen  (Math 103,  Alabama).
Voting and Complexity  by  Rob LeGrand  (qualifying exam, 2005).
Voting Methods  Eric Pacuit  (Stanford Encyclopedia of Philosophy, 2011-08-03).
Economics and Computation (2)  Fabian Kolbe  (under F. Brandt, 2013-05-22).
There are better ways to vote than the system we're stuck with  (2016-11-09).
Mathematicians who want to save democracyCarrie Arnold  (Nature, 2017).
Electoral Reform Society  (ERS).  Oldest extant electoral society  (UK, 1884).
Proportional Representation Society of Australia (PRSA)  &  Hare-Clark STV.
The Center for Election Science  (Electology.org)  Aaron Hamlin.
Voting MattersMcDougall Trust  (Last issue was #30, April 2013).

Wikipedia :   Social choice theory   |   Voting systems   |   Voting paradox (Condorcet)
Arrow's theorem (1951)   |   Gibbard-Satterthwaite theorem (1973, 1975)   |   Schulze method (1997)
Electoral systems   |   Downs paradox   |   Matrix vote
Apportionment paradoxes

Essai sur l'application de l'Analyse à la probabilité
des décisions rendues à la pluralité des voix
M. le Marquis de Condorcet, secrétaire perpétuel de l'Académie des Sciences
Videos :   Voting systems and the Condorcet Paradox  by  Kelsey  (2017-06-15).
Réformons l'élection présidentielle  by  David Louapre  (2016-10-21).
Avez-vous vraiment gagné cette élection?  by  Nathan Uyttendaele  (2016-08-19).


Putting Good and Evil to a Vote

(2016-12-27)   The  Majority Rule  decides between two options  (only).
Whichever option is supported by the most people prevails.

It took a while for this simple assertion to emerge as the  fundamental principle  behind modern democracies.  At first,  voting was reserved to wealthy males.  Then all free men were included.  Then, there were no more slaves...  Finally,  women  could vote too!

However,  that's not the end of it when three options or more are put to a vote.  In that case  the  Majority Rule  is no longer directly applicable:  The  Condorcet paradox  goes to show that the intention of the voters may not at all be  clear when there are at least  3  options to choose from.

Now,  that's no reason to settle for any half-baked arbitrary procedure,  just to reach a quick decision,  or any  ersatz  thereof.

The rest of this page will show that there are pretty good ways to proceed,  in the utmost respect of the  Majority Rule.  It will also show how some popular methods are utterly unacceptable in the computer age.

The analysis of what's at stake was inaugurated by  Condorcet  in 1785 and vigorously revived by  Kenneth Arrow  in 1951.  It's now a whole academic field of study which goes well beyond this introduction.

However,  the basics are fairly easy to grasp and dangerous to ignore.  Ultimately,  the  Will of the People  shouldn't be allowed to be distorted beyond recognition, for something as trivial as a lack of understanding of the mathematical structure involved.

  The only Good is knowledge, the only Evil is ignorance.
Attributed to  Socrates (c.469-399 BC)  by  Diogenes Laërtius

(2010-03-06)   The case for a rigorous approach:
Beware of  simplicity.  Don't rely on intuition alone.

Mathematics is like checkers in being suitable for the young,
not too difficult, amusing, and without peril to the state.

Plato (427-347 BC) 

Mathematics is the art of spelling out everything  (as concisely as possible)  organizing patterns as they emerge.  In practice,  that's also a great way to prevent costly mistakes:  Measure twice,  cut once...

By contrast,  simple-minded methods are often impractical or too costly.  We can't build modern bridges by intuition or trial-and-error.  If anything,  the construction of voting systems is more delicate than the building of bridges,  because the flaws are far less obviously revealed!

The eloquence of the ignorant may only serve the nefarious purpose of discrediting rigor in the public eye.  In particular,  derision may undermine the unfolding of logical arguments for the greater good.  Ultimately,  we may all suffer as a result.  It's true that reasonning alone isn't always able to unearth a  complete  solution to human problems.  However, when it can do so, the public good is never served by suppressing the rational approach.

Also,  mathematical thinking helps counteract the dark side of propaganda.  Rushing something to a vote,  before digesting it,  is  unconscionable.

That's true, in particular, when it comes to the adoption of the voting system itself.  The misguided use of raw arithmetic rules is but a powerful way to hide utterly unacceptable prejudices...

No voting system is perfect and none is valid for all purposes.  However some are extemely bad, according to objective criteria, and we should weed them out.  Let's not shove such things down the throat of the voters, for the sake of expediency.  More often that not, expediency just leads to an ersatz of decision without rartional justification, drowned in decorum.

There's no excuse for accepting, at the core of the democratic process, any procedure, or lack thereof, which promotes dishonesty or unfairness  (whether this was the original intent or not).

The rest of this page provides the basic tools to do the right thing.

Whoever answers before pondering the question is foolish and confused.
Proverbs 18:13

(2017-01-08)   The Simplest Case of  Condorcet's Paradox
A group of voters who prefer  A to B  and  B to C  may  prefer  C to A !

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

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

With  3  options to choose from  (A, B & C)  each voter may express his or her own preferences in one of  6  distinct ways  (there would be  13  ways if  indifference  was allowed).

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.  (x, y and z may be negative).

Order of
 Number of 
C A Ba + x
B A Ca
A B Cb + y
C B Ab
B C Ac + z
A C Bc
Simplest paradoxical cases  (3 voters)
 A > B > C > A 
 a=b=c=0   x=y=z=1 
 A > C > B > A 
 a=b=c=1   x=y=z= -1 

With those notations, the outcomes of the  3  head-to-head 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 advertised paradoxical result  (one of  two)  occurs if and only if  x, y and z  verify  strictly  their  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+(x+y) > x+z > y  which implies  x>0.  The three triangular inequalities are thus satisfied  iff  x, y & z  are the sides of a Euclidean triangle  (hence the name).
The other paradoxical case  (B>A, A>C, C>B)  occurs when the three triangular inequalities are all backwards, which implies that x,y,z are negative and that their  opposites  verify the triangular inequalities.  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.

 Come back later, we're
 still working on this one...

(2016-12-21)   Tallying votes using additive preference points.
Such a tally can  always  conceal the  true  preference of voters.

Although  the  above  shows that voters may not always have a clear collective preference,  any tallying method which is unable to properly detect a clear choice,  when there is one,  is either dubious or fraudulent.

Surprisingly enough,  that applies to the popular method of translating the ranks given by every voter into amounts of  points  to be  added  based on a nonincreasing sequence of coefficients  c≥  c≥  c≥ ...

It won't do to just add  c1  for every first choice,  c2  for every second choice,  cn  for every n-th choice.  Let's prove that :

According to  Condorcet  (or any rational person)  a choice is  clear  among voters when some option is preferred to every possible one by a  majority vote  head-to-head  (by universal agreement,  a majority vote is the only democratic way to decide between  two  options).

When that happens,  the preferences of voters are said to obey  Condorcet's criterion  and the option which is then clearly the best one is called a  Condorcet winner.  A tallying method which picks the Condorcet winner,  whenever  there is one,  is said to be a  conform  Condorcet method.  With that in mind,  we may restate the above claim:

Theorem :   No tallying based on additive preference points can be  conform  (assuming the voters have at least three options to choose from).

Proof :   We only need to establish that inadequacy in cases where just three options are available  (with more than three options,  it may well be that all voters happen to prefer three of those over any other).

If   c1  =  c2  =  c3   then the tally of points is utterly inconclusive,  as the Condorcet winner  (if there is one)  always ties the other two by points.  Otherwise,  we may assume,  without loss of generality,  that:

c1   =   1         c2   =   q         c3   =   0         where   1  ≥  q  ≥ 0

That is so because we obtain an equivalent system of coefficients by subtracting the smallest one from all of them and, then, by dividing all coefficients into the largest one, which is necessarily nonzero.

To establish the claim, all we need is a paradoxical example for every value of  q.  Consider first a special case where only two of the six preference types are present:  Let's say  u  voters have one order of preference and the  v  others all agree on a different one,  according to the following table:

PreferenceNumber of voters Points to APoints to BPoints to C
A > B > Cuuq u0
B > C > Av0vq v
Totalu + vuq u  +  vq v

If  u > v ,  then option  A  is clearly a  Condorcet winner  because it wins over either  B  or  C  by  u  votes against  v.  However, the tally of points does designate  B  (never C)  as the winner over  A  if we also have:

q u  +  v   >   u         or, equivalently,         v   >   (1-q) u

Thus,  A  is the clear Condorcet winner but loses the point tally  if and only if  the following double inequality holds:

u   >   v   >   (1-q) u

As the vote counts are integers,  the largest value of  v  which satisfies the left inequality is   v  =  u-1   in which case the right inequality becomes:

u-1   >   (1-q) u         which is equivalent to         q u   >   1

So, if  q  is nonzero,  we obtain two integers meeting our requirements by letting  u  be any integer greater than  1/q  (since  q  is at most  1,  u  is at least 2)  while  v  is equal to  u-1.  That yields examples where the clear Condorcet winner doesn't prevail according to point totals.

For  q = 0,  we use the case below,  where  A  is the Condorcet winner  (garnering 4 or 5 votes out of 7, against B or C)  but  B  wins by points.

PreferenceNumber of voters Points to APoints to BPoints to C
A > B > C2200
B > A > C3030
C > A > B2002

All told, for any possible  q,  there are always cases where the clear Condorcet winner loses by points.   QED

(2016-12-26)   Plurality Voting   (voter's first choice gets whole vote)
It's just a special case of voting by  points.  Only worse.

Plurality  is the voting system whereby each voter gets to vote for only  one  candidate.  The candidate with the most votes wins.

The French call this  majorité relative.  This is a misnomer, since the winner isn't necessarily supported by the  Majority Rule.  The French call  majorité absolue  the special case when the winner of a plurality vote happens to garner more than half the votes.

That's exactly what you'd get with  points  if you count one point for the first choice of each voter and zero for the other choices.  (Incidentally, that forced single-mindedness would be even more damaging to the election of an ordered list than it is to the election of a single candidate.)

Thus, the  theorem  proven above fully applies to discredit  plurality  voting.  More specifically,  the  case  "q = 0"  (singled out in our proof)  is directly applicable.  So is the 7-voter example which we put forth to resolve it  (just a plurality vote thinly disguised).  Let's modify that example slightly to make flaws even more obvious.  We simply add two more voters who support  C,  and obtain the situation presented in the following table:

An Example of a Paradoxical Plurality Vote
PreferenceNumber of voters Votes for AVotes for BVotes for C
A > B > C2200
B > A > C3030
C > A > B4004

Option  A  is still the clear Condorcet winner  (beating respectively B and C head-to-head with 6 and 5 votes out of 9).  Likewise,  C  is still the definite Condorcet loser  (losing to either A or B, by 4 votes against 5).

Yet, the dubious result of the  plurality vote  is the exact opposite:

  • C wins with 4 votes  (but would lose head-to-head against A or B).
  • B gets 3 votes.
  • A is last with 2 votes  (in spite of being preferred to either B or C).

Real-life electoral commentators might say that  B  robbed  the votes of  A.  Would a primary between A and B have helped avoid the disastrous victory of C?  Well,  surely,  but what kind of primary do you have in mind?

  • A  would have won an  open primary  by  6  votes against  3.
  • B  would have won a  closed  one by 3 to 2  (among opponents to C).

This goes to show that  closed primaries  can help conceal the best choice when there's a clear one.  They may go against the greater public good.

On the other hand,  open primaries  are best integrated into the voting system itself,  according to transparent criteria compatible with the absolute democratic requirement of  conformity  with the Condorcet criterion.  (Separate open primaries are notoriously prone to  tactical voting.)

The above example can also be used as an argument against any form of  runoff election,  including  instant-runoff voting  (IRV).  Indeed,  the clear democratic choice of the voters  (namely, A)  arrives last in the first round and is thus eliminated from the  runoff.

Nonmonotonicity in Alternative Vote (AV)  by  Eivind Stensholt  (2002).
Wikipedia :   First-past-the-post (single voting)   |   Plurality-at-large (unlimited voting)   |   Duverger's law

(2016-12-27)   Runoff Elections:  The Art of Elimination
The result is compelling only if the last round opposes just two options.

Some runoff elections do not insist on having just two candidates in the final round.  One example is French legislative elections,  in which a candidate need not be first or second in the first round to qualify for the final round.

When that happens,  a tactical decision  (whether to fold or not)  must be made between the two rounds by all qualified candidates.  Such decisions are made more complex by the fact that those elections are not isolated.  The political parties of the candidates may trade one position in one district against another one in another district.  Here's one example:

 Come back later, we're
 still working on this one...

Nonmonotonicity in Alternative Vote (AV)  by  Eivind Stensholt  (2002).
Wikipedia :   Contingent voting   |   Instant-runoff voting (IRV)  =  Alternative voting (AV)
Nanson's Method  &  Baldwin's method  (conform runoff elections based on Borda's method)

(2016-12-27)   Qualifying for an Election
A necessary feature prone to abuse.

When the possibility exists that a large number of candidates would be tempted to run for office,  some legal prerequisites may be helpful to avoid swamping the actual election.

 Come back later, we're
 still working on this one...

(2016-12-26)  Why are point systems so popular?  (e.g.,  Borda counting)
Why equate a relative preference to an arbitrary number of points?

Time and time again,  people have adopted quickly such tallying methods for the sake of expediency.  Counting points doesn't seem evil at first and most people will trust the familiarity with  points  which they've acquired in irrelevant non-electoral contexts,  including academic grades, games, sports and entertainment  (e.g., Eurovision Song Contest).

Voting by points was abandoned by the Roman Senate early in the second century  (AD)  not at all for any mathematical reasons but because such voting methods proved far too prone to  unsavory  manipulations  (including political  cloning).

Jean-Charles de Borda (1733-1799)  advocated his own version of the thing in 1770  (the so-called  Borda count  of an option on a ballot is the number of options ranked below it on that ballot).  Borda understood the validity of Condorcet's objections and recognized the lack of fairness of his own method,  but he kept advocating it "for the sake of simplicity"  (which was indeed a serious consideration before the advent of computers).

Unfortunately,  the  Borda method  endures to this day,  especially for fairly casual elections to the boards of many associations in the United States.  It's being gradually replaced by more rational methods  (like the Schulze method)  at least at national levels where computerized ballot counting makes it untenable to retain the Borda method  "for the sake of simplicity".

Some people would rather die than think...  Many do.
  Bertrand Russell  (1872-1970)

(2017-01-04)   Cardinal Voting :  Grading candidates
Aggregating the scores given to candidates by a panel of judges  (voters).

Each judge attributes nonnegative scores to the candidates for a total of  k>0  points  (there may be additional constraints on the scoring to encourage judges to spread their votes).

This unusual type of voting is very different from the more common  ordinal voting systems  which most of this presentation is concerned with  (using only relative preferences without quantifying their strengths).  Interestingly,  the well-known  impossibility theorems  only apply to ordinal voting and the  cardinal voting  discussed here may help circumvent their conclusions.  In that spirit,  cardinal voting  is worth considering,  possibly as part of a  composite systems.

Let  A  be the  rectangular  matrix  whose element  Aij  (at line i and column j)  is the score cast by  judge j for candidate i.  We define two symmetric  square matrices  using  A  and its transpose  A*...

  • The  candidate matrix   C   =   A A*
  • The  judge matrix           J   =   A* A

The row sums of  C  are used to rank the candidates.

 Come back later, we're
 still working on this one...

Voting matrices and tie-breaking  by  Jeffrey L Stuart  &  James R. Weaver.
International Journal of Pure and Applied Mathematics, 54, 3  (January 2009).
Numericana :   Perron vector     Wikipedia :   Cardinal voting systems   |   Edmund Landau (1877-1938)

(2016-12-25)   Conform  Condorcet methods:
They correctly produce the voters' intended choice,  whenever it's  clear.

Fortunately,  there are many ways to tally preference ballots which always produce the  Condorcet winner  if there is one.  (Plurality voting  and any other form of counting by  points  are just  not  among those methods.)

Such methods are said to be  [conform]  Condorcet methods, in the weak sense  (that's the way the term  Condorcet method  is understood in the literature, unless otherwise specified).

Some tallying methods achieve that conformity by using adequate  scores  and ranking the options from highest to lowest score.  (The conform voting system devised by Ramon Llull in 1299 doesn't rely on scoring.)  Before the advent of computers,  such  conform  scores were tedious to compute  (much more so than the aforementioned expedient  additive points).

For example,  we may give to each option a score equal to the number of victories obtained in head-to-head matchups  (counting a tie as a fraction of a victory).  This is obviously a  conform  scoring system because, with  N  options to choose from,  the Condorcet winner would have scored  N-1  victories and all others  N-2  or less.  That's called  Copeland scoring.

Strong Condorcet conformity :

When we ask a voting system to yield an ordered list of the various options,  it's reasonable to demand that such a system would produce the perfect Condorcet  order  whenever it exists.  A Condorcet order is an ordering of the options such that the first one is the Condorcet winner, the second one is the Condorcet winner among all other options,  the third one is the Condorcet winner among all options but the first two, etc.

In the present work,  we say that a tallying method is  strongly conform  (or that it's a [conform] Condorcet method in the strong sense)  when the following conditions are met:

  • The Condorcet winner,  if there is one,  is produced at the first place.
  • The Condorcet loser,  if there is one,  is produced at the last place.
  • The Condorcet order,  if there is one,  is obtained as the full result.

So stated,  the last condition is fairly weak,  since a Condorcet order is a rare thing.  However, it suits our purposes well enough at this time.

We'll see that  Copeland scoring  is strongly conform.  So is  any method of breaking ties between options having equal Copeland scores.  I advocate as a tie-breaker,  the total number of votes garnered by an option in all pairwise confrontations.  (The remaining ties have to be broken with arbitrary time-honored methods,  for example using the ages of the candidates.)

Also,  it's very important to note that  Llull's procedure  transforms a strongly-conform method into another strongly-conform one  (because any  Llull switch  has this property).  That's true for either the  normal procedure  of Llull  (increasing  zig)  or the backward one.  (decreasing  zag).  It also holds for any concatenation of zigs and/or zags.

Those lemmas will ultimately allow us to  construct  a complete voting system which is  strongly conform  in the above sense and also  stable,  which is a requirement of paramount  political  importance  (although it's almost always ignored).

By contrast,  a single Lull  zig-zag  appended to an arbitrary voting system will produce a voting system which satisfies the first and the second of the above conditions,  but not necessarily the third!

(2016-12-22)   Llull's Procedure     (Raymond Lull = Ramon Llull, 1299)
Turning any preliminary order into a conform Condorcet voting system.

Let's use the following graphical convention to describe how a left-to-right ordering of two options is corrected by a possible  switch,  according to the result of the head-to-head vote symbolized by the  checkbox.

First   Llull's procedure Last    An  important  convention is that no switch takes place in case of a draw,  so the initial left-to-right ordering is effectively a tie-breaking rule.
The figure shown at left then describes graphically one way to turn an arbitrary preliminary ordering of any number of options  (top)  into a  conform  Condorcet tallying method  (bottom).

Bowlers will recognize the structure used in the finals of  major bowling tournaments,  as pairwise matches amend an otherwise established ranking.

 Ramon Llull   Coat-of-arms of Ramon Llull  Using some arbitrary preliminary order,  the above was proposed as a voting system for the Church in 1299  (De arte eleccionis)  by  Ramon Llull  (1232-1316)  a  Franciscan tertiary.  His name used to be anglicized as  Raymond Lully  or  Raymond Lull.  He published in Latin as  Raimondus Lullus  and is known to the French as  Raymond Lulle.  He was beatified in 1857 and is revered as a saint in his native  Catalonia.  He is the dominant author of medieval  Catalan literature.

Llull's system is a  conform  Condorcet method,  as it always elects the clear  Condorcet winner,  whenever there is one.  Back in the thirteenth century,  Llull may or may not have realized that there need not always be such a clear winner,  in which case the outcome depends on the preliminary order.

Originally, voters only had to reveal their preferences as needed and they could very well vote for C over A for tactical reasons,  even if they secretly preferred A to B and B to C...  (Llull assumed they would vote sincerely.)

We may as well disallow this type of  manipulation  by specifying that voters cast their ballots only in the form of a linear list of preferences  (which, incidentally, allows voting by mail).  According to the terminology used in the literature,  this is equivalent to stating that all voters are  rational.  (It's rather pointless to study the cases when they're allowed not to be.)

If a spreadsheet  (Excel)  is used to tally such preference ballots automatically  (see example)  then each step of Lull's procedure is implemented by a single line which  normally  reproduces the previous one  except  for a pair of cells identified by a colored background.  The programming of those colored pairs ought to make them  equivalent  (i.e., any such pair can be  cut-and-pasted  from any other).  Only one pair has to be programmed from scratch.  A typical fragment looks like this:

 Llull's method on a spreadsheet.

In this, the numbers represents the numerical identifiers assigned to every option  (or candidate).  A colored pair may either reproduce the cells above it  (as in the case of  [4 vs. 7]  or  [9 vs. 4]  in the above screenshot)  or switch them  (like  [5 vs. 7]  or  [2 vs. 4]  are switched).  The latter is done only when the voters positively prefer the right option to the left one.  Doing nothing in case of equality means that the preliminary order stands  (left to right)  unless contradicted by a vote,  which reduces it to a tie-breaker.

De arte eleccionis (1299-07-01)   |   Ramon Llull (1232-1316)   |   Ramon Llull University (1990-)

(2017-01-02)   Tallies  are the relevant summaries of the votes.
Tallies are  sufficient  data:  Identical tallies yield identical outcomes.

Obtaining tallies is the time-consuming operation.  After that,  the computation of the outcome doesn't depend on the number of voters.

Many voting systems may share the same tallying process.

With manual counting of the ballots,  the simplicity of the tallying was an important consideration.  That's a key aspects which made  plurality voting  so popular.  (In a plurality vote,  the only tally needed is a running count of the votes garnered by each option.)

For related  point systems,  the tally can be the running total of the points garnered by each option,  but it's computationally better to update a  tallying matrix,  telling how many times a given candidate is attributed a given rank.  The total garnered by each option can easily be obtained from that matrix.

For  ranked voting systems  (with ballots consisting of linear lists of preferences)  the tally we need is a  matrix  which gives at line  i  and column  j  the number of votes  Vij  in favor of  j  in a head-to-head confrontation with  i.  (Some authors interchange the rôles of i and j.  The two conventions are related to each other by  transposition.)

If there are  N  options to be ranked and  M  voters to rank them,  then the above tallying can be obtained in a time proportional to  M N2.

Alternatively,  if the number of options is so small that the number  W  of  possible ranking choices  is substantially smaller than the number of voters,  then the tallying can be done in a time at most proportional to  MN+WN2  by first recording  (in a tallying array of size  W)  what type of preference each voter expressed.  The results are then used as coefficients in a weighted sum of the  W  matrices corresponding to every type of preferences.

Wikipedia :   Ranked voting systems   |   Ranked Pairs  (Tideman, 1987)

(2016-12-27)   Nefarious  manipulations  of the electoral process:
Taking advantages of the weaknesses of a voting system.

Back in the thirteenth century,  Ramon Llull  insisted on a certain decorum in his voting proposal  (for Church functions).  He prescribed two things:

  1. Every elector would take an oath that he would vote according only to his best judgment of the qualifications of each candidate.
  2. The vote was public,  so everyone could see how every elector was discharging the duty he was sworn to.

In those days,  an oath meant something and that was a good enough way to prevent a type of  manipulation  of the voting system which I like to call  tactical misvoting  (it's a perverse form of  tactical voting).

I find the  proposed  term of  bribery  rather unfortunate,  as the cause of intentional misvoting need not be a bribe.  Manipulations are not necessarily related to outright corruption.

There are two widely-recognized ways of manipulating a voting system:

  • Strategic nominations :   Adding or withdrawing candidates.
  • Tactical misvoting :   Voting against one's true preferences.

In this context,  misvoting  (or  perverse voting)  consists of casting an insincere vote to take advantage of the idiosyncrasies of the voting system.  It's essentially the practice of voting for an inferior candidate just to increase the chances of one's overall favorite in the final decision.  To do such a thing,  the voter must know something  (not necessarily much)  about the intentions of the other voters.

Misvoting  is opposed to more palatable forms of tactical voting,  like mere  compromising  (which consists in supporting a moderate option in order to defeat an extreme view opposed to one's own).  Misvoting is  immoral  in the sense any student of  Immanuel Kant  will instantly recognize:  If everyone misvoted,  the whole system would collapse and one of the worst options could triumph,  against the true will of an overwhelming majority of voters  (possibly,  even against unanimity).  A weaker Kantian  argument also applies to open primaries;  if both camps always vote for the weaker candidate in the other camp,  the best candidates will never get elected,  in either camp.

Here's one of the simplest examples of  tactical misvoting,  which works like a charm in the case of Llull's system  (at least in the thirteenth century version where the voters were not constrained by a rational order of preferences  a priori):

The supporters of two serious contenders may figure out that one very good candidate stands in the way of both of their respective favorites.  So,  with or without premeditation,  they decide to cast their votes in favor of an inferior fourth candidate to block the good one in the early stages of a Llull vote.  After that nefarious elimination,  everyone forgets about the inferior candidate and the road is wide open for the other contenders...  Bad.

Unfortunately,  it has been shown that no  ordinal voting system  is completely immune to manipulations.  At least,  that's one way to interpret some of the classical  impossibility theorems  pertaining to voting.

A fairly recent fashion has been to observe that,  at least with some  robust  voting systems,  the  computation  of manipulations is provably hard in the convincing sense of  NP-completeness.

NP-completeness means this:  If there was an efficient algorithm to compute all possible manipulations available to a small number ot voters  (assuming a decent voting system)  then there would be an equally efficient way to break the current security of all electronic banking transactions.  (Here "efficient" means a computing time less than some  polynomial function  of the  size of the input;  that's not always a practical criterion but it often is.)

By itself however,  this fact provides little solace,  since all such analyses are ultimately concerned only with worst-case scenarios.  In the above theoretical sense,  manipulation procedures which are effective in almost all configurations may still be computed fairly efficiently...  For example,  the NP-completeness of  some  voting systems is strictly based on the possibility of pairwise ties,  which are extremely unlikely in general elections  (but then,  again,  the fewer the voters, the more effective the voting manipulations).

The computational difficulty of manipulating an election  by  J.J. Bartholdi III,  C.A. Tovey  &  M.A. Trick
Social Choice and Welfare6,  3,  pp. 227-241  (1989).
Llull and Copeland Voting Broadly Resist Bribery and Control   (Rochester CS TR-2007-913,  2007-02-19)
by  Piotr Faliszewski,  Edith Hemaspaandra,  Lane A. Hemaspaandra  &  Jörg Rothe.
When Are Elections with Few Candidates Hard to Manipulate
by   Vincent Conitzer,  Thomas Sandholm  &  Jérôme Lang.   Journal of the ACM, 54, 3, #14  (June 2007).
Wikipedia :   Strategic nominations   |   Independence of irrelevant alternatives   |   Cloning
Tactical voting

(2016-12-30)   The Impossibility Theorems   (lowering expectations)
Voting systems can be downright unacceptable or darn good,  not perfect.

Condorcet's  intransitivity of pairwise voting decisions,  can be construed as the earliest impossibility theorem:  Although pairwise voting is arguably the only sound basis for a democracy,  it's not adequate tu run government on a daily basis.  It would be theoretically impossible to put every decision to a vote,  even if it was practical to so so.

In modern times,  other  impossibility theorems  have been established which show the incompatibility of various desirable features of voting systems.

We're stuck with imperfection,  but that's no reason to settle for mediocrity!

Arrow's impossibility theorem (Ph.D. Thesis, 1951)  by  Kenneth Arrow (b.1921, Nobel 1972)
Gibbard-Satterthwaite theorem (1973, 1975)
"Group Decision Theory", 1974 Ph.D. Thesis (Lund University)  by  Peter Gärdenfors (1949-).

 Gerard Michon (2016-12-28)   Composite Voting Systems
Thinking  out of the box.

Most  vote theorists  will consider only the problem of  aggregating lists of preferences  provided by the voters,  without considering the possibility of requesting  additional  information from them.

In the late 1990's,  I was instrumental in the adoption of a system  (for primaries meant to form  slates  in French senatorial elections)  which required  two  ballots from each voter.  One was a linear preference list,  the other allowed voters to  distribute  (with a few constraints)  a total of  12  points to the candidates.  That allowed the voters to give a clue about the strengths of their relative preferences.  The system lasted for nearly  20  years and helped form successful French senatorial slates.

Points were tallied first,  to obtain a  preliminary  ranking order with a  modicum  of legitimacy.  To that preliminary order,  the  Llull procedure  was applied,  which made the voting system  conform.  There was a provision which made it  stable  with respect to the first three positions  (with manual counting,  it was difficult to go much beyond that and the rest of the slate was of limited importance).

The resulting system accommodated mail voting  (that was a key requirement)  and it was quite resistant to  manipulations.

To my eternal shame,  the earliest version of that voting system was only  cardinal.  Unlike later versions,  that was neither  conform  nor  robust  enough.  At the time,  the very concept of any type of structured primaries was already difficult enough for  incumbents  to accept.  Any lack of aesthetic appeal could have resulted in a stillborn initiative.

Generally speaking,  such a two-ballot system is considered unwieldy,  in spite of its ability to circumvent some  impossibility theorems.

The above system survived until  2014,  when the left-wing government elected in 2012  (and openly irresponsive to the plea of French expatriates)  changed the electoral rules.  The artificial increase in the size of the electoral college for senatorial elections gave back control to the upper management of political parties,  rather than entrust exclusively the representatives of said political parties within the community of French expatriates.  That made this type of carefully-designed primaries less relevant,  if at all.

(2016-12-09)   Stability  is desirable in an aggregation of preferences:
No majority should  ever  contradict the order of two consecutive options!

Stability  so defined is especially important in real-life political primaries  (where the voting system is agreed upon,  rather than imposed by law).  Primaries are useless unless a good number of participants rally behind the result and third parties find the result compelling  (not just by itself but also because of the way it was obtained).

Therefore,  the outcome of slate primaries ought to be justifiable by something other than the arcane peculiarities of a newly-adopted voting system.  Only justifications based on the  Majority Rule  are universally accepted in that context.

If the outcome is  stable  in the above sense,  then any potential disgruntled candidate ends up being ranked just below a  predecessor  against whom he or she is known to be unable to garner a majority of votes.  Any claim for a better position would thus infringe on the rights of that legitimate predecessor and that doesn't seem fair...  That's one way to to put an end to bickering and encourage  rallying  behind the result.

 Gerard Michon

Stabilization by zig-zag :

If a normal application of  Llull's procedure  (in ascending order of pair matching)  is called a  zig,  then the backward procedure  (in descending order of pair matching)  may be called a  zag.  A single  zig  need not yield a  stable  order in the above sense.  The only guarantee after one  zig  is that the top pair is correctly ordered.  Likewise,  one  zag  only guarantees that the bottom pair is correctly ordered.

Now, if we apply  zigs  and/or  zags  repeatedly,  then we shall obtain a  stable  ordering after finitely many steps.

Proof :   Let's call an  inversion,  with respect of the current ordering,  any pair of options  (not necessarily consecutive)  whose relative order would be disavowed by a  majority  vote  (a tied vote doesn't disavow anything).

Clearly, either a full  traversal  (zig or zag)  doesn't switch anything  (which shows that the ordering is now stable)  or it reduces the number of inversions by at least one unit...  Starting with finitely many inversions,  a stable ordering must thus be obtained after finitely many traversals.   QED

We can be more definite than that  (and obtain a more efficient procedure)  if we always  alternate  zigs and zags.  We'll always start with a  zig.

It's more  natural  to do so,  because a  Condorcet winner,  if there's one,  is thus detected as early as possible  (after just one zig).  The Condorcet loser,  if there's one,  is usually of lesser importance to onlookers,  in case of public manual counting.  Also,  truncating the procedure just after the beginning of the first zag makes the first three positions  stable  in the sense discussed here.

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(2016-12-18)   Encouraging or Discouraging Multiple Political Terms
About incumbents.

Incumbents have a vested interest to preserve the  status quo  which got them elected in the first place,  if they want to run again.  Less cynically,  consecutive terms may be beneficial to the general public as it gives us more experienced elected officials.

In any case,  it stands to reason that incumbents shouldn't be judged on the same scale as newcomers.  They have an actual record to defend,  which may be an asset or a liability.  Newcomers have to prove their worth and/or their ability to do better than an incumbent.  The criteria are very different.

At the time of the  aforementioned  two-ballot primaries,  I had devised a preliminary ratification of incumbents,  which they had to win by a strict majority to take part in the primaries.

For completeness,  the rules called for a vote among the incumbents so ratified,  using a rule similar to what was specified for newcomers,  if there was more than two ratified incumbents  (which never happened).

The rules stated that the leading ratified incumbent would lead the slate.  This particular clause won the incumbents over to the new set of rules.  One added bonus is that a majority vote is far more predictable  (for the incumbent)  than a vote with more than two options  (regardless of the voting system).  Thus,  an incumbent who was tempted to run again could renounce honorably rather than face the humiliation of being formally disavowed by the majority of his or her supporters.  That's precisely what one incumbent did  (after being ratified unanimously for the previous term).

However,  that clause was eventually repelled as  unfair  to newcomers,  in part at the instigation of said newcomers  (which was probably a misguided effort on their part,  as the next edition seemed to imply that an incumbent capable of winning a ratification could easily compete successfully among newcomers,  supposedly on an equal footing).

(2017-01-04)   Voter Transition Matrix
From a primary to a general election.

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(2016-12-28)   Aggregating Linear Lists of Voting Preferences
How to make the best of preferences lists when that's all we have.

The double-ballot procedure mentioned  above  is a luxury.  In practice,  it's only applicable to motivated voters insensitive to  Downs paradox.

Failing that,  it's best to accept the conventional wisdom and request from the voters only the linear order of their preferences.

Even if the counting of the votes is computerized,  it's important to be able to describe the procedure in as few words as possible.  This does influence the choice among equally rational methods  (e.g.,  Schulze's method  is difficult to fully describe in just a few words).  For that reason,  I advocate the voting method described as follows  (which is  strongly conform and stable):

A Set of Instructions for a  Fair  and  Robust  Aggregation of Preference Lists

Ballots are printed with short instructions and a list of candidates in a unique order  (predetermined randomly).  The voters are instructed to put  different  ranks  (positive integers which need not be consecutive)  in front of the candidates they choose  (the lower the rank, the more esteemed the candidate).  Unranked candidates are considered ranked last,  at the same level.

The ballots will be  tallied  by first establishing a preliminary order among the candidates according to the  number of victories  each would have won in  pairwise  majority votes  against each other  (a tie being worth half a win).  The total number of votes garnered in those confrontations is used as a tie-breaker  (if the tie remains,  the older candidate prevails).

That preliminary order is then  corrected  by the following procedure:

  • In increasing order, starting with the candidate now ranked last, every candidate is matched against his/her current predecessor and their order is switched if it contradicts a majority vote between them.
  • The same is done in decreasing order, starting with a match-up between the candidates now ranked second and third.
  • The above two steps are iterated alternately  until  one entails no switch.

The  stable  ordering so obtained is the final outcome of the vote.

It is essentiial that the tie-breaking method be rigorously specified in advance.  One advantage of the type of  stabilization  described here is that the tie-breaking method is an essential part of the preliminaries  a priori.  It's not a vague afterthought subject to bickering!

(2016-12-21)   Copeland Scoring   (A. H. Copeland, 1951)
How many times would an option prevail in head-to-head matchups?

When ballots are linear preference lists among  N  options  (or candidates)  we may oppose the options pairwise in a majority vote and record the results.  (There are  N(N-1)/2  of them.)

The  balanced  Copeland score of an option is defined as the number of other options against which it would win a two-way vote  minus  the number of options it would lose a vote against  (ties don't contribute to either number).

Alternately,  we may count one point for a win, half-a-point for a tie and nothing for a loss.

Those are just two different ways of measuring the same reality,  just like  Celsius and Fahrenheit  are two different ways of measuring  temperature.  The conversion between the two flavors of Copeland scores is obtained from a simple observation:  A given candidate is involved in just  N-1  majority votes.  Therefore,  its number of wins  (W)  ties  (T)  and losses  (L)  verify:

W + T + L   =   N-1

Adding the  balanced Copeland score  (W-L)  to both sides, we obtain:

2 (W + T/2)   =   (W - L)  +  (N-1)

That's a fixed relation between the two score flavors,  which may serve as a conversion formula.  For the sake of programming simplicity,  we could also use a third equivalent type of scoring by assigning two points for a win, one point to a tie and zero to a loss.

Ranking according to Copeland scores  (in whichever flavor you choose)  provides a  conform  voting system.  More precisely,  the Condorcet winner,  if there is one,  will be the only option achieving the highest possible Copeland score.  The Condorcet loser,  if there is one,  will be the only option achieving the lowest possible Copeland score.

The most severe limitation is that only  2N-1  different Copeland scores can occur.  In cases where ties are rare or ruled out,  all but  N  of those are unlikely or utterly impossible.  (Ties are rare when there are many ballots and they are impossible when there is an odd number of valid ballots,  assuming the rules demand definite preferences on every ballot.)

Originally,  A.H. Copeland counted only wins and discarded ties and/or losses,  which made only  N  scores possible  (from 0 to N-1).

For completeness,  there are infinitely many non-equivalent scoring methods which differ from  [any flavor of]  the above only by the weight  t  attributed to every tie.  They are all  conform  when  t  is between  0  and  1.  When  t  is irrational,  (N+1)(N+1)/2  different scores are possible.  For any choice of  t  other than  0, 1 and ½ manipulations  are provably difficult to compute  (cf. reference below).  This is,  however,  no guarantee against fraudsters,  since the  heuristic  which ignores the possibility of ties can still compute reasonably fast a manipulation which takex advantage of most weaknesses  (albeit not in the worst case).  I'll ignore all that, since better voting methods are provided by more decisive scoring methods  (including second-order Copeland scoring).

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It's often reported that Copeland scores were put forth by  Ramon Llull  in the thirteenth century.  To the best of my knowledge,  that's not so.

"A reasonable social welfare function"  (1951, University of Michigan)  by  A. H. Copeland (1898-1970).
Copeland Voting:  Ties Matter  Piotr Faliszewski, Edith Hemaspaandra, Henning Schnoor  (2007-12-04).
Copeland's method   |   Arthur Herbert Copeland (1898-1970,  PhD 1926)  of  Copeland-Erdös constant fame.
Numericana :   Pigeonhole principle   |   What's an NP-complete problem?

(2016-12-21)   Second-Order Copeland Method
A voting system whose  manipulation  is an  NP-complete problem.

The second-order Copeland score of a candidate is the sum of the  Copeland scores  of all the candidates he would defeat face-to-face  (and half the Copeland scores of the candidates he would tie with).

So defined,  Second-order Copeland scoring does depend on the particular flavor of first-order Copeland scoring it is based on.  What remains invariant is the entire  class  of second-order Copeland scoring where the score of an option is defined as a fixed linear combination of its first-order score and the associated second-order Copeland score defined above.

In that class,  the  conform  scoring methods form a  convex set.  The scoring methods on the  border  of that set are said to be  purely  second-order.

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Manipulation of Second-Order Copeland Elections  by  Ramoni O. Lasisi  (FLAIRS-29 Conference, 2016).

(2016-12-29)   The Schulze Method
Deriving a  transitive relation between options  from preference lists.

The fact that voting preferences are not  transitive  is equivalent to the  Condorcet paradox  in the form stated  above.

Therefore,  a good basis for the design of a rational voting system would be to derive from the wishes expressed by the voters a  transitive relation  between the options they have to choose from.  Once such a relation is obtained,  the ordering of the options is trivial  (unless ties subsist).

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Wikipedia :   Schulze method

(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.

Ideally,  quotas are only a temporary mean to repair a grave injustice due to a long discrimination toward a group of people.  It's a way to forcibly end the effects of that discrimination.  Once enough time has passed to distance the heirs of that discrimination from its effects,  there is no longer any justification for maintaining any legal distinction between this group and the rest of society.  The normal state of the law is that all individuals should be treated equally.  Any deviation from that norm can only be  temporary.

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

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

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(2013-09-18)   Weighting Conflicting Interests
The proper political domain is only the  boundary  of a convex set.

Very often, choices have to be made between incommensurable things.  To simplify,  let's say we have just two such incommensurable criteria against which possible decisions could be measured.  Financial considerations and human lives at stake,  for example.

Obviously,  when a decision is inferior to another according to  both  criteria,  it should be ruled out.  However,  other decisions can be ruled out too,  because...

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

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(2017-01-02)   Apportioning Integers  in lieu of  Fractional Values
This  ubiquitous  political conundrum has a simple mathematical solution.

The practical  apportionment problem  is to divide a given integer  P  (number of seats, say)  into  B  integers corresponding to as many  buckets  (e.g.,  political parties,  running slates, territories to be represented, etc.)  so that each gets a number  Pi  [roughly]  proportional to  its  merit  Mi  (e.g.,  number of votes received,  total population)  under the  constraint:

P   =   P1  +  P2  +  P3  +  ...  +  PB

The Largest-Remainder Method :

That  simplistic method,  defined below,  has no theoretical justification.  It was once popular,  in spite of its many flaws.  The rule was abandoned in 1911 for the apportionment of U.S. representatives,  mostly because it allows the  unconscionable  Alabama paradox.

The  Alabama paradox  is the name now given to an anomaly of the largest-remainder method which was observed about the number of U.S. representatives to be allotted to the state of Alabama after the U.S. Census of 1880:  C.W. Seaton  (chief clerk of the  U.S. Census Bureau)  pointed out that Alabama would have  8  seats in a House of  299  representatives,  but only  7  seats in a larger House of  300...

This dubious apportionment rule also goes by several other names,  referring to various people who once advocated it  "for the sake of simplicity":  Hare-Niemayer method,  Hamilton-Vinton method,  etc.

The  largest-remainder rule  consists of the following three steps:

  1. Let  Q  be the total voting population  M  divided by  P.
  2. First assign to bucket  i  (e.g.,  i-th state or slate)  the quotient  Mi / Q  rounded down,  denoted  [ Mi / Q ].
  3. This leaves  k  unassigned seats.  Assign one of them to each of the  k buckets yielding the highest  remainders  in the above divisions.

When all divisions are exact,  k  is zero.  This isn't an exception to the above wording if "one of zero" is understood to mean "none".  There may be a need for an arbitrary tie-breaking rule in the third step.

The procedure seems fairly innocent until you are aware of its many flaws,  including the fatal  Alabama paradox  mentioned above.  The French call it  proportionnelle au plus fort reste  (it was once part of the French electoral code but it's now outlawed in favor of the method introduced next).

The deprecated  largest remainder method  is classified as a  ranking method.  All such methods are considered inferior for apportionment,  in part because they allow the  Alabama paradox.

A Rational Approach :

Instead of looking too hard at the problem at hand  (for a fixed number  P  of seats to assign)  we ignore the particular value of  P  and see how the apportionment changes when the  divisor  Q  varies from very large values  (when no seats are assigned)  down to tiny ones  (for which too many seats are assigned).  If we denote by  [x]  the largest integer not exceeding  x  (the value of  x  rounded down)  then we assign this many seats:

[ M1 / Q ]  +  [ M2 / Q ]  +  [ M3 / Q ]  +  ...  +  [ MB / Q ]

That sum takes on nonnegative integer values  (from zero to infinity)  as  Q  varies as stated.  Some values can be skipped  (which correspond to the possible existence of ties)  Otherwise,  this sum is equal to the given  P  for some value of  Q  (known in this context as the  largest divisor).  When that happens,  the right number of seats has been assigned according to what's known as  Jefferson's method   Arms of Thomas Jefferson (which is immune to the Alabama paradox).  It's also called  method of the largest divisor.  The French call it  proportionnelle à la plus forte moyenne  (it's now the only legal apportionment system in France).

It's very effectively described as an unending sequence of seat assignments:  Once  k  seats have been assigned  (starting with  k = 0)  the  (k+1)-st  seat is assigned to whatever bucket yields the highest value for the following expression of the  average  (moyenne  in French)  which depends on the number  ki  of seats assigned so far to that particular bucket:

Mi / (ki+1)

The French legal rules are entirely equivalent to that method but they save some time,  on election day,  by apportioning the first seats quickly with the  same first two steps  as the deprecated  largest-remainder method.

Because of  convexity  considerations,  the same shortcut could be used with other ways of defining an average  (there are  many,  the  arithmetic mean  underlying the above is just the simplest).

Apportionment methods based on variants of the above scheme  are sometimes collectively known as  rounding methods.  (All  rounding methods  avoid the  Alabama paradox.)  They can differ in two respects:

  • Are breaking points rounded up or down?
  • What constitutes  averaging?

Rounding up is ideally suited when there is a legal requirement to apportion at least one seat to every bucket.  The U.S. Constitution does impose at least one representative per state.  The resulting method  (with arithmetic averaging)  is called  Adam's method  or the  smallest-divisor method.  It has never been used.

Four Historical Methods of apportionment:  United States Census Bureau.
The House of Representatives Apportionment Formula  by  David C. Huckabee   (2001-08-10).
Wikipedia :   Proportional representation   |   Apportionment   |   Largest-remainder   |   Highest-average

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

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(2016-12-31)   "Election" of the Doge of Venice
Procedures gone wild.

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Doge of Venice   |   Sortition

(2016-12-25)   Purpose.
On the necessity of proper elections.

If you are right and you know it,  speak your mind.
Even if you are a minority of one,  the truth is still the truth.

 Mahatma Gandhi  (1869-1948)
The best argument against democracy is a
five-minute conversation with the average voter.

 Winston Churchill  (1874-1965)

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What voting should not be used for :

Putting things to a vote is first and foremost an admission of failure.  At the very least,  a failure to reach a consensus.  However.  it could also be a failure to grasp the fact that a complicated issue has components which can be optimized without infringing on the interests of  anyone.  It's rarely the case that human affairs can be settled by reasoning alone,  but when that happens it's a darn shame not to do so.

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Majority Rule  &  Minority Rights  Democracy Web

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) 
visits since June 27, 2011
 (c) Copyright 2000-2018, Gerard P. Michon, Ph.D.