**Implications of Arrow's "Impossibility Theorem"
for Voting Methods **

Kenneth Arrow proved no voting method can satisfy a certain set of desirable

criteria, implying no voting method is ideal. But this does not mean we

should abandon the search for the best (non-ideal) voting method, and

in particular, since the set of nominees is endogenous the effect of the

voting method on the set of nominees should be included in the analysis.

There are often gains to be had by an organization or
society by making a collective choice

from a set of alternatives available to them, rather than having each
individual act independently

(uncertain how others will act). However, since there are many
ways to aggregate individuals'

reports of their preferences in order to reach a collective choice, the
gain (or loss) may depend

on the procedure by which the collective choice is made. To study
this we need to model
the

nature of individuals' preferences and consider various criteria by which various aggregation

methods can be compared.

We make some useful abbreviations. We use
letters like *i*, *j*, etc., to denote individuals who

vote. Assume the group is choosing from a (possibly
large) set of possible alternatives, which

we call *X*. We use letters like *x*, *y*, *z*, etc.,
as abbreviations for alternatives in *X*. Assume

the
alternatives are mutually exclusive, in that at most one can be elected, and
assume *X* is

complete,
in that it includes all possible outcomes. Thus one and only one
alternative in *X*

will be elected. The individuals might not be asked to consider every alternative in *X*,

particularly if *X* is large, so we refer to
the alternatives under consideration as the "agenda"

and call them *A*.
We can also think of *A* as
the set of "nominated" alternatives, those which

appear on the
ballot. *A* is not determined by nature
but is affected by nomination decisions

made by individuals--perhaps only a
small number of
individuals are required to add an

alternative to the agenda. Whether or not
individuals have incentives to
nominate certain

alternatives will depend on their beliefs about how the action would affect the outcome

in the short and long term.

We model each individual as behaving as if she has "preferences" regarding
alternatives.

Every preference is a relative comparison of some pair of alternatives. That is, for any

pair
of alternatives, say *x* and *y*, each individual has a preference for
*x* over *y* or has a

preference
for *y* over *x* or is indifferent between *x* and *y*.
We assume each individual's

preferences
are "self-consistent": Each individual who prefers *x* over *y* and *y*
over *z*

also prefers *x* over *z*,
and each individual who is indifferent between *x* and *y* and

between
*y* and *z* is also indifferent
between *x* and *z*. Such self-consistent preferences

are called
"orderings" of the alternatives,
in the same sense that numbers can be ordered

from largest to smallest.
Alas, no individual's
preferences can be directly observed;

all we can observe are behaviors such as how they choose from a set of options,

or how they answer polls (not necessarily
honestly), or how they mark ballots.

We don't attempt here to model the educational
processes by which individuals

acquire preferences, nor how preferences may change with time; we
are concerned

mostly with preferences as they are when society votes
(hopefully after due
deliberation,

but not necessarily).

Individuals' preferences may be intense, or mild, or in-between. Depending on the

criteria we impose on the voting method, information about preference
intensities

might not
be admissible when voting, or might be ignored when tallying the outcome.

Without loss of generality, we assume that when society votes, individuals mark ballots.

The collection of all ballots is input to a tallying procedure, called a
"choice function,"

which we will call *C*. To avoid overly constraining the analysis, we will not assume *C*

always chooses a single winner; in the cases where *C* chooses more than one we assume

a subsequent procedure,
such as flipping a coin, will be used to pick one of those chosen

by *C*. Thus our first
criterion is simply the following:

Prime directive: The choice function must choose one or more of the

nominated alternatives (if at least one alternative has been nominated)

and not choose any non-nominated alternatives.

The *prime directive* should not be interpreted
as banning "write-in" candidates, which

we would treat as "just-in-time" nominees. Besides ensuring that
at least one of the

nominees will be chosen, its purpose is to ensure that no alternative left
unranked

by every voter will be chosen.

The next two criteria are straightforward and very mild constraints:

Unanimity: No alternative that is ranked by all voters below another

alternative, sayx, may be chosen ifxis one of the nominees.

Non-dictatorship: No voter may be so privileged that, regardless of the

other voters' votes, the choice is always his top-ranked nominee (or from

among his top-ranked nominees, when he votes indifference at the top).

Our next criterion serves to limit the amount of information that must be
elicited from

the voters, so they only need to express preferences regarding nominated alternatives (*A*).

This is justifiable since the set of possible alternatives *X* might be very large, so a voting

method that needs preference information regarding all of *X* would exhaust all participants.

Or,
if the voting method needs info about some non-nominated alternatives (in addition
to

info about the
nominees) then it would not be obvious which alternatives outside *A* should

be included, and if
any individuals are given the power to decide which other alternatives

will be voted on, they might be able to
manipulate the outcomes in their favor. Also,

game theory predicts that this constraint
is actually quite mild, since if the voters know

that alternatives outside *A* cannot be chosen then
their optimal voting strategies would

elect the same alternatives as would be elected if those outside
alternatives could not

appear in their votes.

Independence from Irrelevant Alternatives(IIA): The choice function

must neglect all information about non-nominated alternatives.

The next criterion further constrains the information that may be considered by *C*.

Specifically, we require *C* to ignore information about the intensity of voters' preferences,

so
"mild" preferences will be treated the same as "intense" preferences. In other words,

two ballots that rank the alternatives in the same order must be treated the same.
The

justification for this is
that, if intensity information were not ignored, it would create a

strong incentive for each voter to
exaggerate her intensities by dividing the alternatives

into two groups and voting the maximum possible
intensity between the two groups

(and indifference within each group). To see this, suppose pre-election polls indicate

the two likely front-running candidates are *x* and *y*. Then
each voter who prefers *x* over *y*

has an incentive to report the maximum possible intensity for *x* over *y*,
to avoid partially

wasting the power of her vote. Similarly, those who prefer *y* over *x* have an
incentive to

report the maximum possible intensity for *y* over *x*. If the voters who prefer *x*
over *y*

believe those who prefer *y* over *x* will vote the maximum intensity for *y* over *x*, they would

be foolish not to
vote the maximum intensity for *x* over *y*, etc. While doing so, it would

be most effective for those
voting *x* over *y* to also cast the maximum possible vote for

candidates preferred over *x* (in other words,
indifference between them and *x*, since *x*

already is getting their maximal vote) and the
minimal possible vote for candidates less

preferred than *y* (indifference between them and *y*),
etc. This may not be obvious at first,

but we presume most voters would quickly learn the strategy since it is so
easy. The result

would be that voters would express much less information in their votes than if the
choice

function ignores all intensity information. Thus, we have our next criterion:

Ordinality: The choice function must neglect all "intensity" information.

In other words, only "ordinal" information may affect the choice.

The next criterion requires that the choice function accept a considerable
amount and

diversity of information from each voter about her preferences, if she wishes to
express it.

Since Kenneth Arrow was analyzing whether and how voters' preferences might
be

aggregated to reach a collective decision, and since there is no *a priori*
reason to expect

voters' preferences to adhere to any pre-ordained pattern, it makes sense to
require the

method of aggregating preferences to work no matter what the voters' preferences
may be.

Universal Domain: The choice function must accept from each voter

any ranking of the alternatives.

On the other hand, we are not really limited to
Arrow's framework, which was designed

merely to try to aggregate voters' (sincere) preferences. Although it is reasonable
to require

the voting method to work for any collection of preferences the voters may have, it
does

not necessarily follow that no constraints should be placed on the expressions
voters may

make when voting. For instance, the so-called Approval voting method
constrains each

voter to partitioning the alternatives into two subsets, which is
equivalent to a non-strict

ordering that has at most two "indifference classes."
It is not *a priori* obvious that the use

of voting methods such as Approval, which constrain the
voters from completely expressing

their preference orderings, are worse for society, so
the *universal domain* criterion should

be considered controversial until other
arguments not explored by Arrow are examined

(assuming those arguments support the
conclusion that it is better not to constrain the voters

from expressing orderings). In
other words, other criteria for comparing voting methods,

in addition to Arrow's criteria,
need to be evaluated. (My own conclusion is that there are

solid reasons why it is better not to
constrain the voters' expressions, but that is beyond

the scope of this
document.)

Kenneth Arrow's theorem [1951, 1963] states that, if *X* includes at
least 3 alternatives

then no choice
function that satisfies all of the criteria listed above also satisfies the

following criterion:

Choice consistency: For all pairs of alternatives, sayxandy, if the votes

are such thatxbut notywould be chosen from some set of nominees that

includes both, thenymust not be chosen from any set of nominees that

includes both. (The literature usually calls thisrationality, but I prefer

the less loaded termchoice consistency.)

(A proof of Arrow's theorem is provided in the appendix.)

*Choice consistency* is demanding, and requiring that it be satisfied is
controversial. It requires

the same self-consistency of the social choice function as we observe in individuals'
choices.

If *choice consistency* is satisfied then it would be impossible to manipulate outcomes by

manipulating the agenda or by misrepresenting preferences when voting. (That claim may

not be
obvious.) But it is easy to construct an example that shows that for most voting

methods, in particular all
methods that reduce to majority rule when there are only two

nominees, *choice consistency* is too
demanding to satisfy:

Suppose the nominees are

x,yandz. Suppose about a third of the voters rank

"xoveryoverz" and about a third rank "yoverzoverx" and the rest rank

"zoverxovery." Each voter's preferences are self-consistent (an ordering of

the alternatives), but the social choice is not: About 2/3 of the voters rankxovery,

so the choice by majority rule from the agenda {x,y} would bex, which means

choice consistencyrequires thatynot be chosen from {x,y,z}. Also, about 2/3

rankyoverz, so the choice by majority rule from {y,z} would bey, which means

choice consistencyrequires thatznot be chosen from {x,y,z}. And about 2/3

rankzoverx, so the choice by majority rule from {x,z} would bez, which means

choice consistencyrequires thatxnot be chosen from {x,y,z}. Thuschoicerequires that none be chosen from {

consistencyx,y,z}. But theprime directiverequires that at least one must be chosen.

(The example also illustrates a fundamental principle of which few people are
aware:

Whenever there are more than two alternatives, there is more than one majority.)

Arrow's result spawned a great deal of activity researching implications and
variations of his

theorem. Many social scientists seem to have interpreted the literature to mean
no voting

method is reasonable, when it actually only means that no voting method is
ideal. The search

for the best (non-ideal) method is still important, and no doubt much of the effort will need

to be empirical work, since otherwise it will remain disputed which of the many possible

weakenings of *choice consistency*
are the most important to satisfy. One approach might

be to calculate for each
possible voting method the fraction of the possible voting scenarios

in which *choice consistency* is
violated, seeking voting methods that minimize this fraction.

Another approach stems from the argument that in large public elections,
where many voters

are not strategically minded, the easiest manipulations are accomplished (if the voting
method

is not immune to this) by finding and nominating candidates that are similar to and/or
inferior

to other nominees. By requiring satisfaction of *independence from clone alternatives*

(defined below), this would avoid the incentive for
parties to nominate as many candidates

per office as possible (which would make a farce of elections)
and also significantly reduce

incentives for parties to nominate only one candidate (or no
candidates) per office. Instead,

parties might have a net incentive to try to increase the turnout of
their supporters on election

day by nominating a diversity of candidates, each of whom inspires enthusiasm amongst

one or
more factions of supporters. (While turning out to vote for a favorite candidate,

many of those voters would
presumably also rank their party's other nominees over other

parties' candidates.)

Independence from clone alternatives: Call a group of alternatives "clones" if

no voter ranks any nominee outside the group between any members of the group.

The election chances of every candidate outside a group of clones must be the

same when one of the clones is nominated as when two or more of the clones

are nominated. (See Tideman 1987.)

Another potentially important criterion is motivated by the observation that
people do not

like to compromise, and hate compromising unnecessarily, and may be unsure how much

they need to compromise. Thus it may be easier to persuade
voters to lower "greater evil"

candidates than to persuade voters to raise "compromise" candidates
equal to or over more

preferred candidates. This difference matters in scenarios where a
minority who prefer a

"greater evil" candidate over a "compromise" candidate can
cause the greater evil to be

elected by strategically misrepresenting their preferences, since it bears on
the ability of

the majority who prefer the compromise to thwart (and deter) that
strategy. It also matters

in scenarios where a minority-preferred "greater evil" candidate would
be elected even

without strategic misrepresentation (which can easily occur using some voting
methods,

such as plurality rule, majority runoff, and instant runoff) since it bears on
the ability of

the majority to defeat the greater evil; if the majority cannot reliably
coordinate their

voting strategies, then the only option remaining to them is to deter some of
their favorite

candidates from competing (for instance by forming a large party and choosing
only one

nominee using a primary election, which may not nominate the best candidate),
which may

depress voter turnout. The *minimal defense* criterion embodies
the solution to these

problems, by making a lowering strategy effective when wielded by a majority:

Minimal defense: The voting method must satisfy both of the following properties:

(1) Each voter must be allowed to vote any ordering of the alternatives,

with indifferences allowed (at least at the bottom of each ordering).

(2) For all pairs of nominees, sayxandy, if the number of voters who rank both

"xovery" and "yno higher than tied for bottom" exceeds the number of

voters who rank "yhigher than tied for bottom" thenymust not be chosen.

Use of the voting strategy enabled by satisfaction of *minimal
defense* (that is, downranking

"greater evil" alternatives to tied for bottom) may create an equilibrium that elects the same

alternative that would be chosen if no one
misrepresented any preferences. In such cases

the strategy should be deemed benevolent, not manipulative, and provides an
argument

why the voting method should allow voters to express indifference in their
orderings,

at least at the bottom. (A second argument is that when the number of nominees
is large,

it would be tedious to force each voter to strictly rank every alternative.
Allowing the

voter to leave some nominees unranked, then treating those left unranked as if
they

had been ranked below those explicitly ranked, would be a useful
shortcut.)

Many criteria have been advocated in the social choice literature. Some
appear to have only

aesthetic value, such as requiring that the outcome must not change if two
additional opposite

orderings are added to the set of votes. Another criterion, *reinforcement*,
requires that if the

same alternative would be chosen by two separate groups of voters then it must
be chosen

if both groups' votes are combined together. In principle, voting methods
that do not satisfy

*reinforcement* might be manipulated by individuals who have the power to
decide whether

and how the voters are partitioned, but it is simple for the rules to prevent
this by not granting

such power to a minority. So *reinforcement* seems much less
important than *independence
from clones* and

Another criterion that seems less important is *participation*, which
requires that the winning

alternative must not be ranked by any voter below the alternative that would
have won had

that voter abstained. Besides its aesthetic appeal, the argument may be
made that voter

turnout is already so low that there should be no new incentives to
abstain. There are two

flaws with this argument. One flaw is that the argument neglects the
option of misrepresenting

some of the voters' preferences: if a voter has the information about other
voters' likely votes

that leads her to believe she would prefer the outcome if she abstains more than
if she votes

her sincere preferences, that information points the way to a strategic
vote that would

result in an outcome she would prefer at least as much as if she abstains.
The other flaw is

that it is an "all else being equal" argument and all else is not
equal. That is, there may be

voting methods that do not satisfy *participation* yet create much greater
incentives for

non-voters to vote than do methods that do satisfy *participation*.
For instance, the

nominations may depend on the voting method, and voters' decisions whether or
not to

vote may depend on which alternatives are nominated; some individuals might vote
only

if they enthusiastically support at least one nominee. There might even be
some individuals

who will not vote if aware the voting method is easily manipulated by strategic
nomination

of clones or inferior alternatives.

**Appendix - A brief rigorous proof of Arrow's theorem**

Note: A proof written in more concise mathematical notation is provided in
the

document "Proof
of Arrow's Theorem."

(We patterned this after a proof by John
Geanakoplos [2001].) Assume *X* includes at

least 3 alternatives and *C* is a social choice function that satisfies all of Arrow's criteria

except perhaps *non-dictatorship*. We will show
that *C* violates *non-dictatorship*.

Call any collection of votes, one vote per voter, a
"profile." Without loss of generality

due to satisfaction of *universal domain* and *ordinality*, assume each
vote in a profile

may be any
ordering of *X*.

First we establish the following claim:

(1) For any alternative, say

x, given any profile in which each vote that does not

rankxstrictly top ranksxstrictly bottom, one of the following conditions holds:

(1.1)Cchoosesxalone from every agenda that includesx.

(1.2)Cdoes not choosexfrom any agenda that includes another alternative.Proof of 1: Pick any alternative

x. Pick any profile such that each vote that does

not rankxalone at the top ranksxalone at the bottom, and call this profileV.

Suppose 1.1 does not hold givenV. We must show 1.2 holds. Since 1.1 does not

hold givenV, there must be at least one alternative distinct fromx, sayy, such that

Cchoosesy(and perhaps alsox) from some agenda that includesx, givenV.

Bychoice consistencyand theprime directive, the following must hold:

(1.3) GivenV,Cchoosesy(and perhaps alsox) from {x,y}.

Since there are at least three alternatives, we can arbitrarily pick an alternative distinct

fromxandy, sayz. Construct another profile, call itV', that is the same asVexcept

zis moved immediately overyin every vote inV'. That is, the following holds:

(1.4) Every vote inV'rankszovery.

Sincexis at the top or bottom of each vote inV, both of the following hold:

(1.5) For each voter, the relative ordering ofxandyis the same inV'as inV.

(1.6) For each voter, the relative ordering ofxandzis the same inV'as inV.

By 1.4 andunanimity, the following must hold:

(1.7) GivenV',Cdoes not chooseyfrom {x,y,z}.

By 1.7 and theprime directive, the following must hold:

(1.8) GivenV',Cmust choosexand/orzfrom {x,y,z}.

Next we show the following statement holds:

(1.9) GivenV',Cdoes not choosexfrom {x,y,z}.

Suppose to the contraryCchoosesxfrom {x,y,z} givenV'. By 1.7 andchoice,

consistencyCdoes not chooseyfrom {x,y} givenV'. By 1.5 andIIA,

Cdoes not chooseyfrom {x,y} givenV. But this contradicts 1.3,

so the contrary assumption cannot hold, establishing 1.9.

By 1.7, 1.9 and theprime directive,Cchooseszalone from {x,y,z} givenV'.

Bychoice consistency,Cchooseszalone from {x,z} givenV'. By 1.6 andIIA,

the following holds:

(1.10)Cchooseszalone from {x,z} givenV.

Bychoice consistency,Cdoes not choosexfrom any agenda that includesz,

givenV. Since z was any arbitrary alternative distinct fromxandy,

the following holds:

(1.11) GivenV,Cdoes not choosexfrom any agenda that includes

at least one alternative that is notxory.

Statement 1.11 is close to statement 1.2, which we have been aiming to establish;

it remains only to show that, givenV,Cdoes not choosexfrom any agenda that

includesy. By 1.10, 1.3 would still hold if we swapped the labels ofyandz.

Therefore, by the same reasoning that followed 1.3 and led to 1.11, the

following statement (like 1.11 withyandzswapped) must also hold:

(1.12) GivenV,Cdoes not choosexfrom any agenda that includes

at least one alternative that is notxorz.

Together, 1.11 and 1.12 imply 1.2. Sincexwas picked arbitrarily andVwas

any set of votes in whichxis ranked strictly top or bottom by every voter,

claim (1) is established.

Next we establish the following claim:

(2) For any alternative, say

x, there is an individual voter, call herd, who "dictates"_{x}

over all pairs of alternatives distinct fromx. (By "dictating over a pair," for instance

yandz, we mean that ifdranks_{x}yoverzthenCdoes not choosezfrom any agenda

that includesy, no matter how the other voters vote.)Proof of 2: Let

ndenote the number of voters (assumed to be at least 1) and label

the voters from 1 ton. Pick any alternative, sayx. Consider any set of admissible

votes, call itV^{0}, such that the following condition holds:

(2.1) Each vote inV^{0}ranksxstrictly bottom (below all other alternatives).

Byunanimity, the following statement must hold:

(2.2) GivenV^{0},Cdoes not choosexfrom any agenda that includes

another alternative.

Construct a sequence ofnadmissible sets of votes, labeledV^{1}toV, such that^{n}

all three of the following conditions hold for each integerkfrom 1 ton:

(2.3) For all pairs of alternativesyandzdistinct fromx, the relative ordering of

yandzis the same in each vote inVas in the corresponding vote in^{k}V^{0}.

(2.4) For each voter fromk+1 ton, her vote inVranks^{k}xstrictly bottom.

(2.5) For each voter from 1 tok, her vote inVranks^{k}xstrictly top.

By 2.5, every vote inVranks^{n}xstrictly top. Byunanimityand theprime, the following statement must hold:

directive

(2.6) GivenV,^{n}Cchoosesxalone from every agenda that includesx.

By 2.1, 2.4 and 2.5,xis ranked either strictly top or strictly bottom by every vote in

each ofV^{0},V^{1},V^{2}, ...,V. Therefore, by (1) one of the following two statements^{n}

must hold for each integerkfrom 0 ton:

(2.7) GivenV,^{k}Cchoosesxalone from every agenda that includesx.

(2.8) GivenV,^{k}Cdoes not choosexfrom any agenda that includes

another alternative.

By 2.6, 2.8 does not hold fork=n. Thus we can letddenote the smallest integer

between 0 andn(inclusive) such that 2.8 does not hold fork=d. By 2.2,dis not 0.

Thus the following two statements hold:

(2.9) GivenV,^{d}Cchoosesxalone from every agenda that includesx.

(2.10) GivenV^{d}^{-1},Cdoes not choosexfrom any agenda that includes

another alternative.

Now pick any pair of distinct alternatives distinct fromx, sayyandz(which we can

do since there are at least 3 alternatives). Construct a set of votes, call itV', that is

the same asVexcept voter^{d}dranksytop. (Thus voterdranksxnext-to-top inV'.)

InV', voters 1 tod-1 rankxoveryand votersdtonrankyoverx, as inV^{d}^{-1}.

Thus, byIIAand 2.10,Cdoes not choosexfrom {x,y} givenV'.

By theprime directive,Cchoosesyalone from {x,y} givenV'.

Bychoice consistency,Cdoes not choosexfrom {x,y,z} givenV'. (2.11)

InV', voters 1 todrankxoverzand votersd+1 tonrankzoverx, as inV.^{d}

Thus, byIIAand 2.9,Cchoosesxalone from {x,z} givenV'.

Bychoice consistency,Cdoes not choosezfrom {x,y,z} givenV'. (2.12)

By 2.11, 2.12 and theprime directive,Cchoosesyalone from {x,y,z} givenV'.

Bychoice consistency, the following must hold:

(2.13) GivenV',Cdoes not choosezfrom any agenda that includesy.

Since the relative orderings ofyandzinV^{0}were arbitrary, this means the relative

orderings ofyandzinV'were arbitrary except for voterdwho ranksyoverz. It

follows byIIAand 2.13 that given any set of votes in which voterdranksyoverz,

Cdoes not choosezfrom any agenda that includesy. Thus voterddictates over

yandz. Sinceyand z were picked arbitrarily, being any two alternatives distinct

fromx, it follows that voterddictates over every pair of alternatives distinct fromx.

Sincextoo was picked arbitrarily, this means we can find a voter likedfor any

alternative, establishing claim (2).

Since *X* includes at least three alternatives, we can arbitrarily pick three
distinct alternatives

and label them *x*, *y* and *z*. By (2), there exist voters
*i*, *j* and *k* (not necessarily distinct)

such
that *i* dictates over *y* and *z*, *j* dictates over *x*
and *z*, and *k* dictates over *x* and *y*.

We will show *i* = *j* = *k*.
Suppose the contrary, meaning we are dealing with two or

three "dictatorial" voters. By *universal domain* these two or three voters (like every voter)

can vote
any ordering of the
alternatives, so we can find a set of admissible votes in which

*i* ranks
*y*
over *z* and *j* ranks *z* over *x* and *k*
ranks *x* over *y*. Since the two or three voters

dictate over their
respective pairs of alternatives, this means *C*
chooses no alternative from

{*x,y,z*} given these votes, which contradicts the *prime directive*. Thus the
contrary

assumption cannot hold, so *i* = *j* = *k*.
Since *x*, *y* and *z* were picked arbitrarily, it follows

that this voter *i* = *j* = *k* is a unique voter who dictates over all pairs of
alternatives. That is,

for all pairs of alternatives, for instance *x*
and *y*, if voter *i* ranks *x* over *y* then *C*
does

not choose *y* from any agenda that includes *x*,
regardless of how the other voters vote.

Thus *C* cannot choose any alternative
that voter *i* ranks below some nominee. By the

*prime directive*, *C* always chooses from among voter *i*'s
top-ranked nominees, which

means *C* violates *non-dictatorship*. Thus, we have
established that no social choice

function satisfies all of Arrow's criteria when there are at least three alternatives. Ω

**References**

[1] Arrow, Kenneth J (1951, 1963). *Social Choice and
Individual Values*.

New York: John Wiley and Sons.

[2] Condorcet (1785). *Essai
sur l'application de l'analyse à la probabilité des
décisions rendues à la pluralité des voix*. Paris.

[3] Geanakoplos, John (2001). *Three Brief Proofs of Arrow's
Impossibility Theorem*.

Cowles Foundation discussion paper No.1123RRR. Cowles Foundation for
Research

in Economics, Yale University, New Haven, Connecticut. (http://cowles.econ.yale.edu)

[4] Tideman, TN (1987). *Independence of Clones as a Criterion
for Voting Rules*.

Social Choice and Welfare, 4: 185-206.