Impossibility theorems and voting paradoxes in collective choice theory
Forthcoming in The Wiley Encyclopedia of Operations Research and Management Science
Elizabeth Maggie Penn
Associate Professor of Political Science
Washington University in St.Louis
In the ﬁeld of positive political theory,the term “impossibility theorem” refers to a collection of
results that take as given a set of normatively appealing criteria and deductively prove that such a
set of criteria are inconsistent:that is it impossible for a voting system – any voting system – to
simultaneously satisfy the given set of criteria.This approach to studying voting systems is called
axiomatic because the normative criteria that we might like our voting systems to satisfy are used
as starting points in these results,or taken as a given.In other words,these normative criteria are
axioms,which in mathematics are deﬁned as statements that are assumed solely for the sake of
studying the consequences that follow fromthem.
The mathematical study of voting systems is motivated by the fact that any group seeking to
make a collective decision must choose some method of translating the preferences of the group
into a societal outcome.Moreover,there are an inﬁnite number of such methods:from simple
majority rule,to weighted voting,to choosing a “dictator” to unilaterally make societal decisions.
A brief consideration of the types of voting rules that have been used throughout the course of
human history reveals an assortment that varies not only with respect to procedure and formula,
but also with respect to the goals that each system was intended to further.In ancient Athens,for
example,the election of delegates was done by lottery,and it is estimated that half of all citizens
over the age of 30 served as public ofﬁcers at some point in their lives.This process formed the
basis of Aristotle’s concept of democracy,which espoused “ruling and being ruled by turns.”[1,
pp.2022] Conversely,many modern elections utilize runoffs to ensure that electoral winners
have received the support of a majority of voters.An advisory board may utilize weighted voting
to ensure that more knowledgeable experts have more say in a ﬁnal decision.
The concepts of “ruling by turns,” “winners should receive the support of a majority,” and
“greater voting weight to more knowledgeable individuals” can all be thought of as decision
making desiderata of the particular groups described above.Axiomatic approaches to studying
1
voting systems are informative and useful because they take these types of goals of a voting sys
tem as a given,and focus on whether any voting system is capable of satisfying the goals and,if
so,what such a voting system must look like.For example,suppose a group wishes to utilize a
voting system in which no individual has an incentive to misrepresent his or her true preferences;
in other words,a voting systemin which all individuals are always best served by voting truthfully.
The GibbardSatterthwaite Theorem,an important impossibility theorem,would then tell us that
this goal is impossible to attain with any reasonably democratic voting system.
The remainder of this article will present a brief survey of several of the most wellknown
impossibility theorems,including Arrow’s Theoremand the GibbardSatterthwaite Theoremmen
tioned above.I will also discuss various widely known voting paradoxes,or particular examples
of howseemingly sensible voting procedures can produce undesirable outcomes.These paradoxes
can be thought of as impossibility results in their own right,as they demonstrate an incompatibility
between sensibility in procedure and sensibility in outcome.
Notation and deﬁnitions
The most wellknown impossibility theorems in positive political theory involve voting systems,or
systems of preference aggregation.Thus,the theorems concern procedures that take a collection
of individuals with heterogeneous preferences as an input,and produce some collective prefer
ence ordering or collective choice as an output.It should be noted,however,that there are many
other impossibility theorems in positive political theory that do not explicitly involve preference
aggregation,per se.
To present these results,some simple notation is needed.First,we will assume that there exists
a collection of individuals seeking to make a collective choice.This group is termed N,and it is
of size n.We will assume that N contains at least two individuals,or that n 2.Second,we
will assume that there is a ﬁnite set of alternatives,or policies,under consideration by the group.
We will call this set X,and will assume (to begin with) that X contains at least three different
alternatives.And third,we will assume that each individual has his or her own preference ordering
of the alternatives under consideration.This preference relation is denoted
i
for person i.Thus,
if person i likes alternative x more than alternative y,it is denoted x
i
y.
We will assume that each individual’s preference relation
i
is a weak order,which means
that it is a way of comparing pairs of alternatives that is reﬂexive,transitive,and complete.The
2
i
relation is intended to be similar to ,the “greater than or equal to” relation,which we use to
compare numbers on the real line.Reﬂexivity means that for any alternative x,it is the case that
x
i
x,or that i likes x at least as much as x.Completeness means that for any two alternatives,
x and y,it is the case that either x
i
y,or that y
i
x,or both.In other words,i either likes x at
least as much as y,or y at least as much as x,or is indifferent between the two.Last,transitivity
means that if x
i
y and y
i
z,then x
i
z.This ﬁnal condition is a rationality condition;voters
cannot have preferences that “cycle” in the sense of ordering the alternatives x y,y z,and
z x.
At times we will decompose
i
into two parts:
i
and
i
.While
i
represents Person i’s
preference relation,
i
represents his strict preferences,and
i
represents indifference.Thus,
x
i
y implies that x
i
y,but not y
i
x,or that i strictly prefers x to y.Similarly,x
i
y
implies that both x
i
y and that y
i
x,or that i is indifferent between x and y.Last,we will use
the term to describe the entire collection of individual preferences.Thus, = (
1
;
2
;:::;
n
).
More concretely,suppose that we have two people in our group,so N = f1;2g,and that we
have three alternatives under consideration,so X = fx;y;zg.Suppose Person 1 prefers x
1
y
1
z and Person 2 has preferences y
2
x
2
z.In words,Person 1 strictly prefers x to y and z,
and strictly prefers y to z,and Person 2 strictly prefers y to z and x,and z to x.Then
=
x
1
y
1
z
y
2
z
2
x
(1)
Thus, characterizes our group and the preferences of its members.
Now that we have described the collection of alternatives,people,and their preferences over
the alternatives,we can begin to consider various ways of conceptualizing collective choice.We
will consider two different types of mechanisms for generating a group choice.The ﬁrst is termed
a preference aggregation rule,and is denoted f.This mechanism takes a preference proﬁle as
an input,and generates a group preference relation,,over all alternatives.This group preference
relation will be reﬂexive and complete,but not necessarily transitive.
As an example,suppose that f is the method of Borda count (let’s call it f
B
),and is the
proﬁle described above in Equation 1.As there are three alternatives under consideration,Borda
count works as follows:an alternative that a voter ranks ﬁrst receives two points,an alternative
he ranks second receives one point,and an alternative he ranks third receives zero points.The
social ranking is then the sum of these scores across individuals.Thus,given the in Equation 1,
3
x receives two total points (two from Voter 1 and zero from Voter 2),y receives three total points
(one from Voter 1 and two from Voter 2),and z receives one total point (from Voter 2).It follows
that Borda count ranks the alternatives y x z,or more formally,
f
B
() = y x z:
While a preference aggregation rule determines a social preference ordering of the entire col
lection of alternatives under consideration,a choice function,F,takes a preference proﬁle as an
input and simply generates a single winner.We can also think of Borda count as choice function,
F
B
.In this case,our function would return the alternative with the highest Borda score,breaking
a tie if need be.Using our proﬁle in Equation 1,
F
B
() = y:
A choice function or preference aggregation rule satisﬁes a property termed unrestricted do
main if its domain is the set of all possible preference proﬁles.In other words,unrestricted domain
requires that these collective choice mechanisms be capable of yielding a social choice or a social
preference relation when provided with any possible collection of preference orderings of the in
dividuals;we cannot restrict the preferences that individuals may have.For the purposes of this
article,we will always assume that our preference aggregation rules and choice functions satisfy
this property.
A survey of impossibility theorems in voting theory
This section will begin by brieﬂy presenting three impossibility theorems concerning preference
aggregation rules:Arrow’s Theorem,May’s Theorem,and Sen’s Theorem.Arrow’s Theorem is
the most wellknown of the impossibility theorems,and partially motivates the theorems that fol
low.Arrow lays out four simple axioms that any reasonable aggregation rule should (arguably)
satisfy,and then proves that these axioms are incompatible with each other;no rule can simulta
neously satisfy all four.May looks speciﬁcally at majority rule over two alternatives,and shows
that it is the only voting systemthat can simultaneously satisfy several desirable axioms including
anonymity and neutrality,the properties of treating all voters and candidates equally.Sen demon
strates that a very minimal requirement that a voting rule be responsive to the preferences of more
than one individual is incompatible with that rule being capable of generating a social ranking of
the alternatives that is both efﬁcient and capable of yielding a “best” alternative.
4
This section concludes with a discussion of the GibbardSatterthwaite Theorem,an impossibil
ity theorem concerning choice functions.Loosely speaking,this theorem tells us that it is impos
sible to design a system for generating a collective choice that both takes the preferences of more
than one individual into account,and that provides no incentive for individuals to misrepresent
their preferences.
Arrow’s Theorem
We will begin by laying out Arrow’s four axioms,or conditions that any reasonable aggregation
rule should satisfy.Perhaps one of the least demanding requirements a group would seek to impose
on its voting system is that a group decision is minimally responsive to the preferences of the
members of that group.To this end,Arrow ﬁrst deﬁnes the condition of Pareto efﬁciency.
An aggregation rule f is Pareto efﬁcient if,for any proﬁle ,if x
i
y for all individuals i in
N,then x y.In words,this condition says that if every individual i strictly prefers x to y,then
our aggregation rule f should generate a collective ranking of the alternatives that ranks x strictly
higher than y.This condition rules out aggregation rules that,for example,always rank x y,
regardless of the group’s x;y preferences.
Second,we might want our voting system to not consider “irrelevant” alternatives when gen
erating a social ranking between two alternatives.In other words,the group members’ preferences
between alternatives c and d should not affect the social ranking of two different alternatives,a and
b.This leads to Arrow’s second condition of independence of irrelevant alternatives.
An aggregation rule f is independent of irrelevant alternatives (IIA) if for any two proﬁles,
and
0
,if every individual’s x;y ranking under agrees with their x;y ranking under
0
,then the
social x;y ranking generated by f() should agree with the social x;y ranking generated by f(
0
).
The example belowis intended to make this concept more concrete,and to illustrate why the Borda
count procedure violates IIA.Consider the following two proﬁles, and
0
:
=
x
1
y
1
z
y
2
z
2
x
0
=
x
1
z
1
y
y
2
x
2
z
(2)
As discussed earlier,Borda count produces the social ranking y x z when applied to
proﬁle .However,at proﬁle
0
,f
B
(
0
) generates the social ranking x y z,because at this
proﬁle x receives three combined points,y receives two,and z receives one.If we look solely at
5
the two individuals’ x;y rankings,these two proﬁles look identical:Voter 1 prefers x
1
y under
both and
0
,and Voter 2 prefers y
2
x under both and
0
.However,f
B
() generates y x,
while f
B
(
0
) generates x y.This is a violation of IIA.
Arrow’s third axiomconcerns the ability of a preference aggregation to generate an unambigu
ous winner,or collection of winners,if there is a tie.A procedure that,for example,generates the
social ranking x y,y z,and z x might not be particularly useful to a group seeking to
make a collective choice,because it does not generate an unambiguously “best” collection of alter
natives.Any given alternative is strictly worse than another.This leads to Arrow’s third condition
of transitivity.
An aggregation rule f is transitive if it always produces a transitive ordering of the alternatives.
Thus,if f produces an ordering in which x y and y z,then it must also be the case that
x z.This condition guarantees that the social ordering generated by f cannot cycle,in the
sense that it cannot produce a ranking in which x y,y z,and z x;it guarantees that
the social preference ordering satisﬁes the same rationality condition as the individual preference
orderings it was constructed from.Moreover,it ensures the existence of an alternative or collection
of alternatives that are not strictly worse than anything else.
Last,Arrow’s axiom of no dictator concerns the responsiveness of the preference aggregation
rule to the preferences of more than one individual.An aggregation rule is dictatorial if there is
one particular voter whose individual preferences always determine the social preference ordering,
irrespective of the preferences of the other voters.Formally,this condition says that there exists
one voter i,so that every time x
i
y,the aggregation rule f produces the ranking x y.An
aggregation rule f satisﬁes no dictator if it is not dictatorial.We are now ready to state Arrow’s
Theorem.
Theorem1.(Arrow,1950 &1963).With three or more alternatives,any aggregation rule satisfy
ing unrestricted domain,Pareto efﬁciency,IIA and transitivity is a dictator.
Arrow’s Theorem then tells us that if a group wishes to design a preference aggregation rule
that is Pareto efﬁcient,transitive and independent of irrelevant alternatives,and if we place no
restrictions on the preferences that individuals may have,then the rule must grant all decision
making authority to a single individual.[2,3] Thus,any aggregation rule that satisﬁes no dictator
must violate either transitivity,or Pareto efﬁciency,or IIA.And practically speaking,it will violate
either transitivity or IIA,as the only nondictatorial aggregation rules that are ruled out by the
6
addition of Pareto efﬁciency are those rules that are either null (generate a tie over all alternatives)
or inverse dictatorships.This extension of Arrow’s Theorem to nonPareto efﬁcient rules was
proved by Wilson [4],and is known as Wilson’s Impossibility Theorem.
May’s Theorem
May’s Theorem comes as a response to the axiom of independence of irrelevant alternatives uti
lized by Arrow,which requires that group choices between pairs of alternatives depend only on
individual preferences over those pairs.May argues that this condition implies that knowing a
group’s choice between every pair of alternatives is akin to knowing the group’s entire preference
relation;in this sense,Arrow’s result can be thought of as a problemconcerning choice over pairs.
Thus,while Arrow’s Theorem requires at least three alternatives in the group’s choice set,May
focuses solely on the case in which there are only two alternatives under consideration.
May lays out three axioms that any preference aggregation rule should satisfy when there are
two alternatives under consideration.His ﬁrst condition is anonymity,which states that the pref
erence aggregation rule f should treat every voter equally.This means that in f(
1
;
2
;:::;
n
)
any two arguments (e.g.
5
and
3
) could be interchanged without changing the group preference
relation produced by f.In other words,changing the voters’ names (as deﬁned by the subscripts
on
i
) can have no effect on how f ranks the alternatives.
May’s second condition is neutrality,which says that the aggregation rule f does not favor
either alternative.In other words,the labeling of the alternatives cannot matter to f.May’s ﬁnal
condition is positive responsiveness,which says that the group preference relation should respond
to individual preferences in a positive way.Thus,if we have a preference proﬁle in which f()
generates x y,then changing a single individual’s preferences such that he is more favorable to
x (i.e.changing his preferences fromy
i
x to x
i
y,or fromx
i
y to x
i
y) should make the
group strictly favor x over y (i.e.change the group preference from x y to x y,or keep the
group at x y if that was the group’s ordering before i’s preferences changed).
Last,deﬁne the method of majority rule to be a preference aggregation rule,f
M
over a pair of
alternatives,x and y,in which x y if more people strictly prefer x to y than y to x,and y x if
more people strictly prefer y to x than x to y,and x y if the same number of people prefer x to
y as prefer y to x.We are now ready to state May’s Theorem.
Theorem 2.(May,1952) The only preference aggregation rule that is anonymous,neutral and
7
positively responsive is the method of majority rule.
Aparticularly succinct explanation of May’s Theoremis provided by May himself:“In Arrow’s
terms our theorem may be expressed by saying that any [preference aggregation rule] that is not
based on simple majority decision,i.e.,does not decide between any pair of alternatives by majority
vote,will either...favor one individual over another,favor one alternative over the other,or fail to
respond positively to individual preferences.”[5]
Sen’s Theorem (The Liberal Paradox)
Like May’s Theorem,Sen’s Theorem can also be viewed as building upon Arrow’s results.Sen
argues that certain kinds of choices are personal,and that individuals should have the freedom to
make these kinds of choices for themselves.In this vein,Sen deﬁnes an axiom termed minimal
liberalism,which states that there are at least two individuals and two pairs of distinct alternatives
(x;y) and (w;z) such that Person 1 is free to decide between (x;y) and Person 2 is free to decide
between (w;z).Minimal liberalism is a particular version of Arrow’s condition of no dictator,as
it characterizes a situation in which decisionmaking power is minimally decentralized.Sen shows
that even a weak condition such as minimal liberalism is incompatible with a procedure being
capable of both generating transitive outcomes and being Pareto efﬁcient.
Theorem3.(Sen) No aggregation rule that produces transitive outcomes can simultaneously sat
isfy unrestricted domain,Pareto efﬁciency,and minimal liberalism.
Sen provides the following illustration of his theorem.Consider a social choice over three
alternatives:either Mr.A can read a copy of Lady Chatterly’s Lover (Alternative a),or Mr.B
can read it (Alternative b),or neither of them can read it (Alternative c).Mr.A,the prude,prefers
neither of them reading it to himself reading it (in order to protect the impressionable Mr.B),to
Mr.B reading it.Thus he prefers c
A
a
A
b.Mr.B,the lascivious,prefers Mr.A reads it
(in order for Mr.A to be exposed to Lawrence’s prose),to himself reading it,to neither of them
reading it,and thus prefers a
B
b
B
c.
An argument can be made that Mr.A should decide between the (a;c) states of the world,
the states in which he reads the book or neither reads the book.Similarly,Mr.B should decide
between the (b;c) states of the world.Thus,the condition of minimal liberalism is satisﬁed,and
yields the social ordering b c (by Mr.B’s preferences),and c a (by Mr.A’s preferences).
8
However,both individuals prefer a to b,and so Pareto efﬁciency along with minimal liberalism
yields the cycle a b;b c;and c a.Thus,the procedure is incapable of generating an optimal
choice.[6,pp.8081]
The GibbardSatterthwaite Theorem
Our ﬁnal impossibility theorem,proved independently by Gibbard [7] and Satterthwaite [8],dif
fers from the previous theorems in several important ways.First,it concerns choice functions
rather than preference aggregation rules (i.e.voting systems that produce a single winner,as op
posed to a social ordering of the alternatives),and second,it does not assume that a voting system
takes a “true” collection of preferences as an input.Rather,it considers voting systems that take
individuals’ reported preferences,or ballots,as an input.
Gibbard and Satterthwaite are concerned with the strategyproofness of a choice function,or
the ability of a choice function to discourage insincere behavior by a voter.Suppose that = (
1
;:::;
n
) is a “true,” or “sincere,” preference proﬁle,whereas
0
= (
1
;:::;
0
i
;:::;
n
) is a proﬁle
that only differs from in that Voter i reports the “insincere” preferences
0
i
,as opposed to his
true preferences
i
.Then a choice function F is strategyproof if F(
0
) will never generate an
outcome that Voter i strictly prefers to the outcome generated by F(),the outcome he would
have received by sincerely reporting his preferences.Thus,F being strategyproof implies that
no voter can ever strictly beneﬁt by claiming to have preferences that are different than what they
actually are.Unfortunately,Gibbard and Satterthwaits demonstrate that strategyproofness and
Pareto efﬁciency are not attainable with a nondictatorial procedure.Note that the deﬁnitions of
“dictator” and “Pareto efﬁciency” used in this theorem are modiﬁed slightly from our previous
deﬁnitions to accommodate the fact that we are considering choice functions.
Theorem4.(Gibbard,1973;Satterthwaite 1975) With three or more alternatives and unrestricted
domain,any choice function that is strategyproof and Pareto efﬁcient is dictatorial.
The GibbardSatterthwaite Theorem proves that the possibility of strategic voting,or voting
against one’s true preferences,is endemic to the vast majority of voting systems.Despite their
differences,the GibbardSatterthwaite Theorem is mathematically similar to Arrow’s Theorem,
and it has been demonstrated that the two results can be derived from what is essentially a single
proof.[9] It should also be noted that Pareto efﬁciency can be replaced by the far weaker condition
of nonimposition,which states that any alternative can be achieved as a social choice,given some
9
collection of ballots.
Voting paradoxes
Impossibility theorems are paradoxical,in the sense that they prove that seemingly sensible condi
tions are incompatible with each other.While these results are quite general,in that they typically
apply to a wide range of voting systems,it is also wellknown that speciﬁc voting systems can
suffer from speciﬁc anomalies.A voting paradox occurs when a particular voting system yields
an end result that is highly counterintuitive.The paradoxes discussed here are wellknown prob
lems stemming from widelyused voting rules,and represent only a handful of the documented
paradoxes.
Condorcet’s Paradox
The most famous voting paradox of all is known as Condorcet’s paradox,and was introduced by
the Marquis de Condorcet,a French mathematician and philosopher,in his 1785 treatise Essay on
the Application of Analysis to the Probability of Majority Decision.This paradox demonstrates that
majority rule with three or more alternatives may yield intransitive outcomes.It can be summed
up by the following preference proﬁle:
=
x
1
y
1
z
y
2
z
2
x
z
3
x
3
y
Given proﬁle ,it is clear that a majority vote between x and y would yield the social ranking
x y (by the votes of persons 1 and 3);a majority vote between y and z would yield y z (by the
votes of persons 1 and 2);and that a majority vote between x and z would yield the ranking z x
(by the votes of persons 2 and 3).Thus,majority voting over pairs of alternatives is not guaranteed
to yield transitive outcomes when there are three or more alternatives,even when we assume that
individual preferences are transitive.In this case,majority voting yields the cycle x y,y z,
and z x.
Ostrogorski’s Paradox
Ostrogorski’s paradox involves the role of political parties as intermediaries in the process of vot
ing.His paradox stems from the fact that the following two procedures can yield different out
10
comes:(1) Each voter votes for the party whose stand is closest to his own on the most issues.
The winner is the party with the most votes.(2) On a series of issues,each voter votes for the
party whose stance is closest to his own.The winner is the party that wins the most issues.The
following table demonstrates this paradox.It is taken from[10,p.72].
Favorite Party
Favorite Party
Favorite Party
Voter
on Issue 1
on Issue 2
on Issue 3
Party Supported
Voter 1
X
Y
Y
Y
Voter 2
Y
X
Y
Y
Voter 3
Y
Y
X
Y
Voter 4
X
X
X
X
Voter 5
X
X
X
X
The above table shows that a majority of voters prefer Party Y to Party X,overall.At the same
time,if we examine the table issuebyissue,we can see that Party X defeats Party Y on every
issue.Thus,a party winning an election (Y,in this case),could fail to represent the views of a
majority on every single issue.
The Additional Support (Monotonicity) Paradox
One straightforward criterion that we might like a voting systemto satisfy is that increased support
for a winning candidate does not turn that candidate into a loser.It is wellknown,however,that
many tworound voting systems do not satisfy this property,and thus suffer from the additional
support,or monotonicity,paradox.The following two tables demonstrate this phenomenon in a
majority runoff election,or an election in which,if no candidate receives a majority,the top two
votegetters proceed to a runoff.
Number of Voters
Preferences
6
a b c
7
b a c
8
c a b
In the above election,11 of 21 votes are required to win.Since no candidate receives a majority,a
runoff is held between b and c (the two highest votegetters),and as the a voters prefer b to c,b wins
11
the runoff.Now suppose that three voters switch their preferences from c a b to b c a.
This is represented in the following table:
Number of Voters
Preferences
6
a b c
10
b a c
5
c a b
In this case,a and b proceed to the runoff,where a ends up defeating b by 11 to 10.Thus,receiving
strictly greater support in the second election caused b to change from a winning candidate to a
losing candidate.
The Alabama Paradox
In the United States,congressional seats are awarded to the ﬁfty states following every decennial
census.The Constitution speciﬁes that these seats should be awarded in proportion to state pop
ulations,but does not specify a particular formula by which the seats should be distributed.Prior
to 1911,the total number of congressional seats was readjusted every ten years to account for
population growth.It was because of such a readjustment that in 1882 the Alabama paradox was
discovered:as the total number of seats to be distributed increased,the state of Alabama ended up
losing a seat.This paradox is a consequence of the particular apportionment formula being used
at the time,called Hamilton’s method,which worked as follows:First,a quota is determined to
represent the ideal number of people to be represented by a single congressional seat.If 435 seats
are to be distributed we would get
Quota =
Population of the United States
435
:
Second,a state’s ideal number of seats is determined by dividing the state’s population by this
quota:
State’s ideal number of seats =
Population of the state
Quota
:
In general,however,a state’s ideal number of seats (its entitlement) will involve a fractional part.
Hamilton’s method speciﬁes that each state should be awarded the whole number portion of its
entitlement,and that any remaining seats to be distributed go to the states with the largest fractional
portions of their entitlements.The following tables demonstrate how Hamilton’s method works,
and its vulnerability to the Alabama paradox.
12
Seat allocations with 10 seats to distribute
State
Population
State’s Ideal#of Seats
Seats Awarded
A
1200
4.29
4
B
1200
4.29
4
C
400
1.42
2
In the above table there was one remaining seat to distribute after the states were awarded their
integer amounts.This seat went to State C,because it had the largest fractional entitlement (0.42).
The next table shows what happens when the total number of seats to be distributed increases from
ten to eleven.
Seat allocations with 11 seats to distribute
State
Population
State’s Ideal#of Seats
Seats Awarded
A
1200
4.71
5
B
1200
4.71
5
C
400
1.57
1
The above table shows that increasing the number of seats to be distributed causes State C’s frac
tional entitlement to dip below the fractional entitlements of States A and B.Thus,even though
there are more seats to be awarded,State C ends up losing a seat.
Further readings
The ﬁeld of voting theory is vast and growing,with many topics for the interested reader to dis
cover.AustenSmith and Banks [11,12] provide an extremely thorough and technical presentation
of this material.A less technical introduction to this material with many realworld applications
can be found in Saari [13].Nurmi [10] provides an excellent overview of the various voting
paradoxes,and a classiﬁcation of their types.Balinski and Young [14] provide a history of rep
resentation paradoxes that have occurred in the context of United States congressional apportion
ment.Taylor [15] focuses exclusively on the manipulability of voting systems,and in particular
the GibbardSatterthwaite Theoremand its many extensions.In addition,the interested reader will
ﬁnd the classic texts of Arrow,May,Sen,Gibbard and Satterthwaite to be readable and thought
provoking.
13
References
[1] Colomer,J.(2004) The strategy and history of electoral system choice In Josep Colomer,
(ed.),Handbook of Electoral SystemChoice,pp.3–73 Palgrave Macmillan New York.
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