Strategyproofness of voting systems
Nora Boeke
April 26, 2013
Bachelorproject
Begeleiding: dr. Ulle Endriss
Korteweg-De Vries Instituut voor Wiskunde
Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Universiteit van Amsterdam
Abstract
The Gibbard-Satterthwaite Theorem states that the only resolute strategyproof voting
system is dictatorschip. Duggan and Schwartz proved that every non-dictatorial surjective voting system among three or more candidates, is manipulable. Recently, Brandt
and Brill proved that there exists a class of non-dictatorial social choice functions that
is strategyproof. In this thesis the different articles and their concepts are placed next
to each other in order to understand what causes the different keyresults . It becomes
clear that this difference lies in the notion of calling a voting system strategyproof or
not. The varying concepts of strategyproofness and the preferences over sets, will be
analyzed, explained and discussed.
Titel: Strategyproofness of voting systems
Auteur: Nora Boeke, [email protected], 6041892
Begeleiding: dr. Ulle Endriss
Einddatum: April 26, 2013
Korteweg-De Vries Instituut voor Wiskunde
Universiteit van Amsterdam
Science Park 904, 1098 XH Amsterdam
http://www.science.uva.nl/math
2
Contents
1 Introduction
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2 Theory
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Strategyproofness in the resolute context . . . . . . .
2.3 Strategyproofness in the non-resolute context, Taylor
2.4 Strategyproofness in the non-resolute context, Brandt
2.5 Systematic Comparative Review . . . . . . . . . . . .
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3 Conclusion
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4 Popular Summary
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Bibliography
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3
1 Introduction
Social choice theory is a branch of Mathematical Economics that studies the mechanisms
of voting procedures. When a group of individuals has to take a collective decision, different procedures can be used to aggregate the different preferences of the group members
into this common decision. Usually, during elections each individual gives his vote to
the candidate he most prefers. Out of all given votes the candidate who gathers most
votes becomes the winner. This method of choosing a winner is called ‘the plurality
rule’. There are alternative ways to choose a winner out of the different preferences in a
population. The construction of how to aggregate the preferences into a collective choice
is called a voting system or a voting rule. The preferences of each voter can be presented
as a row of candidates in order from the one which is preferred most to the one who is
least favoured. So a voting rule can therefore be seen as a function in which the input
is a bunch of preference rankings and the outcome can vary between one candidate to
a set of candidates. When the outcome of such a voting rule is restricted to only one
winner, we call it a resolute system. Irresolute voting rules are mechanisms where ties
are allowed and the winning sets can therefore consist of more candidates.
In September 2012 PvdA and VVD became the biggest parties in the Netherlands and
thereby formed a coalition cabinet. Polls showed us that 35% of the PvdA voters would
have voted alternatively if there had not been a neck and neck race between PvdA and
VVD. Similarly, 25% of the VVD-voters said that they voted strategically. So there
was a considerable number of individuals that did not vote for the party they most
preferred. They adapted their vote in order to get a better outcome compared to the
situation where they would have voted for the candidate they truly wanted. This phenomenom is what we call strategic voting. Here I give another simple example:
In this election there are 3 candidates and 14 individuals that give their votes. In
the following table the true preferences of the voters are presented. For example, 6 voters rank A over B over C.
How many voters Preference
6
A>B>C
7
B>A>C
1
C>A>B
If the social choice function is the plurality rule it is easy to see that candidate C has
no chance to win. If everybody votes for the candidate he truly wants, according to the
plurality rule B becomes the winner. Since C is not a winner, this is an incentive for the
voter who ranks C as the best candidate, to misrepresent his preference in order to get a
4
better outcome. If this voter changes his vote by choosing for candidate A instead of C,
A comes in the winning set. Since A becomes a winner when this voter misrepresents his
preference, and therefor the outcome has become better, this is an example of strategic
manipulation.
It is desirable to work with voting systems in which people always present their true
preferences and not have the opportunity to gain a better result when they manipulate
their votes. This thesis is about the existence and requirements to construct such desirable strategyproof functions.
Allan Gibbard and Mark Satterthwaite independently published in respectively 1973
and 1975 articles, both written about the impossibility to find resolute reasonable strategyproof social choice functions[5] [9]. In this article a social choice function is called
strategyproof if it does not allow for strategic manipulation (if no voter can ever get
a better election outcome by misrepresenting her true preferences when voting). The
GibbardSatterthwaite Theorem states that for resolute voting rules (among three or
more candidates) under certain assumptions, the only way to get strategyproofness is
by dictatorship. Dictatorship is a voting rule where the winner becomes the one that
is most preferred by one particular voter (the dictator itself). In Section 2.1 of this
thesis I will introduce some basic notation about social choice functions. Besides that, I
will come up with some definitions about basic properties of voting systems. Once this
common ground is created I will in Section 2.2 guide you through the proof of the the
impossibility to find another rule than dictatorship to get strategyproofness.
Although the Gibbard-Satterthwaite Theorem can be seen as one of the keyresults in
social choice theory, some critics argue that the resolute context in which it has been
attained is somewhat unnatural. For instance, consider an election where two parties
get the most and exactly the same number of votes. If we use the plurality rule it is clear
that these two parties battle for the win. But since a tie is not an option the SCF has to
be ‘biased towards an alternative or a voter (or both)’ [2]. This shows that voting rules
in the resolute context cannot always be seen as ‘fair’ systems. In 2000, John Duggan
and Thomas Schwartz published an article about voting rules where the outcome is not
restricted to be one winner [3]. And not very surprisingly they state that it is impossible to find another strategyproof SCF than dictatorship for three or more candidates.
Since the original paper of Duggan and Schwartz is quite hard to read I have studied
an article of Taylor about this Duggan-Schwartz Theorem [8]. In the resolute context
it was straightforward to compare different outcomes with each other: A candidate x is
regarded better then candidate y by voter i if x is above y in the linear ballot. It is a
more complex task to compare sets of winners with each other. In Section 2.3 I start
by explaining how Duggan, Schwartz and Taylor compare winning sets with each other
and what it means to be a strategyproof voting system. After that I will prove some
lemmas in order to set out the proof of the Duggan-Schwartz Theorem. After this proof
I will give two specific examples of social choice functions; these are not strategyproof
according to the definition that is given by Duggan and Schwartz but they are in a sense
5
related to that definition. Although this may not seem quite helpful right now, once
you are at Section 2.3, you will understand why these two examples are relevant to gain
comprehension about the different sorts of manipulability.
Recently an article by Brandt and Brill, [2], was published that tells us that it is achievable to construct a irresolute voting rule that is strategyproof and is not dictatorship.
This result seems to contradict with the previous papers of Duggan, Schwartz and Taylor. In this thesis I want to investigate what the different results between the last two
papers are based on. Brandt and Brill worked with another representation of preferences
than linear ballots. Since they have used a different way of describing someone’s preferences, the proofs and lemmas in their article are embedded in a completely different
notational framework tcompared to the previous papers. In Section 2.4 I will introduce
this other notation and will give some definions about basic properties of social choice
functions within the new framework. Brandt and Brill chose to re-define the concept
of whether one winning set is preferred to another. Therefore they used the theories
of Barbera and Kelly, formulated in 1977 [1]. In Section 2.4 these re-definitions and
subsequently strategyproofness according to these newer concepts will be discussed. At
the end of this section I will explain how Brandt and Brill constructed their proof about
the class of strategyproof social choice functions. The last part of my thesis, Section 2.5,
is about the different frameworks that are used by Duggan and Schwartz and Brandt
and Brill. Once you comprehend in which parts these frameworks and their embedded
definitions, like preferences over sets and strategyproofness, vary from one another, you
will at the end understand why the two articles have different keyresults.
6
2 Theory
2.1 Notation
The set of voters is denoted by N = {i1 , i2 , . . . , in } and X = {x1 , x2 , . . . , xm } stands
for the set of candidates. In this thesis we only consider finite and nonempty sets of
candidates.
U. Endriss, [4], discusses the Gibbard-Satterthwaite Theorem and represents the preferences of a voter i as a linear order Ri . Let L(X) be the set of all linear orders on X. Ri
is a vector with on top the candidate which is preferred the most and at the bottom the
one who is the least favoured. Note that a linear order is a strict binary relation while
a voter in reality can appreciate two or more candidates equally. Although Taylor also
makes use of linear orders, the proofs can easily be extended to the version where voters
can equally like candidates [8].
In the article by Brandt and Brill the preference of a voter i are expressed by a preference
relation Ri . In this and in the coming two sections, we will refer to preferences as linear
orders. Another word for these preference vectors are ballots of voters.
A profile R ∈ L(X)N consists of the preferences of all the voters and is therefore a
vector of ballots (R = (R1 , R2 , . . . , Rn )). We denote the set of voters that prefers x
above y as Nx>y .
Definition 2.1. Social Choice Function (SCF)
A social choice function or a voting rule is a function F that maps profiles to a nonempty
set of candidates. So: F : L(X)N → P(X) \ {∅}. A single-valued SCF is called resolute.
Now we know what a social choice function is I will give two examples to get a better
understanding.
Example 2.2. The Condorcet Rule
Under the Condorcet Rule, a candidate x is the winner if in every one-to-one comparison with another alternative it is preferred by more voters. So x ∈ F (R) if ∀y 6= x,
|Nx>y | > |Ny>x |.
Look at the following example:
Voter Preference
One
A>B>C
Two
B>A>C
Three C>B>A
Since |NB>A | > |NA>B | and |NB>C | > |NC>B | it follows that B is the Condorcet winner.
7
Now look at the following case:
Voter Preference
One
A>B>C
Two
B>C>A
Three C>A>B
It is easy to see that there is no Condorcet winner in this case: Because |NA>B | > |NB>A |,
candidate B cannot be the winner. Since |NA>C | < |NC>A | also A cannot be the winning
candidate and C is also a losing candidate because |NC>B | < |NB>C |. The cyclic phenomenon where there does not exist a unique winner is called the ‘Condorcet Paradox’.
Since a social choice function always has to have a winning set we add something to the
previous definition; F (R) = X iff 6 ∃x such that ∀y 6= x, |Nx>y | > |Ny>x |.
Example 2.3. Borda Count
According to the preference of each voter the candidates get points for there position in
the ballots. Suppose we have m candidates then when a candidate is at the topranked
position of one’s ballot he receives m−1 points. When a candidate is at the r’th position
he will get m − r points. When we add up all the points that the candidates receive
from the different voters, the candidate with the most points wins.
Look at the following case:
Number of Voters Preference
7
B>D>C>A
6
C>B>D>A
5
A>C>B>D
3
A>B>C>D
Voter A receives 4 ∗ 8 + 3 ∗ 0 + 2 ∗ 0 + 1 ∗ 13 = 45 points.
Voter B receives 4 ∗ 7 + 3 ∗ 9 + 2 ∗ 5 + 1 ∗ 0 = 65 points.
Voter C receives 4 ∗ 6 + 3 ∗ 5 + 2 ∗ 10 + 1 ∗ 0 = 59 points.
Voter D receives 4 ∗ 0 + 3 ∗ 7 + 2 ∗ 6 + 1 ∗ 8 = 41 points
Herefore B is the winner according to the Borda Count Rule.
In the Introduction I explained that the winner under the Plurality Rule is the candidate who gatherers the most votes. To be precize; the winner is the candidate who is
top ranked by the most voters. In this preference profile candidate A is top ranked by 8
voters, B is top ranked by 7 voters, C is top ranked by 6 voters and nobody has candidate D on the top of his ballot. This means that A is the winner under the Plurality Rule.
Since |NC>A | = 13 > 8 = |NA>C |, NC>B | = 11 > 10 = |NB>C | and NC>D | = 14 >
7 = |ND>C | it follows that C is the Condorcet winner.
In the introduction it is stated that a strategyproof social choice function under some
simple assumptions must be dictatorship. I will now introduce some properties of social
choice functions in order to make the different proofs less complex.
8
Definition 2.4. Surjectivity
We call a function surjective when for every candidate, there exists a profile under which
this candidate will be in the winning set of the SCF.
For every x ∈ X there exists a profile R such that x ∈ F (R)
Definition 2.5. Weak monotonicity
A SCF is weakly monotone if a winning candidate remains a winner if he gets additional
support while leaving the relative ranking of the other alternatives unchanged.
Suppose candidate x is the winner. If we move x above another candidate y in one or
more ballots, x has to remain the winner.
0
and
x ∈ F (R) implies x ∈ F (R’) for distinct profiles R and R’ with Nx>y ⊆ Nx>y
0
Ny>z = Ny>z for all y, z ∈ X \ {x}.
Weak monotonicity can also be defined as follows: A winner stays in the winning set one
voter changes his ballot by strengthening a winning candidate one spot and everybody
else keeps the same preference.
It is easy to see that the first definition can be obtained by the latter one by applying
induction on the number of voters and the number of spots you strengthen the winner.
Definition 2.6. Strong monotonicity
A SCF is strongly monotone if the winning candidate remains the winner if he gets
additional support. So compared to weak monotonicity it is not neccesarry to hold the
relative ranking of non-winning candidates.
0
x ∈ F (R) implies x ∈ F (R’) where R and R’ are distinct profiles with Nx>y ⊆ Nx>y
for all y ∈ X \ {x}.
Strong monotonicity can also be defined as follows: A winner remains a winner if one
voter changes his ballot by weakening another candidate than the winner, one spot and
everybody else keeps the same preference.
It is easy to see that the first definition can be obtained by the latter one by applying induction on the number of voters and the number of spots you weaken the other
candidate.
Definition 2.7. Pareto
A social choice function satisfies the Pareto condition if, whenever all voters prefer x
over y, y cannot be among the set of winners. Formal: Nx>y = N implies y 6= F (R)
Definition 2.8. Decisiveness
A coalition G ⊆ N is called decisive on (x, y) if G ⊆ Nx>y implies {y} 6= F (R). Notice
that whether G is decisive or not is dependent on the SCF F that is used. When G is
decisive on (x, y) we can simply write down xGy. If a coalition is decisive on all pairs
we just call the coalition decisive. Note that when a SCF satisfies the Pareto condition,
the set of all voters together is desicive.
Definition 2.9. Topset
A topset P ⊆ X of a profile R is a set of candidates that each voter prefers to every
candidate that is not in P . So the candidates of the topset are topranked in each ballot.
9
2.2 Strategyproofness in the resolute context
In this section the Gibbard-Satterthwaite Theorem will be proved. This theorem shows
us the impossibility to find another rule than dictatorship to get strategyproofness.
In order to prove the Gibbard-Satterthwaite Theorem I will use the construction that
Endriss designed in order to prove the Gibbard-Satterthwaite theorem in his article [4].
A social choice function where the outcome is always the topranked candidate of one
particular person is known as dictatorship. The Muller-Satterthwaite Theorem states
that whenever a resolute SCF for three or more candidates, is surjective and satisfies
strong monotonicity, it must be a dictatorship [6]. In order to prove this theorem I will
first go through some definitions and lemmas.
Since we are working in the resolute context we simply write down F (R) = x instead of
F (R) = {x}. If in the following sections explicitly is said that a SCF is single valued we
use this notation F (R) = x.
Definition 2.10. Dictatorship
A SCF where the winner is always the topranked candidate of one particular voter, is
called dictatorship.
Lemma 2.11. Strong monotonocity implies the independence property (if x 6= y, F (R) =
0
x, and Nx>y = Nx>y
, then F (R’) 6= y).
0
Proof. Suppose x 6= y, F (R) = x, and Nx>y = Nx>y
and F (R’) = y. Take a profile
00
R” where x and y are topranked by all the voters and Nx>y = Nx>y
. When moving
from profile R to R”, x and y are strengthened with respect to the other candidates
and the relative ranking of these two are unchanged. Therefore it follows by strong
monotonicity that F (R”) = x. Since we have assumed that F (R’) = y it follows again
by strong monotonicity that F (R”) = y. Since we already found that F (R”) = x this
is a contradiction. So therefore F (R’) 6= y.
Lemma 2.12. Surjectivity and strong monotonicity together imply the Pareto condition.
Proof. Take x,y, such that x 6= y. Since the SCF is surjective there exists a profile R
such that F (R) = x. If we move x above y in all ballots (we get a new profile R’ in
0
which Nx>y
= N ), strong monotonicity implies that candidate x is still the winner. Now
00
00
0
= N out of R’ with Nx>y
= Nx>y
= N.
we can construct any profile R” such that Nx>y
The independence property tells us that F (R”) 6= y.
Theorem 2.13. Muller-Satterthwaite Theorem, 1977:
A resolute SCF for three or more candidates that is surjective and strongly monotonic
is a dictatorship.
Proof. By Lemma 2.11 it follows that such a SCF satisfies the independence property.
Then 2.12 implies that we get Pareto efficiency. So if we now prove that any resolute
SCF for three or more candidates that is independent and satisfies the Pareto condition,
is a dictatorship, we are done.
10
First of all I claim the following:
G = Nx>y → F (R) 6= y, implies that G is decisive on any given pair (x0 , y 0 ).
Proof. Assume that G = Nx>y → F (R) 6= y. Now take a profile R in which {x, y, x0 , y 0 }
is a topset and in G everyone ranks x0 > x > y > y 0 and all other voters rank x0 > x,
y > y 0 and y > x. Since everybody in G ranks x above y it follows by the assumption
that y can not be a winner and because everybody ranks x0 above x and y above y 0 , the
Pareto condition implies that x and y 0 can nott be winners. And since {x, y, x0 , y 0 } is a
topset, no other candidate than these can win, hence x0 is the winner.
Now take any profile R’ where G ⊆ Nx0 0 >y0 . Since in R the relative ranking of x0 and y 0
for the voters outside of G was not defined we can w.l.o.g. assume that Nx0 >y0 = Nx0 0 >y0 .
Since also x = F (R) it follows by independence that y 0 can not be the winner, so G is
decisive on (x0 , y 0 )
If G is a coalition that is decisive on all pairs and |G| ≥ 2 and G1 and G2 are coalitions such that G = G1 ∪G2 and G1 ∩G2 = ∅, then either G1 or G2 is decisive on all pairs.
Proof. Consider a profile R where {x, y, z} is the topset and G is decisive. In G1 the
voters rank x > y > z and the people in G2 rank y > z > x and everybody else ranks
z > x > y. Since G is decisive, z cannot win. So either x or y wins.
• Suppose x wins. Notice that G1 = Nx>z . Since the independence property holds,
it follows that F (R”) 6= z for any R” where G1 = Nx>z . By the claim that we
proved it follows that G1 is decisive on any pair.
• Suppose y wins. Notice that G2 = Ny>x . Since the independence property holds,
it follows that F (R”) 6= x for any R” where G2 = Ny>x . By the same claim it
follows that G2 is decisive on any pair.
Since the coalition of all voters N is decisive. It follows by induction that a singleton
is decisive. And when a singleton is decisive on all pairs it means that the topranked
candidate of this voter is the winning candidate, which means that this singleton is the
dictator.
Definition 2.14. Strategyproofness
A resolute SCF F is strategyproof if for no individual i ∈ N there exists a profile
R (which contains the true preferences Ri of the i’th voter) and a linear order Ri0
(a disingenuous vote of i) such that F (R−i , Ri0 ) is positioned above F (R) in the true
preference ballot of voter i.
11
In words this means that a SCF is strategyproof if there is no incentive for a voter
to misrepresent his true preference in order to get a better outcome. If a SCF is not
strategyproof we can also say that this voting system can be manipulated by a voter.
Theorem 2.15. Gibbard-Satterthwaite Theorem, 1973/1975
Any resolute SCF for three or more candidates that is surjective and strategyproof must
be a dictatorship.
Proof. If we prove that a strategyproof SCF is strongly monotone we can apply the
Muller-Satterthwaite Theorem whereby dictatorship is implied. By showing that the
assumption of strategyproofness and strong monotonicity will lead to a contradiction,
the Gibbard-Satterthwaite Theorem is established.
Consider a SCF F that is strategyproof but does not satisfy strong monotonicity. Then
0
there exist profiles R, R’ such that F (R) = x and Nx>y ⊆ Nx>y
, ∀y 6= x, although
0
0
F (R’) = x , x 6= x.
If we now change profile R to R’ then at a sudden moment the winner changes from x
to x0 . So after the change of a particular voter’s ballot the winner changes from x to x0 .
W.l.o.g. we can assume that these two profiles R and R’ differ in only one ballot due
which they have a different outcome. Call the voter of this ballot, voter i.
0
• Suppose i ∈ Nx>x
0 . So if profile R’ is seen as the profile corresponding to voter i’s
true preference then it’s better for him to misrepresent his preference as in R to
get a better outcome, namely x which is rated higher than x0 .
0
0
• Suppose i 6∈ Nx>x
0 . By the assumption of Nx>y ⊆ Nx>y ∀y 6= x, it follows that
0
i∈
/ Nx>x
0 and thus i ∈ Nx0 >x . Consider R as the profile with the true preference
of voter i, then it is better to change the ballot as in R’ to obtain x0 as the winner,
which is ranked above x by voter i.
12
2.3 Strategyproofness in the non-resolute context,
Taylor
In this section the Duggan-Schwartz Theorem will be proved [3]. This theorem shows
us the impossibility to find another rule than dictatorship to get strategyproofness. As
I already said, the original article of Duggan and Schwartz is constructed in a complex
way, whereby I have studied an article of Taylor about the Duggan-Schwartz theorem,
which is less hard to read [8]. In order to prove this theorem I will use the construction
that is made by Taylor.
The definition of strategyproofness that was given in Section 2.1 has been constructed
in a context where the outcome is a singleton. When the outcome of a SCF consists
of a winning set, the concept of strategyproofness is therefore slightly more complex.
Resolute voting rules are strategyproof if no voter i can misrepresent his vote in order
to get a better outcome (a candidate that is ranked higher in the ’true’ preference ballot
that corresponds to voter i). Since we now have to compare winning sets instead of just
two winning candidates we introduce the new terms: optimistic voter and pessimistic
voter.
Definition 2.16. Strategyproofness for non-resolute SCF’s
A voting system can be manipulated by optimistic voters if there exists a profile R (which
includes the true preferences Ri of the i’th voter) and a linear order Ri0 (a disingenuous
ballot of i) such that at least one candidate of F (R−i , Ri0 ) by voter i is ranked above
all candidates of F (R). When a SCF can be manipulated by an optimistic voter the
maximum of the winning set according to his preference Ri , can be improved by the
misrepresentation of his true ballot.
A voting system can be manipulated by a pessimistic voter if there exists a profile R
(which consists of the true preferences Ri of the i’th voter) and a linear order Ri0 (a
disingenuous ballot of i) such that all the candidates of F (R−i , Ri0 ) by voter i are ranked
above at least one candidate of F (R) . So in this case the minimum of the set according
to a voter i can be improved by i to show a disingenuous ballot.
From now on we will call a SCF F strategyproof if it can not be manipulated by
optimistic or pessimistic voters.
Because we are going to work in a broader context (non-resolute SCF’s) we have to extend some of our defintions in order to understand what the Duggan-Schwartz Theorem
is about.
Definition 2.17. Strong Surjectivity
A non-resolute SCF is called stronlgy surjective if for every candidate there exists a
profile where this candidate is the only winner according to this SCF. So for every
x ∈ X there exists a profile R such that F (R) = {x}.
Definition 2.18. Dictatorship for non-resolute SCF’s
A SCF where the topranked candidate of one particular voter is always in the winningset,
is called dictatorship.
13
The Duggan-Schwartz Theorem tells us that when a SCF for three or more candidates
is strategyproof and satisfies strong surjectivity it has to be a dictatorship. We will now
go through some lemmas in order to simplify the final proof of the Duggan-Schwartz
Theorem.
Lemma 2.19. Consider a SCF F that is strategyproof and a profile R for which P is a
topset. If there is at least one profile R’ sucht that F (R’) ⊆ P , then F (R) ⊆ P
Proof. Assume the opposite, so that F (R) 6⊆ P . We change the ballots one by one to
switch from R to R’ and call the voter due which the winning set shifts to a subset of
P , voter i. We will consider the profile right before the change of the winning set occurs
and refer to this profile as Rt , the profile concerning the true preference of voter i. We
call the first profile where the winning set is contained in P , Rs and refer to this profile
as the one containing the disingenous ballot of i. For voter i it is better to misrepresent
his preference as in Rs because the minimum of the winning set is improved. So if we
assume that F (R) 6⊆ P the SCF can not cope with pessimistic voters. Hereby the lemma
is proved.
Definition 2.20. Down monotonicity for singleton winners
If |F (R)| = 1 then F satisfies down monotonicity for singleton winners if F (R) = {x} =
0
0
, y ∈ X, z ∈ X \{x} and Nz>w = Nz>w
F (R’) for all profiles R and R’ with Ny>z ⊆ Ny>z
for all z, w ∈ X \ {y}.
So a SCF that satisfies down monotonicity for singleton winners can also be defined as
follows: when there is only one winning candidate this candidate stays the sole winner
when some voter moves a losing candidate one spot. Or: when there is only one winning
candidate this candidate stays the sole winner when somebody moves a candidate y
above a losing candidate z in his ballot.
Lemma 2.21. If a SCF is strategyproof, it satisfies down monotonicity for singleton
winners.
Proof. Suppose the SCF F does not satisfy down monotonicity for singleton winners.
Then there exist profiles R with F (R) = {x} and R’ where one voter i has weakened a
losing alternative y one spot, but F (R’) = W 6= {x}. Now take z ∈ W . Then we can
distinguish three cases:
• In both profiles x > z. Then consider R’ as the profile concerning i0 s true preference and R the one with the misrepresentation of i0 s preference. When i changes
from R’ to R he improves his minimum.
• In both profiles z > x. Then consider R as the profile concerning i0 s true preference
and R’ the one with the misrepresentation of i0 s preference. When i changes from
R to R’ he improves his maximum.
• In one profile z > x and in the other x > z. Since we have weakened a losing
alternative y one spot to get from R to R’ it follows that z = y and z > x in R
14
and x > z in R’. So if we consider R’ as the profile concerning i0 s true preference,
it is better for this voter to misrepresent his ballot as in R to improve his minimum.
When we had assumed that R corresponds to the true preference of i, there was
also an incentive for this voter to show the disingenous ballot as in R’ to improve
his maximum.
Lemma 2.22. If a SCF satisfies down monotonicity for singleton winners then if you
can find a profile R that satisfies the following conditions, it implies that G is decisive
on (x, y).
• {x, y} is a topset of R
• G = Nx>y
• x ∈ F (R)
Proof. Assume that the four conditions are satisfied by a profile R but xGy fails. Consider there exists a profile R’ whereas in R, G ⊆ Nx>y but F (R’) = {y}. Candidate x
is then a losing candidate and we can weaken x in all the ballots of the voters that are
not in G such that G = Nx>y . If we put down all the other candidates than x and y
in the ballots of the voters, {x, y} becomes a topset and we obtain the profile R that
is assumed to exist. Since we have got this by the weakening of losing candidates, the
winner stays the same as in R’ (due to down monotonicity for singleton winners). This
gives us a contradiction.
Lemma 2.23. If a SCF is strongly surjective and down monotone for singleton winners,
then the set of all voters is decisive, i.e. satisfies the Pareto condition.
Proof. Consider a SCF that is strongly surjective and satisfies down monotonicity for
singleton winners, but N is not decisive. Then there exist candidates x and y such that
N = Nx>y but {y} = F (R). Due to strong surjectivity there exists a profile R’ such that
F (R’) = {x}. Since by down monotonicity it is not necessary to preserve the relative
ranking of losing candidates we can change R’ into R to still hold {x} as the winning
candidate. But then we have a contradiction.
Lemma 2.24. Consider x SCF F that is strategyproof and strongly surjective. Then the
following holds:
Suppose G is x coalition decisive on (x, z), |G| ≥ 2. Now assume that y 6= x and y 6= z,
then if G1 and G2 are coalitions such that G = G1 ∪ G2 and G1 ∩ G2 = ∅, it follows that
either xG1 y or yG2 z.
Proof. To prove that either xG1 y or yG2 z we use Lemma 2.22. So we want to prove
that there exists a profile R such that it satisfies the three conditions given in Lemma
2.22. Take a look at the following profile R.
In G1 everyone’s ballot looks like (x, z, y, . . .)
15
In G2 everyone’s ballot looks like (y, x, z, . . .)
The rest of the voter’s ballot looks like (z, y, x, . . .)
Since {x, y, z} is a topset of R and F is strongly surjective, it follows by lemma 2.19
that F (R) ⊆ {x, y, z}. because G is a coalition decisive on (x, z), z is not a winner. So
therefore x ∈ F (R) or y ∈ F (R).
• Suppose that y ∈ F (R).
If we now change profile R into R’ by moving x below z in G2 we still obtain y as
a winner. The new profile R’ looks as follows:
In G1 everyone’s ballot looks like (x, z, y, . . .)
In G2 everyone’s ballot looks like (y, z, x, . . .)
The rest of the voter’s ballot looks like (z, y, x, . . .)
To show that y remains a winner, suppose y would not be in the winning set
anymore. Consider R’ as the profile concerning the true preferences of a voter
i ∈ G2 . Since y is the topranked candidate in both ballots it is better to change
to profile R in order to improve the maximum.
If we now change profile R’ into R” by moving x below y in G1 we still obtain y
as a winner. The new profile R” looks as follows:
In G1 everyone’s ballot looks like (z, y, x, . . .)
In G2 everyone’s ballot looks like (y, z, x, . . .)
The rest of the voter’s ballot looks like (z, y, x, . . .)
If y would not be in the winning set anymore and we consider R’ as the profile
concerning the true preferences of a voter i ∈ G1 , it should be better to switch toR”
since then the minimum of the winning set is improved. So we have constructed a
profile R” wherefore
– {z, y} is a topset
– G2 = Ny>z
– y ∈ F (R”)
So by Lemma 2.22 it follows that yG2 z
• Suppose that x ∈ F (R). Just as we did in the preceding case, we will show that
under some modifications of this profile, x remains a winner.
If in G1 we move z below y, x remains a winner because of the same reason as
above (if not, consider the new profile as the one concerning the true ballots of
G1 . It is better to switch to R to improve the maximum.) If we now put z under
x in N \ G we still get x as a winner (if not, consider the R containing the true
preference of voters in N \ G, then there is an incentive to switch in order to
improve the minimum).
16
Due the changes we have constructed a new profile R’ wherefore
– {y, x} is x topset
– G1 = Nx>y
– x ∈ F (R’)
So by Lemma 2.22 it follows that xG1 y
Lemma 2.25. If xGz then
• xGy ∀y 6= x
• yGz ∀y 6= z
Proof. First note that the empty set can never be decisive on a pair (a, b). Suppose that
it is possible, then if ∅ ⊆ Na>b then b 6∈ F (R). And since the empty set is contained in
every set it follows that for all profiles b 6∈ F (R). But then the voting system does not
satisfy strong surjectivity which is a contradiction.
The first implication is proved by choosing G1 = G and G2 = ∅ and then applying lemma
2.24. Since the emptyset can not be decisive over any pair it therefore follows that xGy.
The second result is proved in a similar way; applying lemma refdisjoint decisive and
taking G1 = ∅ and G2 = G gives us yGz.
Lemma 2.26. If G is decisive over (x, z) then G is decisive over all pairs.
Proof. Take arbitrary a,b. We want to prove that aGb. We use Lemma 2.25 and can
distinquish four cases:
• a 6= x,b = z. Then xGz = xGb, implies aGb
• a 6= x,b 6= z. Then xGz implies xGb, implies aGb
• a = x,b 6= z. Then xGz = aGz, implies aGb
• a = x,b = z. Then xGz = aGb
Theorem 2.27. (The Duggan-Schwartz Theorem) A strategyproof SCF for three or more
candidates that is strongly surjective, has to be a dictatorship.
Proof. By Lemma 2.21 we know that this SCF satisfies down monotonicity for singleton
winners. Then because of this property and by strong surjectivity, Lemma 2.23 tells
us that this SCF satisfies the Pareto condition. Lemma 2.24 and Lemma 2.26 together
imply that if we partion N in disjoint sets N1 and N2 , that one of these coalitions is
decisive. We can repeat this argument till we get one voter i that is decisive. This means
that the topranked candidate of this voter i is the winner whenever the outcome is a
singleton.
So now we have to prove that the topranked candidate of voter i is always in the winning
set. In order to prove this we assume that the topchoice x of i is not always a winner,
which will lead us to a contradiction.
17
We now take the profile R such that x 6∈ F (R) and |F (R)| is as small as possible. Note
that the set F (R) consists of at least two candidates since once we have a singleton as a
winner this one has to be x. Suppose the winning set of this profile is {w1 , w2 , . . . , wp },
p ≥ 2. We have ordered the winning set in such a way that voter i ranks w1 over w2 over
w3 etc. Now consider a profile R’ where the ballot of i is the same as in R and the rest
of the voters have ranked {w1 , w2 , . . . , wp } above the other candidates and in the same
order (w1 over w2 over w3 etc.). If we switch from profile R to R’ the winning set doesn’t
change. Namely, suppose a candidate y is added to the winning set. Then consider R’
as the profile containing the true ballot of a voter j. If j gives the indigineous ballot as
in R he improves his minimum, since the candidate y is not in the winning set anymore.
Since we have chosen F (R) as the smallest winning set that does not contain x, no
candidate can be taken out of the winning set when we move to R’.
If we now take R’ and let voter i change his ballot by only putting w1 as the top ranked
candidate, then {w1 } becomes a topset. Because of this and the fact that this SCF is
strongly surjective it follows by Lemma 2.19 that w1 is the singleton winner. This shows
us that the SCF F is not strategyproof because if we take R as the profile concerning
the true preference of voter i then there is an incentive for him to misrepresent his true
preference by putting w1 on the top of his ballot. When he does so he improves his
minimum. So we showed that if the topchoice x of i is not always in the winning set,
the SCF is not strategyproof, which is a contradiction.
We have shown that when a SCF is strongly surjective and immune to strategic manipulation of pessimistic and optimistic voters, the winning set always contains the topchoice
of a particular voter. I will now give two examples of non-dictatorial SCF’s that are
strongly surjective and can not be manipulated by pessimistic or can not be manipulated
by optimistic voters. So in fact I give examples which show that if we only assume that
the SCF has to be immune for one type of strategic manipulation, other voting systems
than dictatorships exist.
Example 2.28. A voting rule that only satisfies strategyproofness for pessimistic voters
The winning set of a profile R under this SCF F consists of the candidates that are at
the topposition for at least two voters. Note that when we assume that we have more
voters than candidates the outcome can never be the empty set.
Consider the profile where every voter puts x at the top ranked position of their ballot.
Clearly in this profile x is the only winner. This shows that for every candidate x there
exists a profile wherefore x is the sole winner. Hereby this SCF F is strongly surjective.
We want to show that pessimistic voters cannot manipulate within this voting system:
As a pessimistic voter i you change your ballot when you can improve the minimum of
the winningset. Now suppose that F (P) = {x, . . . } and min{F (P)} = x according to
voter i0 s ballot. Now we can distinguish two cases:
18
• x is the top ranked candidate of voter i (this is only possible if x is the only winner).
In this case there is no incentive for voter i to change his ballot.
• x is not the top choice of voter i. In order to improve the minimum x has to be
removed from the winning set. But a voter is only able to eliminate a winner if
that candidate is the first one in his ballot.
Because of our earlier observation that this SCF satisfies strong surjectivity it follows by
the Duggan-Schwartz Theorem that (because in the voting system that we constructed
the topchoice of a particular candidate is not always in the winning set) this SCF F can
be manipulated by optimistic voters.
Here is an example of manipulation by an optimistic voter: Profile R looks as follows:
How many voters Preference
6
x>y>z
F (R) = {x}.
1
z>y>x
1
y>z>x
Suppose the voter that ranks z > y > x changes his ballot into y > z > x. Then
we get the following profile R’:
How many voters Preference
6
x > y > z F (R’) = {x, y}.
2
y>z>x
Clearly the maximum of the winning set is improved according to this voter who misrepresented his preference.
Example 2.29. A voting rule that only satisfies strategyproofness for optimistic voters
n
Suppose we have n voters (|N | = n) and m candidates (|X| = m). Then we compute m
and define v as follows:
n
n
+ 1 if m
is an integer
m
v=
n
n
d m e if m is not an integer
Candidate x is in the winning set of this SCF F if there are less then v voters who
ranked x as the worst candidate. Note that this SCF F does not have the emptyset as
a possible outcome and every alternative is viable by the construction of v.
First of all I want to show that the winning set never equals the empty set. Because if
so, this means that no candidate is a winner and therefore every candidate has at least
v voters that ranked this candidate as the least one in their ballot. But this implies that
n
there are v × m voters. And since v × m > m
× m this would mean that there are more
then n voters, which is contradictory to the fact that we have n voters. This shows that
the empty set is not among the possible winning sets.
Now I want to prove that this SCF F is strongly surjective. So I want to show that for
every candidate x there exists a profile R such that F (R) = {x}. Now take the profile
R wherein less than v voters rank x as the least favored candidate. Then we want that
19
each of the other m − 1 candidates are ranked as the least favored by at least v voters.
n
This is definitily reached when we assume that m ≤ m
+ 1. Because if so
n
n
(m − 1)(˙ + 1) = n + m −
−1
m
m
≤n
This shows us that there is enough space to put the candidates other than x at bottom
of ballots in order to rule these candidates out of the winning set. Hereby it follows that
n
+ 1, which is satisfied if the set of voters is much bigger (more then the square
if m ≤ m
of the number of candidates), the SCF F is strongly surjective.
We want to show that optimistic voters cannot manipulate within this voting system:
As an optimistic voter i you change your ballot when you can improve the maximum of
the winningset. Now suppose that F (P) = {x, . . . } and max{F (P)} = x according to
voter i’s ballot. Now we can distinguish two cases:
• Candidate x is the top choice voter i. In this case there is no incentive for voter i
to change his ballot.
• Canidate x is not the top choice of voter i. In order to improve the maximum
voter i wants to add a candidate y in the winning set. This y is ranked above x
in the linear ballot of voter i. But since it is only possible to add a candidate to
the winningset by removing the candidate that is at the bottom of your ballot, it
is not possible to make y a winner.
Since, in this SCF F , the top choice of a particular candidate is not always among the
winners and because F satisfies strong surjectivity, it follows that F is not strategyproof
(Duggan-Schwartz Theorem). We have proved that F is not manipulable by optimistic
voters so therefore pessimistic voters can misrepresent their preference in order to get
a better outcome. Although we know this for sure by the Duggan-Schwartz Theorem I
will give an example of manipulation by an pessimistic voter: Profile R looks as follows:
How many voters Preference
5
x>y>z
F (R) = {x, y}.
3
x>z>y
2
y>z>x
Consider a voter that ranks x > y > z. Then the minimum of the winning set according to his preference is y. Suppose that this voter changes his ballot into x > z > y.
Then we get the following profile R’:
How many voters Preference
4
x>y>z
F (R’) = {x}
4
x>z>y
2
y>z>x
This shows us that a voter i can improve his minimum by representing a disingenous
preference, which tells us that the SCF F can not cope with pessimistic voters.
20
In this section we proved the Duggan-Schwartz Theorem. This theorem tells us that if
we consider a strategyproof voting system that is strongly surjective for three or more
candidates, it has to be dictatorship. Dictatorship in the irresolute context means that
the top choice of a particular candidate must always be in the winning set. However
this is a useful result, it is not true that we can assume that a SCF is strategyproof if
it always returns the top choice of a particular candidate as a winner. So if we want
to know whether a SCF is strategyproof or not, we first have to check if the top choice
of a candidate is always among the set of winners. If this is not the case, it can never
be a strategyproof voting system. If it is true, we still have to prove that the system is
strategyproof.
Example 2.30. A strategyproof votingrule
In Section 2.1, I mentioned the Pareto condition. We can also see this as a rule: A
candidate x is Pareto dominated if there exists a candidate y such that y is ranked
above x by all voters. This SCF returns all candidates as winners if they are not Pareto
dominated. So if there does not exists a candidate y such that y is ranked above x by
all voters. This can also be stated as follows: x ∈ F (R) ⇐⇒ ∀y ∃i such that he ranks
x above y.
I will explain that this SCF is strategyproof according to Duggan and Schwartz. First
I am going to show that this SCF can not be manipulated by an optimistic voter. We
observe the winning set V and define x as the maximum of this winning set according
to the preference of voter i. Since the top ranked position can never be dominated by
another candidate it follows that the maximum of the winning set according to voter
i, wil allways be the top ranked candidate of this voter i. Therefore the maximum can
never be improved.
Now I want to show that this SCF cannot be manipulated by a pessimistic voter. Again
we look at the winning set V and now define x as the minimum of this winning set
according to the preference of voter i. We distinguish two cases:
• x is the top ranked candidate of voter i. Then the outcome cannot be improved.
• x is not the topranked candidate of voter i. If this voter i wants to improve his
minimum, he has to get rid of the candidate x that is in the winning set. In order
to put x out of the winning set, it has to be dominated by another candidate y due
to manipulation of the preference of i. So that means that in the profile according
the true preference of i, y is above x in the preference ballots of all candidates
except of voter i. If voter i manipulates his preference by placing y above x, x is
dominated and not in the winning set anymore. We can distinguish two cases:
21
– Candidate y is not dominated by another candidate in the profile containing
the true preference of i. This means that y is in the winning set V . Since y
is ranked below x by voter i and we assumed that x was the minimum of V ,
we have a contradiction.
– Candidate y is dominated by another candidate in the profile containing the
true preference of i. If this candidate is again dominated by another candidate
then choose the candidate that is not dominated and dominates y. We call
this candidate z and this candidate z is contained in the winning set because
it is not dominated. Since we assumed that x was the minimum of the winning
set, z is ranked above x, by voter i. By transitivity it follows that z is also
ranked above y, in the ballot of i. But then x is dominated by z and therefore
not in the winning set. This is a contradiction.
22
2.4 Strategyproofness in the non-resolute context,
Brandt and Brill
In the preceding sections we have proved all theorems with the assumption that the
preferences of the voters are presented as lineor orders. In the end, these theorems
could be extended to versions where candidates could be equally liked by voters. Brandt
and Brill do not make use of ballots as linear orders [2]. Instead of representing the
preferences of a voter i in one vector they use a preference relation Ri ⊆ X × X,
whereby people can equally like candidates.
If (x, y) and (y, x) are in Ri then x is equally liked as y by voter i.
If (x, y) in Ri and (y, x) not in Ri then x is ranked above y by voter i.
If (y, x) in Ri and (x, y) not in Ri then y is ranked above x by voter i.
If as in the first example, x and y are equally liked by voter i, we say that i is indifferent
about x and y and write (x, y) ∈ I ⊆ R. When a voter prefers x over y we notate:
(x, y) ∈ P ⊆ R and when he ranks y over x, (y, x) ∈ P ⊆ R. As you can see the order
of the candidates in a pair is essential to know their relation. Ri |{x,y} ⊆ Ri is the subset
of the preference relation that contains x and y. So Ri |{x,y} gives us information about
the relation between x and y.
In order to understand the Gibbard-Satterthwaite Theorem and the Duggan-Schwartz
Theorem we have used different articles with varying symbols and definitions. Nevertheless, we succeeded up to now, in constructing a common notation. But since Brandt
and Brill represent someone’s preferences in a relation, and this representation is crucial
for the construction of definitions and proofs, we have to redefine some concepts.
As a first a profile R consists now of the preference relations R of all the voters. So from
now on a profile is not a vector of ballots anymore but a set of preference relations. Note
that transitivity does not always hold in this form of representing someones preferences.
(x,y)
Ri
is the same profile as R except that voter i strengthens x with respect to y.
(x,y)
So Ri
= (R−i , Ri0 ) where

Ri0 = Ri

0
R0 = Ri \ {(y, x)}
Ri =
 0i
Ri = Ri ∪ {(x, y)}
if x is ranked above y in Ri
if x is equally liked as y in Ri
if x is ranked beneath y in in Ri
Also when Ri is a transitive preference, Ri0 doesn’t satisfy transitivity in almost all cases.
23
Definition 2.31. Weak monotonicity
A winner remains a winning candidate if a voter changes his preference relation R by
strengthening this winner and leaving the relation between the other candidates unchanged.
(x,y)
∀i, x ∈ F (R) implies x ∈ F (Ri
) ∀y ∈ X
Definition 2.32. Down monotonicity
A winner stays in the winning set if a voter changes his preference relation R by weakening another candidate and leaving the relation between the other candidates unchanged.
(y,z)
∀i, x ∈ F (R) implies x ∈ F (Ri
) ∀y ∈ X and z 6= x
Definition 2.33. Set-monotonicity
The winning set remains the same if a voter changes his preference relation Ri by weakening a losing candidate and leaving the relation between the other alternatives unchanged.
(y,z)
∀i, F (R) = F (Ri
)∀y ∈ X, ∀z ∈ X \ F (R)
Definition 2.34. Set-independence
The winning set remains the same if a voter changes his preference relation Ri by weakening a losing candidate with respect to another losing candidate and leaving the relation
between other alternatives unchanged.
(y,z)
∀i, F (R) = F (Ri
)∀y, z ∈ X \ F (R)
In the following lemmas the relation between the previous definitions will become clear.
Lemma 2.35. Set-monotonicity implies weak monotonicity and set-independence
Proof. That set-monotonicity implies set-independence follows directly from the definitions. Now we will prove that set-monotonicity implies weak-monotonicity. So suppose
(x,y)
x ∈ F (R). We have to show that ∀i, x ∈ F (Ri ) ∀y ∈ X.
(x,y)
If y 6∈ F (R), then by set-monotonicity it follows that ∀i, x ∈ F (Ri
(x,y)
)
(x,y)
If y ∈ F (R), then suppose x 6∈ F (Ri ). Now if we change profile Ri
into R then
x is weakened w.r.t. y. Because we assumed that x is a losing candidate when
(x,y)
(x,y)
the profile is Ri , it follows by set-monotonicity that F (R) = F (Ri ). Then
x 6∈ F (R) which is a contradiction. So therefore also when y ∈ F (R) it follows
(x,y)
that x ∈ F (Ri )
24
Lemma 2.36. Down monotonicity and set-independence imply set-monotonicity
(x,y)
Proof. We have to prove that F (R) = F (Ri )∀x ∈ X and y ∈ X \ F (R). Down
(x,y)
monotonicity implies that F (R) ⊆ F (Ri ). Now suppose there exists a candidate x0
(x,y)
such that x0 ∈ F (Ri ) and x0 6∈ F (R). Since down monotonicity is satisfied we know
that if we weaken another candidate than a winner,this winner remains in the winning
(x,y)
set. Therefore since, x0 ∈ F (Ri ) \ F (R) there has to be a candidate y 0 such that if
(x,y)
(x,y)
(x0 ,y 0 )
we transform Ri
into R, y 0 is strenghtened w.r.t. x0 . So therefore Ri
= Ri
(x,y)
and x = x0 and y = y 0 . By assumption x0 ∈ F (Ri ) \ F (R) so x 6∈ F (R). And since
we started with the assumption that y ∈ X \ F (R) it follows by set-independence that
(x,y)
F (R) = F (Ri ). But x 6∈ F (R), so we have a contradiction and thereby the lemma is
proven.
By this time we always take the whole set of candidates X as the set where voters can
determine their preference relation over. It is also possible to give the voters a smaller
set than X and let them choose between a smaller number of candidates. For the next
definition we have to take this set from choosable candidates, into account. From now
on, we will write down F (R, A). Where F is the SCF, A is the feasible set of candidates
we are looking at and R is the profile that contains the preference over the candidates
in A. Note that a feasible set can never be the empty set.
Definition 2.37. Strong superset property (SSP)
A SCF satisfies the strong superset property if ∀A, B and profile R,
F (R, A) ⊆ B ⊆ A implies F (R, A) = F (R, B)
In words the strong superset property says that the winning set does not change whenever
we remove losing candidates from the feasible set. Up to now we took every time the
whole set of candidates X as the feasible set. So therefore we can not compare this new
definition with other concepts we have previously made.
Lemma 2.38. Weak monotonicity and SSP imply set-monotonicity.
Proof. We assume that the SCF F is weakly monotone and satisfies SSP. We will now
(y,x)
proof that for a feasible set A, F (R, A) = F (Ri , A), x, y ∈ X \ F (R). As a first we
(y,x)
will show that x 6∈ F (Ri , A).
(y,x)
Suppose that x ∈ F (Ri , A), than to get to profile R, x is strengthned and than it
should follow by weak monotonicity that x ∈ F (R). Since this is a contradiction it
(y,x)
follows that x 6∈ F (Ri , A).
(y,x)
(y,x)
Then SSP implies that F (R, A) = F (R, A \ {x}) and F (Ri , A) = F (Ri , A \ {x})
(y,x)
and because profile Ri
only differs from R in the way where x is putted down,
(y,x)
the profiles R and Ri
are identical if we only look at them within the feasible set
(y,x)
A \ {x}. So therefore F (Ri , A \ {x}) = F (R, A \ {x}). Consequently F (R, A) =
(y,x)
F (Ri , A).
25
Since Brandt and Brill use preference relations instead of linear ballots, it is possible
that someone equally likes candidates. The authors distinguish two sorts of preferences
over sets; the R-variant and the P-variant (the strict version of the previous one).
Preference
Set A is R-preferred over set B by voter i ⇐⇒ a ≥ b ∀a ∈ A, ∀b ∈ B according to the
preference relation Ri . We can also simply write down AR̂i B.
Set A is P-preferred over set B by voter i ⇐⇒ a ≥ b ∀a ∈ A ∀b ∈ B and a > b
for one pair (a, b) with a ∈ A, b ∈ B according to the preference relation Ri . We can
also simply write down AP̂i B.
Strategyproofness
From the above it follows that a SCF is called R-strategyproof by Brandt and Brill if
there is no voter who can misrepresent his true preference relation in order to get a
winning set that is R-preferred over the winning set of the profile containing the true
preference relation. Note that these two winning sets have to differ from each other in
order to prefer one over the other.
Brandt en Brill introduce a new concept according to strategyproofness. When a group
of voters G can misrepresent their preference relation in order to get a better outcome,
a SCF F is called group-strategyproof.
Definition 2.39. Group-strategyproofness
In a more formal way it says that a SCF F is R-group-strategyproof if for no group
G ⊆ N there exists a profile R (which consists of the true preferences of group G) and
a profile R’ that can contain disingenuous preference relations Ri for i ∈ G, such that
F (R) 6= F (R’) and F (R’) is R-preferred over F (R) ∀i ∈ G.
In the coming lemmas and theorems we will use this new notion; R-group-strategyproofness.
Note that when a SCF F is R-group-strategyproof, there is certainly no voter that can
misrepresent their preference relation in order to get a better outcome. So when a voting system is R-group-strategyproof it implies that it is R-strategyproof in general (for
single voters).
In the paper of Brandt and Brill emerged that a class of voting systems that does
not satisfy R-group-strategyproofness can only be manipulated by voters who are eleminating pairs of candidates from the set I. So such a SCF can only fail to be R-groupstrategyproof when it can be manipulated by breaking ties. Therefore, Brandt and Brill
introduced a subdivision of manipulability; weak- and strong manipubility.
26
Definition 2.40. Weakly manipulability
Voters i ∈ G obtain a better outcome by misrepresenting their true preferences by
breaking ties. In formal notation:
∃G ⊆ N such that there exist profiles R and R’ with Ii0 ⊂ Ii ∀i ∈ G and Ri = Ri0
∀i 6∈ G, for which holds that F(R’) is R-preferred over F(R) by all the voters in G.
Definition 2.41. Strongly manipulability
Voters i ∈ G obtain a better outcome by misrepresenting their true preferences by
misrepresenting their strict preferences. In formal notation:
∃G ⊆ N such that there exist profiles R and R’ with Ii ⊆ Ii0 ∀i ∈ G and Ri = Ri0
∀i 6∈ G, for which holds that F(R’) is R-preferred over F(R) by all the voters in G.
Definition 2.42. Weakly R-group-strategyproofness
A SCF is R-group-strategyproof if it is not strongly manipulable by any group of voters
G ⊆ N.
Note that when we only consider strict preferences, weakly R-group-strategyproofness
and R-group-strategyproofness are exactly the same concepts. Since the indifference
relation I is the empty set, the extra requirement falls out.
Theorem 2.43. Every SCF that satisfies set-monotonicity is weakly R-group-strategyproof.
Proof. Consider a SCF F that satisfies set-monotonicity but is not weakly R-strategyproof.
We will show that this leads to a contradiction. Since F is strongly manipulable by a
group G, there exist profiles R and R’ with Ri = Ri0 ∀i 6∈ G and Ii ⊆ Ii0 ∀i ∈ G,
such that F (R’) is R-preferred over F (R) ∀i ∈ G. We take the the profiles R and R’
that coincide as much as possible. So their symetric difference R4R’ is minimal. Note
that this difference can not be the empty set, because then R and R’ would be exactly
the same. Since this SCF is strongly manipulable there exists a voter i and candidates
x, y ∈ X, such that (x, y) ∈ Pi and (x, y) ∈ Ii0 . So (y, x) is now in the preference relation
Ri0 . In the following part I want to show that such candidates x and y do not exist,
whereby we have proved that no profile R exists that can be strongly manipulated.
Now consider the following profiles:
We can distinguish three cases:
• x ∈ F (R) and y ∈ F (R’). Since F(R’) is R-preferred over F(R) by all voters in
G, it follows that (y, x) ∈ Ri ∀i ∈ G. This is a contradiction to the assumption
that (x, y) ∈ Pi .
(y,x)
• x 6∈ F (R). Since x is a losing candidate under profile R and in profile Ri
candidate x is weakened with respect to y by voter i, it follows from setmono(y,x)
(y,x)
tonicity that F (R) = F (Ri ). So therefore F (R’) is R-preferred over Ri
by
(y,x)
voter i and since |Ri 4R’| < |R4R’| we have found a contradiction (seeing the
assumption that the symmetric difference of R and R’ was minimal).
27
• y 6∈ F (R’). By the same argument as in the preceding case it follows by set(x,y)
monotonicity that F (R’) = F (R0 i ), whereby we have founded a smaller counterexample.
Now we know that every set-monotonic SCF is weakly strategyproof it will be useful to
see a voting system that is set-monotonic and therefore weakly strategyproof.
Example 2.44. Top Cycle
To obtain the winning set according to this rule we first have to define a new relation
RM ⊆ X × X. (x, y) ∈ RM ⇐⇒ |{i ∈ N |(x, y) ∈ Ri }| ≥ |{i ∈ N |(y, x) ∈ Ri }|.
Now we take the transitive closure R∗ M of RM : (x, y) ∈ R∗ M ⇐⇒ ∃k ∈ N and
a1 , a2 . . . , ak ∈ X where a1 = x and ak = y such that (ai , ai+1 ) ∈ RM . Now the winning
set = {x ∈ X|(x, y) ∈ R∗ M ∀y ∈ X}.
This SCF is set-monotone because the winning set does not change if you weaken a
(y,z)
losing candidate. We have to prove that F (R) = F (Ri )∀y ∈ X, z ∈ X \ F (R). Con(y,z)
sider F (Ri ) with y ∈ X and z ∈ X \ F (R). Since in this profile a losing candidate
is weakened, the candidates who are ranked first do not change. Therefore, the winning
(y,z)
set of Ri ) is the same as the winning set of R.
By Theorem 2.43 it follows that the Top Cycle is a weakly R-strategyproof social choice
function and therefore not not manipulable by voters who misrepresent their strict preferences. I will now show that the Top Cycle is not strategyproof in general because it
can be manipulated by breaking ties.
Suppose there are three candidates and two of them equally like x,y and z. The other
candidate equally likes x and y but ranks z beneath the other two candidates. The majority relation consists of all the possible pairs (x, y),(y, x),(y, z),(z, y),(x, z) and (z, x).
Then clearly x,y, and z are all winners. Now if one voter i who equally likes all the candidates misrepresents his preference by putting z beneath the other candidates. Then
(x, y),(y, x),(y, z) and (x, z) are still in the majority relation, but (z, y) and (z, x) are
removed from it. The transitive closure will be exactly the same as the majority relation itself, whereby the winning set consists of x and y. Since {x, y} is R-preferred
over {x, y, z} by voter i, this shows that the Top Cycle can be manipulated by breaking ties. Hence, the Top Cycle is not strategyproof in general but only satisfies weakly
R-strategyproofness.
Lemma 2.45. A SCF that satisfies down monotonicity and set-independence is weakly
R-group-strategyproof
Proof. This is a collorary of Lemma 2.36 and Lemma 2.43.
Lemma 2.46. A SCF that satisfies weak monotonicity and SSP is weakly R-groupstrategyproof
28
Proof. This is a collorary of Lemma 2.38 and Lemma 2.43.
So when a SCF is set-monotone it follows that this SCF is weakly R-strategyproof.
The other direction does not always hold; not every weakly R-strategyproof SCF is setmonotone. We will now introduce another property of voting systems that we can use
to define a class of SCF’s that is set-monotone under the assumption of weak strategyproofness.
Definition 2.47. Pairwise Social Choice Functions
When two profiles R and R’ differ but have the same amount of netto plurality that
prefers x over y, ∀x, y ∈ X. Then we call a SCF F pairwise if the winning set is the
same for such profiles that have the same amount of netto plurality.
∀R, R0 F (R) = F (R0 ) ⇐⇒ ∀x, y ∈ X|{i ∈ N |xPi y}| − |{i ∈ N |yPi x}| =
|{i ∈ N |xPi0 y}| − |{i ∈ N |yPi0 x}|
Theorem 2.48. Every weakly R-group-strategyproof pairwise SCF satisfies set-monotonicity.
Proof. In order to prove this, we will show that whenever a pairwise SCF is not setmonotone it can be strongly manipulated. So if the SCF F does not satisfy set(x,y)
monotonicity then there exists a voter i, preference profiles R and R’ = Ri
and
candidates x ∈ X and y 6∈ F (R), such that F (R) 6= F (R’). It follows that (y, x) ∈ Ri
(x,y)
would be exactly the same profile as
(because if this pair would not be in Ri then Ri
Ri and therefore the winningset of these two profiles could not differ).
We will now show that this SCF is strongly R-manipulable by adding one voter to the
system. Now suppose that the extra voter n + 1 is indifferent between all candidates
except x and y. We define the following preference relations:
Rn+1 = (U × U ) \ {(x, y)} ∪ Ri0 |{x,y}
0
= (U × U ) \ {(y, x)} ∪ Ri |{x,y}
Rn+1
Consider the following preference profiles:
Q = R1 , R2 , . . . , Ri−1 , Ri ∪ {(x, y)}, Ri+1 , . . . , Rn , Rn+1 )
0
0
Q’ = R10 , R20 , . . . , Ri−1
, Ri ∪ {(y, x)}, Ri+1 , . . . , Rn , Rn+1
)
Note that voter i is now indifferent between candidate x and y and therefore the
only difference between the profiles Q and Q’ lies in the behaviour of the n + 1’th
voter. By the pairwise property of this SCF it follows that F (R) = F (Q) = V and
F (R’) = F (Q’) = W
Remind that y 6∈ F (R) = V . Since (y, x) ∈ Ri , we can distinguish two cases:
• (y, x) ∈ Pi . We know that voter n + 1 is indifferent to all pairs except {x, y} in
profile Q. So since y 6∈ V the maximum of this set is not y. So therefore the
minimum of W is at least the maximum of V and therefore W is R-preferred over
29
V by the n+1’th voter. So therefore it is possible for voter n + 1 to move from
0
in order to get a better outcome.
preference relation Rn+1 to Rn+1
• (y, x) ∈ Ii . Then voter n+1 is indifferent to all pairs except {x, y} in profile Q.
0
Since (x, y) ∈ Rn+1
it follows that V is R’-preferred over W by voter n + 1. this
0
to Rn+1 in order to get a
F can be strongly manipulated by moving from Rn+1
better outcome.
So a pairwise F that does not satisfy set-monotonicity can be strongly manipulated.
Therefore it follows that every weakly R-group-strategyproof pairwise SCF satisfies setmonotonicity.
Lemma 2.49. Every weakly R-group-strategyproof pairwise SCF satisfies set-independence
and weak monotonicity
Proof. This is a collorary of Theorem 2.48 and Lemma 2.35.
30
2.5 Systematic Comparative Review
This section is about the different frameworks that are used by Duggan and Schwartz and
Brandt and Brill. I will set out where these frameworks and their embedded definitions,
like preferences over sets and strategyproofness, differ from one another. Once their is a
full understanding of where these concepts differ, I will make some assumptions whereby
it is possible to compare the definitions of preferences over sets and strategyproofness.
Once this comparison has been made it will become clear why the articles of Taylor and
Brand and Brill have different keyresults.
Frameworks
The Duggan-Schwartz Theorem states that a social choice function for three or more
candidates, that is strategyproof and satisfies strong surjectivity must be dictatorship.
In Section 2.3 we have proved this theorem for strict linear orders. So we assumed that
the preferences where transitive and candidates could not be equally liked by the individuals. Althought we have only discussed the strict preferences in order to prove the
theorem, this proof can easily be extended for the non-strict case [8].
Brandt and Brill showed that every set-monotonic function is weakly R-strategyproof
and every pairwise weakly strategyproof SCF is set-monotonic. They proved these statements while they have used a preference relation as the representation of the preference
of an individual. Since they have used preference relations in their framework it was
possible for individuals to equally like candidates. The first theorem is proved for general preferences. They state that this theorem holds for transitive and intransitive
preferences[1]. However, in my opinion, this does not seem to be evident. I will now
explain why I do not totally agree with that:
Brandt and Brill make use of the concept ‘set-monotonicity’. This means that a winning
set is invariant under the weakening of a losing candidate by one voter. I want to point
out that this property only makes sense if we consider non-transitive systems. Take for
(x,y)
is only transitive in two
example a transitive preference profile R. Then clearly R0 i
situations: x and y are ranked next under each other and do not have any other candidates that are ranked equal to them, or if they are equally prefered and without other
(x,y)
candidates best or worst ranked by the voter i. So in almost all cases R0 i
is not a
transitive preference profile. This shows that we cannot really talk about set-monotone,
transitive voting systems. Let alone prove that a transitive, set-monotone function is
strategyproof.
The second theorem however is proved for intransitive preferences. Brandt and Brill
state in their article that the second theorem also holds for transitive preferences, but
much more work is required to prove that part.
31
Preferences over sets & Strategyproofness
Duggan & Schwartz
Duggan and Schwartz distinguished two sorts of extreme types of voters; optimistic and
pessimistic voters. Preference over sets depends on the type of person you are:
A set is preferred by an optimistic voter i if the maximum of the set is improved according to his true preference Ri .
A set is preferred by a pessimistic voter i if the minimum of the set is improved according
to his true preference Ri .
According to the article of Taylor, a SCF is strategyproof when there is no voter who
can improve the maximum or minimum of the winning set by manipulating his ballot.
Brandt & Brill
Brandt and Brill work with the same concept of whether a voting system is strategyproof
or not; when none of the voters can manipulate their vote in order to get a better result.
The difference with the article of Taylor lies in the notion of preferences over winning sets.
Set A is R-preferred over set B by voter i ⇐⇒ a ≥ b ∀a ∈ A, ∀b ∈ B according to the
preference relation Ri . We can also simply write down AR̂i B.
Set A is P-preferred over set B by voter i ⇐⇒ a ≥ b ∀a ∈ A ∀b ∈ B and a > b
for one pair (a, b) with a ∈ A, b ∈ B according to the preference relation Ri . We can
also simply write down AP̂i B.
In order to compare the different concepts by Taylor and Brandt and Brill of whether
a winning set is preferred, I will now explain in words what it means when a set is Ror P-preferred over another set. Note that since transitivity does not always hold in a
preference relation, there are situations in which it is impossible to compare two winning
sets with each other in order to say which one is R-preferred.
AR̂i B: the minimum of A is at least the maximum of B according to the preference
relation Ri . The maximum is therefore unchanged or improved.
AP̂i B: the minimum of A is at least the maximum of B and the maximum of A is bigger
then the maximum of B according to the preference relation Ri .
From the above it follows that a SCF is called R-strategyproof by Brandt and Brill
if there is no voter who can change the minimum, by misrepresenting his preference relation, in such a way that this becomes at least the maximum of the preceding winning
set. So a SCF is not R-strategyproof if there exists a voter who can misrepresent his
true preference relation in order to get an outcome that he equally likes or prefers. Note
that whether a SCF is R-strategyproof, we can also say that none of the voters can
R-manipulate this SCF.
32
Recapitulatory, a pessimistic voter prefers a set where the worst ranked candidate in
the winning set is improved and the optimistic voter prefers the set that has the best
ranked candidate. So if for example a winner has to be picked out of the winning set, the
pessimistic voter believes that the worst case scenario will happen: the lowest ranked
candidate will be picked, and the optimistic voter believes that the best canidate will
be picked. I want to point out that this is of course a way to divide all voters into
two groups, but this distinction is quite rough. For instance, consider someone who is
neither a pessimist nor an optimist, but gives equal weigth to all candidates that are in
the winning set. Now consider the following preference ballot of voter i:
A > B > C > D > ....... > V > W > Y > Z
When voter i gives his true preference as his vote, the winning set will be:
Set 1 = {B, C, D, E, Z}
When voter i gives is a disingenous ballot, the following winning set is obtained:
Set 2 = {A, V, W, X, Y }
Since min{Set 2} > min{Set 1} and max{Set 2} > max{Set 1} according to the preference ballot of voter i, Set 2 is preferred over Set 1 if voter i was a pessimistic or an
optimistic voter. However if voter i considers every winner and gives equal weight to
all the candidates in the winning set, than Set 2 is not preferred over Set 1, since Set 2
contains a lot more high ranked candidates in the winning set than Set 1.
This example shows that although a strategyproof SCF by Duggan and Schwartz, does
not allow manipulation by optimistic or pessimistic voters, there are plausible alternatives for defining whether someone prefers a set over another.
Brandt and Brill do also not make use of the principle where the voters give equal weight
to all canidates that are in the winning set. Instead, they assume that a voter only Rprefers a set A over a set B, when every candidate in A is better than or equally ranked
over, every candidate in B. If for example a winner has to be picked out of the winning
set then the voter is unaware of this mechanism.
In the following part I will take a closer look at the concept of strategyproofness by
Brandt and Brill. In their article they call R-strategyproofness of a SCF quite a strong
requirement since none of the manipulable voters has to be strictly better off in the new
preference profile [1]. What they want to make clear, is that it in the resolute context it
can be hard to find a SCF that matches up to a voter that prefers a winner he obtained
by manipulating his vote, to an equally ranked candidate he attained by giving his true
preference.
Suppose for example we have a SCF F and there exists a voter i, who equally likes candidates x and y. Consider the preference profiles R which contains the true preference of
voter i and R’ = (R−i , Ri0 ) the untruthful preference relation. Suppose that F (R) = {x}
and F (R’) = {y}. Because yRi x, it follows that this SCF F is R-manipulable and therefore not strategyproof by Brandt and Brill. However the preceding statement does not
say that this SCF F is manipulable by the definition of Taylor, since in that case the
33
winner corresponding to profile R’ has to be strictly better than the winner of profile
R, according to Ri .
Also an example in the irresolute context:
Suppose for example we have a SCF F and there exists a voter i and preference profiles R
wherein voter i equally likes x and y. R’ = (R−i , Ri0 ), F (R) = {x, y} and F (R’) = {x}.
This example shows that the SCF F is manipulable acorrding to Brandt and Brill, since
the min{F (R’)} = min{F (R)}.
So when winning sets consist of candidates that are equally liked by a voter, this voter
can prefer the winning set that is obtained by the manipulated vote. Therefore I agree
with Brandt and Brill that strategyproofness in their article can be a strong requirement.
But this only holds very occasionaly: In the resolute context and in a few cases in the
irresolute context.
Now we have seen the difference between the definitions of preferences over winning sets
and strategyproofness given in the two papers, we will now consider the two concepts
in the same environment whereby it is possible to really compare the different concepts.
From now on we will consider a framework where the preferences are presented as linear
orders. So for the following lemmas we assume that transitivity is satisfied and the
candidates in the preference ballots are related to each other in a strict way.
From now, I do not have to make the distinction between R and P-strategyproofness
anymore, snce R- and P-strategyproofness only differ from each other when we consider
profiles where it is possible to equally prefer candidates.
Lemma 2.50. In the resolute context, strategyproofness by Brandt and Brill and strategyproofness by Duggan and Schwartz are the same.
Proof. Brandt and Brill call a SCF strategyproof if the outcome can not be improved
in such a way that the minimum becomes at least the maximum of the preceding set.
Since we only consider outcomes where there is a sole winner, we can just say that a SCF
is strategyproof when none of the voters can improve his outcome by manipulating his
preference. So this is exactly the same as the definition of strategyproofness by Duggan
and Schwartz.
Since we now only consider strict transitive preferences, the condition to be strategyproof according to Brandt and Brill is weaker then the one by Duggan and Schwartz.
This relation is stated in the following lemma:
Lemma 2.51. Strategyproofness by Duggan and Schwartz implies strategyproofness by
Brandt and Brill.
Proof. If we can find a strategyproof SCF by Duggan and Schwartz then no voter can
improve his maximum or minimum by manipulating his preference. So certainly no
voter can misrepresent his preference in order to improve his maximum and improve
his minimum in such a way that it becomes at least the maximum of the preceding
34
winning set. So if a SCF is strategyproof according to Duggan and Schwartz, it is also
R-strategyproof by Brandt and Brill.
The previous lemma and proof can be presented in an easy way:
Strategyproof by Duggan and Schwartz → strategyproof for optimistic voter =
not manipulable by optimistic voter → P-strategyproof by Brandt and Brill
Because the SCF F in Example 2.29 is strategyproof for optimistic voters it is not
possible to improve the maximum of the winning set by someone who manipulates his
vote. This SCF F is therefore also strategyproof by Brandt and Brill. Since the SCF is
not strategyproof by Duggan and Schwartz, this shows that the implication in the other
direction of Lemma 2.51 does not hold.
By now, we have seen that when preferences are represented as linear orders, strategyproofness by Duggan and Schwartz implies strategyproofness by Brandt and Brill. If
we consider the environment where people can equally like candidates but still have to
give their preferences in a transitive way, we have to distinguish P- and R-strategyproofness
from each other. Lemma 2.51 still holds in this context when we substitute P-strategyproofness.
So if we find a strategyproof voting system by Duggan and Schwartz, it is in this context
also P-strategyproof. I will now give the same sort of example as 2.29, but now wherein
people can equally like candidates and represent their preference in a transitive way. So
a SCF that is not manipulable by an optimistic voter and therefore P-strategyproof, but
can be manipulated by a pessimistic voter.
We consider the following SCF F :

X if there is no candidate that is top ranked by all voters

{x} if there is one candidate x that is top ranked by all voters
F (R) =

{x, y, ..} if there are candidates x,y, .. that are top ranked by all voters
First I will show that this SCF satisfies strategyproofness by Brandt and Brill. The
winning set will always contain the best ranked candidate(s) from each voter:
• If there is no candidate that is top ranked by all voters, than all candidates are
contained in the winning set. And therefore also the top ranked candidate of each
voter.
• If the winning set is smaller than X, everybody has these winners on the top ranked
position of their ballot.
Since the best possible result is already in the winning set, the maximum can not be
improved. This SCF F is therefore not manipulable by an optimistic voter according to
the definition of Taylor. It follows by the previous lemma that F is P-strategyproof by
Brandt and Brill.
I will now show that F is not strategyproof for pessimistic voters:
Consider a profile R in which voter i ranks x over y and all the other candidates are
35
ranked below x and y by voter i. Everybody else rates y as the best candidate. Then
F (R) = X. Now suppose voter i change his preference relation in which he strenghtens
y with respect to x. Then y is moved above x in the preference ballot wherefore y
becomes the top ranked candidate by all voters. In the new profile y is the sole winner.
Because the minimum of the winningset of someone is improved by the manipulation
of his preference, this SCF is manipulable by a pessimistic voter. This shows that the
implication in the other direction of Lemma 2.51 does not hold.
Finally, I want to show that this SCF does not satisy R-strategyproofness. Consider
the profile R where everybody rhas x and y at the top position of their ballot. Then the
winning set is {x,y}. If one voter changes his ballot by removing y from the top of his
ballot, x becomes the sole winner. Since {x}R̂i {x, y}, this example shows that someone
can misrepresent his preference in order to get an outcome that is R-preferred. Hence,
F is not R-strategyproof.
36
3 Conclusion
In this thesis we have placed the articles of Taylor and Brandt and Brill next to each other
in order to analyze where the different keyresults of their articles are based on. It became
clear that the authors give different meaning to the concept strategyproofness. This
difference in concepts stems from the varying definitions of whether a set is preferred over
another one. Duggan and Schwartz distinguish two types of voters, who prefer a set over
another one when the minimum or the maximum is improved. They call a voting system
strategyproof if it can not be manipulated by pessimistic or optimisitic voters. On the
other hand, Brandt and Brill say that a set is R-preferred over another if the minimum
is improved in such a way that it becomes at least the maximum of the preceding set.
The strict version of this kind of preference over sets, is called P-preference: additionally
to the previous definition the maximum has to be improved. Brandt and Brill refer to
R-strategyproof social choice functions as voting systems that are not manipulable by
voters who R-prefer winning sets that can be obtained by a manipulated vote. When we
consider voting systems where preferences are transitive, strategyproofness by Duggan
and Schwartz implies P-strategyproofness by Brandt and Brill. P-strategyproofness can
therefore be interpreted as a weak property. R-strategyproofness, on the other hand,
can in some cases be seen as a strong requirement: R-strategyproofness is strong in the
sense that due to manipulation, also none of the voters can obtain exclusively equal
ranked candidates with the winning set concerning the truthfull vote. Since Duggan
and Schwartz and Brandt and Brill have defined strategyproofness in a different way it
is not surprising that their statements about the existence of classes of strategyproof
voting systems differ from each other.
37
4 Popular Summary
‘Social Choice Theory’ is een tak van Mathematische Economie die zich bezig houdt
met de mechanismen rondom stemprocedures. Als een groep mensen een gezamenlijke beslissing moet nemen kunnen stemsystemen gebruikt worden om op grond van de
voorkeuren van de groepsleden een collectief besluit te nemen. In deze scriptie gaan we
uit van een groep stemmers die tot een keuze moet komen over een groep verkiesbare
kandidaten. Stemsystemen zijn functies die uit deze waarderingen, een of meerdere winnaars selecteren. Deze functies noemen we ook wel ‘social choice functions’ (SCF’s).
De bekendste stemprocedure geeft als winnaar de kandidaat die de meeste stemmen
krijgt. Wanneer dit stemsysteem wordt gebruikt is alleen de eerste voorkeur van de
kiezers van belang. In andere stemprocedures is ook de tweede en derde keus, of soms
de waardering van alle kandidaten, van betekenis. De stem van een kiezer kan op verschillende manieren gepresenteerd worden. De waardering van en tussen de kandidaten
kan bijvoorbeeld vastgelegd worden als een rangorde van kandidaten in de volgorde van
waardering.
Uit strategisch oogpunt is het dan mogelijk dat iemand zijn werkelijke waardering van
de kandidaten niet uitbrengt als stem. Dit is denkbaar wanneer middels een valse stem,
die niet de daadwerkelijke waardering van een stemmer bevat, een beter resultaat te
behalen is. Neem bijvoorbeeld de situatie waarin 3 kandidaten zich verkiesbaar stellen.
De waardering van stemmer s is als volgt: A > B > C. Dit houdt in dat stemmer
s, kandidaat A het hoogst waardeert en kandidaat C het laagst. De stemprocedure is
‘meeste stemmen gelden’ wat inhoudt dat de kandidaat of kandidaten die de meeste
eerste-stemmen krijgen, wint of winnen. We nemen nu aan dat B en C de meeste
eerste stemmen hebben en daarom strijden voor de winst. Als B en C precies evenveel
eerste stemmen hebben is het voor stemmer s gunstiger om zijn stem te manipuleren
in B > C > A. Als s dit doet zal B de meeste eerste stemmen hebben waardoor B de
winnaar wordt. Doordat stemmer s zijn echte waardering heeft gemanipuleerd heeft hij
dus een voordeligere uitkomst gekregen.
Het is wenselijk om stemsystemen te gebruiken waarbij mensen altijd hun werkelijke
waardering als stem uitbrengen en niet de mogelijkheid hebben om een betere uitslag
te krijgen wanneer zij hun stem manipuleren. Deze scriptie gaat over het bestaan en de
voorwaarden van zulke strategyproof stemsystemen (stemsystemen die opgewassen zijn
tegen de manipulatie van kiezers).
De Gibbard-Satterthwaite Theorem en de Duggan-Schwartz Theorem zijn twee stellingen die zeggen dat elk strategyproof stemsysteem een dictatorschap moet zijn. Dicta-
38
torschap is een SCF waarbij de hoogst gewaardeerde kandidaat van één stemmer altijd
een winnaar is. Dit betekent dat als deze stemmer, kandidaat A als hoogst waardeert
dat A sowieso één van de winnaars is. Mocht B als hoogst door hem gewaardeerd zijn,
dan is B per definitie één van de winnaars. De Duggan-Schwartz Theorem kan gezien
worden als een uitbreiding van de Gibbard-Satterthwaite Theorem aangezien de laatst
genoemde stelling alleen geldt voor stemsystemen waarbij er maar één winnaar is. De
Duggan-Schwartz Theorem daarentegen laat ook zien dat een strategyproof SCF, waar
meerdere winnaars mogelijk zijn, altijd dictatorschap is.
In 2011 publiceerde Brandt en Brill een artikel waarin zij bewezen dat er naast dictatorschap nog een andere klasse aan strategyproof SCFs bestaat. Dit lijkt te conflicteren
met de eerder genoemde stellingen. Het doel van deze scriptie was om te onderzoeken
waar het verschil tussen deze stellingen op berust.
Al snel werd duidelijk dat Duggan en Schwartz en Brandt en Brill de stemmen op een
andere manier presenteerden. Hierdoor is het kader waarbinnen de stellingen zijn geconstrueerd totaal verschillend. Daarnaast kwam naar voren dat er een verschil bestaat
tussen de betekenis die wordt gegeven aan het al dan niet strategyproof zijn van een
stemsysteem. Dit verschil stoelt op het feit dat Duggan en Schwartz en Brandt en Brill,
een ander criterium hanteren om te bepalen wanneer een uitkomst als beter wordt ervaren.
Wanneer een stemsysteem altijd maar één winnaar kent is het duidelijk welke uitslag
een stemmer de voorkeur geeft. Stel dat stemmer s de volgende waardering heeft:
A > B > C, dan is het duidelijk dat de meest gunstige uitslag kandidaat A is en
kandidaat C de meest onvoordelige uitkomst is. Wanneer we te maken hebben met
een stemsysteem waar meerdere winnaars mogelijk zijn, kunnen verschillende methodes
worden gebruikt om te bepalen wanneer een uitslag wordt verkozen boven een andere.
Zoals al werd gezegd zijn de kaders waarbinnen de stellingen geconstrueerd zijn verschillend van elkaar. Hierdoor was het niet direct mogelijk om de verschillende definities
van strategyproofness naast elkaar te leggen en zo met elkaar te vergelijken. Nadat ik een
frame heb geconstrueerd waarin beide definities van strategyproofness betekenis hebben,
kon ik deze met elkaar vergelijken. In de laatste paragraaf heb ik dan ook kunnen laten
zien hoe de verschillende soorten strategyproofness zich tot elkaar verhouden.
Het is dus gebleken dat de Duggan-Schwartz Theorem en de stelling van Brandt en
Brill, niet met elkaar in strijd zijn omdat ze beide in een andere context zijn opgesteld.
Aangezien Duggan en Schwartz en Brandt en Brill het begrip strategyproofness op een
andere manier gedefinieerd hebben is het niet verwonderlijk dat hun uitspraken over het
bestaan van klassen aan strategyproof SCFs, van elkaar verschillen.
39
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