Multidisciplinary Workshop on Advances in Preference Handling: Papers from the AAAI-14 Workshop
Justified Representation in Approval-Based Committee Voting
Haris Aziz
Toby Walsh
NICTA and UNSW,
Sydney 2033, Australia
NICTA and UNSW,
Sydney 2033, Australia
Abstract
ity’ (Laslier and Sanver 2010). However for the case of multiple winners, the merits of AV are ‘less clear’ (Laslier and
Sanver 2010). In particular, for the multi-winner case, AV
does not address concerns such as proportional representation. For example, if 51% of the agents approve the same k
candidates, and the rest of the 49% agents another disjoint
set of k candidates, then the agents in minority do not get
any of their approved candidate selected!
We consider approval-based committee voting in which
agents approve a subset of candidates and based on the approvals, a set of winners is selected. We propose a natural axiom called justified representation that is a strengthening of unanimity. The axiom ensures that a group of agents
that is large enough should have at least one approved candidate in the winning set. The concept has some better computational, fairness, and proportionality properties than some
of the previous approaches for proportional voting using approval ballots. We show that some standard approval-based
multi-winner rules — AV (approval voting), SAV (satisfaction approval voting) and MAV (minimax approval voting)
— do not satisfy justified representation. We suggest natural modifications of MAV as well as PAV (proportional approval voting) that satisfy justified representation. We then
present a simple linear-time algorithm that satisfies justified
representation and constitutes a reasonable approval-based
rule in its own right. Finally, we show how justified representation can be used to formulate other attractive approvalbased multi-winner rules.
Over the years, multi-winner rules based on approval ballots have been introduced in the literature (Kilgour 2010).
In the case of Proportional Approval Voting (PAV ), one
reduces the weight of an approval from a particular agent
based on how many other candidates the agent approves of
have been elected. Another way to modulate the approvals is
through computing a satisfaction score for each agent based
on their ratio of approved candidates appearing in the committee to their total number of approved candidates (Satisfaction Approval Voting (SAV )). Finally, an approval-based
rule that received considerable attention in the literature is
MAV (minimax approval voting). Minimax approval voting selects the set of k candidates that minimizes the maximum Hamming distance from the submitted ballots. All
the rules informally described above have a more egalitarian objective than AV . Steven Brams the main proponent of
AV in single winner elections has argued for example that
SAV is more suitable for more equitable representation in
multiple winner elections (Brams and Kilgour 2010). Based
on their relative merits, approval-based multi-winner rules
have been examined in great detail in both economics and
computer science in recent years (Brams and Fishburn 2007;
LeGrand, Markakis, and Mehta 2007; Meir, Procaccia, and
Rosenschein 2008). The Handbook of Approval Voting discusses various approval-based multi-winner rules including
SAV , PAV and MAV (Kilgour 2010). However, there has
been limited axiomatic analysis of the rules from the perspective of representation. Approaches to generalize approval voting to the case of multiple winners each have their
own benefits and drawbacks. One way to compare the various rules in a principled manner is by an axiomatic analysis.
Introduction
The aggregation of preferences is a central problem in artificial intelligence (Conitzer 2010). In this paper we consider
the setting in which agents express preferences over candidates and the goal is to select a specified number of candidates that are deemed winning. Such multi-winner settings
cover domains such as committee selection, parliamentary
elections, the hiring of faculty members or automated agents
deciding on a set of plans (Elkind, Lang, and Saffidine 2011;
LeGrand, Markakis, and Mehta 2007).
In particular, we focus on approval-based multi-winner
rules in which each agent approves of (votes for) a subset of the candidates instead of submit a full-blown ranking over the candidates. Eliciting approvals is simple and
has less information requirements than eliciting rankings.
The most straightforward way to aggregate approvals is to
have every approval for a candidate contribute one point to
that candidate and select those candidates that get the highest number of approvals. This rule is called Approval Voting (AV ). AV has many desirable properties in the single
winner case (Brams, Kilgour, and Sanver 2006), including
its ‘simplicity, propensity to elect Condorcet winners (when
they exist), its robustness to manipulation and its monotonic-
In this paper, we formalize justified representation in
approval-based voting. We show that some prominent rules
including AV , SAV , and MAV do not satisfy justified representation. We show that MAV and PAV can be suitably modified so as to satisfy justified representation. We
20
as |Q \ T | + |T \ Q| = |Q| + |T | − 2|Q ∩ T |. Minimax
approval voting was proposed by Brams, Kilgour, and Sanver (2007). It is known that computing the outcome of MAV
is NP-hard (LeGrand, Markakis, and Mehta 2007).
then present linear-time algorithms to verify and compute
outcomes that satisfy justified representation. Finally, we
show how justified representation can be used to formulate other attractive approval-based multi-winner rules. The
area of multi-winner approval voting is related to the study
of proportional representation when selecting a committee (Procaccia, Rosenschein, and Zohar 2008). However,
the results in the proportional representation literature have
largely been negative even for approval ballots (Procaccia,
Rosenschein, and Zohar 2008). In this paper, we show that
justified representation seems to be a meaningful property
for approval-based rules: although it gives justifiably fair solutions, unlike previous approaches, it is not so constraining
as to become computationally infeasible or highly manipulable.
Observation 1 For k = 1, AV , SAV , PAV , and MAV
coincide.
Justified Representation
We first define a minimal requirement that any reasonable
approval based voting rule should satisfy: unanimity prescribes that if there exists a set of candidates that are approved by all the agents, then at least one of these candidates is selected. It is easy to identify a candidate that is
unanimously approved. Hence unanimity is computationally easy to achieve. Note that for k = 1, our definition of
unanimity coincides with definition of unanimity used for
social choice functions: if agents have strict preferences and
maximally prefer a certain alternative, then that particular
alternative should be selected. We now introduce the main
concept discussed in the paper.
Setup and voting rules
We consider the social choice setting (N, C) where N =
{1, . . . , n} is the set of agents and C = {c1 , . . . , cm }
is the set of candidates. Each agent expresses an approval ballot Ai ⊂ C that represents the subset of candidates that he approves of, yielding a set of approval ballots
A = {A1 , . . . , An }. We will consider approval-based multiwinner rules that take as input (N, C, A, k) and return the
subset W ⊆ C of size k that is the winning set (Marshall
and Kilgour 2012). 1
Definition 1 (Justified representation) An
approvalbased rule satisfies justified representation if there does not
exist a profile of ballots
and a set N ∗ of at least d|N |/ke
T
agents
such that i∈N ∗ Ai 6= ∅, and no candidate in
S
i∈N ∗ Ai is selected in the winning set.
Approval Voting (AV ) AV finds a set W
P ⊆ C of size k
that maximizes the total score App(W ) = i∈N |W ∩ Ai |.
AV has been adopted by several academic and professional
societies such as the Institute of Electrical and Electronics
Engineers (IEEE) and the International Joint Conference on
Artificial Intelligence.
The idea is that if k candidates are to be selected, then set
N ∗ that is completely unrepresented can demand that at least
one of their unanimously approved candidates should be selected. We say that N ∗ is a violation of justified representation.
Satisfaction Approval Voting (SAV ) An agent’s satisfaction is the fraction of his or her approved candidates that
are elected. SAV maximizes the sum of such scores. Formally, SAV finds a set W ⊆ C of size k that maximizes
P
i|
Sat(W ) = i∈N |W|A∩A
. The rule was proposed with the
i|
aim of representing more diverse interests than AV (Brams
and Kilgour 2010).
Theorem 1 Justified representation =⇒ unanimity.
Proof: We show that a rule that does not satisfy unanimity does not satisfy justified representation. Assume that a
rule is not unanimous. Then, all the agents approve of a
certain candidate but no such candidate is selected. Since
∗
0
|N
T | ≥ |N |/k, there exists a set N = N such that C =
A
=
6
∅,
and
no
candidate
that
has
been
approved
by
i∈N ∗ i
any agent in N ∗ has been selected. Thus justified representation is violated.
2
Proportional Approval Voting (PAV ) In PAV , an
agent’s satisfaction score is 1 + 1/2 + 1/3 + · · · + 1/j where
j is the number of his or her approved candidates that are
selected in W . Formally, PAV finds a set W P
⊆ C of size k
that maximizes the total score PAV (W ) = i∈N r(|W ∩
Pp
Ai |) where r(p) = j=1 1j . It has recently been shown that
computing PAV is NP-hard (Aziz et al. 2014).
Since justified representation is based on a highly natural idea of proportionality, it bears similarities to the Droop
proportionality criterion and different variants of the solid
coalition property (Dummett 1984; Tideman and Richardson 2000; Elkind et al. 2014). However, these proportionality notions are designed for elections for strict preferences
and focus on the election of certain candidate(s) which have
sufficient support.
Minimax Approval Voting (MAV ) MAV (minimax approval voting) returns a set W that minimizes the maximum distance between W and approvals of agents. The
distance d(Q, T ) between any two sets Q and T is defined
Justified representation of
Approval-based Rules
1
If more than one outcome is possible, then ties are broken according to some preset tie-breaking over the candidates.
We first observe that all the rules considered in the paper
satisfy unanimity.
21
Theorem 2 AV , SAV , PAV , RAV , and MAV satisfy
unanimity.
Quest for rules satisfying justified
representation
Although they satisfy unanimity, some of the prominent
approval-based multi-winner rules do not satisfy justified
representation.
We wonder whether any of rules discussed so far that do not
or may not satisfy justified representation can be modified so
as to satisfy justified representation. We propose adaptations
of MAV and PAV .
Theorem 3 AV and SAV do not satisfy justified representation.
• Modified Proportional Approval
). MaxP Voting (MPAV
0
imize score mpav(C 0 ) =
r(|A
∩
A
|)
where
i
i∈N
Pj
1
r(1) = k + 1 and r(j) = k + 1 + i=1 kj−2
for j ≥ 2.
Proof: Consider the following profile and let k = 3.
A1
A3
A5
A7
A8
A9
:
:
:
:
:
:
{a, b, d}
{a, b, d}
{a, b, d}
{c, x1 , x2 , x3 , x4 , x5 , x6 , x7 }
{c, x1 , x2 , x3 , x4 , x5 , x6 , x7 }
{c, x1 , x2 , x3 , x4 , x5 , x6 , x7 }
A2 : {a, b, d}
A4 : {a, b, d}
A6 : {a, b, d}
• k-Minimax (k-MAV ). Requires agents to only approve k
candidates and then use MAV .
Theorem 5 MPAV satisfies justified representation.
Proof Sketch: Let us consider a situation in which MPAV
violates justified representation. This happensTwhen there
exists a S
set N ∗ such that |N ∗ | ≥ |N |/k, C 0 = i∈N ∗ Ai 6=
∅, and i∈N ∗ Ai does not intersect with the winning set.
We know that |N ∗ | = c ≥ n/k. Then |N \ N ∗ | = n − c ≤
ck −c = c(k −1). Now let us look at the P
winning set B. The
∗
total MPAV score of B is as follows:
i∈N \N ∗ r(|N ∩
Ai |). We need to place the candidate in positions for each
agent in N \ N ∗ in a way so as to maximize the contributed
score of the candidates that contributed the least total score.
Such a score is maximized if each candidate in the winning
is approved by each agent in N \N ∗ . The total MPAV score
of the candidates is less than or equal to
Then {a, b, d} is selected by AV as well as SAV .
But N ∗
=
{7, 8, 9} can demand for one of
{c, x1 , x2 , x3 , x4 , x5 , x6 , x7 } to be selected.
Hence,
justified representation is violated due to N ∗ .
2
Next, we show that even an outcome of MAV may not
satisfy justified representation.
Theorem 4 An outcome of MAV may not satisfy justified
representation.
Proof: Consider the following profile and let k = 3.
A1
A2
A3
A4
A5
A6
A7
A8
A9
:
:
:
:
:
:
:
:
:
k−1
1(1 − k1
c(k − 1)(k + 1) + c(k − 1)
1 − k1
k
< c(k − 1)(k + 1) + c(k − 1)
k−1
= c(k − 1)(k + 1) + ck.
{a, b, d, x1 , x2 , x3 , x4 }
{a, b, d, x4 , x6 , x7 , x8 }
{a, b, d, x9 , x10 , x11 , x12 }
{a, b, d, x13 , x14 , x15 , x16 }
{a, b, d, x17 , x18 , x19 , x20 }
{a, b, d, x21 , x22 , x23 , x24 }
{c}
{c}
{c}
)
We want to show that the contribution of (k + 1)c of candidate x is greater than the average contribution of candidate
in B i.e,
(k + 1)c >
c(k − 1)(k + 1) + ck
k
This holds iff
Then a, b, d are selected by MAV since the maximum hamming distance of {a, b, c} is 6 where as for {a, b, d}, the
maximum hamming distance is 4. But N ∗ = {7, 8, 9} can
ask for c to be selected. Hence, justified representation is
violated.2
2
k(k + 1)c > c(k − 1)(k + 1) + ck
iff k 2 + k > (k + 1)(k − 1) + k
iff k 2 > (k + 1)(k − 1)
iff k 2 > k 2 − 1.
As for PAV , it is not clear whether it satisfies justified
representation or not. The counter examples that we constructed for other rules do not work to show that PAV does
not satisfy justified representation. We conjecture that PAV
does satisfy justified representation.
We have shown that the average contributed MPAV
T score
of candidate in B is less than the contribution x ∈ i∈N ∗ Ai
can make. Hence, there exists at least one candidate b
that has a contributed MPAV score of less than x which
means that the total MPAV score can be increased by removing
T such a candidate b and replacing it with candidate
x ∈ i∈N ∗ Ai . But this is a contradiction because B maximizes the MPAV total score.
2
2
The example in this proof shows that agents 1 to 6 manage to
get {a, b, d} selected by simply approving too many dummy candidates of the type x. This shows that MAV can in some situations
be easily manipulated.
22
sets of agents of cardinality at least n/k, justified representation can be checked in linear time.
Corollary 1 There always exists an outcome that satisfies
representation.
Next we turn our attention to k-MAV .
Theorem 7 It can be checked in linear time whether a committee satisfies justified representation or not.
Lemma 1 For k-MAV , if there is a solution in which each
agent has one approved candidate then k-MAV will select
such a solution.
0
Proof: We
Sidentify the set N ⊂ NSof unrepresented agents.
Consider i∈N 0 Ai . For each a ∈ i∈N 0 Ai , check whether
there exist at least n/k agents in N 0 that approve of a. If
there exists such an a, then the committee does not satisfy
justified representation.
2
Proof: Such a solution will lead to a maxmin score of at
most 2k − 1 whereas for a solution in which an agent is not
represented at all has a maxmin score of 2k.
2
Next, we show that a committee that satisfies justified representation can be computed in linear time as well via a
greedy algorithm as follows. We refer to the greedy algorithm as GRAV (greedy representation approval voting).
Theorem 6 At least one solution of k-MAV satisfies justified representation.
Proof Sketch: The maximum possible Hamming distance of
an agent’s ballot from the solution can be 2k (when no approved candidates of an agent get selected). If the maxmin
score of an MAV solution is less than 2k, then we know that
each agent has one approved candidate as a winner. Hence,
justified representation is trivially satisfied by a solution for
which the maximum Hamming distance is less than 2k.
Now let us consider the case when even the k-MAV solution has a maxmin value of 2k. Thus by Lemma 1, there
exists no solution for which each agent is represented. We
show that the violations can be removed. The distance of 2k
remains constant since it is upper-bounded by 2k. Assume
that there exists an S a violation of justified
representation.
T
|S| = c ≥ n/k. Thus a candidate x ∈ i∈N ∗ Ai represents
c agents. Now let us compute the average number of agents
a candidate b ∈ B represents. The maximum number of
agents that B can represent is n − c. Therefore the average
number of agents represented by candidates in B is less than
or equal to (n − c)/k. Hence, there exists at least one candidate b whose removal will lead to less than (n − c)/k agents
in N \ N ∗ not being represented anymore. But then at least
c ≥ n/k additional agents can be represented by replacing b
with x. Thus, whenever there is a violation of justified representation, it can be removed and the total number of agents
who are represented by at least one candidate in the winning set increases. The process of resolving violations by
swapping b with x can involve n/k iterations. After this we
know that each agent is represented. But in this case, justified representation is satisfied by definition since there exists
no unrepresented coalition S with a justified complaint. 2
Input: N, C, (A1 , . . . , An ), k.
Output: Committee S
0
1 k ←− 0
0
0
0
2 S ←− ∅; N ←− N ; C ←− C; V ←− V ;
0
3 while k < k do
4
Find a candidate c with the highest approval score
with respect to (N 0 , C 0 , V 0 ) and put it in S
5
k 0 ←− k 0 + 1
6
Remove all agents from N 0 who approve of c and
remove c from V 0
7
if N 0 = ∅ then
8
N 0 ←− N
9 return S
Algorithm 1: GRAV
Theorem 8 There exists a linear-time algorithm GRAV to
compute a representative committee.
Proof: Let N ∗ be the final assignment of variable set N 0
under GRAV . We claim that the S satisfies justified representation. Assume for contradiction that S does not satisfy
justified representation. This means there exists a candidate
b∈
/ S that is approved by at least n/k agents in N ∗ who are
not represented in S. Since b is unrepresented, this also implies that N 0 was never reset to N . Observe that each time a
candidate a0 is placed in S, it has approval score (restricted
to the preference profile when it was removed) that is at most
the approval score (restricted to the preference profile when
a was removed) of any candidate a that was selected in S before a0 . Since b that was not even selected in S is approved
by at least n/k agents in N ∗ , this means that each candidate
in S has approval score at least n/k in the preference profile
when it was removed. Hence, each time a candidate a is removed from C 0 , there are also at least n/k agents removed
from N 0 . But this implies that N ∗ is empty and each agent
is represented.
2
The result for k-MAV is somewhat unsatisfactory because agents are forced to approve exactly k candidates even
if they actually approve of less or more than k candidates.
Complexity of Justified representation
We note that k-MAV and MPAV , the two rules that we
showed satisfy justified representation, are both NP-hard to
compute. The fact that MPAV outcome is NP-hard to compute follows from the proof of NP-hardness for PAV in
(Aziz et al. 2014). This raises the question whether there exists a rule that is polynomial-time and satisfies justified representation. We first point out that although the naive way of
checking justified representation requires going through all
Observation 2 For k = 1, GRAV is equivalent to AV .
GRAV also has some pitfalls. If a voter is sure that a
candidate a will be selected because it has a high enough
23
Example 9 Consider the following profile and let k = 2.
approval score, it may have an incentive not to approve of it
so that it can affect the selection of some candidate after a.
Consider the following profile and let k = 2.
A1 : {a, b}
A3 : {a}
A1 : {a, c}
A3 : {a, b}
A5 : {a}
A2 : {c}
A4 : {a}
A2 : {b, c}
A4 : {a, b}
A6 : {a}
Then {a, b} and {a, c} are two committees that satisfy representation. However agent 1 can ensure {a, c} is selected
by reporting only c has his approved candidate. However, if
the rule only satisfies justified representation, agent 1 cannot
ensure the selection of {a, c}.
Then GRAV returns {a, b} as the outcome. However if
agent 1 reports only c as his approved candidate, then by
neutrality {a, b} and {a, c} have the same chance of being
selected. If ties are broken in favour of b over c, then agent
1 certainly has an incentive to only approve b.
Finally, we note that as long as each agent has an approved
candidate elected, the resultant committee also satisfies justified representation. However, if no such committee exists,
then justified representation can still help refine which committee to select.
Previous Approaches
to Approval-based Representation
Constrained approval voting
Although voting rules such as PAV have proportional representation as one of the goals (as implied by the name),
some other approaches in approval-based voting were explicitly designed for better ‘representation’. However, the
approaches are computationally demanding. Justified representation has more bite than unanimity but is still computational easy to satisfy.
There is another approval-based method that concerns a
different setting in which there are lower bound and upper bound constraints on the number of candidates that can
be elected from different categories of candidates and approval voting is implemented while taking into account these
quota constraints for categories (Brams and Pothoff 1990;
Straszak et al. 1993). Implementing such constraints again
results in NP-hardness of computing such solutions. Moreover, we are interested in the the outcome being a function
of the ballots and not on other additional side constraints.
Proportional Representation and Approval ballots
Proportional representation has been proposed by Monroe (1995) in which agents should be partitioned into k
equally-sized groups, and each group assigned to a candidate so as to minimize dissatisfaction of agents from their
associated candidates. Chamberlin and Courant (1983) used
a similar approach but did not impose the condition of equal
sized groups and compensated it by attributing weights to
groups of different sizes. Unfortunately, even with approval
ballots, finding an optimal representative committee is NPhard in both the Monroe as well as the model by Chamberlin
and Courant (Procaccia, Rosenschein, and Zohar 2008).
Discussion
Unanimity
Justified representation
+
+
+
+
+
+
+
+
+
?
+
+
+
+
+
Rule
AV
SAV
MAV
PAV
k-MAV
MPAV
GRAV
U JRAV
EJRAV
Threshold approaches
Kilgour (2010) mentioned another kind of (threshold) representation which if possible will return a winning set in which
for each agent, at least one approved candidate is selected.
We note this version of representation is overly strong and
unreasonable. If k = 2 and 99 agents prefer the committee
{a, b} and the only one agent 100 approves of {c, d}, then
threshold representation requires either c or d to be selected.
In this sense, threshold representation is overly egalitarian
and ignores the relative numbers of agents supporting different candidates.
Threshold-based representation gives too much flexibility
to agents to manipulate an outcome by only approving their
most preferred candidate that has not been selected if they
are sure that other approved candidates will be selected in
any case. On the other hand, if a rule only satisfies justified
representation, then an agent can only get a particular candidate selected if he has sufficient support from other agents.
Table 1: Satisfaction of justified representation by approvalbased voting rules.
In this paper, we initiated the axiomatic analysis of an important fairness aspect of approval-based multi-winner rules.
We identified a useful axiom called justified representation
that is a generalization of unanimity. Justified representation is also computationally easier to achieve than trying to
represent each agent.
It will be interesting to check whether other approvalbased rules satisfy justified representation. A promising rule
is PAV (Kilgour 2010). Other rules based on approval may
also be worth considering (Betzler, Slinko, and Uhlmann
2013; Skowron, Faliszewski, and Slinko 2013).
24
Using justified representation to devise new rules Justified representation can also be used to formulate new
approval-based rules. We mention a couple of rules that
seem particularly attractive:
Conitzer, V. 2010. Making decisions based on the preferences of multiple agents. Communications of the ACM
53(3):84–94.
Dummett, M. 1984. Voting Procedures. Oxford University
Press.
Elkind, E.; Faliszewski, P.; Skowron, P.; and Slinko, A.
2014. Properties of multiwinner voting rules. In Proceedings of the 13th International Conference on Autonomous
Agents and Multi-Agent Systems (AAMAS), 53–60.
Elkind, E.; Lang, J.; and Saffidine, A. 2011. Choosing collectively optimal sets of alternatives based on the condorcet
criterion. In IJCAI, 186–191.
Kilgour, D. M. 2010. Approval balloting for multi-winner
elections. In Handbook on Approval Voting. Springer. chapter 6.
Laslier, J.-F., and Sanver, M. R., eds. 2010. Handbook on
Approval Voting. Studies in Choice and Welfare. SpringerVerlag.
LeGrand, R.; Markakis, E.; and Mehta, A. 2007. Some
results on approximating the minimax solution in approval
voting. In Proceedings of the 6th International Conference
on Autonomous Agents and Multi-Agent Systems (AAMAS),
1193–1195.
Marshall, E., and Kilgour, D. M. 2012. Approval balloting
for fixed-size committees. In Electoral Systems, Studies in
Choice and Welfare. Springer. chapter 12, 305–326.
Meir, R.; Procaccia, A. D.; and Rosenschein, J. S. 2008.
A broader picture of the complexity of strategic behavior in
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Systems (AAMAS), 991–998.
Monroe, B. L. 1995. Fully proportional representation. The
American Political Science Review 89(4):925–940.
Procaccia, A. D.; Rosenschein, J. S.; and Zohar, A. 2008.
On the complexity of achieving proportional representation.
Social Choice and Welfare 30:353–362.
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proportional representation as resource allocation: Approximability results. In Proceedings of the 23rd International
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Straszak, A.; Libura, M.; Sikorski, J.; and Wagner, D. 1993.
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methods through technology: The refinement-manageability
trade-off in the single transferable vote. Public Choice
103(1-2):13–34.
• Utilitarian justified representation (UJRAV) rule: the rule
that among all committees that satisfy justified representation, returns a committee that maximizes the sum of approvals.
• Egalitarian justified representation (EJRAV) rule: the rule
that among all committees that satisfy justified representation, returns a committee that maximizes the the number
of representatives of the agent who has the least number
of representatives in the winning committee.
The computational complexity of winner determination
for these rules is an interesting problem. Finally, analyzing the compatibility of justified representation with other
important properties such as strategyproofness for dichotomous preferences is another avenue of future research.
Acknowledgements
The authors acknowledge the helpful comments of Steven
Brams.
This material is based upon work supported by the Australian Government’s Department of Broadband, Communications and the Digital Economy, the Australian Research
Council, the Asian Office of Aerospace Research and Development through grant AOARD-12405.
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