Piotr Faliszewski AGH U niversity of Science and Technology, Krakow
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Transcript of Piotr Faliszewski AGH U niversity of Science and Technology, Krakow
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The Shield that Never Was:Societies with Single-Peaked Preferences are More Open to Manipulation and Control
Piotr FaliszewskiAGH University of Science and
Technology, Krakow
Jörg RotheHeinrich-Heine-Universität
Düsseldorf
Lane A. HemaspaandraUniversity of Rochester
Edith HemaspaandraRochester Institute of
Technology
Moscow, SCW 2010
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Outline Introduction
Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation
Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences
NP-Hardness Shields: Removing them Leaving them in Place Erecting them
A Dichotomy Result for 3-Candidate Scoring Protocols
A Sample Proof Sketch
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CFSC Project Participants
Principal Investigators: Felix Brandt (München) Ulle Endriss (Amsterdam) Jeffrey Rosenschein (Jerusalem) Jörg Rothe (Düsseldorf) Remzi Sanver (Instanbul)
Associated Partners: Vincent Conitzer (Duke University) Edith Elkind (Singapore/Southampton) Edith Hemaspaandra (Rochester) Lane Hemaspaandra (Rochester) Jerome Lang (Paris/Toulouse) Jean-François Laslier (Paris) Nicolas Maudet (Paris)
AI TCS
AI LOG
AI
TCS
ECON
AI ECON
TCS
TCS LOG
TCS
AI LOG
ECON
AI
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What did the Düsseldorf Group do in 2009?
This is Nadja Betzler from Jena,not Magnus Roos from D’dorf.
ClaudiaDoro Gábor
Jörg
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What did the Düsseldorf Group do in 2009?
Magnus
Jörg
Gábor Frank
Doro Claudia
Düsseldorf
Felix
Ulle
Jeff
Piotr
Remzi
Edith H.Lane
Vince Edith E.
Jérôme Yann
NicolasJean-François The Shield that Never Was: Societies
with Single-Peaked Preferences are MoreOpen to Manipulation and Control.
TARK’09
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Introduction Computational Social Choice
Applications in AI Multiagent systems Multicriteria decision making Meta search-engines Planning
Applications in social choice theory and political science Computational barrier to prevent cheating in elections
Manipulation Control Bribery
Computational agents can systematically
analyze an election to find the optimal
behavior.
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Introduction
Computational agents can systematically
analyze an election to find the optimal
behavior.
Using the power of NP-hardness, vulcans have created complexity shields to
protect elections against many types of manipulation and procedural control.
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Introduction
Computational agents can systematically
analyze an election to find the optimal
behavior.
Using the power of NP-hardness, vulcans have created complexity shields to
protect elections against many types of manipulation and procedural control.
Our Main Theme: Complexity shields may
evaporate in single-peaked societies
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Elections An election is a pair (C,V) with
candidate set C = {c1, ..., cm}:
and a list of votes V = (v1, ..., vn):
Each vote vi is represented via its preferences over C: Either linear orders:
> > > >
Or approval vectors: (1,1,0,0,1)
An election system aggregates the preferences and outputs the set of winners.
Hi v7, I hope you are not one of those awful people who
support Mr. Smith!
Hi, my name is v7.
How will they aggregate our
votes?!
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Election Systems Approval (any number of candidates): Every
vote is an approval vector from All candidates with the most points are winners. Example:
C1,0
v1 1 1 0 0 1
v2 0 1 1 0 0
v3 1 1 0 0 1
v4 0 0 0 1 0
v5 1 0 0 1 1
v6 1 0 0 0 1
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Election Systems Approval (any number of candidates): Every
vote is an approval vector from All candidates with the most points are winners. Example:
C1,0
v1 1 1 0 0 1
v2 0 1 1 0 0
v3 1 1 0 0 1
v4 0 0 0 1 0
v5 1 0 0 1 1
v6 1 0 0 0 1
∑ 4 3 1 2 4
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Election Systems Approval (any number of candidates): Every
vote is an approval vector from All candidates with the most points are winners. Example:
Winners:
C1,0
v1 1 1 0 0 1
v2 0 1 1 0 0
v3 1 1 0 0 1
v4 0 0 0 1 0
v5 1 0 0 1 1
v6 1 0 0 0 1
∑ 4 3 1 2 4
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Election Systems Approval (any number of candidates): Every
vote is an approval vector from All candidates with the most points are winners. Scoring protocols for m candidates are specified
by scoring vectors with where each voter‘s i-th candidate gets points: m-candidate plurality:
m-candidate j-veto:
Borda: Plurality (any number of candidates): Veto (any number of candidates):
),...,,( 21 m m ...21
)0,...,0,1(1
m
)0,...,0,1,...,1( jjm
)0,...,2,1( mm)0,...,0,1(
)0,1,...,1(
C1,0
i
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
A voter‘s preference curve on galactic taxes
low galactic taxes high galactic taxes
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A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
A voter‘s > > > preference curve on galactic taxes
low galactic taxes high galactic taxes
Single-Peaked Preferences
Single-peaked preference consistent with linear order of candidates
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A collection V of votes is said to be single-peaked if there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
A voter‘s > > > preference curve on galactic taxes
low galactic taxes high galactic taxes
Single-Peaked Preferences
Preference that is inconsistent with linear order of candidates
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
If each vote vi in V is a linear order >i over C, this means that for each triple of candidates c, d, and e:
(c L d L e or e L d L c) implies that for each i,if c >i d then d >i e.
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
If each vote vi in V is a linear order >i over C, this means that for each triple of candidates c, d, and e:
(c L d L e or e L d L c) implies that for each i,if c >i d then d >i e.
Bartholdi & Trick (1986); Escoffier, Lang & Öztürk (2008): Given a collection V of linear orders over C, in polynomial time we can produce a linear order L witnessing V‘s single-peakedness or can determine that V is not single-peaked.
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
If each vote vi in V is an approval vector over C, this means that for each triple of candidates c, d, and e:
c L d L e implies that for each i,if vi approves of both c and e then vi approves of d.
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Single-Peaked Preferences A collection V of votes is said to be single-peaked if
there exists a linear order L over C such that each voter‘s „degree of preference“ rises to a peak and then falls (or just rises or just falls).
If each vote vi in V is an approval vector over C, this means that for each triple of candidates c, d, and e:
c L d L e implies that for each i,if vi approves of both c and e then vi approves of d.
Fulkerson & Gross (1965); Booth & Lueker (1976): Given a collection V of approval vectors over C, in polynomial time we can produce a linear order L witnessing V‘s single-peakedness or can determine that V is not single-peaked.
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Control and Manipulation The bad guy wants to make someone win (constructive)
or prevent someone from winning (destructive). The bad guy knows everybody else’s votes. In control, the chair modifies an election‘s structure by:
Adding candidates (limited/unlimited number) Deleting candidates Partition of candidates with/without runoff Adding/deleting voters Partition of voters
In manipulation, a coalition of agents change their votes to obtain their desired effect. Both nonmanipulators and manipulators are weighted. In the single-peaked case, both nonmanipulators and
manipulators are single-peaked w.r.t. the same order L. See Bartholdi, Tovey & Trick (1989; 1992), Conitzer, Sandholm
& Lang (2007), Hemaspaandra, Hemaspaandra & Rothe (2007).
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Outline Introduction
Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation
Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences
NP-Hardness Shields: Removing them Leaving them in Place Erecting them
A Dichotomy Result for 3-Candidate Scoring Protocols
A Sample Proof Sketch
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Control Results: Approval Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
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Control Results: Approval Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
For comparison: Among all types of control by adding/deleting either candidates or voters, the above two cases are the only two resistances in the general case.
(Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; Artificial Intelligence 2007)
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Control Results: Approval Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
For comparison:
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Approval Voting (general case)
constructive destructiveAdding Candidates (limited) Immune Vulnerable
Adding Candidates (unlimited) Immune Vulnerable
Deleting Candidates Vulnerable Immune
Adding Voters Resistant Vulnerable
Deleting Voters Resistant Vulnerable
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Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is
vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates
in the unique-winner and the nonunique-winner model.
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Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is
vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates
in the unique-winner and the nonunique-winner model. For comparison:
For each of these six types of candidate control plurality voting is resistant in the general case, but is vulnerable to the four types of control involving adding/deleting voters.
(Bartholdi, Tovey & Trick, 1992; Hemaspaandra, Hemaspaandra & Rothe, 2007)
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Control Results: Plurality Theorem 3: For the single-peaked case, plurality voting is
vulnerable to constructive and destructive control by adding candidates, adding an unlimited number of candidates, and deleting candidates
in the unique-winner and the nonunique-winner model. For comparison:
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Plurality (general case)
constructive destructiveAdding Candidates (limited) Resistant Resistant
Adding Candidates (unlimited) Resistant Resistant
Deleting Candidates Resistant Resistant
Adding Voters Vulnerable Vulnerable
Deleting Voters Vulnerable Vulnerable
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Outline Introduction
Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation
Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences
NP-Hardness Shields: Removing them Leaving them in Place Erecting them
A Dichotomy Result for 3-Candidate Scoring Protocols
A Sample Proof Sketch
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Manipulation: Removing NP-Hardness Shields
Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P:
The scoring protocol , i.e., 3-candidate Borda. Each of the scoring protocols , . Veto.
)0,1,2()0,...,0,1,...,1(
ji
ji
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Manipulation: Removing NP-Hardness Shields
Theorem 4: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) for each of the following election systems is in P:
The scoring protocol , i.e., 3-candidate Borda. Each of the scoring protocols , . Veto.
For comparison: 3-candidate Borda, Veto, and the „ “ cases of , , are NP-complete in the general case (and the rest is in P).
(Hemaspaandra & Hemaspaandra, 2007; Procaccia & Rosenschein, 2007; Conitzer, Sandholm & Lang, 2007).
)0,1,2()0,...,0,1,...,1(
ji
ji
12 ji )0,...,0,1,...,1( ji
ji
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Manipulation: Removing NP-Hardness Shields
Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates.
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Manipulation: Removing NP-Hardness Shields
Theorem 5: For the single-peaked case, the constructive coalition weighted manipulation problem for m-candidate 3-veto is in P for m in {3,4,6,7,8,…} and is resistant (indeed, NP-complete) for m=5 candidates.
For comparison: m-candidate 3-veto is in P for m in {3,4} and is resistant (indeed, NP-complete) for five or more candidates.
(Hemaspaandra & Hemaspaandra; Journal of Computer and System Sciences 2007).
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Manipulation: Leaving them in Place
Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol ,
i.e., 4-candidate Borda.
)0,1,3()0,1,2,3(
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Manipulation: Leaving them in Place
Theorem 6: For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) for the scoring protocol and the scoring protocol ,
i.e., 4-candidate Borda.
For comparison: These problems are known to be NP-complete also in the general case.
(Hemaspaandra & Hemaspaandra, 2007)
These results are particularly inspired by Walsh (2007) who proved the same for Single Transferable Voting.
)0,1,3()0,1,2,3(
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Manipulation: Erecting NP-Hardness Shields
Can restricting to single-peaked preferences ever erect a complexity shield?
General case
Single-peakedcase
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Manipulation: Erecting NP-Hardness Shields
Can restricting to single-peaked preferences ever erect a complexity shield?
Theorem 7: There exists an election system, whose votes are approval vectors, for which constructive size-3-coalition unweighted manipulation is in P for the general case but is NP-complete in the single-peaked case.
General case
Single-peakedcase
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Manipulation: A Dichotomy Result
Theorem 8: Consider a 3-candidate scoring protocol
For the single-peaked case, the constructive coalition weighted manipulation problem (in both the unique-winner and the nonunique-winner model) is resistant (indeed, NP-complete) when
and is in P otherwise.
),,,( 321 .321
02 3231
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Outline Introduction
Computational Foundations of Social Choice, an ESF Project Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation
Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences
NP-Hardness Shields: Removing them Leaving them in Place Erecting them
A Dichotomy Result for 3-Candidate Scoring Protocols
A Sample Proof Sketch
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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
We focus on: constructive control by adding voters in the unique-winner model for the succinct input model.
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A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is
vulnerable to constructive control by adding voters and constructive control by deleting voters,
in the unique-winner and the nonunique-winner model, for the standard and the succinct input model.
We give a poly-time algorithm that, given collections V and W of votes over candidate set C and single-
peaked w.r.t. order L, a designated candidate p in C, and an addition limit k,
decides if by adding at most k votes from W we can make p the unique winner.
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A Sample Proof Sketch
1
1
473
95
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities)
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A Sample Proof Sketch
1
1
473
95
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Which vote
types from W should we add? Especially if they are incomparable?
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A Sample Proof Sketch
1
1
473
95
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) We‘ll handle
this by a „smart greedy“
algorithm.
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A Sample Proof Sketch
1
1
473
95
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Why are F, C,
B, c, f, and j dangerous but the remaining candidates can be ignored?
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A Sample Proof Sketch
1
1
473
95
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) First, each
added vote will be an interval
including p. So drop all others.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) First, each
added vote will be an interval
including p. So drop all others.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Now, if adding
votes from W causes p to
beat c then p must also beat
a and b.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Thus, c is a
dangerous rival for p
but a and b can safely
be ignored.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Likewise, f is
dangerous but d and e
can safely be ignored.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Likewise, j is
dangerous but g, h, and i can safely be
ignored.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) Hey, why do
you do that step by step?Just say j is dangerous and ignore a, …, i.
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) No! Look what
happens if we add 6 votes of the type with multiplicity 7!
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A Sample Proof Sketch
1
1
413
2votes in W that can be added (withmultiplicities) No! Look what
happens if we add 6 votes of the type with multiplicity 7!
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A Sample Proof Sketch
1
1
473
2
number of approvals from voters in V for candidates that are
votes in W that can be added (withmultiplicities) OK, that‘s not
illogical. But how does your „smart greedy“ algorithm work?
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Smart Greedy Algorithm OK, first I need more space for that!
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Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous
rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in [p,c)
can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left
endpoint. This is a perfectly safe strategy!
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Smart Greedy Algorithm
1
1
473
2votes in W that can be added (withmultiplicities)
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Smart Greedy Algorithm
1
1
2votes in B that can be added (withmultiplicities)
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Smart Greedy Algorithm
1
0
2votes in B that can be added (withmultiplicities)
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Smart Greedy Algorithm
1
First rival
defeated
1votes in B that can be added (withmultiplicities)
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Smart Greedy Algorithm OK, first I need more space for that! In smart greedy, we eat through all dangerous
rivals to the right of p starting with the leftmost: c. To become the unique winner, p must beat c. Only votes in W whose right endpoints fall in [p,c)
can help. Let B be the set of those votes. Choose votes from B starting with the rightmost left
endpoint. This is a perfectly safe strategy! Iterate. If you run out of dangerous candidates on the right
of p, mirror image the societal order (i.e., reverse L) and finish off the remaining dangerous candidates until you either succeed or reach the addition limit.
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Summary and Future Directions Single-peakedness
removes many complexity shields against control and manipulation
leaves others in place can even erect complexity shields
When choosing election systems for single-peaked electorates, one must not rely on such shields.
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Summary and Future Directions Single-peakedness
removes many complexity shields against control and manipulation
leaves others in place can even erect complexity shields
When choosing election systems for single-peaked electorates, one must not rely on such shields.
Do such shield removals hold in two-dimensional (or k-dimensional) analogues of our unidimensional single-peakedness?
Can our results be extended to „very nearly“ single-peaked societies?
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COMSOC 2010
Third International Workshop on Computational Social ChoiceDüsseldorf, Germany, September 13–16, 2010
Important Dates•Paper submission deadline: May 15, 2010 •Notification of authors: July 1, 2010 •Camera-ready copies due: July 15, 2010 •Early registration deadline: July 15, 2010 •LogICCC Tutorial day: September 13, 2010 •Workshop dates: September 14–16, 2010
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Thank you!
Stop it!No more
questions please!
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