jörg rothe

Overview of Overview of CComputationalomputationalFFoundations of oundations of SSocial ocial CChoicehoice

GASICS WorkshopAachen, October 2009

Jörg Rothe

2

Computational Social Choice What is computational social choice?

A new interdisciplinary field of study at the interface of social choice theory and computer science

What is social choice theory? Social choice theory studies the aggregation of

individual preferences Key concepts

Preference relation: typically transitive and complete Set of preference relations over a given set of

alternatives : Social welfare function Social choice function Social choice correspondence

3

Computational Social Choice

Bidirectional transfer Computer science ➠ Social choice

Apply complexity theory, algorithms, learning theory to problems of social choice

Social choice ➠ Computer science Import concepts from social choice

to solve questions arising in AI (e.g., in societies of autonomous software agents), webpage ranking, or collaborative filtering

Social Choice Theory

Computer Science

Computational Computational Social ChoiceSocial Choice

4

Game Theory Social Choice Theory

precursorsCournot (1801-1877)

Borel (1871-1956)

Llull (1232-1316)

Condorcet (1743-1794)

Borda (1733-1799)

Dodgson / Carroll (1832-1898)

early positive results

2-Player zero-sum games: security level (Minimax Theorem, v. Neumann, 1928)

Voting among 2 alternatives: majority rule (May’s Theorem, 1952)

seminal monographTheory of Games and Economic Behavior (v. Neumann & Morgenstern, 1944)

Social Choice and Individual Values (Arrow, 1951)

Various “solution concepts”

recent trend “Algorithmic Game Theory”

“Computational Social Choice”

Theorem (Arrow, 1951): There is no nondictatorial social welfare function satisfying Pareto-optimality and independence of irrelevant alternatives.

Theorem (Arrow, 1951): There is no nondictatorial social welfare function satisfying Pareto-optimality and independence of irrelevant alternatives.

I‘ll show you some solution concepts!

5

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

6

Aims & Objectives Social choice and theoretical computer science

To deepen our understanding of algorithmic and complexity-theoretic issues in social choice

Social choice and logic To develop logic-based languages for modeling and

reasoning about social choice problems and preference structures

Social choice and artificial intelligence To apply established techniques from AI, such as

preference elicitation and learning, to problems of social choice

7

The Community Where do we meet?

International Workshop on Computational Social Choice (COMSOC), coordinated by Ulle Endriss & Jérôme Lang)

1st COMSOC, Amsterdam, 6-8 December 2006 2nd COMSOC, Liverpool, 2-5 September 2008 3rd COMSOC, Düsseldorf, 13-16 September 2010

Dagstuhl Seminars Computational Issues in Social Choice, 21-26 October 2007 Computational Foundations of Social Choice, 7-12 March 2010

Where do we publish? Conferences: AAAI, AAMAS, IJCAI, SODA, TARK, WINE, ... Journals: AIJ, IC, JACM, JAIR, MSS, SCW, TCS, TOCS, ... MLQ special issue (edited by Paul Goldberg and Jörg Rothe): “Logic and Complexity within Computational Social Choice”

8

Main Topics Computational aspects of evaluating voting rules

Theorem (Bartholdi et al., 1989): There is no social welfare function that is neutral, consistent, Condorcet, and efficiently computable (unless P=NP).

Other issues: efficient algorithms, approximation, exact computational complexity, etc.

Computational hardness of manipulation Theorem (Bartholdi et al., 1989): There is a social

welfare function that is easy to compute, but not efficiently manipulable (unless P=NP).

Moreover, this function is neutral, Condorcet, Pareto-optimal, etc.

Other issues: few alternatives, weighted voting, typical-case, approximation, heuristics, other types of manipulation (control, bribery, ...), etc.

9

Main Topics (cont.) Computational aspects of fair division

How to fairly divide divisible goods or resources among several agents or players e.g., cutting a cake

Indivisible goods (multiagent resource allocation) e.g., complexity of social welfare optimization e.g., (combinatorial) auctions and mechanism design

Social choice in combinatorial domains Combinatorial structure gives rise to exponential growth

multiple referenda, committee election Representation of preferences (graphical or logical)

CP-nets, weighted propositional formulas important factors: compactness, expressiveness,

computational properties

I‘ll show you how to cut a cake!

10

Main Topics (cont.) Computational aspects of coalitional voting games

Voting settings are often modeled as cooperative games e.g., weighted voting games: compact representation

Complexity of game-theoretic solution concepts e.g., the core, the Shapley-Shubik and Banzhaf power index

Manipulation and control e.g., false identities/splitting weight, changing threshold,

adding/deleting voters

Epistemic issues in social choice Incomplete preferences Elicitation of preferences

Communication complexity

11

What did the Düsseldorf Group do in 2009?

This is Nadja Betzler from Jena,not Magnus Roos from D’dorf.

ClaudiaDoro Gábor

Jörg

12


Magnus

Jörg

Gábor Frank

DoroEdith E.

Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince

Jérôme Yann

NicolasJean-François

The Cost of Stability in Coalitional Games.Joint with Y. Bachrach, R. Meir, D. Pasechnik, M. Zuckerman.To appear at SAGT’09; extended abstract: AAMAS’09

13


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann


The Complexity of Probabilistic Lobbying.Joint with H. Fernau J. Goldsmith, N. Mattei, D. Raible.

To appear at ADT’09

14


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann


Generalized Juntas and NP-Hard Sets.Joint with H. Spakowski.

FCT’07; Theoretical Computer Science 2009

15


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Frequency of Correctness

versus Average Polynomial Time.Joint with H. Spakowski.

FCT’07; Information Processing Letters 2009

16


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Satisfiability Parsimoniously Reduces to

the Tantrix(TM) Rotation Puzzle Problem.MCU’07; Fundamenta Informaticae 2009

17


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François The Three-Color and Two-Color Tantrix(TM)

Rotation Puzzle Problems are NP-CompleteVia Parsimonious Reductions.

LATA’08; Information & Computation 2009

18


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François The Complexity of Computing Minimal

Unidirectional Covering Sets.Joint with F. Fischer and J. Hoffman.

Submitted

19


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Degrees of Guaranteed Envy-Freeness in

Finite Bounded Cake-Cutting Protocols.WINE’09

20


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Complexity of Social Welfare Optimization

in Multiagent Resource Allocation.Submitted

21


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann


Hybrid Elections Broaden Complexity-Theoretic Resistance to Control.

IJCAI’07; Mathematical Logic Quarterly 2009

22


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Llull and Copeland Voting Computationally

Resist Bribery and Constructive Control.AAAI’07; AAIM’08;

Journal of Artificial Intelligence Research 2009

23


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François Sincere-Strategy Preference-Based Approval

Voting Fully Resists Constructive Control and Broadly Resists Destructive Control.

Joint with M. Nowak.MFCS’08; Mathematical Logic Quarterly 2009

24


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann


Control Complexity in Fallback Voting.To appear at CATS’10

25


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Jérôme Yann

NicolasJean-François A Richer Understanding of the

Complexity of Election Systems.In „Fundamental Problems in Computing:

Essays in Honor of Professor Daniel J. Rosenkrantz“S. Ravi & S. Shukla, eds., Springer, 2009

26


Magnus

Jörg

Gábor Frank

Doro Claudia

Düsseldorf

Felix

Ulle

Jeff

Piotr

Remzi

Edith H.Lane

Vince Edith E.

Nicolas

Jérôme Yann

Jean-François Computational Aspects of Approval Voting.

To appear in „Handbook of Approval Voting“J.-F. Laslier & R. Sanver, eds., Springer

27


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

The Shield that Never Was:Societies with Single-Peaked Preferences are More Open to Manipulation and Control

Piotr FaliszewskiAGH University of Science and

Technology

Jörg RotheHeinrich-Heine-Universität

Düsseldorf

Lane A. HemaspaandraUniversity of Rochester

Edith HemaspaandraRochester Institute of

Technology

TARK XII, Palo Alto, USA, July 2009

Outline Introduction

Elections and Single-Peaked Preferences. Thanks, Toby! Control and Manipulation

Overview of Results Control: Single-Peakedness Removing NP-Hardness Shields Manipulation: Single-Peaked Preferences

Removing NP-Hardness Shields Leaving them in Place Erecting them Giving a Dichotomy for 3-Candidate Scoring Protocols

A Sample Proof Sketch Single-Peaked Approval Voting is Vulnerable to Constructive

Control by Adding Voters

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.

Introduction



behavior.

Using the power of NP-hardness, vulcans have created complexity shields to

protect elections against many types of manipulation and procedural control.

Introduction



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

Elections Candidates and voters:

C = {c1, ..., cm}

V = {v1, ..., vn}

Each voter vi is represented via his or her preferences over C: Linear orders: c > e > a > b > d Approval vectors: (0,1,1,0,1)

Election system aggregates these preferences and outputs the set of winners.

Hi v7, I hope you are not one of those awful

people who support c3!

Hi, my name is

v7.

How will they aggregate our

votes?!

Election Systems Approval (any number of candidates): Every

vote is an approval vector from 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):All candidates with the most points are winners.

),...,,( 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

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

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


Single-Peaked Preferences

Single-peaked preference consistent with linear order of candidates

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


Single-Peaked Preferences

Preference that is inconsistent with linear order of candidates



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.



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.



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.



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.

Theorem 1: 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.

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).


Elections and Single-Peaked Preferences Control and Manipulation



A Sample Proof Sketch Single-Peaked Approval Voting is Vulnerable to

Constructive Control by Adding Voters

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.

Control Results: Approval Theorem 2: For the single-peaked case, approval voting is



For comparison: Among all types of control by adding/deleting candidates/voters, the above two cases are the only two resistances in the general case.

(Hemaspaandra, Hemaspaandra & Rothe, AAAI’05; Artificial Intelligence 2007)

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.

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)

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


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


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.


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).

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(

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(

Manipulation: Erecting NP-Hardness Shields

Can restricting to single-peaked preferences ever erect a complexity shield?

General case

Single-peakedcase

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

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

A Sample Proof Sketch Theorem 2: For the single-peaked case, approval voting is






We focus on: constructive control by adding voters in the unique-winner model for the succinct input model.




We focus on: constructive control by adding voters in the unique-winner model for the succinct input model. We give a poly-time algorithm that, given collections V and V‘

of votes over candidate set C and single-peaked w.r.t. order L, a candidate p in C, and an addition limit k, decides if by adding at most k votes from V‘ we can make p the unique winner.

A Sample Proof Sketch

1

1

473

95

2

number of approvals from voters in V for candidates that are

votes in V‘ that can be added (withmultiplicities)


1

1

473

95

2


votes in V‘ that can be added (withmultiplicities) Which vote

types from V‘ should we add? Especially if they are incomparable?


1

1

473

95

2


votes in V‘ that can be added (withmultiplicities) We‘ll handle

this by a „smart greedy“

algorithm.


1

1

473

95

2


votes in V‘ that can be added (withmultiplicities) Why are F, C,

B, c, f, and j dangerous but the remaining candidates can be ignored?


1

1

473

95

2


votes in V‘ that can be added (withmultiplicities) First, each

added vote will be an interval

including p. So drop all others.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) First, each

added vote will be an interval

including p. So drop all others.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) Now, if adding

votes from V‘ causes p to

beat c then p must also beat

a and b.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) Thus, c is a

dangerous rival for p

but a and b can safely

be ignored.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) Likewise, f is

dangerous but d and e

can safely be ignored.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) Likewise, j is

dangerous but g, h, and i can safely be

ignored.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) Hey, why do

you do that step by step?Just say j is dangerous and ignore a, …, i.


1

1

473

2


votes in V‘ that can be added (withmultiplicities) No! Look what

happens if we add 6 votes of the type with multiplicity 7!


1

1

413

2votes in V‘ that can be added (withmultiplicities) No! Look what

happens if we add 6 votes of the type with multiplicity 7!


1

1

473

2


votes in V‘ that can be added (withmultiplicities) OK, that‘s not

illogical. But how does your „smart greedy“ algorithm work?

Smart Greedy Algorithm OK, first I need more space for that!

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 V‘ whose right endpoints fall in

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!

cp,

Smart Greedy Algorithm

1

1

473

2votes in V‘ that can be added (withmultiplicities)


1

1

2votes in B that can be added (withmultiplicities)


1

0



1

First rival

defeated


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 V‘ whose right endpoints fall in

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.

cp,

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.

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?

... and a Call for (Reading) Papers

„„Logic and Complexity within Computational Social Logic and Complexity within Computational Social Choice“Choice“

Special issue of Special issue of Mathematical Logic QuarterlyMathematical Logic Quarterly, August 2009, August 2009Edited by Paul Goldberg and Jörg RotheEdited by Paul Goldberg and Jörg Rothe

... and a Call for (Writing) Papers

Third International Workshop on Computational Social ChoiceDüsseldorf, Germany, September 13–16, 2010

Important Dates•Paper submission deadline: June 1, 2010 •Notification of authors: July 15, 2010 •Camera-ready copies due: August 1, 2010 •Early registration deadline: August 1, 2010 •Tutorial day: September 13, 2010 •Workshop dates: September 14–16, 2010

Thank you!

Stop it!No more

questions please!

jörg rothe

Documents

social choice problems

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given set of alternatives