My notations Set of voters {1,...,n} Set of m candidates {a,b,c...} Preference profile: a vector of rankings a a b b c a c b c Voter 1 Voter 2 Voter 3
Download ReportTranscript My notations Set of voters {1,...,n} Set of m candidates {a,b,c...} Preference profile: a vector of rankings a a b b c a c b c Voter 1 Voter 2 Voter 3
My notations
Set of voters {1,...,n}
Set of m candidates {a,b,c...}
Preference profile: a vector of rankings
a
a
b
b
c
a
c
b
c
Voter 1
Voter 2
Voter 3
Charles Dodgson
English author and
mathematician, better
known as Lewis Carroll
Suggested choosing a
candidate “as close as
possible” to a Condorcet
winner
Dodgson’s rule
Score of x = minimum # of exchanges
between adjacent candidates needed to
make x a Condorcet winner
Dodgson score example
a
b
c
d
e
Voter 1
e
c
d
a
b
Voter 4
b
a
c
d
e
Voter 2
b
e
d
a
c
Voter 5
e
b
c
a
d
Voter 3
P(a,b)
P(a,c)
P(a,d)
P(a,e)
3
2
3
3
4
2
3
Dodgson’s rule
Score of x = minimum # of exchanges
between adjacent candidates needed to
make x a Condorcet winner
Alternatively: total number of positions that
the voters push x
Elect candidate with minimum score
Complexity of Dodgson
DODGSON-SCORE: given candidate x, a
preference profile, and a threshold k, is the
Dodgson score of x at most k ?
[Bartholdi et al, SCW 89] DODGSON-SCORE is
NP-complete, DODGSON-WINNER is NP-hard
[Hemaspaandra et al., JACM 97] DODGSONWINNER is complete for Parallel access to NP
What can we do?
Heuristics [McCabe-Dansted et al., COMSOC
06; Homan&Hemaspaandra, MFCS 06]
Fixed parameter tractable algorithms
[Betzler et al., I&C 10]
Approximation [Cargiannis et al., SODA 09;
Caragiannis et al., EC 10]
-approximation = solution that is at most
times the Dodgson score
Greedy algorithm
Given candidate x and pref profile
y is alive if it beats x in pairwise elections,
otherwise dead
Cost-effectiveness of push = ratio between #
of live candidates overtaken and # of
positions pushed
Greedy Algorithm: while live candidates,
perform the most cost-effective push
It’s alive!
d
d
d
c
b
a
c
c
ed4
ec9
eb13
ax
b
b
e5
e10
x
e16
a
a
e6
x
e14
e17
e1
x
e11
e15
e2
e7
e12
e3
e8
x
x
Greed pays off
Theorem [Caragiannis et al., SODA
09]: The greedy alg has an approx ratio
of Hm-1
Best ratio possible in polytime
Proof relies on the dual fitting
technique [Vaz 01], inspired by
constrained set multicover
Primal solution found by algorithm
upper-bounded by infeasible dual
assignment
Divide dual assignment by Hm-1 and show
that shrunk assignment is feasible
INF
ALG
OPT
INF/Hm-1
ALG/Hm-1
Approximation algs as voting rules
Does it make sense to approximate a voting
rule?
Approximation algorithm is a new voting rule
Should satisfy desirable social choice
properties - possibly not satisfied by Dodgson!
E.g., monotonicity
Designing a polytime monotonic O(log m)approx algorithm is difficult but possible
[Caragiannis et al., EC 10]
Greedy is nonmonotonic
d
d
d
c
b
a
c
d
c
e4
e9
e13
e16
b
bc
e5
e10
e14
e17
a
b
a
e6
e11
e15
x
ea1
e7
e12
x
e2
e8
x
e3
x
x
x
Greedy is nonmonotonic
d
d
d
c
b
a
c
c
e4
e9
e13
e16
b
b
e5
e10
e14
e17
a
a
e6
e11
e15
x
e1
e7
e12
x
e2
e8
x
e3
x
x
x
Greedy is nonmonotonic
d
d
d
c
b
a
c
c
ed4
ec9
eb13
ax
b
b
e5
e10
x
e16
a
a
e6
x
e14
e17
e1
x
e11
e15
e2
e7
e12
e3
e8
x
x
Computational hardness as a
barrier to manipulation
Manipulation
Often it is in the
voters’ interest to
reveal false
preferences
Example: Borda
May lead to the
election of a socially
bad candidate
b
a
c
b
a
c
d
d
Voter 1
Voter 2
a
b
c
d
Voter 3
Gibbard-Satterthwaite Theorem
If m=2, Plurality is nonmanipulable
Let m3. The following properties are
incompatible:
1.
2.
3.
Onto: any candidate can be elected
Nondictatorship: there is no single voter who
dictates the outcome of the election
Nonmanipulability
Circumventing GibbardSatterthwaite
Mechanism Design: aligning incentives
using money
Restricting preferences
Single Peaked Preferences
A grocery store is being built. Each voter (resident of
the street) wants it as close as possible to his own
house. Need to choose a spot
Suggestion: choose the median peak
Onto and nondictatorial
This is also the Condorcet winner
The median is nonmanipulable!
Circumventing GibbardSatterthwaite
Mechanism Design: assuming money is
available and preferences are quasi-linear
Restricting preferences
Computational Complexity
Complexity of manipulation
Manipulation is always possible in theory
But can we design voting rules where it is
difficult in practice?
Are there “reasonable” voting rules where
manipulation is a hard computational
problem?
The computational problem
R-MANIPULATION
problem:
b
a
c
Given votes of
d
nonmanipulators and
a preferred candidate Voter 1
p
Can manipulator cast
vote that makes p
(uniquely) win under
R?
Example: Borda
b
a
c
d
Voter 2
a
c
d
b
Manip.
A greedy algorithm
Algorithm:
Rank p in first place
While there are
unranked candidates:
If there is a candidate
that can be placed in
next spot without
preventing p from
winning, place this
candidate.
Otherwise return
`false’.
b
a
c
b
a
c
d
d
Voter 1
Voter 2
a
c
b
d
b
b
Manip.
Example: Copeland
a
b
c
d
e
Voter 1
e
c
b
a
d
Voter 4
b
a
d
e
c
Voter 2
a
c
b
d
b
e
b
b
Manip.
e
c
b
a
d
Voter 3
a b c d e
a -
2 3 5 3
b 3 -
2 4 2
c 2 3
2 -
3 2
4
1
d 0 1
0 1 -
2
3
e 2 3
2 3 2 -
When does the alg work?
Theorem [Bartholdi et al., SCW 89]: Let R be a rule
s.t. function s(<,x) such that:
For every < chooses a candidate that maximizes s(<,x)
{y: y < x} {y: y <‘ x} s(x,<) s(x,<‘)
Then the alg always decides R-MANIPULATION
correctly
Captures:
All scoring rules, e.g., Borda
Copeland: s is number of pairwise elections x wins
Maximin: s is the worst pairwise election of x
Voting rules that are hard to
manipulate
Natural rules
Copeland with second order tie breaking [Bartholdi
et al., SCW 89]
STV [Bartholdi&Orlin, SCW 91]
Ranked Pairs [Xia et al., IJCAI 09]
Order pairwise elections by decreasing strength of
victory
Successively lock in results of pairwise elections unless it
leads to a cycle
Winner is the top ranked candidate in final order
Example: Ranked Pairs
6
12
8
10
2
4
Voting rules that are hard to
manipulate
Natural rules
Copeland with second order tie breaking [Bartholdi
et al., SCW 89]
STV [Bartholdi&Orlin, SCW 91]
Ranked Pairs [Xia et al., IJCAI 09]
Order pairwise elections by decreasing strength of
victory
Successively lock in results of pairwise elections unless it
leads to a cycle
Winner is the top ranked candidate in final order
Can also “tweak” easy to manipulate voting rules
[Conitzer&Sandholm, IJCAI 03]
Coalitional manipulation
R-UNWEIGHTED-COALITIONAL-MANIPULATION (UCM)
problem:
R-WCM: the same with weights
Given votes of nonmanipulators and a preferred
candidate p
Can k manipulators cast votes that make p (uniquely)
win under R?
Voters can be weighted by shares in a company or seats
in an assembly
WCM is NP-complete in a variety of voting rules,
even for a constant m [Conitzer et al., JACM 07]
Example: WCM in veto
We want: given the nonmanipulators’ votes
… it is NP-hard to find votes for the manipulators
to achieve their objective
Simple example: veto rule, 3 candidates
Suppose, from the given votes, p has received t-1
more vetoes than a, and t-1 more than b
The manipulators’ combined weight is 2t
The only way for p to win is if the manipulators
veto a with t weight, and b with t weight
But this is doing PARTITION NP-hard!
Hardness of UCM
R-UCM is known to be NP-complete under:
Copeland [Faliszewski et al., AAMAS 08,10]
Maximin [Xia et al., IJCAI 09]
Weird scoring rule [Xia et al., EC 10]
Even with only two manipulators!
Open problem: complexity of UCM under
Borda
Beyond worst-case hardness
Results such as NP-hardness suggest that
the runtime of any successful manipulation
algorithm grows dramatically on some
instances
But there may be algorithms that usually
solve the problem
Can we design rules where manipulable
instances are usually hard to solve?
Uh, no?
Three approaches
The “fraction of manipulators” approach
[Procaccia&Rosenschein, AAMAS 07;
Xia&Conitzer, EC 08; Walsh, IJCAI 09]
The “axiomatic” approach
[Conitzer&Sandholm, AAAI 06; Friedgut et al.,
FOCS 08; Xia&Conitzer, EC 08;
Dobzinski&Procaccia, WINE 08]
The “Window of error” approach
[Procaccia&Rosenschein, JAIR 07; Zuckerman
et al., AIJ 09; Xia et al., EC 10]
What is a “window of error”?
We will focus on WCM and UCM
We will consider algorithms that can
incorrectly decide instances
Return “false” when there is a successful
manipulation
“Window of error” = instances that are
incorrectly decided by the algorithm
Can we say something that would hold for
many distributions over instances?
Window of error illustrated
Algorithm
fails
“Yes”
instances
All
instances
The greedy algorithm revisited
Reminder: R-WCM problem
Given weighted votes of nonmanipulators and a
preferred candidate p
Can weighted manipulators cast votes that make p
(uniquely) win under R?
Greedy algorithm for WCM under scoring rules
[Procaccia&Rosenschein, JAIR 07]: each
manipulator ranks p first, and the other
candidates by inverse score
One manipulator coincides with previous alg
Example: Algorithm is correct
5
5
10
p
0
10
20
30
40
a
0
10
20
30
40
b
0
10
20
30
40
Example: Algorithm is wrong
5
10
p
0
10
20
30
40
a
0
10
20
30
40
b
0
10
20
30
40
Theorem: WCM in Borda
Theorem [Zuckerman et al., AIJ 09]:
1.
2.
If the alg returns “true” then there is a
successful manipulation
If the alg incorrectly returns “false” then it
would find a successful manipulation given an
extra manipulator with max weight
In UCM this is just one extra manipulator
Example for the theorem
5
10
10
p
0
10
20
30
40
a
0
10
20
30
40
b
0
10
20
30
40
Generalization
Zuckerman et al. design algorithms for
specific voting rules
Xia et al. [EC 10] give a more general but
weaker theorem for scoring rules, using a
connection to scheduling
Junta Distributions
Cool animation!
Procaccia and
Rosenschein [JAIR
07]: find hard
distributions over
instances
Other work
Control [Bartholdi et al., 92; Faliszewski et
al., JAIR 09]
Bribery [Faliszewski et al, JAIR 09b, Elkind
et al., SAGT 09]
Manipulation in multi-winner elections
[Meir et al., JAIR 08]
...
Getting involves in this community
Community mailing list:
https://lists.duke.edu/sympa/subscribe/comsoc
Computational Social Choice (COMSOC)
workshop
http://ccc.cs.uni-duesseldorf.de/COMSOC-2010/
A few useful overviews
Y. Chevaleyre, U. Endriss, J. Lang, and N. Maudet. A Short Introduction
to Computational Social Choice. In Proc. 33rd Conference on Current
Trends in Theory and Practice of Computer Science (SOFSEM-2007),
LNCS 4362, Springer-Verlag, 2007.
V. Conitzer. Making decisions based on the preferences of multiple
agents. Communications of the ACM, 53(3):84–94, 2010.
V. Conitzer. Comparing Multiagent Systems Research in Combinatorial
Auctions and Voting. To appear in the Annals of Mathematics and
Artificial Intelligence.
P. Faliszewski, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe. A
richer understanding of the complexity of election systems. In S. Ravi
and S. Shukla, editors, Fundamental Problems in Computing: Essays in
Honor of Professor Daniel J. Rosenkrantz, chapter 14, pages 375–406.
Springer, 2009.
P. Faliszewski and A. D. Procaccia. AI's War on Manipulation: Are We
Winning? To appear in AI Magazine.
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