Transcript Slide 1

Ariel D. Procaccia
Jeffrey S. Rosenschein
Junta Distributions and the
Average-Case Complexity of
Manipulating Elections
A presentation by
Jeremy Clark
Outline
Introduction
•
Manipulability
•
Design Goals
Paper Theorems
•
Preliminaries
•
Junta Distribution
•
Proof of Theorems
Concluding Remarks
Jeremy Clark
2
Introduction
This paper considers the computational
complexity of manipulating an election
outcome
A manipulatable election is one where the
addition of a set number of votes will change
the election outcome to a preferred outcome
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Manipulability
The ability to manipulate an election depends
on the current results (whether exactly
known or not) and the weight of the votes at
the manipulator’s disposal
Given these, we can form a decisional problem
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Manipulation can be constructive or destructive
Constructive: make a candidate win
Destructive: make a candidate lose
Constructive is equivalent to multiple destructive
manipulations: one for each candidate ahead of
your preferred candidate
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In real elections
Strategic voting (destructive)
You are a Liberal and a federalist in a Quebec
riding. Current polls have the Bloc in first,
Conservatives in second, and the Liberals
trailing far behind.
A manipulative vote: vote Conservative to
prevent the Bloc from winning
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In real (US) elections
Gerrymandering (Constructive)
You are a Democrat in charge of election zoning.
The Republicans beat you marginally in two
neighbouring districts. You restructure the
districts by packing Democratic voters in one
of the regions.
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Goal
Design a voting system such that manipulability
is impossible
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Goal
Design a voting system such that manipulability
is impossible
Gibbard-Satterthwaite Theorem: Any
deterministic, non-dictatorial voting system
contain manipulatable instances
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Goal
Design a voting system such that manipulability
is intractable
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Goal
Design a voting system such that manipulability
is intractable
Lots of interesting systems where manipulability
is NP-Hard
However is worst-time complexity the right
metric?
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Goal
Design a voting system such that manipulability
is average-case intractable
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Goal
Design a voting system such that manipulability
is average-case intractable
This paper examines average-case complexity on
manipulation problems
It proves that general classes of NP-hard
manipulation problems are polynomial in the
average-case
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Outline
Introduction
•
Manipulability
•
Design Goals
Paper Theorems
•
Preliminaries
•
Junta Distribution
•
Proof of Theorems
Concluding Remarks
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Preliminaries
Election has m candidates
Election has n+N voters: n manipulatable voters
and N non-manipulatable voters
Voters can have different weights (reduces to a
voter having multiple votes)
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Preliminaries
A vote is an ordered list of candidates that gives i
points to the ith candidate.
A scoring protocol,  = <1, …, m>, is a vector of
scores for each position where i ≥ i+1.
1. Plurality:
2. Veto:
3. Borda:
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<1, 0, … , 0, 0>
<1, 1, … , 1, 0>
<m-1, m-2, … , 2, 1, 0>
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Preliminaries
A voting protocol uses multiple contests, each
decided with a scoring protocol
For example, Exhaustive Ballot is an iterated
plurality protocol where a candidate with over
50% of the vote wins. If no candidate wins, then
the last place candidate is eliminated and the
election is rerun.
Others include Copeland, Maximin, and STV
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Sensitive Scoring Protocol
In sensitive scoring protocols, m=0 and
m-1 > m
<3,2,1,0>
<1,0,0,0>
<3,3,3,3> → <0,0,0,0>
<4,3,2,1> → <3,2,1,0>
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Manipulation Problems
Individual Manipulation (IM): Given knowledge
of all other votes, can I cast my vote for my
preferred candidate such that she wins? Note:
ties are considered losses
P-Time in most scoring protocols (can be hard in
voting protocols with unbounded candidates)
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Manipulation Problems
Coalitional-Weighted-Manipulations (CWM):
Given knowledge of all other votes, can I cast
a set of votes for my preferred candidate such
that she wins?
NP-Hard in sensitive scoring protocols with just
3 candidates. Why? You are increasing the
score of more than one candidate.
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Manipulation Problems
Score-CWM (SCWM): Given the tally of all other
candidates, can I cast a set of votes for my
preferred candidate such that she wins?
Assumptions:
Weights are linear in precision
Output is a linear (decisional)
Score determination is linear/P-time
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Junta Distribution
Hardness: instances are full-sized and hard
Balance: both yes and no instances exist
Dichotomy: instances can be impossible or have
non-negligible probability. Ignore negligible
cases
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Junta Distribution
Symmetry: instance is unbiased toward any
candidate
Refinement: Manipulation fails if all
manipulative votes are identical
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
m-1>m=0 such as Borda but not Plurality
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
Fixed number of candidates
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
p is candidate to manipulate, ci are others
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
There exists a heuristic polynomial time
algorithm A to solve decisional problem M
with a junta distribution  over set of inputs
to M
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Proposition 1
Let P be a sensitive scoring protocol. Then CWM
in P is NP-Hard (with m3)
Sketch of proof:
CWM P Partition
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Proposition 1
Partition: given a set of integers that sum to 2K,
does there exist a subset that sums to K?
Let m=3. Set n~2K. Structure N such that CWM
is true iff exactly K vote p>a>b and K vote
p>b>a. If, say, K+1 vote p>a>b and K-1 vote
p>b>a, then CWM is false.
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Corollary
Let P be a sensitive scoring protocol. Then
SCWM in P is NP-Hard (with m3)
Sketch:
If CWM is NP-Hard, then SCWM is as well as
partitioning does not depend on generating
tally from votes
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Proposition 2
Let P be a sensitive scoring protocol. Then * is
a junta distribution for SCWM in P with
C={p,c1,c2,…,cm-1} and m=O(1).
Where * is the following distribution:
1. Independently randomly choose w(v) from
[0,1] (with discrete precision).
2. Independently randomly choose S[ci] from
[W,(m-1)W].
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Is this Junta?
Hard? Yes
Balance? Authors calculate bounds using
Chernoff’s bounds
Dichotomy? First discrete step is non-negligible
Symmetry? Invariant to candidates
Refinement? 2nd ranked candidate will at least
tie p
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Greedy Algorithm
1. Sort candidates from lowest score to highest
2. Choose p as first choice, and rest in sorted
order
3. Recalculate scores and repeat for each vote
4. When finished, return true iff p has highest
score
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Example
Borda: <3,2,1,0>, n=5
S[Con]
= 20
S[Lib]
= 19
S[NDP]
= 17
S[Gre]
= 10  p
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Example
S[Con]
S[Lib]
S[NDP]
S[Gre]
= 20
= 19
= 17
= 10
t1 : Gre<NDP<Lib<Con
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Example
S[Con]
S[Lib]
S[NDP]
S[Gre]
= 20 + 0 = 20
= 19 + 1 = 20
= 17 + 2 = 18
= 10 + 3 = 13
t1 : Gre<NDP<Lib<Con
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Example
S[Con]
S[Lib]
S[NDP]
S[Gre]
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= 20
= 20
= 18
= 13
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Example
S[Con] = 20, 20 , 20 , 21 , 23 , 23
S[Lib]
= 19, 20 , 21 , 21 , 22 , 24
S[NDP] = 17, 18 , 20 , 22 , 22 , 23
S[Gre]
= 10, 13 , 16 , 19 , 22 , 25
t1 : Gre<NDP<Lib<Con
t2 : Gre<NDP<Lib<Con
t3 : Gre<NDP<Con<Lib
t4 : Gre<Con<Lib<NDP
t5 : Gre<Lib<NDP<Con
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Greedy Properties
Greedy is P-time
Greedy never issues false positives
Greedy does issue false negatives, however
these are bounded to Pr[err]1/p(n)
Therefore Greedy is deterministic heuristic
polynomial time
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Theorem
Let P be a sensitive scoring protocol. If m=O(1)
then P, with candidates C={p,c1,c2,…,cm-1}, is
susceptible to SCWM.
There exists a heuristic polynomial time
algorithm A to solve decisional problem M
with a junta distribution  over set of inputs
to M 
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Theorem 2
The paper contains a second theorem, related to
the first, regarding uncertainty about the
other votes
We are allowed to sample the distribution of the
other votes
Essentially, we try every (m+1)! orders of
candidates and sample the distribution
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Outline
Introduction
•
Manipulability
•
Design Goals
Paper Theorems
•
Preliminaries
•
Junta Distribution
•
Proof of Theorems
Concluding Remarks
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Conclusions
Complexity is best considered in the average-case, not
worst-case
Manipulation problems have been demonstrated to be
worst-case intractable and average-case tractable
This is bad news if it generalizes to any NP-Hard
manipulation problem
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There is still hope
These results are for scoring protocols. Voting protocols
may offer intractable manipulation.
Large number of candidates may increase average case
complexity (intuitively seems the case with Theorem
2: (m+1)! grows very fast)
Junta distributions may be too permissible to easy
instances
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Questions?
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Discussion
What if we make manipulability as easy as possible
and let voters adapt to voting strategically?
What happens with (non-sensitive) cardinal voting
schemes instead of ordinal ones, such as range
voting?
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