Junta Distributions and the Average-Case Complexity of Manipulating Elections

A presentation by Jeremy Clark

Ariel D. ProcacciaJeffrey S. Rosenschein

Outline

Introduction• Manipulability • Design GoalsPaper Theorems• Preliminaries• Junta Distribution• Proof of TheoremsConcluding Remarks

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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

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 GoalsPaper Theorems• Preliminaries• Junta Distribution• Proof of TheoremsConcluding 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.

• Plurality: <1, 0, … , 0, 0>• Veto: <1, 1, … , 1, 0>• Borda: <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 precisionOutput 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 m3)

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 m3)

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:• Independently randomly choose w(v) from

[0,1] (with discrete precision).• Independently randomly choose S[ci] from

[W,(m-1)W].Jeremy Clark 33

Is this Junta?

Hard? YesBalance? Authors calculate bounds using

Chernoff’s bounds Dichotomy? First discrete step is non-negligibleSymmetry? Invariant to candidatesRefinement? 2nd ranked candidate will at least

tie p

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Greedy Algorithm

1. Sort candidates from lowest score to highest2. Choose p as first choice, and rest in sorted

order3. Recalculate scores and repeat for each vote4. When finished, return true iff p has highest

score

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Example

Borda: <3,2,1,0>, n=5S[Con] = 20S[Lib] = 19S[NDP] = 17S[Gre] = 10 p

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Example

S[Con] = 20S[Lib] = 19S[NDP] = 17S[Gre] = 10

t1 : Gre<NDP<Lib<Con

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Example

S[Con] = 20 + 0 = 20S[Lib] = 19 + 1 = 20S[NDP] = 17 + 2 = 18S[Gre] = 10 + 3 = 13

t1 : Gre<NDP<Lib<Con

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Example

S[Con] = 20S[Lib] = 20S[NDP] = 18S[Gre] = 13

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ExampleS[Con] = 20, 20 , 20 , 21 , 23 , 23 S[Lib] = 19, 20 , 21 , 21 , 22 , 24S[NDP] = 17, 18 , 20 , 22 , 22 , 23S[Gre] = 10, 13 , 16 , 19 , 22 , 25

t1 : Gre<NDP<Lib<Cont2 : Gre<NDP<Lib<Cont3 : Gre<NDP<Con<Libt4 : Gre<Con<Lib<NDPt5 : 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 GoalsPaper Theorems• Preliminaries• Junta Distribution• Proof of TheoremsConcluding 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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