Selecting Voting Locations for Fun and Profit Omer Lev Zack - - PowerPoint PPT Presentation

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Selecting Voting Locations for Fun and Profit Omer Lev Zack - - PowerPoint PPT Presentation

Selecting Voting Locations for Fun and Profit Omer Lev Zack Fitzsimmons College of the Holy Cross Ben-Gurion University Worcester, MA USA Beersheba, IL MPREF 2020; August 2020 Introduction Manipulative attacks on elections are well-


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SLIDE 1

Selecting Voting Locations for Fun and Profit

MPREF 2020; August 2020

Zack Fitzsimmons

College of the Holy Cross Worcester, MA USA

Omer Lev

Ben-Gurion University Beersheba, IL

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SLIDE 2

Introduction

Manipulative attacks on elections are well- studied problems. Only recently has geographic information been used (e.g., gerrymandering [Lewenberg et al., 2017]). We consider how selecting where voters can cast their votes can be used to manipulate.

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SLIDE 3

Motivation

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SLIDE 4

Motivation

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SLIDE 5

Geographic Elections

Voters and voting locations are distributed on a metric space. Each voter has a distance-bound and casts their vote only if they are within this bound to a voting location. Each voter has preferences over a given set of candidates.

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SLIDE 6

Example

Plurality election with candidates: {🍐, 🍈}

v4 v2 v5 v1 v3

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SLIDE 7

Example

Plurality election with candidates: {🍐, 🍈}

v4 v2 v5 v1 v3

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SLIDE 8

Example

Plurality election with candidates: {🍐, 🍈}

v4 v2 v5 v1 v3

🍐 wins!

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SLIDE 9

Polling Place Control

Voters and voting locations are distributed on a metric space. Ea Each vote ter r has a dis ista tance it it is is wi willing to

  • go
  • to
  • vot

vote. For an election and a set of possible voting

  • locations. Does there exist a s

a set o

  • f at

at l leas ast k vot voting loca

  • cation
  • ns, such that a given candidate

wins?

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SLIDE 10

Two Parties: Voters on a line

Select at least 2 polling places such that 🍐 wins?

v1

x3 x6

v6

x2

v4

x2

v14

x3

v10 v12 v8

Greedy approach Margin for 🍐, L1: -1, L2: -2, L3: -3.

L1 L2

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L3

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SLIDE 11

Two Parties: Voters on a line

Select at least 2 polling places such that 🍐 wins?

v1

x3 x6

v6

x2

v4

x2

v14

x3

v10 v12 v8

Greedy approach Margin for 🍐, L1: -1, L2: -2, L3: -3. Would incorrectly return no solution!

L1 L2

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L3

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SLIDE 12

Two Parties: Voters on a line

Select at least 2 polling places such that 🍐 wins?

v1

x3 x6

v6

x2

v4

x2

v14

x3

v10 v12 v8

Greedy approach Margin for 🍐, L1: -1, L2: -2, L3: -3. Would incorrectly return no solution! 🍐 wins when choosing L2 and L3.

L1 L2

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L3

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SLIDE 13

Two Parties: Voters on a line

Select at least 2 polling places such that 🍐 wins?

v1

x3 x6

v6

x2

v4

x2

v14

x3

v10 v12 v8

Greedy approach In P using dynamic programming Margin for 🍐, L1: -1, L2: -2, L3: -3. Would incorrectly return no solution! 🍐 wins when choosing L2 and L3.

L1 L2 L3

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SLIDE 14

Two Parties: Voters on the Plane

NP-completeness result on the plane is shown by a reduction from Cubic Planar Vertex Cover [Garey and Johnson, 1977].

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SLIDE 15

Two Parties: Voters on the Plane

We show a variant with all edges on integer gridlines (of length 1 or 1.5) is NP-complete.

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SLIDE 16

Two Parties: Voters on the Plane

We show a variant with all edges on integer gridlines (of length 1 or 1.5) is NP-complete.

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SLIDE 17

Two Parties: Voters on the Plane

We show a variant with all edges on integer gridlines (of length 1 or 1.5) is NP-complete.

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SLIDE 18

Two Parties: Voters on the Plane

We show a variant with all edges on integer gridlines (of length 1 or 1.5) is NP-complete.

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SLIDE 19

Two Parties: Voters on the Plane

We show a variant with all edges on integer gridlines (of length 1 or 1.5) is NP-complete.

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SLIDE 20

Two Parties: Voters on the Plane

For each edge we construct:

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u

v1 0.375 0.375 0.375

v

v2 v4 v5 v3

Add an additional polling place > 1.5 from the constructed graph with k voters for 🍐. Ask if there exists a way to select at least #edges + 1 polling places such that 🍐 wins.

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SLIDE 21

Two Parties: Results

In P on the line using dynamic programming. NP-complete on the plane even when voters can vote at most at 3 locations with same distance-bound. In P on the plane for some natural restrictions (e.g., vote at most at 1 location).

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SLIDE 22

Two Parties: Results

In P on the line using dynamic programming. NP-complete on the plane even when voters can vote at most at 3 locations with same distance-bound. In P on the plane for some natural restrictions (e.g., vote at most at 1 location).

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Also holds for destructive cases

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SLIDE 23

Multi-Party: Results

For more than two candidates, even on the line, polling place control for plurality is NP-complete. Moreover, the optimization version of this problem is inapproximable within any constant factor.

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SLIDE 24

Other control actions

Attempting to change winner by changing voters’ distance-bound. In P for plurality by adapting the result for priced adding/deleting voter control [Miasko and Faliszewski, 2016].

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SLIDE 25

Other control actions

Attempting to change winner by changing voters’ distance-bound. In P for plurality by adapting the result for priced adding/deleting voter control [Miasko and Faliszewski, 2016].

Possible application: Buses?

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SLIDE 26

What’s next?

Complexity of polling place control where voters can vote at most at two locations. Experimental study of our polling place control problem. New models that include geographic information.

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SLIDE 27

What’s next?

Complexity of polling place control where voters can vote at most at two locations. Experimental study of our polling place control problem. New models that include geographic information.

Thank you!

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