Evaluating Resistance to False- Name Manipulations in Elections - - PowerPoint PPT Presentation

Evaluating Resistance to False- Name Manipulations in Elections

Vincent Conitzer Lirong Xia Bo Waggoner

March 2012 1

Thanks to Hossein Azari and Giorgos Zervas for helpful discussions!

Outline

March 2012 2

• Background and motivation: Why study

elections in which we expect false-name votes?

• Our model
• How to select a false-name-limiting method?
• How to evaluate the election outcome?
• Recap and future work

Motivating Challenge:

Poll customers about a potential product

March 2012 3

Preventing strategic behavior

Deter or hinder misreporting

• Restricted settings (e.g., single-peaked

preferences)

• Use computational complexity

March 2012 4

• False-name-proof voting mechanisms?
• Extremely negative result for voting [C., WINE’08]
• Restricting to single-peaked preferences does not

help much [Todo, Iwasaki, Yokoo, AAMAS’11]

• Assume creating additional identifiers comes at a cost [Wagman & C., AAAI’08]
• Verify some of the identities [C., TARK’07]
• Use social network structure [C., Immorlica, Letchford, Munagala, Wagman, WINE’10]

Overview article [C., Yokoo, AIMag 2010] Common factor: false-name-proof

March 2012 6

False-name manipulation

Let’s at least put up some obstacles

March 2012 7

140.247.232.88 jmhzdszx@sharklasers.com

Issues:

• 1. Some still vote multiple times
• 2. Some don’t vote at all

March 2012 8

Approach

Suppose we can experimentally determine how many identities voters tend to use for each method.

March 2012 8

140.247.232.88 jmhzdszx@sharklasers.com

20 40 60 80 1 2 3 4 5

% of people

20 40 60 80 1 2 3 4 5

# of votes

20 40 60 80 1 2 3 4 5

Outline

March 2012 9

• Background and motivation: Why study

elections in which we expect false-name votes?

• Our model
• How to select a false-name-limiting method?
• How to evaluate the election outcome?
• Recap and future work

Model

• For each false-name-limiting method, take the

individual vote distribution 𝜌 as given

• Suppose votes are drawn i.i.d.

March 2012 10

# of votes

0.2 0.4 0.6 0.8 1 2 3 4 5

Probability

• Single-peaked preferences (here: two alternatives)

March 2012 11

Model

March 2012 11

𝝆𝑵

False- name- limiting method

Su Supporters 𝒐𝑩 𝒐𝑪 Votes Cast 𝑾𝑩 𝑾𝑪 Observed 𝒘 𝑩 𝒘 𝑪

Outline

March 2012 12

• Background and motivation: Why study

elections in which we expect false-name votes?

• Our model
• How to select a false-name-limiting method?
• How to evaluate the election outcome?
• Recap and future work

Example

• Is the choice always obvious?

March 2012 13

20 40 60 80 1 2 3 4 5

percent of eligible voters

Votes cast

Actual (in-person)

20 40 60 80 1 2 3 4 1000

percent of eligible voters

Votes cast

Hypothetical (online)

• Individual vote distribution for 2010 U.S.

midterm Congressional elections:

Problem statement

March 2012 14

voters 𝑜𝐵 > 𝑜𝐶 Pr[correct | 𝜌1] Pr[correct | 𝜌2]

> ?

𝝆𝟐 𝝆𝟑

(Pr[correct ] = Pr[𝑊

𝐵 > 𝑊 𝐶])

Our results

• Setting: sequence of growing supporter

profiles (𝑜𝐵, 𝑜𝐶) where:

March 2012 15

• We show: which of 𝜌1 and 𝜌2 is preferable as

elections grow large

• 1. 𝑜𝐵 − 𝑜𝐶 ∈ 𝑃( 𝑜) (elections are “close”)
• 2. 𝑜𝐵 − 𝑜𝐶 ∈ 𝜕 1 (but not “dead even”)

Selecting a false-name-limiting method

March 2012 16

Theorem 1.

Suppose

𝜈1 𝜏1 > 𝜈2 𝜏2 . Then eventually

Pr[correct |𝜌1] > Pr[correct |𝜌2].

“For large enough elections, the ratio of mean to standard deviation is all that matters.”

Selecting a false-name-limiting method

Intuition.

• Distributions approach Gaussians

March 2012 17

𝜈2 𝜏2 𝜈1 𝜏1

• Pr[correct] = Pr[𝑊

𝐵 > 𝑊 𝐶] = Pr[𝑊 𝐵 - 𝑊 𝐶 > 0]

approaches Φ

𝜈 𝜏 𝑜𝐵−𝑜𝐶 𝑜

.

Question 1 Recap

March 2012 18

voters 𝑜𝐵 > 𝑜𝐶

𝝆𝟑

𝝂𝟑 𝝉𝟑

𝝆𝟐

𝝂𝟐 𝝉𝟐

• Takeaway: choose highest ratio!
• Inspiration for new methods?

Outline

March 2012 19

• Background and motivation: Why study

elections in which we expect false-name votes?

• Our model
• How to select a false-name-limiting method?
• How to evaluate the election outcome?
• Recap and future work

Analyzing election results

• Observe votes 𝑤

𝐵 > 𝑤 𝐶

• One approach: Bayesian

Requires a prior, which may be

• costly/impossible to obtain
• biased or open to manipulation
• Our approach: statistical hypothesis testing

March 2012 20

Pr[𝑜𝐵, 𝑜𝐶] (𝑤 𝐵, 𝑤 𝐶) Pr[𝑜𝐵, 𝑜𝐶 | 𝑤 𝐵, 𝑤 𝐶]

Prior Evidence Posterior

𝜸

“test statistic”

Statistical hypothesis testing

March 2012 21

Observed 𝒘 𝑩 > 𝒘 𝑪

𝝆𝑵

Conclusion 𝒐𝑩 > 𝒐𝑪 Pr[𝜸 ≥ 𝜸 ]

𝝆𝑵

Null ll hypothesis 𝒐𝑩 = 𝒐𝑪

“p-value”

𝜸

Statistical hypothesis testing

March 2012 22

Observed

𝝆𝑵

Conclusion 𝒐𝑩 > 𝒐𝑪 p-value Pr[𝜸 > 𝜸 ]

𝝆𝑵

Null ll hypothesis 𝒐𝑩 = 𝒐𝑪

p-value > .05

• bserved is not unlikely

under null hypothesis “accept” null p-value < .05

• bserved is unlikely

under null hypothesis reject null

Complication

Null hypothesis: 𝑜𝐵 = 𝑜𝐶 = 1, 2, 3, 4, ⋯ We can compute a p-value for each one.

March 2012 23

p-value 𝒐𝑩

Reject (max-p < R)

p-value 𝒐𝑩

Accept (min-p > R)

p-value 𝒐𝑩

Unclear

March 2012 24

Procedure:

March 2012 24

• 1. Select significance level R (e.g. 0.05).
• 2. Observe votes 𝑤

𝐵 > 𝑤 𝐶 .

• 3. Compute 𝛾

.

• 4. If max

𝑜𝐵=𝑜𝐶 𝑞-value < R, reject.

• 5. If min

𝑜𝐵=𝑜𝐶 𝑞-value > R, don’t reject.

• 6. Else, inconclusive whether to reject or not.

Our statistical test

Example and picking a test statistic

Supporters 𝝆𝑵 Observed 𝑜𝐵 (?) 92 = 𝑤 𝐵 𝑜𝐶 (?) 80 = 𝑤 𝐶 𝛾(𝑤 𝐵, 𝑤 𝐶) = ?

March 2012 25

False-name- limiting method M

Selecting a test statistic

Difference rule: 𝛾 = 𝑤 𝐵 − 𝑤 𝐶 = 12

March 2012 26

Percent rule: 𝛾 = 𝑤

𝐵−𝑤 𝐶 𝑤

≈ 0.07 General form: 𝛾 = 𝑤

𝐵−𝑤 𝐶 𝑤 𝛽

=

12 172𝛽

Observed: 𝑤 𝐵 = 92, 𝑤 𝐶 = 80.

(Adjusted margin of victory)

Test statistics that fail

March 2012 27

Theorem 2.

Let the adjusted margin of victory be 𝛾 =

𝒘 𝑩−𝒘 𝑪 𝒘 𝛽 .

Then 1. For any 𝛽 < 0.5, max-p = ½: we can never be sure to reject. (Type 2 errors) 2. For any 𝛽 > 0.5, min-p = 0: we can never be sure to “accept”. (Type 1 errors)

Test statistics for an election

March 2012 28

p-value

The “right” test statistic

March 2012 29

Theorem 3.

Let the adjusted margin of victory formula be 𝛾 =

𝑤 𝐵−𝑤 𝐶 𝒘 0.5 .

Then 1. For a large enough 𝛾 , we will reject. (Declare the outcome “correct”.) 2. For a small enough 𝛾 , we will not reject. (Declare the outcome “inconclusive”.)

Test statistics for an election

March 2012 30

p-value

We can usually tell whether to reject or not

March 2012 31

March 2012 32

• 1. Select significance level R (e.g. 0.05).
• 2. Observe votes 𝑤

𝐵 > 𝑤 𝐶 .

• 3. Compute 𝛾

= 𝑤

𝐵−𝑤 𝐶 𝒘 0.5 .

• 4. If max

𝑜𝐵=𝑜𝐶 𝑞-value < R, reject: high confidence.

• 5. If min

𝑜𝐵=𝑜𝐶 𝑞-value > R, don’t: low confidence.

• 6. Else, inconclusive whether to reject or not.

(rare!)

Use this test!

Outline

March 2012 33

• Background and motivation: Why study

elections in which we expect false-name votes?

• Our model
• How to select a false-name-limiting method?
• How to evaluate the election outcome?
• Recap and future work

March 2012 34

• How to select a false-name-limiting method?

A: Pick the method with the highest 𝜈

𝜏 .

• How to evaluate the election outcome?

A: Statistical significance test with 𝛾 = 𝑤

𝐵−𝑤 𝐶 𝑤0.5

using max p-value and min p-value.

Summary

• Model: take 𝜌 as given, draw votes i.i.d.

Future Work

• Single-peaked preferences (done)
• Application to real-world problems
• Other models or weaker assumptions
• How to actually produce distributions 𝜌?

– Experimentally – Model agents and utilities

March 2012 35

Thanks!