[PPT] - CSC2556 Lecture 2 Manipulation in Voting Credit for many visuals: PowerPoint Presentation

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CSC2556 Lecture 2 Manipulation in Voting

CSC2556 - Nisarg Shah 1

Credit for many visuals: Ariel D. Procaccia

SLIDE 2

Recap

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Voting

➢ 𝑜 voters, 𝑛 alternatives ➢ Each voter 𝑗 expresses a ranked preference ≻𝑗 ➢ Voting rule 𝑔

Takes as input the collection of preferences ≻
Returns a single alternative
A plethora of voting rule

➢ Plurality, Borda count, STV, Kemeny, Copeland, maximin,

…

SLIDE 3

Incentives

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Can a voting rule incentivize voters to truthfully

report their preferences?

Strategyproofness

➢ A voting rule is strategyproof if a voter cannot submit a

false preference and get a more preferred alternative (under her true preference) elected, irrespective of the preferences of other voters.

➢ Formally, a voting rule 𝑔 is strategyproof if there is no

preference profile ≻, voter 𝑗, and false preference ≻𝑗

′ s.t.

𝑔 ≻−𝑗, ≻𝑗

′

≻𝑗 𝑔 ≻

SLIDE 4

Strategyproofness

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None of the rules we saw are strategyproof!
Example: Borda Count

➢ In the true profile, 𝑐 wins ➢ Voter 3 can make 𝑏 win by pushing 𝑐 to the end

1 2 3 b b a a a b c c c d d d 1 2 3 b b a a a c c c d d d b Winner a Winner b

SLIDE 5

Borda’s Response to Critics

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Random 18th century French dude

My scheme is intended only for honest men!

SLIDE 6

Strategyproofness

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Are there any strategyproof rules?

➢ Sure

Dictatorial voting rule

➢ The winner is always the most

preferred alternative of voter 𝑗

Constant voting rule

➢ The winner is always the same

Not satisfactory (for most cases)

Dictatorship Constant function

SLIDE 7

Three Properties

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Strategyproof: Already defined. No voter has an

incentive to misreport.

Onto: Every alternative can win under some

preference profile.

Nondictatorial: There is no voter 𝑗 such that 𝑔 ≻

is always the alternative most preferred by voter 𝑗.

SLIDE 8

Gibbard-Satterthwaite

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Theorem: For 𝑛 ≥ 3, no deterministic social choice

function can be strategyproof, onto, and nondictatorial simultaneously 

Proof: We will prove this for 𝑜 = 2 voters.

➢ Step 1: Show that SP implies “strong monotonicity”

[Assignment]

➢ Strong Monotonicity (SM): If 𝑔 ≻ = 𝑏, and ≻′ is such that

∀𝑗 ∈ 𝑂, 𝑦 ∈ 𝐵: 𝑏 ≻𝑗 𝑦 ⇒ 𝑏 ≻𝑗

′ 𝑦, then 𝑔 ≻′ = 𝑏.

If 𝑏 still defeats every alternative it defeated in every vote in ≻, it

should still win.

SLIDE 9

Gibbard-Satterthwaite

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Theorem: For 𝑛 ≥ 3, no deterministic social choice

function can be strategyproof, onto, and nondictatorial simultaneously 

Proof: We will prove this for 𝑜 = 2 voters.

➢ Step 2: Show that SP+onto implies “Pareto optimality”

[Assignment]

➢ Pareto Optimality (PO): If 𝑏 ≻𝑗 𝑐 for all 𝑗 ∈ 𝑂, then

𝑔 ≻ ≠ 𝑐.

If there is a different alternative that everyone prefers, your choice

is not Pareto optimal (PO).

SLIDE 10

Gibbard-Satterthwaite

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Proof for n=2: Consider problem instance 𝐽(𝑏, 𝑐)

≻𝟐 ≻𝟑 a b b a

Say 𝑔 ≻1, ≻2 = 𝑏

≻𝟐 ≻𝟑

′

a b b a

𝑔 ≻1, ≻2

′

= 𝑏

PO: 𝑔 ≻1, ≻2

′

∈ {a, b}

SP: 𝑔 ≻1, ≻2

′

≠ 𝑐

≻𝟐

′′

≻𝟑

′′

a A N Y A N Y

𝑔 ≻′′ = 𝑏

Due to strong

monotonicity 𝐽(𝑏, 𝑐) 𝑔 ≻1, ≻2 ∈ {𝑏, 𝑐}

➢ PO

SLIDE 11

Gibbard-Satterthwaite

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Proof for n=2:

➢ If 𝑔 outputs 𝑏 on instance 𝐽(𝑏, 𝑐), voter 1 can get 𝑏

elected whenever she puts 𝑏 first.

In other words, voter 1 becomes dictatorial for 𝑏.
Denote this by 𝐸(1, 𝑏).

➢ If 𝑔 outputs 𝑐 on 𝐽(𝑏, 𝑐)

Voter 2 becomes dictatorial for 𝑐, i.e., we have 𝐸(2, 𝑐).
For every (𝑏, 𝑐), we have either 𝐸 1, 𝑏 or 𝐸 2, 𝑐 .

SLIDE 12

Gibbard-Satterthwaite

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Proof for n=2:

➢ Fix 𝑏∗ and 𝑐∗. Suppose 𝐸 1, 𝑏∗ holds. ➢ Then, we show that voter 1 is a dictator.

That is, 𝐸(1, 𝑑) holds for every 𝑑 ≠ 𝑏∗ as well.

➢ Take 𝑑 ≠ 𝑏∗. Because 𝐵 ≥ 3, there exists 𝑒 ∈ 𝐵\{𝑏∗, 𝑑}. ➢ Consider 𝐽(𝑑, 𝑒). We either have 𝐸(1, 𝑑) or 𝐸 2, 𝑒 . ➢ But 𝐸(2, 𝑒) is incompatible with 𝐸(1, 𝑏∗)

Who would win if voter 1 puts 𝑏∗ first and voter 2 puts 𝑒 first?

➢ Thus, we have 𝐸(1, 𝑑), as required. ➢ QED!

SLIDE 13

Circumventing G-S

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Restricted preferences (later in the course)

➢ Not allowing all possible preference profiles ➢ Example: single-peaked preferences

Alternatives are on a line (say 1D political spectrum)
Voters are also on the same line
Voters prefer alternatives that are closer to them
Use of money (later in the course)

➢ Require payments from voters that depend on the

preferences they submit

➢ Prevalent in auctions

SLIDE 14

Circumventing G-S

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Randomization (later in this lecture)
Equilibrium analysis

➢ How will strategic voters act under a voting rule that is

not strategyproof?

➢ Will they reach an “equilibrium” where each voter is

happy with the (possibly false) preference she is submitting?

Restricting information

➢ Can voters successfully manipulate if they don’t know the

votes of the other voters?

SLIDE 15

Circumventing G-S

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Computational complexity

➢ So we need to use a rule that is the rule is manipulable. ➢ Can we make it NP-hard for voters to manipulate?

[Bartholdi et al., SC&W 1989]

➢ NP-hardness can be a good thing!

𝑔-MANIPULATION problem (for a given voting rule 𝑔):

➢ Input: Manipulator 𝑗, alternative 𝑞, votes of other voters

(non-manipulators)

➢ Output: Can the manipulator cast a vote that makes 𝑞

uniquely win under 𝑔?

SLIDE 16

Example: Borda

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Can voter 3 make 𝑏 win?

1 2 3 b b a a c c d d 1 2 3 b b a a a c c c d d d b

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A Greedy Algorithm

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Goal: The manipulator wants to make alternative 𝑞

win uniquely

Algorithm:

➢ Rank 𝑞 in the first place ➢ While there are unranked alternatives:

If there is an alternative that can be placed in the next spot

without preventing 𝑞 from winning, place this alternative.

Otherwise, return false.

SLIDE 18

Example: Borda

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1 2 3 b b a a a c c d d 1 2 3 b b a a a b c c d d 1 2 3 b b a a a c c c d d 1 2 3 b b a a a c c c b d d 1 2 3 b b a a a c c c d d d 1 2 3 b b a a a c c c d d d b

SLIDE 19

Example: Copeland

19

1 2 3 4 5 a b e e a b a c c c d b b d e a a e c d d a b c d e a

2

3 5 3 b 3

2

4 2 c 2 2

3

1 d 1

2

e 2 2 3 2

Preference profile

Pairwise elections

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SLIDE 20

Example: Copeland

20

1 2 3 4 5 a b e e a b a c c c c d b b d e a a e c d d a b c d e a

2

3 5 3 b 3

2

4 2 c 2 3

4

2 d 1

2

e 2 2 3 2

Preference profile

Pairwise elections

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SLIDE 21

Example: Copeland

21

1 2 3 4 5 a b e e a b a c c c c d b b d d e a a e c d d a b c d e a

2

3 5 3 b 3

2

4 2 c 2 3

4

2 d 1 1

3

e 2 2 3 2

Preference profile

Pairwise elections

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SLIDE 22

Example: Copeland

22

1 2 3 4 5 a b e e a b a c c c c d b b d d e a a e e c d d a b c d e a

2

3 5 3 b 3

2

4 2 c 2 3

4

2 d 1 1

3

e 2 3 3 2

Preference profile

Pairwise elections

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SLIDE 23

Example: Copeland

23

1 2 3 4 5 a b e e a b a c c c c d b b d d e a a e e c d d b a b c d e a

2

3 5 3 b 3

2

4 2 c 2 3

4

2 d 1 1

3

e 2 3 3 2

Preference profile

Pairwise elections

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SLIDE 24

When does this work?

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Theorem [Bartholdi et al., SCW 89]:

Fix voter 𝑗 and votes of other voters. Let 𝑔 be a rule for which ∃ function 𝑡(≻𝑗, 𝑦) such that:

1. For every ≻𝑗, 𝑔 chooses a candidate 𝑦 that uniquely maximizes 𝑡(≻𝑗, 𝑦). 2. 𝑧 ∶ 𝑦 ≻𝑗 𝑧 ⊆ 𝑧 ∶ 𝑦 ≻𝑗

′ 𝑧 ⇒ 𝑡 ≻𝑗, 𝑦 ≤ 𝑡 ≻𝑗 ′, 𝑦

Then the greedy algorithm solves 𝑔-MANIPULATION correctly.

Question: What is the function 𝑡 for plurality?

SLIDE 25

Proof of the Theorem

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Say the algorithm creates a partial

ranking ≻𝑗 and then fails, i.e., every next choice prevents 𝑞 from winning

Suppose for contradiction that ≻𝑗

′

could make 𝑞 uniquely win

𝑉 ← alternatives not ranked in ≻𝑗
𝑣 ← highest ranked alternative in 𝑉

according to ≻𝑗

′

Complete ≻𝑗 by adding 𝑣 next, and

then other alternatives arbitrarily

𝑐

≻𝑗

′

𝑞 𝑏 𝑒 𝑑 𝑞

≻𝑗

𝑐 𝑒 𝑏 𝑑 Output of algo

𝑣 𝑉 = {𝑏, 𝑑}

SLIDE 26

Proof of the Theorem

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𝑡 ≻𝑗, 𝑞 ≥ 𝑡(≻𝑗

′, 𝑞)

➢ Property 2

𝑡 ≻𝑗

′, 𝑞 > 𝑡(≻𝑗 ′, 𝑣)

➢ Property 1 & 𝑞 wins under ≻𝑗

′

𝑡 ≻𝑗

′, 𝑣 ≥ 𝑡(≻𝑗, 𝑣)

➢ Property 2

Conclusion

➢ Putting 𝑣 in the next position wouldn’t

have prevented 𝑞 from winning

➢ So the algorithm should have continued

𝑐

≻𝑗

′

𝑞 𝑏 𝑒 𝑑 𝑞

≻𝑗

𝑐 𝑒 𝑏 𝑑 Output of algo

𝑣 𝑉 = {𝑏, 𝑑}

SLIDE 27

Hard-to-Manipulate Rules

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Natural rules

➢ Copeland with second-order tie breaking

[Bartholdi et al. SCW 89]

In case of a tie, choose the alternative for which the sum of

Copeland scores of defeated alternatives is the largest

➢ STV [Bartholdi & Orlin, SCW 91] ➢ Ranked Pairs [Xia et al., IJCAI 09]

Iteratively lock in pairwise comparisons by their margin of victory

(largest first), ignoring any comparison that would form cycles.

Winner is the top ranked candidate in the final order.
Can also “tweak” easy to manipulate voting rules

[Conitzer & Sandholm, IJCAI 03]

SLIDE 28

Example: Ranked Pairs

28

8 6

12

2

10

4

a b d c

SLIDE 29

Example: Ranked Pairs

29

8 6 2

34

a b d c

SLIDE 35

Randomized Voting Rules

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Take as input a preference profile, output a

distribution over alternatives

To think about successful manipulations, we need

numerical utilities

≻𝑗 is consistent with 𝑣𝑗 if

𝑏 ≻𝑗 𝑐 ⇔ 𝑣𝑗 𝑏 > 𝑣𝑗(𝑐)

Strategyproofness: For all 𝑗, 𝑣𝑗, ≻−𝑗, and ≻𝑗

′

𝔽 𝑣𝑗 𝑔 ≻ ≥ 𝔽 𝑣𝑗 𝑔 ≻−𝑗, ≻𝑗

′

where ≻𝑗 is consistent with 𝑣𝑗.

SLIDE 36

Randomized Voting Rules

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A (deterministic) voting rule is

➢ unilateral if it only depends on one voter ➢ duple if its range contains at most two alternatives

A probability mixture 𝑔 over rules 𝑔

1, … , 𝑔 𝑙 is a rule

given by some probability distribution (𝛽1, … , 𝛽𝑙) s.t. on every profile ≻, 𝑔 returns 𝑔

𝑘 ≻ w.p. 𝛽𝑘.

SLIDE 37

Randomized Voting Rules

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Theorem [Gibbard 77]:

A randomized voting rule is strategyproof only if it is a probability mixture over unilaterals and duples.

Example:

➢ With probability 0.5, output the top alternative of a

randomly chosen voter

➢ With the remaining probability 0.5, output the winner of

the pairwise election between 𝑏∗ and 𝑐∗

Question: What is a probability mixture over

unilaterals and duples that is not strategyproof?

SLIDE 38

Approximating Voting Rules

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Idea: Can we use strategyproof voting rules to

approximate popular voting rules?

Fix a rule (e.g., Borda) with a clear notion of score

denoted sc ≻, 𝑏

A randomized voting rule 𝑔 is a 𝑑-approximation to

sc if for every profile ≻ 𝔽[sc ≻, 𝑔 ≻ max𝑏 sc ≻, 𝑏 ≥ 𝑑

SLIDE 39

Approximating Borda

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Question: How well does choosing a random

alternative approximate Borda?

1. Θ( Τ

1 𝑜)

2. Θ( Τ

1 𝑛)

3. Θ( Τ

1 𝑛)

4. Θ(1)
Theorem [Procaccia 10]:

No strategyproof voting rule gives Τ

1 2 + 𝜕

ൗ

1 𝑛

approximation to Borda.

SLIDE 40

Interlude: Zero-Sum Games

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1

1 1

1

SLIDE 41

Interlude: Minimiax Strategies

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A minimax strategy for a player is

➢ a (possibly) randomized choice of action by the player ➢ that minimizes the expected loss (or maximizes the

expected gain)

➢ in the worst case over the choice of action of the other

player

In the previous game, the minimax strategy for

each player is ( Τ 1 2 , Τ 1 2). Why?

SLIDE 42

Interlude: Minimiax Strategies

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In the game above, if the shooter uses (𝑞, 1 − 𝑞):

➢ If goalie jumps left: 𝑞 ⋅ −

1 2 + 1 − 𝑞 ⋅ 1 = 1 − 3 2 𝑞

➢ If goalie jumps right: 𝑞 ⋅ 1 + 1 − 𝑞 ⋅ −1 = 2𝑞 − 1 ➢ Shooter chooses 𝑞 to maximize min

1 −

3𝑞 2 , 2𝑞 − 1

− ൗ 1 2

1 1

1

SLIDE 43

Interlude: Minimax Theorem

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Theorem

[von Neumann, 1928]: Every 2-player zero-sum game has a unique value 𝑤 such that

➢ Player 1 can guarantee value at

least 𝑤

➢ Player 2 can guarantee loss at

most 𝑤

SLIDE 44

Yao’s Minimax Principle

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Rows as inputs
Columns as deterministic algorithms
Cell numbers = running times
Best randomized algorithm

➢ Minimax strategy for the column player

min

𝑠𝑏𝑜𝑒 𝑏𝑚𝑕𝑝 max 𝑗𝑜𝑞𝑣𝑢 𝐹[𝑢𝑗𝑛𝑓] =

max

𝑒𝑗𝑡𝑢 𝑝𝑤𝑓𝑠 𝑗𝑜𝑞𝑣𝑢𝑡

min

𝑒𝑓𝑢 𝑏𝑚𝑕𝑝 𝐹[𝑢𝑗𝑛𝑓]

SLIDE 45

Yao’s Minimax Principle

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To show a lower bound 𝑈 on the best worst-case

running time achievable through randomized algorithms:

➢ Show a “bad” distribution over inputs 𝐸 such that every

deterministic algorithm takes time at least 𝑈 on average, when inputs are drawn according to 𝐸

min

𝑠𝑏𝑜𝑒 𝑏𝑚𝑕𝑝 max 𝑗𝑜𝑞𝑣𝑢 𝐹[𝑢𝑗𝑛𝑓] =

max

𝑒𝑗𝑡𝑢 𝑝𝑤𝑓𝑠 𝑗𝑜𝑞𝑣𝑢𝑡

min

SLIDE 47

Randomized Voting Rules

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Rows = unilaterals and duples
Columns = preference profiles
Cell numbers = approximation ratios
The expected ratio of the best strategyproof rule

(by Gibbard’s theorem, distribution over unilaterals and duples) is at most…

➢ The expected ratio of the best unilateral or duple rule

when profiles are drawn from a “bad” distribution 𝐸

SLIDE 48

A Bad Distribution

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𝑛 = 𝑜 + 1
Choose a random alternative 𝑦∗
Each voter 𝑗 chooses a random

number 𝑙𝑗 ∈ 1, … , 𝑛 and places 𝑦∗ in position 𝑙𝑗

The other alternatives are ranked

cyclically

1 2 3 c b d b a b a d c d c a 𝑦∗ = 𝑐 𝑙1 = 2 𝑙2 = 1 𝑙3 = 2

SLIDE 49

A Bad Distribution

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Question: What is the best lower bound on

sc ≻, 𝑦∗ that holds for every profile ≻ generated under this distribution?

1. 𝑜 2. 𝑛

3. 𝑜 ⋅ 𝑛 −

𝑛

4. 𝑜 ⋅ 𝑛

SLIDE 50

A Bad Distribution

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How bad are other alternatives?

➢ For every other alternative 𝑦, sc ≻, 𝑦 ~

𝑜 𝑛−1 2

How surely can a unilateral/duple rule return 𝑦∗?

➢ Unilateral: By only looking at a single vote, the rule is

essentially guessing 𝑦∗ among the first 𝑛 positions, and captures it with probability at most 1/ 𝑛.

➢ Duple: By fixing two alternatives, the rule captures 𝑦∗

with probability at most 2/𝑛.

Putting everything together…

SLIDE 51

Quantitative GS Theorem

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Regarding the use of NP-hardness to circumvent GS

➢ NP-hardness is hardness in the worst case ➢ What happens in the average case?

Theorem [Mossel-Racz ‘12]:

For every voting rule that is at least 𝜗-far from being a dictatorship or having range of size 2, the probability that a profile chosen uniformly at random admits a manipulation is at least 𝑞 𝑜, 𝑛, Τ

1 𝜗 for some polynomial 𝑞.

SLIDE 52

Coalitional Manipulations

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What if multiple voters collude to manipulate?

➢ The following result applies to a wide family of voting

rules called “generalized scoring rules”.

Theorem [Conitzer-Xia ‘08]:

Coalition of Manipulators

Θ 𝑜

Powerful Powerless

Powerful = can manipulate with high probability

SLIDE 53

Interesting Tidbit

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Detecting a manipulable profile versus finding a

beneficial manipulation

Theorem [Hemaspaandra, Hemaspaandra, Menton ‘12]

If integer factoring is NP-hard, then there exists a generalized scoring rule for which:

➢ We can efficiently check if there exists a beneficial

manipulation.

➢ But finding such a manipulation is NP-hard.

SLIDE 54

Next Lecture

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Frameworks to compare voting rules

➢ Even if we assume that voters will reveal their true

preferences, we still don’t know if there is one “right” way to choose the winner.

➢ There are reasonable profiles where most prominent