CMU 15-896 Social choice 3: Advanced manipulation Teacher: Ariel - - PowerPoint PPT Presentation

CMU 15-896

Social choice 3: Advanced manipulation

Teacher: Ariel Procaccia

15896 Spring 2016: Lecture 3

Recap

• A Complexity-theoretic barrier to

manipulation

• Polynomial-time greedy alg can decide

instances of -MANIPULATION for = scoring rules, Copeland, Maximin,...  these rules are easy to manipulate in practice

• Some rules are NP-hard to manipulate:

STV, ranked pairs,...

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15896 Spring 2016: Lecture 3

Criticisms

• What is the complexity of the

Dictatorship-MANIPULATION problem?

• NP-hardness is worst-case, but perhaps a

manipulator can usually succeed

• Approaches:
• Algorithmic: for specific voting rules but

works for every reasonable distribution

• Quantitative G-S: for a specific distribution

but works for every reasonable voting rule

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15896 Spring 2016: Lecture 3

Quantitative G-S

• We’ll do this roughly, to capture intuitions

rather than aiming for accuracy

• The distance between two voting rules is the

fraction of inputs on which they differ where the Pr is over uniformly random preference profiles

• For a set ,

∈

• = set of dictatorships,
• 4

15896 Spring 2016: Lecture 3

Quantitative G-S

• is a manipulation pair for

if

• Theorem [Mossel and Racz 2012]:

, is onto,

• . Then
• manip. pair for

for a polynomial , where and

• are

chosen uniformly at random

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15896 Spring 2016: Lecture 3

Randomized voting rules

• Randomized voting rule: outputs a distribution
• ver alternatives
• To think about successful manipulations we need

utilities (assume strict preferences)

• is consistent with if
• Strategyproofness:
• where

is consistent with

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15896 Spring 2016: Lecture 3

Randomized voting rules

• A (deterministic) voting rule is
• unilateral if it only depends on one voter
• duple if its range is of size at most 2
• A randomized rule is a probability mixture
• ver rules

if there exist

• such that for all ,
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15896 Spring 2016: Lecture 3

Randomized voting rules

• Theorem [Gibbard 1977]: A randomized

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

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Mixture over unilaterals and duples that is not SP?

15896 Spring 2016: Lecture 3

Randomization+approximation

• Idea: can strategyproof randomized rules

approximate popular rules?

• Fix a rule with a clear notion of score

(e.g., Borda) denoted

• Randomized rule

is a -approximation if for every preference profile ,

∈

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15896 Spring 2016: Lecture 3

Approximating Borda

• Theorem [P 2010]: No strategyproof randomized

voting rule can approximate Borda to a factor of

• 10

Poll 1: What is the approximation ratio to Borda from randomly choosing an alternative?

1.

Θ 1/

2.

Θ 1/

3.

Θ 1/

4.

Θ1

15896 Spring 2016: Lecture 3

Interlude: Zero-sum games

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• 1

1 1

• 1

15896 Spring 2016: Lecture 3

Interlude: Minimax strategies

• Minimax (randomized) strategy minimizes worst-

case expected loss (or maximizes the expected gain)

• In the penalty shot game, minimax strategy for

both players is playing

• In the game below, if shooter uses
• Jump left:
• 1 1
• Jump right: 1 2 1
• Maximize min1
• , 2 1 over

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15896 Spring 2016: Lecture 3

Interlude: The Minimax Theorem

• Theorem [von Neumann,

1928]: Every 2-player zero-sum game has a unique value such that:

• Player 1 can guarantee

value at least

• Player 2 can guarantee

loss at most

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15896 Spring 2016: Lecture 3

Yao’s minimax principle

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• 1

15 2 21

• 7

15 5 21

• 4

15 8 21

• 13

15 17 21

Approximation ratio

15896 Spring 2016: Lecture 3

Yao’s minimax principle

• Maximin Theorem  The expected ratio of the best

distribution over unilateral rules and duples against the worst preference profile is equal to the expected ratio of the worst distribution over profiles against the best unilateral rule or duple

• An upper bound on the approximation ratio of the best

distribution over unilateral rules and duples is given by some distribution over profiles against the best unilateral rule or duple

• Gibbard’s Theorem  this is also an upper bound on the

best randomized strategyproof rule

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15896 Spring 2016: Lecture 3

A bad distribution

• Choose

∗

uniformly at random

• Each voter chooses a random

number and puts

∗ in position

• The other alternatives are

ranked cyclically

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

∗ 2 1 2

15896 Spring 2016: Lecture 3

A bad distribution

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Poll 2: What is the best feasible lower bound on

1. 2. 3. 4.

15896 Spring 2016: Lecture 3

A bad distribution

• For

∗

• Unilateral rule: by looking at one vote

there is no way to tell who

∗ is; need to

“guess” among first alternatives

• Duple: by fixing only two alternatives the

probability of getting

∗ is

• 18