CMU 15-251 Social Choice Teachers: Anil Ada Ariel Procaccia (this - PowerPoint PPT Presentation

CMU 15-251 Social Choice Teachers: Anil Ada Ariel Procaccia (this time)

Social choice theory • • • • • 2

The voting model 𝑂 = {1, … , 𝑜} • 𝐵, |𝐵| = 𝑛 • • • 3

Voting rules • • o o o 4

More voting rules • 𝑛 − 𝑙 o 𝑙 o o o o 5

More voting rules • (𝑡 1 , … , 𝑡 𝑛 ) o (1,0, … , 0) (𝑛 − 1, 𝑛 − 2, … , 0) o • 𝑦 𝑧 𝑦 𝑧 • o o 6

More voting rules • 𝑛 − 1 o o o o 7

STV: example 8

Social choice axioms • • 𝑦 𝑦 • 9

Marquis de Condorcet • • • • 10

Condorcet winner 𝑦 𝑧 • 𝑦 𝑧 • • 11

Condorcet consistency • • 12

Condorcet consistency • ⇒ ⇒ 13

More voting rules • • 𝑦 = 𝑛 − 1 o < 𝑛 − 1 o 14

Dodgson’s Rule 𝑦 • 𝑦 • • 15

Awesome example 𝑏 • 𝑐 • • 𝑑 𝑒 • • 𝑓 16

Manipulation • 𝑐 • 𝑏 • • • 17

Strategyproofness • 𝑛 • 𝑛 = 1 1. 𝑛 = 2 2. 𝑛 = 3 3. 𝑛 = ∞ 4. 18

Strategyproofness • • • 19

Gibbard-Satterthwaite • • 𝑛 ≥ 3 • 20

Complexity of manipulation • • • 21

The computational problem • o 𝑞 o 𝑞 𝑞 = 𝑏 • 22

A greedy algorithm 𝑞 • • o 𝑞 o 23

Example: Borda 24

When does the alg work? • 𝑆 𝑆 ∈ { } • 25

Is social choice practical? • • • • 26

Computational social choice • o o • 27

What we have learned • o o o o o • o 28