SOCIAL CHOICE THEORY A mathematical theory that deals with - - PowerPoint PPT Presentation

ALGOS TRUTH JUSTICE

Social Choice I: Basic Concepts

Teachers: Ariel Procaccia (this time) and Alex Psomas

SOCIAL CHOICE THEORY

• A mathematical theory that

deals with aggregation of individual preferences

• Origins in ancient Greece
• Formal foundations: 18th

Century (Condorcet and Borda)

• 19th Century: Charles

Dodgson

• 20th Century: Nobel prizes to

Arrow and Sen

THE VOTING MODEL

• Set of voters 𝑂 = {1, … , 𝑜}
• Set of alternatives 𝐵; denote |𝐵| = 𝑛
• Each voter has a ranking 𝜏𝑗 ∈ L over

the alternatives; 𝑦 ≻𝑗 𝑧 means that voter 𝑗 prefers 𝑦 to 𝑧

• A preference profile 𝝉 ∈ L𝑜 is a

collection of all voters’ rankings

• A voting rule is a function 𝑔: L𝑜 → 𝐵

VOTE OVER CUISINES

VOTING RULES

• A positional scoring rule is defined by a

score vector (𝑡1, … , 𝑡𝑛)

• Each voter gives 𝑡𝑙 points to the

alternative ranked in position 𝑙

• Alternative with most points wins
• Examples:
• Plurality: (1,0, … , 0)
• Borda: (𝑛 − 1, 𝑛 − 2, … , 0)

Lordi Eurovision 2006 winners

MORE VOTING RULES

• 𝑦 beats 𝑧 in a pairwise election if

the majority of voters prefer 𝑦 to 𝑧, i.e., 𝑗 ∈ 𝑂: 𝑦 ≻𝑗 𝑧 > 𝑜/2

• Plurality with runoff
• First round: two alternatives with

highest plurality scores survive

• Second round: pairwise election

between these two alternatives

MORE VOTING RULES

• Single Transferable vote (STV)
• 𝑛 − 1 rounds
• In each round, alternative with least

plurality votes is eliminated

• Alternative left standing is the winner
• Used in:
• Ireland, Malta, Australia, and New Zealand
• US: Maine (governor, US congress), cities

like San Francisco and Cambridge

STV: EXAMPLE

2 voters 2 voters 1 voter 𝑏 𝑐 𝑑 𝑐 𝑏 𝑒 𝑑 𝑒 𝑐 𝑒 𝑑 𝑏 2 voters 2 voters 1 voter 𝑏 𝑐 𝑑 𝑐 𝑏 𝑐 𝑑 𝑑 𝑏 2 voters 2 voters 1 voter 𝑏 𝑐 𝑐 𝑐 𝑏 𝑏 2 voters 2 voters 1 voter 𝑐 𝑐 𝑐

MARQUIS DE CONDORCET

• 18th Century French

Mathematician, philosopher, political scientist

• One of the leaders of the

French revolution

• After the revolution

became a fugitive

• His cover was blown and he

died mysteriously in prison

CONDORCET WINNER

• Recall: 𝑦 beats 𝑧 in a pairwise election if a

majority of voters rank 𝑦 above 𝑧

• Condorcet winner beats every other

alternative in pairwise election

• The Condorcet Paradox: There may be a

cycle in the majority preference relation

1 2 3 𝑏 𝑑 𝑐 𝑐 𝑏 𝑑 𝑑 𝑐 𝑏

𝑏 𝑐 𝑑

CONDORCET CONSISTENCY

• A voting rule is Condorcet consistent if it

selects a Condorcet winner whenever one exists

Which rule is Condorcet consistent?

• Plurality
• Both rules
• Borda count
• Neither one

Poll 1

?

CONDORCET CONSISTENCY

• Theorem: No positional scoring rule is

Condorcet consistent

• Proof:
• Assume for ease of

exposition that 𝑡𝑗 > 𝑡𝑗+1 for all 𝑗

• Consider the profile on

the right

• 𝑏 is a Condorcet winner
• Scores are 3𝑡1 + 2𝑡2 + 2𝑡3 for 𝑏, 3𝑡1 + 3𝑡2 + 𝑡3

for 𝑐, so 𝑐 is selected ∎

3 voters 2 voters 1 voter 1 voter 𝑏 𝑐 𝑐 𝑑 𝑐 𝑑 𝑏 𝑏 𝑑 𝑏 𝑑 𝑐

CONDORCET CONSISTENCY

• Copeland
• Alternative’s score is #alternatives it

beats in pairwise elections

• Why does Copeland satisfy the Condorcet

criterion?

• Maximin
• Score of 𝑦 is min𝑧 |{𝑗 ∈ 𝑂: 𝑦 ≻𝑗 𝑧}|
• Why does Maximin satisfy the Condorcet

criterion?

DODGSON’S RULE

• Distance function between profiles: #swaps

between adjacent alternatives

• Dodgson score of 𝑦 is the min distance from

a profile where 𝑦 is a Condorcet winner

• Dodgson’s rule: select alternative that

minimizes Dodgson score

• The problem of computing the Dodgson

score is NP-complete!

DODGSON UNLEASHED

Voter 1

𝑏 𝑐 𝑑 𝑒 𝑓

Voter 2

𝑐 𝑏 𝑑 𝑒 𝑓

Voter 3

𝑓 𝑐 𝑑 𝑏 𝑒

Voter 4

𝑓 𝑑 𝑒 𝑐

Voter 5

𝑐 𝑓 𝑒 𝑑 𝑏 𝑏

MONOTONICITY

• We say that 𝝉′ is obtained from 𝝉 by pushing 𝑦 ∈ 𝐵

upwards if for all 𝑗 ∈ 𝑂 and 𝑧 ∈ 𝐵, 𝑦 ≻𝑗 𝑧 ⇒ 𝑦 ≻𝑗

′ 𝑧, and for all 𝑧, 𝑨 ≠ 𝑦, 𝑧 ≻𝑗 𝑨 ⇔ 𝑧 ≻𝑗 ′ 𝑨

• A voting rule is monotonic if whenever 𝑔(𝝉) = 𝑦,

and 𝝉′ is obtained from 𝝉 by pushing 𝑦 upwards, then 𝑔(𝝉′) = 𝑦

Which rule is not monotonic?

• Plurality
• STV
• Borda count
• Copeland

Poll 2

?

STV IS NOT MONOTONIC

• 𝑑 is the winner in the following profile:
• But 𝑐 becomes the winner if the rightmost voters

push 𝑑 upwards:

6 voters 2 voters 3 voter 4 voter 2 voters 𝑑 𝑐 𝑐 𝑏 𝑏 𝑏 𝑏 𝑑 𝑐 𝑑 𝑐 𝑑 𝑏 𝑑 𝑐 6 voters 2 voters 3 voter 4 voter 2 voters 𝑑 𝑐 𝑐 𝑏 𝑑 𝑏 𝑏 𝑑 𝑐 𝑏 𝑐 𝑑 𝑏 𝑑 𝑐

AWESOME EXAMPLE

33 voters 16 voters 3 voters 8 voters 18 voters 22 voters 𝑏 𝑐 𝑑 𝑑 𝑒 𝑓 𝑐 𝑒 𝑒 𝑓 𝑓 𝑑 𝑑 𝑑 𝑐 𝑐 𝑑 𝑐 𝑒 𝑓 𝑏 𝑒 𝑐 𝑒 𝑓 𝑏 𝑓 𝑏 𝑏 𝑏

Different rules select different winners: Plurality (𝑏), Borda count (𝑐), Copeland and Maximin (𝑑 is a Condorcet winner), STV (𝑒), and Plurality with runoff (𝑓)

IS SOCIAL CHOICE PRACTICAL?

• UK referendum (2011):

Choose between plurality and STV as a method for electing MPs

• Academics agreed STV is

better...

• ... but STV seen as beneficial

to the hated Nick Clegg

• Hard to change political

elections!

COMPUTATIONAL SOCIAL CHOICE

However, in emerging paradigms of democracy and tools for group decision making, the designer is free to choose any voting rule!

LIQUID DEMOCRACY

Monarchy or dictatorship Direct democracy Representative democracy Liquid democracy

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VIRTUAL DEMOCRACY