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

T RUTH J USTICE A LGOS 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: 18 th Century (Condorcet and Borda) • 19 th Century: Charles Dodgson • 20 th 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 𝜏 𝑗 ∈ L over the alternatives; 𝑦 ≻ 𝑗 𝑧 means that voter 𝑗 prefers 𝑦 to 𝑧 • A preference profile 𝝉 ∈ L 𝑜 is a collection of all voters’ rankings • A voting rule is a function 𝑔: L 𝑜 → 𝐵

VOTE OVER CUISINES

VOTING RULES • A positional scoring rule is defined by a score vector (𝑡 1 , … , 𝑡 𝑛 ) • Each voter gives 𝑡 𝑙 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 2 1 2 2 1 voters voters voter voters voters voter 𝑏 𝑐 𝑑 𝑏 𝑐 𝑑 𝑐 𝑏 𝑒 𝑐 𝑏 𝑐 𝑑 𝑒 𝑐 𝑑 𝑑 𝑏 𝑒 𝑑 𝑏 2 2 1 2 2 1 voters voters voter voters voters voter 𝑏 𝑐 𝑐 𝑐 𝑐 𝑐 𝑐 𝑏 𝑏

MARQUIS DE CONDORCET • 18 th 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 𝑏 1 2 3 𝑏 𝑑 𝑐 𝑐 𝑏 𝑑 𝑑 𝑐 𝑑 𝑐 𝑏

CONDORCET CONSISTENCY • A voting rule is Condorcet consistent if it selects a Condorcet winner whenever one exists ? Poll 1 Which rule is Condorcet consistent? • Plurality • Both rules • Borda count • Neither one

CONDORCET CONSISTENCY • Theorem: No positional scoring rule is Condorcet consistent • Proof: 3 2 1 1 ◦ Assume for ease of voters voters voter voter exposition that 𝑡 𝑗 > 𝑡 𝑗+1 𝑏 𝑐 𝑐 𝑑 for all 𝑗 𝑐 𝑑 𝑏 𝑏 ◦ Consider the profile on 𝑑 𝑏 𝑑 𝑐 the right ◦ 𝑏 is a Condorcet winner ◦ Scores are 3𝑡 1 + 2𝑡 2 + 2𝑡 3 for 𝑏 , 3𝑡 1 + 3𝑡 2 + 𝑡 3 for 𝑐 , so 𝑐 is selected ∎

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 min 𝑧 |{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 𝑏 𝑐 𝑓 𝑓 𝑐 𝑐 𝑏 𝑐 𝑑 𝑓 𝑑 𝑑 𝑑 𝑒 𝑒 𝑒 𝑒 𝑏 𝑏 𝑏 𝑓 𝑓 𝑒 𝑐 𝑑 Voter 1 Voter 2 Voter 3 Voter 4 Voter 5

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 ⇒ ′ 𝑧 , and for all 𝑧, 𝑨 ≠ 𝑦 , 𝑧 ≻ 𝑗 𝑨 ⇔ 𝑧 ≻ 𝑗 ′ 𝑨 𝑦 ≻ 𝑗 • A voting rule is monotonic if whenever 𝑔(𝝉) = 𝑦 , and 𝝉′ is obtained from 𝝉 by pushing 𝑦 upwards, then 𝑔(𝝉 ′ ) = 𝑦 ? Poll 2 Which rule is not monotonic? • Plurality • STV • Borda count • Copeland

STV IS NOT MONOTONIC • 𝑑 is the winner in the following profile: 6 2 3 4 2 voters voters voter voter voters 𝑑 𝑐 𝑐 𝑏 𝑏 𝑏 𝑏 𝑑 𝑐 𝑑 𝑐 𝑑 𝑏 𝑑 𝑐 • But 𝑐 becomes the winner if the rightmost voters push 𝑑 upwards: 6 2 3 4 2 voters voters voter voter voters 𝑑 𝑐 𝑐 𝑏 𝑑 𝑏 𝑏 𝑑 𝑐 𝑏 𝑐 𝑑 𝑏 𝑑 𝑐

AWESOME EXAMPLE 33 16 3 8 18 22 voters voters voters voters voters 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 Direct Representative Liquid dictatorship democracy democracy democracy 6 2

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