REMINDER: THE VOTING MODEL Set of voters = {1, , } Set of - - PowerPoint PPT Presentation

ALGOS TRUTH JUSTICE

Social Choice II: Implicit Utilitarian Voting

Teachers: Ariel Procaccia (this time) and Alex Psomas

REMINDER: 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

UTILITIES AND WELFARE

• The voting model assumes ordinal preferences, but it

is plausible that they are derived from underlying cardinal preferences

• Assume that each voter 𝑗 has a utility function 𝑣𝑗: 𝐵 →

[0,1], such that σ𝑦∈𝐵 𝑣𝑗 𝑦 = 1

• Voter 𝑗 reports a ranking 𝜏𝑗 that is consistent with his

utility function, denoted 𝑣𝑗 ⊳ 𝜏𝑗: 𝑦 ≻𝑗 𝑧 ⇒ 𝑣𝑗 𝑦 ≥ 𝑣𝑗(𝑧)

• As usual, the (utilitarian) social welfare of 𝑦 ∈ 𝐵 is

sw 𝑦, 𝒗 = σ𝑗∈𝑂 𝑣𝑗(𝑦)

• Our goal is choose an alternative that maximizes social

welfare, even though we cannot observe the utilities directly

DISTORTION

• We want to quantify how much social

welfare a voting rule loses due to lack of information

• The distortion of voting rule 𝑔 on 𝝉 is

dist 𝑔, 𝝉 = max

𝒗 ⊳ 𝝉 max

𝑦∈𝐵 sw(𝑦,𝒗)

sw(𝑔(𝝉),𝒗)

• The distortion of voting rule 𝑔 is

dist 𝑔 = max

𝝉

dist 𝑔, 𝝉

DISTORTION

• Consider the preference profile

1 2 3 𝑏 𝑏 𝑐 𝑑 𝑑 𝑏 𝑐 𝑐 𝑑

Distortion of Borda count on this profile?

• 3/2
• 2
• 5/3
• 5/2

Poll 1

?

DISTORTION

• Consider the preference profile

1 2 … 𝑛 − 1 𝑏1 𝑏2 … 𝑏𝑛−1 𝑦 𝑦 … 𝑦 ⋮ ⋮ … ⋮

Distortion of plurality on this profile?

• Θ(1)
• Θ(𝑛)
• Θ

𝑛

• Θ 𝑛2

Poll 2

?

DETERMINISTIC LOWER BOUND

• Theorem: Any deterministic voting rule 𝑔 has

distortion at least 𝑛

• Proof:
• Partition 𝑂 into two subsets with 𝑂𝑙 = 𝑜/2, and let

the profile 𝝉 be such that voters in 𝑂1 rank 𝑏1 first, and voter in 𝑂2 rank 𝑏2 first

• W.l.o.g. 𝑔 𝝉 = 𝑏1
• Let 𝑣𝑗 𝑏2 = 1, 𝑣𝑗 𝑏𝑘 = 0 for 𝑗 ∈ 𝑂2, 𝑣𝑗 𝑏𝑘 = 1/𝑛 for

all 𝑗 ∈ 𝑂1

• It holds that

dist 𝑔, 𝝉 ≥ 𝑜 2 𝑜 2𝑛 = 𝑛 ∎

RANDOMIZED UPPER BOUND

• Under the harmonic scoring rule, each voter gives 1/𝑙

points to alternative ranked 𝑙-th

• Denote the score of 𝑦 under 𝝉 as sc 𝑦, 𝝉
• Why is this useful? Because

sw 𝑦, 𝒗 ≤ sc 𝑦, 𝝉 for any 𝐯 ⊳ 𝝉

• Theorem [Caragiannis et al. 2015]: The randomized

voting rule that, with prob. ½, selects 𝑦 ∈ 𝐵 with prob. proportional to sc(𝑦, 𝝉), and selects a uniformly random alternative with prob. ½, has distortion 𝑃 𝑛 log 𝑛

• Discussion: In what sense is this result practical?

PROOF OF THEOREM

• Case 1: The welfare-maximizing 𝑦∗ satisfies

sw 𝑦∗, 𝒗 ≥ 𝑜 (ln 𝑛 + 1)/𝑛

• Then sc 𝑦∗, 𝝉 ≥ 𝑜 (ln 𝑛 + 1)/𝑛
• σ𝑦∈𝐵 sc 𝑦, 𝝉 = 𝑜 σ𝑙=1

𝑛

1/𝑙 ≤ 𝑜(ln 𝑛 + 1)

• 𝑦∗ is selected with prob. at least

1 2 ⋅ 𝑜 ln 𝑛 + 1 𝑛 𝑜 ln 𝑛 + 1 = 1 2 𝑛 (ln 𝑛 + 1)

• Now,

𝔽[sw 𝑔 𝝉 , 𝒗 ≥ Pr 𝑔 𝝉 = 𝑦∗ sw 𝑦∗, 𝒗 ≥ 1 2 𝑛 (ln 𝑛 + 1) sw 𝑦∗, 𝒗

PROOF OF THEOREM

• Case 2: For every 𝑦 ∈ 𝐵 it holds that

sw 𝑦, 𝒗 < 𝑜 (ln 𝑛 + 1)/𝑛

• Uniformly random selection gives expected social

welfare 1 2 1 𝑛 ෍

𝑦∈𝐵

෍

𝑗∈𝑂

𝑣𝑗 𝑦 = 1 2 1 𝑛 ෍

𝑗∈𝑂

෍

𝑦∈𝐵

𝑣𝑗(𝑦) = 𝑜 2𝑛

• Distortion is at most

sw(𝑦∗, 𝒗) 𝔽 sw 𝑔 𝝉 , 𝒗 ≤ 𝑜 ln 𝑛 + 1 𝑛 𝑜 2𝑛 = 2 𝑛(ln 𝑛 + 1) ∎

RANDOMIZED LOWER BOUND

• Theorem [Caragiannis et al. 2012]: Any randomized voting

rule 𝑔 has distortion Ω 𝑛

• Proof:
• Partition 𝑂 into subsets with 𝑂𝑙 = 𝑜/ 𝑛, and let the profile be
• W.l.o.g. 𝑏1 is selected with prob. ≤

1 𝑛

• Let 𝑣𝑗 𝑏1 = 1, 𝑣𝑗 𝑏𝑘 = 0 for 𝑗 ∈ 𝑂1, 𝑣𝑗 𝑏𝑘 = 1/𝑛 otherwise
• 𝑜/ 𝑛 ≤ sw 𝑏1, 𝒗

≤ 2𝑜/ 𝑛, whereas sw 𝑏𝑘, 𝒗 ≤ 𝑜/𝑛 for 𝑘 ≠ 1

• Distortion is at least

𝑜 𝑛 1 𝑛 ⋅ 2𝑜 𝑛 + 1 − 1 𝑛 ⋅ 𝑜 𝑛 ≥ 𝑛 3 ∎

𝑂1 𝑂2 … 𝑂 𝑛 𝑏1 𝑏2 … 𝑏 𝑛 ⋮ ⋮ ⋮ ⋮

PARTICIPATORY BUDGETING

Porto Alegre Brazil Since 1989 Paris France €100M (2016) Madrid Spain €24M (2016) New York USA $40M (2017)

THE MODEL

• The total budget is 𝐶
• Each alternative 𝑦 has a cost 𝑑𝑦
• For 𝑌 ⊆ 𝐵, the cost 𝑑(𝑌) is additive
• Utilities are also additive, that is,

𝑣𝑗 𝑌 = σ𝑦∈𝑌 𝑣𝑗(𝑦)

• The goal is to find 𝑌 ⊆ 𝐵 that

maximizes the social welfare sw 𝑌, 𝒗 = σ𝑗∈𝑂 𝑣𝑗(𝑌) subject to the budget constraint 𝑑 𝑌 ≤ 𝐶

Utility 6 Cost 6 Utility 3 Cost 2 Utility 2 Cost 1 Utility 8 Cost 9 Ranking by value Ranking by VFM Knapsack voting Threshold approval

INPUT FORMATS

≻ ≻ ≻ ≻ ≻ ≻

Budget 9 Threshold 5

DISTORTION REDUX

• Distortion allows us to objectively compare

input formats, by associating an input format with the distortion of the best voting rule

• Theorem [Benade et al. 2017]: Any randomized

voting rule has distortion at least Ω(𝑛) under knapsack votes

• Proof:
• Let 𝐶 = 1, 𝑑 𝑏𝑘 = 1 for all 𝑏𝑘 ∈ 𝐵
• Define 𝝉: For each 𝑏𝑘 ∈ 𝐵 we have 𝑜/𝑛 voters 𝑂

𝑘

who choose 𝑦

• W.l.o.g. 𝑏1 is selected with prob. ≤ 1/𝑛, then let

𝑣𝑗 𝑏1 = 1 for all 𝑗 ∈ 𝑂1, and 𝑣𝑗 𝑏𝑘 = 𝑣𝑗 𝑏1 = 1/2 for all 𝑗 ∈ 𝑂

𝑘, 𝑘 ≠ 1 ∎

RANDOMIZED BOUNDS

$ Threshold approval 𝑃(log2 𝑛) Knapsack voting Θ(𝑛) Ranking by VFM ෩ Θ 𝑛 Ranking by value ෩ Θ 𝑛

[Benade et al., 2017]

METRIC PREFERENCES

• Assume a metric space with metric 𝑒 on space of

voters and alternatives

• Preferences are defined by

𝑒 𝑗, 𝑦 < 𝑒 𝑗, 𝑧 ⇒ 𝑦 ≻𝑗 𝑧

• Now we want to minimize the social cost, defined

as sc 𝑦, 𝑒 = σ𝑗∈𝑂 𝑒(𝑗, 𝑦)

𝑏 𝑑 𝑐

𝑏 ≻ 𝑐 ≻ 𝑑

LOWER BOUND

• Theorem [Anshelevich et al. 2015]: The

distortion of any deterministic rule under metric preferences is at least 3

• Proof:
• Theorem [Anshelevich et al. 2015]: The

distortion of Copeland under metric preferences is at most 5