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

T RUTH J USTICE A LGOS 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 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

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 𝑣 𝑗 : 𝐵 → [0,1] , such that σ 𝑦∈𝐵 𝑣 𝑗 𝑦 = 1 • Voter 𝑗 reports a ranking 𝜏 𝑗 that is consistent with his utility function, denoted 𝑣 𝑗 ⊳ 𝜏 𝑗 : 𝑦 ≻ 𝑗 𝑧 ⇒ 𝑣 𝑗 𝑦 ≥ 𝑣 𝑗 (𝑧) • As usual, the (utilitarian) social welfare of 𝑦 ∈ 𝐵 is = σ 𝑗∈𝑂 𝑣 𝑗 (𝑦) sw 𝑦, 𝒗 • 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 𝑦∈𝐵 sw (𝑦,𝒗) max dist 𝑔, 𝝉 = max sw (𝑔(𝝉),𝒗) 𝒗 ⊳ 𝝉 • The distortion of voting rule 𝑔 is dist 𝑔 = max dist 𝑔, 𝝉 𝝉

DISTORTION • Consider the preference profile 1 2 3 𝑏 𝑏 𝑐 𝑑 𝑑 𝑏 𝑐 𝑐 𝑑 ? Poll 1 Distortion of Borda count on this profile? • 3/2 • 2 • 5/3 • 5/2

DISTORTION • Consider the preference profile 1 2 … 𝑛 − 1 𝑏 1 𝑏 2 … 𝑏 𝑛−1 𝑦 𝑦 … 𝑦 ⋮ ⋮ … ⋮ ? Poll 2 Distortion of plurality on this profile? • Θ(1) • Θ(𝑛) • Θ 𝑛 2 • Θ 𝑛

DETERMINISTIC LOWER BOUND • Theorem: Any deterministic voting rule 𝑔 has distortion at least 𝑛 • Proof: ◦ Partition 𝑂 into two subsets with 𝑂 𝑙 = 𝑜/2 , and let the profile 𝝉 be such that voters in 𝑂 1 rank 𝑏 1 first, and voter in 𝑂 2 rank 𝑏 2 first ◦ W.l.o.g. 𝑔 𝝉 = 𝑏 1 ◦ Let 𝑣 𝑗 𝑏 2 = 1 , 𝑣 𝑗 𝑏 𝑘 = 0 for 𝑗 ∈ 𝑂 2 , 𝑣 𝑗 𝑏 𝑘 = 1/𝑛 for all 𝑗 ∈ 𝑂 1 ◦ It holds that 𝑜 2 dist 𝑔, 𝝉 ≥ = 𝑛 ∎ 𝑜 2𝑛

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 𝑛 • σ 𝑦∈𝐵 sc 𝑦, 𝝉 = 𝑜 σ 𝑙=1 1/𝑙 ≤ 𝑜(ln 𝑛 + 1) • 𝑦 ∗ is selected with prob. at least 𝑜 ln 𝑛 + 1 1 1 𝑛 2 ⋅ 𝑜 ln 𝑛 + 1 = 2 𝑛 (ln 𝑛 + 1) • Now, ≥ Pr 𝑔 𝝉 = 𝑦 ∗ sw 𝑦 ∗ , 𝒗 𝔽[sw 𝑔 𝝉 , 𝒗 1 sw 𝑦 ∗ , 𝒗 ≥ 2 𝑛 (ln 𝑛 + 1)

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 1 𝑣 𝑗 𝑦 = 1 1 = 𝑜 𝑛 ෍ ෍ 𝑛 ෍ ෍ 𝑣 𝑗 (𝑦) 2 2 2𝑛 𝑦∈𝐵 𝑗∈𝑂 𝑗∈𝑂 𝑦∈𝐵 • Distortion is at most 𝑜 ln 𝑛 + 1 sw(𝑦 ∗ , 𝒗) 𝑛 ≤ = 2 𝑛(ln 𝑛 + 1) ∎ 𝑜 𝔽 sw 𝑔 𝝉 , 𝒗 2𝑛

RANDOMIZED LOWER BOUND • Theorem [Caragiannis et al. 2012]: Any randomized voting rule 𝑔 has distortion Ω 𝑛 • Proof: ◦ Partition 𝑂 into subsets with 𝑂 𝑙 = 𝑜/ 𝑛 , and let the profile be 𝑂 𝑛 𝑂 1 𝑂 2 … 𝑏 𝑛 𝑏 1 𝑏 2 … ⋮ ⋮ ⋮ ⋮ 1 ◦ W.l.o.g. 𝑏 1 is selected with prob. ≤ 𝑛 ◦ Let 𝑣 𝑗 𝑏 1 = 1 , 𝑣 𝑗 𝑏 𝑘 = 0 for 𝑗 ∈ 𝑂 1 , 𝑣 𝑗 𝑏 𝑘 = 1/𝑛 otherwise ◦ 𝑜/ 𝑛 ≤ sw 𝑏 1 , 𝒗 ≤ 2𝑜/ 𝑛 , whereas sw 𝑏 𝑘 , 𝒗 ≤ 𝑜/𝑛 for 𝑘 ≠ 1 ◦ Distortion is at least 𝑜 𝑛 𝑛 ≥ ∎ 1 𝑛 ⋅ 2𝑜 𝑛 + 1 − 1 𝑛 ⋅ 𝑜 3 𝑛

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

THE MODEL • The total budget is 𝐶 • Each alternative 𝑦 has a cost 𝑑 𝑦 • For 𝑌 ⊆ 𝐵 , the cost 𝑑(𝑌) is additive • Utilities are also additive, that is, 𝑣 𝑗 𝑌 = σ 𝑦∈𝑌 𝑣 𝑗 (𝑦) • The goal is to find 𝑌 ⊆ 𝐵 that maximizes the social welfare sw 𝑌, 𝒗 = σ 𝑗∈𝑂 𝑣 𝑗 (𝑌) subject to the budget constraint 𝑑 𝑌 ≤ 𝐶

INPUT FORMATS Ranking Knapsack Budget by value voting ≻ ≻ ≻ 9 Ranking Threshold Threshold by VFM approval ≻ ≻ ≻ 5 Utility 3 Utility 6 Utility 2 Utility 8 Cost 6 Cost 2 Cost 1 Cost 9

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 𝑏 𝑘 = 1 for all 𝑏 𝑘 ∈ 𝐵 ◦ Define 𝝉 : For each 𝑏 𝑘 ∈ 𝐵 we have 𝑜/𝑛 voters 𝑂 𝑘 who choose 𝑦 ◦ W.l.o.g. 𝑏 1 is selected with prob. ≤ 1/𝑛 , then let 𝑣 𝑗 𝑏 1 = 1 for all 𝑗 ∈ 𝑂 1 , and 𝑣 𝑗 𝑏 𝑘 = 𝑣 𝑗 𝑏 1 = 1/2 for all 𝑗 ∈ 𝑂 𝑘 , 𝑘 ≠ 1 ∎

RANDOMIZED BOUNDS ෩ Ranking by value Θ 𝑛 $ ෩ Ranking by VFM Θ 𝑛 Θ(𝑛) Knapsack voting Threshold approval 𝑃(log 2 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 = σ 𝑗∈𝑂 𝑒(𝑗, 𝑦) 𝑏 𝑐 𝑏 ≻ 𝑐 ≻ 𝑑 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