Characteriza racterization tion of Gr Group up-Stra Strate - - PowerPoint PPT Presentation

Characteriza racterization tion of Gr Group up-Stra Strate tegyp gyproof

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Mechan chanis isms ms for Facili ility ty Lo Locat cation ion in Str trictly ictly Convex vex Space ce

Shengyu Zhao

IIIS, Tsinghua University

Joint work with Pingzhong Tang

IIIS, Tsinghua University

Dingli Yu

Computer Science Department, Princeton University

Settings

• Facility location game of 𝑜 agents
• Profile: 𝒚 = (𝑦1, … , 𝑦𝑜)
• Mechanism: 𝑔 𝒚 ; possibly random
• Strategyproofness / Group-strategyproofness
• Strictly convex space
• 𝑦 ≠ 𝑧, 𝑦

= 𝑧 ⟹ 𝑦 + 𝑧 > 𝑦 + 𝑧

• E.g., Euclidean space, 𝑀𝑞-norm vector space 𝑞 ∈ 1, ∞
• Unanimity: 𝑦1 = ⋯ = 𝑦𝑜 = 𝑦 ⟹ 𝑔 𝒚 = 𝑦 (the same as Arrow’s)
• Translational invariance: 𝑔 𝒚 + 𝑏 = 𝑔 𝒚 + 𝑏 (𝑏 is any constant)

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Our Results

• Theorem 1 (Informal). Deterministic, unanimous,

group-strategyproof ⟹ dictatorial.

• Dictatorship: 𝑔 𝒚 = 𝑦𝑗 for any 𝒚; agent 𝑗 is the

dictator.

• Theorem 2 (Informal). Unanimous, translation-

invariant, group-strategyproof ⟹ 2-dictatorial.

• 2-Dictatorship: 𝑔(𝒚) always lies between 𝑦𝑗 and 𝑦𝑘;

agents 𝑗 and 𝑘 are the 2-dictators.

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Our Results

Deterministic Randomized Maximum cost [2, 2] 3/2𝑏, 3/2 when 𝑜 = 2, 2𝑐, 2 when 𝑜 ≥ 3 Social cost [𝑜 − 1, 𝑜 − 1] [𝑜/2 – 1𝑐, 𝑜/2]

a Proved by (Procaccia and Tennenholtz, 2009) in the one-dimensional case. b Requires translational invariance.

• Theorem 1 and 2 implies almost tight bounds of

approximately optimal mechanisms under both maximum and social cost.

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