Characteriza racterization tion of Gr Group up-Stra Strate tegyp gyproof oof Me 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 Dingli Yu IIIS, Computer Science Department, Tsinghua University Princeton University
Settings β’ Facility location game of π agents β’ Profile: π = (π¦ 1 , β¦ , π¦ π ) β’ Mechanism: π π ; possibly random β’ Strategyproofness / Group-strategyproofness β’ Strictly convex space β’ π¦ β π§, π¦ = π§ βΉ π¦ + π§ > π¦ + π§ β’ E.g., Euclidean space, π π -norm vector space π β 1, β β’ Unanimity: π¦ 1 = β― = π¦ π = π¦ βΉ π π = π¦ (the same as Arrowβs) β’ Translational invariance: π π + π = π π + π ( π is any constant) 2
Our Results β’ Theorem 1 (Informal). Deterministic, unanimous, group-strategyproof βΉ dictatorial . β’ Dictatorship: π π = π¦ π for any π ; agent π is the dictator. β’ Theorem 2 (Informal). Unanimous, translation- invariant, group-strategyproof βΉ 2-dictatorial . β’ 2-Dictatorship: π(π) always lies between π¦ π and π¦ π ; agents π and π are the 2-dictators. 3
Our Results β’ Theorem 1 and 2 implies almost tight bounds of approximately optimal mechanisms under both maximum and social cost. Deterministic Randomized Maximum cost 3/2 π , 3/2 when π = 2, 2 π , 2 when π β₯ 3 [2, 2] Social cost [π/2 β 1 π , π/2] [π β 1, π β 1] a Proved by (Procaccia and Tennenholtz, 2009) in the one-dimensional case. b Requires translational invariance. 4
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