Single-Minded Agents KIRA GOLDNER, COLUMBIA UNIVERSITY NIKHIL - PowerPoint PPT Presentation

Optimal Mechanism Design for Single-Minded Agents KIRA GOLDNER, COLUMBIA UNIVERSITY NIKHIL DEVANUR RAGHUVANSH SAXENA ARIEL SCHVARTZMAN MATT WEINBERG AMAZON PRINCETON PRINCETON -> RUTGERS PRINCETON C ONTACT : kgoldner@cs.columbia.edu | AR X IV : 2002.06329 | P APER : www.kiragoldner.com 1

The Single-Minded Model Results bundle G = service or set of Service options goods desired General case: C Characterization via dual properties. Menu complexity unbounded . (But finite!) A 𝒘, 𝑯 ∼ 𝑮 For any 𝑵 , ∃𝐺 over (𝑤, 𝐻) s.t. the optimal mechanism has ≥ 𝑵 different options. value v = B how much getting DMR: 𝑤𝑔 𝑤 − 1 − 𝐺 𝑤 their bundle is worth = 𝑔(𝑤)𝜒(𝑤) increasing 𝑮 is DMR: Algorithmic characterization, deterministic. FedEx options Out-degree ≤1: 1 day 2 days 3 days FedEx solution [FGKK ‘16] applies. 2

Complexity Spectrum: Characterization of the Optimal Mechanism Number of Distinct Options to the Buyer Charact- closed explicit (open, seems erization dual properties none form dual harder) DMR => deterministic Menu Complexity unbounded countably uncountably 𝑃(2 ! ) 1 ≥ unbounded (but finite) infinite infinite 2 items 1 item Single-Minded [MV ‘07, DDT ‘15] [Mye ’81] [ DGSSW ‘20 ] Budgets: 3 0 2 !"# − 1 Multi-Unit Pricing $5, $10, $12 budgets 1,2,3-cap for documents [DW ‘17] [DHP ‘17, DGSSW ‘20 ] FedEx: 2 ! − 1 Coordinated Valuations 1, 2, 3-day shipping Wifi, +TV, +Cable [w/ g(v)] [FGKK ‘16, SSW ‘18] [ DGSSW ‘20 ] 3

A Key Idea for Menu Complexity Bound C B A B For any M: > M distinct options. a + ¯ 𝑏 $ 𝑦 # = 1 𝑏 % 𝑦 # > 0 r A + x 1 ¯ r B > 𝑏 $ 𝑦 & 𝑏 $ 𝑦 & > 0 0 x 2 v r B r A x 2 𝑏 % 𝑦 ' > 0 > 𝑏 % 𝑦 ' x 3 Master Theorem: 𝑏 $ 𝑦 ) > 0 > 𝑏 $ 𝑦 ) For any dual given only by signs ( + / − ) and x 4 > 𝑏 % 𝑦 ( … nonnegative variables ( 𝟏 and 𝜷 ← ), there … exists a distribution that causes this dual. … … Corollary: The “bad dual” exists. x M − 1 x M 𝑏 % 𝑠 % > 0 r A Upper Bound: − 𝑏 $ 𝑠 $ > 0 > 𝑏 $ 𝑠 $ r B Our algorithm gives menu complexity length − of the sequence (𝑦 1 , 𝐵), (𝑦 2 , 𝐶), (𝑦 3 , 𝐵), … + ⟹ 𝑏 = 1 and − ⟹ 𝑏 = 0 Complementary An infinite such sequence: 0 𝝁 𝑯 𝒘 > 0 ⟹ allocation constant Bounded and monotone sequence Slackness ← 𝜷 𝑫,𝑩 𝒘 > 0 ⟹ A is preferable to B at v Converges to 𝑦 ∗ , can set this price. (at least as much area under A than B) 4