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

Optimal Mechanism Design for

Single-Minded Agents

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MATT WEINBERG

PRINCETON

RAGHUVANSH SAXENA

PRINCETON

ARIEL SCHVARTZMAN

PRINCETON -> RUTGERS

NIKHIL DEVANUR

AMAZON

KIRA GOLDNER, COLUMBIA UNIVERSITY

CONTACT: kgoldner@cs.columbia.edu |

ARXIV: 2002.06329

| PAPER: www.kiragoldner.com

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Service options

bundle G = service or set of goods desired value v = how much getting their bundle is worth

𝒘, 𝑯 ∼ 𝑮

3 days 2 days 1 day

FedEx options

General case: Characterization via dual properties. Menu complexity unbounded. (But finite!) For any 𝑵, ∃𝐺 over (𝑤, 𝐻) s.t. the optimal mechanism has ≥ 𝑵 different options. 𝑮 is DMR: Algorithmic characterization, deterministic. Out-degree ≤1: FedEx solution [FGKK ‘16] applies.

A B C

The Single-Minded Model

Results

DMR: 𝑤𝑔 𝑤 − 1 − 𝐺 𝑤 = 𝑔(𝑤)𝜒(𝑤) increasing

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Complexity Spectrum:

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

DMR => deterministic

Characterization of the Optimal Mechanism Number of Distinct Options to the Buyer

Key Idea for Menu Complexity Bound

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Master Theorem: For any dual given only by signs (+/−) and nonnegative variables (𝟏 and 𝜷 ←), there exists a distribution that causes this dual. Corollary: The “bad dual” exists. Upper Bound: Our algorithm gives menu complexity length

• f the sequence (𝑦1, 𝐵), (𝑦2, 𝐶), (𝑦3, 𝐵), …

An infinite such sequence: Bounded and monotone sequence Converges to 𝑦∗, can set this price.

> M distinct options.

A B

…

+ +

…

x1 x2 x3 x4 xM xM−1

¯ rB rB rA ¯ rA

… …

For any M:

𝑏$ 𝑦# = 1 𝑏% 𝑦# > 0 𝑏$ 𝑦& > 0 𝑏% 𝑦' > 0 𝑏% 𝑠

% > 0

𝑏$ 𝑠$ > 0 > 𝑏$ 𝑠$ > 𝑏% 𝑦( > 𝑏$ 𝑦) > 𝑏$ 𝑦& > 𝑏% 𝑦' 𝑏$ 𝑦) > 0

− −

C B A

a

v

x2 rB rA

+ ⟹ 𝑏 = 1 and − ⟹ 𝑏 = 0 0 𝝁𝑯 𝒘 > 0 ⟹ allocation constant ← 𝜷𝑫,𝑩 𝒘 > 0 ⟹ A is preferable to B at v (at least as much area under A than B)

Complementary Slackness