Private capacities in mechanism design

Private capacities in mechanism design Vincenzo Auletta Paolo Penna Giuseppe Persiano Università di Salerno, Italy

In practice… Ideal World

Different “entities” which • have their own goal • may not follow the “protocol” Selfish agents The Internet Self organized, no central authority, anarchic

Link down Example: BGP Routing AS1 destination source AS2 An Autonomous System may report false link status to redirect traffic to another AS

 0  0 Toy problem Routing cost (time/latency) t1 source destination t2 Algorithm: smallest cost Payment (large enough)

Toy problem Routing cost (time/latency) C1 source destination C2 Algorithm: smallest cost Payment: 2nd smallest cost

Truthful mechanism Toy problem 7  10 – 7 8  10 – 7 11  0 – 0 Routing cost (time/latency) 7 source destination 10 Algorithm: smallest cost Payment: 2nd smallest cost A truthtelling agent maximizes his/her utility

Mechanism design Mechanism = Algorithm + Payments When are truthful mechanisms possible? How to design them? Focus on the Algorithm!! truthful mechanism  monotone algorithm

tn tn r1 t1 ri ti … … t1 ti tn utilityi = paymenti – worki ti work1 workn worki payment1 paymenti paymentn One-parameter setting Mechanism = Algorithm + Payments per-unit cost

speed One-parameter setting Mechanism = Algorithm + Payments • P2P, Grid computing, scheduling • Routing • Wireless communications utilityi = paymenti – worki ti

worki truthful  ti [Myerson81] [Archer&Tardos01] One-parameter setting Mechanism = Algorithm + Payments • P2P, Grid computing, scheduling • Routing • Wireless communications battery utilityi = paymenti – worki ti Well understood…but assumes infinite capacities!

ALGORITHMIC!! Our contribution • New model (one-parameter + capacities) • Characterization of truthfulness • Few applications (optimal mechanisms) • Open questions

t1 c1 ti ci tn cn … … utilityi = –  utilityi = paymenti – worki ti when “work  capacity” when “work > capacity” Private capacities per-unit cost max work that can effort

ti ci t1 t1 c1 c1 ti ci tn tn cn cn … … … … algorithm algorithm worki worki Characterization of truthfulness per-unit cost increases too much work less work Weakly monotone algorithm: ti >ti  worki worki or worki > ci truthful mechanism  weakly monotone algorithm

fixed capacities NO! x x2 5 10 6 x 1 10 6 5 Example: Greedy Algorithm Weakly monotone algorithm: ti >ti  worki worki or worki > ci 5 10 6

 ti•(work i worki) ti ci worki t i c i work i t i•(workiwork i) … nonnegative length nonnegative length k different inputs for agent i Hard to understand/use simple Characterization of truthfulness: proof ingredients truthfulness  cycle monotonicity [Rochet 87]  weak monotonicity worki worki or worki > ci how worki changes (t i ti)•(workiwork i)

obvious … nonnegative length nonnegative length hard Hard to understand/use simple Characterization of truthfulness: proof ingredients truthfulness  cycle monotonicity [Rochet 87]  weak monotonicity

Algorithm Algorithm + Payments Truthful mechanism weakly monotone polytime optimal cost not always [Archer&Tardos 01] yes [this work] How to design the mechanism Without capacities [AT 01] Can we have all of them? min-max problems (max congestion, makespan, fairness)

Algorithm weakly monotone polytime optimal cost How to design the mechanism Scheduling: truthful PTAS for O(1) machines [this work] Without capacities [Andelmann et al 07] exact + rounding min-max problems (max congestion, makespan, fairness)

Open questions • PTAS for any number of machines? • Withoutcapacities: Yes [Christodoulou & Kovacs 09] • Complexity of “truthful” algorithms • BGP routing • Network lifetime • Multi-parameter settings? • No simple characterization [this work] Private capacities

Thank You

Private capacities in mechanism design