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

Ideal World In practice…

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

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

Toy problem sourcedestination Routing cost (time/latency) t1t1 t2t2 0 0 Payment (large enough) Algorithm: smallest cost

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

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

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

One-parameter setting Mechanism = Algorithm + Payments t 1 t i t n utility i = payment i – work i t i r 1 r i t n work n work 1 work i … … t1t1 titi tntn payment i payment 1 payment n per-unit cost

One-parameter setting Mechanism = Algorithm + Payments utility i = payment i – work i t i P2P, Grid computing, scheduling Routing Wireless communications speed

One-parameter setting Mechanism = Algorithm + Payments utility i = payment i – work i t i P2P, Grid computing, scheduling Routing Wireless communications Well understood…but assumes infinite capacities! battery work i titi truthful [Myerson81] [Archer&Tardos01]

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

Private capacities … … t1t1 titi tntn c1c1 cici cncn per-unit cost max work that can effort utility i = payment i – work i t i when work capacity utility i = – when work > capacity

Characterization of truthfulness Weakly monotone algorithm: t i > t i work i work i or work i > c i truthful mechanism weakly monotone algorithm … … t1t1 titi tntn c1c1 cici cncn algorithm work i … … t1t1 t i tntn c1c1 c i cncn algorithm work i per-unit cost increases less work too much work

Example: Greedy Algorithm Weakly monotone algorithm: t i > t i work i work i or work i > c i fixed capacities x 2 x NO! 1 x

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

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

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

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

Open questions PTAS for any number of machines? –Without capacities: Yes [Christodoulou & Kovacs 09] Complexity of truthful algorithms –BGP routing –Network lifetime Multi-parameter settings? –No simple characterization [this work] Private capacities

Thank You