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

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Ideal World In practice…

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

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

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

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

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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 – 7 8 10 – 7 11 0 – 0

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

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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

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

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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]

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

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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

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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

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Example: Greedy Algorithm 5 6 10 Weakly monotone algorithm: t i > t i work i work i or work i > c i fixed capacities x 2 x 10 6 5 NO! 1 x 10 6 5

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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 )

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

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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]

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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]

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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

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Thank You

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