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Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents

Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents. Carmine Ventre Joint work with A. Ferrante, G. Parlato and F. Sorrentino. Selfish agents. An Autonomous System may report false link status to redirect traffic to another AS Different “components” which

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Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents

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  1. Improvements for Truthful Mechanisms with Verifiable One-Parameter Selfish Agents Carmine Ventre Joint work with A. Ferrante, G. Parlato and F. Sorrentino

  2. Selfish agents • An Autonomous System may reportfalse link status to redirect traffic to another AS • Different “components” which • have their own goal • may not follow the “protocol” The Internet Link down AS1 Selfish agents source destination AS2

  3. One-Parameter Selfish Agents • Selfish agents own the links and privately know their speeds(one single number) • How to compute opt(s1,…, sm)? Routing/Scheduling s1 b1 source destination s2 b2 bm sm J1, …, Jn Unsplittable traffic (jobs) GOAL: compute opt(s1,…, sm) e.g. minimize the makespan (maxi worki/si)

  4. Mechanism design Mechanism: M=(A, P) Computes a schedule X = A(b1, …, bm) Provides a payment Pi(b1, …, bm) Agents’ GOAL: maximize their own utility ui(b1, …, bm) := Pi(b1, …, bm) – costi(X)

  5. Truthful Mechanisms for One-Parameter Selfish Agents A(b1, …, bm) solution … … wm(b1,…,bm) w1(b1,…,bm) wi(b1,…,bm) work ti = 1/si is the i’s type cost wi(b1,…,bm) ¢ti ui(b1, …, bm) = Pi(b1,…, bm) – wi(b1,…,bm) ¢ti utility Truthfulness: ui(ti,b-i) ¸ ui(bi,b-i) 8 b-i, bi with bi2i i’s type set

  6. Prior Works • Concept of mechanism with verification (observe jobs’ release time) [Nisan & Ronen, 99] • Truthful mechanisms for one-parameter selfish agents [Archer & Tardos, 01] • Truthful mechanisms with verification for one-parameter selfish agents [Auletta et al, 04]

  7. Our main contribution [Prior] • Payments computation depends by i • No polynomial-time in general • It works only in some case • Payments computation does not depend on i • It does not require finite I • Polynomial-time • Preserve approximation ratio • Solves for the continuous case [Ours]

  8. Verifiable One-Parameter Selfish Agents Verification = observe jobs’ release time 3 Verification is impossible! costi(X, ti) = wi(X) ¢ti i={1/2, 1, 2} 1 i underbids 1/2 i’s release time should be 2 but… … i’s finishing time is 4 ti= 1 i can wait 2 time slots delivering the results in the right time 1 i overbids 1 2 IDEA ([NR99]): No payment for underbidding agents

  9. Weakly-Monotone Algorithms Truthful mechanism with verification for one-parameter agents must use weakly-monotone algorithms ([ADPP04]) wi(bi, b-i) bi

  10. An easier and more powerful payment function speeds are integers A weakly-monotone (A, P(1)) is truthful ) Pi(1)(bi, b-i)=Wmax/ bi (= Wmax¢ s’i) Payment ¸ Cost Proof idea: si2 N ) Wmax¢ si¸ Wmax ¢ si-1(*) ¸1 ·1 Verification  No payment by (*) ui(bi) · 0, ui(ti)¸ 0 true - false (utility) = (payment) - (cost) (payment) ¸(cost) Wmax is an upper bound to the work assigned by A bi ti bi

  11. (payment) ¸(cost) “Proof” ti = 1/si & bi = 1/(si-1) (payment) = Wmax Pi(ti) = Wmax¢si Pi(bi) = Wmax¢ (si-1) ) Pi(ti) - Pi(bi) = Wmax wi(si) – wi(si-1) wi(si) (cost) · Wmax (cost) = · si si ·Wmax/si· Wmax

  12. Generalization of P(1) • Upper-bounded and discrete type sets: • Pi(2)(bi, b-i)= Pi(1)(bi, b-i) ¢ ci(2) • ci(2) is a suitable constant • Applications • CPU speeds are expressed as multiple of (M)Hz (discrete) • It does not exist CPU of 100.83 Mhz • “good” solution don’t use very slow machines (upper-bounded)

  13. P(2) and our results

  14. Continuous type sets: generalization of P(2) rounding (b1, …, bm) (b1R, …, bmR) “discrete” “continuous” (-r1, …, -rm) si siR ri-1 ri Pi(3)(bi, b-i)= Wmax/ biR¢ ci(3) ci(3) constant (value depends by rmin and ) rmindepends by the minimum possible speed

  15. P(3) and our results • computation time is independent from the chosen  • smaller ) better apx ratio but larger payments • bigger upper bound ) larger payments

  16. Makespan problem in general environments i’s discrete but not finite? i’s finite & discrete Use P(2) Are i’s upper bounded? [ADPP04] characterization i’s continuous? don’t care Use P(3) “good” algorithms induces upper bounded type sets

  17. Conclusions • Power of verification • Payments more efficient then the previous ones • In (many) real life applications we can preserve the approximation ratio of existing algorithms • Weak-monotonicity suffices (for truthful mechanisms) in more general settings • Open problems • Unbounded continuous type sets in general • Running time vs Amount of money (Tradeoff)

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