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Pal-Tardos Mechanism

This seminar will discuss the cost-sharing mechanism in algorithmic game theory, focusing on the Moulin & Shenker mechanism and known cross-monotonic functions. It will also cover easy facts, constructing a solution, and cost recovery.

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Pal-Tardos Mechanism

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  1. Pal-Tardos Mechanism Seminar on Algorithmic Game Theory

  2. Cost sharing function • ξ : 2UU ℛ+ • ξ(S,j) – cost share of user j, given set S • Competitiveness: ΣjSξ(S,j)≤c*(S) • Cost recovery: c(S)/β≤ΣjSξ(S,j) • Voluntary participation: ξ(S,j) = 0 if jS • Cross-monotonicity: for jST • ξ(S,j) ≥ ξ(T,j) Seminar on Algorithmic Game Theory

  3. The Moulin&Shenker mechanism • S := U • repeat • ask each user i • “is ξ(S,i)≤ ui ?” • drop all iS who say NO • until all iS say YES • Output: set S; prices pi = ξ(S,i) Theorem: [Moulin&Shenker]: ξ(.) cross-monotonic  group strategyproof mechanism Seminar on Algorithmic Game Theory

  4. Known cross-monotonic functions • Exact cross-monotonic sharing exists if c*() submodular (Shapley value; as in the Multicast example) • [Moulin&Shenker 98] • Exact cost sharing for spanning tree • [Kent&Skorin-Kapov 96], [Jain&Vazirani 01] • Facility location and ROB games • [Pal&Tardos 03] implies 2-approx. cost sharing for Steiner tree Seminar on Algorithmic Game Theory

  5. cost shares previous algorithms... each user j raises its j until connected j pays for connection first, then for facility if facility paid for, declared open =5 =5 =6 Seminar on Algorithmic Game Theory

  6. ...do not yield cross-monotonic shares previously, ()=6 with , ()=8 helped  to stop earlier  failed to help  =3 =3 =8 Seminar on Algorithmic Game Theory

  7. Ghost shares After i freezes, continue growing its ghosti ghosts keep growing forever =3 =3 =5.5 Seminar on Algorithmic Game Theory

  8. Easy facts Fact 1: cost shares j are cross-monotonic. Pf: More users opens facilities faster  each j can only stop growing earlier. Fact 2 [competitiveness]: ΣjSj≤c*(S). Pf: j is a feasible LP dual. Hard part: cost recovery. Seminar on Algorithmic Game Theory

  9. q tq ≤tp/3 r tr ≤tq/3 Constructing a solution (1) Sp: set of users contributing to p at time of opening – “contributor set” tp: time of opening facility p tp Sp p facility p is well funded, if j≥tp/3 for every jSp Close down all facilities that are not well funded Lemma: For every facility p there is a nearby well funded facility r s.t. dist(p,r) ≤ 2(tp- tr) Seminar on Algorithmic Game Theory

  10. well-funded open p r q ≤ ≤2(tp- tr) ≤2tr Constructing a solution (2) Problem: user contributing to multiple well-funded facilities Solution: close all of them but one (process by increasing tp) tp Sp p Lemma: For every well funded facility p there is a nearby open q such that dist(p,q) ≤ 2tp tq Sq q Seminar on Algorithmic Game Theory

  11. tp Sp p well-funded open p r q ≤ ≤2(tp- tr) ≤2tr Cost recovery Fact 3: p open  clients in Sp can pay 1/3 their connection + facility cost Pf: fp = ΣjS(p)tp – cjp and j≥tp/3 Fact 4: j is in no Sp can pay for 1/3 of connection Pf: Seminar on Algorithmic Game Theory

  12. Pal-Tardos Summary • Cost shares can pay for 1/3 of solution constructed • Never pay more than cost of the optimum • With increasing # of users, individual share only decreases – cross monotonicity • But yet doesn’t have the strongest property... Seminar on Algorithmic Game Theory

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