1. 22/6/04Seminar on Algorithmic Game Theory 1 Pal-Tardos Mechanism

2. 22/6/04Seminar on Algorithmic Game Theory 2 Cost sharing function ξ : 2 U  U  ℛ + ξ(S,j) – cost share of user j, given set S Competitiveness: Σ j  S ξ(S,j) ≤ c * (S) Cost recovery: c(S)/β ≤ Σ j  S ξ(S,j) Voluntary participation: ξ(S,j) = 0 if j  S Cross-monotonicity: for j  S  T ξ(S,j) ≥ ξ(T,j)

3. 22/6/04Seminar on Algorithmic Game Theory 3 The Moulin&Shenker mechanism S := U repeat ask each user i “is ξ(S,i) ≤ u i ?” drop all i  S who say NO until all i  S say YES Output: set S; prices p i = ξ(S,i) Theorem: [Moulin&Shenker]: ξ(.) cross-monotonic  group strategyproof mechanism

4. 22/6/04Seminar on Algorithmic Game Theory 4 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] Known cross-monotonic functions implies 2-approx. cost sharing for Steiner tree

5. 22/6/04Seminar on Algorithmic Game Theory 5 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  =6 cost shares

6. 22/6/04Seminar on Algorithmic Game Theory 6...do not yield cross-monotonic shares with,  ( )=8 helped to stop earlier failed to help  =3  =8 previously,  ( )=6

7. 22/6/04Seminar on Algorithmic Game Theory 7 Ghost shares After  i freezes, continue growing its ghost  i ghosts keep growing forever  =3  =5.5

8. 22/6/04Seminar 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]: Σ j  S  j ≤ c * (S). Pf:  j is a feasible LP dual. Hard part: cost recovery.

9. 22/6/04Seminar on Algorithmic Game Theory 9 Constructing a solution (1) S p : set of users contributing to p at time of opening – “contributor set” t p : time of opening facility p facility p is well funded, if  j ≥t p /3 for every j  S p SpSp tptp p 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(t p - t r ) q t q ≤t p /3 r t r ≤t q /3

10. 22/6/04Seminar on Algorithmic Game Theory 10 Constructing a solution (2) Problem: user contributing to multiple well-funded facilities Solution: close all of them but one (process by increasing t p ) SpSp tptp p SqSq tqtq q Lemma: For every well funded facility p there is a nearby open q such that dist(p,q) ≤ 2t p p r ≤≤ ≤2(t p - t r )≤2t r q well-fundedopen

11. 22/6/04Seminar on Algorithmic Game Theory 11 Cost recovery Fact 3: p open  clients in S p can pay 1/3 their connection + facility cost Pf: f p = Σ j  S(p) t p – c jp and  j ≥t p /3 Fact 4: j is in no S p  can pay for 1/3 of connection Pf: p r ≤≤ ≤2(t p - t r )≤2t r q well-fundedopen SpSp tptp p

12. 22/6/04Seminar 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...