Free-Riders in Steiner Tree Cost-Sharing Games Paolo Penna and Carmine Ventre Università di Salerno

Cost-Sharing Games U Q 1.Which customers to service? 2.At which price? S Service providerCustomers user i wants to pay at most v i

Cost-Sharing Games U Q S Service providerCustomers S Multicast: S wiredwireless

Cost-Sharing Games U Q S Service providerCustomers 1. Budget balance: Cost(Q) =  P i 2. Users can form coalitions  Group strategyproof mechanisms

Group Strategyproof Mechanisms user i wants to pay at most v i Private knowledge Service i and charge P i Don’t service i 0 v i - P i Utility u i Pi’Pi’ Pi’Pi’ ui’ui’ riri

Group Strategyproof Mechanisms Customers U C uiui Truth-telling Lieui’ui’ None gains i At least one looses Coalition is useless u i ’ < u i Breaks off the coalition

Requirements 1.Budget Balance 2.Group Strategyproof 3.Voluntary Participation: P i  v i (unless i lies) 4.Consumer Sovereignity: r i “large enough”  service i 5.No Positive Transfer: P i  0 (do not pay customers)

Steiner Tree Game Given network G = (U  s, E, w), with s source node s Cost(Q) = cost of opt Steiner tree connecting s to Q

Related Work Polytime mechanisms: 2-APX budget balance [Jain & Vazirani, STOC01] Budget balance [Penna &Ventre, WAOA04] Mechanisms and free-riders: No free-riders [Immorlica et al, SODA05] Relax budget balance Ignores free-riders Ignores computational issues Budget balance + polytime + free-riders issue ?

Our Contribution N-L free-riders Less free-riders?Polytime? Polytime, budget balance, N-L free-riders (L = #leaf nodes) Budget balance, no free-riders  NP-hard No free-riders [IMM04] N-1 free-riders fairness [PV04]  -APX budget balance, no free-riders  NP-hard for some  > 1 (1+  )-APX BB Wireless case: similar results (BB  6-APX BB)

How to Build Mechanisms U Q   (Q,i) = Cost(Q) Cost-sharing methods: distribute Cost(Q) among users in Q  (Q,i)  0  (Q,i) = 0, i  Q Idea: associate prices to service set

How to Build Mechanisms Cost-sharing method  (, )  Mechanism M(  )  (Q,i) > r i U Drop i Q

U Q 1 =U How to Build Mechanisms Cost-sharing method  (, )  Mechanism M(  ) Q3Q3 QkQk … Q2Q2 Prices do not decrease Group Strategyproof  (Qk,i) (Qk,i)  (Q2,i) (Q2,i)  (Q3,i) (Q3,i) P i =  (Q k,i) Changes Monotonicity [Moulin & Shenker ’97] [PV04]

Our Method  no Steiner nodes  MST(U) is optimal s G = (U  s, E, w) Easy case: Q = U Hard case: Q “any”

U s Our Method MST pay s prune Q MST(Q) opt Steiner tree T +  = opt s u Q v  s u Q T s T*T* > Q u v  v  + 

Our Method Monotonicity: Prices do no lower  u v   / (L v -1) 0   / (L v -1) s  / L v LvLv

Q Consumer Sovergnity Hardness AnyU Budget balance, no free-riders  NP-hard vivi  0 Voluntary Participation P i  v i = 0 Free-riders

Consumer Sovergnity Hardness U Budget balance, no free-riders  NP-hard  -APX budget balance, no free-riders  NP-hard for some  > 1 Voluntary Participation Any Q = Budget balance  Must compute Cost(Any)

Open Questions Trade-offs (Polytime mechanisms) a-APX Budget balance + bN-free-riders Complete Metric Graphs Budget balance + (N/2) -free-riders?

Thank You