1. Sharing the cost of multicast transmissions in wireless networks Carmine Ventre Joint work with Paolo Penna University of Salerno, WP2

2. Wireless transmission Power(i)= d(i,j) α = range(i) α, α>1 (empty space α = 2) A message sent by station i to j can be also received by every station in transmission range of i i j d(i,j) α

3. Wireless multicast transmission Who receives Roma-Juventus How to transmit Goal: maximize Benefit – Cost i.e. the social welfare Paolo 1€ 10€1€ 3€ Carmine 1€Christos 10€Andrea 30€ Pino 50€ known private source

4. Selfish agents COST = 10 + 5 = 15 WORTH = 50 + 30 = 80 NET WORTH = 80 – 15 = 65 source 10 5 Pino 50 € Andrea 30 € Paolo 9 € 0 € Pino says 0 € and gets Roma – Juventus for free 5.1 € Andrea says 5.1 € and gets Roma – Juventus for a lower price Andrea says 5.1 € Pino says 0 € Nobody gets Roma - Juventus NW’ = 0 WYSWYP (What You Say What You Pay)

5. Graph model A complete directed weighted communication graph G=(S,E,w) w(i,j) = cost of link (i,j)  w(1,4) = d(1,4) 2.1  w(1,2) = d(1,2) 5  w(2,4) = ∞  w(4,2) = d(4,2) 2.1 A source node s v i = private valuation of agent i 21 43v4v4 v3v3 v1v1 v2v2

6. Mechanism design: model Design a mechanism M=(A,P)  Each agent declares b i  Algorithm A selects, based on (b 1, …, b n ), a set of receivers a subset of connection T  E  Agent i must pay P i (b 1, …, b i-1, b i, b i+1..., b n ) Utility of the agent u i (b i )= Goal of agent i: maximize u i (b i )

7. Mechanism’s desired properties No positive transfer (NPT)  Payments are nonnegative: P i  0 Voluntary Participation (VP)  User i is charged less then his reported valuation b i (i.e. b i ≥ P i ) Consumer Sovereignty (CS)  Each user can receive the transmission if he is willing to pay a high price.

8. Mechanism’s desired properties: Incentive Compatibility Strategyproof (truthful) mechanism  Telling the true v i is a dominant strategy for any agent Group-strategyproof mechanism  No coalition of agents has an incentive to jointly misreport their true v i Stronger form of Incentive Compatibility.

9. Mechanism’s desired properties Budget Balance (BB)   P i = COST(T) (where T is the solution set) Efficiency (NW)  the mechanism should maximize the NET WORTH(T) := WORTH(T)-COST(T) where WORTH(T):=  i  T v j Mutually exclusive!! Efficiency  No Group strategy-proof

10. Previous work Wireless broadcast  1d: COST opt in polynomial time [Clementi et al, to appear]  2d: NP-hard, MST is an O(1)-apx [Clementi et al, ‘01]  On graphs:  (log n)-apx [Guha et al ‘96, Caragiannis et al, ‘02]  Many others… Wired cost sharing (selfish receivers)  Distributed polytime truthful, efficient, NPT, VP, and CS mechanism for trees (no BB) [Feigenbaum et al, ‘99]  Budget balance, NPT, VP, CS and group strategy-proof mechanism (no efficiency) [Jain et al, ‘00]  No α-efficiency and β-BB for each α, β > 1 [Feigenbaum et al, ‘02]  polytime algorithm  no R-efficiency, for each R > 1 [Feigenbaum et al, ‘99]

11. Our results G is a tree  NW opt in polytime distributed algorithm  Polytime mechanism M=(A,P) truthful, NPT, VP and CS  Extensions to “metric trees” graphs G is not a tree  2d: NP-hard to compute NW opt  1d: Polytime mechanism M=(A,P) truthful, NPT, VP, CS and efficient (i.e. NW is maximized)  Precompute an universal multicast tree T  G A polytime truthful, NPT, VP and CS mechanism O(1) or O(n)-efficiency, in some cases  polytime algorithm  no R-efficiency, for every R > 1

12. VCG Trick (marginal cost mechanism) Utilitarian problem:   X  sol, measure(X)=  i valuation i (X) A opt computes X  sol maximizing measure(X)  P VCG : M=(A opt, P VCG ) is truthful

13. VCG Trick (marginal cost mechanism) Making our problem utilitarian: measure(X) valuation i (X) WORTH(X)-COST(X) =  i iXiX vivi = WORTH(X) vivi cici Initially, charge to every receiver i the cost c i of its ingoing connection - c i - COST(X) P i = c i + P VCG

14. Free edges on Trees 21 4 3 5 s graph tree 21 4 3 5 s RECURSION? NO! YES! 34 45 45 43

15. Trees algorithm: recursive equation It is easy to see that the best solution has an optimal substructure It is simple to compute NW opt (s) in distributed bottom-up fashion O(n) time, 2 msgs per link k s.t. c k ≤ c j i j cjcj vivi

16. Trees with metric free edges Path(i,4)=w(i,1)+w(1,4) w(i,3) ≥ path(i,4) (i,4) metric free edge 21 4 3 5 i 15 7 5 6

17. Tree with metric free edge: idea A node k reached for free gets some credit i j cjcj k gets c j -c k units of credit k ckck

18. Tree with metric free edge: credit usage k can use its credit to reach all of its children If there is a child l s.t. c l > credit(k) and NW opt (l)>0 then credit(k) is useless  For each r Є ch(k): c l – c r > credit(k) – c r Paying a free edge is not a good solution (i.e. we have a smallest credit and a greater cost) k k l r credit(r)=c l -c r r credit(r) = credit(k)-c r credit(l)=0

19. Tree with metric free edge: recursive equations We have two contributions:  the nodes whose ingoing edge is paid  the nodes with credit c whose ingoing edge is free NOTE: the optimum is NW opt (s,0)

20. The one dimensional Euclidean case Stations located on a line (linear network) s ij 1 n receivers Clementi et al algo

21. (Some) Open problems 2d Euclidean case:  O(1)-APX multicast algorithm  “Good” universal Euclidean multicast trees  Truthful mechanism with O(1)-APX  BB truthful mechanisms