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Sharing the Cost of Multicast Transmissions J. Feigenbaum, C. Papadimitriou, S. Shenker Hong Zhang, CIS620, 4/24.

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Presentation on theme: "Sharing the Cost of Multicast Transmissions J. Feigenbaum, C. Papadimitriou, S. Shenker Hong Zhang, CIS620, 4/24."— Presentation transcript:

1 Sharing the Cost of Multicast Transmissions J. Feigenbaum, C. Papadimitriou, S. Shenker Hong Zhang, CIS620, 4/24

2 Problem Outline  S,  1, …,  6 : Network Nodes 33 SS 11 55 66 22 44

3 Problem Outline 33 SS 11 55 66 22 44 Live Concert, Movie, etc.

4 Problem Outline 33 SS 11 55 66 22 44 : Users

5 Problem Outline 33 SS 11 55 66 22 4 Network Link Costs Users’ Utilities

6 Problem Outline 33 SS 11 55 66 22 4 Unicast vs. Multicast

7 Problem Outline 33 SS 11 55 66 22 4 Which users receive services? 2.How much do receivers pay?

8 Problem Notations (1) N = {  S,  1,  2, …,  n }: Set of network nodes. L: Set of bi-directional network links. P = {1, 2, … i, …, p }: The user population. u i : User i ‘s utility.  i : User i receives service (  i = 1) or not (  i = 0). x i : User i ‘s shared cost. w i : User i ‘s individual welfare ( w i =  i u i – x i ). Unknown

9 Assumption 0: Multicast Tree 33 SS 11 55 66 22 4 Source T( i ): fixed path from source to i Simplify problem

10 Problem Notations (2) N, L, P, u i,  i, x i, w i =  i u i – x i. R  P: Receiver set Construct a multicast tree T(R) =  i  R T( i )  L c(T(R)): The cost of the tree T(R) reaching R, c(T(R)) =  l  T(R) c( l ) Total Welfare NW(R) = u R – c(T(R)) = Σ i  R u i – c(T(R))

11 Assumptions 0. Nondecreasing: c(T(R + i ))  c(T(R)) Submodular: c(T(R 1 ))+c(T(R 2 ))  c(T(R 1  R 2 )) + c(T(R 1  R 2 )) 1.No Positive Transfers (NPT): shared costs are positive ( x i ( u )  0 ) 2.Voluntary Participation (VP): reporting u i = 0 ensures  i = 0 ( w i ( u )  0 ) 3.Consumer Sovereignty (CS): reporting a high u i ensures  i = 1

12 Incentive Compatible Strategyproof mechanism –Telling the true u i is a dominant straegy for any user.  u, u i ', w i ( u 1, u 2, … u i, …, u p )  w i ( u 1, u 2, … u i ', …, u p )

13 Desired Properties Under incentive compatible mechanism Budget Balance:  i  P x i = c(T(R)) –The money raised from receivers covers the cost of transmission exactly. Efficiency: NW(R * ) = [ u R – c(T(R)) ] –The receiver set maximizes the overall benefit of the network. Notice Total Welfare (NW(R)) and Efficiency does not depend on shared costs x i

14 Desired Properties - Example Source 3, 32, 42, Source 3, 32, 42, Source 3, 32, 42, Budget Balanced Link Cost Utility Shared Cost Efficiency Source 3, 32, 42,

15 Desired Properties Under incentive compatible (strategyproof) mechanism Budget Balance & Efficiency are mutually exclusive. Only one strategyproof cost-sharing mechanism is efficient: Marginal Cost Mechanism. –Maximize overall benefit. There are many possible mechanisms for budget balance, among which the most efficient one: Shapley Value Mechanism. –Cover the cost.

16 Marginal Cost Mechanism R*( u ): The largest efficient receiver set W( u ) = NW(R*( u )) Each receiver pays marginal cost: x i = u i  i ( u ) – (W( u ) – W( u | u i = 0)) Source 3, 34, 22, , 1 3 3, 1 Link Cost Utility Shared Cost

17 Marginal Cost Mechanism Theorem 3.1, MC cost sharing requires exactly two messages per link. W  ( u ) : welfare from the subtree rooted at  W  ( u ) = u  + [  W  ( u ) ] - c  –child(  ) is all the child nodes in the subtree –u  is the sum of the utilities of the user in  –C  the cost of the link between  and its parent Source 14 2  child(  ) | W  (u)  0 3, 34, 22, C  = 3 

18 Marginal Cost Mechanism If W  ( u )  0, then  i ( u ) = 1; else  i ( u ) = 0. y i ( u ) = min w  ( u ) If u i  y i ( u ), then x i ( u ) = 0; If u i  y i ( u ), then x i ( u ) = u i - y i ( u ), ;  node on the path from i to the root Source , 34, C  = 3  1, 13, 1 2, 2 5

19 Marginal Cost Mechanism Exactly 2 messages per link 1. Bottom Up: Calculate W  ( u ) for each node. 2. Top Down: Propagate  i ( u ) y i ( u ) and x i ( u ), allocation and cost. Source , 34, C  = 3  1, 13, 1 2, 2 5

20 Shapley Value Mechanism The cost of a link l is shared equally by all receivers who are downstream of the link. Receiver set is the largest possible. Source 3, 34, 12, Link Cost Utility Shared Cost

21 Shapley Value Mechanism In each iteration, users with u i < x i are dropped and other users’ prices are recomputed

22 Shapley Value Mechanism n = | Network Nodes |; p = | Population | Theorem 5.1, The algorithm (brute-force) requires O( n  p ) message exchanges. Theorem 5.2, There is an infinite family of multicast computations, with n nodes and O( n ) users, such that any linear distributed algorithm that implements the SH mechanism requires in the worst case O( n 2 ) message exchanges

23 Conclusion Sharing multicast cost No Positive Transfers, Voluntary Participation, and Consumer Sovereignty Strategyproof (incentive compatible) mechanism Efficiency vs. Budget Balance Marginal Cost – Efficiency, 2 Messages Implementing, but Budget Deficit Shapley Value – Budget Balanced, O( n 2 ) Complexity, Feasible Problem


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