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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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Problem Outline S, 1, …, 6 : Network Nodes 33 SS 11 55 66 22 44
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Problem Outline 33 SS 11 55 66 22 44 Live Concert, Movie, etc.
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Problem Outline 33 SS 11 55 66 22 44 : Users
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Problem Outline 33 SS 11 55 66 22 44 33 1 5 5 2 6 6 Network Link Costs 1 2 3 0 10 67 1 2 Users’ Utilities
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Problem Outline 33 SS 11 55 66 22 44 33 1 5 5 2 6 6 1 2 3 0 10 67 1 2 Unicast vs. Multicast
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Problem Outline 33 SS 11 55 66 22 44 33 1 5 5 2 6 6 1 2 3 0 10 67 1 2 1.Which users receive services? 2.How much do receivers pay?
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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
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Assumption 0: Multicast Tree 33 SS 11 55 66 22 44 33 1 5 5 2 6 61 2 3 0 10 67 1 2 Source T( i ): fixed path from source to i Simplify problem
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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))
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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
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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 )
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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
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Desired Properties - Example Source 3, 32, 42, 2 4 5 5 Source 3, 32, 42, 2 4 5 5 Source 3, 32, 42, 2 4 5 5 Budget Balanced Link Cost Utility Shared Cost Efficiency Source 3, 32, 42, 2 4 5 5
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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.
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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, 2 2 5 5 1, 1 3 3, 1 Link Cost Utility Shared Cost
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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, 2 2 5 5 C = 3
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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 14 2 3, 34, 2 2 5 C = 3 1, 13, 1 2, 2 5
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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 14 2 3, 34, 2 2 5 C = 3 1, 13, 1 2, 2 5
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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, 2 2 2 5 3 3 Link Cost Utility Shared Cost
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Shapley Value Mechanism In each iteration, users with u i < x i are dropped and other users’ prices are recomputed
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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
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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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