Sharing the Cost of Multicast
Transmissions
J. Feigenbaum, C. Papadimitriou, S. Shenker
Hong Zhang, CIS620, 4/24
Problem Outline
S, 1, … ,6:
Network Nodes
S
1
2
3
5
4
6
Problem Outline
Live Concert,
Movie, etc.
S
1
2
3
5
4
6
Problem Outline
: Users
S
1
2
3
5
4
6
Problem Outline
3
3
5
2
3
S
1
1
Network Link Costs
Users’ Utilities
3
2
0
6
2
4
5
1
6
10
5
6
6
1
7
2
Problem Outline
3
3
5
2
3
S
1
1
Unicast vs. Multicast
3
2
0
6
2
4
5
1
6
10
5
6
6
1
7
2
Problem Outline
1. Which users
receive services?
2. How much do
receivers pay?
3
S
1
1
3
5
2
3
3
2
0
6
2
4
5
1
6
10
5
6
6
1
7
2
Problem Notations (1)
Unknown
N = {S, 1, 2, …, n}: Set of network nodes.
L: Set of bi-directional network links.
P = {1, 2, … i, …, p }: The user population.
ui: User i‘s utility.
3
3
i: User i receives
S
1
service (i = 1)
6
3
1
or not (i = 0).
5
5
xi: User i‘s shared cost. 2
1
6
wi: User i‘s individual
6
5
welfare (wi = i ui – xi). 10
6
3
2
0
2
4
1
7
2
Assumption 0: Multicast Tree
Source
T(i): fixed path
from source to i 3
S
1
1
3
5
2
3
3
2
0
6
2
4
5
1
6
10
5
Simplify 
6
problem
6
1
7
2
Problem Notations (2)
• N, L, P, ui, i, xi, wi = i ui – xi.
• R  P: Receiver set
• Construct a multicast tree T(R) = iRT(i)  L
• c(T(R)): The cost of the tree
T(R) reaching R,
1
c(T(R)) = lT(R)c(l)
Source
3
S
10
2
3
5
2
3
0
1
• Total Welfare
NW(R) = uR – c(T(R))
= ΣiRui – c(T(R))
3
2
4
5
1
5
6
6
1
7
2
Assumptions
0. Nondecreasing: c(T(R + i))  c(T(R))
Submodular: c(T(R1))+c(T(R2)) 
c(T(R1  R2)) + c(T(R1  R2))
1. No Positive Transfers (NPT):
shared costs are positive ( xi(u)  0 )
2. Voluntary Participation (VP):
reporting ui = 0 ensures i = 0 ( wi(u)  0 )
3. Consumer Sovereignty (CS):
reporting a high ui ensures i = 1
Incentive Compatible
• Strategyproof mechanism
– Telling the true ui is a dominant straegy
for any user.
 u, ui', wi(u1, u2, … ui, …, up) 
wi(u1, u2, … ui', …, up)
Desired Properties
Under incentive compatible mechanism
• Budget Balance: iP xi = c(T(R))
– The money raised from receivers covers the cost
of transmission exactly.
• Efficiency: NW(R*) = max [ uR – c(T(R)) ]
R P
– The receiver set maximizes the overall benefit of
the network.
Notice Total Welfare (NW(R)) and Efficiency
does not depend on shared costs xi
Desired Properties - Example
Link Cost
Utility
Source
Source
Shared Cost
4
3, 3
5
2, 4
5
4
2, 2
3, 3
5
5
2, 4
2, 2
Budget Balanced
Efficiency
2, 2
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.
Marginal Cost Mechanism
• R*(u): The largest efficient
receiver set
• W(u) = NW(R*(u))
• Each receiver pays marginal
cost: xi = ui i(u) –
(W(u) – W(u | ui = 0))
2
Link Cost
Utility
Shared Cost
Source
3
5
3, 3
4, 2
1, 1
3, 1
5
2, 2
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() | W (u)  0
– child() is all the child
nodes in the subtree
Source
C = 3
– u is the sum of the
utilities of the user in 
–
C
the cost of the link
between  and its parent
2
2
4
3, 3
5
1
4, 2

5
-1
2, 2
Marginal Cost Mechanism
• If W(u)  0, then i(u) = 1; else i(u) = 0.
•
yi(u) =
min
 node on the path
Source
w(u)
from i to the root
C = 3
2
• If ui  yi(u), then xi(u) = 0;
2
• If ui  yi(u), then
xi(u) = ui - yi(u), ;
4
3, 3
1, 1
5
1
4, 2
3, 1

5
-1
2, 2
Marginal Cost Mechanism
Exactly 2 messages per link
1. Bottom Up: Calculate W(u)
for each node.
2. Top Down: Propagate i(u)
yi(u) and xi(u), allocation
and cost.
Source
C = 3
2
2
4
3, 3
1, 1
5
1
4, 2
3, 1

5
-1
2, 2
Shapley Value Mechanism
| R |!(| R | - | R | -1)!
xi (R) = 
[c(T(R  i )) - c(T(R))]
| R |!
R R- i
Source
•
•
The cost of a link l is
shared equally by all
receivers who are
downstream of the link.
Receiver set is the
largest possible.
Link Cost
Utility
Shared Cost
3
2
2
3, 3
4, 1
2, 2
3
5
2, 2
Shapley Value Mechanism
6
2
4
6
6
4
2
4
5, 5
4, 1
2, 2
5, 5
4, 1
2, 2
3, 3
3, 3
3, 3
5
4
2
2, 2
5, 5
4
4, 1
5
2, 2
4, 4
In each iteration, users with ui < xi are dropped and other
users’ prices are recomputed
Shapley Value Mechanism
•
n = | Network Nodes |;
•
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(n2) message exchanges
p = | Population |
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(n2)
Complexity, Feasible Problem