1. Profit Maximizing Mechanisms for the Multicasting Game
Shuchi Chawla
Carnegie Mellon University
Joint work with
David Kitchin, Uday Rajan, R. Ravi, Amitabh Sinha

2. The Multicasting Game Nodes with utilities ui root 6 12 30 10 20
Shuchi Chawla, Carnegie Mellon University

3. The Multicasting Game Edges with costs ce 6 4 30 5 6 10 15 3 18 6 1619 10 30 12 20 14 8
Shuchi Chawla, Carnegie Mellon University

4. The Multicasting Game Not served 6 4 30 5 6 10 15 3 18 6 16 19 10 3012 20 14 8
Shuchi Chawla, Carnegie Mellon University

5. The Multicasting Game Task: Efficiency max u(T) - c(T)
select tree T in polynomial time
assign payments pi to nodes and pe to edges
Efficiency max u(T) - c(T)
Budget balance Spi  Spe
Profit Maximization max Spi – Spe
Shuchi Chawla, Carnegie Mellon University

6. Previous Work Known edge costs No node utilities – connect all nodes
BB, but no guarantee of efficiency
Shapley value, Jain-Vazirani
Efficiency – Marginal cost
Budget imbalanced, computationally inefficient
No node utilities – connect all nodes
Simple mechanism based on Vickrey prices
[Bikhchandani et al]
Shuchi Chawla, Carnegie Mellon University

7. Achieving Budget Balance
Compute the MST
Use some cost division mechanism to distribute Vickrey costs among nodes
Prune the tree if necessary
Vickrey-MST stays truthful even if pruning is done.
Shuchi Chawla, Carnegie Mellon University

8. Profit Maximization Why is this problem hard?
Profit  Efficiency
Profit maximization requires good Efficiency and Budget Balance
Efficiency and Budget Balance cannot be simultaneously approximated
[Feigenbaum et al]
Shuchi Chawla, Carnegie Mellon University

9. The optimal solution serves both clients
Any approximation to efficiency must do the same
u+d
u , u
Strategyproofness  cannot charge more than d from either client
 Budget imbalance of u-d
Shuchi Chawla, Carnegie Mellon University

10. Computational Issues Efficiency is inapproximable in polynomial time
Determining whether there exists a non trivial positive efficiency solution is NP-hard
By reduction from decision version of Prize Collecting Steiner Tree (PCST)
Shuchi Chawla, Carnegie Mellon University

11. (a,b)-Profit Guarantee
If T* with f(T*)>(1-a)U, we find T with profit > k(a)U
If every tree T has c(T)>bu(T), we demonstrate that there is no positive efficiency solution
Else, we output a non negative profit solution.
Shuchi Chawla, Carnegie Mellon University

12. An example Payment functions are bid independent
(Assume u>c)
pe = pe(u)
pi = pi(c)
Payment functions are bid independent
pi and pe are increasing functions.
Keeping c constant, increase u
Efficiency of solution increases
Our profit decreases
Shuchi Chawla, Carnegie Mellon University

13. What went wrong? Need competition among nodes and among edges
Example generalizes to the case of many nodes and clients if pi depends only on c or pe only on u.
Shuchi Chawla, Carnegie Mellon University

14. A candidate mechanism Run an auction at every node to generate revenue
Select a set of nodes and edges based on true edge costs and node revenue
Pay edges their Vickrey costs
(analogous to running an auction at edges)
Shuchi Chawla, Carnegie Mellon University

15. The details Auction at nodes
Some nodes are not selected in the solution
How do we figure out where to run the auction?
“Cancelable auctions”
Fiat et al give a 4-approximate c.a.
Shuchi Chawla, Carnegie Mellon University

16. The details Selecting the final solution set
We use a well known 2-approximation for the Prize Collecting Steiner Tree problem
[Goemans Williamson]
Gives a profit guarantee when f(T*) > ¾U
Shuchi Chawla, Carnegie Mellon University

17. The details Vickrey auction at edges
If we assume that there is sufficient competition among edges, we pay only a factor of (1+e) extra
Obtain a=1/16(1+e)
Shuchi Chawla, Carnegie Mellon University

18. Future directions Improve the (a,b) guarantee Lower bound
e.g. Improving a or b would give a better approximation to PCST
Shuchi Chawla, Carnegie Mellon University