Profit Maximizing Mechanisms for the Multicasting Game

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

3. The Multicasting Game 6 4 30 5 6 10 15 3 18 6 Edges with costs ce 16 19 10 30 12 20 14 8 Shuchi Chawla, Carnegie Mellon University

4. Not served The Multicasting Game 6 4 30 5 6 10 15 3 18 6 16 19 10 30 12 20 14 8 Shuchi Chawla, Carnegie Mellon University

5. The Multicasting Game • Task: 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 Maximizationmax Spi– Spe Shuchi Chawla, Carnegie Mellon University

6. Previous Work • Known edge costs • 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. u+d u , u The optimal solution serves both clients Any approximation to efficiency must do the same 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. c u pi = pi(c) pe = pe(u) An example (Assume u>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 • Obtaina=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