Multicasting Game Profit Maximization Mechanisms

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