Profit Guaranteeing Mechanisms for Multicast Networks

1. Profit Guaranteeing Mechanisms for Multicast Networks Shuchi Chawla, David Kitchin, Uday Rajan, R. Ravi, Amitabh Sinha Carnegie Mellon University

2. The Multicasting Game Given: Root node r Edges with private costs ce Vertices with private utilities ui 6 4 30 6 5 10 15 3 r 18 6 16 19 10 Not served 30 12 20 14 8 Tree T of served vertices and chosen edges

3. The Multicasting Game • Task: Select tree T rooted at r in polynomial time Charge fees fi from nodes Assign payments fe to edges • Goals: • Strategyproofness, Individual Rationality • Efficiency maxT u(T) - c(T) • Budget surplus STfiSTfe • Profit MaximizationmaxT (STfi– STfe)

4. Previous Work • Known edge costs – agents only at nodes • Budget surplus, but no guarantee of efficiency eg. Shapley value [MS’01], primal-dual [JV’01] • Efficiency – Marginal cost [MS’01] Budget deficit, computationally inefficient • No node utilities – agents only at edges • connect all nodes • Simple polytime mechanism – MST based on Vickrey costs charged to edges [BVSV’02]

5. Profit Maximization: Why is it hard? • Efficiency is inapproximable in polytime • NP-hard to determine whether a positiveefficiency solution exists • Reduction from decision version of Prize Collecting Steiner Tree (PCST) • Game theoretic complexity • Profit maximization requires good Efficiency and Budget Surplus • Efficiency and Budget Balance cannot be simultaneously approximated [FKSS’02]

6. (a,b)-Profit Guarantee • a  [0,1];b  1 • If T* with efficiency= dU > (1-a)U, we find T with profit = k(d)U(kis increasing in d) • 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 with no guarantee on profit

7. c u fi = fi(c) fe = fe(u) Importance of competition (Assume u>c) • Functions fi and fe are bid-independent and non-decreasing. • Keeping c constant, increase u • Efficiency of solution increases • Our profit decreases or stays the same What went wrong? Need competition among nodes and among edges Example generalizes to the case of many nodes and clients if fi depends only on c or fe only on u Henceforth, we assume sufficient competition among agents

8. A candidate mechanism • Run an auction at every node to generate revenue • Select a set of nodes and edges based on revealed edge costs and node revenue • Pay edges their Vickrey costs • (analogous to running an auction at edges)

9. The details.. auctions at nodes & edges • Auction at nodes • Some nodes are not selected in the solution • How do we figure out where to run the auction? • “Cancelable auctions” • [FGHK’02] give a 4-approximate cancelable auction • Vickrey auction at edges • If we assume that there is sufficient competition among edges, we pay only a factor of (1+e) extra

10. The details.. constructing the tree • Selecting the final solution set • We use a well known 2-approximation for the Prize Collecting Steiner Tree problem [GW’95] • If f(T*) = (1-g)U, we get f(T) = (1-2g)U • We lose a factor of 4(1+e) from the auctions • Obtain a=⅛(1+e), k(d) = 1 - 8(1+e)(1-d) • The efficiency of the solution is at least (2d-1)U

11. The details.. constructing the tree • If there is no profitable solution… • If GW returns empty tree, rerun with u’=2u • If every tree T has c(T) > 2u’(T) = 4u(T), then GW with u’=2u returns empty solution • If GW with u’=2u returns empty solution, then there is no tree with c(T) < u(T) • Obtainb = 4

12. Future directions • Improve the (a,b) guarantee • Extend our technique and the definition of profit guarantee to other similar problems • For example, agents own subsets of edges, rather than individual edges