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

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

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

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

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

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

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

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

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

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

(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

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

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

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

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

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

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

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