1. On Cost Sharing Mechanisms in the Network Design Game Rohit Khandekar IBM Watson Joint work with Baruch Awerbuch JHU TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AAAA A

2. Network Design Problem c e = cost

3. Network Design Problem Connect all pairs at minimum total cost.

4. Network Design Problem

5. Selfish Interactions and Cost Sharing Path between each pair is chosen by a greedy agent. Cost sharing mechanism: how these agents split the cost of the shared edges. Fair share: if e is used by n agents, each pays [Anshelevich et al. 2004] Under “fair share”, the greedy moves may lead to equilibria that are factor worse that the optimum solution. (Here k = # agents.)

6. Our Goal How to remedy this situation? Do there exist other cost sharing mechanisms that induce greedy agents to a near-optimal solution quickly? (We have a partial positive answer.)

7. Fractional Network Design Each agent splits its flow along several paths between its source-sink pair. Our main result: There is a cost-sharing mechanism such that starting from an arbitrary configuration, the concurrent greedy moves of the agents converge to approximation in time

8. Our Cost Sharing Mechanism Let denote the flow of agent i on edge e. We split the cost of edge e as: where Thus agent i reroutes its flow to minimize his cost

9. Proof of Convergence The cost of the solution is This can be approximated by the following potential where is a good approximation for max if ® is large. log

10. Proof of Convergence We argue that the potential decreases due to greedy moves of the agents. It decreases fast if we are far from the optimum.

11. Thank You