The price of anarchy in loss systems

The price of anarchy (PoA) measures the loss of social inefficiency due to customers behaving selfishly. We bound the PoA in parallel M/M/1/1 loss systems where each server is associated with a service rate and a social service completion reward.  The decision is how to route a Poisson stream of customers among the servers so that their expected long run reward is maximized.  We characterize (1) the symmetric Nash equilibrium solution under steady state conditions, where each customer maximizes his own reward; and (2) the solution that maximizes the expected social reward per customer.  We show that in both cases the set of open servers consists of servers that are associated with the largest rewards, independently of the servers' capacity. In addition, the set of open servers in equilibrium is a subset of the set of open servers in social optimization. The PoA is defined as the ratio between the solutions of social optimization and equilibrium.  Finally, we present some upper bounds on the PoA .
This research is a joint work with Moshe Haviv from the Hebrew University.

Shoshana Anily is a professor of optimization of logistics system and head of the Operational Strategy Department at the Coller School of Management of Tel Aviv University, Israel. She holds her Ph.D. in Management Science from the Columbia Business School.  She  held visiting positions at Columbia Business School in the US,  Chair of Prof. Leibling at the EPFL, Switzerland,  and Chair of Prof. Laporte at the CRT, Montreal University, Canada. Her research interests include cost allocation in cooperative games in supply chains and service systems, and performance analysis of heuristics in routing, inventory, and scheduling and production management