Large population games in radial loss networks: Computationally tractable equilibria for distributed network admission control

The computational intractability of the dynamic programming (DP) equations associated with optimal admission and routing in stochastic loss networks of any non-trivial size (Ma et al, 2006, 2008) leads one to consider suboptimal distributed game theoretic formulations of the problem. The special class of radial networks with a central core of infinite capacity is considered, and it is shown (under adequate assumptions) that an associated distributed admission control problem becomes tractable asymptotically, as the size of radial network grows to infinity. This is achieved by following a methodology first explored by M. Huang et. al. (2003, 2006–2008) in the context of large scale dynamic games for sets of weakly coupled linear stochastic control systems. At the established Nash equilibrium, each agent reacts optimally with respect to the average trajectory of the mass of all other agents; this trajectory is approximated by a deterministic infinite population limit (associated with the mean field or ensemble statistics of the random agents) which is the solution of a particular fixed point problem. This framework has connections with the mean field models studied by Lasry and Lions (2006, 2007) and close connections with the notion of oblivious equilibrium proposed by Weintraub, Benkard, and Van Roy (2005, 2008) via a mean field approximation.