On A Locally Minimum Cost Forwarding Game

We consider the problem of all-to-one (reverse multicast) selfish routing in the absence of a payment scheme in wireless networks, where a. natural model for cost is the power required to forward. Whereas each node requires a path to the destination, it does not care how long that path is, so long as its own individual or local forwarding cost is minimized. Thus, we refer to this setting as a Locally Minimum Cost Forwarding Game (LMCF). From a system-wide perspective, short paths are clearly desirable, yielding two related social objectives of finding topologies that minimize: (i) the maximum stretch factor, and (ii) the directed weighted diameter. We prove that Nash equilibria always exist for LMCF, in particular the directed MST always being one, and we analyze the ratio of the social cost of Nash equilibria to the global optimum. The worst (maximum) possible value of this ratio is called the price, of anarchy (PoA), and the best (minimum) possible value is called the price of stability (PoS). For the maximum stretch factor we present a Omega(n) worst-case bound on PoA and PoS, and for the directed weighted diameter we present a omega(n(c)) worst-case bound on PoA and PoS for all c < 1, even when restricted to Euclidean instances. We prove hardness of computing the optimal Nash equilibrium in three-dimensional Euclidean instances as well as approximation hardness in arbitrary instances. Finally, we propose a heuristic for finding Nash equilibria and analyze, via simulations and probabilistic arguments, the social costs given by the heuristic and by the MST. These results suggest that for random Euclidean power instances, the expected PoA is omega(1) while the expected PoS is Theta(1).