The essence of the routing problem in real networks is that the traffic demand from a source to destination must be satisfied by choosing a single path between source and destination. The splittable version of this problem is when demand can be satisfied by many paths, namely a flow from source to destination. The unsplittable, or discrete version of the problem is more realistic yet is more complex from the algorithmic point of view; in some settings optimizing such unsplittable traffic flow is computationally intractable. In this paper, we assume this more realistic unsplittable model, and investigate the "price of anarchy", or deterioration of network performance measured in total traffic latency under the selfish user behavior. We show that for linear edge latency functions the price of anarchy is exactly $2.618$ for weighted demand and exactly $2.5$ for unweighted demand. These results are easily extended to (weighted or unweighted) atomic "congestion games", where paths are replaced by general subsets. We also show that for polynomials of degree $d$ edge latency functions the price of anarchy is $d^{\Theta(d)}$. Our results hold also for mixed strategies. Previous results of Roughgarden and Tardos showed that for linear edge latency functions the price of anarchy is exactly $\frac{4}{3}$ under the assumption that each user controls only a negligible fraction of the overall traffic (this result also holds for the splittable case). Note that under the assumption of negligible traffic pure and mixed strategies are equivalent and also splittable and unsplittable models are equivalent.

Joint work with B. Awerbuch and A. Epstein