Network congestion games are robust to variable demand

We consider a non-atomic network congestion game with incomplete information in which nature decides which commodities travel. The users of a commodity do not know which other commodities travel and only have distributional information about their presence. Our main result is that the price of anarchy bounds known for the deterministic demand game also apply to the Bayesian game with random demand, even if the travel probabilities of different commodities are arbitrarily correlated. Moreover, the extension result of price of anarchy bounds for complete information games to incomplete information games in which the set of players is randomly determined can be generalized to the class of smooth games