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We consider an atomic congestion game in which each player $i$ participatesin the game with an exogenous and known probability $p_{i}\in(0,1]$,independently of everybody else, or stays out and incurs no cost. We computethe parameterized Price of Anarchy (PoA) to characterize the impact of demanduncertainty on the efficiency of selfish behavior, considering two differentnotions of a social planner. A prophet planner knows the realization of therandom participation in the game; the ordinary planner does not. As aconsequence, a prophet planner can compute an adaptive social optimum thatselects different solutions depending on the players that turn out to beactive, whereas an ordinary planner faces the same uncertainty as the playersand can only minimize the expected social cost according to the playerparticipation distribution. For both type of planners we obtain tight boundsfor the PoA, by solving suitable optimization problems parameterized by themaximum participation probability $q=\max_{i} p_{i}$. In the case of affinecosts, we find an analytic expression for the corresponding bounds.

Ordinary and Prophet Planning under Uncertainty in Bernoulli Congestion Games