  We consider an atomic congestion game in which each player $i$ either
participates in 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 compute the parameterized price of anarchy to characterize the impact
of demand uncertainty on the efficiency of selfish behavior, considering two
different notions of a social planner. A prophet planner knows the realization
of the random participation in the game; the ordinary planner does not. As a
consequence, a prophet planner can compute an adaptive social optimum that
selects different solutions depending on the players that turn out to be
active, whereas an ordinary planner faces the same uncertainty as the players
and can only compute social optima with respect to the player participation
distribution. For both planners, we derive the precise price of anarchy, which
arises from an optimization problem parameterized by the maximum participation
probability $q=\max_{i} p_{i}$. In the case of affine costs, we provide an
analytic expression for the ordinary and prophet price of anarchy,
parameterized as a function of $q$.
