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 $q=\max_{i} p_{i}$. For the case of affine costs, we provide an analytic expression for the ordinary and prophet price of anarchy, parameterized as a function of $q$.

翻译：我们考虑一种原子拥堵博弈，其中每位玩家$i$以外部给定的已知概率$p_{i}\in(0,1]$独立于其他玩家决定是否参与博弈，若不参与则不产生任何成本。通过引入参数化无政府状态价格，我们刻画了需求不确定性对自私行为效率的影响，并考察了两种不同的社会规划者概念。先知规划者知晓博弈中随机参与的实现结果；而普通规划者则不知晓。因此，先知规划者能够计算自适应社会最优解，根据实际激活的玩家群体选择不同策略；而普通规划者与玩家面临相同的不确定性，仅能基于玩家参与分布计算社会最优解。针对两类规划者，我们推导出精确的无政府状态价格，该价格源于以最大参与概率$q=\max_{i} p_{i}$为参数的优化问题。在线性成本情形下，我们给出了普通与先知规划者无政府状态价格的解析表达式，该表达式以$q$为参数。