# The Value of Nonatomic Games Arising from Certain Noncooperative Congestion Games

For a class of nonatomic congestion games, two solution concepts, a noncooperative one and a cooperative one, are compared. Each player in the game chooses one of several common facilities. The player's payoff is the difference between the reward and the cost associated with choosing that facility. The rewards are fixed and player-specific. The costs are uniform, but variable: they strictly increase with the measure of the set of players using the facility. The noncooperative solution of the game is the (unique) Nash equilibrium outcome. The cooperative one is the Aumann-Shapley value of the cooperative game that results when players are willing to cooperate in order to minimize the total utility. Using a new result in the theory of values of nonatomic games, we derive a formula for the value. We show that there is exactly one case in which the Nash equilibrium outcome andthe value always coincide: this is the case in which the costs increase logarithmically with the measure of the set of users.