Network Games with Atomic Players

We study network games with atomic players that can split their flow. Some errors in the literature led to incorrect bounds on the price of anarchy of these games. We correct past results with a new bound for arbitrary sets of cost functions. In the case of affine cost functions, this bound implies that the price of anarchy is at most 3/2 and the cost of an equilibrium is not larger than that of a social optimum of an instance with demands multiplied by 4/3. We also present an improved version of this bound for the case of a single OD pair that depends on the variability of the market power of each of the players. In addition, for the case in which all players share the same OD pair and control the same amount of flow, we provide a new potential function formulation. It implies that the cost of a Nash equilibria is no more than that of the corresponding nonatomic game, ruling out paradoxical phenomena that may arise in the general case. For instance, the cost of an equilibrium increases as the number of players increases as it is more difficult to coordinate players. All our results extend to the more general atomic congestion games with divisible demands, and to mixed atomic and nonatomic games.