Two-terminal routing games with unknown active players

We analyze 2-terminal routing games with linear cost functions and with unknown number of active players. We deal with both splittable and unsplittable models. We prove the existence and uniqueness of a symmetric safety-level equilibrium in such games and show that in many cases every player benefits from the common ignorance about the number of players. Furthermore, we prove new theorems on existence and uniqueness of equilibrium in 2-terminal convex routing games with complete information.