Traffic flow assignment system that includes two navigation providers is considered. Navigation providers predict future traffic conditions and provide travel guidance for given original-destination demand and network capacity. Three classes of drivers are introduced. The users of navigation service from the first and the second classes are supplied with the shortest path guidance by navigation providers «Navigator 1» and «Navigator 2» correspondingly. The third class of drivers is characterized by user-optimal behavior without current guidance information about traffic conditions. The main goals of Navigators are to assign the traffic flow of their customers (users) so as to maximize appropriate pay-off functions which depend on original estimation of traffic volume delay and are influenced by decision of both of them. Existence and uniqueness of Nash equilibrium in considered non-zero sum game are proven and an explicit form of Nash equilibrium is presented.