  We study a heterogeneous routing game in which vehicles might belong to more
than one type. The type determines the cost of traveling along an edge as a
function of the flow of various types of vehicles over that edge. We relax the
assumptions needed for the existence of a Nash equilibrium in this
heterogeneous routing game. We extend the available results to present
necessary and sufficient conditions for the existence of a potential function.
We characterize a set of tolls that guarantee the existence of a potential
function when only two types of users are participating in the game. We present
an upper bound for the price of anarchy (i.e., the worst-case ratio of the
social cost calculated for a Nash equilibrium over the social cost for a
socially optimal flow) for the case in which only two types of players are
participating in a game with affine edge cost functions. A heterogeneous
routing game with vehicle platooning incentives is used as an example
throughout the article to clarify the concepts and to validate the results.
