An Equilibrium Problem with an Equilibrium Constraint (EPEC) is a mathematical construct that can
be applied to private competition in highway networks. In this paper we consider the problem of finding a
Nash Equilibrium in a situation of competition in toll pricing on a network utilising two alternative
algorithms. In the first algorithm, we utilise a Gauss Seidel fixed point approach based on the cutting
constraint algorithm for toll pricing. The second algorithm that we propose, a novel contribution of this
paper, is the extension of an existing sequential linear complementarity programming approach for
finding the competitive Nash equilibrium when there is a lower level equilibrium constraint. Finally we
develop an intuitive approach to represent collusion between players and demonstrate that as the level of
collusion goes from none to full collusion so the solution maps from the Nash to monopolistic solution.
However we also show that there may be local solutions for the collusive monopoly which lie closer to
the second best welfare toll solution than does the competitive Nash equilibrium.
