This work studies a recently proposed model of the Braess paradox, which has four parameters and is defined piecewise continuous on three regions. The Braess paradox describes the fact that the introduction of a new choice into a system of rational agents may not improve the situation but can make it even worse. To investigate the system and its undergoing border collision bifurcations, we are going to fix some parameters and thereby get new results on the overall dynamics. Also part of the coexistence and the period adding scenario between the cycles are analyzed.