Stackelberg-Nash Equilibrium in competitive facility location game among a franchisor and two investors

In the competitive location problems, the matter of the optimal location of single or multiple facilities are in a condition in which competitors exist as well. This paper deals with a type of a competitive location on which a leader possibly uses the investment of other investors through concession and receives a percentage of their income. He also can place his own facilities on potential locations that are available. In fact there are three decision-makers, one as a leader, others as followers who get in the game of facility location for placing their facilities. The location of facilities is a simultaneous game in which decisions are made in two levels, in the first level they are asynchronous and sequential, in the second level decisions are synchronous, namely the leader selects places for his own facilities, where other decision-makers (investor) play the simultaneous and non-cooperative game, and according to Nash equilibrium, select the optimal location. The leader uses new locations to optimize his objective function. Eventually, a numerical example is presented to test the model and a comprehensive sensitivity analysis is carried out to extract some managerial insights.