Algorithms for voting and competitive location on a network

Consider a network with weights associated to its vertices. When a facility must be located through a voting procedure, these weights may be interpreted as the numbers of users located on the vertices. When a facility must be located in a competitive setting, i.e., when a concurrent facility will be located later and will capture all clients closer to it, these weights may be viewed as the purchasing power of the clients. A Condorcet point is any point of the network, i.e., a vertex or a point on an edge, such that the set of vertices closer to any other point has a total weight not larger than the half sum of the weights of all the vertices. A Simpson point is a point of the network which minimizes the largest total weight of the set of vertices closer to another point. Polynomial algorithms are provided to determine the sets of Condorcet and of Simpson points of a network.