Comparison of α-Condorcet points with median and center locations

The usual concept of solution in single voting location is the Condorcet point. A Condorcet solution is the location such that no other location is preferred by a strict majority of voters; i.e. a half of them. It is assumed that each user always prefers closer locations. Because a Condorcet point does not necessarily exist, the α-Condorcet point is defined in the same way but assuming that two locations are indifferent for a user if the distances to both differ at most in α. We give bounds for the value of the objective function in an α-Condorcet point in the median and center problems. These results, for a general graph and for a tree, extend previous bounds for the objective function in a Condorcet point. We also provide a set of instances where these bounds are asymptotically reached.