Note on the Hardness of Bounded Budget Betweenness Centrality Game with Path Length Constraints

In this technical report, we generalize the betweenness definition in Bounded Budget Betweenness Centrality Game (called B3C game) introduced in [1] to only count shortest paths with a length limit `. We denote this game `-B3C game. We prove that the hardness results in [1] about nonuniform game still hold in this generalized version. In Section 2, we provide the detailed definition of the `-B3C game. In Section 3, we prove that there exists an instance of `-B3C game such that it does not have any maximal Nash equilibrium. In Section 4, we prove that it is NP-hard to decide whether an instance of `-B3C game has a maximal or strict Nash equilibrium.