The paper considers a game theory approach to calculating the centrality value of the vertices in a directed graph, based on the number of vertex occurrences in fixed length paths. It is proposed to define vertex centrality as a solution of a cooperative game, where the characteristic function is given as the number of simple paths of fixed length in subgraphs corresponding to coalitions. The concept of integral centrality is introduced as the value of a definite integral of the payoff function. It is shown that this centrality measure satisfies the Boldi–Vigna axioms.