We consider the Cauchy problem for an elliptic operator, formulated as a Nash game. The over specified Cauchy data are split among two players : the first player solves the elliptic equation with the Dirichlet part of the Cauchy data prescribed over the accessible boundary, and a variable Neumann condition (which we call first player's strategy) prescribed over the inaccessible part of the boundary. The second player makes use correspondingly of the Neumann part of the Cauchy data, with a variable Dirichlet condition prescribed over the inaccessible part of the boundary. The first player then minimizes the gap related to the non used Neumann part of the Cauchy data, and so does the second player with a corresponding Dirichlet gap. The two costs are coupled through a distributed field gaps. We prove that there exists always a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion algorithm is stable with respect to noise. Some numerical 2D and 3D experiments are provided to illustrate the efficiency and stability of our algorithm.