<p>We study the inverse problem of determining the diffusion parameter in a parabolic operator when only partial and over-specified boundary measurements are available. Such problems are typically ill-posed and highly sensitive to incomplete data and measurement noise. To overcome these challenges, we develop a game-theoretic approach that jointly reconstructs the diffusion parameter and the missing boundary data. The proposed method is formulated as a three-player game, where each player minimizes a specific criterion: the first two players adopt Dirichlet and Neumann boundary conditions, respectively, to address the data completion task, while the third player optimizes the diffusion coefficient via a dedicated cost functional. We prove the existence and uniqueness of the Nash equilibrium for the two-player data-completion subgame, providing a well-posed foundation for its numerical implementation. To evaluate the performance of the proposed strategy in solving this ill-posed inverse problem, we consider two numerical experiments: experiment 1 focuses on the two-player algorithm for boundary data completion, validating its convergence and robustness, while experiment 2 addresses the coupled inverse problem, where the diffusion parameter and the missing boundary data are reconstructed simultaneously using the three-player framework. The results demonstrate the effectiveness and stability of our game-theoretic approach in handling incomplete and noisy boundary measurements.</p>