Pareto optimality in the infinite horizon cooperative difference game

This study is concerned with the necessary and sufficient conditions for the existence of Pareto solutions in the infinite horizon cooperative difference game. Based on the assumption about the Lagrange multipliers, utilising the equivalent characterisation of the Pareto optimality, the necessary conditions for the existence of the Pareto solutions are put forward. Furthermore, two conditions are presented to guarantee that zero does not belong to the Lagrange multiplier set. In addition, it is shown that the necessary conditions are also sufficient under certain convexity assumptions and a transversality condition. Next, the indefinite linear quadratic case is discussed. For a fixed initial state, under the condition of controllability, the necessary conditions are put forward. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional as well as a transversality condition provide the sufficient conditions for a control to be Pareto optimal. For an arbitrary initial state, if the system is stabilisable, then the solvability of the related algebraic Riccati equation provides a sufficient condition under which all Pareto optimal strategies are obtained by the weighted sum minimisation method.