Necessary/sufficient conditions for Pareto optimum in cooperative difference game

Abstract

SummaryThis paper is concerned with the necessary/sufficient conditions for the Pareto optimum of the cooperative difference game in finite horizon. Utilizing the necessary and sufficient characterization of the Pareto optimum, the problem is transformed into a set of constrained optimal control problems with a special structure. Employing the discrete version of Pontryagin's maximum principle, the necessary conditions for the existence of the Pareto solutions are derived. Under certain convex assumptions, it is shown that the necessary conditions are sufficient too. Next, the obtained results are extended to the linear‐quadratic case. For a fixed initial state, the necessary conditions resulting from the maximum principle and the convexity condition on the cost functional provide the necessary and sufficient description of the well‐posedness of the weighted sum optimal control problem. For an arbitrary initial state, the solvability of the related difference Riccati equation provides a sufficient condition under which the Pareto‐efficient strategies are equivalent to the weighted sum optimal controls. In addition, all Pareto solutions are derived based on the solutions of a set of difference equations. Two examples show the effectiveness of the proposed results.