Two-Agent Noncooperative Dynamical Systems With Quadratic Vector-Valued Payoff Functions and Weak Pareto Improvement

Definition of the weak Pareto improvement and the trap of weak Pareto improvement are given for noncooperative dynamical systems with vector-valued payoff functions. Specifically, we develop an incentive mechanism that satisfies sustainable budget constraint. We assume that there is a system manager who is authorized to design the incentive functions in order to maximize the social welfare of her choice. Specifically, the socially maximum state is predicated on the weighted sum of all the payoff functions. It turns out that depending on the choice of the design parameters, we may observe that some of the agents can be trapped in a weak Pareto improvement even though the system trajectory is weakly Pareto improving. Our results is a generalized version of the results for scalar-valued payoff functions.