Stability and Stabilization of Nash Equilibrium for Noncooperative Systems With Vector-Valued Payoff Functions

A zero-sum tax/subsidy approach and a necessary condition for stabilizing unstable Nash equilibria in pseudo-gradient-based noncooperative dynamical systems with vector-valued payoff functions are proposed. Specifically, we first present a necessary and sufficient condition for the Nash equilibrium of the noncooperative game with vector-valued payoff functions to be bounded. Then we give a sufficient condition for such Nash equilibrium to be stable. After that, we develop a framework where a system manager constructs a zero-sum tax/subsidy incentive structure by collecting taxes from one agent and giving the same amount of subsidy to the other agent to make the incentivized Nash equilibrium stable and bounded, which can make the trajectories converge to the interior of original Nash equilibrium set. Finally, we present a numerical example to illustrate the utility of the zero-sum tax/subsidy approach.