Incorporation of Likely Future Actions of Agents Into Pseudo-Gradient Dynamics of Noncooperative Games

In this article, we connect cognitive hierarchy theory with the pseudo-gradient dynamics in noncooperative systems to extend the pseudo-gradient dynamics with some prediction behaviors under level-$k$ thinking. In this framework, each agent believes that he is the most sophisticated person in the noncooperative system and makes the proactive decision according to some strategic reasoning of the other agents' likely actions. Depending on a knowledge network of payoff functions, the modified pseudo-gradient dynamics are presented under the assumption that the agents may be able to reason the other agents' best-response states and use these predicted states in the pseudo-gradient dynamics. Some sufficient conditions are provided to guarantee stability of a Nash equilibrium with uncertain sensitivity parameters or uncertain knowledge network. The transition of the agents' target state while increasing the cognitive levels for a two-agent noncooperative system with quadratic payoff functions is characterized. We present the utility of our approach for homogeneous and differentiated oligopoly markets. Our result indicates that to ensure asymptotic stability of the differentiated oligopoly markets with Cournot competition under pseudo-gradient dynamics, a larger market with more firms requires more differentiated products.