Time-Inconsistent LQ Games for Large-Population Systems and Applications

In this paper, we formulate a time-inconsistent linear-quadratic (LQ) mean-field game problem for large-population systems. By the Nash certainty equivalence methodology, we construct an auxiliary problem consisting of time-inconsistent LQ control problems for every agent, respectively. An equilibrium, instead of optimal solution, is presented explicitly in the form of linear feedback for each agent in this auxiliary problem. Inspired by the framework of tackling time-inconsistency for control problems, we generalize the definition of ϵ -Nash equilibrium for large-population games to the time-inconsistent case. Then, the set of linear feedback controls obtained above can be proved to be an ϵ -Nash equilibrium for our original problem. Moreover, we solve a resource investment problem and provide a numerical simulation to illustrate the application of our theoretic results.