Semi-Explicit Solutions to some Non-Linear Non-Quadratic Mean-Field-Type Games: A Direct Method
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This article examines the solvability of mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical LQ game problems. It includes quadratic-quadratic games, power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as control-dependent regime switching, quadratic state, cotangent state and hyperbolic cotangent state are considered. Both equilibrium strategies and equilibrium costs are given in a semi-explicit way. The optimal strategies are shown to be in state-and-conditional mean-field-type feedback form. It is shown that a simple direct method can be used to solve a broader classes of non-quadratic mean-field-type games under jump-diffusion-regime switching Gauss-Volterra processes which include fractional Brownian motion and multi-fractional Brownian motion. We provide semi-explicit solutions to the fully cooperative, noncooperative nonzero-sum, and adversarial game problems.
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