# Semi-Explicit Solutions to some Non-Linear Non-Quadratic Mean-Field-Type Games: A Direct Method

## TL;DR

This paper introduces a direct method to find semi-explicit solutions for a broad class of non-linear, non-quadratic mean-field-type games, including complex dynamics and payoff functions, expanding solvable models beyond classical cases.

## Contribution

It presents a simple direct approach to solve complex mean-field games with non-linear dynamics and payoffs, including jump-diffusion and regime switching processes.

## Key findings

- Derived semi-explicit solutions for various non-quadratic mean-field games.
- Extended solvability to models with complex dynamics like log-state and hyperbolic functions.
- Provided solutions for cooperative, noncooperative, and adversarial game settings.

## Abstract

This article examines mean-field-type game problems by means of a direct method. We provide various solvable examples beyond the classical linear-quadratic game problems. These include quadratic-quadratic games and games with power, logarithmic, sine square, hyperbolic sine square payoffs. Non-linear state dynamics such as log-state, control-dependent regime switching, quadratic state, cotangent state and hyperbolic cotangent state are considered. We identify equilibrium strategies and equilibrium payoffs in state-and-conditional mean-field type feedback form. It is shown that a simple direct method can be used to solve broader classes of non-quadratic mean-field-type games under jump-diffusion-regime switching Gauss-Volterra processes which include fractional Brownian motions and multi-fractional Brownian motions. We provide semi-explicit solutions to the fully cooperative, noncooperative nonzero-sum, and adversarial game problems.