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

arXiv: https://arxiv.org/abs/1812.06695 · HTML: https://mlsanity.com/paper/1812.06695/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [Reinforcement Learning for Discrete-time LQG Mean Field Social Control Problems with Unknown Dynamics](https://mlsanity.com/paper/2507.01420/) (2025)

## Abstract (arXiv)

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.