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In this paper we study a class of matrix-valued linear-quadraticmean-field-type games for both the risk-neutral, risk-sensitive and robustcases. Non-cooperation, full cooperation and adversarial between teams aretreated. We provide a semi-explicit solution for both problems by means of adirect method. The state dynamics is described by a matrix-valued linearjump-diffusion-regime switching system of conditional mean-field type where theconditioning is with respect to common noise which is a regime switchingprocess. The optimal strategies are in a state-and-conditional mean-fieldfeedback form. Semi-explicit solutions of equilibrium costs and strategies arealso provided for the full cooperative, adversarial teams, risk-sensitive fullcooperative and risk-sensitive adversarial team cases. It is shown that fullcooperation increases the well-posedness domain under risk-sensitivedecision-makers by means of population risk-sharing. Finally, relationshipbetween risk-sensitivity and robustness are established in the mean-field typecontext.

Matrix-Valued Mean-Field-Type Games: Risk-Sensitive, Adversarial, and Risk-Neutral Linear-Quadratic Case