  In this paper we study a class of matrix-valued linear-quadratic
mean-field-type games for both the risk-neutral, risk-sensitive and robust
cases. Non-cooperation, full cooperation and adversarial between teams are
treated. We provide a semi-explicit solution for both problems by means of a
direct method. The state dynamics is described by a matrix-valued linear
jump-diffusion-regime switching system of conditional mean-field type where the
conditioning is with respect to common noise which is a regime switching
process. The optimal strategies are in a state-and-conditional mean-field
feedback form. Semi-explicit solutions of equilibrium costs and strategies are
also provided for the full cooperative, adversarial teams, risk-sensitive full
cooperative and risk-sensitive adversarial team cases. It is shown that full
cooperation increases the well-posedness domain under risk-sensitive
decision-makers by means of population risk-sharing. Finally, relationship
between risk-sensitivity and robustness are established in the mean-field type
context.
