Uncertainty quantification in mean-field-type teams and games

This article studies uncertainty quantification methodologies in team and strategic decision-making problems of mean-field type. Considering McKean-Vlasov state dynamics are that square integrable over a finite horizon, we use Kosambi-Karhunen-Loeve expansion which is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function over a bounded domain. The mean-field-type team and game problems are then transformed into equivalent formulations with series expansions. By identification of coefficients, these mean-field-type problems become interactive systems of deterministic state variables over multiple indexes. We illustrate some situations where these deterministic control and game problems can be handled. In the general setting, approximation methods such as truncature techniques are proposed, and their challenges and limitations are examined.