This chapter introduces optimal recovery games on Banach spaces, presents their natural lift to mixed strategies, and then characterizes their saddle points interms of Gaussian measures, cylinder measures, and fields. The canonical Gaussian field is shown to be a universal field in the sense that its conditioningwith respect to linear measurements producesoptimal strategies. When those measurements form a nested hierarchy, hierarchies of optimal approximations form a martingale obtained by conditioningthe Gaussian field on the filtration formed by those measurements.

This chapter introduces two-person zero-sum games, optimal recovery games, and their lifts tomixed extended games and defines saddle points and minmax solutions.The optimal mixed strategy for the mixed extension of the optimal recovery games is generated by conditioning the canonical Gaussian field associated with the energy norm. Since the dependence ofthese optimal solutions on the measurement functions is through the conditioning process only, the canonical Gaussian field is referred to as a universal field. This fact demonstrates that the optimal solutions generated from a nested hierarchy of measurement functions form a martingale.