A remarkable result from Game Theory (as developed by Von Neumann [@VNeumann28], Von Neumann and Morgenstern [@VonNeumann:1944] and Nash [@Nash:1951]) is that optimal strategies for deterministic zero sum finite games are mixed (i.e. randomized) strategies. Although the information game described above is zero sum, it is not finite. Nevertheless, as in Wald’s Decision Theory [@Wald:1945], under sufficient regularity conditions it can be made compact and therefore approximable by a finite game. Therefore although the information game described above is purely deterministic (and has no a priori connection to statistical estimation), under compactness (and continuity of the loss function), the best strategy for Player I is to play at random by placing a probability distribution $\pi_{I}$ on the set of candidates for $g$ (and select $g$ as a sample from $\pi_{I}$) and the optimal strategy for Player II is to place a probability distribution $\pi_{II}$ on the set of candidates for $g$ and approximate the solution of by the expectation of $u$ (under $\pi_{II}$ used as a prior distribution) conditioned on the measurements $\int_{\Omega} u\phi_i$.

Although the estimator employed by Player II may be called Bayesian, the game described here is not (i.e. the choice of Player I might be distinct 