Finite- and Infinite-Horizon Shapley Games with Nonsymmetric Partial Observation

We consider asymmetric partially observed Shapley-type finite-horizon and infinite-horizon games where the state, a controlled Markov chain $\{X_t\}$, is not observable to one player (minimizer) who observes only a state-dependent signal $\{Y_t\}$. The maximizer observes both. The minimizer is informed of the maximizer's action after (before) choosing his control in the MINMAX (MAXMIN) game. A nontrivial open problem in such situations is how the minimizer can use this knowledge to update his belief about $\{X_t\}$. To address this, the maximizer uses off-line control functions which are known to the minimizer. Using these, novel control-parameterized nonlinear filters are constructed which are proved to characterize the conditional distribution of the full path of $\{X_t\}$. Using these filters, recursive algorithms are developed which show that saddle-points exist in both behavioral and Markov strategies for the finite-horizon case in both games. These algorithms are extended to prove saddle-points in M...