Discrete game problem with incomplete information (Q1078532)

The paper describes the mathematical set up of a zero-sum discrete time game problem over a given finite horizon. One player wants to maximize the probability P that the state of the game enters a given subset M of the state space during the game whereas the other player wants to minimize this probability. The initial state is given probabilistically; the game starts from a given countable subset of the state space \({\mathbb{R}}^ n\), with given probabilistics assigned to all elements of this subset. With respect to the information available to the players several cases are distinguished. (1) Majorant and minorant program strategies which correspond to respectively, min max P and max min P, where the minimization and maximization are taken with respect to the decisions of the whole time horizon; and (2) majorant and minorant positional strategies which correspond to respectively min max...min max P and max min...max min P, where each minimization and maximization refers to the decision of one time instant. The four possible orderings of max and min just mentioned are considered with respect to both pure and mixed decisions. Some assertions are given with respect to lower- and/or upper bounds which one player can assure himself concerning the probability of entering M. An elucidating two-stage game (for which the state space is the set of integers) is given. It clearly shows that different information (and hence different strategies) may lead to different outcomes (the difference in outcome is a measure for information).