Some problems in decision making associated with random walks (Q1284332)

Problems of the decision making theory related to the motion of a point or an object on an integer lattice are generally connected with an analysis of asymptotics of the probability distribution of finding the point in the knots of the lattice. The asymptotics are considered for a case when the time of the motion tends to infinity. The analysis of a motion control and a design of the optimal control strategy are presented in the article. It is supposed that a transfer probability in relation to one of the coordinates defined by an operating side while transfer probabilities in relation to other coordinates are noncontrollable factors. Additionally it is supposed that in the knots of the lattice a gain function is given. Using different assumptions some optimal strategies of the operating side are defined. The following situations are considered: 1) choice of the transfer probability in relation to the first coordinate comes true when the value of the noncontrollable factor is known; 2) as to transfers in relation to the second coordinate it is known only its possible values in the next period. The optimal critaria are: a) mean value of the gain; b) the maximal value of the gain which can be guaranteed with a given confidence probability; c) the probability of getting of a gain not less that the given one . The research is presented for two classes of the gain function which are especially interesting for auctions of shares etc.