Stochastic Linearisation of Indeterminateness

Abstract

The starting point of our considerations is the experience that, in most practical cases of decision-making, the information available is partial in the sense that the decision-maker is neither in the situation of complete uncertainty, nor in the situation of complete information. We try to provide a theory for decision-making under partial information, mainly by means of stochastic linearisation of inde-terminateness. We define the notion of the decision value whose determination is possible and seems reasonable in games against nature without information (application of the maximin — or minimax operator), as well as in games against nature with some information (application of the Fax E_min — operator). We define the notion of Linear Partial Information (LPI) and introduce the notion of LPI-structure. The latter leads to relatively simple game models. We prove that the decision value in LPI-situations is equal to the value of the corresponding LPI-zero sum game. Those games form the basis for some applications which we consider: Linear and non-linear stochastic programmes and some statistical inferences. In the algorithmic respect we propose an iterative method of fictitious playing.