Welldeened Decision Scenarios

Innuence diagrams serve as a powerful tool for modelling symmetric decision problems. When solving an innuence diagram we determine a set of strategies for the decisions involved. A strategy for a decision variable is in principle a function over its past. However , some of the past may be irrelevant for the decision, and for computational reasons it is important not to deal with redundant variables in the strategies. We show that current methods (e.g. the Decision Bayes-ball algorithm Shachter, 1998]) do not determine the relevant past, and we present a complete algorithm. Actually, this paper takes a more general outset: When formulating a decision scenario as an innuence diagram, a linear temporal ordering of the decisions variables is required. This constraint ensures that the decision scenario is welldeened. However, the structure of a decision scenario often yields certain decisions conditionally independent, and it is therefore unnecessary to impose a linear temporal ordering on the decisions. In this paper we deal with partial innuence diagrams i.e. innuence diagrams with only a partial temporal ordering speciied. We present a set of conditions which are necessary and suucient to ensure that a partial innuence diagram is welldeened. These conditions are used as a basis for the construction of an algorithm for determining whether or not a partial innu-ence diagram is welldeened.