An optimal information gathering algorithm

The current paper defines the optimal sequential information gathering structure of a rational utility maximiser decision maker in the simplest non-trivial theoretical scenario, where the decision maker is allowed to acquire only two pieces of information from a set of multidimensional goods. We show how this problem, hardly ever considered in the literature, does not admit a simple or intuitive solution. Indeed, while the standard sequential search and information gathering algorithms presented in the literature are identified with optimal stopping rules, we analyse explicitly the behaviour of the decision maker when choosing which piece of information to acquire. We show that the decision of how to optimally allocate the second available piece of information depends on two well-defined real-valued expected utility functions. The crossing points between the graphs of both functions correspond to optimal thresholds for the information gathering process that define the dynamic behaviour of the algorithmic search structure. We characterise explicitly the behaviour and the value of these thresholds through the properties of the utility functions and probability densities inherent to the decision maker. The results are illustrated numerically for a variety of utility functions commonly used in decision theory.