Stopping a Size Dependent Exploration Process.
Morrey Kramer · Deep Blue (University of Michigan) · 1983
We study a process in which N objects of r and om size A(,i) are discovered or otherwise identified at r and om time T(,i). For concreteness we imagine the objects to be hydrocarbon deposits grouped together in a homogeneous geological configuration. We require that our model have a "sampling proportional to size property." That is, given the sizes of the deposits A(,1) = a(,1),...,A(,N) = a(,N) the probability that they are discovered in that order is (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) The model implies that large deposits tend to be found first. Under assumptions we characterize the family of distributions having the sampling proportional to size property. For an exploration process terminated at time t we will regard our payoff as the aggregate reward from discoveries up to time t less a cost of exploration. We find a stopping time which maximizes the expected payoff of the discovery process when N is known and formulate and evaluate an adaptive strategy when it is not. We also consider a related problem: the payoff at time t is the largest discovery made by time t less a search cost.