On a multidimensional oil exploration problem

This paper is concerned with optimal strategies for drilling in an oil exploration model. An exploration area contains n 1 large and n 2 small oilfields, where n 1 and n 2 are unknown, and represented by a two-dimensional prior distribution π . A single exploration well discovers at most one oilfield, and the discovery process is governed by some probabilistic law. Drilling a single well costs c , and the values of a large and small oilfield are v 1 and v 2 respectively, v_2>c>0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> v 1 > v 2 > c > 0 . At each time t = 1 , 2 , … , the operator is faced with the option of stopping drilling and retiring with no reward, or continuing drilling. In the event of drilling, the operator has to choose the number k , 0 ≤ k ≤ m ( m fixed), of wells to be drilled. Rewards are additive and discounted geometrically. Based on the entire history of the process and potentially on future prospects, the operator seeks the optimal strategy for drilling that maximizes the total expected return over the infinite horizon. We show that when π ≻ π ′ in monotone likelihood ratio, then the optimal expected return under prior π is greater than or equal to the optimal expected return under π ′ . Finally, special cases where explicit calculations can be done are presented.