How long to eat a cake of unknown size? Optimal time horizon under uncertainty

This paper is concerned with the determination of the optimal time horizon for the cake-eating problem under uncertainty. It is shown that if the uncertain exhaustible resource stock is a discrete random variable admitting at most a finite number of values, the optimal planning horizon is infinite (finite) according as the marginal utility of extraction-cum-consumption is infinite (a finite positive value) as the latter approaches zero, thereby extending the scope of the similar result under perfect certainty. Other results show that uncertainty will generally lengthen the planning horizon, implying a more conservative extraction policy under uncertainty, and that the extraction policy aimed at extracting an amount equal to the expected value of the uncertain resource stock takes longer than the expected value of the optimal planning horizon.