An Intergenerational Cake Eating Game

A close reading of Hotelling's famous article on exhaustible resources reveals the following passage: 'If a mine-owner produces too rapidly, he will depress the price, perhaps to zero. If he produces too slowly, his profits, though larger, may be postponed farther into the future than the rate of interest warrants. Where is hisgolden mean? And how does this most profitable rate of production vary as exhaustion approaches?' [Hotelling (1931, p. 139)]. Clearly his reference to the 'golden mean' is only figurative, referring to the most profitable extraction rate. The literal definition of the 'golden mean' is given by the requirement that (a + b)/a = a/b = (1 + .\/5)/2, where a and b are positive numbers. The esthetic and mathematical properties of this ratio, also referred to as the Golden Section, the Divine Proportion, the Divine Section, the Golden Ratio, and the Golden Proportion, have been studied for centuries [Runion (1972)l and it has found application in sequential search methods [Saaty (1970, pp. 27-30)]. In terms of the exhaustion of a fixed stock of a depletable resource, that we will call a 'cake', a 'golden mean7 extraction rate is one in which the total amount extracted in any two successive periods is in the same proportion to the amount extracted in one of them as the amount extracted in that period is to the amount extracted in the other. The question we address is whether there exists a situation under which a golden mean extraction path would obtain. In what follows we describe a game played between succeeding generations in which the 'golden mean' extraction rate is a minmax solution.