Laws of large numbers for non-additive probabilities

We apply the concept of exchangeable random variables to the case of non-additive robability distributions exhibiting ncertainty aversion, and in the lass generated bya convex core convex non-additive probabilities, ith a convex core). We are able to rove two versions of the law of arge numbers (de Finetti's heorems). By making use of two efinitions. of independence we rove two versions of the strong law f large numbers. It turns out that e cannot assure the convergence of he sample averages to a constant. e then modal the case there is a true" probability distribution ehind the successive realizations of the uncertain random variable. In this case convergence occurs. This result is important because it renders true the intuition that it is possible "to learn" the "true" additive distribution behind an uncertain event if one repeatedly observes it (a sufficiently large number of times). We also provide a conjecture regarding the "Iearning" (or updating) process above, and prove a partia I result for the case of Dempster-Shafer updating rule and binomial trials.

Extinction, pollution, and the sustainable use of exhaustible and renewable resources

Single ownership of natural resources is conunon in many developing countries and socialist economies. The sole owner is usually the .state or society at large, and governments are responsible for either distributing exploitation rights or engaging in exploitation through their own corporations. • Under this circumstance, the notion of externality may not fully explain pollution problems existent in these nations. This paper studies the case where a single agent owns both exhaustible and renewable resources, and attempts to maximize its welfare. The resources are either perfect or imperfect substitutes. Initially, exhaustible resource extraction does not affect the renewable resource, and sustainable growth is attainable. A lactor of pollution flowing from the extraction of the nc.nrenewable resource into the growth of the renewable resource is introduced. The continuous exploitation of the exhaustible resource leads to the " optimal " extinction of the renewable resource, and sustainable growth is no longer reached. Regulation from a supra governmental agency such as an multinational institution may prove to be of utmost importance, if sustainability is to be achieved. The paper is divided into five sections. Section two provides a brief survey of the relevant literature. Section three presents the model without pollution. This factor is introduced in section four. The final section discusses some possible approaches for attaining sustainable growth, and contains the concluding remarks .