An exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners' dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be approximated by an equilibrium of a finitely repeated version of that game. The departure from common knowledge is small in the following sense:(1) the players know T with precision +/-K; (2) with probability 1 - epsilon, the players know T precisely; moreover, this knowledge is mutual of order epsilon T; and (3) the deviation of T from its finite expectation is exponentially small.

Analytic decision rules and horizon rules are developed for the warehousing problem. A stochastic warehousing problem is defined and solved, first by backwards induction, then by means of a forward working algorithm. The influence of holding costs and discount rates on forecasting horizons is illustrated.

Edgeworth's conjecture that as the number of traders in an exchange economy increases the core approaches the set of competitive equilibria has been formalized both as a theorem about a sequence of finite economies, and as a theorem about an economy having an infinite number of agents. This paper, using nonstandard analysis, provides a synthesis of these two approaches. It is shown that the core and the set of competitive equilibria are equivalent within a non-standard exchange economy. This theorem implies a asymptotic theorem concerning the core and competitive equilibria of sequences of finite economies. (Из Ebsco)