A production set with indivisibilities is described by an activity analysis matrix with activity levels which can assume arbitrary integral values. A neighborhood system is an association with each integral vector of activity levels of a finite set of neighboring vectors. The neighborhood relation is assumed to be symmetric and translation invariant. Each such neighborhood system can be used to define a local maximum for the associated integer programs obtained by selecting a single commodity whose level is to be maximized subject to specified factor endowments of the remaining commodities. It is shown that each technology matrix (subject to mild regularity assumptions) has a unique, minimal neighborhood system for which a local maximum is global. The complexity of such minimal neighborhood systems is examined for several examples.

[Part I of this paper introduces a general framework for the discussion of discrete production sets and the associated programming problems which arise when a particular endowment of factors is specified. In this part of the paper we shall apply these ideas to integer programming problems with two activities and bring to bear some of the basic considerations of the theory of computational complexity. The numbering of sections, figures, and equations will follow those used in Part I.]

THE PROBLEMS of distribution in an economic system may be analysed either by means of the behavioral assumptions of a competitive model or by the more flexible techniques of n person game theory. In the competitive model, consumers are assumed to respond to a set of prices by maximizing utility subject to a budget constraint and producers by maximizing profit. Consistent production decisions and an allocation of commodities are obtained by the determination of a set of prices at which all markets are in equilibrium. The analysis of these problems by means of n person game theory requires us to specify the production and distribution activities that are available to an arbitrary coalition of economic agents. It is frequently sufficient to summarize the detailed strategic possibilities open to a coalition by the set of possible utility vectors that can be achieved by the coalition. For example, in a pure exchange economy each coalition will have associated with it the collection of all utility vectors that can be obtained by arbitrary redistributions of the resources of that coalition. The core of an n person game is a generalization of Edgeworth's contract curve. A vector of utility levels is suggested which is feasible for all of the players acting collectively, and an arbitrary coalition is examined to see whether it can provide higher utility levels for all of its members. If this is possible, the utility vector which was originally suggested is said to be blocked by the coalition. The core of the n person game consists of those utility vectors which are feasible for the entire group of players and which can be blocked by no coalition. As we have seen during the last several years, there is an intimate connection between these two methods of analysis. If the conventional assumptions of the competitive model are made, such as convexity of preferences and convexity and constant returns to scale for the production set, then there will be a price system at which all markets are in equilibrium and a resulting assignment of commodity bundles to consumers. The utility vector associated with this competitive equilibrium may be shown to be in the core. Even further, if the number of consumers tends