The pits and the core: Simple collective decision problems with concave preferences

This essay extends the theory of simple collective decision problems to spatial games in which (contrary to the traditional assumption) each agent's preferences are concave, in the sense that the alternatives that the agent does not prefer to any particular reference alternative together constitute a convex set. Such concave preferences might characterize decision problems in which, say, a site must be selected for some obnoxious facility, such as a prison, garbage dump, or facility for managing hazardous materials. The results indicate that, under these conditions, the (weak -)core can be structurally unstable, changing discontinuously with apparently minor perturbations of the decision problem. The main theorem identifies a curious property of the core when the set of feasible alternatives is compact and convex and each agent's preferences are strictly concave. Namely, a point in the feasible set's interior can belong to the core only if there is no feasible alternative that makes every member of any winning coalition strictly worse off. In this sense, an interior point belongs to the core only if it lies in the pits.A preliminary version of this essay was presented at the West Coast Conference on Small Groups Research, Stanford University, 17 April 1985, and the Experimental Social Choice Workshop, Utah State University, Logan, Utah, 20–21 June 1985. This material is based on work supported by grants from the National Science Foundation (Grants SES 83-12123 and SES 84-10094), the Real Estate Center and Department of Decision Sciences of The Wharton School, and the Research Fund of the University of Pennsylvania.