Published online by Cambridge University Press: 13 October 2009
The competitive equilibrium, core, and value are solution notions widely used in economics and are based on disparate ideas. The competitive equilibrium is a notion of noncooperative equilibrium based on individual optimization. The core is a notion of cooperative equilibrium based on what groups of individuals can extract from society. The value can be interpreted as a notion of fair division based on what individuals contribute to society. It is a remarkable fact that, under appropriate assumptions, these solution notions (nearly) coincide in large economies. The (near) coincidence of the competitive equilibrium and the core for large exchange economies was first suggested by Edgeworth (1881) and rigorously established by Debreu and Scarf (1963) in the context of replica economies and by Aumann (1964) in the context of continuum economies. This pioneering work has since been extended to much wider contexts; see Hildenbrand (1974) and Anderson (1986) for surveys. The (near) coincidence of the value and the competitive equilibrium (and hence the core) for large exchange economies was first suggested by Shubik and rigorously established by Shapley (1964) in the context of replica economies with money. This pioneering work, too, has since been extended to much wider contexts; see, for example, Shapley and Shubik (1969), Aumann and Shapley (1974), Aumann (1975), Champsaur (1975), Hart (1977), Mas-Colell (1977), and Cheng (1981).