From weakest-link to best-shot: The voluntary provision of public goods

It has traditionally been assumed that the socially available amount X,of a public good is the simple sum of the separate amounts x i produced by the i = 1, . . . , I members of the community. But there are many other possibilities of practical importance. Among them are: (i) Weakest-link rule, where the socially available amount is the minimum of the quantities individually provided, and (ii) Best-shot rule, where the socially available amount is the maximum of the individual quantities. The former tends to arise in linear situations, where each individual has a veto on the total to be provided (e.g., if each is responsible for one link of a chain); the latter tends to arise when there is a single prize of overwhelming importance for the community, with any individual's effort having a chance of securing the prize. In comparison with the standard Summation formula of ordinary public-good theory, it is shown that underprovision of the public good tends to considerably moderated when the Weakest-link function is applicable, but aggravated when the Best-shot function is applicable. In time of disaster, where the survival of the community may depend upon each person's doing his duty, the conditions for applicability of the Weakest-link rule are approximated. This circumstance explains the historical observation that disaster conditions tend to elicit an extraordinary amount of unselfish behavior. Anarch ia is a perfectly circular island, and each citizen owns a wedgeshaped slice (not all equal) f rom the center to the sea. Like the Nether lands, Anarch ia is protected by dikes f rom occasional s torms that threaten to f lood the land. But since Anarch ia has no government , everyone makes his own decision as to how high a dike to build. While the height of each ci t izen's dike is perfectly visible to all, the customs of Anarch ia forbid enforcement of any threat , inducement , or contract whereby some parties might inf luence the choices of others. In times of f lood the sea will penetrate the sector belonging to whichever citizen has constructed the lowest dike, bu t the topography of Anarch ia is such that no mat ter where the sea enters, damage will be suffered equally over the whole island. The economists of Anarch ia have long realized that f lood-protect ion for their island is a public good. M a n y centralized schemes for mot iva t ing individuals to bui ld dikes of the socially opt imal height have been discussed, but Ana rch ia ' s citizens f ind any such social p lann ing intolerable. It so *Department of Economics, University of California, Los Angeles, CA 90024.