The Absorbing Sets Solution in a Public Goods Economy

Resumo

The formation of coalitions to provide a public good creates positive externalities on nonmember players. Players which sign an agreement to provide a public good increase their total contributions to the provision of the good. Therefore some players may prefer to free-ride, that is to benefit from the increased supply of the public good without increasing their own contributions. This free-riding effect induces a high instability in the agreements, since, once a coalition has been formed, members of the coalition obtain a lower profit than outsiders and thus have an incentive to break the agreement. There are some works in the literature on noncooperative theory that analyze the stability of coalition structures. Some of them are Bloch [1995, 1996], Iñarra, Kuipers and Olaizola [2000], Ray and Vohra [2001], and Yi [1996]. Although this works share a common objective of analyzing equilibrium coalition structures, each adopts a different notion of stability. In this work, we apply the framework developed in Iñarra, Kuipers and Olaizola [2000] to a problem of international environmental agreements between countries introduced by Barrett [1994]. The aim of the paper is to provide a complete characterization of the equilibrium coalition structures for this model. Let us briefly explain our process of coalition formation: We allow for the formation of several coalitions, which give rise to coalition structures (partitions of the set of countries). Given a coalition structure, some countries may be satisfied but others may tray to force transition to another coalition structure. We assume that the transition from one coalition structure to another is motivated by the payoffs that countries can obtain in them. These payoffs are provided by a profit function derived from the model. In the paper we describe precisely how transitions take place. Moreover we consider that players are myopic, that is when considering a movement they do not wonder about further consequences that this movement can provoke. And we do not consider any stopping criteria to end the transitional process from one coalition structure to another. Therefore, countries may move successively in discrete steps unless they converge upon some coalition structures which are stable. The stability concept we use is the absorbing sets solution. Each absorbing set coincides with the elementary dynamic solution, Shenoy [1979]. And the absorbing sets solution is the collection of all the absorbing sets. The notion of stability lying on the absorbing sets solution may be understood as follows. Suppose that at some point in time an alternative in an absorbing set is reached, then alternatives in the absorbing set will be visited an infinite number of times, while no alternative outside the set will ever be visited again. Our main result demonstrates that for the environmental agreement model there is a unique absorbing set and this set contains the grand coalition. We pay special attention to stability of the grand coalition since it is the unique efficient coalition structure. Moreover we identify the unique absorbing set. The equilibrium coalition structures belonging to the absorbing set are described as follows: Only one group of countries will decide to sing an agreement while the others free-ride. But whenever the number of coalitions is smaller than a certain critical number at least one country will find it profitable to break the agreement, while up to this critical number all countries agree on signing the agreement for the grand coalition.