Nash Implementation and Opportunity Equilibrium

An economic institution in equilibrium analysis is formulated by a system “opportunities” the institution offers to its members. Extending this formulation, an opportunity system is a non-empty set of profiles of individual opportunity sets (subsets of alternatives). An alternative is an opportunity equilibrium under a system if there is an opportunity profile in the system such that the alternative maximizes each agent’s well-being over his opportunity set in the profile. Examples are Walrasian equilibrium, Lindahl equilibrium, valuation equilibrium by Mas-Colell (1980), equal opportunity equilibrium by Thomson (1994), etc. The main results show that this formulation of economic institutions by opportunity systems is closely related with the alternative formulation by game forms in Implementation Theory and that in some well-known environments, they are equivalent. A useful by-product is a decomposition of implementation procedure into two steps: the first step is to identify an opportunity system supporting a rule as the opportunity equilibrium correspondence, and the second step is to use this system to design a game form implementing the rule. Thus, informational efficiency in the opportunity system, if any, can be embedded in the game form implementing the rule.