In view of these difficulties, it is desirable to design mechanisms in which all equilibrium outcomes are optimal for the given goal function. The quest for this property is known as the implementation problem. Groves and Ledyard (1977) and Hurwicz and Schmeidler (1978) showed that, in certain situations, it is possible to construct mechanisms in which all Nash equilibria are Pareto optimal, while Eric Maskin (1977) gave a general characterization of Nash implementable social-choice functions. He showed that Nash implementation requires a condition now known as Maskin monotonicity (see Section 3.3 for an illustration of this property). Maskin (1977) also showed that if Maskin monotonicity and a condition called no-veto-power are both satisfied, and if there are at least three agents, then implementation in Nash equilibrium is possible.
Maskin considered Nash equilibria in games of complete information, but his results have been generalized to Bayesian Nash equilibria in games of incomplete information (see Postlewaite and Schmeidler, 1986, Palfrey and Srivastava, 1989, Mookherjee and Reichelstein, 1990, and Jackson, 1991). For example, Palfrey and Srivastava (1991) show how the double auction can be modified so as to render all equilibria incentive efficient.
Maskin’s results have also been extended in many other directions, such as virtual (or approximate) implementation (Matsushima, 1988, Abreu and Sen, 1991), implementation in renegotiation-proof equilibria (Maskin and Moore, 1999) and by way of sequential mechanisms (Moore and Repullo, 1988). Implementation theory has played, and continues to play, an important role in several areas of economic theory, such as social choice theory (Moulin, 1994) and the theory of incomplete contracts (Maskin and Tirole, 1999).