Virtual Implementation in Nash Equilibrium

Statistical Utilitarianism

“…19 Meanwhile, the central premise of our results in Sections 2 to 5 is that it is enough to obtain an arbitrarily high probability of selecting a utilitarian optimum, rather than certainty. This is exactly the same premise as the theory of virtual implementation introduced by Matsushima (1988) and Abreu and Sen (1991). Virtual implementation is an extremely powerful and versatile implementation technology.…”

“…Virtual implementation is an extremely powerful and versatile implementation technology. For example, if the voters have complete information about one another, then any social choice rule can be virtually implemented in Nash equilibrium (Abreu and Sen, 1991) or iterated undominated strategies (Abreu and Matsushima, 1992). Even with incomplete information, a very large class of social choice rules can be virtually implemented in Bayesian Nash equilibrium (Serrano and Vohra, 2005), or even robustly virtually implemented (Artemov et al, 2013).…”

Virtual repeated implementation

“…The proof strategy is simple and is essentially a reinterpretation of virtual implementation (Abreu and Sen, 1991;Matsushima, 1988). Any alternative stream which realizes all alternatives can be thought of as being in the ''interior'' of the set of alternative streams.…”

“…there exists a game form for which agents playing a Nash equilibrium select the desired social alternative and for which every desired social alternative can be realized by some Nash equilibrium). See Matsushima (1988) and Abreu and Sen (1991).…”

Bayesian implementation in exchange economies with state dependent preferences and feasible sets

“…For complete information environments, characterization results were given by Maskin [14], Hurwicz, Maskin, Postlewaite [12], Repullo [28], Sajio [29], Moore and Repullo [19], Dutta and Sen [5], Danilov [4], and others for Nash implementation; Moore and Repullo [18], Abreu and Sen [2] and others for implementation using re®ne-ments of Nash equilibrium; Matsushima [15] and Abreu and Sen [3] for virtual Nash implementation. For incomplete information environments, characterization results were given by Postlewaite and Schmeidler [25], Palfrey, and Srivastava [20,21], Mookherjee and Reichelstein [17], Jackson [13], Hong [9] among many others for Bayesian implementation; by Palfrey and Srivastava [23] and Mookherjee and Reichelstein [17] for implementation in using re®nements of Bayesian equilibrium; Abreu and Matsushima [1], Matsushima [16], Duggan [7], and Tian [35] for virtual Bayesian implementation.…”