Subgame perfect implementation: A full characterization

Moore and Repullo [Subgame perfect implementation, Econometrica 56 (1988) 1191–1220], and Abreu and Sen [Subgame perfect implementation: a necessary and almost sufficient condition, J. Econ. Theory 50 (1990) 285–299] introduce distinct necessary and sufficient conditions for SPE implementation, when the number of players is at least three. This paper closes the gap between the conditions—a complete characterization of the SPE implementable choice rules is provided. The characterization consists of α*α*, which strengthens αα of Abreu–Sen by adding it a restricted veto-power condition, and the unanimity   condition. Under strict preferences α*α* is equal to αα.