The paper fully characterizes the class of Nash implementable social choice correspondences (SCCs) by mechanisms endowed with Saijo’s message space specification—<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$s$$</EquationSource> </InlineEquation>-mechanisms. This class of SCCs is equivalent to the class of Nash implementable SCCs, though any game form involving...</equationsource></inlineequation>