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>

Three alternative characterizations of the proportional solution defined on compact and comprehensive bargaining problems with claims are provided. Two new contraction-type and expansion-type axioms are used. Moreover, the single-valuedness axiom is dispensable if the classical symmetry axiom is...