Nash and strongly consistent two-player game forms

A two-player game form is Nash-consistent if and only if it is tight (Gurvich). Therefore Nash-consistency of two-player game forms depends only on the effectivity structure. This fact is no longer true for strong consistency. In this paper we introduce a new object called the joint effectivity structure and define the exact joint effectivity set. These notions are similar though more sophisticated than the usual effectivity functions. We prove that a two-player game form is strongly consistent if and only if it is tight and jointly exact. Joint exactness is a property of the exact joint effectivity set which basically requires that the joint exact effectivity set coincides with the classical effectivity function. As a corollary we have a characterization of two-player strongly implementable social choice correspondences.