An axiomatization of the inner core using appropriate reduced games

I adapt a reduction process introduced by Serrano and Volij [Serrano, R., Volij, O., 1998. Axiomatization of neoclassical concepts for Economies. Journal of Mathematical Economics 30, 87-108] so that the reduced games of convex-valued games are convex-valued. I use the corresponding consistency property and its converse to axiomatize the inner core for games that are convex-valued, non-level and smooth.