Reduced games and egalitarian solutions

For a class of reduced games satisfying a monotonicity property, we introduce a family of set-valued solution concepts based on egalitarian considerations and consistency principles, and study its relation with the core. Regardless of the reduction operation we consider, the intersection between both sets is either empty or a singleton containing the lexmax solution Arin et al. (Math Soc Sci 46:327-345, 2003). This result induces a procedure for computing the lexmax solution for a class of games that contains games with large core Sharkey (Int J Game Theory 11:175-182, 1982).