The problem of the existence of countably additive core for cooperative games with countable set of players is solved. The existence theorem for Neumann-Morgenstern solution in each 4-person cooperative game was proved (with O.N.Bondareva and T. E. Kulakovskaya). A number of existence theorems for bargaining sets $M$ in cooperative games were obtained under the assumption that objections and counterobjections are admitted among members of special collections of coalitions. All social welfare orderings on the entire space $R^n$ satisfying scale independence and preserving in the limit conditions are described (with E. B. Yanovskaya). For a fixed arbitrary orthant in $R^n$ each of these orderings is representable by a lexicographical ordering defined by a collection of Cobb–Douglas functions. Vectors from different orthants are compared by a rule based on a linear ordering on the set of orthants and a special number ("depth of comparison") for these orthants. The conditions for commutation of mappings convolving rows and columns of matrices with integer elements and integer values of mappings are obtained. The results generalize Ostrogorski paradox. A number of papers were devoted to axiomatical justification of solutions of bargaining problem with claim point. A complete description of strictly monotonic, consistent and path independent solutions for allocation problem with claims was obtained. This result was applied for axiomatical justification of a class of solutions of bargaining problem with claim points and convex feasible sets including the least square and the maximal weighted entropy solutions.
Main publications:
Naumova N. I., Yanovskaya E. B. Nash social welfare orderings // Mathematical Social Sciences, 2001, 42(3), 203–231.
Naumova N. I. Nonsymmetric equal sacrifice solutions for claim problem // Mathematical Social Sciences, 2002, 43(1), 1–18.
