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The extension of Dutta-Ray solution to convex NTU games
The most of solutions for games with non-transferable utilities (NTU) are NTU extensions of solution concepts defined for games with transferable utilities (TU). For example, there are three NTU versions of the Shapley value due to Aumann(1985), Kalai--Samet(1977), and Maschler--Owen(1992). The Shapley value is {\it standard} for two-person games. An NTU analog of standard solution is called the {\it symmetric proportional solution (SP)} (Kalai 1977), and the most of NTU solutions are SP solutions for two-person games. Another popular TU game solution which is not standard for two-person case is the {\it egalitarian Dutta-Ray solution (Dutta, Ray (1989), Dutta 1990). It was defined for the class of convex TU games and then extended to the class of all TU games (Branzei et al. 2006). . The DR solution for superadditive two-person TU games is the solution of constrained egalitarianism, it chooses the payoff vectors the closest to the diagonal of the space R^N. Its extension to superadditive two-person NTU games and then to n-person bargaining problems is the lexicographically maxmin solution}: for each game/bargaining problem it is the individually rational payoff vector which is maximal w.r.t. the lexmin relation. This solution if positively homogenous, but is not covariant w.r.t. shifts of individual payoffs. In the presentation this solution is extended to the class of NTU non-levelled games which are both ordinal and cardinal convex. Since convex TU games considered in NTU setting are ordinal and cardinal convex, the NTU DR solution is, in fact, an extension of the original TU version to the mentioned class of NTU games. It turns out that in this class the DR solution is single-valued and belongs to the core. A result similar to that of Dutta for TU convex games is obtained: the DR solution for the class of non-levelled ordinal and cardinal convex games is the single solution being the lexicographically maxmin solution for two-person games and consistent in (slightly modified) Peleg's definition (Peleg 1985) of the reduced games.