Bargaining with ordinal and cardinal players

We consider bargaining problems with at least one cardinal player and with ordinal players, and provide a complete description of utility invariant solutions of such problems for two players. For the n-person case we provide a procedure that: (i) returns a given cardinal solution if there are only cardinal players; (ii) is based on the ordinal solution for gradual bargaining problems, introduced by O'Neill et al. [O'Neill, B., Samet, D., Wiener, Z., Winter, E., 2002. Bargaining with an agenda. Games Econ. Behav., in press], for the ordinal players. Finally, we introduce the so-called cardinal concession solution as another example of a utility invariant solution.