We propose new axiomatizations of values of cooperative games where traditional properties connected with special players (dummy, null or zero) are replaced with weaker properties relating to such participants of the game. We assume that the change of payoff of a player when combining the game with another game where this player is special is constant. Using such axioms with an additional assumption that a value is odd and-if necessary-the fairness axioms holds, one can obtain axiomatizations without additivity where not only classical dummy, null or zero players axioms but even equal treatment can be redundant. These properties are used to construct new axiomatizations of the Shapley, Banzhaf and Deegan-Packel values. Some of them contain a new mirror game axiom.