We offer a general derivation of axiomatizations for allocation rules, referred to as "duality" and "anti-duality" approach. We show basic properties of duality and anti-duality approach. Using these properties, we can derive axiomatizations of allocation rules by taking (anti-)dual of axioms involved in axiomatizations of their self-(anti-)dual rules. As an illustration, we derive a new axiomatization of the Shapley value for bidding ring problems from using the notion of duality and axioms involved in axiomatizations of the Shapley value for airport problems. As another illustration, we derive a new axiomatization of the nucleolus for bidding ring problems from using the notion of anti-duality and axioms involved in axiomatizations of the nucleolus for airport problems.