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The solution for the Contested Garment Problemproposed in the Babylonic Talmud,one of the most important sources of inspiration for solving situations where demandovercomes supply of some resources, suggests that each agent should receive at least somepart of the available amount when facing these situations. This idea has underlied thetheoretical analysis of bankruptcy problems from its beginning (ONeill, 1982) to presentday (Dominguez and Thomson, 2006). In this context, starting from the fact that a societyestablishes its own set of Commonly Accepted Equity Principles, we propose a new lowerbound on awards de…ned, for each agent, as the minimum amount she gets according toall the admissible rules for such a society. Moreover, we analyze the recursive applicationof this new bound, since it will not exhaust the resources, in general.