Fairness and efficiency for allocations with participation constraints

We propose a notion of fairness for allocation problems in which different agents may have reservation utilities, stemming from outside options, or property rights. Fairness is usually understood as the absence envy, but this can be incompatible with utilities. It possible that Alice's envy Bob's assignment cannot remedied without violating participation constraint. Instead, we seek to rule out justified defined remedy would not violate any agent's show fairness, meaning achieved together efficiency and individual rationality. introduce competitive equilibrium approach price-dependent incomes obtaining desired properties.