Impossibility results for parametrized notions of efficiency and strategy-proofness in exchange economies

We study a standard model of exchange economies with individual endowments. It is well known that no rule is individually rational, efficient, and strategy-proof. In order to quantify the extent of this impossibility, we parametrize axioms on allocation rules. Given an axiom A, a parametrization of A is a continuum of axioms {δ-A}δ∈[0,1] such that (i) δ-A is equivalent to A if and only if δ=1; (ii) δ-A is vacuous if and only if δ=0; and (iii) for each pair δ,δ′∈[0,1] with δ