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 < ^{δ '}, ^{δ '}-A implies δ-A. Thus, as δ decreases from 1 to 0, δ-A weakens monotonically, eventually to a vacuous requirement. We consider two parametrizations _{{δ-efficiency}δ ∈[0,1]} and _{{δ-strategy-proofness}δ ∈[0,1]}, and investigate their compatibility with individual rationality for the class of two-agent economies defined on the linear preference domain. We show that (i) for each δ ∈ (0, 1], no rule is individually rational, δ-efficient, and strategy-proof; and (ii) for each δ ∈ (0, 1], no rule is individually rational, efficient, and δ-strategy-proof.