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, e cient, and strategyproof. 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 only if δ = 1; (ii) δ-A is vacuous 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-e ciency}δ∈[0,1] and {δ-strategy-proofness}δ∈[0,1], and investigate their compatibility with individual rationality for the class of two-agent economies de ned on a domain containing linear preference relations. We show that (i) for each δ ∈ (0, 1], no rule is individually rational, δ-e cient, and strategy-proof ; and (ii) for each δ ∈ (0, 1], no rule is individually rational, e cient, and δ-strategy-proof. These results strengthen existing impossibility theorems in two directions that have not been explored so far. Journal of Economic Literature Classi cation Numbers: D71