On the Trade off Between Deficit and Inefficiency and the Double Auction with a Fixed Transaction Fee

In many trading environments, any incentive compatible and individually rational market mechanism will be either inefficient or will run a deficit. We prove that as the market size m gets large, for any fixed surplus (or deficit) x, m times the minimal absolute inefficiency converges to c(x) where c(·) is essentially a quadratic function of x. We introduce a new mechanism, the double auction with a fixed transaction fee. By choosing the size of the fee appropriately, any level of deficit or surplus can be implemented and the resulting mechanisms achieve the above bound. Corollaries include: an asymptotic version of the Myerson-Satterthwaite Impossibility Theorem; a description of the minimal subsidy required to implement the efficient trading rule; a characterization of the minimal inefficiency obtainable with budget-balanced market mechanisms; recommendations on the optimal organization of trade; and insights on the effects of taxation.