A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities

Using the powerful machinery of the linear complementarity problem and Lemke's algorithm, we give a practical algorithm for computing an equilibrium for Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities, despite the PPAD-completeness of this case. As a corollary, we obtain the first elementary proof of existence of equilibrium for this case, i.e., without using fixed point theorems. In 1975, Eaves [10] had given such an algorithm for the case of linear utilities and had asked for an extension to the piecewise-linear, concave utilities. Our result settles the relevant subcase of his problem as well as the problem of Vazirani and Yannakakis of obtaining a path following algorithm for SPLC markets, thereby giving a direct proof of membership of this case in PPAD.