A Complementary Pivot Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities

Using Lemke's scheme, we give a complementary pivot algorithm for computing an equilibrium for Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities. Despite the polynomial parity argument on directed graphs (PPAD) completeness of this case, experiments indicate that our algorithm is practical-on randomly generated instances, the number of iterations it needs is linear in the total number of segments (i.e., pieces) in all the utility functions specified in the input. Our paper settles a number of open problems: (1) Eaves (1976) gave an LCP formulation and a Lemke-type algorithm for the linear Arrow-Debreu model. We generalize both to the SPLC case, hence settling the relevant part of his open problem. (2) Our path following algorithm for SPLC markets, together with a result of Todd (1976), gives a direct proof of membership of such markets in PPAD and settles a question of Vazirani and Yannakakis (2011). (3) We settle a question of Devanur and Kannan (2008) of obtaining a "systematic way of finding equilibrium instead of the brute-force way" for the separable case and we obtain a strongly polynomial algorithm if the number of goods or agents is constant. (4) We give a combinatorial way of interpreting Eaves' algorithm for the linear case, hence answering Eaves' question (1976), "That the algorithm can be interpreted as a 'global market adjustment mechanism' might be interesting to explore."