This paper studies the equilibrium property and algorithmic complexity of the exchange market equilibrium problem with concave piece-wise linear functions, which include linear and Leontief’s utility functions as special cases. We show that the Fisher model again reduces to the weighted analytic center problem, and the same linear programming complexity bound applies to computing its equilibrium. However, the story for the Arrow-Debreu model with Leontief’s utility becomes quite different. We show that, for the first time, solving this class of Leontief exchange economies is equivalent to solving a linear complementarity problem whose algorithmic complexity is finite but not polynomially bounded. 1

Optimization in the Private Value Model: Competitive Analysis Applied to Auction Design by Jason D. Hartline Chair of Supervisory Committee: Professor Anna R. Karlin Computer Science and Engineering We consider the study of a class of optimization problems with applications towards profit maximization. One feature of the classical treatment of optimization problems is that the space over which the optimization is being performed, i.e., the input description of the problem, is assumed to be publicly known to the optimizer. This assumption does not always accurately represent the situation in practical applications. Recently, with the advent of the Internet as one of the most important arenas for resource sharing between parties with diverse and selfish interests, this distinction has become more readily apparent. The inputs to many optimizations being performed are not publicly known in advance. Instead they must be solicited from companies, computerized agents, individuals, etc. that may act selfishly to promote their own self-interests. In particular, they may lie about their values or may not adhere to specified protocols if it benefits them.

Abstract. We introduce a new family of utility functions for exchange markets. This family provides a natural and ”continuous ” hybridization of the traditional linear and Leontief utilities and might be useful in understanding the complexity of computing and approximating of market equilibria. We show that a Fisher equilibrium of an exchange market with m commodities and n traders and hybrid linear-Leontief utilities can be found in O ( √ mn(m + n) 3 L) time. Because this family of utility function contains Leontief utility functions as special cases, finding an approximate Arrow-Debreu equilibria with hybrid linear-Leontief is PPAD-hard in general. In contrast, we show that, when the linear component is wellconditioned and the Leontief components are structured and finite, we can efficiently compute an approximate Arrow-Debreu equilibrium. 1