A rational convex program for linear Arrow-Debreu markets

We present a new flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known ([Nenakov and Primak 1983; Jain 2007; Cornet 1989]), our program exhibits several new features. It provides a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani [2012]. As a consequence, we also obtain a simple new proof of the result in Mertens [2003] that the equilibrium prices form a convex polyhedral set

Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model

Recent results, establishing evidence of intractability for such restrictive utility functions as additively separable, piecewise-linear and concave, under both Fisher and Arrow-Debreu market models, have prompted the question of whether we have failed to capture some essential elements of real markets, which seem to do a good job of finding prices that maintain parity between supply and demand. The main point of this paper is to show that even non-separable, quasiconcave utility functions can be handled efficiently in a suitably chosen, though natural, realistic and useful, market model; our model allows for perfect price discrimination. Our model supports unique equilibrium prices and, for the restriction to concave utilities, satisfies both welfare theorems

Markets and games: a simple equivalence among the core, equilibrium and limited arbitrage

This note provides simple proofs of the equivalence among the core, equilibrium and limited arbitrage in markets with short sales, and with uniform strictly convex preferences.mathematical proof; modeling; limited arbitrage; core; equilibrium

A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities.Comment: Preliminary version in ICALP 201

The Complexity of Non-Monotone Markets

We introduce the notion of non-monotone utilities, which covers a wide variety of utility functions in economic theory. We then prove that it is PPAD-hard to compute an approximate Arrow-Debreu market equilibrium in markets with linear and non-monotone utilities. Building on this result, we settle the long-standing open problem regarding the computation of an approximate Arrow-Debreu market equilibrium in markets with CES utility functions, by proving that it is PPAD-complete when the Constant Elasticity of Substitution parameter \rho is any constant less than -1

Market Equilibrium in Exchange Economies with Some Families of Concave Utility Functions

We present explicit convex programs which characterize the equilibrium for certain additively separable utility functions and CES functions. These include some CES utility functions that do not satisfy weak gross substitutability.Exchange economy, computation of equilibria, convex feasibility problem

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.Comment: An extended abstract will appear in SODA 201