# A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities

@article{Jain2004APT,
title={A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities},
author={Kamal Kumar Jain},
journal={45th Annual IEEE Symposium on Foundations of Computer Science},
year={2004},
pages={286-294}
}
• K. Jain
• Published 2004
• Mathematics, Computer Science
• 45th Annual IEEE Symposium on Foundations of Computer Science
We provide the first polynomial time exact algorithm for computing an Arrow-Debreu market equilibrium for the case of linear utilities. Our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation. As a side result, we prove that the set of assignments at equilibria is convex and the equilibria prices themselves are log-convex. Our convex program is explicit and intuitive, which allows maximizing a concave function over the set of… Expand
123 Citations
A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market
• Mathematics, Computer Science
• ICALP
• 2013
We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities. Our algorithm views the allocation of money as flowsExpand
A complementary pivot algorithm for markets under separable, piecewise-linear concave utilities
• Mathematics, Computer Science
• STOC '12
• 2012
A practical algorithm for computing an equilibrium for Arrow-Debreu markets under separable, piecewise-linear concave (SPLC) utilities is given, despite the PPAD-completeness of this case, and the first elementary proof of existence of equilibrium for this case is obtained, without using fixed point theorems. Expand
A combinatorial polynomial algorithm for the linear Arrow-Debreu market
• Computer Science, Mathematics
• Inf. Comput.
• 2015
This work presents the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities, and develops new methods to carefully deal with the flows and surpluses during price adjustments. Expand
Rational Convex Programs, Their Feasibility, and the Arrow-Debreu Nash Bargaining Game
This work provides a formal context to this activity by introducing the notion of {\em rational convex programs} and obtains primal-dual algorithms for determining feasibility, as well as giving a proof of infeasibility and finding an equilibrium. Expand
A Perfect Price Discrimination Market Model with Production, and a Rational Convex Program for It
• Mathematics, Computer Science
• Math. Oper. Res.
• 2011
This paper shows that introducing perfect price discrimination into the Fisher model with PLC utilities renders its equilibrium polynomial time computable, and gives an application of the price discrimination market model to online display advertising marketplaces. Expand
A strongly polynomial algorithm for linear exchange markets
• Computer Science, Mathematics
• STOC
• 2019
This work presents a strongly polynomial algorithm for computing an equilibrium in Arrow-Debreu exchange markets with linear utilities based on a variant of the weakly-polynomial Duan-Mehlhorn (DM) algorithm, and shows that it can be approximated by a simpler LP with two variables per inequality that is solvable in stronglyPolynomial time. Expand
An Alternating Algorithm for Finding Linear Arrow-Debreu Market Equilibrium
• Computer Science, Mathematics
• ArXiv
• 2019
This paper designs an algorithm for computing linear bijective market equilibrium, based on solving the rational convex program formulated by Devanur et al, that repeatedly alternates between a step of gradient-descent-like updates and a distributed step of optimization exploiting the property of such conveX program. Expand
An Arrow-Debreu Extension of the Hylland-Zeckhauser Scheme: Equilibrium Existence and Algorithms
• Computer Science, Mathematics
• ArXiv
• 2020
The $\epsilon$-approximate ADHZ model fills a void in the space of general mechanisms for one-sided matching markets, and satisfies Pareto optimality, approximate envy-freeness and incentive compatibility in the large. Expand
A path to the Arrow–Debreu competitive market equilibrium
• Y. Ye
• Mathematics, Computer Science
• Math. Program.
• 2008
A continuous path leading to the set of the Arrow–Debreu equilibrium, similar to the central path developed for linear programming interior-point methods is presented, derived from the weighted logarithmic utility and barrier functions and the Brouwer fixed-point theorem. Expand
Combinatorial Algorithms for General Linear Arrow-Debreu Markets
• Mathematics, Computer Science
• FSTTCS
• 2018
A combinatorial algorithm for determining the market clearing prices of a general linear Arrow-Debreu market, where every agent can own multiple goods, and refines the iterative algorithm of Duan, Garg and Mehlhorn using several new ideas. Expand

#### References

SHOWING 1-10 OF 30 REFERENCES
A Polynomial Time Algorithm for Computing the Arrow-Debreu Market Equilibrium for Linear Utilities
• K. Jain
• Mathematics, Computer Science
• FOCS
• 2004
The main idea in this generalization is to allow ellipsoids not to contain the whole convex region but a part of it, which makes a powerful theorem even more powerful in the area of geometric algorithms and combinatorial optimization. Expand
On the polynomial time computation of equilibria for certain exchange economies
• Mathematics, Computer Science
• SODA '05
• 2005
The first polynomial-time algorithms for exchange markets under the general setting of weak gross substitutability are shown, and this approach does not make use of the specific form of these utility functions and is in this sense more general. Expand
An Improved Approximation Scheme for Computing Arrow-Debreu Prices for the Linear Case
• Mathematics, Computer Science
• FSTTCS
• 2003
This paper gives a strongly polynomial time approximation scheme for the problem of computing a market equilibrium in the Arrow-Debreu setting, when the utilities are linear functions. Expand
Auction algorithms for market equilibrium
• Computer Science
• STOC '04
• 2004
It is shown that finding the market equilibrium is the same as finding a linear-program from the family of programs where the optimal dual solution satisfies certain properties, and the algorithm obtained outperforms previously known methods. Expand
Market equilibria for homothetic, quasi-concave utilities and economies of scale in production
• Mathematics, Computer Science
• SODA '05
• 2005
The notion of a trading cone is introduced which enables us to compute market equilibrium in the presence of economies of scale in production provided differential pricing is allowed. Expand
On the complexity of equilibria
• Mathematics, Computer Science
• STOC '02
• 2002
A polynomial-time algorithm is proved that approximates the market equilibrium arbitrarily closely when the number of goods is bounded and the utilities are linear, implying that the ideal informational economy of a market with unique individual optima is unattainable in general. Expand
Frontiers in Applied General Equilibrium Modeling: Mathematical Programs with Equilibrium Constraints: Automatic Reformulation and Solution via Constrained Optimization
• Mathematics
• 2002
Constrained optimization has been extensively used to solve many large scale deterministic problems arising in economics, including, for example, square systems of equations and nonlinear programs. AExpand
EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY
• Economics
• 1954
A. Wald has presented a model of production and a model of exchange and proofs of the existence of an equilibrium for each of them. Here proofs of the existence of an equilibrium are given for anExpand
Market equilibrium via a primal-dual-type algorithm
• Mathematics, Computer Science
• The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
• 2002
This work provides the first polynomial time algorithm for the linear version of a problem defined by Irving Fisher in 1891, modeled after Kuhn's primal-dual algorithm for bipartite matching. Expand
A converging algorithm for a linear exchange model
Abstract An exchange model with linear utilities functions is considered. An elementary proof of the inequality of revealed preference is described. The model is equivalent to an infinite system ofExpand