A strongly polynomial algorithm to solve combinatorial linear programs
Operations Research
Convex Optimization
Market equilibria for homothetic, quasi-concave utilities and economies of scale in production
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Exchange market equilibria with Leontief's utility: Freedom of pricing leads to rationality
Theoretical Computer Science
Theoretical Computer Science
Settling the complexity of computing two-player Nash equilibria
Journal of the ACM (JACM)
On the Complexity of Nash Equilibria and Other Fixed Points
SIAM Journal on Computing
2-Player nash and nonsymmetric bargaining games: algorithms and structural properties
SAGT'10 Proceedings of the Third international conference on Algorithmic game theory
IEEE Transactions on Information Theory
A Perfect Price Discrimination Market Model with Production, and a Rational Convex Program for It
Mathematics of Operations Research
The notion of a rational convex program, and an algorithm for the arrow-debreu Nash bargaining game
Journal of the ACM (JACM)
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Inspired by the convex program of Eisenberg and Gale which captures Fisher markets with linear utilities, Jain and Vazirani [K. Jain and V. V. Vazirani, Games and Economic Behavior, 70 (2010), pp. 84-106] introduced the class of Eisenberg-Gale (EG) markets. We study the structure of EG(2) markets, the class of EG markets with two agents. We prove that all markets in this class are rational, that is, they have rational equilibrium, and they admit strongly polynomial time algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatorial linear program (LP). This helps positively resolve the status of two markets left as open problems by Jain and Vazirani: the capacity allocation market in a directed graph with two source-sink pairs and the network coding market in a directed network with two sources. Our algorithms for solving the corresponding nonlinear convex programs are fundamentally different from those obtained by Jain and Vazirani; whereas they use the primal-dual schema, our main tool is binary search powered by the strong LP-duality theorem.