Algorithms, games, and the internet
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
On the complexity of equilibria
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Market Equilibrium via a Primal-Dual-Type Algorithm
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Auction algorithms for market equilibrium
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
A Polynomial Time Algorithm for Computing the Arrow-Debreu Market Equilibrium for Linear Utilities
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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
Eisenberg-Gale markets: algorithms and structural properties
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On competitiveness in uniform utility allocation markets
WINE'07 Proceedings of the 3rd international conference on Internet and network economics
New results on rationality and strongly polynomial time solvability in eisenberg-gale markets
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
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We provide the first strongly polynomial time exact combinatorial algorithm to compute Fisher equilibrium for the case when utility functions do not satisfy the Gross substitutability property. The motivation for this comes from the work of Kelly, Maulloo, and Tan [15] and Kelly and Vazirani [16] on rate control in communication networks. We consider a tree like network in which root is the source and all the leaf nodes are the sinks. Each sink has got a fixed amount of money which it can use to buy the capacities of the edges in the network. The edges of the network sell their capacities at certain prices. The objective of each edge is to fix a price which can fetch the maximum money for it and the objective of each sink is to buy capacities on edges in such a way that it can facilitate the sink to pull maximum flow from the source. In this problem, the edges and the sinks play precisely the role of sellers and buyers, respectively, in the Fisher’s market model. The utility of a buyer (or sink) takes the form of Leontief function which is known for not satisfying Gross substitutability property. We develop an O(m3) exact combinatorial algorithm for computing equilibrium prices of the edges. The time taken by our algorithm is independent of the values of sink money and edge capacities. A corollary of our algorithm is that equilibrium prices and flows are rational numbers. Although there are algorithms to solve this problem but they are all based on convex programming techniques. To the best of our knowledge, ours is the first strongly polynomial time exact combinatorial algorithm for computing equilibrium prices of Fisher model under the case when buyers’ utility functions do not satisfy Gross substitutability property.