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
On the complexity of price equilibria
Journal of Computer and System Sciences - STOC 2002
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
On the polynomial time computation of equilibria for certain exchange economies
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A path to the Arrow–Debreu competitive market equilibrium
Mathematical Programming: Series A and B
Exchange market equilibria with leontief’s utility: freedom of pricing leads to rationality
WINE'05 Proceedings of the First international conference on Internet and Network Economics
Rationality and Strongly Polynomial Solvability of Eisenberg-Gale Markets with Two Agents
SIAM Journal on Discrete Mathematics
How profitable are strategic behaviors in a market?
ESA'11 Proceedings of the 19th European conference on Algorithms
Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems - Volume 2
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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 [F.P. Kelly, A.K. Maulloo, D.K.H. Tan, Rate control for communication networks: Shadow prices, proportional fairness and stability, Journal of Operational Research (1998)] and Kelly and Vazirani [F.P. Kelly, Vijay V. Vazirani, Rate control as a market equilibrium (2003) (in preparation)] 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 that 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 Fisher's market model. The utility of a buyer (or sink) takes the form of a Leontief function which is known for not satisfying Gross substitutability property. We develop an O(m^3) 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 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's model under the case when buyers' utility functions do not satisfy gross substitutability property.