Synonyms
Algorithmic game theory; Arrow-debreu market; Max-min utility; Walras equilibrium
Years and Authors of Summarized Original Work
2005; Codenotti, Saberi, Varadarajan, Ye
2005; Ye
Problem Definition
The Arrow-Debreu exchange market equilibrium problem was first formulated by Léon Walras in 1954 [7]. In this problem, everyone in a population of m traders has an initial endowment of a divisible goods and a utility function for consuming all goods – their own and others’. Every trader sells the entire initial endowment and then uses the revenue to buy a bundle of goods such that his or her utility function is maximized. Walras asked whether prices could be set for everyone’s goods such that this is possible. An answer was given by Arrow and Debreu in 1954 [1] who showed that, under mild conditions, such equilibrium would exist if the utility functions were concave. In general, it is unknown whether or not an equilibrium can be computed efficiently; see, e.g., General Equilibrium.
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Notes
- 1.
The reader may want to read Brainard and Scarf [2] on how to compute equilibrium prices in 1891; Chen and Deng [2] on the most recent hardness result of computing the bimatrix game; Cottle et al. [5] for literature on linear complementarity problems; and all references listed in [4] and [8] for the recent literature on computational equilibrium.
Recommended Reading
The reader may want to read Brainard and Scarf [2] on how to compute equilibrium prices in 1891; Chen and Deng [2] on the most recent hardness result of computing the bimatrix game; Cottle et al. [5] for literature on linear complementarity problems; and all references listed in [4] and [8] for the recent literature on computational equilibrium.
Arrow KJ, Debreu G (1954) Existence of an equilibrium for competitive economy. Econometrica 22:265–290
Brainard WC, Scarf HE (2000) How to compute equilibrium prices in 1891. Cowles Foundation Discussion Paper 1270
Chen X, Deng X (2005) Settling the complexity of 2-player Nash-equilibrium. ECCC TR05-140
Codenotti B, Saberi A, Varadarajan K, Ye Y (2006) Leontief economies encode nonzero sum two-player games. In: Proceedings SODA, Miami
Cottle R, Pang JS, Stone RE (1992) The linear complementarity problem. Academic, Boston
Gilboa I, Zemel E (1989) Nash and correlated equilibria: some complexity considerations. Games Econ Behav 1:80–93
Walras L (1954) [1877] Elements of pure economics. Irwin. ISBN 0-678-06028-2
Ye Y (2005) Exchange market equilibria with Leontief’s utility: freedom of pricing leads to rationality. In: Proceedings WINE, Hong Kong
Zhu Z, Dang C, Ye Y (2012) A FPTAS for computing a symmetric Leontief competitive economy equilibrium. Math Program 131:113–129
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Ye, Y. (2016). Leontief Economy Equilibrium. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_201
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