Efficient Algorithms for Market Equilibria

Combinatorial;Network flow;Utilities;Arrow-Debreu;Fisher

Devanur, Nikhil Rangarajan ; Computing

University:Georgia Institute of Technology

Department:Computing

关键词: Combinatorial;    Network flow;    Utilities;    Arrow-Debreu;    Fisher;   

Others  :  https://smartech.gatech.edu/bitstream/1853/16282/1/devanur_nikhil_200708_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

The mathematical modelling of a market, and the proof of existence ofequilibriahave been of central importance in mathematical economics.Since the existence proof is non-constructive in general,a natural question is if computation of equilibria can be done efficiently. Moreover, the emergence of Internet and e-commerce has given rise to new markets that have completelychanged the traditional notions. Add to this the pervasiveness of computing resources, and an algorithmic theory of market equilibrium becomes highly desirable.The goal of this thesis is toprovide polynomial time algorithms for various market models. Two basic market models are the Fisher model: one in which there is ademarcation between buyers and sellers, buyers are interested in thegoods that the sellers possess, and sellers are only interested in themoney that the buyers have; and the Arrow-Debreu model: everyone hasan endowment of goods, and wants to exchange them for other goods. We give the first polynomial time algorithm for exactly computing an equilibrium in the Fisher model with linear utilities. We also show that the basic ideas in this algorithm can be extended to give a stronglypolynomial time approximation scheme in the Arrow-Debreu model.We also give several existential, algorithmic and structural results for new market models: - the *spending constraint* utilities (defined by Vazirani) that captures the "diminishing returns" property while generalizing the algorithm for the linear case.- thecapacity allocation market (defined by Kelly), motivated by the study of fairness and stability of the Transmission Control Protocol (TCP)for the Internet, and more generally the class of Eisenberg-Gale (EG) markets (defined by Jain and Vazirani). In addition, we consider the adwords market on search engines and show thatsome of thesemodels are a natural fit in this setting. Finally, this line of research has given insights into the fundamental techniques in algorithm design. Theprimal-dual schema has been a great success in combinatorial optimization and approximation algorithms. Our algorithms use thisparadigm in the enhanced setting ofKarush-Kuhn-Tucker (KKT) conditions and convex programs.