In his seminal 1950 paper, John Nash defined the bargaining problem; the ensuing theory of bargaining lies today at the heart of game theory. In this work, we initiate an algorithmic study of Nash bargaining problems.
We consider a class of Nash bargaining problems whose solution can be stated as a convex program. For these problems, we show that there corresponds a market whose equilibrium allocations yield the solution to the convex program and hence the bargaining problem. For several of these markets, we give combinatorial, polynomial time algorithms, using the primal-dual paradigm.
Unlike the traditional Fisher market model, in which buyers spend a fixed amount of money, in these markets, each buyer declares a lower bound on the amount of utility she wishes to derive. The amount of money she actually spends is a specific function of this bound and the announced prices of goods. Over the years, a fascinating theory has started forming around a convex program given by Eisenberg and Gale in 1959. Besides market equilibria, this theory touches on such disparate topics as TCP congestion control and efficient solvability of nonlinear programs by combinatorial means. Our work shows that the Nash bargaining problem fits harmoniously in this collage of ideas.
Vijay Vazirani got his Bachelor's degree in Computer Science from MIT in 1979 and his Ph.D. from the University of California at Berkeley in 1983. His research has spanned a broad range of themes within the design of efficient algorithms - combinatorial optimization, approximation algorithms, randomized algorithms, parallel algorithms, and most recently algorithmic issues in game theory and mathematical economics. He has also worked in complexity theory, cryptography and information theory.
In 2001 he published what is widely regarded as the definitive book on Approximation Algorithms. This book has been translated into Japanese,Polish and French. Last year, he co-editing a comprehensive volume on Algorithmic Game Theory. He is a Fellow of the ACM.