## Proportional response dynamics leads to market equilibrium

Venue: | In STOC |

Citations: | 18 - 1 self |

### BibTeX

@INPROCEEDINGS{Wu_proportionalresponse,

author = {Fang Wu and Li Zhang},

title = {Proportional response dynamics leads to market equilibrium},

booktitle = {In STOC},

year = {}

}

### OpenURL

### Abstract

One of the main reasons of the recent success of peer to peer (P2P) file sharing systems such as BitTorrent is its built-in tit-for-tat mechanism. In this paper, we model the bandwidth allocation in a P2P system as an exchange economy and study a tit-for-tat dynamics, namely the proportional response dynamics, in this economy. In a proportional response dynamics each player distributes its good to its neighbors proportional to the utility it received from them in the last period. We show that this dynamics not only converges but converges to a market equilibrium, a standard economic characterization of efficient exchanges in a competitive market. In addition, for some classes of utility functions we consider, it converges much faster than the classical tâtonnement process and any existing algorithms for computing market equilibria. As a part of our proof we study the double normalization of a matrix, an operation that linearly scales the rows of a matrix so that each row sums to a prescribed positive number, followed by a similar scaling of the columns. We show that the double normalization process of any non-negative matrix always converges. This complements the previous studies in matrix scaling that has focused on the convergence condition of the process when the row and column normalizations are considered as separate steps. 1