## Proportional response dynamics leads to market equilibrium

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Venue: | In STOC |

Citations: | 20 - 1 self |

### BibTeX

@INPROCEEDINGS{Wu_proportionalresponse,

author = {Fang Wu and Li Zhang},

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

booktitle = {In STOC},

year = {}

}

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### 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

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Citation Context ... limit matrix represents the bottleneck decomposition of the corresponding graph. Related work. The study of market equilibrium has a long and distinguished history in Economics. We refer to the book =-=[17]-=- for a comprehensive exposition. In [2], Arrow and Debreu established the existence of market equilibrium for a large family of exchange economies with concave utility functions. In a series of papers... |

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Citation Context ... Proportional response has appeared in a number of economics models of non-price-taking behavior. It has also been studied by Computer Science community in the context of network bandwidth allocation =-=[15, 14]-=- and computing resource allocation [11]. In our convergence proof for ρ = 1, we connect proportional response to the matrix scaling problem, which asks that whether one can transform a matrix, through... |

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Citation Context ...ies satisfying weak gross substitutability (WGS). There have also been attempts at computing market equilibria for economies with linear utility functions [22, 10, 20]. In Computer Science community, =-=[19, 8]-=- initiated the study of the computation and approximation of market equilibria. [9] proposed a polynomial time algorithm for computing the market equilibrium 2sfor linear utility functions in Fisher’s... |

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Citation Context ...ies satisfying weak gross substitutability (WGS). There have also been attempts at computing market equilibria for economies with linear utility functions [22, 10, 20]. In Computer Science community, =-=[19, 8]-=- initiated the study of the computation and approximation of market equilibria. [9] proposed a polynomial time algorithm for computing the market equilibrium 2sfor linear utility functions in Fisher’s... |

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Citation Context ... Proportional response has appeared in a number of economics models of non-price-taking behavior. It has also been studied by Computer Science community in the context of network bandwidth allocation =-=[15, 14]-=- and computing resource allocation [11]. In our convergence proof for ρ = 1, we connect proportional response to the matrix scaling problem, which asks that whether one can transform a matrix, through... |

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Citation Context ... does not involve any auxiliary price vector. The proportional response mechanism has also been studied in the context of relating non-pricetaking and price-taking behaviors in exchange economies. In =-=[23]-=-, Shapley and Shubik proposed the trading post game as a model to “explore the transition zone between perfect competition and oligopolistic competition”. In the game, each trader places a monetary bi... |

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Citation Context ...he bottleneck decomposition of a graph and the double normalization of a matrix. The bottleneck decomposition is similar to the characterization of market equilibrium in the linear Fisher’s market in =-=[9]-=- — it is a decomposition obtained by repeatedly removing the bottleneck, the set of nodes that have the smallest expansion rate, and its neighbors from the remaining graph. We show that in a market eq... |

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Citation Context ...s shown that tâtonnement converges for economies satisfying weak gross substitutability (WGS). There have also been attempts at computing market equilibria for economies with linear utility functions =-=[22, 10, 20]-=-. In Computer Science community, [19, 8] initiated the study of the computation and approximation of market equilibria. [9] proposed a polynomial time algorithm for computing the market equilibrium 2s... |

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Citation Context ...for a comprehensive exposition. In [2], Arrow and Debreu established the existence of market equilibrium for a large family of exchange economies with concave utility functions. In a series of papers =-=[3, 1, 4, 18, 27]-=-, several economists considered the problem of the stability of equilibrium and the convergence to equilibrium by the continuous tâtonnement dynamics, and it was shown that tâtonnement converges for e... |

61 |
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Citation Context ... for computing the market equilibrium 2sfor linear utility functions in Fisher’s market. The work was later extended to approximating general market equilibrium with linear utility functions [13]. In =-=[12]-=-, a polynomial time algorithm was proposed for computing the market equilibrium with linear utility function, and an interior point method was proposed by [28]. For more general utility functions, [7,... |

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Citation Context ...lt was later generalized to non-negative matrices [26] and to more general sums [25, 5, 21]. Matrix scaling has many applications in linear algebra and numerical computing. More recently, the work in =-=[16]-=- produced a strongly polynomial approximation algorithm for matrix scaling and, based on which, an approximation algorithm for calculating the permanent of non-negative matrices. 2 Preliminaries Bandw... |

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Citation Context ... [24] showed that for any positive matrix, alternative normalizing its rows and columns to 1 makes it converge to a doubly stochastic matrix. His result was later generalized to non-negative matrices =-=[26]-=- and to more general sums [25, 5, 21]. Matrix scaling has many applications in linear algebra and numerical computing. More recently, the work in [16] produced a strongly polynomial approximation algo... |

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Citation Context ...mber of economics models of non-price-taking behavior. It has also been studied by Computer Science community in the context of network bandwidth allocation [15, 14] and computing resource allocation =-=[11]-=-. In our convergence proof for ρ = 1, we connect proportional response to the matrix scaling problem, which asks that whether one can transform a matrix, through row and column normalization, to a mat... |

35 | A Path to the Arrow-Debreu Competitive Mar- ket Equilibrium
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Citation Context ...m with linear utility functions [13]. In [12], a polynomial time algorithm was proposed for computing the market equilibrium with linear utility function, and an interior point method was proposed by =-=[28]-=-. For more general utility functions, [7, 6] showed that the discretized tâtonnement process converges for WGS utility functions. Compared to the computation of market equilibrium, its convergence dyn... |

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Citation Context ...12], a polynomial time algorithm was proposed for computing the market equilibrium with linear utility function, and an interior point method was proposed by [28]. For more general utility functions, =-=[7, 6]-=- showed that the discretized tâtonnement process converges for WGS utility functions. Compared to the computation of market equilibrium, its convergence dynamics is more appealing as trading usually h... |

24 | Approximating market equilibria
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(Show Context)
Citation Context ...algorithm for computing the market equilibrium 2sfor linear utility functions in Fisher’s market. The work was later extended to approximating general market equilibrium with linear utility functions =-=[13]-=-. In [12], a polynomial time algorithm was proposed for computing the market equilibrium with linear utility function, and an interior point method was proposed by [28]. For more general utility funct... |

23 | Market equilibrium via the excess demand function
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Citation Context ...12], a polynomial time algorithm was proposed for computing the market equilibrium with linear utility function, and an interior point method was proposed by [28]. For more general utility functions, =-=[7, 6]-=- showed that the discretized tâtonnement process converges for WGS utility functions. Compared to the computation of market equilibrium, its convergence dynamics is more appealing as trading usually h... |

21 | The diagonal equivalence of a nonnegative matrix to a stochastic matrix
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Citation Context ...tive matrix, alternative normalizing its rows and columns to 1 makes it converge to a doubly stochastic matrix. His result was later generalized to non-negative matrices [26] and to more general sums =-=[25, 5, 21]-=-. Matrix scaling has many applications in linear algebra and numerical computing. More recently, the work in [16] produced a strongly polynomial approximation algorithm for matrix scaling and, based o... |

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Citation Context ...tive matrix, alternative normalizing its rows and columns to 1 makes it converge to a doubly stochastic matrix. His result was later generalized to non-negative matrices [26] and to more general sums =-=[25, 5, 21]-=-. Matrix scaling has many applications in linear algebra and numerical computing. More recently, the work in [16] produced a strongly polynomial approximation algorithm for matrix scaling and, based o... |

13 |
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Citation Context ...s shown that tâtonnement converges for economies satisfying weak gross substitutability (WGS). There have also been attempts at computing market equilibria for economies with linear utility functions =-=[22, 10, 20]-=-. In Computer Science community, [19, 8] initiated the study of the computation and approximation of market equilibria. [9] proposed a polynomial time algorithm for computing the market equilibrium 2s... |

13 |
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(Show Context)
Citation Context ...s shown that tâtonnement converges for economies satisfying weak gross substitutability (WGS). There have also been attempts at computing market equilibria for economies with linear utility functions =-=[22, 10, 20]-=-. In Computer Science community, [19, 8] initiated the study of the computation and approximation of market equilibria. [9] proposed a polynomial time algorithm for computing the market equilibrium 2s... |

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Citation Context ...for a comprehensive exposition. In [2], Arrow and Debreu established the existence of market equilibrium for a large family of exchange economies with concave utility functions. In a series of papers =-=[3, 1, 4, 18, 27]-=-, several economists considered the problem of the stability of equilibrium and the convergence to equilibrium by the continuous tâtonnement dynamics, and it was shown that tâtonnement converges for e... |

9 |
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Citation Context ...decomposition of the corresponding graph. Related work. The study of market equilibrium has a long and distinguished history in Economics. We refer to the book [17] for a comprehensive exposition. In =-=[2]-=-, Arrow and Debreu established the existence of market equilibrium for a large family of exchange economies with concave utility functions. In a series of papers [3, 1, 4, 18, 27], several economists ... |

5 |
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Citation Context ...for a comprehensive exposition. In [2], Arrow and Debreu established the existence of market equilibrium for a large family of exchange economies with concave utility functions. In a series of papers =-=[3, 1, 4, 18, 27]-=-, several economists considered the problem of the stability of equilibrium and the convergence to equilibrium by the continuous tâtonnement dynamics, and it was shown that tâtonnement converges for e... |

4 |
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Citation Context ...tive matrix, alternative normalizing its rows and columns to 1 makes it converge to a doubly stochastic matrix. His result was later generalized to non-negative matrices [26] and to more general sums =-=[25, 5, 21]-=-. Matrix scaling has many applications in linear algebra and numerical computing. More recently, the work in [16] produced a strongly polynomial approximation algorithm for matrix scaling and, based o... |

2 |
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