### Abstract

A major goal in Algorithmic Game Theory is to justify equilibrium concepts from an algorithmic and complexity perspective. One appealing approach is to identify natural distributed algorithms that converge quickly to an equilibrium. This paper established new convergence results for two generalizations of proportional response in Fisher markets with buyers having CES utility functions. The starting points are respectively a new convex and a new convex-concave formulation of such markets. The two generalizations correspond to suitable mirror descent algorithms applied to these formulations. Several of our new results are a consequence of new notions of strong Bregman convexity and of strong Bregman convex-concave functions, and associated linear rates of convergence, which may be of independent interest. Among other results, we analyze a damped generalized proportional response and show a linear rate of convergence in a Fisher market with buyers whose utility functions cover the full spectrum of CES utilities aside the extremes of linear and Leontief utilities; when these utilities are included, we obtain an empirical O(1/T) rate of convergence.

Original language | English (US) |
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Title of host publication | ACM EC 2018 - Proceedings of the 2018 ACM Conference on Economics and Computation |

Publisher | Association for Computing Machinery, Inc |

Pages | 351-368 |

Number of pages | 18 |

ISBN (Electronic) | 9781450358293 |

DOIs | |

State | Published - Jun 11 2018 |

Event | 19th ACM Conference on Economics and Computation, EC 2018 - Ithaca, United States Duration: Jun 18 2018 → Jun 22 2018 |

### Other

Other | 19th ACM Conference on Economics and Computation, EC 2018 |
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Country | United States |

City | Ithaca |

Period | 6/18/18 → 6/22/18 |

### Fingerprint

### Keywords

- Bregman divergence
- Fisher market
- Mirror descent
- Proportional response

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Statistics and Probability
- Computational Mathematics
- Economics and Econometrics

### Cite this

*ACM EC 2018 - Proceedings of the 2018 ACM Conference on Economics and Computation*(pp. 351-368). Association for Computing Machinery, Inc. https://doi.org/10.1145/3219166.3219189

**Dynamics of distributed updating in fisher markets.** / Cheung, Yun Kuen; Cole, Richard; Tao, Yixin.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ACM EC 2018 - Proceedings of the 2018 ACM Conference on Economics and Computation.*Association for Computing Machinery, Inc, pp. 351-368, 19th ACM Conference on Economics and Computation, EC 2018, Ithaca, United States, 6/18/18. https://doi.org/10.1145/3219166.3219189

}

TY - GEN

T1 - Dynamics of distributed updating in fisher markets

AU - Cheung, Yun Kuen

AU - Cole, Richard

AU - Tao, Yixin

PY - 2018/6/11

Y1 - 2018/6/11

N2 - A major goal in Algorithmic Game Theory is to justify equilibrium concepts from an algorithmic and complexity perspective. One appealing approach is to identify natural distributed algorithms that converge quickly to an equilibrium. This paper established new convergence results for two generalizations of proportional response in Fisher markets with buyers having CES utility functions. The starting points are respectively a new convex and a new convex-concave formulation of such markets. The two generalizations correspond to suitable mirror descent algorithms applied to these formulations. Several of our new results are a consequence of new notions of strong Bregman convexity and of strong Bregman convex-concave functions, and associated linear rates of convergence, which may be of independent interest. Among other results, we analyze a damped generalized proportional response and show a linear rate of convergence in a Fisher market with buyers whose utility functions cover the full spectrum of CES utilities aside the extremes of linear and Leontief utilities; when these utilities are included, we obtain an empirical O(1/T) rate of convergence.

AB - A major goal in Algorithmic Game Theory is to justify equilibrium concepts from an algorithmic and complexity perspective. One appealing approach is to identify natural distributed algorithms that converge quickly to an equilibrium. This paper established new convergence results for two generalizations of proportional response in Fisher markets with buyers having CES utility functions. The starting points are respectively a new convex and a new convex-concave formulation of such markets. The two generalizations correspond to suitable mirror descent algorithms applied to these formulations. Several of our new results are a consequence of new notions of strong Bregman convexity and of strong Bregman convex-concave functions, and associated linear rates of convergence, which may be of independent interest. Among other results, we analyze a damped generalized proportional response and show a linear rate of convergence in a Fisher market with buyers whose utility functions cover the full spectrum of CES utilities aside the extremes of linear and Leontief utilities; when these utilities are included, we obtain an empirical O(1/T) rate of convergence.

KW - Bregman divergence

KW - Fisher market

KW - Mirror descent

KW - Proportional response

UR - http://www.scopus.com/inward/record.url?scp=85050159797&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050159797&partnerID=8YFLogxK

U2 - 10.1145/3219166.3219189

DO - 10.1145/3219166.3219189

M3 - Conference contribution

SP - 351

EP - 368

BT - ACM EC 2018 - Proceedings of the 2018 ACM Conference on Economics and Computation

PB - Association for Computing Machinery, Inc

ER -