### Abstract

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters for the One-Time Market. Our second protocol, which is analyzed for the One-Time Market only, assumes no global knowledge whatsoever.

Original language | English (US) |
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Title of host publication | STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing |

Pages | 315-324 |

Number of pages | 10 |

State | Published - 2008 |

Event | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada Duration: May 17 2008 → May 20 2008 |

### Other

Other | 40th Annual ACM Symposium on Theory of Computing, STOC 2008 |
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Country | Canada |

City | Victoria, BC |

Period | 5/17/08 → 5/20/08 |

### Fingerprint

### Keywords

- Market equilibria
- Tatonnement

### ASJC Scopus subject areas

- Software

### Cite this

*STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing*(pp. 315-324)

**Fast-converging tatonnement algorithms for one-time and ongoing market problems.** / Cole, Richard; Fleischer, Lisa.

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

*STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing.*pp. 315-324, 40th Annual ACM Symposium on Theory of Computing, STOC 2008, Victoria, BC, Canada, 5/17/08.

}

TY - GEN

T1 - Fast-converging tatonnement algorithms for one-time and ongoing market problems

AU - Cole, Richard

AU - Fleischer, Lisa

PY - 2008

Y1 - 2008

N2 - Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters for the One-Time Market. Our second protocol, which is analyzed for the One-Time Market only, assumes no global knowledge whatsoever.

AB - Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper formalizes the setting of Ongoing Markets, by contrast with the classic market scenario, which we term One-Time Markets. The Ongoing Market allows trade at non-equilibrium prices, and, as its name suggests, continues over time. As such, it appears to be a more plausible model of actual markets. For both market settings, this paper defines and analyzes variants of a simple tatonnement algorithm that differs from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point. Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence. Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking: 1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price. 2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth. 3. The closeness of the market to a Fisher market (a market with buyers starting with money alone). We give two types of protocols. The first type assumes global knowledge of only (an upper bound on) the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters for the One-Time Market. Our second protocol, which is analyzed for the One-Time Market only, assumes no global knowledge whatsoever.

KW - Market equilibria

KW - Tatonnement

UR - http://www.scopus.com/inward/record.url?scp=57049087604&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57049087604&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9781605580470

SP - 315

EP - 324

BT - STOC'08: Proceedings of the 2008 ACM Symposium on Theory of Computing

ER -