### Abstract

Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we de- fine a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. • CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. • The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semi-definite Jacobian. • We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. • A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. - Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an ε fraction of its initial value in O(1/ε) rounds of price updates. - Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the equilibrium is reduced to an ε fraction of its initial value in O(log(1/ε)) rounds of price updates. This shows that tatonnement converges for the entire range of Fisher markets when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range.

Original language | English (US) |
---|---|

Title of host publication | STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing |

Pages | 191-200 |

Number of pages | 10 |

DOIs | |

State | Published - 2013 |

Event | 45th Annual ACM Symposium on Theory of Computing, STOC 2013 - Palo Alto, CA, United States Duration: Jun 1 2013 → Jun 4 2013 |

### Other

Other | 45th Annual ACM Symposium on Theory of Computing, STOC 2013 |
---|---|

Country | United States |

City | Palo Alto, CA |

Period | 6/1/13 → 6/4/13 |

### Keywords

- Equilibria
- Gradient descent
- Market
- Tatonnement

### ASJC Scopus subject areas

- Software

### Cite this

*STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing*(pp. 191-200) https://doi.org/10.1145/2488608.2488633

**Tatonnement beyond gross substitutes? Gradient descent to the rescue.** / Cheung, Yun Kuen; Cole, Richard; Devanur, Nikhil.

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

*STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing.*pp. 191-200, 45th Annual ACM Symposium on Theory of Computing, STOC 2013, Palo Alto, CA, United States, 6/1/13. https://doi.org/10.1145/2488608.2488633

}

TY - GEN

T1 - Tatonnement beyond gross substitutes? Gradient descent to the rescue

AU - Cheung, Yun Kuen

AU - Cole, Richard

AU - Devanur, Nikhil

PY - 2013

Y1 - 2013

N2 - Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we de- fine a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. • CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. • The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semi-definite Jacobian. • We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. • A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. - Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an ε fraction of its initial value in O(1/ε) rounds of price updates. - Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the equilibrium is reduced to an ε fraction of its initial value in O(log(1/ε)) rounds of price updates. This shows that tatonnement converges for the entire range of Fisher markets when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range.

AB - Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we de- fine a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. • CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. • The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semi-definite Jacobian. • We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. • A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. - Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an ε fraction of its initial value in O(1/ε) rounds of price updates. - Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the equilibrium is reduced to an ε fraction of its initial value in O(log(1/ε)) rounds of price updates. This shows that tatonnement converges for the entire range of Fisher markets when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range.

KW - Equilibria

KW - Gradient descent

KW - Market

KW - Tatonnement

UR - http://www.scopus.com/inward/record.url?scp=84879827386&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879827386&partnerID=8YFLogxK

U2 - 10.1145/2488608.2488633

DO - 10.1145/2488608.2488633

M3 - Conference contribution

SN - 9781450320290

SP - 191

EP - 200

BT - STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing

ER -