Yun Kuen Cheung

Tatonnement beyond gross substitutes?: gradient descent to the rescue

Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we define 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.

Tatonnement in ongoing markets of complementary goods

This paper continues the study, initiated by Cole and Fleischer in [Cole and Fleischer 2008], of the behavior of a tatonnement price update rule in Ongoing Fisher Markets. The prior work showed fast convergence toward an equilibrium when the goods satisfied the weak gross substitutes property and had bounded demand and income elasticities. The current work shows that fast convergence also occurs for the following type of markets: All pairs of goods are complements to each other, and the demand and income elasticities are suitably bounded. In particular, these conditions hold when all buyers in the market are equipped with CES utilities, where all the parameters ρ, one per buyer, satisfy -1 < ρ ≤ 0. In addition, we extend the above result to markets in which a mixture of complements and substitutes occur. This includes characterizing a class of nested CES utilities for which fast convergence holds. An interesting technical contribution, which may be of independent interest, is an amortized analysis for handling asynchronous events in settings in which there are a mix of continuous changes and discrete events.

Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds

Given a graph where vertices are partitioned into

Dynamics of Distributed Updating in Fisher Markets

k

Multidimensional Divide-and-Conquer and Weighted Digital Sums (Extended Abstract)

terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The distortion of a compressed graph is the maximum multiplicative blow-up of distances between all pairs of terminals. We study the trade-off between the number of non-terminals and the distortion. This problem generalizes the Steiner Point Removal (SPR) problem, in which all non-terminals must be removed. We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem. This allows us to show that there exist graphs such that every minor with distortion less than

Amortized Analysis on Asynchronous Gradient Descent

2~/~2.5~/~3

Multidimensional divide-and-conquer and weighted digital sums

must have

Steiner point removal: distant terminals don't (really) bother

\Omega(k^2)~/~\Omega(k^{5/4})~/~\Omega(k^{6/5})

Better strategyproof mechanisms without payments or prior: an analytic approach

non-terminals, plus more trade-offs in between. The black-box reduction has an interesting consequence: if the tight lower bound on distortion for the SPR problem is super-constant, then allowing any

Analyzing a Weighted Digital Sum Variant

O(k)