A Consumer-Theoretic Characterization of Fisher Market Equilibria

Denizalp, Goktas,; Viqueira, Enrique Areyan; Greenwald, Amy

doi:10.1007/978-3-030-94676-0_19

Cited by 3 publications

(13 citation statements)

References 16 publications

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“…P L 3.1. Recall from Goktas et al [Goktas et al 2022] that for homogeneous utility functions, the Hicksian demand is homogeneous in , i.e., for all ≥ 0, ( , ) = ( , ). Hence, we have:…”

Section: A Ommited Results and Proofsmentioning

confidence: 99%

“…Additionally, looking back at Equation ( 12), since Hicksian demand is homogeneous of degree 1 for homogeneous utility function (see Lemma 5 of Goktas et al [Goktas et al 2022]), by Euler's theorem for homogeneous functions [Border 2017], we have: ℎ ( ,1) ℎ ( , 1) = ℎ ( ,1) ℎ ( ,1) = 1 .…”

Section: A Ommited Results and Proofsmentioning

confidence: 99%

“…We then showed that together with the KL divergence associated with a change in prices, we can use it to bound the percentage change in the expenditure of one unit of utility, assuming bounded price changes. This observation motivated us to consider analyzing the convergence of mirror descent with KL divergence on a recently proposed [Goktas et al 2022] convex potential, making use of the expenditure function to characterize competitive equilibrium prices in homothetic Fisher markets. An important property of this convex potential is that its gradient is equal to the negative excess demand in the market, implying that mirror descent on it is equivalent to tâtonnement, an observation used to prove previous convergence results regarding tâtonnement in Fisher markets [Cheung et al 2013] .…”

Section: Discussionmentioning

confidence: 99%

“…Recently, Goktas et al [2022] proposed a convex program that is equivalent to the Eisenberg-Gale convex program, but whose optimal value differs from that of the Eisenberg-Gale convex program by an additive constant. The authors state their results only for continuous, concave, and homogeneous Fisher markets; however, their proof is valid for all homothetic Fisher markets, including those in which the buyers' utility functions are non-concave: .…”

Section: Homothetic Fisher Marketsmentioning

confidence: 99%

“…Unfortunately, tâtonnement does not converge to equilibrium prices in all homothetic Fisher markets, e.g., linear Fisher markets [Cole and Tao 2019], which suggests the need for additional restrictions on the class of homothetic Fisher markets. Goktas et al [2022] suggest the maximum absolute value of the Marshallian price demand elasticity, i.e., = max , , max ( , ) ∈Δ ×Δ × [ ] , ( , ) , as a possible market parameter to use to establish a convergence rate of ( (1+ ) / ). However, Cole and Fleischer's [2008] results suggest that it is unlikely that Marshallian demand elasticity could be enough, since the proof techniques used in work that makes this assumption require one to quantify the direction of the change in demand as a function of the change in the prices of the other goods, and hence only apply when one assumes WGS or WGC [Cole and Fleischer 2008].…”

Section: Cheung Et Almentioning

confidence: 99%

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Tâtonnement in Homothetic Fisher Markets

Denizalp

Zhao

Greenwald

2023

Proceedings of the 24th ACM Conference on Economics and Computation

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A prevalent theme in the economics and computation literature is to identify natural price-adjustment processes by which sellers and buyers in a market can discover equilibrium prices. An example of such a process is tâtonnement, an auction-like algorithm first proposed in 1874 by French economist Walras in which sellers adjust prices based on the Marshallian demands of buyers, i.e., budget-constrained utility-maximizing demands. A dual concept in consumer theory is a buyer's Hicksian demand, i.e., consumptions that minimize expenditure while achieving a desired utility level. In this paper, we identify the maximum of the absolute value of the elasticity of the Hicksian demand, i.e., the maximum percentage change in the Hicksian demand of any good w.r.t. the change in the price of some other good, as an economic parameter sufficient to capture and explain a range of convergent and non-convergent tâtonnement behaviors in a broad class of markets. In particular, we prove the convergence of tâtonnement at a rate of ( (1+ 2 ) / ), in homothetic Fisher markets with bounded price elasticity of Hicksian demand, i.e., Fisher markets in which consumers have preferences represented by homogeneous utility functions and the price elasticity of their Hicksian demand is bounded, where is the maximum absolute value of the price elasticity of Hicksian demand across all buyers. Our result not only generalizes known convergence results for CES Fisher markets, but extends them to mixed nested CES markets and Fisher markets with continuous, possibly non-concave, homogeneous utility functions. Our convergence rate covers the full spectrum of nested CES utilities, including Leontief and linear utilities, unifying previously existing disparate convergence and non-convergence results. In particular, for = 0, i.e., Leontief markets, we recover the best-known convergence rate of ( 1 / ), and as → ∞, e.g., linear Fisher markets, we obtain non-convergent behavior, as expected.CCS Concepts: • Theory of computation → Exact and approximate computation of market equilibria.

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“…P L 3.1. Recall from Goktas et al [Goktas et al 2022] that for homogeneous utility functions, the Hicksian demand is homogeneous in , i.e., for all ≥ 0, ( , ) = ( , ). Hence, we have:…”

Section: A Ommited Results and Proofsmentioning

confidence: 99%

Section: A Ommited Results and Proofsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Homothetic Fisher Marketsmentioning

confidence: 99%

Section: Cheung Et Almentioning

confidence: 99%

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Tâtonnement in Homothetic Fisher Markets

Denizalp

Zhao

Greenwald

2023

Proceedings of the 24th ACM Conference on Economics and Computation

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Fisher Markets with Social Influence

Zhao

Denizalp

Greenwald

2023

AAAI

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A Fisher market is an economic model of buyer and seller interactions in which each buyer’s utility depends only on the bundle of goods she obtains. Many people’s interests, however, are affected by their social interactions with others. In this paper, we introduce a generalization of Fisher markets, namely influence Fisher markets, which captures the impact of social influence on buyers’ utilities. We show that competitive equilibria in influence Fisher markets correspond to generalized Nash equilibria in an associated pseudo-game, which implies the existence of competitive equilibria in all influence Fisher markets with continuous and concave utility functions. We then construct a monotone pseudo-game, whose variational equilibria and their duals together characterize competitive equilibria in influence Fisher markets with continuous, jointly concave, and homogeneous utility functions. This observation implies that competitive equilibria in these markets can be computed in polynomial time under standard smoothness assumptions on the utility functions. The dual of this second pseudo-game enables us to interpret the competitive equilibria of influence CCH Fisher markets as the solutions to a system of simultaneous Stackelberg games. Finally, we derive a novel first-order method that solves this Stackelberg system in polynomial time, prove that it is equivalent to computing competitive equilibrium prices via tâtonnement, and run experiments that confirm our theoretical results.

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Building Chinese interdisciplinary research centers : the case of Tsinghua University

Li¹,

李蓉暉²

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Graph machine learning has been extensively studied in both academia and industry. However, in the literature, most existing graph machine learning models are designed to conduct training with data samples in a random order, which may suffer from suboptimal performance due to ignoring the importance of different graph data samples and their training orders for the model optimization status. To tackle this critical problem, curriculum graph machine learning (Graph CL), which integrates the strength of graph machine learning and curriculum learning, arises and attracts an increasing amount of attention from the research community. Therefore, in this paper, we comprehensively overview approaches on Graph CL and present a detailed survey of recent advances in this direction. Specifically, we first discuss the key challenges of Graph CL and provide its formal problem definition. Then, we categorize and summarize existing methods into three classes based on three kinds of graph machine learning tasks, i.e., node-level, linklevel, and graph-level tasks. Finally, we share our thoughts on future research directions. To the best of our knowledge, this paper is the first survey for curriculum graph machine learning.

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A Consumer-Theoretic Characterization of Fisher Market Equilibria

Cited by 3 publications

References 16 publications

Tâtonnement in Homothetic Fisher Markets

Tâtonnement in Homothetic Fisher Markets

Fisher Markets with Social Influence

Building Chinese interdisciplinary research centers : the case of Tsinghua University

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