摘要翻译：
线性Fisher市场是一种基本的经济模型，在公平分工和大规模互联网市场中都有应用。在有限维的$N$购买者和$M$物品的情况下，可以用Eisenberg-Gale凸规划计算市场均衡。在大规模网络广告和公平分割应用的推动下，本文考虑了一个线性Fisher市场的推广，其中有一个有限的购买者集和一个连续的商品集。我们将Eisenberg-Gale凸规划及其对偶推广到这个无穷维集合，从而得到Banach空间优化问题。证明了最优解的存在性、强对偶性以及KKT型条件的必要性和充分性。所有这些性质都是通过非标准变元建立的，从而规避了对偶理论在无限维Banach空间上最优化的局限性。此外，我们还证明了存在一个纯均衡分配，即项目空间的一个划分。当项目空间为闭区间且购买者具有分段线性估价时，我们证明了无限维分配上的Eisenberg-Gale型凸规划可转化为有限维凸锥规划，并利用基于原-对偶内点法的现有优化软件有效地求解该规划。在凸圆锥曲线重构的基础上，我们开发了第一个多项式时间切饼算法，该算法实现了Pareto最优性、无嫉妒性和比例性。对于一般的买方估值或买方数量非常大的情况，我们提出了用随机对偶平均计算市场均衡，它以高概率找到近似均衡价格。最后，我们讨论了上述结果如何容易地推广到拟线性效用的情形。
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英文标题：
《Infinite-Dimensional Fisher Markets and Tractable Fair Division》
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作者：
Yuan Gao, Christian Kroer
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最新提交年份：
2021
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分类信息：

一级分类：Computer Science        计算机科学
二级分类：Computer Science and Game Theory        计算机科学与博弈论
分类描述：Covers all theoretical and applied aspects at the intersection of computer science and game theory, including work in mechanism design, learning in games (which may overlap with Learning), foundations of agent modeling in games (which may overlap with Multiagent systems), coordination, specification and formal methods for non-cooperative computational environments. The area also deals with applications of game theory to areas such as electronic commerce.
涵盖计算机科学和博弈论交叉的所有理论和应用方面，包括机制设计的工作，游戏中的学习（可能与学习重叠），游戏中的agent建模的基础（可能与多agent系统重叠），非合作计算环境的协调、规范和形式化方法。该领域还涉及博弈论在电子商务等领域的应用。
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一级分类：Economics        经济学
二级分类：Theoretical Economics        理论经济学
分类描述：Includes theoretical contributions to Contract Theory, Decision Theory, Game Theory, General Equilibrium, Growth, Learning and Evolution, Macroeconomics, Market and Mechanism Design, and Social Choice.
包括对契约理论、决策理论、博弈论、一般均衡、增长、学习与进化、宏观经济学、市场与机制设计、社会选择的理论贡献。
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一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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英文摘要：
  Linear Fisher markets are a fundamental economic model with applications in fair division as well as large-scale Internet markets. In the finite-dimensional case of $n$ buyers and $m$ items, a market equilibrium can be computed using the Eisenberg-Gale convex program. Motivated by large-scale Internet advertising and fair division applications, this paper considers a generalization of a linear Fisher market where there is a finite set of buyers and a continuum of items. We introduce generalizations of the Eisenberg-Gale convex program and its dual to this infinite-dimensional setting, which leads to Banach-space optimization problems. We establish existence of optimal solutions, strong duality, as well as necessity and sufficiency of KKT-type conditions. All these properties are established via non-standard arguments, which circumvent the limitations of duality theory in optimization over infinite-dimensional Banach spaces. Furthermore, we show that there exists a pure equilibrium allocation, i.e., a division of the item space. When the item space is a closed interval and buyers have piecewise linear valuations, we show that the Eisenberg-Gale-type convex program over the infinite-dimensional allocations can be reformulated as a finite-dimensional convex conic program, which can be solved efficiently using off-the-shelf optimization software based on primal-dual interior-point methods. Based on our convex conic reformulation, we develop the first polynomial-time cake-cutting algorithm that achieves Pareto optimality, envy-freeness, and proportionality. For general buyer valuations or a very large number of buyers, we propose computing market equilibrium using stochastic dual averaging, which finds approximate equilibrium prices with high probability. Finally, we discuss how the above results easily extend to the case of quasilinear utilities.
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PDF链接：
https://arxiv.org/pdf/2010.03025