Motivated by the convergence result of mirror-descent algorithms to market equilibria in linear Fisher markets, it is natural for one to consider designing dynamics (specifically, iterative algorithms) for agents to arrive at linear Arrow-Debreu market equilibria. Jain (SIAM J. Comput. 37(1), 303–318, 2007) reduced equilibrium computation in linear Arrow-Debreu markets to the equilibrium computation in bijective markets, where everyone is a seller of only one good and a buyer for a bundle of goods. In this paper, we design an algorithm for computing linear bijective market equilibrium, based on solving the rational convex program formulated by Devanur et al. The algorithm repeatedly alternates between a step of gradient-descent-like updates and a distributed step of optimization exploiting the property of such convex program. Convergence can be ensured by a new analysis different from the analysis for linear Fisher market equilibria.

受镜像下降算法对线性 Fisher 市场中市场均衡的收敛结果的启发，人们很自然地考虑为代理设计动力学（特别是迭代算法）以达到线性 Arrow-Debreu 市场均衡。耆那教 (SIAM J. Comput. 37(1), 303–318, 2007) 将线性 Arrow-Debreu 市场中的均衡计算简化为双射市场中的均衡计算，其中每个人都只是一种商品的卖家和一捆商品的买家。在本文中，我们设计了一种计算线性双射市场均衡的算法，基于求解 Devanur 等人提出的有理凸程序。该算法在类似梯度下降的更新步骤和利用这种凸程序特性的分布式优化步骤之间反复交替。可以通过不同于线性费雪市场均衡分析的新分析来确保收敛。