[论文解读] When does the Price of Anarchy tend to 1 in Large Walrasian Auctions and Fisher Markets

Richard Cole, Yixin Tao|arXiv (Cornell University)|Aug 28, 2015

Auction Theory and Applications被引用 4

一句话总结

本文表明，在满足温和条件（如替代品需求和商品供应不确定性）的情况下，大型瓦尔拉斯拍卖与费舍尔市场中的纳什均衡价格（PoA）趋于1。随着市场规模扩大（即参与者与商品数量增加），由策略性行为导致的效率损失逐渐减小，极限情况下趋近于社会效率。

ABSTRACT

As is well known, many classes of markets have efficient equilibria, but this depends on agents being non-strategic, i.e. that they declare their true demands when offered goods at particular prices, or in other words, that they are price-takers. An important question is how much the equilibria degrade in the face of strategic behavior, i.e. what is the Price of Anarchy (PoA) of the market viewed as a mechanism? Often, PoA bounds are modest constants such as 4/3 or 2. Nonetheless, in practice a guarantee that no more than 25% or 50% of the economic value is lost may be unappealing. This paper asks whether significantly better bounds are possible under plausible assumptions. In particular, we look at how these worst case guarantees improve in the following large settings. Large Walrasian auctions: These are auctions with many copies of each item and many agents. We show that the PoA tends to 1 as the market size increases, under suitable conditions, mainly that there is some uncertainty about the numbers of copies of each good and demands obey the gross substitutes condition. We also note that some such assumption is unavoidable. Large Fisher markets: Fisher markets are a class of economies that has received considerable attention in the computer science literature. A large market is one in which at equilibrium, each buyer makes only a small fraction of the total purchases; the smaller the fraction, the larger the market. Here the main condition is that demands are based on homogeneous monotone utility functions that satisfy the gross substitutes condition. Again, the PoA tends to 1 as the market size increases. Furthermore, in each setting, we quantify the tradeoff between market size and the PoA.

研究动机与目标

• 研究在市场较大时，尤其是当参与者采取策略性行为而非如实申报时，纳什均衡价格（PoA）是否改善。
• 确定在何种条件下，随着市场规模扩大，PoA趋近于1，表明即使存在策略性行为，效率也接近最优。
• 量化瓦尔拉斯拍卖与费舍尔市场中市场规模与PoA之间的权衡关系。

提出的方法

• 分析具有大量商品副本和大量参与者的大型瓦尔拉斯拍卖，假设商品供应数量存在不确定性。
• 对需求函数应用替代品条件，以确保均衡的稳定性和收敛性。
• 将分析扩展至费舍尔市场，其中随着市场规模扩大，每位买家的总购买份额趋于可忽略。
• 使用渐近分析证明，在给定条件下PoA收敛于1。
• 推导PoA作为市场规模和参与者在总需求中占比的函数的定量边界。
• 证明替代品条件是必要条件——若无此条件，PoA不必然趋于1。

实验结果

研究问题

• RQ1 在具有策略性参与者的大型瓦尔拉斯拍卖中，纳什均衡价格在何种条件下趋于1？
• RQ2 随着市场规模扩大且单个买家份额缩小，费舍尔市场中的PoA行为如何变化？
• RQ3 替代品条件在确保大型市场中PoA收敛于1的过程中起什么作用？
• RQ4 是否可以量化瓦尔拉斯拍卖与费舍尔市场中市场规模与PoA之间的权衡关系？
• RQ5 为使PoA趋近于1，是否必须存在供应或需求的某种不确定性？还是其可确定性成立？

主要发现

• 在具有供应不确定性与替代品需求的大型瓦尔拉斯拍卖中，随着市场规模扩大，纳什均衡价格趋于1。
• 瓦尔拉斯拍卖中PoA收敛于1的条件依赖于替代品条件与供应不确定性；若缺少这些条件，该界不成立。
• 在具有同质单调效用函数且满足替代品条件的大型费舍尔市场中，随着市场扩大，PoA同样趋于1。
• PoA收敛于1的速度可量化，且取决于任一买家所占总购买份额的最大值。
• 替代品条件不仅充分，而且在这些设定中是必要条件，以使PoA趋近于1。
• 结果表明，大型市场本身可内在地缓解策略性行为带来的效率损失，即使缺乏强协调或激励机制。