[论文解读] Bayesian Games and the Smoothness Framework

Vasilis Syrgkanis|arXiv (Cornell University)|Mar 23, 2012

Economic theories and models参考文献 13被引用 55

一句话总结

本文将光滑性框架从完全信息博弈扩展到具有独立私有类型的贝叶斯博弈，表明若一个博弈在完全信息设定下是光滑的，则其对价均衡（PoA）界可推广至贝叶斯-纳什设定。关键贡献在于提出了一种普遍光滑性条件，该条件表明努力市场博弈的贝叶斯-纳什对价均衡至多为2，且将第一价格物品出价拍卖的对价均衡界从4改进至 e/(e−1) ≈ 1.58。

ABSTRACT

We consider a general class of Bayesian Games where each players utility depends on his type (possibly multidimensional) and on the strategy profile and where players' types are distributed independently. We show that if their full information version for any fixed instance of the type profile is a smooth game then the Price of Anarchy bound implied by the smoothness property, carries over to the Bayes-Nash Price of Anarchy. We show how some proofs from the literature (item bidding auctions, greedy auctions) can be cast as smoothness proofs or be simplified using smoothness. For first price item bidding with fractionally subadditive bidders we actually manage to improve by much the existing result \cite{Hassidim2011a} from 4 to $\frac{e}{e-1}\approx 1.58$. This also shows a very interesting separation between first and second price item bidding since second price item bidding has PoA at least 2 even under complete information. For a larger class of Bayesian Games where the strategy space of a player also changes with his type we are able to show that a slightly stronger definition of smoothness also implies a Bayes-Nash PoA bound. We show how weighted congestion games actually satisfy this stronger definition of smoothness. This allows us to show that the inefficiency bounds of weighted congestion games known in the literature carry over to incomplete versions where the weights of the players are private information. We also show how an incomplete version of a natural class of monotone valid utility games, called effort market games are universally $(1,1)$-smooth. Hence, we show that incomplete versions of effort market games where the abilities of the players and their budgets are private information has Bayes-Nash PoA at most 2.

研究动机与目标

• 将光滑性框架扩展至具有独立私有类型的贝叶斯博弈。
• 建立在完全信息下光滑性可推出贝叶斯-纳什对价均衡界成立的条件。
• 统一完全信息与不完全信息博弈的效率分析。
• 利用扩展框架改进特定拍卖与博弈设定中的现有界。
• 证明完全信息下光滑性证明的紧致性意味着在不完全信息设定下的鲁棒性。

提出的方法

• 提出一种在所有类型配置和策略配置下均成立的普遍光滑性条件。
• 通过在类型分布上分析期望效用，将光滑性证明技术适配至贝叶斯设定。
• 将该框架应用于具有分数次可加性投标者的首价物品出价拍卖，推导出更紧致的PoA界。
• 利用普遍光滑性条件证明加权冲突游戏与努力市场博弈在不完全信息下具有鲁棒性。
• 将有效效用博弈的光滑性证明适配至具有私有预算与能力的贝叶斯设定。
• 运用凹函数性质与边际贡献不等式，以界定光滑性条件中社会福利差异。

实验结果

研究问题

• RQ1 光滑性框架能否扩展至具有独立私有类型的贝叶斯博弈？
• RQ2 一个在完全信息下光滑的博弈是否意味着其具有有界的贝叶斯-纳什对价均衡？
• RQ3 能否强化光滑性条件，以确保在所有类型配置下均成立普遍光滑性？
• RQ4 在不完全信息下，具有次可加性投标者的首价物品出价拍卖的贝叶斯-纳什对价均衡是多少？
• RQ5 具有私有能力与预算的努力市场博弈在贝叶斯设定下是否仍保持有界的对价均衡？

主要发现

• 具有分数次可加性投标者的首价物品出价拍卖的贝叶斯-纳什对价均衡从4改进至 e/(e−1) ≈ 1.58。
• 加权冲突游戏满足更强的普遍光滑性条件，因此其低效性界可推广至具有私有权重的不完全信息版本。
• 具有私有能力和预算的努力市场博弈是普遍(1,1)-光滑的，意味着其贝叶斯-纳什对价均衡至多为2。
• 光滑性框架统一了完全信息与贝叶斯不完全信息下的低效性分析，表明完全信息下的紧致性意味着不完全信息下的鲁棒性。
• 该框架简化并强化了对贪心机制与物品出价拍卖的现有证明。
• 本文证明，私有信息不会恶化光滑博弈的效率保证，因为在不完全信息下PoA保持不变。