[论文解读] Budget Feasible Mechanism Design: From Prior-Free to Bayesian

Xiaohui Bei, Ning Chen+2 more|arXiv (Cornell University)|Mar 20, 2012

Auction Theory and Applications参考文献 36被引用 60

一句话总结

本文在无先验和贝叶斯框架下，为次可加和XOS估值函数提出了真实预算可行机制。通过线性规划整数规划间隙分析，实现了对次可加函数的O(log n)近似；在贝叶斯设置下，对XOS和次可加函数实现了常数近似，解决了关于次可加估值下常数因子机制的开放问题。

ABSTRACT

Budget feasible mechanism design studies procurement combinatorial auctions where the sellers have private costs to produce items, and the buyer(auctioneer) aims to maximize a social valuation function on subsets of items, under the budget constraint on the total payment. One of the most important questions in the field is "which valuation domains admit truthful budget feasible mechanisms with `small' approximations (compared to the social optimum)?" Singer showed that additive and submodular functions have such constant approximations. Recently, Dobzinski, Papadimitriou, and Singer gave an O(log^2 n)-approximation mechanism for subadditive functions; they also remarked that: "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive functions." We address this question from two viewpoints: prior-free worst case analysis and Bayesian analysis. For the prior-free framework, we use an LP that describes the fractional cover of the valuation function; it is also connected to the concept of approximate core in cooperative game theory. We provide an O(I)-approximation mechanism for subadditive functions, via the worst case integrality gap I of LP. This implies an O(log n)-approximation for subadditive valuations, O(1)-approximation for XOS valuations, and for valuations with a constant I. XOS valuations are an important class of functions that lie between submodular and subadditive classes. We give another polynomial time O(log n/loglog n) sub-logarithmic approximation mechanism for subadditive valuations. For the Bayesian framework, we provide a constant approximation mechanism for all subadditive functions, using the above prior-free mechanism for XOS valuations as a subroutine. Our mechanism allows correlations in the distribution of private information and is universally truthful.

研究动机与目标

• 解决在无计算约束下，次可加函数是否存在常数因子预算可行机制的开放问题。
• 弥合机制设计中预算约束下无先验最坏情况分析与贝叶斯分析之间的差距。
• 为次可加和XOS估值函数开发普遍真实的机制，实现较小的近似比。
• 通过保持单调性的变换，将结果扩展到非单调次可加函数。
• 探讨XOS与次可加函数在预算约束下可近似性方面的差异。

提出的方法

• 使用一个建模估值函数分数覆盖的线性规划（LP），以整数规划间隙I来界定近似比。
• 设计一种无先验机制，对次可加函数实现O(I)近似，其中I为LP最坏情况整数规划间隙。
• 提出第二种无先验机制，对次可加函数实现O(log n / log log n)近似，优于之前的O(log²n)界。
• 构建一种贝叶斯机制，将无先验的XOS机制作为子程序，实现在已知成本分布下对次可加函数的常数近似。
• 在两种框架下，采用随机抽样和阈值技术，确保机制的真实性和预算可行性。
• 通过定义单调版本v̂(S) = max_{T⊆S} v(T)对非单调次可加函数进行变换，保持近似保证。

实验结果

研究问题

• RQ1 在无先验设置下，次可加函数是否存在常数因子预算可行机制？
• RQ2 在已知成本分布下，能否使用贝叶斯机制对次可加函数实现常数因子近似？
• RQ3 LP整数规划间隙在决定次可加函数近似比方面起什么作用？
• RQ4 在预算约束下，XOS与次可加函数在可近似性方面有何差异？
• RQ5 能否将为单调函数设计的机制扩展到非单调次可加函数？

主要发现

• 本文提出一种无先验机制，对次可加函数实现O(log n)近似比，优于先前的O(log²n)界。
• 第二种无先验机制对次可加函数实现次对数级O(log n / log log n)近似，进一步提升了当前技术水平。
• 对于XOS函数，本文在无先验设置下实现了常数因子近似，以LP整数规划间隙作为性能界。
• 在贝叶斯框架下，本文构建了一种普遍真实的机制，对所有次可加函数实现常数近似，解决了Dobzinski等人提出的一个开放问题。
• 通过v̂(S) = max_{T⊆S} v(T)的变换，机制对非单调次可加函数依然有效，该变换保持了近似保证。
• 研究结果揭示了XOS与次可加函数在可近似性方面存在根本性差异，其关联于估值在均值附近的指数集中。