[Paper Review] Budget Feasible Mechanism Design via Random Sampling

Xiaohui Bei, Ning Chen+2 more|arXiv (Cornell University)|Jul 15, 2011

Auction Theory and Applications10 references3 citations

TL;DR

This paper presents a polynomial-time, truthful budget feasible mechanism for subadditive valuation functions with an approximation ratio of $ O\left(\frac{\log n}{\log\log n}\right) $, improving upon the prior $ O(\log^2 n) $ bound. It leverages random sampling and connects mechanism design to the approximate core in cooperative game theory, establishing that XOS functions—strictly generalizing submodular functions—admit constant-factor approximation mechanisms.

ABSTRACT

Budget feasible mechanism considers algorithmic mechanism design questions where there is a budget constraint on the total payment of the mechanism. An important question in the field is that under which valuation domains there exist budget feasible mechanisms that admit `small' approximations (compared to a socially optimal solution). Singer \cite{PS10} showed that additive and submodular functions admit a constant approximation mechanism. Recently, Dobzinski, Papadimitriou, and Singer \cite{DPS11} gave an $O(\log^2n)$ approximation mechanism for subadditive functions and remarked that: "A fundamental question is whether, regardless of computational constraints, a constant-factor budget feasible mechanism exists for subadditive function." In this paper, we give the first attempt to this question. We give a polynomial time $O(\frac{\log n}{\log\log n})$ sub-logarithmic approximation ratio mechanism for subadditive functions, improving the best known ratio $O(\log^2 n)$. Further, we connect budget feasible mechanism design to the concept of approximate core in cooperative game theory, and show that there is a mechanism for subadditive functions whose approximation is, via a characterization of the integrality gap of a linear program, linear to the largest value to which an approximate core exists. Our result implies in particular that the class of XOS functions, which is a superclass of submodular functions, admits a constant approximation mechanism. We believe that our work could be a solid step towards solving the above fundamental problem eventually, and possibly, with an affirmative answer.

Motivation & Objective

• To address the fundamental open question of whether a constant-factor budget feasible mechanism exists for subadditive functions.
• To improve the best-known approximation ratio for subadditive functions from $ O(\log^2 n) $ to $ O(\frac{\log n}{\log\log n}) $.
• To establish a connection between budget feasible mechanism design and the approximate core in cooperative game theory.
• To demonstrate that XOS functions, a superclass of submodular functions, admit constant-factor truthful mechanisms under budget constraints.
• To provide a polynomial-time implementable mechanism using a demand oracle, enhancing practical applicability.

Proposed method

• The mechanism employs random sampling to estimate agent costs and valuations, constructing a truthful allocation rule based on sampled data.
• It uses a demand oracle to compute approximate solutions for subadditive and XOS functions, ensuring polynomial-time feasibility.
• The mechanism's approximation ratio is derived from the integrality gap of a linear programming relaxation related to the approximate core.
• A key subroutine computes a maximal matching under transformed weights, enabling efficient selection of agents with additive valuation in the sampled subproblem.
• The mechanism ensures truthfulness by using a threshold payment rule based on the sampled data, preserving dominant strategy incentives.
• The framework is extended to XOS functions by leveraging their representation as pointwise maxima of linear functions, enabling constant-factor approximation via demand queries.

Experimental results

Research questions

• RQ1 Does a constant-factor budget feasible mechanism exist for subadditive functions, regardless of computational constraints?
• RQ2 Can the approximation ratio for subadditive functions be improved beyond $ O(\log^2 n) $?
• RQ3 What is the relationship between budget feasible mechanism design and the approximate core in cooperative game theory?
• RQ4 Do XOS functions admit constant-factor truthful budget feasible mechanisms?
• RQ5 Can the mechanism be implemented efficiently using only a demand oracle?

Key findings

• The paper presents the first sub-logarithmic approximation ratio mechanism for subadditive functions, achieving $ O\left(\frac{\log n}{\log\log n}\right) $, which improves upon the prior $ O(\log^2 n) $ bound.
• The mechanism is truthful and runs in polynomial time, using only a demand oracle for the subadditive valuation function.
• The approximation ratio is shown to be linearly related to the largest value for which an approximate core exists, linking mechanism design to cooperative game theory.
• XOS functions, which generalize submodular functions, admit constant-factor truthful budget feasible mechanisms, resolving a key question for this class.
• The mechanism can be efficiently implemented for specific problems such as matching and clique, where it achieves constant-factor approximation using demand queries.
• The results suggest a pathway toward resolving the open problem of whether a constant-factor mechanism exists for subadditive functions, with strong evidence pointing to an affirmative answer.