Technology Category
Game Theory and Economic Paradigms: Mechanism DesignMultiagent Systems: Mechanism DesignReasoning under Uncertainty: Stochastic Optimization
Evaluating the Performance of Approximation Mechanisms under Budget Constraints
🤖 AI Summary
This study addresses revenue-maximizing mechanism design in a single-item monopoly setting where buyers possess both private valuations and private budgets. Leveraging tools from mechanism design theory, probabilistic distribution analysis, and menu complexity measures—alongside metrics such as the Gap from Optimal Revenue (GFOR), Maximal Value Ratio (MVR), and the revenue non-monotonicity gap—the work systematically evaluates the robustness of approximately optimal mechanisms under budget constraints. The main contributions show that, under bounded-support value distributions, simple mechanisms with polylogarithmic menu size can arbitrarily approximate the optimal revenue. However, for unbounded or unit-square concentrated distributions, no finite or sublinear-menu mechanism can guarantee a constant fraction of the optimal revenue, revealing a fundamental limitation of simple mechanisms in achieving robust approximation guarantees.
Technology Category
Application Category
📝 Abstract
We study revenue maximization in a buyer-seller setting where the seller has a single object and the buyer has both a private valuation and a private budget. The presence of private budgets complicates the classic single-product monopoly problem, making optimal mechanisms difficult to analyze. To overcome this, we evaluate the robust performance of approximation mechanisms relative to optimal mechanisms. We work with three measures of performance: the guaranteed fraction of optimal revenue (GFOR) for restricted classes of mechanisms, the maximal value of relaxation (MVR) for relaxed classes, and a revenue non-monotonicity gap for either relaxed or restricted classes. Our analysis reveals sharp contrasts. On the positive side, we show that for distributions with bounded support, simple mechanisms with poly-logarithmic menu size can approximate optimal revenue arbitrarily well, regardless of correlation between valuations and budgets. On the negative side, we establish strong impossibility results: for distributions with unbounded support, or even bounded distributions concentrated in the unit square, no simple mechanism - or indeed any mechanism with a finite or sublinear menu - can guarantee a positive fraction of the optimal revenue. We also demonstrate unbounded revenue gains from certain relaxations when valuations and budgets are negatively correlated, and highlight cases of revenue non-monotonicity. Taken together, our results underscore the fragility of approximation approaches in the presence of private budgets: except for a narrow set of conditions, approximation mechanisms incur large revenue losses, pointing to fundamental limits of simplicity and robustness in mechanism design. Our analysis highlights that approximation results are highly sensitive to details of the design environment.
Problem
Research questions and friction points this paper is trying to address.
budget constraints
revenue maximization
approximation mechanisms
private budgets
mechanism design
Innovation
Methods, ideas, or system contributions that make the work stand out.
approximation mechanisms
private budgets
revenue maximization
mechanism design
menu complexity
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2024-07-16arXiv.orgCitations: 5
