Evaluating the Performance of Approximation Mechanisms under Budget Constraints
Juan Carlos Carbajal, Ahuva Mualem
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.