Many large cities have adopted a policy of restricting the number of new vehicle licenses in order to address the challenges of traffic and air pollution problems. An important question that then arises is how to allocate these limited new license quotas among a large set of demanders while taking both social efficiency and equality into consideration. Inspired by this practical problem, we study the problem of designing simple and optimal mechanisms to tradeoff between efficiency and equality in a similar public goods allocation problem. We first propose a truthful two-group mechanism framework that is both general and simple and then use it to compute the optimal solution that maximizes social efficiency while guaranteeing a certain level of equality. Interestingly and surprisingly, under some natural conditions regarding the players’ private values, we show that the optimal mechanisms within the proposed framework are always the mechanisms that first run an auction and then run a lottery (ATL for short). In addition, beyond the framework and those conditions, we prove that the ATL mechanism can always guarantee at least Formula: see text (Formula: see text is the minimum equality level) of the optimal solution efficiency. This approximation ratio can in practice be further improved to near-optimal when resources are scarce and the natural conditions hold. We further establish a theoretical connection between these natural conditions and common value distributions, demonstrating that these conditions are widely satisfied in practice. Finally, we implement several experiments to verify the ATL mechanism’s optimality and robustness. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the National Natural Science Foundation of China Grants 72401252, 72192830, 72192832, 62472428, 72192804, 72301235, 72394360 and 72394363; the Public Computing Cloud, Renmin University of China, the fund for building world-class universities (disciplines) of Renmin University of China; the Beijing Natural Science Foundation Grant Z220001. Supplemental Material: The online appendices are available at https://doi.org/10.1287/mnsc.2022.02785 .
Chen et al. (2026) studied this question.