Designing incentives for a multi-agent system to induce a desirable Nash equilibrium is both a crucial and challenging problem appearing in many decision-making domains, especially for a large number of agents $N$. Under the exchangeability assumption, we formalize this incentive design (ID) problem as a parameterized mean-field game (PMFG), aiming to reduce complexity via an infinite-population limit. We first show that when dynamics and rewards are Lipschitz, the finite-$N$ ID objective is approximated by the PMFG at rate $\mathscr{O}(\frac{1}{\sqrt{N}})$. Moreover, beyond the Lipschitz-continuous setting, we prove the same $\mathscr{O}(\frac{1}{\sqrt{N}})$ decay for the important special case of sequential auctions, despite discontinuities in dynamics, through a tailored auction-specific analysis. Built on our novel approximation results, we further introduce our Adjoint Mean-Field Incentive Design (AMID) algorithm, which uses explicit differentiation of iterated equilibrium operators to compute gradients efficiently. By uniting approximation bounds with optimization guarantees, AMID delivers a powerful, scalable algorithmic tool for many-agent (large $N$) ID. Across diverse auction settings, the proposed AMID method substantially increases revenue over first-price formats and outperforms existing benchmark methods.

翻译：为多智能体系统设计激励以诱导期望的纳什均衡，是众多决策领域中至关重要且极具挑战性的问题，尤其当智能体数量$N$较大时。在可交换性假设下，我们将该激励设计问题形式化为参数化平均场博弈，旨在通过无限群体极限降低问题复杂度。我们首先证明，当动力学与奖励函数满足Lipschitz连续性时，有限$N$激励设计目标可由PMFG以$\mathscr{O}(\frac{1}{\sqrt{N}})$的速率近似。此外，在非Lipschitz连续的一般情形下，我们针对具有重要应用价值的序列拍卖场景，通过定制化的拍卖专用分析，证明了尽管动力学存在间断性，仍能保持相同的$\mathscr{O}(\frac{1}{\sqrt{N}})$衰减速率。基于这些新颖的近似结果，我们进一步提出了伴随平均场激励设计算法，该算法通过对迭代均衡算子的显式微分实现高效梯度计算。通过将近似界与优化保证相结合，AMID为大规模智能体激励设计提供了强大且可扩展的算法工具。在多种拍卖场景中，所提出的AMID方法相较于首价拍卖形式显著提升了收益，并优于现有基准方法。