Scalable Neural Incentive Design with Parameterized Mean-Field Approximation

TL;DR

The paper addresses incentive design in large, exchangeable multi-agent systems by introducing parameterized mean-field games (PMFG) to circumvent the curse of dimensionality. It proves an $O(1/\,sqrt{N})$ approximation guarantee between the PMFG-based design and the original $N$-player problem under Lipschitz dynamics, and extends to non-Lipschitz auction settings via a batched auction MF framework (BA-MFG) under a no-zero-dominance condition. The algorithmic core, Adjoint Mean-Field Incentive Design (AMID), differentiates through iterated equilibrium operators to obtain scalable first-order gradients with memory efficiency, enabling optimization in complex scenarios such as congestion pricing and revenue-maximizing auctions. Empirical results demonstrate revenue improvements over first-price mechanisms and evidence that MF-based designs achieve near-optimal performance for large $N$, with favorable scalability in time and space. Overall, the work provides a principled, scalable framework for designing incentives in many-agent contexts, with broad applicability to large-scale auctions and dynamic policy design.

Abstract

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.

Figures (9)

• Figure 1: Payment design with AMID in $\mathcal{M}\xspace_{\text{bb}}$. Left: objective and exploitability throughout training iterations. Middle-right: population flow in time before and after learning payments.
• Figure 2: $g_{\text{rev}}$ throughout iterations of AMID and baseline algorithms in settings (A1-3), left to right.
• Figure 3: Left: deviation in revenue in $\mathcal{M}\xspace_\text{mfa}$ vs $N$-player $\mathcal{G}\xspace_\text{auc}$ at $\theta^*$ as functions of $N$, and middle: exploitability curve of OMD iterations $F_{\text{omd}}^{(T)}(\theta\xspace^*, \cdot)$ at optimized $\theta\xspace^*$ in (A1), (A2), (A3). Right: mean bids of NE computed by $F_{\text{omd}}^{(T)}$ for $h\in [4]$before and after optimization with AMID in $(A1)$.
• Figure 4: Payment function $s\rightarrow\theta_s$ learned after training with AMID, where payments are bounded on $[0, 1/2]$.
• Figure 5: Payment design with AMID in $\mathcal{M}\xspace_{\text{bb}}$ with larger payments. Left:objective and exploitability throughout training iterations. Middle: learned payment rule after training with AMID. Right: population flow in time after learning payments.

Theorems & Definitions (36)

• Definition 1: Parameterized Dynamic Games
• Definition 2: Parameterized Mean-Field Games
• Theorem 1
• Lemma 1: Differentiability of $F_{\text{omd}}^\infty$
• Lemma 2: Adjoint method
• Remark 1
• Definition 3: BA-MFG
• Definition 4: No zero-dominance (NZD)
• Theorem 2: Approximation for BA-MFG
• Remark 2