We thank the reviewer for recognizing the strengths of the work and the insightful questions. We summarize our responses below.

W1:

We respond to the concerns of the reviewer for Sections 2-3 jointly.

Theorems 1 and 2: Theorem 1 uses the proof techniques developed for Lipschitz continuous MFGs (as acknowledged in Appendix D.2) with modifications to handle entropy regularization and the incentive function $g$. The theorem justifies our PMFG approximation for $N$-player ID in the first place. However, our main contribution to MFG approximation comes from Theorem 2, which incorporates non-Lipschitz (in fact, not continuous) dynamics and requires special use of the properties of auctions. To the best of our knowledge, no existing approximation result applies to this case (see our discussion on related works between lines 83-93). Critically, both theorems are required to justify our overall framework and demonstrate the well-posedness of incentive design via MFGs.

Lemma 1: We disagree that the lemma is not relevant for the algorithm, as it provides two important conclusions. First, it justifies the use of our explicit differentiation scheme by showing that the scheme approximates $\partial_\theta F_{omd}^{\infty}$ under reasonable assumptions, which is not clear a priori. Second, it provides intuition on how to tune the parameters $T,\eta,\tau$ and their impact on the quality of explicit differentiation. Lemma 1 also provides two conditions that can be evaluated empirically: while $\partial_\theta F_{omd}^{\infty}$ is difficult to evaluate, one can easily verify the requirements 1 and 2 of the lemma.

Significance of AMID: The main significance of AMID in our context is that it reduces an expensive (in fact, intractable) computation into an equivalent but tractable one. The key innovations associated with it are presented in Theorems 1 and 3 (showing that PMFGs form a low-bias approximation to ID) and Lemmas 1 and 2 (showing that PMFGs are also efficiently solvable). Without the PMFG model, many-player incentive design will be intractable.

In principle, apart from the PMFG approximation that is specific to the incentive design problem, the technique could be applied to other settings where there are repeated evaluations of an operator. However, PMFG incentive design can be considered unique since the operator $F_{omd}^\tau$ itself is quite complicated and involves forward-backward iterations on the MFG. For instance, the explicit differentiation scheme of [2] does not require our adjoint optimization, since the iterated operator is much simpler. Our technique permits explicit differentiation of Nash of MFGs, to the best of our knowledge, for the first time. We have seen tangential ideas employed in Neural ODEs [1] with possible applications to recurrent neural networks, although the works do not consider incentive design for MFGs and diverge from our goals and unique challenges of ID.

We also refer to our explicit measurements on various sizes of auctions (see the Table below) in our response to reviewer DFyx, copied for reference. The measurements were conducted in the setting (A2) and demonstrate the effectiveness of AMID even for small problem sizes: in all cases, AMID requires less memory and computational time than a naive gradient backpropagation. Here, ❌ indicates that the naive backprop method was not able to run on a single GPU, and the time ranges indicate one standard deviation. The table (which will be incorporated in the form of a plot in the paper) demonstrates that for explicit differentiation of PMFGs, the AMID approximation is crucial.

W2 and relevant questions:

We thank the reviewer for the questions. We address these below, and will incorporate additional explanations in the final version using our extra page.

Q1

There is no standard formulation to the best of our knowledge when there are multiple equilibria. Most works implicitly or explicitly assume uniqueness [2,3] and incorporate regularization when it does not hold [3]. Some works sidestep the issue by adopting a Nash oracle [4] or restriction to the local implicit map [5]. Our approach is similar to [5], where we differentiate with respect to the implicit map induced by $F_{omd}^\tau$, under the conditions of Lemma 1.

Q2

These were already alluded to in Remark 2, which will be expanded and clarified. In the structured setting of auctions, for many relevant cases, the Lipschitz constant $L$ is upper bounded by 1, therefore, the dependence on the time horizon $H$ scales with $O(H^2)$ with no explicit dependence on $L$ arising. For instance, if the valuations of agents are fixed across rounds (or evolve with a Markov kernel), our results imply $L\leq 1$, see Appendix E.

Our framework is general enough to incorporate complex models where valuations of the bidders can evolve depending on the valuations of others. In this case, the Lipschitz constant $L$ of the valuation dynamics $\omega$ in $\mu$ will incur an $O(\frac{L^H}{\sqrt{N}})$ approximation error, which is the best possible upper bound matching the lower bound of (Yardim et al., 2024).

Q3

We agree with the reviewer that $\mathcal{G}$ is precursive and $\mathcal{M}$ is a useful algorithmic construct. The reason the theorem is stated in the mentioned way is the fact that it holds true for all $\mathcal{G}$ with $N$ players with asymptotic rates on $N$, and the mathematical statement of the inverse problem could obfuscate the main message (cf. [4]). However, for the theorem to be useful, an appropriate MFG formulation $\mathcal{M}$ must be provided, as the reviewer has noticed.

This becomes clearer in Section 3: Definition 3 (of BA-MFGs) provides the heavy-lifting of defining the appropriate MFG, consequently, Theorem 2 provides the approximation guarantees for all relevant $N$-player auctions. More generally, our work is easily compatible with “inverse” problem designs in MFG (e.g. [6,7]), however, we leave the formal development of this as future work.

Q4

For the purpose of the lemma, it is sufficient that the “composite” map $(\theta, \zeta) \rightarrow q^\tau_h(\cdot, \cdot | \Lambda(\operatorname{softmax}(\zeta)), \operatorname{softmax}(\zeta), \theta)$ must be $C^1$ in the neighborhood of $(\theta, \zeta^*)$. This holds true in the specific case $\Lambda$ is $C^1$ and $q_h^\tau$ is $C^1$ in $L, \pi, \theta$, but for the composition to be $C^1$, this is not a necessary condition, hence the current statement of the lemma.

We have noticed that the statement of the lemma (in particular, $q_h^\tau$ being $C^1$ in the neighborhood) could be improved for legibility, which will be done in the final submission.

Q5

For the gradient computation at some $\theta$, $\theta’ \rightarrow \pi^{\theta’}$ must be a map that such that $\pi^{\theta’}$ is the Nash corresponding to design $\theta’$ whenever $\theta’ \in U$ for some open set containing the point of differentiation. The current statement is to ensure clear distinction between computing gradients (which is a local operation) and computing Nash for all $\theta$: Computing a Nash equilibrium for all $\theta$ might be intractable, but a map that locally computes Nash is sufficient for first-order information.

W3:

To assist reading our paper and help readers who are used to other conventions, we have provided a full notation table in Appendix A. This table was moved to the appendix due to space limitations. For the notation on $\rho$ and $\mathbb{e}$ and potentially others, we refer the reviewer to the appendix.

We thank the reviewer for pointing out the typos on lines 146 and the usage of $\bar{F}_{omd}$. These will be fixed in the final submission.

References:

[1] Chen, et al. Neural ordinary differential equations. NeurIPS 2018.

[2] Li et al. End-to-end learning and intervention in games. NeurIPS 2020.

[3] Liu et al. Inducing Equilibria via Incentives: Simultaneous Design-and-Play Ensures Global Convergence. NeurIPS 2022.

[4] Wang, el al. Coordinating followers to reach better equilibria: End-to-end gradient descent for Stackelberg games. AAAI 2022.

[5] Fiez, el al. Implicit Learning Dynamics in Stackelberg Games: Equilibria Characterization, Convergence Analysis, and Empirical Study. ICML 2020.

[6] Yardim et al., Exploiting Approximate Symmetry for Efficient Multi-Agent Reinforcement Learning. L4DC 2025.

[7] Chen at al., Individual-Level Inverse Reinforcement Learning for Mean Field Games. AAMAS 2022

We thank the reviewer for recognizing the strengths of the work and the insightful questions. We summarize our responses below.

W1, Q1, and the discussion of our limitations:

In the main paper, the assumptions (in particular, NZD) as well as their consequences have been discussed (see for instance lines 253-255, lines 245-246, 264-266, and Remark 2). Here, we provide further commentary that will be incorporated into the paper.

Lipschitz continuity is natural in many settings, as it implies that small changes in population behaviour does not cause major changes in game dynamics. It is is a standard and minimal assumption in MFG literature, as it ensures the existence and stability of solutions. In fact, without Lipschitz continuity, even basic properties such as approximation of finite-player Nash equilibria with explicit bounds might not hold (see Yardim et al., 2024). Theorem 1 could be considered therefore as stated under the most relaxed conditions.

That said, a key contribution of our work is showing that such assumptions can be relaxed in structured settings. Specifically, we performed a dedicated analysis for the auction setting, which is inherently non-Lipschitz due to discontinuities in the reward and allocation functions. Despite this, we were able to prove the same approximation results without relying on Lipschitz continuity (Theorem 2). This demonstrates that our framework can handle important real-world cases where standard assumptions do not hold.

The NZD (No zero-dominance) property is realistic in auction settings. When optimizing with entropy regularization, NZD is guaranteed: every entropy-regularized MFG-NE $\pi^*$ assigns non-zero probability to all actions (see lines 253-255).

Even without entropy regularization, the NZD property can naturally arise in auction MFGs due to the structure of the reward and transition dynamics. When the item allocation function depends only on time, all losing actions yield the same immediate reward, zero, and the next state depends solely on the bidder’s valuation, not on the specific action taken. As a result, all losing actions are indistinguishable in value: for all states $s$ and all losing actions $a, a', q_h(s, a | L, \pi, \theta) = q_h(s, a' | L, \pi, \theta)$, and any MFG-NE can be perturbed into an equivalent equilibrium that assigns non-zero probability to the highest losing action, satisfying NZD.

This is also reflected in the behavior of Online Mirror Descent (OMD) when used to approximate MFG-NEs. Since all losing actions have equal expected rewards, OMD assigns equal probability among them, including the highest losing action. Thus, if OMD converges to a MFG-NE, the resulting policy satisfies NZD. The same reasoning extends to many auction settings where allocation depends on the bid distribution. For instance, in reserve-price auctions where goods are allocated proportionally to the number of agents bidding above a threshold, the learned MFG-NE still satisfies NZD due to the equal treatment of losing actions in the optimization dynamics.

Moreover, in our experimental setting, we observed that the NZD property consistently holds across the learned policies. In particular, the policies obtained by AMID and the other baselines assign non-zero probability to the highest losing actions, consistent with the theoretical argument above. To support this, we will include plots in the camera-ready version that illustrate the action distributions and confirm that NZD is satisfied in all experiments.

We thank the reviewer for recognizing the strengths of the work and the insightful questions. We summarize our responses below, by numbering the weaknesses suggested by the reviewer W1 to W5.

(W1): To support readers unfamiliar with our notation, we included a full table of symbols in Appendix A. Due to space constraints, this was placed in the appendix, but we hope it to assist the reading of our paper. An explicit reference to Appendix A (which was missing in our submission) will be added in the camera-ready, before the start of Section 2.

In particular, the quantity $L$ is the idealised, infinite player distribution of the population over states and actions in time, at the infinite-player limit. Mean-field exploitability can be considered an infinite-player extension of exploitability in classical game theory, which quantifies how much each player is incentivised to deviate from a strategy profile (which must be 0 for exact Nash). We will incorporate the corresponding explanations in our final version to assist the understanding of our work.

(W2): Theorem 1 builds on previous results to prove the convergence result for our setting: it relies on two extensions to handle exploitability analysis in the entropy-regularized objective function and the approximation in terms of the design objective $G$. A detailed proof is provided in Appendix E. While Theorem 1 is an extension of existing analysis, our key contribution is Theorem 2, which relies on an entirely novel analysis going beyond the Lipschitz continuous case.

(W3): The NZD (No zero-dominance) property is realistic in auction settings. When optimizing with entropy regularization, NZD is guaranteed: every entropy-regularized MFG-NE $\pi^*$ assigns non-zero probability to all actions (see lines 253-255). Even without entropy regularization, the NZD property can naturally arise in auction MFGs due to the structure of the reward and transition dynamics. When the item allocation function depends only on time, all losing actions yield the same immediate reward, zero, and the next state depends solely on the bidder’s valuation, not on the specific action taken. As a result, all losing actions are indistinguishable in value: for all states $s$ and all losing actions $a, a'$, it holds that $q_h(s, a | L, \pi, \theta) = q_h(s, a' | L, \pi, \theta)$, and any MFG-NE can be perturbed into an equivalent equilibrium that assigns non-zero probability to the highest losing action, satisfying NZD.

This is also reflected in the behavior of Online Mirror Descent (OMD) when used to approximate MFG-NEs. Since all losing actions have equal expected rewards, OMD assigns equal probability among them, including the highest losing action. Thus, if OMD converges to an MFG-NE, the resulting policy satisfies NZD. The same reasoning extends to many auction settings where allocation depends on the bid distribution. For instance, in reserve-price auctions where goods are allocated proportionally to the number of agents bidding above a threshold, the learned MFG-NE still satisfies NZD due to the equal treatment of losing actions in the optimization dynamics.

Moreover, in our experimental setting, we observed that the NZD property consistently holds across the learned policies. In particular, the policies obtained by AMID and the other baselines assign non-zero probability to the highest losing actions, consistent with the theoretical argument above. To support this, we will include plots in the camera-ready version that illustrate the action distributions and confirm that NZD is satisfied in all experiments.

(W4): We would like to emphasize that in many auctions, a discrete action space is natural, either due to the discrete nature of currency or due to technical limitations, particularly in online auctions. As such, discrete auctions have been a topic of study in literature (e.g. [1,2]), and in fact, can introduce unique theoretical difficulty compared to the continuous setting. In order to encapsulate realistic discrete bid auctions, in our experiments, we take a relatively fine discretization ($|\mathcal{S}| = 100$).

Separately, it is possible to extend our work into the continuous action setting by replacing the discrete state-action MFG formalism standard in e.g. [3], which we leave for future work.

(W5): To illustrate the effectiveness of AMID, we provide the following measurements on single iteration time and memory usage of naive backpropagation and AMID using the JAX implementation (on the setting (A2)). The results indicate clearly that naive backpropagation has intractable growth in memory and time as a function of $H$, and in fact becomes prohibitively expensive on a single GPU for $H$ as small as $H=25$. Here, ❌ indicates that we were not able to run the naive algorithm.

We agree that these figures will contribute to the paper significantly, and plan to incorporate these measurements in our final submission, in Section 4. Currently, the computational benefits of AMID are highlighted in the main paper, specifically in Contribution 3 (line 65) and in Remark 1 (line 189). In fact, the experimental results in the paper are already in the regime where the improvement due to AMID is significant.

References:

[1] David et al., 2011. Optimal design of english auctions with discrete bid levels, ACM Transactions on Internet Technology.

[2] Chwe, 1989. "The discrete bid first auction." Economics Letters.

[3] Saldi et al., 2018. Markov--Nash equilibria in mean-field games with discounted cost. SIAM Journal on Control and Optimization.

We thank the reviewer for recognizing the strengths of the work and the insightful questions. We respond to the weaknesses and questions in order.

Weakness 1 and Question:

The PMFG framework makes it possible to solve the incentive design (ID) problem independently of the number of agents. Without the MFG approximation, the problem quickly becomes intractable even for moderate $N$, due to the high computational complexity of computing and differentiating through Nash equilibria. The PMFG construction helps handle this complexity while still providing theoretical guarantees. For the bias-efficiency trade-off due to the PMFG approximation and corresponding AMID algorithm, we refer the reviewer to Figure 2 and Figure 3 (left plot) in our submission.

All the methods compared in the experimental section are built on the PMFG framework to address the ID problem. The per-iteration cost of AMID is similar to that of the other methods, as AMID requires one forward and one backward pass, both of which have similar computational cost. In contrast, zero-order methods require at least two forward passes to approximate the gradient. As a result, while the per-iteration cost is comparable, AMID converges faster because it uses the exact gradient rather than approximating it.

That said, we agree that a numerical comparison of optimization efficiency would be helpful to have in the main paper. In the camera-ready version, we will update the appendix to include actual optimization runtimes for each algorithm, giving a clearer picture of their practical performance.

To illustrate the effectiveness of AMID, we provide the following measurements on single iteration time and memory usage of naive backpropagation and AMID using the JAX implementation (on the setting (A2)). The results indicate clearly that naive backpropagation has intractable growth in memory and time as a function of $H$, and in fact becomes prohibitively expensive on a single GPU for moderate $H$. Here, ❌ indicates that we were not able to run the naive algorithm.

Weakness 2:

We thank the reviewer for pointing this out. While we are not sure which specific spacing inconsistency was referred to between lines 295–299, we will carefully review this section and ensure that any formatting issues, including spacing and alignment, are corrected in the camera-ready version.