__ Summary and Contributions__: This paper proposes to improve multi-agent imitation learning with two enhancements: 1) learn a distribution over phi, the parameter of the discriminator used to estimate the reward function, 2) improve on the multi-type mean field approximation for multi-agent RL by allowing a different number of agents per type. The results show that this leads to faster learning in Rover Tower and transportation network tasks.

__ Strengths__: The empirical evaluation of this paper is strong, in that it benchmarks against 3 relevant and recent baselines for multi-agent imitation learning, and clearly exceeds them.
The paper provides theoretical grounding in the mean field approximation and its convergence.

__ Weaknesses__: A weakness of the paper is the novelty of this approach above prior work; Section 4 states that [3] also used a multi-agent multi-type mean field approximation, only it did not allow types to have a different number of agents. Is this the only difference from this prior work? This should be clarified.

__ Correctness__: I did not notice any major errors.
Figures 3 & 4 should include a description of how the error bars were calculated.

__ Clarity__: The paper is reasonably clear, but there are some typos and errors. For example, the sentence describing Rover Tower in line 250, "The goal is to let the towers to navigate the rovers to arrive at their destinations" is extremely unclear, and interferes with understanding the experiment. It would also help to provide a citation or reference to the Berlin transportation environment.
In addition, I felt that the clarity of the paper could be improved by devoting more explanation to the new mean field formulation, why/how it is different from prior work, and why some of the assumptions are justified. For example, why is it reasonable to assume that it is only necessary to model interactions within one type of agent (lines 151-152)? To save space for this discussion, well-known equations (e.g. 8 and 12) could be removed.

__ Relation to Prior Work__: As mentioned above, the distinction between this paper and [3] should be made more clear.
Lines 232-233 state that MA-DAAC is not scalable because it uses attention to consider every agent's actions. Why is this not scalable? Attention can summarize a variable number of agents' actions in a meaningful way, and attention is often promoted as a mechanism for dealing with increased numbers of agents.

__ Reproducibility__: Yes

__ Additional Feedback__: - The contributions section (lines 41-46) appears to re-state the same contribution several times in different ways. Perhaps if each contribution were made more concise, the differences would be more clear.
- Line 66: "Nash equilibrium described" -> describes
- Line 189: "we introduces" -> we introduce
** I have read the authors' rebuttal and appreciated the clarifications provided, so I am not changing my review score. I hope the authors will work these clarifications into the final version of the paper.

__ Summary and Contributions__: The paper propose to improve the sample efficiency and scalability of MA-GAIL algorithm by introducing a Bayesian formulation as well as a new multi-type mean field approximation. Some theoretical analysis on the proposed multi-type mean field approximation was provided and experimental results on some simulated environments are provided to justify the above claims.

__ Strengths__: The paper contains rich discussions on preliminaries and detailed derivations of the proposed algorithm.
The proposed multi-type mean-filed approximation seems to have the potential of improving the scalability of existing multi-agent reinforcement learning algorithms. The authors also made efforts to analyze the approximation theoretically and show that the approximation error will not lead to large deviations from optimal solutions under certain conditions.

__ Weaknesses__: - I think this paper is a natural extension/combination of previous works [1] and [2], especially the bayesian formulation and the resulting algorithm. And the paper does not give enough insights/analysis of why such a bayesian framework provides special advantages in the context of multi-agent learning. For example, the authors may try to create some toy examples/experiments to demonstrate why the bayesian formulation is necessary and why the previous methods failed. Therefore, I'm not sure if the paper provides new insights/substantial technical contributions.
- Following the same line, I think the paper should be better motivated in the next version. Before delving into the derivations (how to do bayesian MAIL), I don't quite understand why we should do bayesian MAIL.
- It seems the theoretical analysis mainly discuss what difference will the approximation make when we do Nash Q learning. Although this is useful, this seems irrelevant to the main topic, i.e. multi-agent imitation learning (hence a gap between theory and practice). So the paper should also discuss more on what difference will it make in multi-agent imitation learning. Otherwise, the main claim should be the mean-field approximation.
- I think a detailed experiment setup (environments illustration, tasks description, evaluation metrics, etc) is missing at the beginning of the experiments section. So I feel this part is a bit hard to follow.
[1] A bayesian approach to generative adversarial imitation learning.
[2] Multi-agent generative adversarial imitation learning.

__ Correctness__: Yes.

__ Clarity__: Yes. Most parts are easy to follow and the paper has a clear structure. See the weakness section for more comments.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: Post-rebuttal feedback:
I have read the author response and I think the authors have partially addressed my concerns, with a bit handwavy arguments. I hope the authors will address them in the next version in a more clear way. Overall, I choose to keep my original rating.

__ Summary and Contributions__: This paper introduces an approach for multiagent imitation learning (MAIL). The authors propose a new algorithm titled Bayesian Multi-type Mean Field Multi-agent Imitation Learning (BM3EIL), combining imitation learning with the multi-type mean field approximation approach of Subramanian et al., (2020). The authors introduce theoretical results establishing convergence guarantees of their approach in certain classes of games. Empirical evaluations are conducted in two domains (the first comparing Bayesian and non-Bayesian counterparts of the existing MA-DAAC algorithm, and the second comparing the proposed BM3EIL approach against several baselines).
Please find my detailed comments below. For convenience, I have numbered the main points that I would appreciate the authors’ feedback on.

__ Strengths__: In settings where access to an expert policy is available, imitation learning bears potential to have a significant impact on policy learning in multiagent settings.
The approach, to my knowledge, is the first combining mean field theory, imitation learning, and attentional mechanisms into a single cohesive algorithm. The empirical results, albeit limited in scope, are promising in the sense that the proposed algorithm both improves final performance and reduces sample complexity of training.

__ Weaknesses__: 1. The evaluation of the proposed approach (BM3IL) against existing baselines is only conducted on a single domain (the “transportation” environment). While the proposed approach exhibits higher learning rate and improved final performance compared to the baselines, the lack of a more thorough comparison makes it very difficult to judge the significance of the work. I am open to increasing my review score if the authors are able to add comparative results across additional new domains (namely, those which are characteristically different from the transportation environment already considered).
2. Moreover, a closely-related work that is not cited nor compared against is “Coordinated multi-agent imitation learning” (Le et al., 2017), and the closely-related “Data-Driven Ghosting using Deep Imitation Learning” (Le et al., 2017). I recommend the authors compare against this approach to better justify the significance of their work.
3. The interdependence of the agent ‘types’ and the resulting policy-level interactions is quite opaque here, and would benefit from significantly broader analysis. A few related remarks:
a) Presumably, increasing the number of types has a negative impact on learning rate (this seems somewhat evident in Figure 4). Does it have a statistically significant impact on the final performance? How should practitioners choose the appropriate number of types?
b) Do agents tend to specialize in one type? I.e., in equation 11, is $\alpha_m$ generally low or high entropy?
c) How does this type-attention evolve throughout training (and likewise, throughout the state-action space for a fixed policy?)

__ Correctness__: The methodology seems generally sound. The empirical claims are difficult to judge, given the B3MEIL approach is only evaluated on a single domain (see ‘weaknesses’ section).

__ Clarity__: The paper is generally well-written, though suffers from a lack of clarity in some important sections:
4. [Equation 1] ] I believe the inner log in the right hand term of Equation (1) should not be present. I assumed it was a typo, but it is present throughout the text, even for the authors’ proposed approach (e.g., in Equation 3). If intentional, why is this necessary?
5. [Section 3.1.1] The paper introduces the problem scenario as a Markov game in Section 2.1; however, it introduces the notion of binary observations (which are a function of rewards here) in Section 3.1.1 (necessary for training a discriminator). This seems to suggest that perhaps the problem formulation should be corrected to a Partially Observable Markov game (POSG).
6. [Figure 4] While the distinction between the first and second subfigures were clear for B3MEIL, it was less so for the baseline algorithms. Are ‘types’ enforced for the other algorithms via simply partitioning the agents (presumably without any hierarchical attentional mechanism overlying those baseline policies?). If so, any conjectures as to why performance for the baselines drops when the number of types increases?
Minor/typos:
[l66] “Nash equilibrium [5] [describes] the situation that...”
[l73] With respect to Nash-Q learning, the authors state that “The Q function will eventually converge to a Nash equilibrium”. However, this only holds for a restricted class of games, which should be clarified as such (e.g., see “Nash Q-Learning for General-Sum Stochastic Games”, Hu & Wellman, 2003 for details).
[Section 2.2] Function D is undefined. (This is the GAIL discriminator, but should be clearly defined for unfamiliar readers).
[l93] “Let the policy [be] parameterized by…”
[Equation 11] Please add \qquads between the 3 expressions in this equation, as they are difficult to parse.
[l199] “The mean field within each type can be computed through averaging the actions of each [agent]“
[l267] “”The experimental results showed that BM3IL outperforms 268 all other algorithms with faster convergence speed”

__ Relation to Prior Work__: The distinction of the proposed work from prior contributions is fairly clear. Please refer to ‘weaknesses’ section regarding missing prior work/necessary comparisons.

__ Reproducibility__: Yes

__ Additional Feedback__: Post-rebuttal feedback:
Overall, the authors have done a solid job of addressing my major concerns. Specifically, they have run new experiments on a new domain, conducted a more thorough analysis of the attention mechanism in their previous experiments, and fixed some noted mistakes in their equations.
The effort the authors have already put into addressing my concerns is strong enough to convince me to increase my score to an accept.

__ Summary and Contributions__: The paper proposes a bayesian formulation for multi-agent imitation learning problem as well as the variant of the mean field approximation to make it scalable.

__ Strengths__: * Strong experiments demonstrating a consistent improvement in sample efficiency for the proposed method against other baselines.
* Theoretical results.
* Good clarity of the paper

__ Weaknesses__: Whereas Berlin seems like a complex environment, other methods do not seem to struggle much to learn it in fewer epochs than other environment. It suggests that the environment may not be that complex. It would be beneficial for the scalability claims to study the method in scenarios, where even the baseline struggles to learn.

__ Correctness__: Yes.

__ Clarity__: Well written

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: **Post-rebuttal**
After having read the authors response as well as the other reviews, I am inclined to keep my current score.