Aligning Individual and Collective Objectives in Multi-Agent Cooperation

Question 1: I don't think this is fair or true. The mentioned papers do provide substantial theoretical analysis. This is not the way to place your paper...

Response: We apologize for the misunderstanding and appreciate your careful review. We have revised the statement as follows to make it more clear: “On the other hand, several studies focus on automatically modifying rewards by learning additional weights to adjust the original objectives [1,2]. Most of these methods leverage Nash equilibria or related concepts from game theory, such as the price of anarchy. The challenge of finding Nash equilibria in nonconvex games is more difficult than finding minima in neural networks [3], and it is not intuitive to explain how individual incentives align with the collective incentives during the optimization process."

[1] I. Gemp, K. R.McKee, R. Everett, E. A. Duéñez-Guzmán, Y. Bachrach, D. Balduzzi, and A. Tacchetti. D3c: Reducing the price of anarchy in multi-agent learning. arXiv preprint arXiv:2010.00575, 2020. [2] M. Kwon, J. P. Agapiou, E. A. Duéñez-Guzmán, R. Elie, G. Piliouras, K. Bullard, and I. Gemp. Auto-aligning multiagent incentives with global objectives. In ICML Workshop on Localized Learning (LLW), 2023. [3] Letcher, A., Balduzzi, D., Racaniere, S., Martens, J., Foerster, J., Tuyls, K. and Graepel, T., 2019. Differentiable game mechanics. Journal of Machine Learning Research, 20(84), pp.1-40.

Weakness 1: AgA introduces an additional adjustment term, which needs to be tuned case by case.

Response: Thank you for your feedback. Our AgA method indeed includes an gradient adjustment component, however, the additional adjustment term is derived automatically based on our theoretical framework, as outlined in Proposition 4.2 and Corollary 4.3, and is implemented in our derivation code. While the adjustment term itself does not require manual intervention, the parameter $\lambda$ does need to be tuned for different tasks to ensure optimal performance.

Weakness 3: It is not clear whether AgA also has the ability to always keep all individual objectives best. For example, in Figure 1(c), Simul-Co could achieve a better reward for Player 2 than AgA while Simul-Co also achieves a higher collective reward than AgA.

Response: In mixed-motive settings, the group’s incentives can sometimes align and sometimes conflict [1][2]. Therefore, it is almost impossible to consistently optimize all individual objectives simultaneously. If all individual objectives could always be optimal, the problem would reduce to a fully cooperative scenario where all incentives are perfectly aligned. In Section 4.1, we emphasize the mixed-motive nature of our proposed differentiable mixed-motive game: "minimization of individual losses can result in a conflict between individuals or between individual and collective objectives (e.g., maximizing individual stats and winning the game are often conflict in basketball matches)."

[1] McKee, Kevin R., Ian Gemp, Brian McWilliams, Edgar A. Duéñez-Guzmán, Edward Hughes, and Joel Z. Leibo. "Social diversity and social preferences in mixed-motive reinforcement learning." arXiv preprint arXiv:2002.02325 (2020). [2]Du, Yali, Joel Z. Leibo, Usman Islam, Richard Willis, and Peter Sunehag. "A review of cooperation in multi-agent learning." arXiv preprint arXiv:2312.05162 (2023).

Question 1: Could the authors compare the running time of AgA with other baselines?

Response: Please see common response.

Question 2: Could the authors provide more discussion about AgA and Simul-Co? For example, if Simul-Co could achieve a higher collective reward than AgA like in Figure 1, one can redistribute the collective reward for each agent so that the redistributed individual rewards for each agent can be better than AgA.

Response: Thank you for the insightful question regarding the comparison between AgA and Simul-Co. While Simul-Co focuses on maximizing collective rewards, it often overlooks individual agent incentives, potentially leading to dissatisfaction among agents with lower individual rewards. Redistributing collective rewards is also a potential method to address the problem. However, redistribution often encounters the credit assignment problem, one of the most challenging issues in the MARL field. It complicates the redistribution process, typically requiring expert manual design and substantial engineering effort. In this work, our goal is to propose an automatic method that modifies the gradient to seamlessly align individual and collective interests. Unlike traditional reward-shaping techniques, our approach doesn’t rely on manual intervention but instead automatically adjusts the gradient to ensure a balance between individual incentives and overall social welfare.

Question 3: In Algorithm 1, are the parameters w=[w_1,…,w_n] configured manually or determined by the task itself?

Response: The parameters $w = [w_1, \dots, w_n]$ are initially defined in Definition 3 as follows: "The parameter set $w=[w_i]^n\in \mathbb{R}^d$ is defined, each with $w_i\in \mathbb{R}^{d_i}$ and $d = \sum_{i=1}^n d_i$. ... Each player $i\in N$ is equipped with a policy, parameterized by $w_i$, aiming to minimize its loss $\ell_i$." In practice, the parameters $w = [w_1, \dots, w_n]$ in Algorithm 1 refer to the neural network parameters, and our method is based on the popular PPO architecture. These parameters are learned and optimized during the training process rather than being manually configured.

Question 1: Figure 1. It’s not clear how to interpret the reward contours graphs...

Response: Thank you for your advice to improve the clarity of the graph description. The x-axis and y-axis represent the actions of the two players, i.e., for player $i$ ={1, 2}, $a_i \in \mathbb{R}$, where $a_i$ is the action of player $i$. In our future version, we will include the definition of the action space in Example 4.1. The contours in the graph represent the same rewards that can be achieved by the players based on different actions. The contours are formed by sampling the actions of the two players at regular intervals and calculating the corresponding rewards for each action pair. Points that yield the same reward are then connected to form the contour lines, representing levels of equal reward that can be achieved by the players based on their actions. Thank you again for your valuable advice. We will incorporate this explanation into the manuscript to make the graph clearer.

Question 2: What is the significance of AGA moving along the summit?

Response: According to the definition of Example 4.1, the reward function for Player 1 has a maximum value of 1. Thus, moving along the summit represents the algorithm's ability to maximize the player's reward during updates, ensuring that the player's interests are not overlooked. On the other hand, the Simul-Co method focuses solely on improving collective rewards and neglect individual player interests.

Question 3: In Proposition 4.1, what is the gradient without the subscript referring to?

Response: The gradient without the subscript is initially defined in Section 3.1 as follows: "We write the simultaneous gradient $\xi(w)$ of a differential game as $\xi(w) = (\nabla_{w_1}\ell_1, \dots, \nabla_{w_n}\ell_n) \in \mathbb{R}^d,$ which represents the gradient of the losses with respect to the parameters of the respective players." Since this definition is located far from the Proposition, we will reiterate it in our future version to provide clarity.

Question 4: L292, how is the collective loss equation determined?

Response: The collective loss function is determined experimentally within the context of mixed-motive problems and is inspired by the SVO method. Our motivation is to achieve equality while maximizing social welfare, ensuring that the loss function guides a fair contributions among participants.

Dear Reviewer,

Thanks for taking the time to review our response and your valuable input. We appreciate your continued engagement with our work.

Best,

All Authors

Weakness 1: ... I am specifically referring to ' While PED-DQN enables age ...

Response: Thank you for your thorough and detailed reviews of my paper. We apologize for the unclear sentence and have rewritten it to improve clarity: To further promote cooperation, the gifting mechanism—a crucial strategy in mixed-motive cooperation[2]—allows agents to influence each other’s reward functions through peer rewarding. Besides, PED-DQN[3] introduces an automatic reward-shaping MARL method that gradually adjusts rewards to shift agents’ actions from their perceived equilibrium towards more cooperative outcomes.

Weakness 2 & Question 1: ...the complexity of this method... & How much slower is AgA compared to the other algorithm...

Response: Please see Common Response.

Weakness 3 & Question 2:There is no discussion of how AgA satisfied individual incentives & Do you have any results which show the individual metrics

Response: Thank you for your valuable advice regarding individual metrics. Your suggestions are helpful and will contribute to improving the quality of our experiments and the overall paper. To evaluate individual incentives, we introduce the Gini coefficient [1], a commonly used measure of income equality, to assess reward equality in cooperative AI settings [4]. We utilized an expedited method for calculating the Gini coefficient and derived an associated equality metric $E$, defined as $E := 1 - G$. The Gini coefficient $G$ is computed from the ranked payoff vector $p$, which arranges each individual's rewards in ascending order, as $G = \frac{2}{n^2 \bar{p}} \sum_{i=1}^n i(p_i - \bar{p})$, where $\bar{p}$ is the mean of the ranked payoff vector $p$ and $n$ is the total number of players. A higher $E$ value indicates greater equality. Table 1 presents the comparison of the mean and standard deviation of the equality metric achieved by different methods in the Harvest and Cleanup environments (note that the selfish-MMM2 environment involves heterogeneous agents, making it challenging to directly compare rewards achieved by different types of agents). A value closer to 1 indicates more equal rewards among agents. As shown in Table 1, our proposed AgA method outperforms the baselines, demonstrating that AgA can effectively consider all interests within the team. Furthermore, as illustrated in Figure 3 of our manuscript, AgA achieves the highest collective rewards. Therefore, our AgA methods adequately address both individual and collective interests.

Table 1: The comparison of the equality metric.

Question 3: Did you run the experiments for Figure 3 for more seeds? Do the results still hold?

Response: We run all algorithms with three seeds and report the mean and variance with a 95% confidence interval. The results across different environments, including a toy game, a two-player public goods game, Harvest, Cleanup, and our developed selfish-MMM2 show that our method, AgA, consistently outperforms the baselines. Therefore, we believe that the results would remain consistent with more seeds.

Reference:

[1] David, H. A. Gini’s mean difference rediscovered. Biometrika, 55(3):573–575, 1968. ISSN 00063444. URL http://www.jstor.org/stable/2334264.

[2] Lupu, A. and Precup, D. Gifting in multi-agent reinforcement learning. In Proceedings of the 19th International Conference on autonomous agents and multiagent systems, pp. 789–797, 2020.

[3] D. E. Hostallero, D. Kim, S. Moon, K. Son, W. J. Kang, and Y. Yi. Inducing cooperation through reward reshaping based on peer evaluations in deep multi-agent reinforcement learning. In A. E. F. Seghrouchni, G. Sukthankar, B. An, and N. Yorke-Smith, editors, Proceedings of the 19th International Conference on Autonomous Agents and Multiagent Systems, AAMAS ’20, Auckland, New Zealand, May 9-13, 2020, pages 520–528. International Foundation for Autonomous Agents and Multiagent Systems, 2020. doi: 10.5555/3398761.3398825. URL https://dl.acm.org/doi/10.5555/3398761.3398825.

[4] Du, Yali, Joel Z. Leibo, Usman Islam, Richard Willis, and Peter Sunehag. "A review of cooperation in multi-agent learning." arXiv preprint arXiv:2312.05162 (2023).