LOPT: Learning Optimal Pigovian Tax in Sequential Social Dilemmas
Authors: Yun Hua
Shang Gao
Wenhao Li
Haosheng Chen
Bo Jin
Xiangfeng Wang
Jun Luo
Hongyuan Zha
Venue: NeurIPS 2025 poster ()
Keywords: Multi-agent Reinforcement Learning
Sequential Social Dilemma
Primary area: reinforcement_learning

Abstract: Multi-agent reinforcement learning (MARL) has emerged as a powerful framework for modeling autonomous agents that independently optimize their individual objectives. However, in mixed-motive MARL environments, rational self-interested behaviors often lead to collectively suboptimal outcomes situations commonly referred to as social dilemmas.
A key challenge in addressing social dilemmas lies in accurately quantifying and representing them in a numerical form that captures how self-interested agent behaviors impact social welfare.
To address this challenge, \textit{externalities} in the economic concept is adopted and extended to denote the unaccounted-for impact of one agent's actions on others, as a means to rigorously quantify social dilemmas.
Based on this measurement, a novel method, \textbf{L}earning \textbf{O}ptimal \textbf{P}igovian \textbf{T}ax (\textbf{LOPT}) is proposed. Inspired by Pigovian taxes, which are designed to internalize externalities by imposing cost on negative societal impacts, LOPT employs an auxiliary tax agent that learns an optimal Pigovian tax policy to reshape individual rewards aligned with social welfare, thereby promoting agent coordination and mitigating social dilemmas. We support LOPT with theoretical analysis and validate it on standard MARL benchmarks, including Escape Room and Cleanup. Results show that by effectively internalizing externalities that quantify social dilemmas, LOPT aligns individual objectives with collective goals, significantly improving social welfare over state-of-the-art baselines.

LOPT: Learning Optimal Pigovian Tax in Sequential Social Dilemmas
Yun Hual; Shang Gao?; Wenhao Li Haosheng Chen',
Bo Jin',
Xiangfeng Wang4.566 Jun Luo', Hongyuan Zha?
Antai College of Economics and Management, Shanghai Jiao Tong University
School of Computer Science and Technology, East China Normal University
School of Computer Science and Technology. Tongji University
4 Key Laboratory of Mathematics and Engineering Applications (MoE)
5 Shanghai Institute of Al for Education, East China Normal University Shenzhen Loop Area Institute (SLAI)
School of Data Science, Chinese University of Hong Kong (Shenzhen) Ohyyh28, juuo j edu. en, {shanggao, hschen) ecnu. edu. en (bjin, whli) edu. cn, xfwangocs. ecnu. edu. cn, zhaby@cuhk. edu, cn
Abstract
Multi-agent reinforcement learning (MARL) has emerged as powerful framework for modeling autonomous agents that independently optimize their individual ob- jectives. However, in mixed-motive MARL environments, rational self-interested behaviors often lead to collectively suboptimal outcomes situations commonly referred to social dilemmas. A key challenge addressing social dilemmas lies in accurately quantifying and representing them in a numerical form that captures how self-interested agent behaviors impact social welfare. To address this chal- lenge, externalities in the economic concept is adopted and extended to denote the unaccounted-for impact of one agent's actions on others, as a means rigorously quantify social dilemmas. Based on this measurement, a novel method, Learning Optimal Pigovian Tax (L.PT)) is proposed. Inspired by Pigovian taxes, which are designed to internalize externalities by imposing cost on negative societal impacts, LOPT employs an auxiliary tax agent that learns an optimal Pigovian tax policy reshape individual rewards aligned with social welfare, thereby promoting agent coordination and mitigating social dilemmas. We support LOPT with theoret- ical analysis and validate it on standard MARL benchmarks, including Escape Room and Cleanup. Results show that by effectively internalizing externalities that quantify social dilemmas, LOPT aligns individual objectives with collective goals, significantly improving social welfare over state-of-the titthhe-n art baselines.
Introduction
Reinforcement learning 42 achieved remarkable efficacy across diverse domains 12 21 and
has been successfully extended multi-agent settings, especially in cully-cooperative scenarios [46] 26 49). Nevertheless, prevalent centralized multi-agent reinforcement learning (MARL) methods that utilize team rewards [13 38 37 7 are encumbered by inherent limitations in their scalability to large agent populations and are fundamentally deemed unsuitable for self-interested agents in
mixed-motivation environments. While decentralized learning paradigms [43 40 2, wherein agents
independently optimize their individual rewards, provide more natural modeling approach for self- interested behavior. Yet, these methods frequently encounter difficulties in facilitating coordination
'Equal Contribution.
Corresponding to: Xiangfeng Wang.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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among agents. In many real-world environments with mixed motives- particullly those involving
exclusionary or subtractive common-pont -pool resources [36 22 23 rational, self-interested behavior often leads to collectively suboptimal outcomes. These situations are known as social dilemmas.
The concept of social dilemma, originating from economics, refers to situations in which individually rational decision-making leads to collectively suboptimal outcomes [19]. Specifically, these scenarios arise when mutual cooperation would generate universal benefits, yet agents are incentivized to defect due to the prospect of greater individual gain from non-cooperative behavior. In the context of mixed-motte motiivvtiivviinnn MARL, social dilemmas are formally characterized as conflicts between individual reward maximization and the optimization of joint or collective returns (22]. This framing reflects a core economic insight: strategies that are rational from an individual agent's 's perspective can produce inefficient or undesirable outcomes at the group level. Accordingly, a central challenge in mixed- motivation MARI, research is the development of theoretically grounded mechanisms to accurately quantify and represent social dilemmas in a numerical form that captures how self-interested agent behaviors impact social welfare. This involves not only assessing the long-term influence of self- interested agent policies social welfare but also designing learning algorithms capable of aligning individual incentives with collective welfare over extended time horizons.
Established economic theory has long applied the concept of externalities to explain social dilem- mas 8 6) An exterallty arises when the actions of one economic agent directly affect the utility or production possibilities of others without these effects being accounted for in market trans- actions [30]. These impacts on individual utility or production capacities collectively contribute to changes in social welfare. Positive externalities arise from actions that benefit social welfare, while negative externalities result from actions that harm Based on this theoretical foundation, many policy instruments both markett-bssed and non-market-based have been developed to mitigate the negative impacts of externalities and resolve corresponding social dilemmas [35] 55. A prominent example is the Pigovian tax/allowance 5 28 such as carbon tax D which levies taxes on any market activity that generates negative externalities and provides allowances to which bring positive externalities 34). thereby incorporating these effects into market prices. This process known as externality internalization.
Inspired by economic theory. we introduce externality into challenge of accurately quantifying and representing social dilemmas in a numerical form. This theoretical perspective offers clear way to describe MARL dilemmas, where an agent's 's actions may impact others without those effects being reflected in its own reward creating externalities. Building on this insight, we further propose learning- -baied solution that leverages Pigovian tax/allowance mechanisms to alleviate these issues by subsidizing behaviors with positive externalities and taxing those with negative ones. The proposed method, Learning Optimal Pigovian Tax (LOPT), introduces a centralized agent, referred to as the tax planner which learns to allocate tax and allowance rates by maximizing the long-term global reward. This learning process is proven to be equivalent to approximating the optimal Pigovian tax, which reflects the value of externalities. The learned rates then used to design novel reward shaping mechanism, termed optimal Pigovian tax reward shaping, which shapes each agent's 's local reward to reflect how its actions impact overall social welfare. Compared to existing handerafted or performance-driven reward shaping methods for addressing social dilemmas, LOPT offers a theoretically sound shaping approach based on optimal Pigovian tax, which is computed by optimizing social welfare. This ensures theoretical guarantees while demonstrates superior empirical effectiveness and adaptability.
MARL to address the
The primary contributions of this paper are as follows:
Externality theory is introduced in MARL to quantify and represent social dilemmas numerically, providing a theoretically grounded framework for capturing the impact of self-interester agent behaviors on social welfare.
centralized tax/allowance mechanism based on reward shaping LOPT is proposed to approximate the optimal Pigovian tax and internalize the externalities of self-interested agents in mixxed-moivvation MARL tasks, thereby aligning individual agent incentives with social welfare and addressing social dilemmas.
"The carbon tax [29 serves as a widely cited Pigovian tax, making the implicit social costs of carbon emissions explicit by pricing them into market transactions.

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Experiments in the Escape Room and challenging Cleanup environments demonstrate the effectiveness of the proposed mechanism in alleviating social dilemmas MARL.
Externality in MARL
Price of Product a.
Social Marginal SMC PMC + MD
Deechreigtt loss
Marginal PMO
Marginal Damage MD
Private Marginal Bonett PMB SMB
Quantity of Product
Figure 1: Externality 33 The gap between social marginal cost and the private cost is externality.
This section illustrates the concept of externality in MARL and introduces formalism for measuring it, enabling the visualization of social dilemmas. First, the concept of externality in economics is explained, with the graphical analysis shown Figure Consider a firm that produces a product a, meet consumer demand. Simultancously, the firm generates pollution, which negatively impacts social welfare. Let us denote the quantity of produced a; as qe- The price of a, determined by function that depends both the quantity produced yi and the market demand for a; Defining the market demand as q., the profit function is represented as P. q:). The target of the firm is to maximize such utility:
Ui (gi, x. q: x P: (c)
(1)
Let us analyze N firms indexed i 1.2...., N, each firm i producing product a;. Naturally, i aims to maximize its profit by following ( However, the production process inevitably generates activities not reflected in market transactions, such pollution (which harms social welfare) or job creation (which benefits social welfare). To properly account for these externalities, social welfare assessments must incorporate these non-market activities. Therefore, we define the impact of such activities for each firm as a function z; (q:) based on the quantity product a; produced. Consequently, social welfare can be represented as:
U U=E,u (1) + v:(4:))
(2)
The externality is caused by these activities that are not reflected in market transactions, with the economic definition as:
Definition 1. An externality occurs whenever one economic actor's activities affect another's activities in ways that are not reflected in market transactions (30).
The influence 2, can be used to measure the externality. When x, > 0, it represents a positive externality. When I; < 0, it represents a negative externality. We can express the Pigovian tax as function (46) based on the quantity of product a; produced. The after-tax utility for firm i is:
U4 (4t, of) qi x P. (af 4:) (q:).
Here, the Pigovian tax (x(gk) is designed to internalize the externality by making the firm's private cost align with the social cost, with the tax value directly proportional to the influence T; This ensures that addressing externalities is rooted in accurately quantifying the impact æ; of an actor's activities on others. Similarly, in the context of multi-agent reinforcement learning (MARL), externalities can also emerge key concept, where agents' actions influence the outcomes or rewards of other agents
"Without considering social costs, the firm will seek to minimize its Private Marginal Costs at the expense of social welfare. By considering social costs, firm can reduce the Social Marginal Cost, thereby promoting social welfare.

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in ways that are not captured by their individual local rewards. By drawing an analogy between economic markets and MARL environments, an agent's action can be viewed as a form of market behavior, while its local reward corresponds to its individual payoff or utility. The externalities in this context then represent the unintended effects of an agent's 's actions on others, which aligns closely with the fundamental economic definition of externality. By extending this analogy, we can formalize the idea of externality in MARL as follows:
Definition 2. An externality occurs whenever an agent's 's actions affect others in ways that not reflected in individual local rewards.
A decentralized MARL scenario is examined with N N-player partially observable general-sum Markov game on a finite set of states S. At each timestep. each agent E {1,.... N} receives a d-dimensional observation o; E IRd from the observation function : {l...., N + R which maps the current environment state E S and agent identity to individual observation. Based on its observation o₁, agent i selects an action a; E A, according to its policy Ti(a; | 0;), where A, denotes the action space of agent i. which transitions to the next state s' according to the transition function ((s'] s, a) where a = (a1.... ..y) denotes the joint action. Agents then receive their individual extrinsic rewards T; R,(s, a). Each agent aims to maximize its long-term y-discounted payoff:
Q' (',s a) E EΣ (s", aa)) s s, a =a]
The social welfare of the scenario is defined as global long-term 7-discount payoff as follows:
Q(s, x)= [E EE E'E (s', + (s",,a))] - s, a. aa=a
(5)
where 2;(5', .f) represents the influence of agent on other agents in the scenario, and x denotes the joint influence { In this setting, each agent's 's behavior inevitably affects the rewards of other agents. Consequently, social welfare is equivalent to:
( a) = E -' EN ri (8',a') | 8000 8, a0 =a
The optimal joint policy yields the following social welfare:
Q' a = E [EL 7d EN ry a²) | a² = a²)
where a" represents the optimal joint action derived from the optimal joint policy. According to Definition the externality of agent can be defined as follows:
Ex (s, a=a" a.) Q" (s,a") - Q (s)a
where a represents the joint optimal action excluding o; and a is the current action of agent i. Based on (1. an Optimal Pigovian Tax reward shaping approach can be proposed to address external- ities in MARL and resolve social dilemmas. The optimal Pigovian tax-based reward shaping can be expressed as:
F. (s, a_;",, a.) = Q" (s, a²) - Q (s, ",,"",;aa) The agent receives modified reward with the reward shaping:
A (5 a²) = ri (8", a') + F, as, as),
(8)
which successfully internaiizes the externality.
The Prisoner's Dilemma, a classic example of social dilemma, is illustrated in Figure 2 In this scenario, two agents must independently choose between cooperation and defection. While the payoff matrix in Figure ) shows that mutual cooperation maximizes collective welfare, defection remains the dominant strategy for each agent under self-interester reasoning. This misalignment between individual rationality and social welfare leads to an outcome with the lowest utility. The core of the Prisoner's is Dilemma lies in the divergence between private incentives and social costs, which can be captured by the concept of externalities. Each agent neglects the negative externality their actions impose on the other. By quantifying these externalities through Equation O and applying Optimal Pigovian Tax reward shaping as defined in Equation 7 we transform the payoff structure into the revised matrix shown in Figure 2(b) In this modified matrix, the dominant strategy shifts to "Cooperate," " demonstrating that by internalizing externalities via Optimal Pigovian Tax reward shaping. the social dilemma inherent in the Prisoner's Dilemma can be resolved.

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Cooporate
Copuany
Cooperae
Dofox Coppase
-18
(a) Original Payoff Matrix
(b) Payoff Matrix after Tax
Figure 2: Pigovian Tax/Allowance for Prisoner's Dilemma.
3 Learning Optimal Pigovian Tax
In this section, LOPT will be explained in detail. As illustrated in Figure 3] it comprises two major components: (1) A centralized agent called Tax Planner that learns to allocate Pigovian tax and allowance rates maximizing the long-tern global rewards; (2) A reward shaping mechanism based on the learned tax/allowance allocation policy that internalizes each agent's externality, thereby aligning individual incentives with social welfare and effectively addressing social dilemmas. LOPT
Local Observation
Environment
Agent
Action
Q22
a
P(s
update
Reward Function
fq
F2
I'
(s'ss)
Pigavian Tax
xp(ay/a, a)
Updan
Social
Tax planner
Figure 3: The Architecture of the LOPT. The centralized agent Tax planner allocate the Pigovian tax/allowance within a functional percentage formulation. Reward shaping is established based on the Pigovian tax/allowance to alleviate the social dilemmas.
designed learn the Optimal Pigovian Tax reward shaping described in ( internalize each agent's 's social cost. The Pigovian tax rewards is reformulated as:
F (s', "s',,a', aL=;" a.)) ΣE Ta (st a -
-
a ,a))
Pigovian tax reward shaping within percentage tax/allowance is formulated as:
Fé.s (st,a'i"
0.rs (st, ai) 5,(5s' a²) Ors (st, raf)
where 0 represents the tax rates for all agents, O, is the specific tax rate for agent i, while ò denotes the allowance rates for all agents, and d; is the specific allowance rate for agent i. The Optimal Pigovian Tax reward shaping can be learned by determining appropriate values for O and 5, such that each Fg a-",, af cquals Fr "s",""," of). However, since tax and allowance rates vary among different agents different situations, it necessary to represent O and å as functions of the current joint state and action. Therefore, the Pigovian tax reward shaping within percentage
5

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tax/allowance is reformulated as:
Lo at)r (st,ati' F (s, = - (s "t, + 8,(5", a¹)
i=0
Theorem 1. If other agents' actions are treated as part of the environment for any agent i at any timestep i, there always exists typical 0c(s", , a²) and 6 at) to let the Fo.s (s', a ,a)) equal to the Ft
This theorem shows that the Pigovian tax reward shaping within percentage tax/allowance reach the optimum in a specific condition. The theorem is proven in Appendix. The reward shaping function could be treated follows:
Fo, (s', Fo.o (s', "aa
The central challenge is how to learn appropriate and allowance rate functions. As shown in Figure we address this by introducing a centralized tax planner that treats tax and allowance rate as its action space and learns to maximize social welfare. The optimal Pigovian tax based on reward shaping is applied to internalize each agent's externality and solve the social dilemmas. In this form, the tax planner aims to learn the tax rates & and allowance rates for all agents within the MARL task.
Theorem 2. If the interactive influences from other agents are not considered, when the policy of tax planner (f, (s',a') a²), 5g (s', a}) maximizes the social welfare, the typical Fj, (s a a.) will qualitatively equivalent to the F. (st, a²" ; ..?).
Theorem provides a key theoretical foundation for our approach, demonstrating that training the tax planner as centralized reinforcement learning agent maximize total social welfare implicitly approximates the optimal Pigovian tax. This theoretical equivalence is particularly significant, as implies that LOPT can explicitly quantify externalities in MARL by capturing social dilemmas and internalize the broader societal impacts of self-interested agent behavior. In doing so, it directly addresses the core challenge of resolving social dilemmas in multi- agent reinforcement learning, as outlined in this paper. The complete proof is presented in Appendix [
Guided by this insight, we formalize the tax planner reinforcement learning agent defined by the tuple (Sp. Op. Ap, Rpp), where at each timestep &: (1). The planner observes global state and all agents' joint actions of (s* a²) a²); (2). selects taxes and allowances for agents aj (0', 83); receives reward equal to the sum of all agents' rewards, rp. Thus, the tax planner optimizes the cumulative social welfare:
max Jp ETp Σ +== Fp
0² rp a/)
In short, by leveraging reinforcement learning to maximize social welfare, our method implicitly derives and implements optimal Pigovian tax-based reward saaping-providingg principled and practical solution to accurately quantify and mitigate social dilemmas in MARL.
In the training process, we the approximated state-action function @p ap) to replace the cumulative reward rp (o at),, and the objective function then becomes:
max Jp == Ex, [Q (p):
Typically, a policy gradient-based optimization 31 method is applied to train the tax planner. The gradient loss is therefore defined follows:
((ep)
log up (af, of)) & (of, ab)
where the tax planner's policy function parameters are represented by op. Additionally, to maintain balance between tax and allowance, the tax planner needs to minimize the following entropy I (TT)) during the learning process:
f(Tp) = ET
a"," "a.)
'erre we assume that the tax only occurs when the agent get an reward r; A 0, because in reinforcement learning, its profit will only be shown in the step where

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As result, the gradient loss L (op) can be denoted as:
log up (a² oo)) Qu " (op, (b) + 1s (mp²);
E
(9)
where 1) is a hyperparameter weighting the entropy f(Tp).
In light of the learning process of the tax planner, other general agents are trained using the approxi-
mated Optimal Pigovian Tax reward shaping as follows:
C (6:) = E
V
log T' (a; |s) (s) (s. (s.
(10)
where function Qua (8, a) is defined as
(s, a) + F* (s,a"* a.) + max Qu (s',
The typical learning process of LOPT is outlined in Algorithm [ (Appendix), and its performance is demonstrated through experiments in the Escape Room and Cleanup environments.
4 Experiment
Environments We conduct experiments on both the ESCAPE ROOM [50 and the CLEANUP 15 environments, the details are summarized as follows:
Agents
Apple
Bean
River
Escape Room (N- 3, M 2)
(b) Cleanup(N 5)
Figure 4: Environment Examples
Escape Room (ER): In an Escape Room game ER(N, M), where N > M, N agents as players aim to escape from the room (Figure ((a).. this environment, there are 3 available states: door, lever, and start (the initial state), where agents are able to take actions to keep change their states. An agent is able to open the door, then receive an extrinsic reward +10, and end the current episode and only if no less than M other agents pull the lever. Otherwise, agents will receive extrinsic penalty of -1 for making any state change. When agents try to maximize their rewards egoistically, they tend to stay in current positions to avoid punishments move to the door and wait for others to pull the lever that will never happen, which creates a social dilemma. In our experiments, settings of (N 2, M 1) and (N 3, . 2) are applied.
Cleanup: Cleanup game with N agents (()) agents get an extrinsic reward of +1 by harvesting an apple and aim to collect as many apples as possible. Apples are spawned at variable rate, which decreases linearly as the aquifer fills with waste over time. If the waste density reaches the depletion threshold, no more apples will spawn, so agents must clean waste without any extrinsic reward, creating social dilemma. At each timestep 4, agents observe their surroundings as an image and perform one of the following actions:
move left, move right, move up, move down, stay, rotate clockwise, rotate counterclockwise, fire cleaning beam
where the move" rotate" actions change the positions/directions of agents in the map, the stay" action waits at the original positions and does nothing, and the fire cleaning beam" action allows agents to fire cleaning beams (with width 3) to clean wastes (the beam cannot penetrate wastes). To verify how the proposed LOPT resolves the social dilemma, we initialize each episode with sufficient wastes and no spawned apple, then experiment with
on 7 map and 10 x map, where the latter applies lower depletion threshold and apple respawn rate. Finally, more complex scenario of N = 5 Cleanup games with a larger x map and much lower apple respawn rate used to explore the generalizability and scalability of our proposed method.
7

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Implementation and Baselines We compared several baseline approaches in our experiments. First, we evaluated standard reinforcement learning algorithms including Policy Gradient (PG) for Escape Room, and Actor-Critic (AC) along with Proximal Policy Optimization (PPO) for Cleanup.
We then examined state-of-the-art methods for addressing social dilemmas: LIO (50) and its decen- tralized variant LIO-dec, which learn to incentivize cooperation through reward-sharing: Inequity Averse (IA) [15, which promotes cooperation via inequily-ave
social preferences; Model of Other Agents (MOA) [7]. which uses counterfactual reasoning to model agent interactions; and Social Curiosity Module (SCM) 14, which combines curiosity and empowerment rewards.
For specific environments, we implemented various method combinations. In Escape Room, we compared LIO, LIO-dec, and Policy Gradient variants with discrete and continuous reward-giving actions (PG-d/c). The Cleanup(N = 2) evaluation included LIO, IA, MOA, SCM, and Actor-Critic variants (AC-d/c), while the more complex Cleanup(N
5) scenario focused on MOA and SCM.
Manddo 2041
(a) N 2
(b) N = 3
(c) =
(d) =
Figure Results on Escape Room Environment. proposed LOPT: which converges to the optimum and successfully solves the Escape Room social
(a) 5(b) shows the learning curves of the
dilemmas. (5(c) 5(d) shows LOPT is able end the episode in single step without any betrayal.
2-4
My
(a) N 2, (7
(b) N (10 10) (c) N (10x 10) Fixed
(d) N (18 25)
Figure 6: Results on Cleanup Environment. (6(a) 6(b) shows the learning curves for the proposed
LOPT in Cleanup(N 2); (6(c)) shows the learning curves for the proposed LOPT in Cleanup(NV =
2) with the ixee-orientated assumption. ((d) scales to more complex environment with N 5.
Winner Apent SRIN-2)
Cooparster Agantel ERIN-21
1
Taxed
Subsidized
2
Shaped
Extrinsic
2-
Rewards
0
2
4
5
0
2
5
Companstor Agani-l
Cooperater
10
2
0
0
Episodes (1e4)
Figure 7: Rewards for Each Agent with Different Behaviors in Escape Room Environment. LOPT internalizes externalities and redistributes rewards among agents with taxes and allowances. Results Our experiments demonstrate that the proposed LOPT successfully resolves social dilem- mas by approximating externalities among agents in MARL problems and modeling the optimal

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Pigovian tax reward shaping. This approach internalizes the externalities, enabling convergence toward optimal solutions even in complex scenarios. In both Escape Room and Cleanup environments, LOPT implements effective tax/allowance schemes and redistributes rewards among agents, thereby internalizing externalities and guiding agents to develop social-good behaviors (both cooperative and competitive), which significantly accelerates learning curves. Additionally, compared to baseline methods, the internalized externalities in our proposed LOPT result in fewer betrayals, leading to a more stable learning process.
Escape Room. In both ER(N M and ER(N- 3, M 2) settings, Figures ((a) and 5(b)
demonstrate that LOPT rapidly converges to optimal values (8 and 9 respectively) by leveraging optimal Pigovian tax incentives. PG agents completely fail due to selfish optimization, while PC-d/c agents exhibit high variance and suboptimal performance. Although LIO and LIO-dec achieve near-optimal results, they display instability and betrayal-related fluctuations are absent to LOPT. The optimal solution requires only 1 step (M agents pull levers, N M open door). Figures 5(c) and ((d) confirm that LOPT consistently achieves this efficiency, unlike other methods. Figure reveals the underlying mechanism: LOPT taxes "Winner" agents (those creating negative externalities) and rewards "Cooperator" agents (those generating positive exteralities), effectively internalizing externalities through Pigovian incentives.
Staps
2008
300t
(a)
Visualization An Example Rollout
(b) Extrinsic and Shaped Rewards for Agents
April tookk
New Olammiii
Tale
1.400
50400
1080
Shope
(c) Tax/Allowance schemes for Each Agent
Figure 8: An Example Rollout for Cleanup(N 5) Environment. ((a) visualizes this example rollout, where agents apply different social-good behaviors and divisions of laborers (cleaner, har- vester, and part-time) emerge. (8()) shows the approximated optimal Pigovian tax reward shaping by the proposed LOPT. (8(c) shows the reward shaping process of the LOPT in this episode, which demonstrates how the LOPT internalizes externalities for agents with different socially contributed behaviors.
Cleanup. We evaluate LOPT on Cleanup with both simple (N = 2) and complex (N = 5) scenarios. For N 2, we remove LIO's rotation-action restriction, testing 7 x 7 and 10 x 10 maps. Figures [[a] and 6(b) show LOPT achieves near-optimal social welfare, while LIO fails to learn efficient policies. AC-d performs well on 7 x 7 but poorly scales 10 10. Other baselines reach near-optimum on 7 x 7, but IA and AC-c degrade severely on 10 x 10 compared to AC, PPO, SCM, and MOA. Even with fixed-orientation (Figure ((c)). LOPT maintains stable performance by properly internalizing externalities, while LIO shows instability due to potential incentive misalignment. We then compare the proposed LOPT with PPO, SCM, and MOA baselines, which have shown better scalability. in the more complex Cleanup( = 5) scenario, where an 18 25 large map and applied apple respawn rate are applied. Figure b shows that our proposed LOPT is able to scale to more complex scenarios and internalize the approximated externalities by learning optimal Pigovian tax reward shaping, which effectively helps agents to learn in social dilemmas. To demonstrate how LOPT estimates externalities and influences agent behaviors, we analyze their actions and reward redistribution. Figure ((a) shows a Cleanup game with agents: Initially, agents 1, 2, and 4 clean waste (exceeding the depletion threshold) to accelerate apple spawning. Agent 4 becomes a full-time cleaner while agent transitions to part-time harvesting. Agent 2 becomes another part-timer timt,, balancing harvesting with waste cleaning, while agents 0 and

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3 remain full-time harvesters. LOPT naturally induces labor specialization (cleaners, harvesters.
and part-timers) by internalizing externalities, effectively addressing the social dilemma. Figure [(c)
reveals the mechanism: Harvesters (0, 3) pay heavy taxes for negative externalities; part-timers (1, 2)
receive allowances for cleaning but pay taxes for harvesting; cleaner gains substantial allowances
for positive externalities. The system provides near-optimal Pigovian tax incentives (Figure ((b) to
guide agents toward superior outcomes, Additional results appear in Appendix D.3
5
Conclusion
In this paper, we introduce externality theory measure the influence of agents' behavior social welfare. Based on this theoretical foundation the MARL domain, we propose the Learning Optimal Pigovian Tax method to address social dilemmas. We construct centralized agent, Tax Planner, which learns the tax/allowance allocation policy for each agent. Through Optimal Pigovian Tax reward shaping, each agent's externality is internalized, encouraging behaviors that benefit social welfare. Our experiments demonstrate the superiority of the proposed mechanism in alleviating social dilemmas in MARL. For future work, aim develop decentralized Pigovian tax/allowance mechanism to learn reward shaping that internalizes agents' externalities while reducing computational complexity.
6 Acknowledge
Xiangfeng Wang is supported by the National Key R&D Program of China (Nos. 2021 1021Y and 2021 1 YFA 1000302), the NSFC (62231019) and SHEITC (222--GZ-RGGZ--BTXX--00 Wenhao Li is supported by the NSFC (62406270) and the STCSM Shanghai Rising-Staa Program (24YF2748800). Jun Luo is supported by the NSFC (72031006).
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A Related Work
Our work, LOPT, motivated by the challenge of fostering cooperation among independently learning agents in intertemporal social dilemmas (ISDs) [22] In ISDs, agents pursue individual long-term returns, but mutual defection often leads to suboptimal collective outcomes and degraded social welfare over time.
A.1 Limitations of Conventional MARL in ISDs
Conventional Multi-Agent Reinforcement Learning (MARL) algorithms designed for fully coop- erative tasks [13] 41 24 25 [0] struggle with ISDs due to their assumption of aligned agent incentives. In contrast, ISDs feature mixed motivations, where agents' local optima may conflict with collective well-being.
Several approaches attempt address this by incorporating reward shaping intrinsic motivation [12 47]. However, these methods often rely on hand-crafted heuristics or evolution based adaptations other agents' behaviors, limiting generality and scalability. More recent approaches, like LIO (50, enable agents to learn incentives for others, while some studies explore mechanism or information design [27] in fully cooperative contexts. Yet, these methods typically lack unified economic rationale for shaping rewards.
A.2 Externality Theory and Economic Inspiration
LOPT is grounded in externality theory [3]. which provides principled framework for aligning individual incentives with social welfare a central challenge in ISDs. both non-marker [ and market economies (35) various mechanisms have been developed to internalize externalities, such as the Pigovian tax 5 which penalizes behaviors that impose social costs.
Our approach adopts learning-base Pigovian tax framework shape agent incentives and mitigate negative externalities. This aligns with economic findings that reward structures significantly influence cooperative behavior in repeated settings. For instance, [39] demonstrates that limited feedback and longer interaction horizons promote cooperation human queueing systems, emphasizing the role of information and interaction design. Similarly, 4 shows that optimal mechanisms in competitive markets are sensitive to network structures, reinforcing the importance of structural design in multi-agent coordination.
Moreover, highlights the theoretical interchangeability of taxes and subsidies under certain conditions, broadening the space of policy tools for influencing agent behavior. While LOPT focuses on tax- tax-based shaping, its theoretical foundation can naturally extend subsidy schemes depending on fairness implementation considerations.
Our design also draws structural inspiration from the Al Economist [5], employing two-stage ar- chitecture to learn tax policies. However, LOPT specifically targets ISDs in MARL and distinguishes itself by leveraging externality theory to inform its reward shaping paradigm.
A.3 Structural Solutions to ISDs: Centralized vs. Decentralized
Beyond reward shaping, recent work has explored structural interventions for ISDs, drawing parallels economic governance models. These can be categorized into:
Centralized boundaries agent behavior.
9 16, which emulate government-like authorities to regulate
Decentralized sanctions
which enable agents to punish others for
socially harmful behavior.
LOPT follows the centralized boundaries paradigm, introducing centralized planner that learns to enforce Pigovian taxes based on global observations. Unlike previous centralized approaches, such D. which uses arbitrary allocation for shared resources, [16), which introduces fixed tax mechanism, LOPT learns dynamic tax policy tailored to the environment. Furthermore, our method theoretically supported by externality theory. providing a principled foundation for shaping agent behavior.
14

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B Proof
Theorem 1. If other agents' actions are treated as part of the environment for any agent i any timestep t, there always exists typical (s(s', a²) and dz(s", ; at) to let the Fe.s (s', a.) equal to the Ft 6',a"", ,a))-
Proof. We make classified discussions for any agent i create negative externality, agent i create positive externality, For any agent which creates negative externality at timestep t: the agent will not receive any allowance, so the allowance rate function & (s",, a.) is equal to O. And the tax rate can be written as:
P.(sh,af. a²," = (' .a)))
(11)
ri(s", a a')""
Q a.)
0:(8", a',"
(12)
T a'i")
And as the interactive influence from other agents is not considered, other agents' optimal action a"= can be seen as a part of the environment, and this optimum has a fixed result. Therefore, like the reinforcement learning method with an advantage function, for each agent the advantage function based on the current joint state and action can also be found in the tax rate, where:
Q a²*) (s's,, a²) Q(s', a'), Q(st, a' at) = A!(st,a") x Q(st, a'), riss,aa, a" = Af ,') x rri a').
(13)
Then the tax rate for agent i becomes:
0,(8", 0, a"," (A9 f a²) - A} (s', a² a²)) Q a²)
Al a²) r2 (3 ( a²)
(14)
(s a') (A)(s', a²) - A) a')) Q( a²)
AP a²) (s
Then is proven that for any agent which generates negative externality, there always exists typical 0 a²) and 6,(s, a²) to let the F2 (s", " equivalent to the F (s', a", ".a)
Similarly, for any agent i which generates positive externality, there also exists typical (s a²) and 6s("", a²) to satisfy the condition above.
This proves that if the interactive influence from other agents is not considered, for any agent at any timestep 1, there always exists typical (s("",a²) and 8, (s, a²) to let the Fé.s (s', ; equivalent to the F "","""""000
Theorem 2. the interactive influences from other agents are not considered, when the policy of tax planner (0, aa), 8, (s", a)) maximizes the social welfare, the typical F (s", a" aff will qualitatively equivalent to the F2 as","", ,a!).
Proof. Here we use the method of "reduction to absurdity." Suppose that there exists an agent which generates negative externality, and the learned Fà (s', a",","" ,af) does not qualitatively equivalent to the F: (s', a", .a))- The reason why agent will choose the selfish behavior which harms social welfare without reward shaping is because its individual reward shows:
ri(s', a', a > r=("""a"")
(15)
And the effect of the Optimal Pigovian Tax reward shaping is to let any of E A, hold the following constraint:
a".",a"i",d F8, "oa,""""","" aaf) (o'6".a"))
(16)
As we suppose that its typically learned reward shaping does not qualitatively equivalent to the Optimal Pigovian Tax reward shaping. That means there exists some of A,, which causes:
rs(s', a², a²)) + Fs" a', a ri(s",a")
(17)
15

---

This means agent within its optimal policy x. would like to choose the behavior of rather than the behavior optimal joint actions a²". Then if use the tax planner's learned policy xp describe the tax rate allocation, which means there exists another tax planner's s policy Ty. letting:
dr
I
Σ Tp (spp app)
Σ TD (spp ag))
(18)
Thus we have shown that if any learned reward shaping of agent is not qualitatively equivalent to the Optimal Pigovian Tax reward shaping, the tax planner's ' learned policy is not optimal.
C Algorithm
Algorithm LOPT:
Learning Optimal Pigovian Tax
1: Initialization: all general agents' policy parameters {di tax planner's policy parameters dp:
2: for each iteration do
3: Generate a joint state-action trajectory with shaped rewards and tax/allowance rates as {r):
4: for each state-action pair with shaped reward for each agent i.e., as,.a, a, + Ft) in {T} do
5: Compute the new oi by gradient ascent TT);
6: end for
7: for each planner state-action pair with global reward (Op, ap, rp) in {r} do
8: Compute the new &p by gradient ascent on
9: end for
10: Φi + Φi, 6p + Sp for all N.
end for
16

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D Experiment
D.1
Implementations
The policy and value functions in LOPT are implemented as neural networks (detailed architecture provided in Appendix. [D22). Training conducted on virtual machine hosted on GPU server equipped with four NVIDIA GTX 2080 Ti GPUs, a 24-core CPU, and 32 GB of DRAM.
We implemented the LOPT in both Escape Room and Cleanup environments. At each timestep the global observation o' ggiobal from the joint state $t, and the joint action a' are fed to the tax planner as input. To better handle our challenging environments, we provide "bank" variable to the tax planner save rewards from taxes available budgets for allowances, which supports the more sophisticated tax/allowance mechanism. Then, the current bank state honk and joint reward r' are also introduced to the observation:
of - -0 a', Obank
The planner outputs the joint tax rate Of and the joint allowance rate 8'. In addition, the tax planner outputs. Also, it outputs percentage for rewards withdrawn from the bank as the budget ratio at bank So, the action for the current time step is:
af (0', 8", .bank)
In addition, the entropy [(Tp) is weighted by a hyperparameter v) in ( Concretely, both en- vironments with N agents, o and a' are scalers, while a', rf, 0e and 8 are N dimensional vectors. In the Escape Room games, the tax planner agent observes a multi-hot vector global states ggobbl E {0, 1]d from the joint state st, where d 3.N. And in the Cleanup games, the global observation ogoobut the global visual normalized RGB observation with the same width and height of the applied map.
In the Escape Room environment, the policy network for the tax planner is defined as follows: 1). dense layer h1, of size 64 takes ogiobul as input and 3 dense layers h1,, i 2,3, 4 of size 32 for a², banker and r' respectively; 2). the outputs of dense layers h1, 2, 4 are concatenated and fed to a dense layer h.2 of size 32; 3). the output of dense layer h2 is fed to dense layers i 1,2, 2 of sizes 1, N. N and activation functions sigmoid, sigmoid, so softma then output as at bank 0*, 8 respectively. While in the Cleanup environment, the policy network for the tax planner is defined as follows: 1). the global observation o' gglobal is firstly fed a convolutional layer conv1 of kernel size 3 x 3, stride and filters; 2). the output of the convolutional layer convl, a', o and r' are fed to two-layer dense layers h2,, 1, 2, 3, 4 of size 32 and 32 respectively: 3). the outputs of dense layers h2,, = 1, 2, 3, 4 are concatenated and fed to an LSTM of cell size 128; 4). at last, the output of the LSTM is fed to the dense layers and output a
bank 0º, 8' respectively.
The settings of hyperparameters for baselines follow their previous work 15 experiments, the tuned hyperparameters of all baselines and LOPT are given in Table, in the appendix D.2, where: a is the learning rate; d'schedule is list that contains the step and weight pairs for the learning rate scheduler; is the weight for the entropy I' [TTp); in (50) decays linearly from stan to fnd by 6div episodes; coefficient for the entropy of the policy.
50
For all
17

---

D.2 Hyperparameter
N =
N ==2,
=
10 10
N = 5,
Parameters
7 x map
10 x 10 map
map fixed orientations
18 x 25 map
appleRespawaPrrbabiltyy
0.5
0.3
0.3
0.05
wasteSpawwnPrbbbbiiiy
0.5
0.5
0.5
0.5
thresholdDeepletion
0.6
0.4
0.4
0.4
thresholdRestorattin
0.0
0.0
0.0
0.0
rotationEnabled
/
/
view_size
4
7
7
7
max_steps
50
50
1000
Table Experiment Settings for Cleanup Environment.
N
Hyperparameters
RG
P
PGrc
L.O
LiO
LOPT
PGi
Ridd
PCre
LIO
deO
LOPT
1e-4
Te-4
le-3
le-4
1e-4
Te-3
le-4
Te
Te-3
le-4
Te-4
Te 3
1
0.95
0.95
0.5
0.5
0.5
0.5
0.5
1.0
0.5
0.5
0.5
Fond
0.05
0.05
0.1
0.1
0.1
0.05
0.05
0.05
0.1
0.3
0.3
0.05
Cdv
100
0.01
0.01
0.1
0.01
0.01
0.01
0.01
0.01
0.1
0.01
0.01
0.01
Table 2: Hyperparameter Settings for Escape Room Environment.
200
TRY
NEW
11PT
[23
3013
8.
63
0.5
13
Fad
106
1900
1000
106
01
81
E1
01
Libe-d Lihe-3 1.0-3
138-8
010
611
01
126-3
17kes
Table 3: Hyperparameter Settings for Cleanup(N 2) Environment.
Hyperparameters
PPO
MOA
SCM
LOPT
1.26e-3
1.26e-3
1.26e-3
1.26e
Osschedule
I (2e7, 1.26e-4), (2e8,
1.26e-5)
[(2.ee7,
1.26e 4)]
0.95
1.76e-3
1.76e-3
1.76e-3
1.76e-3
Table 4: Hyperparameter settings for Cleanup(NV = 5).
18

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D.3 Addtional Experiment Results
this section, additional results from experiments will be demonstrated. As illustrated in Figure. our proposed LOPT is able to internalize externalities in all of our Cleanup experiment settings and provide approximated optimal Pigovian tax reward shaping to greatly alleviate the social dilemmas. And for Cleanup(1V 5) environment, we further show the relationship among the environmental states the numbers of apples and wastes and the tax/allowance schemes given by the LOPT, where proper tax/allowance schemes given for agents with different socially contributed behaviors in Figure (3) Figure [3(b] and Figure [3(c] Also, Figure. [3(d) shows that the LOPT encourages agents clean wastes efficiently and maintains the density of wastes
relatively low level that the apples are spawned at a relatively high rate. Also, we provide visualized and analyzed results
from example rollouts in Cleanup(NV = 2) with both the x 7 and the 10 10 maps.
Apel (1h0100))
Steps
(a) Visualization of The Example Rollout
(b) Extrinsic Rewards and Shaped Rewards for Each Agent
WW
Bomn
(c) Number of Apples and Wastes in the En (d) Tax/Allowance Schemes for Each Agent vironment
Figure 9: An Example Rollout for Cleanup(N 2) Environment with A 7 Map.
Apenl Pullen
Apoil Pardel
Steps
Visualization of The Example Rollout
(b) Extrinsic Rewards and Shaped Rewards Each Agent
(c) Number of Apples and Wastes in the (d) Tax/Allowance Schemes Each Agent vironment
Figure 10: An Example Rollout for Cleanup(N = 2) Environment with A 10 x 10 Map.
19

---

Staps
Visualization of The Example Rollout
Extrinsic Rewards and Shaped Rewards Each Agent
(c) Number of Apples and Wastes in the En-
vironment
Figure 11: Number of Apples and Wastes Environment
Tax/Allowance Schemes for Each Agent
Figure 12: An Example Rollout for Clcanup(N 2) Environment with A 10 x 10 Map and Fixed Orientations.
20

---

Than
(a) Tax/Allowance Schemes with (b) Tax/Allowance Schemes with (c) Tax/Allowance Schemes with Environmental States for Cleaner Environmental States for Harvester Environmental States for Part-time Agents
Agents
Agents
#Apples
20
#Wastos
40
200
400
600
800
1000
Steps
(d) Number of Apples and Wastes the Environment
Figure 13: An Example Rollout for Cleanup(N 5) Environment, supplemental results for Figure (3) 13(b 13e)) illustrate relationship of environmental states (the number of apples/wastes) and the law/allowance schemes given by the LOPT for 3 types of agents with different socially contributed behaviors. (T(dd) shows the amount for apples and wastes during the episode.
21

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NeurIPS Paper Checklist
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Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The main claim made in abstract and introduction is that we introduce external- ities denote quantify social dilemmas in MARL and LOPT is proposed internalize externalities and solve social dilemmas.
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Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
22

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Answer: [Yes]
Justification:
We provide the proof in Appendix. B
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Reviews:
[{"review_id": "wfMGTmsrmk", "reviewer": "Reviewer_jTeR", "summary": "A key challenge in multi-agent settings is how to address social dilemmas - the scenario whereby rational agents acting in their own self interest produce suboptimal social outcomes. The authors aim to address this issue by training an AI agent to apply reward shaping to internalise the social costs produced by each agent, thereby applying a 'pigovian tax'. Under their problem formulation, the authors show the existence of an optimal solution and then demonstrate empirically the performance of a policy gradient method with their propose reward shaping formulation.", "questions": "- I may have missed this, but how many runs are each of the results averaged over?\n- How many samples are required to develop such a tax planning agent? Is this reasonable in real environments?", "limitations": "There is limited discussion on the limitations of their work", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The formulation and proofs of existence seem correct as far as I can tell.\n- The experiments compare to a good range of baselines.\n- \n\nWeaknesses:\n- The paper could be more clearly written", "was_revised": true, "final_justification": "The authors have answered my questions and addressed the concerns of other reviewers. I will maintain my score.", "rebuttal": "We sincerely thank the reviewer for their constructive feedback and valuable questions. Our work presents a **reward shaping method** to solve social dilemmas in MARL, inspired by the economic concept of a **Pigovian tax**. The **\"tax planner\" agent** acts like a government or regulatory body, learning to apply incentives to steer self-interested agents toward a **collective good**. This framework serves as a practical model for real-world problems like **traffic management** or **carbon pricing**, where a **central planner optimizes social welfare**. We will clarify this framing more explicitly in the revision.\n\n---\n\n### 1. How many samples are required to develop such a tax planning agent? Is this reasonable in real environments?\n\nOur tax-planning agent is **highly sample-efficient**. As shown in **Figure 5** and  **Figure 6**:\n\n- It typically **converges within 50,000 (5e4) interaction steps** in the **Escape Room** environment.\n- In the **CleanUP environments**, it converges within:\n  - **10 million (1e7)** steps (N = 2, 7 X 7),\n  - **20 million (2e7)** steps (N = 2, 10 X 10),\n  - **150 million (1.5e8)** steps (N = 5).\n\nThis level of sample complexity is **reasonable in real-world settings**, as it corresponds to the **training phase** of a reinforcement learning (RL) agent. Once trained, the **learned policy can be deployed directly** without further sampling. That is, the trained tax planner can take in current observations and **immediately output appropriate incentives**.\n\nTo improve clarity, we will **enlarge the learning curve plots** in the revised version.\n\n---\n\n### 2. I may have missed this, but how many runs are each of the results averaged over?\n\nWe apologize for this omission and have added the missing details in the **Experimental Setup** section of the revised manuscript.\n\n- **Learning Curves**: All quantitative results, including the learning curves, are **averaged over 5 independent runs** using different random seeds.  \n  The shaded areas in the plots represent the **95% confidence interval**, demonstrating the **stability** of our method.\n\n- **Policy Analysis**: For qualitative analysis of the **emergent strategy**, we selected the **best-performing policy** from the 5 runs.  \n  This approach allows for a **clear and interpretable illustration** of the optimal **Pigovian tax mechanism** discovered by our method.\n\n---"}, {"review_id": "l3PMGcL6jm", "reviewer": "Reviewer_qvx9", "summary": "The paper proposes a MARL paradigm to escape the social dilemma in multi-agent sequential games. The method is inspired by Pigovian taxes theory and build the structure of a high-level tax planner into the MARL algorithm. The paper first established the exisitng theory and built its connection with the MARL setting. Then, with the proposed approach, it conducts experiments on classic social dilemma games. The results show that the MARL can successfully escape such local minimal and explains the implication of the tax framework to the games.", "questions": "1. Is it possible to extend the algorithm to continuous control space?\n2. How is the scalability of the proposed method of like N=20 agents? Does it significantly take more compute?", "limitations": "No", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n\n1. The idea of building the Pigovian taxes bias into the MARL algorithm is interesting and the motivation is easy to understand.\n3. The proof of concept experiment is comprehensive in analysis and is supportive to the pipeline.\n\na. The empirical results suggest that the proposed frame work is very strong, e.g, the Escape Room, where the policy is able to converge in just a few steps while the existing baselines struggles.\n\nb. The method has been shown to be able to generalize to more complicated setting including 5 agents and 18x25 size, which suggests certain scalability.\n\nWeakness:\n\n1. Apart from the existing proof of concept experiments, it is unknown of the scalability of the MARL algorithm. For example, with significantly more agents or the control space is larger or no longer discrete. I would imagine that doing similar experiments with significantly more agents and grid size would make the work much more impactful.\n2. The paper uses very small figure which make the reader hard to read. i would suggest reducing the narrative and use larger figures and more illustrations including the visualization of the policy of the proposed approach.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their positive feedback and insightful comments. We are encouraged that the reviewer found our idea *\"interesting,\"* the motivation *\"easy to understand,\"* and the results *\"very strong.\"* We appreciate the opportunity to clarify the questions regarding the extension to continuous control and scalability.\n\n---\n\n### 1. On Extending the Algorithm to Continuous Control Spaces\n\nWe thank the reviewer for this insightful question.\n\nYes, our framework **can be extended to continuous control spaces**. Our decision to focus on discrete action spaces in this work was primarily motivated by the **established benchmarks in multi-agent social dilemma research**. To ensure a fair and direct comparison, we adopted the same experimental settings used by seminal works in this area, including *Learning with Opponent-Learning Awareness (LIO)* and others.\n\nThese standard environments—such as the **Cleanup** and **Escape Room** games—are predominantly **discrete**, and demonstrating superior performance on them effectively highlights the **core advantages of our tax-based approach** against existing state-of-the-art methods.\n\n---\n\n### 2. On Scalability to a Larger Number of Agents\n\nWe appreciate the reviewer raising this important point about **scalability**, which is indeed a critical aspect for any MARL algorithm.\n\nThe reviewer is correct that **computational complexity increases** as the number of agents (N) grows. As briefly discussed in the paper (Lines **161–163** and the **caption of Figure 3**), the **primary bottleneck** in our current implementation is the **centralized tax planner**. As *N* increases, the planner’s **observation and action spaces grow**, making its optimization task more challenging.\n\nWe explicitly acknowledged this as a limitation in the **conclusion** of our paper. However, we do not see this as a fundamental flaw of the **tax-based paradigm** itself, but rather as a **challenge related to the centralized implementation**.\n\nA promising next step—highlighted as part of our future work—is to develop a **decentralized tax mechanism**. For instance:\n\n- The tax planner could be **decomposed** into multiple, smaller planners operating on **subsets of agents**.\n- Alternatively, agents could learn to **negotiate taxes locally**.\n\nThis would **significantly enhance scalability** and is a **core focus of our ongoing research**.\n\n---\n\nWe hope these clarifications have addressed the reviewer's questions.  \nWe will also ensure that the **figures are enlarged** in the final version for better readability.\n\n**Thank you once again for your valuable and constructive feedback.**"}, {"review_id": "1OXWsBuCQl", "reviewer": "Reviewer_HqTs", "summary": "The paper presents a new model for MARL in which the reward of each agent is shaped to consider social welfare. The new term is inspired by Pigovian tax and is learned using an external agent using gradient descent towards the optimal tax. The paper includes two theorems that together form a theoretical basis for the learned tax to be optimal. Empirical evaluation includes two domains and a variety of agents that are self-interested. The results show that the new method, LOPT, enables the agents to learn a balanced policy that maintains social welfare.", "questions": "1. In the prisoner's dilemma, shouldn't all values be negative except the 0s in the original formulation, and except the 8s in the corners in the matrix after the tax?\n2. What happens if there is some correlation between x_i and x_j? For example, having everyone use the same power grid simultaneously will cause it to collapse, even though the same usage would not incur such a penalty if it were spread out over time.\n3. Given the current explanation for the escape room domain, why would anyone pull the lever?", "limitations": "It can be beneficial to have a short discussion about dual purpose in this research: can someone use the same paradigm as LOPT to *decrease* social welfare to intentionally cause unrest?", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n+ Looking at shaping social welfare into MARL via a Pigovian tax seems novel\n+ The background and the explanations for the new method are clear and informative\n+ Empirical evaluation seems sound, and results are consistent with the claims\n\nWeaknesses:\n- Notation is inconsistent and sometimes unexplained. For example, in line 159 it seems that the last indexing of r should be (-j) rather than j. Similarly, the notation in line 201 with pi^p_{\\Phi_p} but then the same term in Eq. 9 is pi_p^{\\Phi_p}. Also, I'm unsure that the numbers in Figure 2 depicting the prisoner's dilemma are correct.\n- Theorems are considered a part of the main paper, but the proofs are solely in the appendix, so they cannot be considered a fully sound contribution.\n- Related work section is missing on some substantial recent work in the field of solving social dilemmas using MARL, including for example:\n1. Li, Yang, et al. \"Aligning Individual and Collective Objectives in Multi-Agent Cooperation.\" The Thirty-eighth Annual Conference on Neural Information Processing Systems.\n2. Haupt, Andreas, et al. \"Formal contracts mitigate social dilemmas in multi-agent reinforcement learning.\" Autonomous Agents and Multi-Agent Systems 38.2 (2024): 1-38.\n- The explanation of the experimental design is missing and hard to follow if you're not familiar with the domain. For example, can an agent pull the level from every place, or just if they opened the door? Given the current explanation, why would anyone pull the lever?", "was_revised": true, "final_justification": "The authors have responded to my concerns and have provided enough evidence to show they can improve the paper as needed.", "rebuttal": "We sincerely thank the reviewer for their time and for providing such detailed and constructive feedback on our manuscript. We are grateful for the recognition of our work's novelty, clarity of explanation, and the soundness of our empirical results. The reviewer’s insightful comments have highlighted several key areas for improvement. We have carefully considered all points and will revise our manuscript accordingly. We will also cite the suggested papers in our related work section.\n\nBelow, we address each of the reviewer’s concerns in detail.\n\n---\n\n### **1. Notation and Clarity of Figure 2 (Prisoner's Dilemma)**\n\nWe thank the reviewer for their careful reading and for identifying inconsistencies in notation and clarity issues in the payoff matrix of the Prisoner’s Dilemma example.\n\n* **Notation:** We acknowledge inconsistencies (e.g., `r_j` vs. `r_{-j}`, and `\\pi^p_{\\Phi_p}` vs. `\\pi_p^{\\Phi_p}`), and apologize for the oversight. In our revision, we will unify the notation throughout the main text and appendix, and add a **comprehensive notation table in the appendix** for clarity.\n\n* **Payoff Matrix Clarification:** We appreciate the reviewer’s question regarding the derivation of the after-tax payoff matrix. We realize that our original explanation was insufficiently detailed. Below is the derivation that we will include in the Appendix:\n\n#### **Step-by-Step Derivation:**\n\n1. **Original Payoff Matrix:**\n\nWe start with a classic Prisoner’s Dilemma game:\n\n$$\n\\begin{bmatrix}\n(-1, -1) & (0, -10) \\\\\\\\\n(-10, 0) & (-8, -8)\n\\end{bmatrix}\n$$\n\nHere, mutual cooperation yields \\$(-1, -1)\\$, and mutual defection yields \\$(-8, -8)\\$. In the asymmetric case, say (Defect, Cooperate), the defector gains 1 unit (from \\$-1\\$ to \\$0\\$), while the cooperator suffers a loss of 9 (from \\$-1\\$ to \\$-10\\$), resulting in a **net welfare loss of 8** compared to mutual cooperation:\n\n$$\n(0 + (-10)) - (-1 + -1) = -8\n$$\n\n2. **Pigovian Correction:**\n\nTo internalize the externality, we apply a **Pigovian tax** on the defector equal to the social loss they cause (i.e., 8). Symmetrically, we provide an **equivalent subsidy** of 8 to the cooperator, so the social planner remains revenue-neutral.\n\n3. **After-Tax Matrix (with Tax + Subsidy):**\n\n$$\n\\begin{bmatrix}\n(-1, -1) & (-2, -8) \\\\\\\\\n(-8, -2) & (-8, -8)\n\\end{bmatrix}\n$$\n\nExplanation:\n\n* In the (Defect, Cooperate) case:\n\n  * Defector: \\$0 - 8 = -8\\$\n  * Cooperator: \\$-10 + 8 = -2\\$\n\n* In the (Cooperate, Defect) case: symmetric values.\n\n* Mutual cooperation and mutual defection are unchanged, as no externalities arise in those configurations.\n\nThis transformation **shifts the Nash equilibrium** from the socially suboptimal (Defect, Defect) to the socially optimal (Cooperate, Cooperate).\n\nWe will add this derivation and both the original and after-tax matrices to the Appendix, along with an intuitive diagram.\n\n---\n\n### **2. Correlation Between Agent Actions (e.g., Power Grid Scenario)**\n\nWe appreciate this insightful question on the ability of LOPT to handle **correlated externalities** in multi-agent settings.\n\nWe confirm that **LOPT is designed to address such scenarios**, where the joint outcome depends on **non-linear and interdependent agent actions**.\n\nIn the power grid example:\n\n* The collapse is triggered by specific **joint action profiles** that violate stability constraints.\n* LOPT’s planner optimizes **total social welfare** across all agents, not individual objectives.\n* The tax function \\$T(s\\_i, a\\_i)\\$ is conditioned on each agent’s state \\$s\\_i\\$, which can incorporate **global context** or relevant information about others.\n* Through policy gradients based on total welfare, the planner **learns to penalize** harmful joint behaviors and **incentivize coordination** to avoid collapse.\n\nWe will **add a paragraph to the experimental analysis and limitations sections** to emphasize LOPT’s capacity to handle correlated externalities, using the power grid domain as a concrete example.\n\n---\n\n### **3. Motivation to Pull the Lever in the Escape Room Domain**\n\nThank you for the opportunity to clarify this scenario. The Escape Room is explicitly designed to create a **moral hazard** involving positive externalities:\n\n1. **Environment Dynamics:**\n\n   * A large reward is given only to agents that successfully exit the room.\n   * One agent has the option to **open the door** and escape (selfish action) or **pull a lever** located elsewhere (cooperative action) that allows the *other agent* to escape.\n\n2. **Incentive Misalignment:**\n\n   * **Selfish Action:** Allows one agent to leave but locks the other out (negative externality).\n   * **Lever Pulling:** Enables the *other* agent to escape, but yields no immediate reward to the actor (positive externality).\n\n3. **LOPT’s Intervention:**\n\n   * Self-interested agents do not pull the lever under the default reward structure.\n   * The planner in LOPT learns to:\n\n     * **Tax** agents who act selfishly (door-opening that blocks others),\n     * **Subsidize** lever-pullers (to compensate for their altruistic action).\n\nAs a result, the **reshaped rewards make cooperation individually rational**. The agent who pulls the lever receives a reward from the planner, aligned with the social benefit they generate.\n\nWe will **revise the Escape Room experiment description** to clearly articulate this dilemma. The revision will include:\n\n* A step-by-step walkthrough of the environment and reward structure.\n* A diagram to visually illustrate the spatial layout, action possibilities, and social consequences.\n\n---\n\nOnce again, we sincerely thank the reviewer for their valuable and thoughtful feedback. We are confident that these clarifications and planned revisions—on notation, theoretical explanation, and experimental clarity—will significantly improve the quality and accessibility of our paper."}, {"review_id": "WoBV05Lais", "reviewer": "Reviewer_H64g", "summary": "This paper presents Learning Optimal Pigovian Tax (LOPT), a framework to quantitatively address social dilemmas in mixed-motive multi-agent reinforcement learning (MARL). LOPT is inspired from economics, where externalities are effectively modeled into social dilemmas. Specifically, LOPT uses an agent the authors term a tax planner, which learns to allocate allowances to reflect values of externalities. The LOPT framework is validated through theoretical analyses and on MARL benchmarks.", "questions": "Q1: It is not clearly mentioned how the LOPT formulation in Sec. 2-3 is different from other widely examined reward shaping approaches, including techniques such as potential-based reward shaping. Could the authors comment? \n\nQ2: A critical result when using methods like potential based reward shaping is that the identity of optimal policies are preserved- i.e., the optimal policy learned when using reward shaping is the same as the optimal policy in the original environment without reward shaping. Does LOPT provide an analogous guarantee? \n\nQ3: While the experimental evaluations are on intuitive environments, the large body of work examining reward shaping and credit assignment in MARL showcase verification on complex multi-agent domains such as StarCraft. Is there a scope for evaluating LOPT in these domains in order to provide a more thorough comparison?", "limitations": "Addressed in the Conclusion very briefly. The authors could expand on this, and use a separate Limitations section/ subsection.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n(+) The paper is generally well-written and easy to follow. \n\n(+) The notion of introducing externalities from economics to quantify social dilemmas in MARL is novel. \n\n(+) Theoretical analysis to demonstrate that LOPT can effectively quantity externalities. \n\n(+) Intuitive experiment domains to demonstrate the validity of the proposed approach. \n\n(+) Evaluation on multiple baselines showcase improvement of LOPT over other methods. \n\nAreas for improvement: \n\n(-) It is not clearly mentioned how the LOPT formulation in Sec. 2-3 is different from other widely examined reward shaping approaches, including techniques such as potential-based reward shaping. \n\n(-) A critical result when using methods like potential based reward shaping is that the identity of optimal policies are preserved- i.e., the optimal policy learned when using reward shaping is the same as the optimal policy in the original environment without reward shaping. Does LOPT provide an analogous guarantee? \n\n(-) While the experimental evaluations are on intuitive environments, the large body of work examining reward shaping and credit assignment in MARL showcase verification on complex multi-agent domains such as StarCraft. Is there a scope for evaluating LOPT in these domains in order to provide a more thorough comparison? \n\n(-) The readability of Fig. 5 and Fig. 6 can be significantly improved. In the current version, it is impossible to see the legends without zooming in and further, the shaded regions showing variance are difficult to discern.", "was_revised": true, "final_justification": "I thank the authors for their detailed response. The comparison of the proposed LOPT with PBRS was especially insightful and offered a perspective that I had not considered. I also appreciate the presenting of additional experiment results demonstrating effectiveness of LOPT. For these reasons, I will increase my original score.", "rebuttal": "We sincerely thank the reviewer for their detailed feedback and insightful questions. These comments are invaluable for improving our paper. We address the specific points below.\n\n---\n\n### To Q1: On the Distinction Between LOPT and Potential-Based Reward Shaping (PBRS)\n\nWe thank the reviewer for this crucial question. We acknowledge that this distinction was not made sufficiently clear in the current manuscript, and we will significantly revise the **Introduction** and **Related Work** sections to clarify this point in the final version.\n\nThe **fundamental difference** lies in the **objective**. PBRS and its variants are designed to **accelerate the learning process** while guaranteeing that the **optimal policy of the shaped problem remains identical** to that of the original environment (i.e., **policy invariance**). This property is highly desirable for speeding up convergence in standard RL tasks.\n\nHowever, this very guarantee makes PBRS **fundamentally unsuitable** for resolving **social dilemmas**. The core of a social dilemma is that the optimal policy for **self-interested agents** (i.e., maximizing individual rewards) leads to a **socially suboptimal outcome**. The problem is not inefficient convergence to the optimal policy — the problem is that the **original optimal policy is itself socially undesirable**. Preserving it, as PBRS does, fails to resolve the dilemma.\n\nIn contrast, **LOPT is not bound by policy invariance**. Our goal is to **transform the learning problem** itself to find a new, **socially optimal equilibrium**. LOPT achieves this by adding a **learnable reward function (the tax)** designed to bridge the gap between individual agent returns and global social welfare. We formally define this gap as the **externality**. This leads directly to the theoretical guarantees discussed in our response to Q2.\n\n---\n\n### To Q2: On Theoretical Guarantees Analogous to Policy Invariance\n\nThis is an excellent follow-up question. As discussed above, LOPT intentionally forgoes the **policy invariance** guarantee of PBRS because our goal is to **change the suboptimal equilibrium**, not preserve it.\n\nInstead, LOPT provides a different, more relevant theoretical guarantee for **social dilemmas**: we guarantee that the **learned reward shaping (the tax)** creates a new environment where **agents acting in their own self-interest will converge to a policy that is optimal for collective social welfare**.\n\nOur theoretical results directly support this claim:\n\n- **Theorem 1** establishes the **existence** of an optimal tax function. It proves that there is a tax that can perfectly **internalize the externality**, making the sum of individual rewards in the modified environment **equal to the true social welfare**.\n- **Theorem 2** establishes the **learnability** of this tax, demonstrating that it can be **practically found through our proposed framework**.\n\nIn summary:  \nWhile PBRS guarantees **convergence to the original optimal policy**, LOPT provides a guarantee that our method can find a **reward transformation** that makes the **socially optimal policy the new equilibrium** for self-interested agents.\n\n---\n\n### To Q3: On Evaluation in More Complex Domains Like StarCraft\n\nWe appreciate the reviewer’s suggestion to test LOPT’s scalability and performance in more complex environments.\n\nOur primary motivation for this work was to address **social dilemmas**, which are characterized by a **conflict between individual and collective interests**. Environments like the **StarCraft Multi-Agent Challenge (SMAC)** are primarily **fully cooperative**, meaning agents' interests are aligned. Therefore, SMAC is **not the most suitable benchmark** for evaluating a method designed to resolve social dilemmas.\n\nHowever, to address the reviewer’s valid concern about **complexity**, we have conducted **new experiments** on the **Harvest** environment, also a widely used environment for social dilemmas. We compare LOPT against strong baselines, **MOA** and **AC**, with **N = 5** agents. After **1.5M training steps**, our results show:\n\n- **LOPT** achieves a stable **social welfare** of approximately **700**.\n- **MOA** achieves a lower social welfare of around **580**.\n- **AC** is highly unstable and only reaches about **550**.\n\nThese results demonstrate that **LOPT scales effectively** and significantly **outperforms relevant baselines** in a more complex social dilemma setting.  \nDue to the **rebuttal format limitations**, we cannot include the full learning curves here, but we will add a **new appendix section** with these complete experimental results and analysis in the final version of the paper.\n\n**Thank you once again for your valuable and constructive feedback.**"}]

Author rebuttal:

Meta-review: The paper introduces a novel approach to resolving social dilemmas in MARL, grounded in economic theory and backed by theoretical guarantees.  This work will be of high interest to the MARL, game theory, and AI-for-society communities.There were concerns on clarity and the somewhat narrow scope of the experiments. The authors provide strong rebuttal responses, new results, and clear commitments to improve the final version. I recommend acceptance, conditional on revisions for clarity, expanded related work coverage, and inclusion of the new experimental results.
Decision: Accept (poster)