# Weighted QMIX: Expanding Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning

arXiv: https://arxiv.org/abs/2006.10800 · HTML: https://mlsanity.com/paper/2006.10800/

## Key facts

- Cited by 197 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2020.

## Built on by (most cited first, 30 of 197)

- [The Surprising Effectiveness of PPO in Cooperative, Multi-Agent Games](https://mlsanity.com/paper/2103.01955/) (2021)
- [SMACv2: An Improved Benchmark for Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2212.07489/) (2022)
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- [Rethinking the Implementation Tricks and Monotonicity Constraint in Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2102.03479/) (2021)
- [LLM-based Multi-Agent Reinforcement Learning: Current and Future Directions](https://mlsanity.com/paper/2405.11106/) (2024)
- [Cooperative Exploration for Multi-Agent Deep Reinforcement Learning](https://mlsanity.com/paper/2107.11444/) (2021)
- [Contrasting Centralized and Decentralized Critics in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2102.04402/) (2021)
- [An Introduction to Centralized Training for Decentralized Execution in Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2409.03052/) (2024)
- [Episodic Multi-agent Reinforcement Learning with Curiosity-Driven Exploration](https://mlsanity.com/paper/2111.11032/) (2021)
- [Towards a Standardised Performance Evaluation Protocol for Cooperative MARL](https://mlsanity.com/paper/2209.10485/) (2022)
- [Deep Multiagent Reinforcement Learning: Challenges and Directions](https://mlsanity.com/paper/2106.15691/) (2021)
- [Counterfactual Conservative Q Learning for Offline Multi-agent Reinforcement Learning](https://mlsanity.com/paper/2309.12696/) (2023)
- [Applications of Multi-Agent Reinforcement Learning in Future Internet: A Comprehensive Survey](https://mlsanity.com/paper/2110.13484/) (2021)
- [On the Approximation of Cooperative Heterogeneous Multi-Agent Reinforcement Learning (MARL) using Mean Field Control (MFC)](https://mlsanity.com/paper/2109.04024/) (2021)
- [Revisiting Some Common Practices in Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2206.07505/) (2022)
- [RMIX: Learning Risk-Sensitive Policies for Cooperative Reinforcement Learning Agents](https://mlsanity.com/paper/2102.08159/) (2021)
- [Context-Aware Sparse Deep Coordination Graphs](https://mlsanity.com/paper/2106.02886/) (2021)
- [SHAQ: Incorporating Shapley Value Theory into Multi-Agent Q-Learning](https://mlsanity.com/paper/2105.15013/) (2021)
- [Unbiased Asymmetric Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2105.11674/) (2021)
- [What is the Solution for State-Adversarial Multi-Agent Reinforcement Learning?](https://mlsanity.com/paper/2212.02705/) (2022)
- [Measuring Acoustics with Collaborative Multiple Agents](https://mlsanity.com/paper/2310.05368/) (2023)
- [MAC-PO: Multi-Agent Experience Replay via Collective Priority Optimization](https://mlsanity.com/paper/2302.10418/) (2023)
- [RiskQ: Risk-sensitive Multi-Agent Reinforcement Learning Value Factorization](https://mlsanity.com/paper/2311.01753/) (2023)
- [Regularized Softmax Deep Multi-Agent Q-Learning](https://mlsanity.com/paper/2103.11883/) (2021)
- [HAVEN: Hierarchical Cooperative Multi-Agent Reinforcement Learning with Dual Coordination Mechanism](https://mlsanity.com/paper/2110.07246/) (2021)
- [More Centralized Training, Still Decentralized Execution: Multi-Agent Conditional Policy Factorization](https://mlsanity.com/paper/2209.12681/) (2022)
- [A Survey on Reinforcement Learning in Aviation Applications](https://mlsanity.com/paper/2211.02147/) (2022)
- [Contrastive Identity-Aware Learning for Multi-Agent Value Decomposition](https://mlsanity.com/paper/2211.12712/) (2022)
- [Is Centralized Training with Decentralized Execution Framework Centralized Enough for MARL?](https://mlsanity.com/paper/2305.17352/) (2023)
- [Survey of Recent Multi-Agent Reinforcement Learning Algorithms Utilizing Centralized Training](https://mlsanity.com/paper/2107.14316/) (2021)

## Abstract (arXiv)

QMIX is a popular Q-learning algorithm for cooperative MARL in the centralised training and decentralised execution paradigm. In order to enable easy decentralisation, QMIX restricts the joint action Q-values it can represent to be a monotonic mixing of each agent's utilities. However, this restriction prevents it from representing value functions in which an agent's ordering over its actions can depend on other agents' actions. To analyse this representational limitation, we first formalise the objective QMIX optimises, which allows us to view QMIX as an operator that first computes the Q-learning targets and then projects them into the space representable by QMIX. This projection returns a representable Q-value that minimises the unweighted squared error across all joint actions. We show in particular that this projection can fail to recover the optimal policy even with access to Q^*, which primarily stems from the equal weighting placed on each joint action. We rectify this by introducing a weighting into the projection, in order to place more importance on the better joint actions. We propose two weighting schemes and prove that they recover the correct maximal action for any joint action Q-values, and therefore for Q^* as well. Based on our analysis and results in the tabular setting, we introduce two scalable versions of our algorithm, Centrally-Weighted (CW) QMIX and Optimistically-Weighted (OW) QMIX and demonstrate improved performance on both predator-prey and challenging multi-agent StarCraft benchmark tasks.