Reinforcement Learning Approach to Improve Nash Equilibrium Approximation in Multi-Player General-Sum Games

Core Concepts

RENES leverages reinforcement learning to train a single policy that can modify games with different sizes, and then applies existing solvers to the modified games to obtain better approximations of Nash Equilibrium.

Abstract

The paper proposes a novel method called REinforcement Nash Equilibrium Solver (RENES) to improve the approximation of Nash Equilibrium (NE) in multi-player general-sum games. The key ideas are:

Representing the games as α-rank response graphs to handle games of different sizes, and using graph neural networks to process these graphs.
Using tensor decomposition to make the action space of the modification policy fixed-sized, regardless of the game size.
Training the modification policy using proximal policy optimization (PPO), where the policy modifies the original game, and the obtained solution from applying existing solvers (e.g., α-rank, correlated equilibrium, fictitious play, projected replicator dynamics) on the modified game is evaluated on the original game.

Extensive experiments on large-scale normal-form games show that RENES can significantly improve the approximation performance of various existing solvers, and the trained modification policy can generalize to unseen games. This is the first work that leverages reinforcement learning to train a single policy for modifying games to boost the performance of different NE solvers.

Stats

The paper reports the following key metrics:

NashConv values of the solutions obtained by applying different solvers (α-rank, CE, FP, PRD) on the original games.
Relative improvement in NashConv values after applying RENES to modify the games, compared to the original solvers.

Quotes

"RENES can significantly boost the performance of α-rank, i.e., larger than 0.3 over all three seeds, as shown in Figure 4a, and achieve 0.313 and 0.324 on training and testing sets, respectively."
"For the other solvers, RENES can bring notable improvements, i.e., larger than 0.16 for CE and 0.1 for FP over three seeds."

Key Insights Distilled From

Reinforcement Nash Equilibrium Solver

by Xinrun Wang,... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03518.pdf

Deeper Inquiries

How can RENES be extended to handle extensive-form games, which have imperfect information and sequential structure

To extend RENES to handle extensive-form games, which involve imperfect information and a sequential structure, several modifications and enhancements would be necessary.

Representation: Extensive-form games require a different representation compared to normal-form games. RENES would need to incorporate a representation that captures the sequential nature of the game, including information sets and possible actions at each decision point.

Policy Learning: The modification policy learned by RENES would need to account for the sequential decision-making process in extensive-form games. This would involve learning a policy that can modify the game tree structure, potentially by altering payoffs, information sets, or decision nodes.

Sequential Decision Making: RENES would need to learn a policy that can make sequential modifications to the game tree, considering the information available at each decision point. This would involve incorporating reinforcement learning techniques that can handle sequential decision-making processes.

Imperfect Information: Handling imperfect information in extensive-form games would require the modification policy to make decisions based on the available information at each information set. Techniques like deep reinforcement learning could be employed to learn a policy that can effectively modify the game under imperfect information scenarios.

By incorporating these enhancements, RENES could be extended to handle extensive-form games, allowing for the modification of game structures to improve the quality of solutions in games with imperfect information and a sequential structure.

Can the modification policy learned by RENES be used to guide the design of new NE solvers, rather than just improving existing ones

The modification policy learned by RENES can indeed be used to guide the design of new NE solvers, providing insights into how games can be modified to enhance the performance of existing solvers or develop new solution approaches. Here are some ways in which the modification policy can influence the design of new NE solvers:

Insights into Game Structure: The modification policy can reveal patterns in how games can be altered to improve solution quality. These insights can guide the development of new algorithms that directly incorporate these modifications into the solution process.

Feature Engineering: The modifications suggested by the policy can serve as features for new NE solvers. By incorporating these modifications as input features, new solvers can adapt their strategies based on the suggested changes to the game structure.

Hybrid Approaches: The modification policy can be used in conjunction with existing solver algorithms to create hybrid approaches. By combining the insights from the policy with the strengths of traditional solvers, new hybrid algorithms can be developed that leverage the best of both worlds.

Transfer Learning: The learned modification policy can be transferred to new games or domains, providing a starting point for developing tailored NE solvers for specific contexts. This transfer learning approach can accelerate the development of solvers for diverse game settings.

By leveraging the insights and recommendations of the modification policy, researchers can innovate and create novel NE solvers that are informed by the learned strategies for modifying game structures.

What other applications beyond game theory can the core idea of RENES (modifying the problem to improve solution quality) be applied to

The core idea of RENES, which involves modifying the problem to improve solution quality, can be applied to various other domains beyond game theory. Some potential applications include:

Optimization Problems: RENES can be used to enhance optimization algorithms by modifying problem instances to make them more amenable to existing solvers. This approach can improve the efficiency and effectiveness of optimization techniques in various domains.

Machine Learning: In the field of machine learning, RENES can be applied to enhance model training processes by modifying datasets or hyperparameters to improve model performance. This adaptive modification approach can lead to better model outcomes and generalization.

Supply Chain Management: RENES can optimize supply chain operations by modifying parameters such as inventory levels, production schedules, or distribution strategies to improve overall efficiency and cost-effectiveness.

Financial Modeling: In finance, RENES can be used to adjust input parameters in financial models to enhance predictions and risk management strategies. By modifying key variables, the accuracy and robustness of financial models can be improved.

Healthcare Planning: RENES can assist in healthcare planning by modifying resource allocation strategies, patient scheduling, or treatment protocols to optimize healthcare delivery and patient outcomes.

By applying the core concept of RENES to these diverse domains, practitioners can enhance decision-making processes, optimize resource utilization, and improve overall system performance.