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Nash Equilibrium (NE) is the canonical solution concept of game theory, which provides an elegant tool to understand the rationalities. Though mixed strategy NE exists in any game with finite players and actions, computing NE in two- or multi-player general-sum games is PPAD-Complete. Various alternative solutions, e.g., Correlated Equilibrium (CE), and learning methods, e.g., fictitious play (FP), are proposed to approximate NE. For convenience, we call these methods as ``inexact solvers'', or ``solvers'' for short. However, the alternative solutions differ from NE and the learning methods generally fail to converge to NE. Therefore, in this work, we propose REinforcement Nash Equilibrium Solver (RENES), which trains a single policy to modify the games with different sizes and applies the solvers on the modified games where the obtained solution is evaluated on the original games. Specifically, our contributions are threefold. i) We represent the games as alpha-rank response graphs and leverage graph neural network (GNN) to handle the games with different sizes as inputs; ii) We use tensor decomposition, e.g., canonical polyadic (CP), to make the dimension of modifying actions fixed for games with different sizes; iii) We train the modifying strategy for games with the widely-used proximal policy optimization (PPO) and apply the solvers to solve the modified games, where the obtained solution is evaluated on original games. Extensive experiments on large-scale normal-form games show that our method can further improve the approximation of NE of different solvers, i.e., alpha-rank, CE, FP and PRD, and can be generalized to unseen games.
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Optimistic Policy Gradient in Multi-Player Markov Games with a Single Controller: Convergence Beyond the Minty Property
Policy gradient methods enjoy strong practical performance in numerous tasks in reinforcement learning. Their theoretical understanding in multiagent settings, however, remains limited, especially beyond two-player competitive and potential Markov games. In this paper, we develop a new framework to characterize optimistic policy gradient methods in multi-player Markov games with a single controller. Specifically, under the further assumption that the game exhibits an equilibrium collapse, in that the marginals of coarse correlated equilibria (CCE) induce Nash equilibria (NE), we show convergence to stationary ϵ-NE in O(1/ϵ2) iterations, where O(⋅) suppresses polynomial factors in the natural parameters of the game. Such an equilibrium collapse is well-known to manifest itself in two-player zero-sum Markov games, but also occurs even in a class of multi-player Markov games with separable interactions, as established by recent work. As a result, we bypass known complexity barriers for computing stationary NE when either of our assumptions fails. Our approach relies on a natural generalization of the classical Minty property that we introduce, which we anticipate to have further applications beyond Markov games.
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- Award ID(s):
- 1901403
- PAR ID:
- 10549996
- Publisher / Repository:
- AAAI24
- Date Published:
- Format(s):
- Medium: X
- Location:
- Vancouver, Canada
- Sponsoring Org:
- National Science Foundation
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