Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum. The majority of existing results in this field focuses on either symmetric solution concepts (e.g. Nash equilibrium) or zero-sum games. It remains vastly open how to learn the Stackelberg equilibrium -- an asymmetric analog of the Nash equilibrium -- in general-sum games efficiently from samples. This paper initiates the theoretical study of sample-efficient learning of the Stackelberg equilibrium in two-player turn-based general-sum games . We identify a fundamental gap between the exact value of the Stackelberg equilibrium and its estimated version using finite samples, which can not be closed information-theoretically regardless of the algorithm. We then establish a positive result on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above . We show that our sample complexity is tight with matching upper and lower bounds . Finally, we extend our learning results to the setting where the follower plays in a Markov Decision Process (MDP), and the setting where the leader and the follower act simultaneously .
Real world applications such as economics and policy making often involve solving multi-agent games with two unique features: (1) The agents are inherently asymmetric and partitioned into leaders and followers; (2) The agents have different reward functions, thus the game is general-sum. The majority of existing results in this field focuses on either symmetric solution concepts (e.g. Nash equilibrium) or zero-sum games. It remains vastly open how to learn the Stackelberg equilibrium -- an asymmetric analog of the Nash equilibrium -- in general-sum games efficiently from samples. This paper initiates the theoretical study of sample-efficient learning of the Stackelberg equilibrium , in the bandit feedback setting where we only observe noisy samples of the reward. We consider three representative two-player general-sum games : bandit games, bandit-reinforcement learning (bandit-RL) games, and linear bandit games. In all these games, we identify a fundamental gap between the exact value of the Stackelberg equilibrium and its estimated version using finitely many noisy samples, which can not be closed information-theoretically regardless of the algorithm. We then establish sharp positive results on sample-efficient learning of Stackelberg equilibrium with value optimal up to the gap identified above , with matching lower bounds in the dependency on the gap, error tolerance, and the size of the action spaces. Overall, our results unveil unique challenges in learning Stackelberg equilibria under noisy bandit feedback, which we hope could shed light on future research on this topic .
R
in
, in the bandit feedback setting where we only observe noisy samples of the reward. We consider three representative
D
turn-based
R
. We
: bandit games, bandit-reinforcement learning (bandit-RL) games, and linear bandit games. In all these games, we
R
finite
finitely many noisy
R
a positive result
sharp positive results
R
. We show that our sample complexity is tight with matching upper and lower bounds . Finally, we extend our learning results to the setting where the follower plays in a Markov Decision Process (MDP), and the setting where the leader and the follower act simultaneously
, with matching lower bounds in the dependency on the gap, error tolerance, and the size of the action spaces. Overall, our results unveil unique challenges in learning Stackelberg equilibria under noisy bandit feedback, which we hope could shed light on future research on this topic