Principal-Agent Bandit Games with Self-Interested and Exploratory Learning Agents

We study the repeated principal-agent bandit game, where the principal indirectly interacts with the unknown environment by proposing incentives for the agent to play arms. Most existing work assumes the agent has full knowledge of the reward means and always behaves greedily, but in many online marketplaces, the agent needs to learn the unknown environment and sometimes explore. Motivated by such settings, we model a self-interested learning agent with exploration behaviors who iteratively updates reward estimates and either selects an arm that maximizes the estimated reward plus incentive or explores arbitrarily with a certain probability. As a warm-up, we first consider a self-interested learning agent without exploration. We propose algorithms for both i.i.d. and linear reward settings with bandit feedback in a finite horizon $T$, achieving regret bounds of $\widetilde{O}(\sqrt{T})$ and $\widetilde{O}( T^{2/3} )$, respectively. Specifically, these algorithms are established upon a novel elimination framework coupled with newly-developed search algorithms which accommodate the uncertainty arising from the learning behavior of the agent. We then extend the framework to handle the exploratory learning agent and develop an algorithm to achieve a $\widetilde{O}(T^{2/3})$ regret bound in i.i.d. reward setup by enhancing the robustness of our elimination framework to the potential agent exploration. Finally, when reducing our agent behaviors to the one studied in (Dogan et al., 2023a), we propose an algorithm based on our robust framework, which achieves a $\widetilde{O}(\sqrt{T})$ regret bound, significantly improving upon their $\widetilde{O}(T^{11/12})$ bound.