Online Learning for Dynamic Vickrey-Clarke-Groves Mechanism in Sequential Auctions under Unknown Environments

We consider the problem of online dynamic mechanism design for sequential auctions in unknown environments, where the underlying market and, thus, the bidders' values vary over time as interactions between the seller and the bidders progress. We model the sequential auctions as an infinite-horizon average-reward Markov decision process (MDP), where the transition kernel and reward functions are unknown to the seller. In each round, the seller determines an allocation and a payment for each bidder. Each bidder receives a private reward and submits a sealed bid to the seller. The state, which represents the underlying market, evolves according to an unknown transition kernel and the seller's allocation policy. Unlike existing works that formulate the problem as a multi-armed bandit model or as an episodic MDP, where the environment resets to an initial state after each round or episode, our paper considers a more realistic and sophisticated setting in which the market continues to evolve without restarting. We first extend the Vickrey-Clarke-Groves (VCG) mechanism, which is known to be efficient, truthful, and individually rational for one-shot static auctions, to sequential auctions, thereby obtaining a dynamic VCG mechanism counterpart that preserves these desired properties. We then focus on the online setting and develop an online reinforcement learning algorithm for the seller to learn the underlying MDP model and implement a mechanism that closely resembles the dynamic VCG mechanism. We show that the learned online mechanism asymptotically converges to a dynamic mechanism that approximately satisfies efficiency, truthfulness, and individual rationality with arbitrarily high probability and achieves guaranteed performance in terms of various notions of regret.