Lately, the problem of designing multi-stage dynamic mechanisms has been shown to be both theoretically challenging and practically important. In this paper, we consider the problem of designing revenue optimal dynamic mechanism for a setting where an auctioneer sells a set of items to a buyer in multiple stages. At each stage, there could be multiple items for sale but each item can only appear in one stage. The type of the buyer at each stage is thus a multi-dimensional vector characterizing the buyer's valuations of the items at that stage and is assumed to be stage-wise independent. Compared with the single-shot multi-dimensional revenue maximization, designing optimal dynamic mechanisms for this setting brings additional difficulties in that mechanisms can now branch on different realizations of previous random events and secure extra revenue. In particular, we propose a novel class of mechanisms called bank account mechanisms. Roughly, a bank account mechanism is no different from any stage-wise individual mechanism except for an augmented structure called bank account, a real number for each node that summarizes the history so far. This structure allows the auctioneer to store deficits or surpluses of buyer utility from each individual mechanism and even them out on the long run. While such simplicity allows for easy implementation in practice, it turns out it is also sufficient to guarantee optimal revenue. We first establish that the optimal revenue from any dynamic mechanism in this setting can be achieved by a bank account mechanism, and we provide a simple characterization of the set of incentive compatible and ex-post individually rational bank account mechanisms. Based on these characterizations, we then investigate the problem of finding the (approximately) optimal bank account mechanisms. We prove that there exists a simple, randomized bank account mechanism that approximates optimal revenue up to a constant factor. Our result is general and can accommodate previous approximation results in single-shot multi-dimensional mechanism design. Based on the previous mechanism, we further show that there exists a deterministic bank account mechanism that achieves constant-factor approximation as well. Finally, we consider the problem of computing optimal mechanisms when the type space is discrete and provide an FPTAS via linear and dynamic programming.
Mirrokni et al. REF study revenue maximization for bank account mechanisms subject to an ex-post IR constraint.
Optimal dynamic mechanisms with ex-post IR via bank accounts
ArXiv
ArXiv
Mathematics Computer Science