Mechanism design with limited commitment: Markov environments

We provide a revelation principle for a class of single-agent dynamic mechanism design settings in which the agent's private information evolves stochastically over time and the designer can only commit to short-term mechanisms. We restrict attention to Markov environments, in which (i) the agent's type in period t+1 depends only on her period-t type and the period-t allocation, (ii) the designer's and the agent's payoffs are time-separable, and (iii) their period-t payoffs depend only on period-t type and the period-t allocation. We show all equilibrium payoffs can be attained with the designer using flow direct Blackwell mechanisms, which consist of a mapping from the agent's current type report to posterior beliefs about the current type, and a mapping from these beliefs to allocations. Furthermore, all equilibrium payoffs can be attained with strategies in which the agent participates and truthfully reports her type, and the beliefs that result from the mechanism correspond to the designer's equilibrium beliefs. This result greatly simplifies the search of optimal dynamic and sequentially rational mechanisms in dynamic mechanism design problems, which include dynamic Mirrlees and social insurance models.