  This paper studies dynamic mechanism design in a quasilinear Markovian
environment and analyzes a direct mechanism model of a principal-agent
framework in which the agent is allowed to exit at any period. We consider that
the agent's private information, referred to as state, evolves over time. The
agent makes decisions of whether to stop or continue and what to report at each
period. The principal, on the other hand, chooses decision rules consisting of
an allocation rule and a set of payment rules to maximize her ex-ante expected
payoff. In order to influence the agent's stopping decision, one of the
terminal payment rules is posted-price, i.e., it depends only on the realized
stopping time of the agent. We define the incentive compatibility in this
dynamic environment in terms of Bellman equations, which is then simplified by
establishing a one-shot deviation principle. Given the optimality of the
stopping rule, a sufficient condition for incentive compatibility is obtained
by constructing the state-dependent payment rules in terms of a set of
functions parameterized by the allocation rule. A necessary condition is
derived from envelope theorem, which explicitly formulates the state-dependent
payment rules in terms of allocation rules. A class of monotone environment is
considered to characterize the optimal stopping by a threshold rule. The
posted-price payment rules are then pinned down in terms of the allocation rule
and the threshold function up to a constant. The incentive compatibility
constraints restrict the design of the posted-price payment rule by a regular
condition.
