  In dynamic mechanism design literature, one critical aspect has been
typically ignored-the agents' periodic participation, which they can adapt and
plan strategically. We propose a framework for dynamic principal-multiagent
problems, augmenting the classic model by incorporating agents' periodic
coupled decisions on participation and regular action selections. The principal
faces adverse selection and designs a mechanism comprising a task policy
profile (defining evolving agent action menus), a coupling policy profile
(affecting agent utilities), and an off-switch function profile (assigning
rewards or penalties upon agent withdrawal). Firstly, we introduce payoff-flow
conservation-a sufficient condition to ensure dynamic incentive compatibility
for regular actions. Secondly, we formulate a unique process, persistence
transformation, which integrates task policy's implicit functions, enabling a
closed-form off-switch function derivation, hence securing sufficient
conditions for agents' coupled decisions' incentive compatibility, aligning
with the principal's preferences. Thirdly, we go beyond the traditional
envelope theorem by presenting a necessary condition for incentive
compatibility, leveraging the coupled optimality of principal-desired actions.
This approach helps explicitly formulate both the coupling and off-switch
functions. Finally, we establish envelope-like conditions exclusively on the
task policies, facilitating the application of the first-order approach.
