  We study an exit contract design problem, where one provides a universal exit
contract to multiple heterogeneous agents, with which each agent chooses an
optimal (exit) stopping time. The problem consists in optimizing the universal
exit contract w.r.t. some criterion depending on the contract as well as the
agents' exit times. Under a technical monotonicity condition, and by using
Bank-El Karoui's representation of stochastic processes, we are able to
transform the initial contract optimization problem into an optimal control
problem. The latter is also equivalent to an optimal multiple stopping problem
and the existence of the optimal contract is proved. We next show that the
problem in the continuous-time setting can be approximated by a sequence of
discrete-time ones, which would induce a natural numerical approximation
method. We finally discuss the optimaization problem over the class of all
Markovian and/or continuous exit contracts.
