  This paper studies the mean-field Markov decision process (MDP) with the
centralized stopping under the non-exponential discount. The problem differs
fundamentally from most existing studies on mean-field optimal control/stopping
due to its time inconsistency by nature. We look for the subgame perfect
relaxed equilibria, namely the randomized stopping policies that satisfy the
time-consistent planning with future selves from the perspective of the social
planner. On the other hand, unlike many previous studies on time-inconsistent
stopping where the decreasing impatience plays a key role, we are interested in
the general discount function without imposing any conditions. As a result, the
study on the relaxed equilibrium becomes necessary as the pure-strategy
equilibrium may not exist in general. We formulate relaxed equilibria as fixed
points of a complicated operator, whose existence is challenging by a direct
method. To overcome the obstacles, we first introduce the auxiliary problem
under the entropy regularization on the randomized policy and the discount
function, and establish the existence of the regularized equilibria as fixed
points to an auxiliary operator via Schauder fixed point theorem. Next, we show
that the regularized equilibrium converges as the regularization parameter
$\lambda$ tends to $0$ and the limit corresponds to a fixed point to the
original operator, and hence is a relaxed equilibrium. We also establish some
connections between the mean-field MDP and the N-agent MDP when $N$ is
sufficiently large in our time-inconsistent setting.
