  For a discrete time Markov chain and in line with Strotz' consistent planning
we develop a framework for problems of optimal stopping that are
time-inconsistent due to the consideration of a non-linear function of an
expected reward. We consider pure and mixed stopping strategies and a (subgame
perfect Nash) equilibrium. We provide different necessary and sufficient
equilibrium conditions including a verification theorem. Using a fixed point
argument we provide equilibrium existence results. We adapt and study the
notion of the myopic adjustment process and introduce different kinds of
equilibrium stability. We show that neither existence nor uniqueness of
equilibria should generally be expected. The developed theory is applied to a
mean-variance problem and a variance problem.
