  Canonical models of Markov decision processes (MDPs) usually consider
geometric discounting based on a constant discount factor. While this standard
modeling approach has led to many elegant results, some recent studies indicate
the necessity of modeling time-varying discounting in certain applications.
This paper studies a model of infinite-horizon MDPs with time-varying discount
factors. We take a game-theoretic perspective -- whereby each time step is
treated as an independent decision maker with their own (fixed) discount factor
-- and we study the subgame perfect equilibrium (SPE) of the resulting game as
well as the related algorithmic problems. We present a constructive proof of
the existence of an SPE and demonstrate the EXPTIME-hardness of computing an
SPE. We also turn to the approximate notion of $\epsilon$-SPE and show that an
$\epsilon$-SPE exists under milder assumptions. An algorithm is presented to
compute an $\epsilon$-SPE, of which an upper bound of the time complexity, as a
function of the convergence property of the time-varying discount factor, is
provided.
