  Discounting the influence of future events is a key paradigm in economics and
it is widely used in computer-science models, such as games, Markov decision
processes (MDPs), reinforcement learning, and automata. While a single game or
MDP may allow for several different discount factors, discounted-sum automata
(NDAs) were only studied with respect to a single discount factor. For every
integer $\lambda\in\mathbb{N}\setminus\{0,1\}$, as opposed to every $\lambda\in
\mathbb{Q}\setminus\mathbb{N}$, the class of NDAs with discount factor
$\lambda$ ($\lambda$-NDAs) has good computational properties: it is closed
under determinization and under the algebraic operations min, max, addition,
and subtraction, and there are algorithms for its basic decision problems, such
as automata equivalence and containment.
  We define and analyze discounted-sum automata in which each transition can
have a different integral discount factor (integral NMDAs). We show that
integral NMDAs with an arbitrary choice of discount factors are not closed
under determinization and under algebraic operations and that their containment
problem is undecidable. We then define and analyze a restricted class of
integral NMDAs, which we call tidy NMDAs, in which the choice of discount
factors depends on the prefix of the word read so far. Some of their special
cases are NMDAs that correlate discount factors to actions (alphabet letters)
or to the elapsed time. We show that for every function $\theta$ that defines
the choice of discount factors, the class of $\theta$-NMDAs enjoys all of the
above good properties of integral NDAs, as well as the same complexity of the
required decision problems. Tidy NMDAs are also as expressive as deterministic
integral NMDAs with an arbitrary choice of discount factors.
  All of our results hold for both automata on finite words and automata on
infinite words.
