Energy games are a well-studied class of 2-player turn-based games on a
finite graph where transitions are labeled with integer vectors which represent
changes in a multidimensional resource (the energy). One player tries to keep
the cumulative changes non-negative in every component while the other tries to
frustrate this. We consider generalized energy games played on infinite game
graphs induced by pushdown automata (modelling recursion) or their subclass of
one-counter automata. Our main result is that energy games are decidable in the
case where the game graph is induced by a one-counter automaton and the energy
is one-dimensional. On the other hand, every further generalization is
undecidable: Energy games on one-counter automata with a 2-dimensional energy
are undecidable, and energy games on pushdown automata are undecidable even if
the energy is one-dimensional. Furthermore, we show that energy games and
simulation games are inter-reducible, and thus we additionally obtain several
new (un)decidability results for the problem of checking simulation preorder
between pushdown automata and vector addition systems.Comment: 11 page
