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.