We introduce a way to parameterize automata and games on finite graphs with natural numbers. The parameters are accessed essentially by allowing counting down from the parameter value to 0 and branching depending on whether 0 has been reached. The main technical result is that in games, a player can win for some values of the parameters at all, if she can win for some values below an exponential bound. For many winning conditions, this implies decidability of any statements about a player being able to win with arbitrary quantification over the parameter values. While the result seems broadly applicable, a specific motivation comes from the study of chains of strategies in games. Chains of games were recently suggested as a means to define a rationality notion based on dominance that works well with quantitative games by Bassett, Jecker, P., Raskin and Van den Boogard. From the main result of this paper, we obtain generalizations of their decidability results with much simpler proofs. As both a core technical notion in the proof of the main result, and as a notion of potential independent interest, we look at boolean functions defined via graph game forms. Graph game forms have properties akin to monotone circuits, albeit are more concise. We raise some open questions regarding how concise they are exactly, which have a flavour similar to circuit complexity. Answers to these questions could improve the bounds in the main theorem.
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