Abstract argumentation theory is based on the notion that argument x attacks argument y. A set of arguments S is “admissible” if no arguments within the set attack each other, and for every argument y that attacks an argument from set S, there is a counterargument, that is, an argument z from set S that attacks y. Arguments from an admissible set are called “credulously acceptable.”

In practice, such a set is formed by a dispute process: person A selects an argument, person B selects an argument that counters (“attacks”) this argument, then A selects an argument that counters this argument, and so on. The main constraint is that once B uses an argument, A cannot use it; it is as if B poisons the argument by using it, hence why this is called a “poison game.” Person A wins the dispute if s/he can find a counterargument for each argument of B. This poison game has a winning strategy for A if and only if there exists an admissible set of arguments.

The authors show that the existence of a winning strategy can be described in a special modal logic with two possibility operators--crudely speaking, corresponding to what is possible for A and what is possible for B. The authors analyze the properties of this poison modal logic (PML). They conjecture that this logic is undecidable and, so far, prove undecidability for PML with three pairs of possibility operators.