Representation theorems for actual and alpha powers over two-agent general concurrent game frames

Abstract: Concurrent game frames are a standard semantic framework for logics of strategic reasoning. Two notions of coalition power can be derived from such frames: alpha powers and actual powers. An alpha power of a coalition is a set of possible futures such that the coalition has an action that forces the resulting future to lie in that set. An actual power of a coalition is a set of possible futures satisfying the following condition: the coalition has an action such that (1) the action forces the resulting future to lie in the set, and (2) every future in the set is compatible with that action. In two papers, Li and Ju argued that standard concurrent game frames rely on three assumptions that may be too strong: seriality, independence of agents, and determinism. They therefore considered eight classes of general concurrent game frames, determined by which of these three properties hold, and studied the corresponding coalition logics. In this paper, assuming two agents, we prove that for actual powers, the eight classes of general concurrent game frames are representable by eight corresponding classes of neighborhood frames. Building on this result, we show that for alpha powers, the same eight classes of general concurrent game frames are likewise representable by eight corresponding classes of neighborhood frames.