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

TL;DR

This paper establishes representation theorems connecting two types of coalition powers in two-agent concurrent game frames with neighborhood frames, broadening the semantic understanding of strategic reasoning.

Contribution

It proves that for two-agent frames, both actual and alpha powers can be represented by neighborhood frames across eight classes of general concurrent game frames.

Findings

Representation theorems for actual powers

Representation theorems for alpha powers

Broader semantic framework for coalition logics

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.