Representation Theorems for Actual and Alpha Powers over General Concurrent Game Frames without Assuming Independence of Agents

Jul 16, 2026

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Concurrent game frames provide a semantic basis for logics of strategic reasoning and support two coalition-power notions: alpha powers and actual powers. Alpha powers capture sets of futures that a coalition can force using a suitable action, while actual powers require both forcing into a set and compatibility of every future in that set with the coalition’s action. Extending recent generalizations that vary seriality, agent independence, and determinism, the paper proves representation results for these powers via corresponding neighborhood frames, matching four key frame classes.

LanguageEnglish

CategoryResearch & Report

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Pages50

Reading Time2 h 6 min

Introduction

Concurrent game frames, alpha and actual powers, and alpha and actual neighborhood frames

Alpha powers versus actual powers

arXiv :2607 . 10567v 1 [ cs .GT] 12 Jul 2026 Representation theorems for actual and alpha powers over general concurrent game frames without assuming independence of agents Zixuan Chen 1 ,2 , Fengkui Ju3 ,4 ∗, and Thomas Ågotnes5 ,6 ,7 1 Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam The Netherlands [2](2 zixuan.chen21@outlook.com)[ zixuan.chen21@outlook.com](2 zixuan.chen21@outlook.com) 3 School of Philosophy, Beijing Normal University, Beijing, China [4](4 fengkui.ju@bnu.edu.cn)[ fengkui.ju@bnu.edu.cn](4 fengkui.ju@bnu.edu.cn) 5 Department of Information Science and Media Studies, University of Bergen, Bergen, Norway 6 School of Philosophy, Shanxi University, Taiyuan, China [7](7 thomas.agotnes@uib.no)[ thomas.agotnes@uib.no](7 thomas.agotnes@uib.no) 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. Recent generalizations of concurrent game frames separate three structural assumptions built into the standard model: seriality, independence of agents, and determinism. This yields eight classes of general concurrent game frames. In this paper, we prove that for actual powers, the four classes of general concurrent game frames, where independence of agents is not assumed, are representable by four corresponding classes of neighborhood frames. Building on this result, we show that for alpha powers, the same four classes of general concurrent game frames are likewise representable by four corresponding classes of neighborhood frames. 1 Introduction In this section, we begin by recalling concurrent game frames, alpha and actual powers, and neighborhood frames. We then explain the difference between alpha powers and actual powers, formulate the neighborhood-frame representation problem for both notions of power, and discuss the significance of such representation results. After that, we introduce the eight classes of general concurrent game frames obtained by varying seriality, independence of agents, and determinism. The section closes with an outline of the main results of the paper. Much of the background material in this section is drawn from [ CJÅ26]; it is recalled here to make the present paper self-contained. ∗ Corresponding author. 1.1 Concurrent game frames, alpha and actual powers, and alpha and actual neighborhood frames Concurrent game frames are a standard semantic framework for logics of strategic reasoning. Many influential logics in this area are interpreted over concurrent game frames, including Coalition Logic CL [Pau02] and Alternating-time Temporal Logic ATL [AHK02] . Roughly speaking, a concurrent game frame consists of states and agents (forming coalitions), where, at each state, every coalition has available joint actions, and each such action induces a set of possible outcome states. Two notions of coalition power arise from concurrent game 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 every resulting outcome to lie in that set. An actual power of a coalition is a set of possible futures such that the coalition has an action satisfying both of the following conditions: (1) the action forces every resulting outcome to lie in the set, and (2) every future in the set is realizable by the complement coalition (and the environment) . Thus, unlike alpha powers, actual powers explicitly invol

FAQs

What is the difference between alpha powers and actual powers in concurrent game frames?

Alpha powers record guarantees: an action whose outcomes all lie in a set also guarantees any superset. Actual powers record the precise outcome set generated by an action, and require that every future in the set is compatible with the coalition’s action.

How do the paper’s representation theorems relate actual/alpha powers to neighborhood frames?

For the four general concurrent game frame classes where independence of agents is not assumed, actual powers are representable by corresponding classes of neighborhood frames. The same four classes are also representable for alpha powers by corresponding neighborhood frames.

Which structural assumptions are varied to obtain the general concurrent game frame classes studied in the paper?

The framework varies seriality, independence of agents, and determinism, yielding eight classes of general concurrent game frames used to derive the representation results.

Jul 16, 2026

Upload by Miles