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arxiv: 2605.25545 · v1 · pith:FUS6PUGGnew · submitted 2026-05-25 · 💻 cs.LO

Value Coalition Logic: A Typed Assignment-Based Reconstruction of Coalition Logic

Shanxia Wang This is my paper

Pith reviewed 2026-06-29 19:52 UTC · model grok-4.3

classification 💻 cs.LO
keywords value coalition logiccoalition logictyped assignmentsvalue-coherence axiomsquotient game formseffectivity familiesstrategic value-range hypergraphshilbert axiomatisation
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The pith

Value Coalition Logic replaces propositional valuations with typed value assignments at states while preserving the standard strategic semantics of coalition logic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Value Coalition Logic redefines the atomic level of coalition logic so that each state carries a total assignment of values to variables drawn from a finite typed signature. Exhaustivity and mutual exclusion among values are therefore enforced directly by the state representation rather than added as external constraints. Over any fixed finite typed signature the new logic is shown to be truth-equivalent to ordinary coalition logic restricted to coherent valuations. The equivalence immediately supplies a sound and complete Hilbert-style axiomatisation obtained simply by adjoining the finite-domain value-coherence axioms to the usual coalition-logic axioms. The reconstruction also projects ordinary coalitional ability onto single value domains, producing quotient game forms, projected effectivity families and strategic value-range hypergraphs that expose value-level invariants invisible in flat propositional encodings.

Core claim

Value Coalition Logic keeps the standard one-step game-form clause for coalitional ability but replaces flat propositional valuations with total assignments of values to finitely typed variables. Because each state is already a complete, exhaustive and mutually exclusive assignment, the semantics builds value coherence in at the atomic level. For every fixed finite typed signature the resulting logic is truth-equivalent to propositional coalition logic evaluated only on coherent valuations. Adding the finite-domain value-coherence axioms to the standard coalition-logic axioms therefore yields a sound and complete Hilbert-style system. Projecting coalitional ability onto individual value doma

What carries the argument

Typed value assignments at states, which replace propositional valuations and enforce exhaustivity and mutual exclusion by construction while the strategic clause remains the standard one-step game-form interpretation.

If this is right

  • Over each fixed finite typed signature the two logics validate exactly the same formulas.
  • A sound and complete axiomatisation is obtained by adjoining the finite-domain value-coherence axioms to the standard coalition-logic axioms.
  • Projecting coalitional ability onto a single value domain produces quotient game forms and projected effectivity families.
  • These structures support set-valued strategic exclusion, transversal polarity for disjoint coalitions, and exact boundary duality between the empty and grand coalitions.
  • The logic remains conservative in its strategic modality while exposing value-level invariants such as residual value indeterminacy measured by strategic value-range hypergraphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The typed semantics could be used directly to model domains whose value constraints are already exhaustive and exclusive, avoiding the need to encode coherence as separate modal axioms.
  • The quotient structures may supply a uniform way to compute the effectivity of coalitions when outcomes are required to respect a typed partition of the state space.
  • Extending the construction beyond finite signatures would require additional compactness or finiteness conditions that are not addressed in the present equivalence proof.

Load-bearing premise

The interpretation of coalitional ability stays exactly the standard one-step game-form clause even after states are changed to carry total typed value assignments.

What would settle it

A finite typed signature together with a formula that is true under the typed-assignment semantics but false under the coherent propositional semantics (or vice versa) would falsify the claimed truth-equivalence.

read the original abstract

We introduce Value Coalition Logic, a typed assignment-based reconstruction of classical coalition logic. The strategic semantics is unchanged: coalitional ability is still interpreted by the standard one-step game-form clause. The change is at the atomic level. Instead of flat propositional valuations, states carry total assignments of values to finitely typed variables. As a result, exhaustivity and mutual exclusion of alternative values are built into the semantics, rather than imposed as external coherence constraints. We prove that, over each fixed finite typed signature, Value Coalition Logic is truth-equivalent to propositional coalition logic over coherent valuations. This correspondence yields a sound and complete Hilbert-style axiomatisation obtained by adding finite-domain value-coherence axioms to the standard axioms of coalition logic. The main contribution is structural. Projecting ordinary coalitional ability onto a single value domain yields quotient game forms, projected effectivity families, and strategic value-range hypergraphs. These structures support set-valued strategic exclusion, transversal polarity for disjoint coalitions, exact boundary duality between the empty and grand coalitions, and a measure of residual value indeterminacy. Thus the logic is conservative in its strategic modality, but exposes value-level invariants that are hidden in flat propositional encodings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper introduces Value Coalition Logic as a typed assignment-based variant of coalition logic. States are total assignments of values to finitely typed variables rather than flat propositional valuations, while the one-step game-form semantics for the coalitional ability modality [C] is left unchanged. The central claim is that, for any fixed finite typed signature, the logic is truth-equivalent to standard propositional coalition logic restricted to coherent valuations; this equivalence yields a sound and complete Hilbert axiomatization obtained simply by adjoining the finite-domain exhaustivity and mutual-exclusion axioms to the usual coalition-logic axioms. The paper further develops structural notions obtained by projecting coalitional ability onto single value domains: quotient game forms, projected effectivity families, strategic value-range hypergraphs, set-valued strategic exclusion, transversal polarity, exact boundary duality between empty and grand coalitions, and a measure of residual value indeterminacy.

Significance. If the correspondence holds, the work supplies a conservative reconstruction that makes value-level coherence an intrinsic semantic feature rather than an external constraint. The structural constructions (quotient game forms, value-range hypergraphs, boundary duality) are defined by projection and therefore inherit the effectivity relations of the original models; they expose invariants that remain hidden under flat propositional encodings. The paper ships an explicit bijection between total assignments and coherent valuations together with a direct transfer of soundness and completeness, both of which are load-bearing strengths.

minor comments (3)
  1. §3 (or wherever the quotient game form is defined): the notation for the projected effectivity family could be clarified by an explicit commutative diagram relating the original and quotient relations.
  2. The statement of the truth-equivalence theorem would benefit from an explicit sentence confirming that the one-step game-form clause is literally identical on both sides of the correspondence.
  3. A short table comparing the original coalition-logic axioms with the augmented Value Coalition Logic axioms would make the transfer of completeness immediately visible.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation establishes truth-equivalence by exhibiting an explicit bijection between total assignments over a fixed finite typed signature and coherent propositional valuations, with the one-step game-form clause for coalitional ability left literally unchanged. The added finite-domain coherence axioms are the standard exhaustivity and mutual-exclusion axioms that axiomatize precisely those models, so soundness and completeness transfer directly from the known completeness of coalition logic. All structural constructions (quotient game forms, projected effectivity families, value-range hypergraphs) are defined by projection and therefore inherit the same effectivity relations without introducing new assumptions. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain; the central claim is self-contained against the external benchmark of standard coalition logic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard coalition logic axioms plus newly added finite-domain value-coherence axioms; no free parameters or invented entities are described.

axioms (1)
  • domain assumption Finite-domain value-coherence axioms
    Added to the standard axioms of coalition logic to obtain completeness over coherent valuations.

pith-pipeline@v0.9.1-grok · 5730 in / 1300 out tokens · 26542 ms · 2026-06-29T19:52:37.670383+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 17 canonical work pages

  1. [1]

    Pauly (2002): A modal logic for coalitional power in games

    Marc Pauly. A modal logic for coalitional power in games.Journal of Logic and Computation, 12(1):149–166, 2002. doi:10.1093/logcom/12.1.149

    work page doi:10.1093/logcom/12.1.149 2002

  2. [2]

    On the complexity of coalitional reasoning.International Game Theory Review, 4(3):237–254, 2002

    Marc Pauly. On the complexity of coalitional reasoning.International Game Theory Review, 4(3):237–254, 2002. doi:10.1142/S0219198902000677

    work page doi:10.1142/s0219198902000677 2002

  3. [3]

    Journal of the ACM 49(5), pp

    Rajeev Alur, Thomas A. Henzinger, and Orna Kupferman. Alternating-time temporal logic. Journal of the ACM, 49(5):672–713, 2002. doi:10.1145/585265.585270

    work page doi:10.1145/585265.585270 2002

  4. [4]

    Chatterjee and T.A

    KrishnenduChatterjee, ThomasA.Henzinger, andNirPiterman. Strategylogic.Information and Computation, 208(6):677–693, 2010. doi:10.1016/j.ic.2009.07.004

    work page doi:10.1016/j.ic.2009.07.004 2010

  5. [5]

    Fabio Mogavero, Aniello Murano, Giuseppe Perelli, and Moshe Y. Vardi. Reasoning about strategies: On the model-checking problem.ACM Transactions on Computational Logic, 15(4):34:1–34:47, 2014. doi:10.1145/2631917

    work page doi:10.1145/2631917 2014

  6. [6]

    J., and Witteveen, C

    Harrenstein, P., van der Hoek, W., Meyer, J. J., and Witteveen, C. Boolean games. In Proceedings of the 8th Conference on Theoretical Aspects of Rationality and Knowledge, pages 287–298, 2001

    2001

  7. [7]

    Boolean games revisited

    Elise Bonzon, Marie-Christine Lagasquie-Schiex, Jérôme Lang, and Bruno Zanuttini. Boolean games revisited. InECAI 2006: 17th European Conference on Artificial Intelligence, volume 141, page 265. SAGE Publications Limited, 2006

    2006

  8. [8]

    Dunne, Wiebe van der Hoek, Sarit Kraus, and Michael Wooldridge

    Paul E. Dunne, Wiebe van der Hoek, Sarit Kraus, and Michael Wooldridge. Cooperative boolean games. InProceedings of the 7th international joint conference on Autonomous agents and multiagent systems-Volume 2, pages 1015–1022. 2008. 27

    2008

  9. [9]

    Wooldridge

    Wiebe van der Hoek and Michael J. Wooldridge. On the logic of cooperation and proposi- tional control.Artificial Intelligence, 164(1–2):81–119, 2005. doi:10.1016/j.artint.2005.01.003

    work page doi:10.1016/j.artint.2005.01.003 2005

  10. [10]

    Logics of propositional control

    Jelle Gerbrandy. Logics of propositional control. InProceedings of the 5th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2006), pages 193–200. ACM, 2006

    2006

  11. [11]

    On logics of strategic ability based on propo- sitional control

    Francesco Belardinelli and Andreas Herzig. On logics of strategic ability based on propo- sitional control. InProceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI 2016), pages 95–101. IJCAI/AAAI Press, 2016

    2016

  12. [12]

    Strategic games and truly playable effectivity functions.Autonomous Agents and Multi-Agent Systems, 26(2):288–314, 2013

    Valentin Goranko, Wojciech Jamroga, and Paolo Turrini. Strategic games and truly playable effectivity functions.Autonomous Agents and Multi-Agent Systems, 26(2):288–314, 2013. doi:10.1007/s10458-012-9192-y

    work page doi:10.1007/s10458-012-9192-y 2013

  13. [13]

    Fabio Mogavero, Aniello Murano, Giuseppe Perelli, and Moshe Y. Vardi. Reasoning about strategies: On the satisfiability problem.Logical Methods in Computer Science, 13(1):1–37,

  14. [14]

    doi:10.23638/LMCS-13(1:9)2017

    work page doi:10.23638/lmcs-13(1:9)2017 2017

  15. [15]

    Alternating-time temporal logics with irrevocable strategies

    Thomas Ågotnes, Valentin Goranko, and Wojciech Jamroga. Alternating-time temporal logics with irrevocable strategies. InProceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK 2007), pages 15–24, 2007

    2007

  16. [16]

    Comparing variants of strategic ability: How uncertainty and memory influence general properties of games.Autonomous Agents and Multi-Agent Systems, 28(3):474–518, 2014

    Nils Bulling and Wojciech Jamroga. Comparing variants of strategic ability: How uncertainty and memory influence general properties of games.Autonomous Agents and Multi-Agent Systems, 28(3):474–518, 2014. doi:10.1007/s10458-013-9231-3

    work page doi:10.1007/s10458-013-9231-3 2014

  17. [17]

    Raphaël Berthon, Bastien Maubert, Aniello Murano, Sasha Rubin, and Moshe Y. Vardi. Strategy logic with imperfect information. InProceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2017), pages 1–12. IEEE Computer Society,

    2017

  18. [18]

    doi:10.1109/LICS.2017.8005136

    work page doi:10.1109/lics.2017.8005136 2017

  19. [19]

    Practical verification of multi-agent systems against SLK specifications.Information and Computation, 261:588– 614, 2018

    Petr Cermák, Alessio Lomuscio, Fabio Mogavero, and Aniello Murano. Practical verification of multi-agent systems against SLK specifications.Information and Computation, 261:588– 614, 2018. doi:10.1016/j.ic.2017.09.011

    work page doi:10.1016/j.ic.2017.09.011 2018

  20. [20]

    Approximating perfect recall when model checking strategic abilities: Theory and applications.Journal of Artificial Intelligence Research, 73:897–932, 2022

    Francesco Belardinelli, Alessio Lomuscio, Vadim Malvone, and Emily Yu. Approximating perfect recall when model checking strategic abilities: Theory and applications.Journal of Artificial Intelligence Research, 73:897–932, 2022. doi:10.1613/jair.1.12539

    work page doi:10.1613/jair.1.12539 2022

  21. [21]

    STV+FLY: On- the-fly model checking of strategic ability in multi-agent systems

    Damian Kurpiewski, Mateusz Kamiński, and Wojciech Jamroga. STV+FLY: On- the-fly model checking of strategic ability in multi-agent systems. InECAI 2024 — 27th European Conference on Artificial Intelligence, pages 4483–4486. IOS Press, 2024. doi:10.3233/FAIA241035

    work page doi:10.3233/faia241035 2024

  22. [22]

    Scalable verification of strategy logic through three-valued abstraction

    Francesco Belardinelli, Angelo Ferrando, Wojciech Jamroga, Vadim Malvone, and Aniello Murano. Scalable verification of strategy logic through three-valued abstraction. In Proceedings of the 32nd International Joint Conference on Artificial Intelligence (IJCAI 2023), pages 46–54. IJCAI, 2023. doi:10.24963/ijcai.2023/6

    work page doi:10.24963/ijcai.2023/6 2023

  23. [23]

    Strategy logic with simple goals: Tractable reasoning about strategies

    Francesco Belardinelli, Wojciech Jamroga, Vadim Malvone, and Aniello Murano. Strategy logic with simple goals: Tractable reasoning about strategies. In28th International Joint Conference on Artificial Intelligence (IJCAI 2019), pages 88–94. 2019

    2019

  24. [24]

    Multi- valued verification of strategic ability.Fundamenta Informaticae, 175(1–4):207–251, 2020

    Wojciech Jamroga, Beata Konikowska, Damian Kurpiewski, and Wojciech Penczek. Multi- valued verification of strategic ability.Fundamenta Informaticae, 175(1–4):207–251, 2020. doi:10.3233/FI-2020-1955. 28

    work page doi:10.3233/fi-2020-1955 2020

  25. [25]

    Combining quantitative and qualitative reasoning in concurrent multi-player games.Autonomous Agents and Multi-Agent Systems, 36:Article 2,

    Nils Bulling and Valentin Goranko. Combining quantitative and qualitative reasoning in concurrent multi-player games.Autonomous Agents and Multi-Agent Systems, 36:Article 2,

  26. [26]

    doi:10.1007/s10458-021-09531-9

    work page doi:10.1007/s10458-021-09531-9

  27. [27]

    A Modal Logic for Joint Abilities under Strategy Commitments

    Zhaoshuai Liu, Liping Xiong, Yongmei Liu, Yves Lespérance, Ronghai Xu, and Hongyi Shi. A Modal Logic for Joint Abilities under Strategy Commitments. InProceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI 2020), pages 1805–1812. 2020. doi:10.24963/ijcai.2020/250

    work page doi:10.24963/ijcai.2020/250 2020

  28. [28]

    Logic for coalitions with bounded resources.Journal of Logic and Computation, 21(6):907–937, 2011

    Natasha Alechina, Brian Logan, Hoang Nga Nguyen, and Abdur Rakib. Logic for coalitions with bounded resources.Journal of Logic and Computation, 21(6):907–937, 2011

    2011

  29. [29]

    Resource-bounded alternating-time temporal logic

    Natasha Alechina, Brian Logan, Nguyen Hoang Nga, and Abdur Rakib. Resource-bounded alternating-time temporal logic. InProceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1-Volume 1, pages 481–488. 2010. 29

    2010