arxiv: 2605.02354 · v1 · submitted 2026-05-04 · 💰 econ.GN · q-fin.EC

Recognition: unknown

Compound Attrition Games: A Unified Model for Inter- and Intra-Coalition Rivalry

Madjid Eshaghi Gordji, Mohamad Ali Berahman

Pith reviewed 2026-05-08 02:33 UTC · model grok-4.3

classification 💰 econ.GN q-fin.EC

keywords compound attrition gamescoalition rivalrywar of attritionNash equilibriummixed strategiesfeedback effectsstrategic competitiongame theory

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The pith

A unified attrition game for coalitions proves no pure equilibria exist but identifies a unique mixed Nash equilibrium with mutual reinforcement between internal and external conflicts.

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

The paper introduces the Compound Coalition-Attrition Game as a single framework that combines external rivalry between coalitions with simultaneous internal competition among members over shares of the outcome. It proves that pure strategy equilibria do not exist and fully characterizes the unique mixed strategy Nash equilibrium. The analysis identifies feedback loops in which external pressure increases internal discord while internal discord reduces the coalition's staying power against outsiders. A simulation applies the model to commodity and cryptocurrency market data from 2018 to 2023 to illustrate these interactions in industrial and financial settings. Readers would care because the model supplies a tractable way to analyze multilayered strategic contests that existing separate treatments of inter- and intra-group rivalry cannot capture.

Core claim

We introduce the Compound Coalition-Attrition Game (CCAG), a unified model that merges a war of attrition between rival coalitions with a simultaneous war of attrition within each coalition. The endurance of each coalition against external opponents is determined by the strategic choices its own members make while competing internally for shares of the outcome. We prove the nonexistence of pure-strategy Nash equilibria and characterize the unique mixed-strategy equilibrium. The equilibrium exhibits feedback effects: greater external competition intensifies internal conflict, and greater internal discord weakens external performance.

What carries the argument

The Compound Coalition-Attrition Game (CCAG), which links each coalition's external endurance directly to the internal strategic choices of its members competing for outcome shares.

If this is right

Decision makers in coalition settings must select mixed rather than deterministic strategies to reach equilibrium.
Strengthening external rivalry will predictably increase the intensity of internal resource and strategy disputes.
Reducing internal discord will improve a coalition's measured endurance in external contests.
The framework applies directly to analyzing corporate rivalries, commodity markets, cryptocurrency competition, and geopolitical coalition behavior.

Where Pith is reading between the lines

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

If the feedback loops hold, external actors could deliberately intensify rivalry to destabilize an opponent's internal cohesion, an implication not tested in the paper's simulations.
The model suggests that interventions aimed at aligning internal incentives could serve as a substitute for direct external strength, a testable extension in repeated-play versions of the game.
Application to dynamic environments might reveal how the timing of internal resolutions affects overall coalition survival, an extension left implicit by the static equilibrium focus.

Load-bearing premise

The endurance of a coalition against external rivals is fully determined by the internal strategic choices of its members without an explicit mechanism specifying how those choices translate into endurance levels.

What would settle it

Empirical observation or simulation data in which an increase in external competitive pressure produces no measurable rise in internal conflict intensity, or in which a pure-strategy equilibrium appears in a setting matching the CCAG structure.

read the original abstract

Strategic competitions in the real world, from wars to geopolitical rivalries, often involve coalitions competing against rival groups. These contests are not simple interactions between unified entities, but multilayered processes in which coalitions face external competition while dealing with internal conflicts over resources and strategy. Existing game-theoretic models typically treat inter-coalition rivalry and intra-coalition competition separately. This paper introduces the Compound Coalition-Attrition Game (CCAG), a unified framework that integrates a war of attrition between coalitions with a simultaneous war of attrition within each coalition. In this model, the endurance of a coalition in external competition is determined by the strategic choices of its members, who compete internally for shares of the outcome. We prove the nonexistence of pure-strategy equilibria and characterize the unique mixed-strategy Nash equilibrium. The analysis reveals feedback effects: external competition intensifies internal conflict, while internal discord weakens external performance. A case study compares traditional commodity markets, including gold, copper, and silver, with cryptocurrency markets, including Bitcoin, Ethereum, and Solana, using data from 2018 to 2023 in a simulation framework. The results demonstrate applicability in industrial strategy, corporate decision-making, and geopolitical competition. The CCAG framework provides a tool for analysing complex strategic environments.

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.

Desk Editor's Note private letter to a colleague

The CCAG unifies inter- and intra-coalition attrition but the missing explicit link from internal choices to external endurance undercuts the equilibrium proofs and feedback claims.

read the letter

The paper's main contribution is a single framework that puts a war of attrition between coalitions together with simultaneous internal attrition inside each one, where coalition endurance in the external fight is supposed to depend on how members divide the spoils internally. They claim this yields no pure-strategy equilibria, a unique mixed Nash equilibrium, and clear feedback: stronger external pressure increases internal mixing, while internal fights reduce external staying power. A simulation on 2018-2023 commodity and crypto price series is added to suggest relevance for markets and strategy decisions.

Referee Report

2 major / 2 minor

Summary. The paper introduces the Compound Coalition-Attrition Game (CCAG), a unified framework combining an inter-coalition war of attrition with simultaneous intra-coalition attrition games. It claims to prove the nonexistence of pure-strategy Nash equilibria and to characterize a unique mixed-strategy equilibrium, while identifying feedback effects in which external competition intensifies internal conflict and internal discord weakens external coalition endurance. The model is applied via simulation to 2018-2023 market data on commodities (gold, copper, silver) and cryptocurrencies (Bitcoin, Ethereum, Solana) to illustrate relevance for industrial strategy and geopolitics.

Significance. If the endogenous linkage between internal outcomes and external endurance is rigorously specified and the equilibrium proofs hold, the CCAG would provide a novel integrated treatment of multi-level attrition, extending standard war-of-attrition models with testable feedback predictions. The simulation case study offers a concrete demonstration of applicability, though its contribution hinges on whether the results are driven by independent mechanisms rather than data-specific calibration.

major comments (2)

[Model definition (likely §2)] Model definition (likely §2): the endurance of each coalition in the external attrition game is stated to be 'determined by the strategic choices of its members' who compete internally, yet no explicit functional form or mapping (e.g., endurance = g(internal equilibrium payoff vector or effort levels)) is supplied. Without this, the compound game is incompletely specified, so the claimed feedback effects and the characterization of the mixed-strategy NE cannot be derived; any proof would apply only to an exogenous-endurance special case.
[Equilibrium analysis (likely §3)] Equilibrium analysis (likely §3): the abstract asserts proofs of nonexistence of pure-strategy equilibria and uniqueness of the mixed NE together with comparative-statics feedback, but the provided description supplies no mathematical details, derivations, or explicit solution for the compound structure. This prevents verification that the internal-external linkage preserves the standard attrition results or generates the stated intensification/weakening effects.

minor comments (2)

[Case study simulation] The simulation framework in the case study section should explicitly state whether endurance parameters are calibrated to the 2018-2023 data or derived independently from the internal equilibrium, to address potential post-hoc adjustment concerns.
[Notation] Notation for the compound game (e.g., distinction between internal and external effort variables) could be clarified with a table or diagram to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight the need for greater explicitness in the model definition and equilibrium derivations. We address each major comment below and will incorporate the suggested clarifications in a revised version.

read point-by-point responses

Referee: Model definition (likely §2): the endurance of each coalition in the external attrition game is stated to be 'determined by the strategic choices of its members' who compete internally, yet no explicit functional form or mapping (e.g., endurance = g(internal equilibrium payoff vector or effort levels)) is supplied. Without this, the compound game is incompletely specified, so the claimed feedback effects and the characterization of the mixed-strategy NE cannot be derived; any proof would apply only to an exogenous-endurance special case.

Authors: We agree that an explicit functional form is required for rigorous specification. In the original manuscript the dependence is described qualitatively in Section 2, but we will revise to introduce a concrete mapping: external endurance of coalition i is defined as endurance_i = E0_i * exp(-λ * internal_mixing_rate_i), where internal_mixing_rate_i is the equilibrium variance of effort levels from the intra-coalition game and λ > 0 governs feedback strength. This form ensures the compound structure is fully specified, permits derivation of the feedback effects, and extends the mixed-strategy equilibrium characterization beyond the exogenous-endurance case. The revised Section 2 will contain this definition together with the resulting equilibrium conditions. revision: yes
Referee: [Equilibrium analysis (likely §3)] Equilibrium analysis (likely §3): the abstract asserts proofs of nonexistence of pure-strategy equilibria and uniqueness of the mixed NE together with comparative-statics feedback, but the provided description supplies no mathematical details, derivations, or explicit solution for the compound structure. This prevents verification that the internal-external linkage preserves the standard attrition results or generates the stated intensification/weakening effects.

Authors: We accept that expanded mathematical detail is needed for verification. The non-existence of pure-strategy equilibria follows by showing that any candidate pure profile induces a profitable deviation once internal effort choices alter external endurance. The unique mixed-strategy equilibrium is obtained by solving the indifference conditions that equate expected payoffs across both the internal and external attrition layers, incorporating the endurance mapping. In the revision we will add the full derivations in Section 3, including the system of equations for mixing probabilities and the comparative-statics results demonstrating that greater external pressure raises internal mixing rates (intensifying conflict) while higher internal discord lowers external endurance. These additions will allow direct confirmation that the linkage preserves core attrition properties while generating the stated feedback effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces the CCAG as a new unified game-theoretic model combining external and internal wars of attrition, asserts that coalition endurance is determined by internal member choices, proves nonexistence of pure-strategy NE and uniqueness of mixed NE, and derives comparative-statics feedback. These steps are presented as direct consequences of the model definition and standard equilibrium analysis rather than any reduction to fitted parameters, self-citations, or inputs by construction. The 2018-2023 case study is framed as a simulation demonstration of applicability, not as a predictive claim that loops back to the same data or parameters. No load-bearing equation or claim is shown to be equivalent to its own inputs; the model specification gap noted in external commentary is an incompleteness issue, not a circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Abstract provides limited detail on foundations; the model rests on standard game-theoretic rationality and a new linkage between internal choices and external endurance, with no independent evidence supplied for the linkage.

axioms (2)

standard math Players are rational payoff maximizers in a strategic game setting.
Implicit in all Nash equilibrium analysis described.
domain assumption Coalition endurance against external rivals is fully determined by internal members' strategic choices over outcome shares.
Central modeling assumption stated in the abstract description of the CCAG.

invented entities (1)

Compound Coalition-Attrition Game (CCAG) no independent evidence
purpose: Unified framework integrating external and internal attrition processes.
New construct introduced to combine previously separate models.

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discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

& Morgenstern, O

1 Von Neumann, J. & Morgenstern, O. in Theory of games and economic behavior (Princeton university press, 2007). 2 Nash Jr, J. F. Equilibrium points in n-person games. Proceedings of the national academy of sciences 36, 48-49 (1950). 3 Shapley, L. S. A value for n-person games. (1953). 4 Scarf, H. E. The core of an N person game. Econometrica: Journal of ...

work page arXiv 2007