Weak Pareto–Nash equilibria in generalized interval-valued multiobjective games with fuzzy constraint mappings and applications to price competition

Wen Li; Du Zou; Deyi Li; Wen Li; Du Zou; Deyi Li

doi:10.3934/math.2026121

AIMS Mathematics

2026, Volume 11, Issue 1: 3011-3037. doi: 10.3934/math.2026121

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Research article

Weak Pareto–Nash equilibria in generalized interval-valued multiobjective games with fuzzy constraint mappings and applications to price competition

Wen Li ^{1
,
,},
Du Zou ^{1
,},
Deyi Li ^{1,2
,}

1.
School of Mathematics and Systems Science, Wuhan University of Science and Technology, Wuhan 430065, China; zoudu@wust.edu.cn
2.
Department of Public Basic Course, Wuhan Technology and Business University, Wuhan 430065, China; lideyi@wust.edu.cn

Received: 27 October 2025 Revised: 07 January 2026 Accepted: 16 January 2026 Published: 30 January 2026
MSC : 03E72, 52A20, 65G30, 91A10

In this study, we examine a generalized interval-valued multiobjective game with fuzzy constraint mappings (GIMGFCM). By employing interval support functions from convex geometry, the set of all $ d $-dimensional interval vectors is embedded into the space of real-valued continuous functions defined on the unit sphere in $ \mathbb{R}^d $. This embedding yields a closed convex cone $ \mathcal{I}(\mathbb{R}_+^d) $ within that function space. Using this cone, we define a partial order for interval vectors and establish the semi-continuity and generalized $ \mathcal{I}(\mathbb{R}_+^d) $-quasi-concavity of interval-vector-valued functions. On this basis, we propose a weak Pareto–Nash equilibrium concept for a GIMGFCM and prove an existence theorem for such equilibria. Finally, we apply the relevant theoretical framework to analyze a price competition problem between two firms that sell heterogeneous goods.
Citation: Wen Li, Du Zou, Deyi Li. Weak Pareto–Nash equilibria in generalized interval-valued multiobjective games with fuzzy constraint mappings and applications to price competition[J]. AIMS Mathematics, 2026, 11(1): 3011-3037. doi: 10.3934/math.2026121

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Abstract

In this study, we examine a generalized interval-valued multiobjective game with fuzzy constraint mappings (GIMGFCM). By employing interval support functions from convex geometry, the set of all $ d $-dimensional interval vectors is embedded into the space of real-valued continuous functions defined on the unit sphere in $ \mathbb{R}^d $. This embedding yields a closed convex cone $ \mathcal{I}(\mathbb{R}_+^d) $ within that function space. Using this cone, we define a partial order for interval vectors and establish the semi-continuity and generalized $ \mathcal{I}(\mathbb{R}_+^d) $-quasi-concavity of interval-vector-valued functions. On this basis, we propose a weak Pareto–Nash equilibrium concept for a GIMGFCM and prove an existence theorem for such equilibria. Finally, we apply the relevant theoretical framework to analyze a price competition problem between two firms that sell heterogeneous goods.

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