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The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

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Abstract

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.

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Authors and Affiliations

  1. Department of Mathematics, The University of Queensland, Brisbane, QLD, Australia, 4072

    George Xian-Zhi Yuan

  2. Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ONT, Canada, K7K 5L0

    George Isac

  3. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada, B3H J35

    Kok-Keong Tan & Jian Yu

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  1. George Xian-Zhi Yuan
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  2. George Isac
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  3. Kok-Keong Tan
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  4. Jian Yu
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Yuan, G.XZ., Isac, G., Tan, KK. et al. The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria. Acta Applicandae Mathematicae 54, 135–166 (1998). https://doi.org/10.1023/A:1006095413166

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