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Vector-Valued Minimax Theorems In Multicriteria Games

In: New Frontiers Of Decision Making For The Information Technology Era

Author

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  • TAMAKI TANAKA

    (Department of Mathematical System Science, Faculty of Science and Technology, Hirosaki University, Hirosaki 036–8561, Japan)

Abstract

Saddle point theorem in usual game theory says: a real-valued payoff function possesses a saddle point if and only if the minimax value and the maximin value of the function are coincident; and that minimax theorems say: the minimax and maximin values are coincident under certain conditions. These facts are valid based on the total ordering of R, but if we consider more general partial orderings on vector spaces then what kind of results on minimax and maximin of a multiobjective payoff with multiple noncomparable criteria are obtained? In order to answer the question, we adopt the concepts of “cone extreme point” or “non-dominated solution,” which have been proposed by Dr.Yu (1974).63 Under suitable conditions, we observe that vector-valued minimax theorems and saddle point problems are closely connected with each other, whose results are similar to standard ones for scalar games. Dr.Yu's ideas have given many directions to study vector optimization and multicriteria analysis such as this work and its related topics.

Suggested Citation

  • Tamaki Tanaka, 2000. "Vector-Valued Minimax Theorems In Multicriteria Games," World Scientific Book Chapters, in: Yong Shi & Milan Zeleny (ed.), New Frontiers Of Decision Making For The Information Technology Era, chapter 5, pages 75-99, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789812792907_0005
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    Cited by:

    1. Andreas H. Hamel & Andreas Löhne, 2018. "A set optimization approach to zero-sum matrix games with multi-dimensional payoffs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 369-397, December.

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