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.
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