Math Meth Oper Res https://doi.org/10.1007/s00186-018-0639-z ORIGINAL ARTICLE A set optimization approach to zero-sum matrix games with multi-dimensional payoffs 1 2 Andreas H. Hamel · Andreas Löhne Received: 1 October 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract A new solution concept for two-player zero-sum matrix games with multi- dimensional payoffs is introduced. It is based on extensions of the vector order in R to order relations in the power set of R , so-called set relations, and strictly moti- vated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector mini- max and maximin solutions as well as Pareto optimal security strategies are clariﬁed. Keywords Zero-sum game · Multi-dimensional payoff · Multi-objective program- ming · Set relation · Set optimization · Incomplete preference Mathematics
Mathematical Methods of Operations Research – Springer Journals
Published: May 28, 2018
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