A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Methods of Operations Research Springer Journals

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Operations Research/Decision Theory; Business and Management, general
ISSN
1432-2994
eISSN
1432-5217
D.O.I.
10.1007/s00186-018-0639-z
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Mathematical Methods of Operations ResearchSpringer Journals

Published: May 28, 2018

References

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