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

Author

Listed:
  • Andreas H. Hamel

    (Free University Bozen-Bolzano)

  • Andreas Löhne

    (Friedrich Schiller University)

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 $$\mathbb {R}^d$$ R d to order relations in the power set of $$\mathbb {R}^d$$ R d , so-called set relations, and strictly motivated 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 minimax and maximin solutions as well as Pareto optimal security strategies are clarified.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:88:y:2018:i:3:d:10.1007_s00186-018-0639-z
    DOI: 10.1007/s00186-018-0639-z
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    References listed on IDEAS

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    Cited by:

    1. Kuntal Som & V. Vetrivel, 2021. "On robustness for set-valued optimization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 905-925, April.
    2. Kuntal Som & V. Vetrivel, 2023. "Global well-posedness of set-valued optimization with application to uncertain problems," Journal of Global Optimization, Springer, vol. 85(2), pages 511-539, February.
    3. Jaeok Park, 2019. "Decision Making and Games with Vector Outcomes," Working papers 2019rwp-146, Yonsei University, Yonsei Economics Research Institute.

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