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

Andreas H. Hamel () and Andreas Löhne ()
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Andreas H. Hamel: Free University Bozen-Bolzano
Andreas Löhne: Friedrich Schiller University

Mathematical Methods of Operations Research, 2018, vol. 88, issue 3, No 2, 369-397

Abstract: 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.

Keywords: Zero-sum game; Multi-dimensional payoff; Multi-objective programming; Set relation; Set optimization; Incomplete preference; Primary 91A05; Secondary 91A10; 62C20; 91A35 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s00186-018-0639-z

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