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New Approaches to the Solution of Noncooperative Games and Multicriteria Choice Problems

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Book cover The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics

Part of the book series: Static & Dynamic Game Theory: Foundations & Applications ((SDGTFA))

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Abstract

This chapter considers three new approaches to important problems of mathematical game theory and multicriteria choice, which are described in four sections (5.1–5.4). The first approach ensures payoff increase with simultaneous risk reduction in the Savage–Niehans sense in multicriteria choice problems (Sect. 5.1) and noncooperative games (Sect. 5.2). The second approach allows to stabilize coalitional structures in cooperative games without side payments under uncertainty (Sect. 5.3). The third approach serves to integrate the selfish Nash equilibrium with the altruistic Berge equilibrium. Note that the investigations in Sects. 5.2–5.4 involve a special Germeier convolution of criteria and calculation of its saddle point in mixed strategies.

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Notes

  1. 1.

    Edgar Watson Howe, (1853–1937), was an American editor, novelist, and essayist.

  2. 2.

    German “Those who risk win.” This is an analog of the English proverb “Nothing ventured, nothing gained.”

  3. 3.

    French “Put oneself at risk.”

  4. 4.

    This Latin phrase expresses a main postulate of realism: universals exist in reality and independently from consciousness.

  5. 5.

    Latin “Words instruct, illustrations lead.”

  6. 6.

    Cases 1 and 2 will be considered with Pareto optimality, and case 3 with Proposition 5.1.1.

  7. 7.

    An English translation of a statement from [26].

  8. 8.

    Laura Elizabeth Ingalls Wilder (1867–1957) was an American writer known for the Little House on the Prairie series of children’s books.

  9. 9.

    Latin “To inform.”

  10. 10.

    Latin “When two do the same thing, it is not the same thing.” This phrase belongs to Terence, Latin in full Publius Terentius Afer, (195–159? B.C.), after Plautus the greatest Roman comic dramatist. See The Brothers V. 3.

  11. 11.

    Latin “All determination is negation.” This phrase belongs to Benedict de Spinoza, (1632–1677), a Dutch Jewish philosopher.

  12. 12.

    This Latin word combination identifies a starting point, an origin, a source, the heart of the matter.

  13. 13.

    French “All that is needed.”

  14. 14.

    French “That just goes to show!”

  15. 15.

    French “Complaining that the bride is too beautiful.” In our book, the advantages of Proposition 5.2.4 have exceeded all expectations.

  16. 16.

    Latin “Of all things that can be known and all kind of other things.” The first part of this phrase (de omni re scibili, meaning “of all things that can be known”) was the motto of pompous young lad and famous Italian philosopher Pico della Mirandola, who thought this was a fitting description of his encyclopedic knowledge. The second part (et quibusdam aliis, meaning “and even certain other things”) was ironically appended by pompous old and famous French philosopher Voltaire, who was somewhat under the impression he was any less full of himself.

  17. 17.

    French “At the end after all.”

  18. 18.

    French “To be continued.”

  19. 19.

    Claude Elwood Shannon, (1916–2001), was an American mathematician and electrical engineer who laid the theoretical foundations for digital circuits and information theory.

  20. 20.

    The Old Testament, Leviticus 19:9–18.

  21. 21.

    French “Every one is the architect of his own fortune.”

  22. 22.

    Robert Burns, (1759–1796), was a national poet of Scotland, who wrote lyrics and songs in Scots and in English.

  23. 23.

    Ursula K. Le Guin, original name Ursula Kroeber, (1929–2018), was an American writer best known for tales of science fiction and fantasy.

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Authors and Affiliations

  1. Department of Engineering Sciences and Information Technology, Georgian National Academy of Sciences, Tbilisi, Georgia

    Mindia E. Salukvadze

  2. Department of Optimal Control Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

    Vladislav I. Zhukovskiy

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  1. Mindia E. Salukvadze
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  2. Vladislav I. Zhukovskiy
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E. Salukvadze, M., I. Zhukovskiy, V. (2020). New Approaches to the Solution of Noncooperative Games and Multicriteria Choice Problems. In: The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-25546-6_5

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