Abstract
In this chapter, the concept of Berge equilibrium is introduced as a mathematical model of the Golden Rule. This concept was suggested by the Russian mathematician K. Vaisman in 1994. The Berge–Pareto equilibrium is formalized and sufficient conditions for the existence of such an equilibrium are established. As an application, the existence in the class of mixed strategies is proved.
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Notes
- 1.
French “He who thinks he has the power to content the world greatly deceives himself, but he who thinks that the world cannot be content with him deceives himself yet more.” François de La Rochefoucauld (1613–1680) was a French classical writer; a quote from Réflexions ou Sentences et Maximes morales (1665).
- 2.
Latin “To each according to its own merits; to each his/her own.” This phrase appeared in philosophical dialogs and treatises On Duties 1, 5, 14, and Tusculan Disputations, Vol. 22, by Marcus Tullius Cicero (102–43 BC), a Roman statesman, lawyer, scholar, and writer.
- 3.
Latin “Not many, but much,” meaning not quantity but quality. This phrase belongs to Plinius the Younger (62–114 A.D.); see Letters, VII, 9.
- 4.
Osip E. Mandelshtam, (1891–1938), was a major Russian poet, prose writer, and literary essayist.
- 5.
French, meaning narrow-mindedness and a lack of understanding or even interest in the world beyond one’s own town’s boundaries.
- 6.
Latin, meaning a feature by which two subclasses of the same class of named objects can be distinguished.
- 7.
Latin “Every man is the artisan of his own fortune.” This phrase goes back to Appius Claudius Caecus (4–3 centuries BC), an outstanding statesman, legal expert and author of early Rome who was one of the first notable personalities in Roman history.
- 8.
French, meaning that all things in the worlds are interconnected.
- 9.
Latin “From the chair,” used with regard to statements made by people in positions of authority.
- 10.
There also exist other approaches to conflict analysis: descriptive, which is to find the resulting collections of players’ actions (the so-called strategy profiles) in a given conflict; constructive, which is to implement the desired (e.g., optimal) strategies in a given conflict; predictive, which is to forecast the actual result (outcome) of a given conflict.
- 11.
French “Every man for himself, every man to himself.”
- 12.
German “One for all and all for one.”
- 13.
Latin “For discussion.”
- 14.
Latin “Who? What? Where? Who helped? Why? How? When?”; a well-known system of seven questions for crime investigation suggested by Roman rhetorician Quintilian, Latin in full Marcus Fabius Quintilianus, (appr. 35–100 A.D.).
- 15.
French “We would yearn for very few things if we clearly understood what we wanted.” A quote from Réflexions ou Sentences et Maximes Morales by F. de La Rochefoucauld.
- 16.
Latin “To each his own,” or “May all get their due”; also, see the epigraph to Sect. 2.2.1.
- 17.
French “From each according to his ability, to each according to his needs.”
- 18.
Italian, “The policy of his/her own bell tower.” Used to describe narrow-mindness and commitment to local interests.
- 19.
John Forbes Nash, Jr. Born June 13, 1928, in Bluefield, West Virginia. Successfully graduated from the Carnegie Institute of Technology (now, Carnegie Mellon University) with bachelor’s and master’s degrees in mathematics. Richard Duffin, Nash’s undergraduate advisor at the Carnegie Institute of Technology, gave him a brief characterization, “He is a mathematical genius.” In 1948 Nash started his postgraduate study at Princeton University, where he was particularly influenced by International Economy, the faculty course of J. von Neumann, and by the famous book Theory of Games and Economic Behavior (1944), written by von Neumann together with O. Morgenstern. In 1949 Nash presented his thesis on equilibrium solutions of noncooperative games; after 45 years—in 1994—he was awarded the Noble Prize in Economic Sciences for that research. From 1951 to 1959 worked at the Cambridge at Massachusetts Institute of Technology (MIT). In 1958 Fortune called Nash “America’s brilliant young star of the ‘new mathematics.”’ In 1959 moved to California to work for the RAND Corporation and became a leading expert in the Cold War. Since 1959 suffered from a mental disorder (completely overcame the disease by 1980, to the great astonishment of doctors). Since 1980 again worked at Princeton University as a consulting professor. Was killed in a car crash on May, 24, 2015, at the age of 86. Throughout the world, Nash is well-known through R. Howard’s movie A Beautiful Mind (2001, featuring R. Crowe) based on S. Nasar’s book Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash. The movie received four Oscars and the Golden Globe.
- 20.
“And will not let true merit miss its due, Remembering always rather good than evil.” A quote from Tartuffe, Scene VII, a famous theatrical comedy by Molière (1622–1873).
- 21.
Naum Ya. Vilenkin, (1920–1991), was a Soviet mathematician and student of A. G. Kurosh, who contributed to general algebra, topology, real-variable theory and functional analysis. A quote from Kvant, 1975, no. 8, p. 2.
- 22.
James P. Pierpont, (1866–1938), was an American mathematician. Known for research in the field of real and complex variable functions.
- 23.
Veniamin F. Kagan, (1860–1953), was a Russian and Soviet mathematician. A quote translated into English from Kvant, 1975, no. 6, p. 16.
- 24.
Cassius Jackson Keyser, (1862–1947), was an American mathematician of pronounced philosophical inclinations. A quote from On Mathematics and Mathematicians, R.E. Moritz, Ed., New York: Dover, 1958, pp. 360–361.
- 25.
From a PBS interview with John F. Nash.
- 26.
Jean Le Rond d’Alembert, (1717–1783), was a French mathematician, philosopher, and writer.
- 27.
Walter Warwick Sawyer, (1911–2008), was a British mathematician, mathematics educator and author, who popularized mathematics on several continents.
- 28.
Richard Courant, (1888–1972), was a German-born American mathematician, educator and scientific organizer who made significant advances in the calculus of variations. A quote from The Australian Mathematics Teacher, vols. 39–40, Australian Association of Mathematics Teachers, 1983, p. 3.
- 29.
From a letter of French mathematician Charles Hermite, (1822–1901), to Dutch mathematician Thomas Joannes Stieltjes, (1856–1894), written in 1893.
- 30.
Norbert Wiener, (1894–1964), was an outstanding American mathematician and philosopher, the father of cybernetics. A quote from his book I Am a Mathematician: the Later Life of a Prodigy, MIT Press, 1964.
- 31.
David Hilbert, (1862–1943), was a German mathematician who axiomatized geometry and contributed substantially to the establishment of the formalistic foundations of mathematics. Recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. A quote from the book Hilbert by C.B. Reid, Springer, 1996.
- 32.
Pierre-Simon, marquis de Laplace, (1749–1827), was a French scholar who made important contributions to the development of mathematics, statistics, physics and astronomy. An English translation of a quote from his book Théorie Analytique des Probabilités, 1795.
- 33.
Stefan Banach, (1892–1945), was a Polish mathematician who founded modern functional analysis and helped contributed to the development of the theory of topological vector spaces. Generally considered one of the most important and influential mathematicians of the twentieth century.
- 34.
Latin “Words instruct, illustrations lead.”
- 35.
Latin “The burden of proof.”
- 36.
Latin “Negative statements are not proved.”
- 37.
French “For our peace of mind.”
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E. Salukvadze, M., I. Zhukovskiy, V. (2020). Static Case of the Golden Rule. 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_2
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