The Existence of Berge Equilibrium

The sufficient conditions of the existence of Berge equilibrium situation in noncooperative game of many persons in normal form are established. On the basis of these conditions the existence of Berge equilibrium situation in mixed strategies (by compact sets of strategies of players and continuity of their payoff functions) is proved. Let us consider the history of the appearance of the Berge equilibrium notion.
In 1949 the 21-years-old PhD student of Prinston University, John F. Nash (jun.), formalized the notion of “good” solution in noncooperative games (later called “Nash equilibrium”). It has got the broad spectrum of applications in economics, sociology, military sciences. And now after more than 50 years, in any journal of system analysis, game theory, mathematical programming we find the papers devoted to Nash equilibrium (NE). In 1994 John Nash won the Nobel Prize in economics in a common effort with John Harsanyi and R. Selten “for fundamental analysis of equilibria in noncooperative game theory”. Actually 20-years-old Nash developed the foundation of the scientific method that played the great role in the development of world economy.
However “in the sun there are spots” (proverb). And the main of them is “the eqoistic character” of Nash equilibrium concept. It appears in the fact that every player tries to increase only his own payoff, i. e. follows “politica dei campanile”, without considering interests of other participants of the conflict. One of the methods to remove this negative is to use the approach (by formalization of “good” solution of the game), which differs from “dictated” Nash equilibrium. Such approach was proposed in 1994 at the scientific seminar (leader V.I.Zhukovskiy) at discussing the book of C.Berge «Theorie generale des jeux a n personnes games» (this book was published in Paris in 1957 and in 1961 it was translated into Russian [1]). Concretely the criticism of NE was caused by non-existence NE at strongly concave in strategy at least one player his payoff function (but the decision making is necessary!). The sense of the new approach lies in change of condition of solution stability not to deviation of the player whom belongs “payoff function” but to deviation of all players except the one who is “the owner” of this payoff function. We shall note three circumstances. Firstly, we called the proposed new concept “BE”. The term “BE” arouse as the result of reviewing Claude Berge’s book. Secondly, in 1994 K.S.Vaisman (then the post-graduate student of V.Zhukovskiy) was engaged in construction of initial foundations of mathematical BE theory. In 1995 K. Vaisman defended his thesis “Berge equilibrium” (BE) in Leningrad University. (K.Vaisman died in 1998 at the age of 35 years). His sudden death suspended further development of the Berge equilibrium in Russia, but the notion of a Berge equilibrium was “exported from Russia” by Algerian scholars of V.Zhukovskiy M. Radjef and M. Larbani. This notion caused the broad interest of our foreign colleagues. The acquaintance with their publications showed that “par le temps qui ceurt” the most papers of this direction devoted to the properties of Berge equilibrium, singularities, modifications of this notion, relations with Nash equilibrium. It is supposed that in originated theory of Berge equilibrium the stage of formation
of strict mathematical theory becomes nearer. Probably an intensive accumulation of facts will be replaced by the stage of evolutionary internal development. At this stage one should traditionally answer two fundamental questions:
1. Does the Berge equilibrium exist?
2. How one should find this equilibrium?
The present article is just devoted to answers of these both questions. Thirdly, the authors were motivated by the IX Moscow Festival of Science that partially was held in a new building of MSU Fundamental library on October 10, 2014. Apart from lectures of Nobel laureates chemists Kurt Wuthrich (USA, California), Jean-Marie Lehn (France), biochemist Sir Richard Roberts (USA), RAS academician M.Ya.Marov (“The Chelyabinsk meteor”), L.M. Zelenyi (“Exoplanets: Searching for a second Earth”), Doctors of Sciences A.V. Markov (“Why a human has large brain”), Yury I. Aleksandrov (“Neurons, humans and cultures”), the program included the lecture of RAS academician, director of RAS Institute of Philosophy A.A.Guseinov “The Golden Rule of ethics”. Being inspired by lecture the first author of this article addressed the following question to the speaker, “Are you interested in a mathematical theory of the Golden rule?” The answer was confirmative. Now, at our strong belief, the concepts of Berge equilibrium most completely meet main requirement of the Golden Rule of Ethics, “Behave to others as you would like them to behave to you”.
Thus, the article offered to the reader, first, suggests the method of construction of Berge equilibrium situation for finding minimax strategy in specific Germeier convolution, effectively constructed in assumed mathematical model of existence strategies if sets of strategies are compact and payoff function is continuous according to situations.

Keywords: noncooperative game, payoff function, payoff, Nash and Berge equilibrium, Germeier convolution, mixed strategies.